
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00037026/00001
Material Information
 Title:
 An equilibrium model of index futures pricing
 Creator:
 Wu, Soushan, 1950
 Publication Date:
 1984
 Language:
 English
 Physical Description:
 ix, 99 leaves : ; 28 cm.
Subjects
 Subjects / Keywords:
 Dividends ( jstor )
Financial portfolios ( jstor ) Futures contracts ( jstor ) Index numbers ( jstor ) Investors ( jstor ) Market prices ( jstor ) Prices ( jstor ) Standard and Poors 500 Index ( jstor ) Stock market indices ( jstor ) Subject terms ( jstor ) Dissertations, Academic  Finance, Insurance, and Real Estate  UF Finance, Insurance, and Real Estate thesis Ph. D Stock index futures ( lcsh )
 Genre:
 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 1984.
 Bibliography:
 Bibliography: leaves 9498.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Soushan Wu.
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 University of Florida
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 University of Florida
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 The University of Florida George A. Smathers Libraries respect the intellectual property rights of others and do not claim any copyright interest in this item. This item may be protected by copyright but is made available here under a claim of fair use (17 U.S.C. Â§107) for nonprofit research and educational purposes. Users of this work have responsibility for determining copyright status prior to reusing, publishing or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. The Smathers Libraries would like to learn more about this item and invite individuals or organizations to contact the RDS coordinator (ufdissertations@uflib.ufl.edu) with any additional information they can provide.
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AN EQUILIBRIUM MODEL OF INDEX FUTURES PRICING
BY
SOUSHAN WU
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1984
ACKNOWLEDGMENTS
Intellectual development is a contract: it gives you leverage, and like an index futures contract, it is marked to the market with some "saftydeposit" margin. The initializer of my development contract is also the chairperson of my committee, Dr. Robert C. Radcliffe, to whom I wish to express my sincere gratitude for his extensive discussion and encouragement. Furthermore, the initial margin of my development contract is reduced by Dr. M.Y. Tarng, Dr. C.C. Chu, Dr. C.Y. Ho, Dr. Tony Lai, Dr. M.P. Narayanan, Dr. Roger Huang and Dr. Raymond Chiang. Their encouragement, valuable comments, and intellectual stimulation made this dissertation possible. Of course, the highest leverage is further inplemented by Dr. A.A. Heggested, Dr. Heim Levy, Dr. Richard Pettway, Dr. Raymond Chiang, and Dr. Rodger Huang, who have instilled in me knowledge of finance both in the class as well as through projects. Special thanks go to Dr. S.R. Cosslett, from whom I have learned more than econometrics, and to Dr. Raymond Chiang, from whom I have learned more than merely finance.
A special acknowlegment of gratitude is due to the National Science Foundation of the Republic of China, the National ChiaoTung University and the Department of Finance of the University of Florida, which have held a substantial equity interest in this venture. In providing the financial risk capital, these institutions showed overwhelming faith in me.
To my grandfather, Mr. TuChen Wu, I also wish to express my deepest gratitude. His spiritual support and constant encouragement have contributed substantially to my study over the years.
Finally, I would like to thank my wife, MeiLin Chang. Without her understanding and encouragement, my study at the University of Florida would not have been possible.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS .................................................... ii
KEY TO SYMBOLS ..................................................... v
ABSTRACT ........................................................... viii
CHAPTER I MOTIVATION OF THE STUDY ........................... 1
CHAPTER II INTRODUCTION TO STOCK INDEX FUTURES ............... 5
Contract Description .......................... ... . 5
Review of Literature ............................. 11
Note .............................................. 22
CHAPTER III THE PRICING OF THE INDEX FUTURES .................. 23
Notations, Assumptions, and Definitions ........... 23
Model ............................................ 28
Comparative Static Analysis ....................... 34
Summary .......................................... 38
Notes ............................................ 40
CHAPTER IV THE EMPIRICAL STUDY ............................... 41
A Prelimary Result ................................ 43
Basic Discussion: Measurement and Data Sets ...... 44 Test Methodology and Results ..................... 49
Summary ........................................... 64
Notes ........................................... . 66
CHAPTER V CONCLUSION ....................................... 83
APPENDICES
A DATA BASES ....................... . ............ 85
B RESULTS OF FIRST STAGE LEAST SQUARE ............... 88
REFERENCES ................................... 94
BIOGRAPHICAL SKETCH ................................................ 99
KEY TO SYMBOLS
t  Current period T  Maturity of the index futures contract F(t,T)  Current futures price maturity at T. F(T,T)  Maturity futures price S(t)  Value of the indexequivalent portfolio at t (which is
formed as the same value as of the spot value I(t)) I(T)  Value of the spot index at time T D  Total cash received from dividend payments on the indexequivalent portfolio during the period t to T
S(T)  Value of the indexequivalent portfolio at T which is formed
at t, includes dividend, equal to I(T) + D
Cpl  Cost of being long the indexequivalent portfolio C  Cost of being short the indexequivalent (which may
include foregone interest on the short sales proceeds) Cfs  Cost of being long the index futures C fl  Cost of being short the index futures X  Number of endowment units which investor i holds of the
index equivalent portfolio (This is exogenously given) X  Total number of endowment units of the indexequivalent
portfolio in the economy
Xsi  Number of units of the index portfolio held by the investor i Xfi  Net long number of units of index futures contract held by
the investor i
C  Initial cash of investor i b  Number of units of bond held by the investor i rf  Riskfree rate during this period (t,T)
 Risk tolerance of investor i
 Risk tolerance in the economy
Wti  Wealth of the investor i at time t which can be put among
bonds, indexequivalent portfolio and the index futures
WTi E i
0i2
Gi(Ei.2) a2
0 D
02
s
F 2
ï¿½sF
S&P 500
VLA NYSE
NYFE AMEX OTC CRSP
COMPUSTAT
 Terminal wealth of investor i at time T
 Expected value of the terminal wealth of investor i
at time T
 Variance of terminal wealth of investor i at time T
 Preference function of investor i, which is determined
by the expected value and the variance of the terminal wealth
 Variance of the spot index at time T
 Variance of the dividend payout which is carried over in
the spot indexequivalent portfolio
 Covariance of the spot index and the random dividend payout
over period
 Variance of the indexequivalent portfolio price
 Variance of the index futures price
 Covariance of the index futures price and indexequivalent
portfolio price
 Covariance of the index futures price and random dividend
payout
 Percentage price change of the index futures
 Rate of return on the indexequivalent portfolio
 Percentage price change of the index futures in terms
of spot index
 Correlation coefficient of the index futures price and
the indexequivalent portfolio
 Standard and Poor's Composite Average Index
 Value Line Composite Average Index
 New York Stock Exchange
 New York Futures Exchange
 American Stock Exchange
 Over the Counter
 Center of Research in Security Prices,
University of Chicago
 Standard & Poor's Compustat Service, Inc.
FD  Futures Discount FCESP  Future CertaintyEquivalent Spot Price
4
Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
AN EQUILIBRUIM MODEL OF INDEX FUTURES PRICING By
Soushan Wu
December 1984
Chairman: Robert C. Radcliffe
Major Department: Finance, Insurance and Real Estate
The objectives of this study, all pertaining to pricing efficiency, are as follows: (1) to develop a theory of cashfutures price relationships for stock index futures which includes dividend risk within the framework of the partial equilibrium approach; (2) to test whether a dividend risk premium has affected cashfutures price relationships; and (3) to provide a framework for evaluating the efficiency and hedging in the pricing of stock index futures. Two types of evidence are employed to attain these objectives. First, the principles of cashfutures price behavior derived from observation of longestablished futures markets serve as points of departure, with substantial consideration given to certain fundamental differences that arise from the specification of stock index futures. Second, empirical tests of the hypothesized equilibrium pricing are evaluated.
This study demonstrates that when the dividend risk is taken into
consideration, a covariance term between the dividend payout and the indexequivalent portfolio is added to the Perfect Market Model developed by an arbitrage approach. The study also shows that the dividend risk premium
viii
empirically increases with seasonality when time to the maturity increases. Moreover, in the last thirty trading days, the market bears a dividend risk level which is statistically insignificant.
CHAPTER I
MOTIVATION OF THE STUDY
With the existence of exchange markets, individuals are able to hold their personal endowment in terms of current consumption and investment for the future to consume. The exchange can be improved by enlargening the set of feasible patterns of consumption and investment over time and by sharing of uncertainty associated with consumption and investment in the future. With such a market, satisfaction is increased for the individual and for the society as well.
A new financial instrument, the stock index futures contract, has recently been introduced to faciliate intertemporal allocation of resource and risk sharing (since February 1982). This index futures market, like other existing futures markets, claims to provide two major advantages to market participants: (1) risk transfer and (2) price discovery (Garbade & Silber, 1983). Both effects allow the capital market to become more complete (Ross, 1976). Risk transfer refers to a hedger's ability to transfer price risk to another hedger or to a speculator (if no offseting hedger can be found). Specifically, the index futures market can separate market risk from unsystematic risk. Price discovery refers to the information available in explicit prices of the trading transaction. Both the pricing discovery and risk transfer have the ability to extend the range of investment and risk management strategies.
These two functions also result in conceptual economic models which relate existing futures prices to existing spot prices.
Conceptual models together with empirical tests which relate spot price and futures are very common in commodity contracts (although results of such models and tests are debatable). However, within the past few years in which trading has begun on stock index futures contracts, little theoretical modeling or empirical testing has been conducted. The studies of stock index futures to date suggest a puzzle: the actual stock futures prices do not conform with existing theoritical pricing models which are based upon a perfect market pricing model under the assumption of a nonstochastic dividend. The perfect market pricing model implies that the index futures price should be equal to the future certainty equivalent of today's spot price (at the maturity date) minus the accumulated dividend payout over this period (Cornell & French, 1983).
This dissertation is motivated by the fact that existing theoretical models of stock index futures do not stand up to empirical facts. In the earliest period of their trading (February 1982 to September 1983), actual stock index futures consistently sold for less than their "theoretically" derived values based on perfect market model (Figlewski, 1983a). A number of possible explanations have been offered by Modest and Sundaresan (1983), Cornell and French (1983), and Figlewski (1983a). Among these studies, Figlewski (1983a) classified the potential reasons as the socalled (market) equilibrium argument and disequilibrium argument. But research on these new financial instruments has just begun and none of the existing studies has been able to adequately explain the differences between actual stock index futures price and "theoretical" price of perfect market model.
Researchers reach their own conclusions by examining some specific dates. In the most recent study, Figlewski (1984) employs more complete data to argue that this futures market is in a longrun disequilibrium in
its early stage. Both Modest and Sundaresan's "short sales not fully used" argument (1983) and Cornell and French's "taxrelated timing option" (1983) employ a cashfutures arbitrage approach to obtain their pricing relationships of index futures based on the market equilibrium approach. And in both of these equilibrium models, there is an assumption that dividends to be paid on the spot indexequivalent portfolio are known with certainty.
However, if the dividend to be paid on the spot index is unknown in advance, then the strategy of "long indexportfolio and short index futures one unit each" is not sufficient to completely hedge the risk associated with such an unknown dividend payment. If an allowance is made for an unknown dividend payment, it is theoretically plausible that the index futures contract should be priced lower than what the socalled "perfect market" model suggests. This discount occurs because the individual who is long an index futures does not receive dividend payments, but does face a dividend risk. If the dividend payout is stochastic, index futures and the indexequivalent portfolio may not be of the same systematic risk level. Consequently, the arbitrage approach to pricing breaks down. A perfect arbitrage is not possible and index futures pricing models should not be developed around such arbitrage relationships. Instead, a more general equilibrium approach should be employed.
The objectives of this study are two: (1) to derive and (2) to test a closed form equilibrium price for a stock index futures contract, which includes the effect of unknown dividends. This closedform equilibrium futures price will also be linked with traditional Capital Assets Pricing Model (CAPM) framework and conventional hedging strategies after some rearrangement.
Following a basic review of index futures and a synthesis of past
studies in Chapter II, a basic pricing model with comparative statics analysis and interpretation is presented in Chapter III. Chapter IV provides econometric justification for the form of this predictive model and fits the forecast to actual New York Stock Exchange (NYSE) and Standard and Poor's Compositive (S&P 500) index futures based on a sample taken from New York Stock Futures Exchange and The Chicago Board, starting from the first trading contract to the December 1984 contract. Conclusions of this study are presented in Chapter V.
CHAPTER II
INTRODUCTION TO STOCK INDEX FUTURES
In this chapter, the legal characteristics of index futures contract are described in the first section. Following this, the various studies relating to this study are briefly reviewed.
Contract Description
A stock index futures contract is an obligation to buy or sell a hypothetical portfolio of all the stocks in an index at a stated price at a certain date. It can be liquidated before maturity.
At present, there are three main actively traded stock index futures contracts: the Value Line Index traded on the Kansas City Board of Trade, the Standard & Poor's 500 traded on the Index and Option market of the Chicago Mercantile Exchange, and the New York Stock Exchange Composite Index traded on the New York Stock Exchange. In addition, a New York Stock Exchange Financial Index Futures and other contracts, which are on narrower measures of equity market activity such as Utility and Transportation indices, have recently been introduced. Options on these three main indices are also available now.
An index futures contract is similar to other kinds of commodity futures contracts paid for on a unique installment plan. The investor who buys a futures contract agrees to buy one unit of a financial product at a specified maturity time at a specified futures price. The index future price is determined when the contract is written and
is specified in the contract. The stock index futures price should differ from current or expected future (t, or T) stock index prices for at least two reasons. First, the futures price is chosen so that no payment is made when the contract is written; i.e., at the initiation phase the futures contract has zero market value. But as the contract matures, the investor must make or receive daily installment payments toward the eventual purchase of the financial product. This is referred to as "markingtomarket." The total of the daily installments and the payment at maturity will equal the futures price specified when the contract was initiated. Second, the futures trader does not receive the dividends that are paid to the stockholder, but faces a dividend risk.
What makes futures contracts unique among installment plans is
that the daily installments are not specified in advance in the contract, but are determined by the daily change in the futures price. If the futures price rises, then the investor, who is long the futures contract, receives a payment from the investor, who is short. The payment is the rise in the futures from the previous day. On the other hand, if the futures price falls, the long holder pays the short holder the change in the daily futures price.
The effect of markingtomarket is to rewrite the futures contract each day at the new futures price. Hence the value of the futures contract after the daily settlement will always be zero since the value of a newly written futures contract is zero. When the contract matures, the long investor will have already paid or received the difference between the initial futures price and the futures price at maturity time. With these payments to his credit, he will have a balance due equal to the futures price at maturity. But the value of a futures contract
Therefore, at maturity the futures price must equal the current spot price. Thus the balance due is simply the current spot price at the maturity time. Unlike a forward contract, the value of a futures contract after settlementis always zero.
Some special characteristics of index futures contracts are
discussed in what follows. The margin on a futures contract represents a "good faith" deposit on the part of the buyer and seller. Margin requirements on stock index futures approximate 510% of the contract's value. This results in a high degree of leverage for the futures trader. Futures margins differ significantly from spot stock margins. They are much lower than stock margins and involve no extension of credit or expense on any unpaid (borrowed) balance. In addition, the futures margin must be restored daily during adverse price movements (marked to the market). Traders in a profitable position may withdraw any excess margin. In certain cases, margin can be posted in Treasury Bills, so interest can be earned while the trading program is in place. The delivery process in the settlement of futures contracts serves a positive economic function in that it assures a certain level of convergence between the cash and futures prices at the expiration date. This price correction is essential to the development of a successful futures contract. Cash/futures convergence in the stock index futures is virtually assured by the "cash settlement" procedure in these contracts. Rather than require the delivery of actual stock, all open positions are markedd to the market" at expiration. In addition, the delivery price is not determined in the futures pit, but by the actual cash market close on the last day of trading (like S&P 500 index futures), or the
average of the closes of the last and next last day (like the NYSE index futures case). This assures convergence of the futures price with the price for the underlying commodity, the spot index, and prevents any attempts to "squeeze," "corner," or otherwise manipulate the market. On the last trading day of the expiring contract, all three indices futures are traded until 4:00 P.M. (EST) only.
Do prices in one market have an impact on prices in the
other market? With cash settlement, the arbitrage possibility between the cash market and futures market is encouraged. An arbitrager can profit by going long a unit of index futures, short a unit of the indexequivalent portfolio and investing the balance in risk free bonds simultaneously, if the observed index futures price is lower than the "theoretical" price in mind. Meanwhile, he also can profit by shorting a unit of index futures, borrowing at the riskfree rate, and going long a unit of indexequivalent portfolio simultaneously, if the observed index futures price is higher than the "theoretical" price in mind. This cashfutures arbitrage is often referred to as basis speculation and is best represented as a type of "risk arbitrage." It contributes to the economic activity by both the risk transfer and price discovery functions of futures markets.
For taxable investors, under the current tax code, the gain or loss
on any futures transaction entered into after June 23, 1981, is be treated as 60 percent longterm capital gains, and 40 percent as shortterm, without regard to the period of time for which the position was held. It means that the tax rate of index futures trade is 32% currently, ije., (60% x 20) + (40% x 50)  32%.
Value Line Composite Average (VLA)
Published and maintained by Arnold Bernhard and Co., the VIA is an equally weighted geometric average of about 1700 stock prices expressed in index form (June 30, 1961 = 100.0) Nearly 90 percent of the stocks included in the VIA are traded on the New York Stock Exchange. Based on this spot index, a Value Line Index Futures Contract is specified in Table 2.1.
TABLE 2.1
Value Line Composite Index Futures
Exchange: Kansas City Board of Trade
Trading Months: March, June, September, December; 18 months forward
Trading Hours: 10:00 A.M.  4:15 P.M., N.Y. time
Contract Size: Futures price x $500
Minimum Price
Fluctuations: 5 points, equivalent to $25
Daily Price Limits: 500 points ($2,500)
Last Trading Day: Last business day of contract month
Trading began: Feb. 24, 1982
Standard & Poor's 500
This index, commonly called the S&P 500, is a broadly based arithmetic average, utilizing the share prices of 500 different companies: 400 industrials, 40 utilities, 20 transportation and 40 financial companies. The market value of the index is approximately 80% market value of all stocks on the New York Stock Exchange. Each stock in the index is weighted to reflect the stock's total influence on the index relative to its market value. To determine the weighting, the number of shares outstanding is multiplied by the price per share. Thus, a stock's total market value determines its importance in this index.
The S&P 500 has been widely accepted as a benchmark for portfolio manager performance as well as a measure of economic activity. It is one of the components of the Index of Leading Economic Indicators. The S&P 500 is based on the average weekly closing value of 19411943 and indexed to a value of 10. The stock index futures contract on this index is specified in Table 2.2.
TABLE 2.2
Standarded & Poor's 500 Index Futures
Exchange: Index and Options Market
Trading Months: March, June September, December, 18 months forward
Trading Hours: 10:00 A.M.  4:15 P.M., N.Y. time
Contract Size: Futures price x $500
Minimum Price
Fluctuation: 5 points, equivalent to $25
Daily Price Limits: 500 points ($2500)
Last Trading Day: Third Thursday of contract month
Trading began: April 21, 1982
New York Stock Exchange Composite
This index is an arithmetic average consisting of all common stocks listed on the NYSE. As with the S&P 500, each stock in the NYSE index is weighted in proportion to the stock's market value. The index is based on the close price of December 31, 1965, and indexed to a value of 50. The composite futures and financial futures based on this are specified as Table 2.3 and Table 2.4.
TABLE 2.3
New York Stock Exchange Index
Exchange: Trading Hours: Contract Size: Minimum Price
Fluctuation:
Daily Price Limits Last Trading Day:
Trading began:
New York Futures Exchange 10:00 A.M.  4:15 P.M., N.Y. Time Futures x $500
5 points, equivalent to $25 none
Second last business day of contract month
May 6, 1982
TABLE 2.4
New York Stock Exchange Financial Index
Exchange: Trading Months:
Trading Hours: Contract Size: Minimum Price
Fluctuation:
Daily Price Limits: Last Trading Day:
New York Futures Exchange March, June, September, December; 18 months forward
10:00 A.M.  4:15 P.M., N.Y. time Futures price x $1000
1 point, equivalent to $5 none
Second last business day of contract month. However, effective with December 1984 contract and all subsequent contracts, the last business day will be the third friday of the month.
Review of Literature
To date, three basic arguments have been proposed to explain why actual futures prices are lower than what the perfect market pricing model suggests. These include 1) a taxrelated timing option, 2) no full use of the short sales, and 3) longrun disequilibrium in the index futures market in the early stage.
In sum, the taxrelated timing option suggested by Cornell and French (1983), and the no full use of short sales suggested by Modest
and Sundaresan (1983) have been classified as equilibrium arguments by Figlewski (1983a). Furthermore, he believes that the market is in disequilibrium stage in the early stage.
In a market of no transaction cost, no tax, dividend certainty, and no information access cost, a perfect market pricing model is developed as follows:
F(t,T)=I(t) exp((rfd)) ............... (2.1)
where
F(t,T) : the index futures price at time t, maturity at T.
I(t) : the indexequivalent portfolio at time t.
rf : riskfree rate
d : dividend rate (total dividend payout from t to T
divided by the current spot index).
Due to possible imperfections of the capital market, as Modest and Sundaresan suggest, empirically observed futures discount may be explained by the fact that short sellers of spot seldom obtain full use of the proceeds of the short sales. They argue that if the proceeds of the short sale of spot are not available to be invested in the money market, the "short spotlong futures" arbitrage becomes profitable only when the futures price is below the current spot index by an amount greater than the dividend yield on the index portfolio because a short seller must pay the dividends on the shares he has borrowed. An upper bound and a lower bound thus are developed for index futures pricing. For illustrative purpose Modest and Sundaresan present the bounds and observed futures prices for the June 1982 and December 1982 S&P 500 stock index futures contract. The bounds are presented under the
alernative assumptions that short sellers have zero, half, and full use of the proceeds, both in the no dividend adjustment case and in the dividend adjustment case under the assumption of nonstochastic dividend. Thus, for each contract, six sets of bounds and prices are presented. For the cases where dividends are adjusted, the bounds are given by the eq(2.2) which can be written as
l(t) + C + Cfs  D l(t)  Cps  Cfl  D F(t,T) >
exp(rf) exp(rf)
............... (2.2)
where
Cpl : Cost of being long in the indexequivalent portfolio
C : Cost of being short in the spot (which may include
foregone interest on the short sales proceeds)
Cfs : Cost of being short in the index futures
I(t) : Price of one unit of indexequivalent portfolio at time t
D : Known dividend with reinvested in riskfree rate
rf : Riskfree rate
F(t,T) : Index futures price at t, maturity at T
Based on this equation, the figures for December 82 S&P 500 contract are reproduced here as Figure (2.1  2.3) for reference. This argument, according to Figlewski (1983a), only explains why arbitrage might not force the futures discount to disappear once it develops. It gives no insight into why the discount should exist in the first place. This means that the motivation to short the index equivalent portfolio is not clear. Besides, trading index futures involves no investment except some margin requirement which can be met by using securities such
Futures Prices
 '
 .,
    a
a
 a
 a
a
a a a a.
S 
a a a a a
 ~I
a I
I' ID
1rv41 1 rV5 1r0rr r16 1v 2 2TItTr t I rr' 1 t1 Irrr1 VTrr 2rr . 0 2 3 4 0 7 0 9 15 11 12 IU 14 1 10 17 ii 19 20 21 22
Figure 2.1
Source:
S&P 500 Contract Maturiting December 1982. Futures prices and bounds (zero use of proceeds, adjusting for dividends). Modest and Sundaresan (1983) p.30, reproduced by permission.
Week
Futures Price
12V~ 1 1~ ~ 110
  %~
'S
ï¿½ I,
I
Week
"yTr If 1*"v 7#8 eIT "IT IT "~I "rIT o irked, *ITT tr r, , I' s I T I# I v I tvrr oI ,1v 1 ri I o rs I rvrr I w v, rT ~TT"
2 a 4 5 0 7 8 0 10 11 12 13 14 15 10 17 10 10 28 21 22 Figure 2.2 S&P 500 Contract Maturiting December 1982. Futures prices and
bounds (halfuse of proceeds, adjusting for dividends).
Data plotted for April 21  September 15, 1982.
Modest and Sundaresan (1983) p.34, reproduced by permission.
Source:
Price
1 r 'T t I r"uTrrv IW "'I rr lrirlr r r1gi rT rr " 11 rrrjv wrrrr r 'Tr a 1 2 3 4 3 0 7 0 U I 1 12 15 14 13 10 17 10 ig 23 21 22
Figure 2.3
Source:
S&P 500 Contract Maturiting December 1982. Futures price and bounds (full use of proceeds, adjusting for dividends). Data plotted for April 21  September 15, 1982.
Modest and Sundaresan (1983) p.38, reproduced by permission.
Futures
1201 21
I I!%
10L3
Week
as Treasury Bill which would be held regard less of the futures today. Both "long the spotshort the futures" and "short the spotlong the futures" strategies are different in risk level and investment level.
Cornell and French (1983) point to a difference in the way stock and futures returns are taxed. For index futures all paper profits are taxed as if they had been realized by the end of the fiscal year or by the maturity of the index futures, whichever comes first. Returns on a stock portfolio, however, are taxed as short term gains if realized within one year. But if the holding period is extended beyond one year, the tax rate drops to 40% of the short term rate. Therefore, a stock portfolio offers a "taxrelated timing option" that the futures contract does not possess.
According to Cornell and French (1983), the timing option is a right to defer to pay the capital gain taxes. Consider an investor who buys an indexequivalent stock portfolio. If its value goes down in a year, he can sell and deduct the loss at short term rates. On the other hand, if it goes up he has the option to extend his holdings period to take advantage of the long term gains rates. Thus, for a taxable investor who knows the dividend payout in advance, Cornell and French derive a functional form for stock index futures to be priced below their theoretical level by an amount equal to the value of the timing option.I Consequently, the relative timing option is defined as the difference between the "theoretical" price and the observed price relative to the "theoretical" price. The relative timing option of some selected
contract is adapted in Table 2.5 for reference. However, the timing option is difficult, if not impossible, to compute a theoretical value (Wu, 1983), if one can not specify how long investors are going to hold the indexequivalent portfolio. In addition, the timing option must be positive. There exist, moreover, a huge volume of stocks held by tax exempt investors and by taxable investors whose holding periods are already greater than one year. If the effect of the timing option on futures prices were the main cause of the futures discount, these investors should be writing timing options by selling their stock and buying index futures and riskfree securities to increase their returns. Furthermore, if the dividend to be received on the index is unknown in advance, the perfect arbitrage argument used to derive the model breaks down.
Besides these two articles, other researchers which make more
comments but offer fewer convincing explanations on the index futures discount can be found in Figlewski (1983a). They are all branches of the market "equilibrium" argument.
Figlewski (1983a) presents a permanent market disequilibrium
argument. He believes that there exists no temporary short run disequilibrium in the sense that prices do not adjust to equate supply and demand but that there does exist a long run disequilibrium in the index futures market. In other words, actions of investors already in the market create profitable investment opportunities for outside investors who, for one reason or another, are slow to take them up. Reasons for this include an unfamiliarity with the new markets, intertia in developing systems to take advantage of the opportunities they present, legal aspects
TABLE 2.5
Relative Value of the Timing Optiona
S&P 500
Contract Days to Relative Value Maturity Maturity of Option I June 1982
June 18 1.69% September 108 2.77% December 199 3.52% March 83 290 4.00%
NYSE
Days to Maturity
28 120
211 304
Relative Value
of Option
1.95% 2.97% 3.20% 3.73%
September December March 83 June
September December March 83 June
September December March 83 June
77 168 259 350
45 136 227 318
15 106 197 288
1 July 1982
0.01% 0.95% 2.03% 2.93%
2 August 1982
0.82%
0.54% 0.14% 0.72%
1 September 1982
1.12% 1.41% 1.76% 2.33%
Source: Cornell and French (1983) p.691, reproduced by permission.
aThe relative value of the timing option is expressed as a
percentage of the stock price. It is estimated as
c(t)
where g:
i:
d (w):
r: R:
=IF(tT)/{I (t) [e~lirtT TtT (i)d(w)e(i)R(twT) (Tw)dw /(1g)
ft
capital gain tax rate ordinary income rate dividend rate at w bond forward rate loan forward rate
0.13% 0.83% 2.06% 2.97%
90 181 272 363
58 149 240 331
28 119 210 301
0.91%
0.62%
0.19% 0.36%
1.07% 1.39% 1.85% 2.42%
of trading futures for cautious portfolio managers, the time required to intergrate accounting procedures for futures trading into daily operations, etc. He regresses the discounts on a time trend and the number of days to expiration, to adjust for the fact that the discount must go to zero as a futures approaches maturity. He observes that futures discounts have been decreasing over time, after taking account of the effect of the time to expiration and the high degree of serial correlation in the relationship. Finally, he believes that the futures market is slowly coming into equilibrium.
In a more recent article, Figlewski (1984) presents some further results in the hedging performance and basis risk on the stock index futures. This empirical study stands on the ground of a "perfect market" pricing model, which is relevant to our topic. However, he argues that the effects of dividend risk, the length of the holding period, and the time remaining to expiration of the futures contract should be considered. He defines the "return" of index futures in terms of the spot index and takes the variance operator to get the hedge ratio which minimizes risk. Furthermore, he states that the risk minimizing hedge ratio is the portfolio's beta coefficient with respect to the market index if the hedge portfolio is held to the maturity and dividends are not random. He reports the effectiveness of Standard and Poor's 500 index futures in hedging major stock index portfolios over a one week holding period in the sample, July 1, 1982, to September 30, 1983. For a one week holding period, hedging a diversified portfolio weighted toward large capitalization stocks can yield fairly good risk reduction, from about 20 to 30 percent of unhedged portfolio's standard deviation. However, hedging
effectiveness is substantially reduced by the presence of unsystematic risk, even in the amount contained in a broadly diversified portfolio of small stocks like the American Stock Exchange (AMEX) and Over the Counter (OTC) portfolios. He concludes that a short duration hedge for an individual stock or a small portfolio might be quite unsatisfactory. Finally, because of basis risk, the minimum risk hedge ratio was less than the portfolio's beta in every case, with the adverse effects of overhedging being more serious for returns than for risk levels.
To explain the basis, Figlewski uses the perfect market pricing model for the futures price to prevent portfolio arbitrage. Price difference between this "equilibrium" price and the actual price is analyzed. The sample is then split into thirds to show the difference in the market's behavior over time. He shows that underpricing of futures was significant in the first third of the sample, but this was not true of the period as a whole for the nearest contracts.
In considering the sources of basis risk in a hedge of the S&P 500 portfolio itself, Figlewski believes that dividend risk was not an important factor, while hedge duration and time to expiration of the futures contract were, to some extent. With regard to the pricing of stock index futures, he finds that the significant underpricing that was widely remarked in the very early months of trading seems to have disappeared, and that deviations from the theoretical pricing relation have diminished. This implies that underpricing does not reflect an equilibrium differential, which a factor like the value of the tax timing option would cause.
In this dissertation, a dividend risk is considered to be a potential factor to explain the futures discount of perfect market model. Because the dividend might be uncertain before it is announced, a dividend risk premium might be priced in the market. Suppose the investor holds a hedge portfolio which consists of index futures and indexequivalent portfolio one unit each, the dividend risk should be considered. If the dividend is indeed unknown, a dividend risk premium explicit in an equilibrium model should be developed.
The following chapter, Chapter III, presents the theoretical setting in which the effect of an unknown dividend is taken into account so as to determine (1) the price relationship between the spot and the index futures, (2) the systematic risk of index futures contract, and (3) the hedging relationship between the spot and the index futures. All three equations are testable and estimatable.
Note
[1] The value of the timing options increases monotonically with the
variance of return, while return variance generally rises following a stock split. Consequently, Cornell (1984) uses stock splits data to test the theory of the timing option in the index futures market.
He finds that the timing option is not empirically important or that
the expected increase in variance is the same for all stocks.
CHAPTER III
THE PRICING OF THE INDEX FUTURES
In the perfect market pricing model, it is assumed that the dividend for the indexequivalent portfolio underlying the index futures is known in advance. Thus, there is capital gain risk but no dividend risk. The implication of this is that the theoretical equation might systematically overprice the index futures. In this chapter, however, index futures pricing is extended to incorporate stochastic dividend yields under a partial equilibrium approach, assuming that market participants are risk averse. Adopting a set of common assumptions, a closed form solution is found, which can link the pricing relationship with the CAPM and with the hedging strategy as well. We discover that a dividend risk premium is required in evaluating the index futures pricing.
To derive these results, a set of notations, definitions, and
assumptions are needed. These are presented in the first section of this chapter. In the second section we develop the model necessary for the analysis of futures contracts in the comparative statics analysis of these results. Consequently, several interpretations of this comparative statics analysis are provided in the third section. A summary of this chapter is provided in the forth section.
Notation, Assumptions, and Definitions
The notation, definitions and assumptions used here is similar to that used to derive the CAPM model as in Sharpe (1964), Lintner (1965), and Mossin (1966). We only add several assumptions to facilitate the
inclusion of an index futures contract, such as cash settlement, and stochastic dividend effect. We will also assume that the investor determines his current consumption level first and then emphasizes on the composition and size of his investment portfolio in these three assets: index futures contract, a spot indexequivalent portfolio and a risk free bond. We assume the investor preference function is of the form, Gi(Eia21), where Ei and 02i represent the singleperiod expected value and variance of value of the ith investor's portfolio of risky assets. We further assume that
SGi > 0, and 3Gi < O,
that is, greater expected values are preferred and variance of value is not.
Notations
t T
F(t,T) F(T,T)
S(t)
I(T)
D
S(T)
Xi
si
Current period
Maturity of the index futures contract Current futures price maturity at T Maturity futures price Value of the indexequivalent portfolio at t (which is formed as the same as value of 1(t)) Value of the spot index at time T Total cash received from dividend payment on the indexequivalent portfolio during (t,T) period Value of the indexequivalent portfolio at T which is formed at t, include dividend, equal to I(T)+D Number of endowment units which investor i holds of the index equivalent portfolio (This is exogenously given)
X si
Number of units of the index portfolio held by the investor i
Xfi Net long number of units of index futures contract
held by the investor i
Ci Initial cash of investor i bi Number of units of bond held by the investor i rf Riskfree rate during this period (t,T)
Risk tolerance of the investor i
Wti Wealth of the investor i at time t which can be put among
bonds, indexequivalent portfolio and the index futures WTi Terminal wealth of investor i at time T Ei Expected value of the terminal wealth of investor
i at time T
ai 2 Variance of terminal wealth of investor i at time T Gi(Eipa 2 ) Preference functions of investor i, which is determined
by the expected value and the variance of the terminal wealth a12 Variance of the spot index at time T CD2 Variance of the dividend payout which is carried
over in the spot indexequivalent portfolio
aID Covariance of the spot index and the random dividend
payout over period
aS 2 Variance of the indexequivalent portfolio price OF2 Variance of the index futures price aSF Covariance of the index futures price and indexequivalent portfolio price
aDF Covariance of the index futures price and random
dividend payout
r F Percentage price change of the index futures r S Rate of return on the indexequivalent portfolio RPercentage price change of the index futures in
terms of spot index
p Correlation coefficient of the index futures price
and the indexequivalent portfolio
Note: "Tidle" is used to denote random variable notation.
And "bar" is used to denote the mean value of random variable.
Assumptions
Al: The investor i deals with only three assets (i.e. index equivalent portfolio, index futures and bonds). The market of the specification is as follows:
Market Price at t Future Value at T # held Aggregate
Spot Market S(t) S(T) Xsi Xs Index Fututes 0 F(T,T)F(t,T) Xfi 0 Bond exp(rf) 1 bi B
Since this assumption can develop a one period pricing model only, the markedtothemarket effect is thus neglected.
A2: Homogeneous belief on the spot price, S(T) dividend payout, D, and index futures price, F(T,T), is over all investors.
A3: Every investor maximizes his utility function which is
Gi (EiCi2), where Ei and ai2 are the expected value and the variance of terminal wealth, respectively. It is at least second differentiable with respect to Ei and a2". The preference function, Gi, is also an increasing function of Ei and decreasing function of a 2. Furthermore, the marginal utility of the expected terminal wealth is decreasing
3 G i
S < 0)
a Ei
A4: The total units of demand in the spot market, X , is set in the form specified below for all the investors (N) in this market, namely,
EXsi ESsi Xs = Xs (stands for gross supply).
A5: The interest rate over the buying date to the maturity is
treated as constant. Basically, it follows that the price of the future
is the same as the price of the forward contract (Cox, Ingersoll and Ross, 1981)
A6: No taxes, no transaction costs, pricetaking investors, no indivisibility, and costless information are available to everyone.
Definitions
D1: S(T) E I(T) + D The indexequivalent portfolio, S(t), is formed at time t. From time t to time T, this portfolio receives a total dividend payout, D. The value of the indexequivalent portfolio after dividend payout is I(T) at time T. Thus, the index portfolio, S(T), at time T is the sum of I(T) and D.
D2: The risk tolerance of investor i is defined by
3G /DE
= Ti Efi
1
where n is the total risk tolerance in economy.
D3: The percentage of the futures price change (or the "return") of the index futures in terms of index futures is
F(T,T)F(t,T)
r
F F(t,T)
D4: The return of the spot indexequivalent protfolio
S(T)S(t)
r E
s S(t)
D5: The systematic risk of the index futures
Cov( r, F)
F Var(?)
S
D6: The percentage of the futures price change of the index futures in the terms of spot index is
F(T,T)  F(t,T)
R E
F I(t)
D7: The hedge ratio of the index futures against the spot index is
Cov(F(T,T), S(T)) asF
H Var( (T)) a 2
Model
This approach is to determine conditions for equilibrium of exchange of the three assets: the indexequilibrium portfolio, the index futures contract underlying this spot portfolio and the bond. Each individual brings to the market his or her present holdings of these three assets and an exchange takes place. This equilibrium approach needs to know what the price must be in order to satisfy demand schedules and also fulfill the condition that supply and demand be equal for all three assets. To answer this question, two requirements must be met. First relationship describing individual demand must be established. Second, these relations of all investors' demand are incorporated into a system to describe general equilibrium.
Assuming that there are a large number of risk averse investors labeled i (i=1,2,,N), we can consider the behavior of a typical investor. He or she has to form a portfolio by choosing from the following three assets, namely, the indexequivalent portfolio, the index futures, and the riskfree bond. The future value on these three assets is assumed to be a random variable whose distribution is known to
the investor (Assumption AI). Moreover, all investors are assumed to have identical perceptions of these probability distributions (Assumption A2). The future value on a whole portfolio is, of course, a random variable. The portfolio analyses mentioned earlier assume that, in the choice from all the possible combinations, the investor is satisfied to be guided by its expected future value and its variance only (Assumption A3).
It is important to make precise the description of a portfolio in these terms. It is obvious that the holdings of these three assets must be measured in some kind of units. Because the index futures contract is exactly a derivative asset under a particular indexequivalent portfolio, we select one "physical" unit of the spot portfolio as our measure unit and define expected future value and variance of future value relative to this unit.
It is convenient to give an intrepretation to the concept of
"future (dollar) value" by assuming discrete market dates with intervals of one time unit in an equilibrium perfect market (Assumption A4A6). The future value to be considered on any asset on a given market date may then be thought of as the value per unit that the asset will have at the next market date (including accrued dividends).
The main purpose of this dissertation is to compare the relations
between the price and the future value of the indexequivalent portfolio and the index futures. To facilitate such comparisions, a risk free bond is used as yardstick. Now, partial equilibrium conditions are capable of determining relative prices: to fix the indexequivalent portfolio's price and express the index futures on it.
The utility function of investor i is denoted as Gi(Eio1 2). Likewise, the current value of investor i's portfolio, or his budget constraint, is the following eq (3.1). Therefore, the terminal wealth of investor i's portfolio is expressed in eq (3.2).
The Budget Constraint:
Xi Wti = Xsi I(t) + bi exp(r f) + OXfi
.........(3.1)
The Terminal Wealth:
WTi = Xsi s(T) + Xfi [F(T,T)  F(t,T)] + bi
......... (3.2)
The expected terminal wealth and variance of the terminal wealth are listed as eq (3.3) and eq (3.4), respectively.
The Expected Terminal Wealth:
Ei E(WTi) = Xs S(T) + Xfi [F(T,T)  F(t,T)] + b
........ (3.3)
Variance of Terminal Wealth:
Y 12 Var (WTi)
si s 2fi af2 2Xsi Xfi asF
= Xsi2 (aI2 + 2oID + aD2) + x2fi aF2 + 2XsiXfi aIF + 2XsiXfiaDF ........ (3.4)
Formally, then, we postulate that each investor i, who behaves as if attempting to maximize the utility function, is subject to the budget constraint, the expected value and variance of his portfolio to form his or her portfolio. Forming the LAGRANGEAN for each investor in eq (3.5) and differentiating with respect to Xsi, Xfi, and bi yields the first order conditions in eq (3.6) to eq (3.9).
The Lagrangean Form:
Li =Gi (Ei, ai2) + Xi (Wti Xsi I(t)  bi exp(r f))
.. ............. (3.5)
Maximization:
Max Li
XsiXfi bi s.t. (3.3) and (3.4)
First Order Condition2:
3L G. aGi 2
((T) +D) + [2Xsi os + 2Xfi OsF] X i i
Xi I(t) = 0
aLi = aGi ~i = (F(T,T)  F(t,T) +
aXfi aEi
aG [2Xfi F2 + 2Xsi SF 0
ftF3aOF]
...o ........ (3.7)
. .o......... ...(3 .8)
o ......... *.... (3.9)
a Gi
=   Xi exp(r f) = 0
aEi
= Wti  Xsi I(t)  bi exp(rf) = 0
These firstorder conditions may be aggregated to derive the
equilibrium relationship. Substract eq (3.6) from eq (3.7), and rearranging, we obtain eq (3.10). G {[I(T)+D]  [F(T,T)F(t,T)]} Ei
+ 2[1(as CF s OFO2
s 2 sF X + (aSF as) Xfi] X i s(t)  0
i ............... (3.10)
............... (3.6)
Dividing both side of eq (3.10) yields eq (3.11) DG_/3_ [Y(T) +_D  F(T,T) + F(t,T)] + 2[X s(a2 _ a) 3Gi/ i2E
+ Xfi (OsF  OF)] 
aGi/aaEi
exp( rf) l(t) = 0
............... (3.11)
To obtain the pricing relationship, we have to aggregate over all investors.
To determine general equlibrium, we must also specify equality
between demand and supply for each asset. Recall that Xfi, Xsi, bi were defined as the units of (net long) index futures, spot indexequivalent portfolio and the bond held by investor i. These market clearing conditions can be written as the following: EXi0, ' X f X, Zbi = B.
fi ' i sX i,'
This essentially completes the equations describing a partial equilibrium. Also, in equilibruim, eq (3.11) must hold for all investors and if we assume that all investors have homogeneous expectations regarding E i and
2
ai we can sum eq (3.11) over i and define the risk tolerance, n , to arrive at the closedform in eq (3.12).3 F(t,T) = I(t) e rf + 2/n Xs (a s2  sF)  [D  Y(T) + F(T,T)]
2
=I(t) erf + 2/n Xs (as sF)
= (t) erf 2/n X D  D (3.12)
 I M et f + 2 n X s aDF E...............( .12
In eq (3.12), we have a term, 2Xs (a 2 a )/n, to measure the risk premium of the random dividend effect. The sign and value of this Dremium
is thus an empirical question. Dividing eq
yields eq (3.13) and eq (3.14), respectively.
l(t) exp(rf)  I(T) D = 2/n X a 2 s s
F(t,T)  F(T,T) = 2/n Xs sF
( Gi
(3.6) and eq (3.7) by
.......... (3.13)
..... (3.14)
After rearrangement of eq (3.13) and eq (3.14), we have eq(3.15). F(T,T)  F(t,T) = [I(T) + D l(t) exp(rf)] CsF
s
= S(T)  I(t) exp(rf)] asF
=a................ (3.15)
Both sides of eq (3.15) divided by F(t,T) yields eq (3.16)
aSF S(T)  I(t) (] (t)
r% F (exp(rf)  1)
(t) F(t,T)
Coy (r 7 F)
Var( f) (3 1f
8F S  r f ....... .... (3.16)
Equation (3.16) is similar to the conventional CAPM which applies to an asset holding no initial positions. That is, the introduction of index futures market does not change the basic structure of capital asset pricing under conditions of uncertainty. Hereafter, eq (3.16) is referred as LikeCAPM model. COV(r, ')/ VAR(r) is the risk level of the index
s F s
futures relative to the spot index equivalent portfolio. The covariance term, cov( , ZF), is the key element for the risk premium of the index futures contract. This should come as no surprise, since the index futures contracts do include dividend risk. Furthermore, due to no investment in the index futures at the beginning, we have no intercept term. However, if we take margin into account, we should have the full CAPM. This is to say that the margin is treated as the performance bond.
Now, let us examine the conventional hedging strategy: regress the index futures price change on the spot index change to yield the hedge ratio subject to a holding period as the price change duration. After dividing eq (3.15) by I(t), we arrive at eq (3.17).
asF
RF s F rf)
S
(F rf) (3.17)
H s f
With a position of long the spot indexequivalent portfolio and short the index futures at time t, using 1/aH as the hedge ratio, we would have the riskfree rate of return.
In the next section, we intend to demonstrate a comparative statics analysis of this model, to explain who are going to short or long.
Comparative Statics Analysis
The comparative statics of security risk premium in a meanvariance
context has received rigorous treatment in the literature.4 Including the index futures market, the comparative statics analysis of market equilibrium is not examined since the equilibrium price of index futures not yet throughly understood. We are interested in the factors affecting the optimal number of index futures contract. In this section, the types of comparative statics changes considered include the spot index futures price change, spot price change, the interest rate change, the expected dividend yield change, the expected basis change and the relative risk aversion measurement change. Statements thus are made concerning the expected effect on the demand of assets of an increase in these factors.
To facilitate a comparative statics analysis, we need to develop the optimal units of index futures and indexequivalent portfolio held by the investors.
aGi
Dividing eq(3.6) and eq(3.7) by , together with the budget constraint, aai
we have the optimal set as eq (3.18).
X si + Xfi asF = i/2 [I(T) + D  I(t) exp(rf)]
X si* asF + Xfi aF2 Ti/2 [F(T,T)  F(t,T)]
i e f i  Xsi s(t) ............ (3.18)
Rearranging, we find the optimal holdings to be eq (3.19)
* [(T) + D l(t) erf] .F  [F(T,T)  F(t,T)] asF
si Z
, [F(T,T)  F(t,T)] a 2  [Y(T) +  I(t) exp(rf)] asF
.fi
* erf~ * ()( 9
b i = e f (W  X s t . . . . . . . . . . . . . . .
where Z : 2/Ai (02s2F  a sF2)
These are functions of index functions price, spot index price,
interest rate, dividend payout, degree of risk aversion, etc. Then we do some partital derivatives to examine factors which might affect the optimal hold units of index futures.
For risk adverse investors (which we assume), the riskaversion
measure, ni, is positive. Therefore, Z, defined as 2(a s2 aF2 _ asF2)/ni, is positive also. Thus, we have the following results, eq (3.20) to eq(3.26).
s < 0 .... (3.20)
DF(t,T) z
ax i I' Xfi la rf
aXfi
as(t)
ax ( aXfi aD
aXf l aF(T,T) axfi ani
(t) exp (rf) 0sF
>0
z
exp( rf)sF > 0
z
IsF
Z
<0
..... (3.21)
..... (3.22)
..... (3.23)
2
S >0
z
[F(T,T)  F(t,T)]aS2  [j1(T) +D  I(t)exp(rf)] asF
0 if numberator = 0
< <
2
= 2 >0
 s
ni
....... (3.25)
....... (3.26)
A model of demand or supply for the index futures and indexequivalent portfolio in equilibruim stage is shown as eq (3.19). These results are based on previous closedform, pertaining to some restricted assumptions. Here, the optimal number of units of the index futures is derived first, assumed to be net long if positive. Equation (3.20) shows the law of demand in the index futures. The higher the index futures price is, the less one goes long the index futures. The index futures is a normal goods. The impact of the interest rate on the number of units of the index futures is shown in eq(3.21). The higher the interest rate is, the more the long position in the index futures would be. This result means that the index futures contract tends to be a free good if the interest
aF(t,T) asD
rate is very high. Furthermore, equation (3.22) demonstrates the relationship between holding the indexequivalent portfolio being long the index futures. The higher the spot index is, the more to long the index futures would be. So, these two financial products are substitute goods. Equation (3.23) shows the impact of dividend payout on the number of units to long index futures. If the expected dividend increases, the demand to long the index futures decreases. In equation (3.24), it is shown that expected maturity futures price affects the number of units to long the index futures. The higher the expected maturity futures price (or the expected profit to long the index futures) is, the more to long the index futures would be, other factors being constant.
According to the budget constraint and eq (3.25), individual
initial wealth affects the investment decision through the combined effect of the degree of the riskaversion, the expected price and the variance of the index futures price, and the covariance of indexequivalent portfolio price and index futures price. With regard to the performance bond, the performance bond holding is not influenced by the expectation of the index futures contract. It might suggest that bond holding is a supplementary good of the index futures market. In addition, the initial wealth does not directly affect the holding of these assets.
aF(t,T)
Comparative statics indicates that > 0. Thus the index aasD
futures price could increase, according to eq (3.26), when the underlying indexequivalent portfolio dividend yield is stochastic if the covariance of future value on the indexequivalent portfolio and the dividend payout increases. This is due to the covariance of dividend payout and the indexequivalent portfolio being equal to the variance of dividend payout
plus the covariance of dividend payout and the futures price. (i.e., Cov(D, S(T)) = Var(D) + Cov(F(T,T), S(T)). Therefore, if dividend payout increases following the increase of the indexequivalent portfolio, then the index value would also increase following the increase of dividend payout. This means that the index futures price would increase too. Thus, the higher the correlation between the indexequivalent portfolio and dividend payout, the higher the index futures price would be. The index futures price is influenced by the covariance term between dividend payout and indexequivalent portfolio. Intuitively, this has to do with the fact that the index futures price still has to face the dividend risk even if the dividend payout does not benefit the individual who trades in the index futures contracts in long side or in short side.
Summary
We have presented a closed form model for index futures pricing
in a partial equilibrium approach. When we allow for the existence of random dividend yield, this model does bear a risk premium for this random term. A covariance term of dividend payout and indexequivalent portfolio is added to the perfect market model. We can see that to evaluate the index futures contract is the same as a project evaluation. Market equilibrium causes the inflows received from being long a unit of index futures contract, (holding to the maturity i.e., F(T,T)F(t,T),) to be inflows from borrowing fund to purchase a unit of indexequivalent portfolio (valued as $500 times spot index) and hold this to the maturity of its corresponding index futures contract, over this period to collect the expected dividend yield and its I.. t rf 14
associated dividend risk premium (i.e., I?(T)I(t)e  2Xs Cov(DS(T))/n)).
It means that the cash inflow to long an index futures contract is quite similar to implementing a capital investment project with borrowing funds, over the holding period to collect its profit or loss which is marked to the market and including the uncertain dividend payout or earning performance. Meanwhile, with comparative analysis, we know the index futures is a normal good. The demand to be long index futures increases, if the interest rate, the indexequivalent portfolio price, the expectation of the future spot price, or the correlation between indexequivalent portfolio and dividend payout increases. On the contrary, the demand to be long the index futures decreases if expected dividend decreases. Moreover, the more riskadverse the investor is, the less the demand to long the index futures will be.
When we express this model in rate of return terms, somewhat surprisingly, the equilibrium market relationship between risk and "market return" on index futures contract is still of the same general linear form as that of the SharpLintnerMossin model. The primary testable implications of the model are that the linearity of the relationship between risk and "expected return," or the "hedge ratio" is greater than one, or the dividend risk premium exists in the market.
Notes
[1] If preference function is a quadratic function, then
identical to risk tolerance defined by Pratt (1964).
quaratic utility function is only a special function
this measure is Expected of Gi(Ei,a2).
[2] Without loss of generality, the utility function satisfies some
other condition such that second order condition of optimality is
satisfied.
[3] It is consistent with the Ederington (1979) hedge argument. If
dividend payout is a random term indeed, the hedge ratio is not
equal to one as the traditionalists argued. Furthermore, assuming
that the market setting as follows:
Market Price at t Future Value at t+1 # held Aggregate
Spot Market S(t) S(t+1) X . X
Index Futures 0 7(t+1,T)F(t,T) Xfi0
Bond exp(rf) 1 B
then we could have the pricing relation as eq (3.121)
F(t,T) = I(t)erf + 2/n X [Var(S(t+l)  Cov(S(t+l), F(t+1,T))]
+ [Drt+1 F t(t+1,T)  t (t+1)] ....... (3.12.1)
where D it Ft, I are expected value of D, F, I based on the
informahton set at Eime t.
With arrangement, it becomes eq (3.12.2)
[F(t,T)F t(t+l,r)]  (I(t)  I t(t+l,T)]
r
D t +I(t)(e f1) + 2/n X (Var(S(t+l))  Cov (:(t+l), F(t+1),T))]
,t+...... (3 12.2)
It shows the equilibrium basis relationship. The aggregate basis risk thus is affected by the interest rate, the dividend, the risk aversion in the economy, the market return, the total units of indexequivalent
portfolio in the capital market, and the holding period.
[4] The analysis of the spot market is reported by others in detail.
So, we only analyze the index futures case.
CHAPTER IV
THE EMPIRICAL STUDY
In chapter III, the theoretical price of an index futures contract using an equilibrium model was stated in eq(3.12). A major difference between this model and those based upon an arbitrage approach is the potential importance of an unknown dividend. A covariance term between the dividend payout and the indexequivalent portfolio is added to the equilibrium pricing model. In the arbitrage model, the dividend payout is assumed to be known in advance. In this chapter, we investigate whether this covariance term empirically appears to exist in the actual trading market.
The dividend risk argument of this dissertation is contingent on the existence of a covariance between dividend payouts and an indexequivalent portfolio. We have also linked the theoretical index price using the equilibrium approach to the conventional CAPM model and a hedging relationship as shown in eq(3.16) and eq(3.17), respectively. If dividend risk is important in index futures pricing, the LikeCAPM model, i.e., eq(3.16), suggests a null intercept as well as a systematic risk level greater than one. Therefore, the LikeCAPM model can serve as a supportive test. If dividend risk is present and priced in the markets, the LikeCAPM model predicts that the systematic risk is greater than one. In addition, the existence of a nonzero intercept in the LikeCAPM model is important in that it is suggestive of another missing variable, such as the margin requirement. If the intercept of the regression is not significantly
different from zero, we have no evidence that this is an important factor in pricing index futures. Similarly, a conventional hedge relationship between the spot and the index futures is derived, as shown in eq(3.17). This hedge relationship also serves as a supportive test to the dividend risk argument: the slope of the regression is expected to be greater than one if dividend risk exists in the market.
These three equations, eq(3.12), eq(3.16) and eq(3.17), serve as
empirical devices to evaluate the impact of dividend risk and to describe the behavior of any such dividend risk premium. The pricing equations are, strictly speaking, correct only in a one period case. This means that the futures contract is being treated as a forward contract.'
The dividend risk argument has two critical considerations which must be understood. First, when a dividend payment is declared by a company, there is no dividend risk present any longer between the date of the announcement and subsequent payment. However, for the indexequivalent portfolio, it is not easy to eliminate total dividend risk at any specified date, since not all companies in the indexequivalent portfolio announce dividend payments at the same date. A varying degree of dividend risk is therefore expected to exist in the market. The nearer the maturity date of the index futures contract is approaching, the less the corresponding dividend risk would probably be. Therefore, the last several trading days might bear insignificant dividend risk. Second, most companies pay dividends quarterly which causes a seasonality in dividend payout behavior. Since the dividends of most companies are declared before paid, the dividend risk premium of the index futures price might be affected by the behavior of the dividend payout pattern of the spot index.
For these two reasons, we report two tables of dividend payout
distributions. The first table (Table 4.1) describes the frequency of the number of days between the date of a dividend announcement and the eventual exdividend date for all companies listed in the S&P 500 and the NYSE. This table reveals the degree of certainty about future dividend payments at points in time prior to when the dividend actually is paid. The second table (Table 4.2) describes the monthly payout pattern during the years 19811983. These two tables motivate the basic empirical design of this dissertation.
In what follows, we discuss first how the prelimiary results in Table 4.1 and 4.2 were obtained. Following that variable measurement, test methodology, test hypotheses and data set problems are discussed. The main content of this second section is to examine the difficulty of using a direct testing methodology, and, in its place develop a dummy variable approach for indirect testing. In the third section, results of a stricter test methodology as well as tests of the LikeCAPM and the hedging relationship are evaluated. Finally, a brief summary is provided in the fourth section.
A Prelimiary Result
In this section, two tables are presented to highlight the empirical study: (1) a table regarding the frequency of dividend events and (2) a table regarding the seasonality of monthly dividend payments.
Based on the period 19821983, the dividend data on the CRSP tape was used to evalute the degree of dividend certainty over different
lengths of time. This is done by noting the number of days which transpire between the date of a dividend declaration and its eventual payment (actually, the exdividend date.) This was done for (1) all companies included in the S&P 500 and (2) all companies listed on the NYSE. These two indicies were chosen since it is exactly these indices on which the principle stock index futures are written.
The results are reported in Table 4.1. Notice that seventyfive percent of the observations had announcement dates within thirty days from the exdividend date. This is true for both the NYSE and S&P 500. It suggests that dividend risk is not present in the period up to thirty days before the exdividend date for seventy five percent of all firms. Thus, trading in the indexequivalent portfolio and the index futures bear substantially less dividend risk within the thirty days before exdividend date.
In addition, a monthly dividend payout pattern (from 1981 to 1983) is displayed in Table 4.2. There are four peak dividend payout months. They are February, May, August, and November.
These results suggests that the behavior of dividend risk may be characterized by seasonality and by the length of time before the contract matures.
Basic Discussion
Direct testing of eq(3.12) requires data on futures prices,
interest rates, spot price of the index, an expected dividend, and the covariance between dividend payout and the indexequivalent portfolio. Of these, the futures price, spot price, and interest rate are directly observable. The expected dividend could be estimated in a number of
ways. Unfortunately, the covariance term is a cross section term and presents a severe estimation problem. To compute the covariance between the dividend payout and the indexequivalent portfolio requires that we specify the holding period being considered. One can not simply examine the historical covariance between, say, one day spot price changes and dividends paid on that date. First, the dividends would be known on that date since they would have been previously declared. Second, this would focus only on a one day holding period. It would not begin to evaluate the covariance when the holding period is different. And, if we wish to evaluate holding periods longer than one day, it is almost impossible to specify which holding periods should be examined. Directly testing is quite difficult. We turn, instead, to an indirect test which uses a dummy variable approach.
But, it is still necessary to estimate the expected dividend
payout per unit for the indexequivalent portfolio. Denoting D as the expected dividend, the estimation method used is as follows:
Accumulated Dividend of the index
from t to T S(t)
                 x . ... . ........ (4.1)
Market Value of the index at t 500
In addition, we set up a dummy variable approach to represent the covariance term over the holding period. This is done by using the additive property of the covariance term to handle the holding period problem. The dummy variable coding is set to a specified value in an attempt to measure the holding period effect of dividend risk. Two approaches are used. First, a twelvecalendarmonth dummy variable approach captures the monthly effect. Second, an approach based on a threemonth payout pattern is used to capture any seasonal effect in the dividend risk premium.
Equipped with an estimate of the expected dividend, (D as defined
above) and the number of indexequivalent portfolio, (I(t)) in the market, eq(3.12) can be empirically tested as a regression of F(t,T) (adjusted by the expected dividend of indexequivalent) on the future certaintyequivalent portfolio, and the twelve dummy variables. However, to get a conventional Rsquare, we need to subtract from the right hand side the future certaintyequivalent portfolio. This results in a new dependent variable, which we call the "futures discount." This "futures discount" is regressed on the various dummy variables.
Using this model, we might still have two problems. First, a
serial correlation might arise. Second, we have to consider the stationarity of the residuals. In order to assure that the regression residuals are stationary, the regression should be performed on a rate of change or on a generalized difference price change. In fact, it is the rate of change in the futures price, not the price level alone, that is the variance of economic interest. Taking generialized first differences of eq(3.12), we solve the serial residual problem and the serial correlation adjustment could provide a more efficient testing.
In summation, the generalized difference equation states that the change in the index futures price, adjusted for both the expected dividend and the value of the future certaintyequivalent of spot portfolio, is equal to the dividend risk premium as captured by various dummy variables.
Data Sets
Following the preliminary result above, five formal tests are motivated: (1) the use of a calendarmonth dummy variable approach (2) the
use of seasonal dividend dummy variable approach (3) a test of the market pricing model, (4) a test of the LikeCAPM relationship, and (5) a test of the hedge relationship.
These five tests are performed on a data base of the two most commonly traded indices: the S&P 500, and the NYSE. There are six index futures simultaneously traded in the futures market, since every futures contract lasts eighteen months (except in the case of eariest contracts offered) and new contracts have maturity dates which are three months apart. For all futures contracts, daily closing prices (at 4:15 P.M. Chicago Time) are obtained. For the spots, daily closing prices are also observed (at 4:00 P.M. Chicago Time). The Treasury Bill futures series is used as an interest rate series. The sample period is from the first trading date of every futures to March 8, 1984. In all cases, the percentage price change (or the "return") of the index futures is computed as a oneday holding. In addition, the dividend payout is assumed to be reinvested at the riskfree rate daily during the holding period. Since all subsequent statistical measures are assumed to be serially independent price changes, serial correlation coefficients of order one will be computed for each futures price series in order to adjust to the fact that price tend to move together. Since we use generalized first difference least squares to obtain efficient estimators, all observations which involve Monday or the day after a holiday are excluded in order to avoid a different time span problem.
Unexpectedly, the dividend data series is not easily accessible. For New York Index Futures, the daily dividend payout series of the unit indexequivalent portfolio is only available from January 1, 1983.
This was provided by the NYSE statistics department. For the S&P 500, data is aggregated from the CRSP daily tape (from Center Research of Security Prices, University of Chicago) based on the December 1982 COMPUSTAT TAPE (from Standard & Poor's Computing Service, Inc) cusip list. Unfortunately, 20 overthecounter companies are not accessible from the CRSP tape and data from 13 other companies are not fully covered over the sample period. Therefore, 6.6% of the S&P 500 is missing in the data set, meaning that we only create an S&P 467 dividend and market value data set. In terms of both the dividend and market value on December 1982, this S&P 467 dividend payout data series underestimates the total S&P 467 dividend payout data series the total S&P 500 by 6.6%. What we need, however, is the dividend payout per unit of index futures. As a consequence, we divide the total dividend by the market value to get the dividend per unit. This "unit" dividend is still approximately acceptable. The effective sample period is from January 1, 1983 to December 31, 1983 for the NYSE Index Futures and from April 21, 1982 to December 31, 1983 for the S&P 500 Index Futures. The data base is listed in Appendix A.
Seven data subsets are named and tested for each contract. They are follows:
1. WHOLE: all observations of the index futures contracts.
2. LE30: all observations of the index futures contracts,
excluding the last thirty tradingday observations.
3. LAST30: the last thirty tradingday observations of the index
futures contract.
4. LAST90: the last ninety tradingday observations of the index
futures contracts.
5. LE90: all observations on the index futures contracts,
excluding the last ninety tradingday observations.
6. L120LE30: the last one hundred and twenty observations,
excluding the last thirty tradingday observations.
7. LE120: all observations of the index futures contracts,
excluding the last one hundred and twenty tradingday
observations.
The reason for forming these data sets is that they enable us to analyze the dividend risk in different respects. In the first place, WHOLE is necessary for us to have a whole picture of this "futures discount." Second, we need to know what happens to the nearest (to the maturity) contract. Therefore, LAST90 is useful. Third, the trading in the maturity month should have minimal dividend risk exposure. Thus, LAST30 is needed in order for us to see whether the perfect market model holds. Others are examined to see if there is a change in the pattern of dividend risk over different periods.
Test Methodology and Results
In the previous section, five tests related to three equations, eq(3.12), eq(3.16), and eq(3.17), were briefly discussed. In this section, the test methodology of each is presented in more detail. Results are reviewed in the subsequent subsections. The first subsection discusses the use of a calendarmonth dummy variable approach. The second subsection discusses the use of a dividendpayoutpattern dummy variable approach. The above two tests methods essentially employ the same concept except that they use a different dummy variable coding to capture the dividend risk premium. Based on initial results,
the last thirty trading days before the index futures mature might not be exposed to dividend risk. If so, the perfect market model should hold in LAST30 data set. Therefore, the test of the perfect market model is performed in the third subsection. Other related issues, such as the test of LikeCAPM and the test of the hedge relationship, are discussed in the fourth subsection as supportive tests of the dividend risk argument. Finally, the fifth subsection summarizes the findings of this chapter.
Use of a CalendarMonth Dummy Variable Approach
In eq(3.12), the dividend risk adjustment is contingent on the
existence of a covariance term between dividend payout and the indexequivalent portfolio. The dividend payout referred to is the accumulated daily dividend payment over the holding period. Assume that the dividend accumulating process is as follows: Tt1
D = E D t+i,t+i+1
i=O ........... (4.3)
Since the covariance has an additive property, we have the following equation.
Tt1
Coy (D, S(T)) E Cov(Dt+i,t+i+1, (T)) .......... (4.3)
i=O
The covariance term which is of interest depends upon two things:
(1) individual covariance over shorter period of time (say, monthly or seasonal) and (2) the number of such periods which will occur from the date of pricing to maturity, i.e., time.
To capture this covariance relationship, we will first use a monthly dummy variable approach as follows. Define the dummy variable coding, Mi, i=1,2,12, for the dividend risk premium of each calendar month.
Values of any Mi can vary from zero to two, depending on how many times the calendar month is duplicated over the holding period. For example, if an investor buys or sells a unit of index futures and holds from t to T, then the covered calendarmonth dummy variable of the twelve dummy variables is valued as one or two depending on whether the months are duplicated one or two times. Other noncovered calendar month dummy variables are set to be zero. For example, if we buy a December 1984 contract in June 1983, then the dummy variables standing for June through December are two and others are one. Similarly, if it is bought in June 1984, then the dummy variables standing for June through December are one and others are zero.
To facilitate the statement of empirically testable null hypotheses, the ex ante equilibrium relation can be restated as an ex post relation assuming that (1) the independent variables contain no measurement errors, (2) that they are stationary, and (3) that individual expectations defined over the variates are unbiased. For OLS estimators to be efficient, it is required that observations be serially uncorrelated. Unfortunately, serial correlation is usually present in most time series regression in the first step of the estimation process. In our test, if serial correlation occurs, the Durbin procedure is used to adjust the ordinary leastsquares regression procedure to obtain efficient parameter estimates. This is referred to here as the second step of the estimation process. This procedure involves the use of generalized differencing to alter the linear model into one in which the errors are independent. Economically, it implies that the prior price information is carried over to today by the degree of serial correlation. Estimation of this generalized differencing results in a new, more efficient, set of parameter estimates.
Since the empirical model consists of the index futures regressed on the future certaintyequivalent portfolio, and the twelve calendarmonth dummy variables without an intercept term, the Rsquare would be almost invariably be one. This is due to the exclusion of the intercept in the regression. Given this, a new dependent variable, which has been labeled futures discount, is created. Futures discounts is defined to be the value of the index futures price after subtracting the future certaintyequivalent spot portfolio and adding back the expected accumulated dividend payout per unit. In addition, the future certaintyequivalent spot portfolio must be an independent variable in the regression. Hereafter, we use FD and FCESP to indicate futures discount and the future certaintyequivalent spot portfolio, respectively.
To summarize, a regression of FD on M and FCESP is performed.2 The results of the first step regression are reported in Appendix B. The results of the second step regression are interpretated after the mathematical details of this test are examined. Test I: Test of eq(3.12) using monthly dummy variable approach
Step 1: The first step regression
Model: FD(W) = a M (W) + S FCESP(W) + e ...... (4.4)
where
T
1) FD(W) E F(W,T) + Z D(i) exp(rf (Ti)/(Tt)) i=W
 I(W) exp(rf (TW)/(Tt))
[M 1(W) 0
2) M(W) FKM2(W)J M i(W) I
2
MI2(W) i=1,2,..., 12
3) FCESP E I(W) exp(rf(TW)/(Tt))
4) Sw = PwI + w
5) t
(1) a =o (2) = 0
THE ALTERNATIVE HYPOTHESES:
(1) a# 0 (2) #0
These estimators, however, are not efficient due to serial correlation in the error term. To obtain an efficient estimator, a second step regression using the serial correlation estimate of the first step regression, p, is performed.
Step 2: The second step regression
Model: A FD(W) = a A M(W) + B A FCESP(W) + Ew .... (4.5)
where
1) A FD(W) = FD(W)  p FD(W1)
2) A M(W) = M(W)  p M(W1)
3) A FCESP(W) = FCESP(W)  P FCESP(W1)
4) t
(1) a = 0 (2) B =0
THE ALTERNATIVE HYPOTHESES:
(1) a 0 (2) 8' 0
Note that this linear transformation does not change the null and alternative hypotheses.
Results of step one on the data sets of WHOLE and LE30 are reported
in Appendix B, Table B.1B.2, respectively. Results of step two are in Table 4.3 and Table 4.4 for data sets of WHOLE and LE30.
We find that the slope coefficients of the calendarmonth dummy variables are mostly negative with significance at 1% significance level. Not surprisingly, the absolute value of the slopes are larger in the data set LE30 than they are in the data set WHOLE (which includes the last month of trading.) This implies that the earlier the transaction is, the larger the futures discount will be. Intuitively, the accumulated dividend risk (i.e., the sum of the coefficients of several calendarmonth dummy variables) is a potential factor which would cause this discount to decrease over time. It suggests that a transaction in the early trading days of contract is riskier due to the higher uncertainty of total dividend payout. If dividend uncertainty is reduced, such as at the time when many companies submit their 10K form and annual report around February, the dividend risk is not a significant factor in creating futures discounts any more. In some cases, the March dummy variable is positive with significance (at 1% significance level.) This positive effect might be caused by any number of things, a change in the investment opportunity set, for example. Occasionally, multicollinearity does occur in some regressions. This is caused by the fact that the observations all duplicate at the maturity (calendar) month, such as L12OLe3O consisting of maturity month observations, or the effective observations of some months are missing due to no active trading in that month. In this case, we simply drop one variable which is collinear to another one. The slope of the remaining variable in the regression would then be the sum of the slopes of the two which are multicollinear before we drop it. At any rate, it does not affect
results and their implications. There is evidence that dividend risk is priced in the market. Meanwhile, since all observations always duplicate at the maturity month, the calendarmonth dummy variable, which also represents the maturity month, would have a larger coefficient. This larger value actually is mixed by the true value with the intercept.3
Use of a DividendPayoutPattern Dummy Variable Approach
The results of the monthly dummy variable approach discussed above are consistent with the posibility of a dividend risk premium. They are, however, consistent with other explainations. For example, simply time alone (the greater the time to expiration the greater the discount due to whatever reason) could be the cause of these results. The only part of the tests which points strongly towards an exclusion dividend risk impact is the insignificance of February, a month in which corporate reports would tend to reduce risk. Yet the financial literature has not observed a strong February effect on other areas of risk measurement.
What is needed is a test which is more closely associated with dividends themselves, a test which confirms or rejects solely the dividend risk argument. That is precisely what the tests associated with seasonal dummies are designed to do.
In Table 4.2, a dividend payout pattern was seen with peak months in February, May, August, and November. Three dummy variables are defined in order to investigate the possible seasonality of dividend risk premium. These three dummy variables are called the dividendpayoutpattern dummy variables, So, SI, and S Their corresponding values are set as follows.
S = M2 + M5 + M8 + MII
S1 1 1 + M4 + M7 + M10 S0 M12 + M3 + M6 + M9
Since a futures contract is eighteen months forward, the value of the dividendpayoutpattern dummy variables can take on values between one and six.
The basic idea underlying this approach is to examine whether a
seasonality of the futures discount appears corresponding to the dividend payout pattern. In other words, we want to investigate the behavior of the dividend risk premium. In spite of this objective, the behavior of dividend risk might have other complications. The changing pattern along with its magnitude could be further estimated. The test procedure is the same as the first subsection. The mathematical details and hypotheses are as follows. Test II: Test of eq(3.12) in a dividendpayoutpattern dummy variable
approach
Step 1: The first step regression
Model: FD(W)  a S(W) + 8 FCESP(W) + c ...... (4.6)
where
T
1) FD(W) F(W,T) + Z D(i) exp(rf (Ti)/(Tt)) iw
 I(W) exp(rf (TW)/(Tt)) r two month before dividendpeak monthi
2) S(W) = one month before dividendpeakmonthI
 Lduring dividendpeakmonth
S 2(WY
= S
si(w)
6 i = 0,1,2
3) FCESP(W) = I(W) exp (rf (TW)/(Tt)
4) cw =p cwI + w
5) t
(1) =(2) 0
THE ALTERNATIVE HYPOTHESES:
(1) a# 0 (2) $ 0
Step 2: The second step regression
Model: A FD(W)  a A S(W) + 8 FCESP(W) + w
where
1) A FD(W) = FD(W)  p FD(W1)
2) A S(W)  s(W)  p S(W1)
3) A FCESP(W)  FCESP(W)  p FCESP(W1)
4) t
(1) a 0 (2) 0
THE ALTERNATIVE HYPOTHESES:
(1) 0#o (2) 8# 0
The results summarized are derived after performing step one and step two on WHOLE, LE30, LAST30, L120LE30, LE9o, LE120 data sets, respectively. Some of the results are quite significant. The results of the first step regression are presented in Appendix B, Table B.3B.5, respectively. The results of the second step regression are
reported in Table 4.5, Table 4.6, and Table 4.7.
Before examining the results, a review of the test's logic is appropriate. Suppose a quarterly dividend payout is just announced, the dividend risk should be partially or temporarily reduced. Will the announcement affect the structure of the dividend risk premium? This hypothesis expands on the results of Test I (monthly dummies.) The extent to which uncertainty is resovled by the announcement can be understood by examining the trend of the slope coefficient of dividendpayoutpattern dummy variables. If the announcement essentially resolves all uncertainty, the dividend risk premium would be zero. On the other hand, the announcement could represent partial resolution only.
In Table 4.5, Table 4.6 and Table 4.7, we find that the risk premium of two months before and one month before are significant over the following data subsets: WHOLE, LE120, LE90. However, the risk premium of the dividendpeakpayout month is not significantly different from zero. Furthermore, the observations of two months before the dividend peak month bear the sum of the dividend risk premium of two months before and one month before dividendpeakmonth, suggesting that dividend risk decreases when the dividend announcement is forthcoming. Finally, the uncertainty is resolved at the announcement. These results confirm the earlier conclusion that the futures discount can be explained by a factor which has seasonal pattern. Furthermore, since we are also interested in the changing pattern of dividend risk of the nearest contract, we apply the regression on the L120LE30 and LAST90. As shown in Table 4.7, only two months before the matuity is significant (at 1%
significance level). Recall that Table 4.1 shows that 98 percent of dividend events leave no dividend uncertainty within sixty tradingday of the exdividend date. Therefore, the fact that ther is no significance in both one month before and during the dividendpeakmonth dummy variables suggests strongly that the dividend risk is reduced when time to the maturity is within sixty days before dividend announcement. The dividend risk indeed disappears in the market for the very near contract. We thought that perfect market pricing model would hold in LAST30 at least.
Two step least square procedure could provide the better efficient estimators. The coefficients of the dummy variables in Test I and Test II, therefore, provide more reliable estimations of the dividend risk premium of the S&P 500 for every futures contract to the corresponding one of the NYSE futures contract. We find that S&P 500's dividend risk premium (in absolute value) is larger than that of NYSE's. Recall that our model is constructed in terms of a "physical unit." Therefore, the dividend risk premium is measured by a unit. But one should recall that the transaction value of S&P 500 index futures is currently greater than that of the NYSE index futures by approximately fifty percent. Therefore, it is not surprising that the S&P 500 bears larger dividend risk premium.
Test of the Perfect Market Model
Regressing FD on FCESP using the LAST 30 data set, we expect that both the intercepts and the slopes are not significantly different from zero with low Rsquare. This is due to the lack of any explanatory power of the FCESP variable. It means that the following model can
test the perfect market model. Test III: Test of eq(3.12) for LAST30 data set
Step 1: The first step regression Model: FD(W)  a + 8 FCESP(W) + e
w
where
T
1) FD(W) = F(W,T) + E D(i) exp (r, (Ti
.......... (4.8)
)/(Tt)
i=w
 I(W) exp (rf (TW)/(Tt)) 2) FCESP(W) E I(W) exp (rf (TW)/(Tt)
3) r = + E
4) t
(1) ai = 0 (2) =0
THE ALTERNATIVE HYPOTHESES:
(1) a# 0 (2) #o
Step 2: The second step regression
Model: AFD(W) i +8 A FCESP(W) + w
where
1) AFD(W) = FD(W)  P FD(W1)
2) AFCESP(W) = FCESP(W)  p FCESP(WI)
3) t
(1) c 0 (2) 80
THE ALTERNATIVE HYPOTHESES:
(1) a 0 0 (2) 8 # 0
......... (4.9)
The results are shown in Table 4.8. None of the parameters are significant. The perfect market pricing model is confirmed in the LAST30 data set. This is true for all contracts, including the very earlier stage contracts, such as March 1983 contract. This suggests that the index futures market was in equilibrium at an earlier stage than other researchers have suggested.
Supportive Tests: LikeCAPM and the Hedge Relation
Dusak (1973) starts from the CAPM to explain the "return" of a
futures contract in order to study individual commodity futures. The following equation is employed.
E(P(i, l))P' (i,0)
= Bi (E(RF)rf)
p(i,O)
where
R.: the return of the market portfolio
rf: riskfree rate
P(i,O): spot price of ith commodity at period 0 P(i,l): spot price of ith commodity at period 1
p'(i,0): futures price of ith commodity at period 0
The above form is similar to our hedging relation. However, the model was obtained from a heuristic discussion as opposed to an equilibrium derivation which was used in this research.
The test hypotheses and the regression used to test the LikeCAFM and the hedging relation are follows: Test IV: Test of eq(3.16), the LikeCAPM
Model: F + rf) ....... (4.10)
THE NULL HYPOTHESES:
H:c =0, 8=1
THE ALTERNATIVE HYPOTHESIS:
HI: c# 0 or a * 1
Test V: Test of eq(3.17), the hedging relation
Model: R=cz+8( rf)
THE NULL HYPOTHESIS:
H: a=0, B=1
THE ALTERNATIVE HYPOTHESIS:
HI: a # 0 or 8 # 1
Both regressions' null hypotheses state that the intercepts are zero and the slopes are one. Under the dividend risk argument, we also expect the intercepts of eq(4.7) and eq(4.8) to be zero. But the dividend risk argument suggests that the slope of eq(4.7) and eq(4.8) will be significant and greater than one. As noted before, serial correlation would be adjusted if necessary. However, performing ordinary least square at the rate of return level might adequately avoid serial correlation in general.
The above two tests are performed on all of the seven data subsets. The results of the LikeCAPM test are reported in Tables 4.94.12. No intercepts of the regressions are significantly different from zero. The statistical insignificance of the intercept term only indicates that the regression line does go through the origin. This suggests that the margin requirement does not affect the index futures pricing and that no abnormal return is earned corresponding to its systematical risk. The index futures is priced efficiently in general, even in the early stage of the index futures market development. Examining the trend of the slope coefficient over the different data subsets, we find some evidence that
earlier observations have greater slope values. This is consistent with the belief that less dividend risk is priced in the market as time approaches the maturity date of the index futures contract. This result is consistent with Test I suggesting that the effect of dividend risk decrease over time to the maturity. The LikeCAPM test actually is derived from Test I, after we have defined the term "return" of index futures contract. It is therefore not surprising that the results are not changed. In Table 4.12, the slope is not significantly different from one. This would support our conclusion that the perfect market pricing model holds in LAST30.
The results of the hedge relationship are reported in Table 4.134.16. Again, the intercepts are not statistically significant over most contracts. However, the main idea of this test is to examine the behavior of hedging overnight. Thus, our main concern is the slope coefficient. The results, however, do not lend support to the claim of the null hypothesis that the slopes, in general, are equal to one. It means that the overnight hedge ratio is less than one since the slope is greater than one. One day perfect hedge of one index futures to one indexequivalent portfolio is impossible. Dividend risk, along with holding period and interest rate change, appear to be important factors for short term hedgers to consider.
In sum, the extended analysis suggests that the basic findings is robust for many different definitions of the variables used in the analysis. The results are basically suportive of the dividend risk hypothesis since they adequately explain the futures discount, in the sense that the dividend risk premium is priced in the market and is negative for the investor who shorts the index futures.
However, there is one puzzling aspect of the results in most tables. The slope coefficients of the FCESP are significantly different from zero. This variation in the reported value may be caused by four reasons. In the first place, it might be due to a term structure problem; thus a multiperiod pricing model should be developed to guide the further empirical study. In the second place, it might be due to a measurement error of the interest rate series for its unmatched maturity date to futures price series. In the third place, the approximation technique is not totally accurate regarding the collected dividend payout series. Finally, there might exists the liquidity risk associated with the maturity dynamics of the earier days' trading. Future research should be done in this direction.
Summary
This chapter presents an empirical test of whether a dividend risk is priced in the index futures market and whether the pricing model explains the structure under study. We empirically investigate the closed forms developed in Chapter III. The results support the dividend risk argument. Futures discounts could be explained by a factor which increases with seasonality as the time to maturity increases. Dividend uncertainty is resolved by dividend announcements and as the contract approaches maturity. The unit dividend risk premium is estimated using the pricing model developed in this dissertation. Due to the greater transaction value of S&P 500 index futures, S&P 500's unit dividend risk premium is priced more in the market than NYSE index futures'. Of course, one has to interpret these comparison with considerable caution. In sum, the results reported in this chapter may be
65
of use in analyzing historical price change, evaluatin market efficiency, and developing strategies to identify and exploit arbitrage opportunities.
However, the use of models of equilibrium pricing under uncertainty in empirical study of futures markets is very limited. As yet, no precise pricing model of index futures contracts has been found which offers completely satisfactory explanatory power, though some progress has been made here.
66
Notes
[1] Cox, Ingersoll, and Ross (1978) examine the theoretical differences
between forward and futures prices in a variety of contests. However,
Cornell and Reinganum (1981), and Elton, Gruber, and Rentzler (1982)
indicate that the difference is economically insignificant.
[2] The reason to include FCESP in the right hand side is to make the
test completely meet the econometric considerations. See Pindyck
and Rubinfeld (1981), p. 82.
[3] However, if we include the intercept in these regression a new
multicollinerity arises.
TABLE 4.1
Frequency of Days Between the Announcement Date and the Exdate of
Dividend Events in NYSE and S & P 500 19821983*
Days Accumulated Percentage Accumulated Percentage of NYSE of S & P 500
5 0.06 0.06 10 0.34 0.36 15 0.51 0.53 20 0.60 0.61 25 0.68 0.68 30 0.75 0.75
35 0.81 0.81 40 0.85 0.86 45 0.90 0.90 50 0.93 0.93 55 0.95 0.95 60 0.97 0.97 98 1.00 1.00
Note: * Dividend Code of CRSP tape, 1212, 1232, 1239, 3225, 3285,
3723, 2763, 3823, 3825, 3863, 4533, 4822, 5523, 5533, 6521,
are excluded. In addition, 20 over the counter companies of
S & P 500 are not available. In sum, NYSE has 11992 dividend
events and S & P 500 has 4550 dividend events
TABLE 4.2
198183 Monthly Dividend Yields for the NYSE Composite Index#
Month
January February March April May June July August September October November December
1983
3.49% 13.68% 6.69% 4.62% 14.57% 6.38% 3.46% 13.99% 7.70% 4.65% 14.27% 6.50%
1982
2.16% 9.84% 4.92% 3.60% 10.92% 4.08% 2.76% 11.76% 4.08% 3.36% 9.24% 3.00%
1981
2.24% 8.39% 3.83% 2.31% 8.15% 3.87% 2.24% 8.90% 3.72% 4.15% 8.68% 3.63%
Note:
# The dividend yields are measured by the difference between the
monthly valueweighted return, including dividends, and the
valueweighted return, excluding dividends, for the New York
Stock Exchange Stocks. The yields have been converted to
annnual estimates by multiplying the monthly estimates by 12.
The data is from CRSP monthly tape.
Test I: FD = a M + 0 FCESP Second Step Least Square
Data Set: WHOLE
Market Index: S & P 500 NYSE
Contract #
(Maturating) 8206 8209 8212 8303 8306 8309 8312 8312 8403
Ml na na na 0.79*  1.08**  0.89** 1.43** 0.57** 0.47**
M2 na na na 0.71 0.89** 0.90** 0.84** 0.46** 0.46** M3 na na na 7.09 0.05 0.44 0.04 0.01 0.06 M4 0.08 0.53 1.32** 0.38  1.43**  1.31** 1.36 0.53** 0.48**
M5 0.41 0.48 0.38 0.08  0.13  0.65* 0.58** 0.37** 0.27**
M6 2.99 1.53** 2.33** 2.49** 11.63**  1.11** 0.91"* 0.40** 0.47**
M7 na 0.81** 1.10** 1.35** 0.53**  1.19** 0.91** 0.31* 0.15
M8 na 0.09 0.61 0.58 0.70 16.45** 0.93** 0.48** 0.47**
M9 na 4.87 0.75 0.43 0.29 0.54** 0.64** 0.45** 0.37**
ML0 na na 0.61 0.73* 0.28  1.45 0.71** 0.36** 0.39**
MIl na na 0.05 0.28 0.75** na 0.02 0.06 0.18 M12 na na 7.01* 0.82* 0.81**  0.89 3.53 2.93 1.28
FCESP 0.07 0.08 0.07 0.06 0.07** 0.11** 0.03 0.05 0.02
R2 0.29 0.61 0.86 0.91 0.96 0.97 0.98 0.97 0.98 DW 1.69 1.50 1.85 1.99 1.87 2.06 1.90 2.04 2.27 p 0.10 0.24 0.07 0.00 0.03 0.02 0.01 0.01 0.04 df 28 73 118 143 121 114 156 171 171
Note: *
*#
Significant at the 5% level of significance Significant at the 1% level of significance 8206 stands for YYMM, meaning that June 1982 contract. Others are the same format.
Test I: FD = a M + 8 FCESP Second Step Least Square
Data Set: LE30
Market Index: S & P 500 NYSE
Contract
(Maturating) 8206 8209 8212 8303 8306 8309 8312 8312 8403
MI na na na 0.67 1.09** 0.89** 1.43** 0.60** 0.54**
M2 na na na 3.57A 0.89** 0.92** 0.85** 0.48** 0.48** M3 na na na 10.28A 0.04 0.46 0.03 0.01 0.05 M4 0.29 0.38 1.31** 0.37 1.42** 1.29** 1.36** 0.57** 0.55**
M5  3.97A 0.84 0.40 0.08 7.70A 0.65* 0.58** 0.39** 0.31*
M6 10.21A 1.43* 2.32** 2.50** 19.05A 1.12** 0.91** 0.43** 0.50*
M7 na 0.36 1.09** 1.35 0.53 1.16** 0.92** 0.34** 0.18
M8 na 39.18A 0.64 0.59 0.66 17.24 0.92** 0.56** 0.52**
M9 na 48.72A 0.72 0.43 0.32 0.60 0.64** 0.49** 0.43**
ML0 na na 0.69 0.73 0.30 1.45 0.84** 0.41* 0.43**
MIl na na 10.47A 0.28 0.75* na 0.20A 0.76A 0.20 M12 na na 17.67A 0.82* 0.91** 0.88 3.54A 3.89A 1.06
FCESP 0.26 0.18 0.07 0.06 0.07* 0.11** 0.03 0.05 0.01
R2 0.85 0.62 0.87 0.91 0.96 0.97 0.98 0.98 0.98 DW 1.74 1.42 1.82 2.00 1.86 2.06 1.90 1.93 2.08 P 0.01 0.25 0.08 0.01 0.02 0.02 0.02 0.03 0.01 df 10 56 102 126 110 104 140 1.62 171
Note: * Significant at the 5% level of significance
** Significant at the 1% level of significance
A Perfect multicellinearity
TABLE 4.5
Test IV: FD = a S + a FCESP
Second Step Least Square
Data Set: WHOLE
2 month 1 month during FCESP R2 DW p df Contract before before
S & P 500:
8206 2.9925 0.0801 0.4134 0.4134 0.29 1.69 0.10 28 8209 1.4823** 0.7420** 0.2742 0.0216** 0.60 1.47 0.25 76 8212 1.2417** 0.3397 0.0699 0.0113** 0.72 1.98 0.00 124 8303 0.8653** 0.0594 0.1075 0.0072** 0.75 2.19 0.10 152 8306 0.6663** 0.8150** 0.0479 0.0016 0.91 1.87 0.03 130 8309 0.4722** 0.7632** 0.0782 0.0039 0.89 2.28 0.03 122
8312 0.3879** 0.6263** 0.0942 0.0087** 0.91 2.13 0.09 165
NSYE:
8312 0.2124** 0.2077** 0.0540 0.0063** 0.88 2.30 0.11 180
8402 0.2205** 0.0738 0.0511 0.0182** 0.81 2.49 0.16 180
Note: * Significant at the 5% level of significance
ï¿½* Significant at the 1% level of significance
TABLE 4.6
Test IV: FD = a S + 8 FCESP
Second Step Least Square
Data Set: LE30
2 month 1 month during FCESP R2 DW p df Contract before before
S & P 500:
8209 1.3500** 0.6906* 0.3333 0.0231 0.60 1.4140 0.2611 59 8212 0.3498** 0.4474* 0.1540 0.0082 0.72 0.9297 0.0345 108 8303 0.9157* 0.0395 0.0755 0.0056 0.78 2.1905 0.0963 135 8306 0.6659** 0.8098** 0.0433 0.0015 0.92 0.8701 0.0372 119 8309 0.4696* 0.7531** 0.0758 0.0039 0.89 2.2883 0.0291 112 8312 0.3569** 0.6730** 0.1370 0.0099 0.92 2.1349 0.0927 149
NYSE:
8312 0.2293** 0.2387** 0.0674 0.0065* 0.90 2.0932 0.0541 171
8403 0.2389** 0.1097 0.0687 0.0166** 0.87 2.2406 0.0929 180
Note: * Significant at the 5% level of significance
ï¿½* Significant at the 1% level of significance
Test IV: FD = a S + 0 FCESP
Second Step Least Square
2 Month 1 Month 2
Contract before before during FCESP R DW p df Data Set
S&P 500
8212 1.8667** 1.1869** 0.0429 0.0112** 0.75 1.84 0.02 43 LAST90
1.5864** 1.6057** 0.2042 0.0112** 0.86 2.00 0.00 43 L120LE30
1.8085** 0.9664** 0.0422 0.0160* 0.86 1.48 0.25 76 LE90
1.6662** 0.9697** 0.1675 0.0174 0.85 1.44 0.27 60 LE120 8303 0.6858* 1.0411** 0.6211* 0.0017 0.68 1.95 0.00 43 LAST90
1.0860 0.7261 0.6531 0.0144 0.83 1.98 0.00 41 L120LE30
1.0413** 0.1913 0.1612 0.0004 0.82 2.18 0.09 104 LE90
1.1745** 0.2431 0.3182 0.0044 0.83 2.12 0.07 89 LE120
8306 0.0465 1.4533 0.0223 0.0036 0.86 1.78 0.11 39 LAST 90
0.3237 1.3649 1.0094 0.0098 0.89 1.78 0.10 43 L120LE30
1.2497** 0.6957** 0.0401 0.0076** 0.95 1.79 0.04 87 LE90
1.1532** 0.6213* 0.4020 0.0079 0.96 1.78 0.02 71 LE120 8309 0.2531 0.3086 8.8376 0.0695 0.85 2.02 0.03 36 LAST90
0.6485** 0.3852 0.2731 0.0067 0.76 1.92 0.04 42 L12OLE30
0.2636 1.5329 0.2014 0.0057 0.96 1.96 0.06 81 LE90
0.3397 1.7241 0.4360 0.0078 0.97 1.98 0.04 65 LE120 8312 0.3343* 0.5339** 0.0958 0.0029 0.59 1.94 0.01 43 LAST90
0.4086** 0.5969** 0.6680** 0.0128** 0.85 1.92 0.01 43 L120LE30
0.3549* 0.8384** 0.0441 0.0048 0.94 2.05 0.06 117 LE90
0.4295 0.8290 0.0781 0.0028 0.94 2.08 0.08 101 LE120
NYSE
8312 8.5983** 0.1328 0.0531 0.1361** 0.74 1.88 0.09 35 LAST90
0.2972** 0.1952 4.4184 0.0871 0.82 1.60 0.13 42 L120LE30
0.1747** 0.2953** 0.1034 0.0016 0.93 2.05 0.03 141 LE90
0.2468* 0.3247** 0.0064 0.0025 0.94 2.07 0.04 125 LE120
Significant at the 5% level Significant at the 1% level
of significance. of significance.
Note: *
Test V: FD  a + 0 FCESP
Data Set: LAST 30
Regression Type: First Stage Least Square Second Stage Least Square coefficient a 8 R2 DW p df a 8 R2 DW p
S&P 500
8209# 4.34 0.03 0.05 1.65 0.16 14 3.13 0.03 0.02 1.68 0.14
8212# 7.16 0.05 0.05 1.84 0.06 13 5.76 0.04 0.03 1.98 0.00
8303 28.325* 0.19 0.35 1.33 0.27 14 18.60 0.17 0.21 1.75 0.09
8306## 25.99** 0.16 0.41 1.55 0.19 13 19.97* 0.15* 0.32 1.90 0.03
8309 6.36 0.04 0.03 1.38 0.20 14 4.38 0.03 0.04 1.74 0.02
8312 1.64 0.01 0.00 1.69 0.06 13 1.36 0.01 0.00 1.78 0.02
NYSE
8312 6.18 0.08 0.15 1.52 0.04 6 6.21 0.07 0.16 1.54 0.03
Note: *
**
Significant at 5% significance level Significant at 1% significance level Joint tests of null hypotheses, significant at 5% significance level Joint tests of null hypotheses, significant at 1% significance level
TABLE 4.9
Test II: rF
= + 5F (rS  rf)
Data set: WHOLE
Market Contract D F R_2 DW p df
S & P 500
8209 0.0002 1.21** 0.78 2.28 0.16 78 8212 0.0004 1.20** 0.78 2.58 0.29 126 8303 0.0003 1.20** 0.75 2.83 0.42 154 8306 0.0004 1.02 0.70 2.57 0.27 132 8309 0.0005 1.20** 0.74 2.79 0.33 124 8312 0.0000 1.08 0.79 2.70 0.37 167
NYSE
8308 0.0008 1.29** 0.80 2.52 0.24 41 8306 0.0002 1.57 0.20 2.57 0.28 81 8309 0.0001 1.02 0.54 3.08 3.08 127 8312 0.0004 1.16** 0.80 2.51 0.23 182 8403 0.0004 1.16** 0.77 2.67 0.23 182
Note: *
**
Significant at the 5% level of significance Significant at the 1% level of significance
TABLE 4.10
Test II: rF = a + aF (rS  rf)
Data set: LE30
Market Contract M BF R2 DW p df
S & P 500
8209 0.0005 1.36** 0.72 2.19 0.13 61 8212 0.0006 1.23** 0.78 2.56 0.28 110 8303 0.0002 1.21** 0.75 2.87 0.44 137 8306 0.0004 1.01** 0.70 2.57 0.27 121 8309 0.0007 1.23** 0.74 2.83 0.35 114 8312 0.0000 1.08** 0.78 2.70 0.37 151
NYSE 8303 0.0008 1.33** 0.83 2.79 0.41 28
8306 0.0002 1.59 0.20 2.56 0.28 77 8309 0.0001 1.02 0.54 3.05 0.53 126 8312 0.0004 1.16** 0.77 2.45 0.23 173 8403 0.0004 1.16** 0.77 2.57 0.23 182
Note: *
**
Significance Significance
at the 5% level of significance at the 1% level of significance
TABLE 4.11
Test II: rF= +a
(rS  rf)
Contract
S & P 500:
0.0003 0.0014
0.0002
0.0009
0.0004
0.0015
0.0005
0.0000
0.0010
0.0005
0.0003
0.0001
1.23*
1.54**
1.16 0.88
1.18
1.24* 1.23
1.26**
1.27** 1.24** 1.20** 1.23**
0.81 0.74 0.69 0.84
0.77
0.78 0.79 0.78
0.84 0.84 0.73
0.73
0.0009 1.23* 0.84
0.0012 1.23* 0.82
0.0002 0.98 0.68
0.0001 0.97 0.67
0.0002 0.0002
0.0013
0.0015
0.0004 0.0002
0.0001
0.0002
0.96 1.03
1.38** 1.40**
0.94 0.93 1.12*
1.15
0.88
0.84 0.74 0.73
0.86
0.88 0.77 0.76
2.41 2.46 1.74 2.37
2.98
2.87 2.28 2.18
2.48 2.75 2.89 2.85
2.32 2.14 2.58
2.63
2.94 2.64 2.93 3.05
2.49 2.52
2.73 1.17
p
0.25
0.24
0.05
0.30
0.49
0.44
0.14
0. 11
0.27
0.40
0.44
0.44
0.20
0.07
0.27
0.29
0.47
0.33
0.36
0.38
0.25
0.27
0.39
0.00
df Data Set
45 43 106 91
41 45 89 73
38
44 83
67
45 45 119 103
LAST90 L12OLE30 LE90 LE120
LAST90 Li20LE30 LE90 LE120
LAST90
L12OLE30 LE90
LE120
LAST90
L12OLE30 LE90 LE120
LAST90 L12OLE30 LE90 LE120
LAST90 LI20LE30 LE90
LE120
NYSE
8312 0.0004 1.03 0.85 2.19 0.07 37 LAST90
0.0001 0.99 0.85 2.11 0.09 44 L120LE30
0.0005 1.19** 0.79 2.49 0.25 143 LE90
0.0007 1.21 0.80 2.49 0.25 127 LE120
Note: * Siznificant at the ST level of 4 ..
** Significant at the 1% level
8209 8212
8303 8306 8309 8312
of significance
TABLE 4.12
Test II: rF =a + 8F (rS  rf)
Data Set: LAST30
Market Contract a 8F R2 DW p df
S & P 500:
8206 0.0016 1.29 0.73 2.51 0.26 15 8209 0.0018 1.24 0.84 2.44 0.30 14 8212 0.0002 1.08 0.82 2.51 0.30 13 8303 0.0003 1.05 0.81 2.10 0.08 14 8306 0.0000 1.08 0.85 2.42 0.27 13 8309 0.0004 0.92 0.86 2.48 0.28 14 8212 0.0010 1.05 0.79 2.50 0.30 13
NYSE
8312 0.0009 1.01 0.75 1.53 0.05 6
Note: * Significant at the 5% level of significance
ï¿½* Significant at the 1% level of significance
Test III:
TABLE 4.13 RF = a + 8H (rs
 r f)
Data Set: WHOLE
Market Contract a 8H R2 DW p df
S & P 500:
8209 0.0002 1.22** 0.78 2.29 0.16 78 8212 0.0004 1.20** 0.78 2.59 0.30 126 8303 0.0003 1.21** 0.75 2.84 0.42 154 8306 0.0004 1.02 0.70 2.57 0.27 132 8309 0.0006 1.22 0.74 2.78 0.33 124 8312 0.0000 1.10* 0.78 2.69 0.37 167
NYSE
8303 0.0008 1.30** 0.80 2.51 0.24 41 8306 0.0002 1.60 0.20 2.64 0.32 81 8309 0.0001 1.04 0.53 3.09 0.54 127 8312 0.0004 1.18** 0.80 2.50 0.23 182 8403 0.0004 1.18** 0.77 2.67 0.24 182
Note: *
**
Significance at the 5% level of significance Significance at the 1% level of significance
Test III:
TABLE 4.14 R F + 8H (rS  rf
Data Set: LE30
Market Contract a 8 R2 DW p df
S & P 500
8209 0.0005 1.36** 0.72 2.20 0.14 61 8212 0.0006 1.24** 0.78 2.57 0.29 110 8303 0.0003 1.22** 0.75 2.88 0.44 137 8306 0.0004 1.02** 0.70 2.57 0.27 121 8309 0.0007 1.25** 0.74 2.82 0.35 114 8312 0.0000 1.10* 0.78 2.69 0.37 151
NYSE
8303 0.0009 1.34** 0.83 2.79 0.41 28 8306 0.0002 1.62 0.20 2.64 0.32 77 8309 0.0001 1.04 0.54 3.05 0.52 126 8312 0.0004 1.18** 0.80 2.45 0.23 173 8403 .00004 1.19** 0.77 2.58 0.23 182
Note: * Significance at the 5% level of significance
** Significance at the 1% level of significance
TABLE 4.15 R= c + a H (FS  rf)
Contract a OH R2 DW df Data Set
S & P 500 8209 8212 8303 8306 8309 8312
LAST90 L120LE30 LE90 LE120
LAST90 L120LE30 LE90 LE120
LAST90 L12OLE30 LE90 LE120
LAST90 Li20LE30 LE90 LE120
LAST90 L12OLE30 LE90 LE120
LAST90 LI20LE30 LE90 LE120
0.0003 0.0014
0.0002
0.0010
0.0004
0.0016
0.0005
0.0000
0.0010
0.0006
0.0001
0.0001
0.0009
0.0012
0.0003
0.0002
0.0002 0.0002
0.0014
0.0016
0.0004 0.0002
0.0005
0.0002
NYSE
8312 0.0004 1.04 0.85 2.19 0.07 37 LAST90
0.0001 1.00 0.85 2.11 0.10 44 L12OLE30
0.0005 1.21** 0.79 2.49 0.25 143 LE90
0.0007 1.23 0.79 2.49 0.25 127 LE120
Note: * Significance at the 5% level of significance
ï¿½* Significance at the 1% level of significance
Test III:
1.23*
1.54**
1.17 0.90
1.18 1.24*
1.24** 1.27**
1.27**
1.25** 1.20** 1.23**
1.23* 1.23* 1.00 0.99
0.97 1.02
1.40**
1.42
0.94 0.95
1.14* 1.17*
0.81 0.74 0.70 0.85
0.77 0.78 0.79 0.78
0.84 0.83 0.73 0.73
0.83
0.82
0.68 0.67
0.88
0.84 0.74 0.73
0.87 0.88 0.77 0.76
2.41 2.47 1.76 2.37
2.99 2.88
2.29 2.18
2.48 2.76 2.90 2.85
2.33 2.14 2.58
2.63
2.93 2.64 2.93 3.05
2.49 2.53 2.73 2.27
0.25
0.24
0.04
0.30
0.50
0.44
0.15
0.11
0.27
0.40
0.45
0.44
0.20
0.07
0.27
0.29
0.47
0.32
0.36
0.39
0.25
0.27
0.38
0.41
45 45 30 13
45 45 78
62
45 43 106 91
41 45 89
73
38
44 83 67
45 45 119 103
TABLE 4.16
Test III:
R+F a BH (FS
 rf)
Data Set: LAST30
Market Contract a aH R 2 D W df
S & P 500:
8206 0.0015 1.28 0.73 2.51 0.26 15 8209 0.0018 1.24 0.85 2.43 0.30 14 8212 0.0002 1.07 0.82 2.52 0.30 13 8303 0.0003 1.05 0.81 2.08 0.07 14 8306 0.0000 1.08 0.85 2.42 0.27 13 8309 0.0004 0.93 0.86 2.48 0.28 14 8312 0.0010 1.05 0.79 2.50 0.30 13
NYSE
8312 0.0009 1.01 0.75 1.53 0.04 6
Note: * Significant at the 5% level of significance
ï¿½* Significant at the 1% level of significance
CHAPTER V
CONCLUSION
The adjustment for dividend uncertainty, either constant or stochastic, appears to reduce index futures prices. The extent of the difference between the constant and stochatic cases depends on the size of the constant dividend yield, the size of the variance of the dividend process, and the extent correlation between the return on the indexequivalent portfolio and the dividend yield. As time to maturity increases, the difference is magnified.
Using a market equilibrium approach, a model which includes a dividend risk premium was developed. This dividend risk premium can not be hedged away, if we buy a classic riskarbritrage portfolio which goes long one unit of the indexequivalent portfolio and short one unit of the index futures simultaneously. In addition, the risk level of the futures contract relative to the indexequivalent portfolio has been examined.
In Chapter III, three empirically testable equations were derived. The first specifies the price relationship between the spot and the index futures which includes the effect of a random dividend. The second is a LikeCAPM model which shows the corresponding systematic risk level of index futures contracts. Finally, a hedge relationship was examined. Based on these three equations, five tests were performed on seven data sets to examine the dividend risk behavior empirically.
In Chapter IV, the dividend risk argument is examined empirically. The empirical results show that the dividend risk premium with
seasonality the closer the contract is to maturity. The total risk premium two months before the dividendpeakmonth is large; one month before the dividendpeakmonth it is somewhat smaller, but still significant; and during the dividendpeakmonth it is not significantly different from zero. As to the magnitutite, S&P 500 has larger market value with a larger associated dividend risk premium in general. In addition, the nearest contract appears to be priced according to the perfect market pricing model, which implies that the dividend risk is not important for the last thirty tradingdays at least.
Since we have examined the daily price of index futures in New York Futures Exchange and Chicago Board Futures Exchange over a one and a half year time period, inferences drawn from this research must be tentative. However, we have analyzed in excess of four contracts in each market and have found similarities in the results for index futures on these two markets. Therefore, we consider this research to be one of the most extensive empirical examinations of index futures price to be reported in the literature to date.
APPENDIX A DATA BASES
TABLE A.1
Data Base of Index Futures Data
Market
Market Index
NYSE***
S&P 500****
Contract
8206* 8209 8212 8203 8206 8306 8312 8403
8206 8209 8212 8303 8306 8309 8312 8403
StartTrading Date
820506** 820506 820506 820506 820630 820722 820723 820930
820421 820421 820421 820421 820625 820920 821223 830325
EndTrading Date
820629 820929 821230 830330 830629 830929 831229 840329
820617 820916 821216 830317 830616 830915 831215 840308
Number of Observations
38
102 166 229 253
302 364 379
41
104 168 213 202
196 232 200
Note: * 8206 stands for YYMM, meaning index futures contract maturity
on June 1982. Others are the same format
ï¿½* 820506 stands for YYMMDD, meaning the date of May 6, 1982.
Others are the same format
*** The data source of NYSE is directly provided by statistics
department NYSE. Data are in 4:15 P.M. New York time.
ï¿½ * The data source of S&P 500 is from a ticker by ticker tape,
provided by Chicago Board. The close price at 4:15 P.M. is
sorted as the data base.
TABLE A.2
Data other than Index Futures ****
Market Data type StartDate EndDate
NYSE Spot 820104 840329
Dividend* 830101 831231
S & P 500 Spot 820421 840308
Dividend** 820104 831231 TreasuryBill*** 820421 840308
Note: * Dividend payout of market index is provided by statistics
department, NYSE.
** Dividend payout of S & P 500 is collected from CRSP daily
tape, using December 1982 name list in COMPUSTAT tape. This
data series is a approximation value, 6.6% number of companies
series is a approximation value, 6.6% number of companies
missing or 3.6% dividend payout of total dividends missing,
based on the value of December 31, 1982. COMPUSTAT tape.
ï¿½ *TreasuryBill Futures data is sorted from a ticker by ticker
tape, provided by Chicago Board.
**** Market value of S & P 500 and NYSE are computed from CRSP
tape. The market value of S & P 500 underestimates 3.6% value of total value, based on the value of December 31, 1982 COMPUSTAT tape. The names of missing companies are
available upon request.
APPENDIX B
RESULT OF FIRST STEP
LEAST SQUARE
TABLE B.1
Test I: FD = a M + 0 FCESP
First Step Least Square
Data Set: WHOLE
Market Index S&P 500 NYSE
Contract
(Maturity) 8206 8209 8212 8303 8306 8309 8312 8312 8403
MI na na na 0.99** 1.16** 1.07** 1.79** 0.85** 1.00**
M2 na na na 0.85** 0.97** 0.99** 0.93** 0.52** 0.59** M3 na na na 8.81* 0.11 0.52 0.08 0.05 0.08 M4 0.47 1.28** 1.88** 0.48 1.53** 1.49** 1.85** 0.92** 1.04**
M5 0.31 0.45 0.40 0.05 0.18 0.75** 0.79** 0.60** 0.62**
M6 2.10 3.09** 3.34** 3.50** 13.73** 1.25** 1.20** 0.64** 0.76**
M7 na 1.44** 1.48** 1.79** 0.56 1.36** 1.21** 0.49** 0.52**
M8 na 0.30 0.81 0.67 0.86 18.65** 1.20** 0.72** 0.80**
M9 na 4.99 0.92* 0.55 0.16 0.84 0.93** 0.78** 0.85**
MI0 na na 0.88* 1.01** 0.30 1.61 0.93** 0.52** 0.73**
MIl na na 0.39 0.42 0.85** na 0.02 0.12 0.38**
M12 na na 9.99** 1.15** 0.96** 1.01* 4.42 2.67 0.41
FCESP 0.01 0.03 0.07** 0.05*** 0.008** 0.10** 0.03** 0.02 0.00**
R2 0.69 0.78 0.90 0.94 0.97 0.97 0.99 0.98 0.99 DW 0.67 0.98 1.4 1.48 1.76 1.83 1.46 1.26 1.02
0.62 0.50 0.29 0.26 0.08 0.13 0.25 0.38 0.51 df 28 73 118 143 121 114 156 171 171
Note: * Significant at the 5% level of significance
ï¿½* Significant at the 1% level of significance
Test I: FD = a M + 0 FCESP
First Step Least Square
Data Set: LE30
Market Index S&P 500 NYSE
Contract
(Maturity) 8206 8209 8212 8303 8306 8309 8312 8312 8403
Ml na na na 0.79 1.17** 1.06* 1.79** 0.86** 1.00**
M2 na na na 1.37 0.96** 1.01** 0.94** 0.54** 0.60** M3 na na na 9.47* 0.09 0.54 0.09 0.06 0.09 M4 0.34 1.22* 1.84** 0.48 1.50** 1.47** 1.85** 0.92** 1.04**
M5 3.24 0.74 0.37 0.05 7.22 0.75** 0.79** 0.60** 0.62**
M6 9.58 3.14** 3.34** 3.50** 20.58 1.26** 1.20** 0.64** 0.76**
M7 na 1.21 1.49** 1.79** 0.56 1.33** 1.23** 0.50** 0.52**
M8 na 13.15 0.77 0.67 0.80 19.42** 1.20** 0.72** 0.80**
M9 na 22.41 0.95* 0.55 0.20 0.90 0.93** 0.78** 0.85**
MLO na na 1.12* 1.01** 0.32 1.59 1.09** 0.55** 0.73**
MIl na na 6.78 0.42 0.84** na 1.40 0.17 0.38**
M12 na na 15.99 1.15** 0.98** 1.00* 3.34 3.16 0.41
FCESP 0.06 0.08 0.06** 0.05*** 0.08** 0.11** 0.03** 0.03 0.00**
0.88 1.59
0.08
10
0.79 0.86 0.55 56
0.91 1.40 0.30
102
0.94 1.49 0.26 126
0.97 1.76 0.08 110
0.97 1.84 0.12 104
0.99
1.46 0.25 140
0.99 1.32
0.34 171
0.99 1.13
0.45 171
Note: * Significant
** Significant
at the 5% level of significance at the 1% level of significance
TABLE B.3
Test IV: FD S + a
Data set: WHOLE
Variable 2 month
1 month during
FCESP
DW p df
Contract before before
S&P 500:
2.103
3.052**
2.439**
1.894**
0.875**
0.681**
0.860**
0.476
1.399**
0.579* 0.124
1.208**
1.492**
1.597**
0.312
0.399
0.161 0.133
0.057
0.272
0. 444**
0.012
0.022** 0.011** 0.008**
0.001
0.003** 0.008**
0.69 0.78 0.86 0.91 0.95 0.96 0.97
0.605** 0.662** 0.279** 0.006*** 0.96
0.606** 0.724** 0.388** 0.012*** 0.98
0.67 0.70 180 0.42 0.81 180
FCESP
8206 8209 8212 8303 8306 8309 8312
0.686
0.98 0.97 0.89 1.25 1.17 0.75
NYSE:
8312 8403
0.62 0.50
0.51 0.55 0.33 0.48 0.60
28
76
124 152 130
122 165
Note: * Significant at the 5% level of significance
ï¿½* Significant at the 1% level of significance

Full Text 
PAGE 1
AN EQUILIBRIUM MODEL OF INDEX FUTURES PRICING BY SOUS HAN WU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 198^
PAGE 2
ACKNOWLEDGMENTS Intellectual development is a contract: it gives you leverage, and like an index futures contract, it is marked to the market with some "saf tydeposit" margin. The initializer of my development contract is also the chairperson of my committee, Dr. Robert C. Radcliffe, to whom I wish to express my sincere gratitude for his extensive discussion and encouragement. Furthermore, the initial margin of my development contract is reduced by Dr. M.Y. Tarng, Dr. C.C. Chu, Dr. C.Y. Ho, Dr. Tony Lai, Dr. M.P. Narayanan, Dr. Roger Huang and Dr. Raymond Chiang. Their encouragement, valuable comments, and intellectual stimulation made this dissertation possible. Of course, the highest leverage is further inplemented by Dr. A. A. Heggested, Dr. Heim Levy, Dr. Richard Pettway, Dr. Raymond Chiang, and Dr. Rodger Huang, who have instilled in me knowledge of finance both in the class as well as through projects. Special thanks go to Dr. S.R. Cosslett, from whom I have learned more than econometrics, and to Dr. Raymond Chiang, from whom I have learned more than merely finance. A special acknowlegment of gratitude is due to the National Science Foundation of the Republic of China, the National ChiaoTung University and the Department of Finance of the University of Florida, which have held a substantial equity interest in this venture. In providing the financial risk capital, these institutions showed overwhelming faith in me. To my grandfather, Mr. TuChen Wu, I also wish to express my deepest gratitude. His spiritual support and constant encouragement have contributed substantially to my study over the years. ii
PAGE 3
Finally, I would like to thank my wife, MeiLin Chang. Without her understanding and encouragement, my study at the University of Florida would not have been possible. iii
PAGE 4
TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii KEY TO SYMBOLS v ABSTRACT viii CHAPTER I MOTIVATION OF THE STUDY 1 CHAPTER II INTRODUCTION TO STOCK INDEX FUTURES 5 Contract Description 5 Review of Literature 11 Note 22 CHAPTER III THE PRICING OF THE INDEX FUTURES 23 Notations, Assumptions, and Definitions 23 Model 28 Comparative Static Analysis 34 Summary 38 Notes 40 CHAPTER IV THE EMPIRICAL STUDY 41 A Prelimary Result 43 Basic Discussion: Measurement and Data Sets 44 Test Methodology and Results 49 Summary 64 Notes 66 CHAPTER V CONCLUSION 83 APPENDICES A DATA BASES 85 B RESULTS OF FIRST STAGE LEAST SQUARE 88 REFERENCES 94 BIOGRAPHICAL SKETCH 99 TV
PAGE 5
KEY TO SYMBOLS t Current period T Maturity of the index futures contract F(t,T) Current futures price maturity at T. F(T,T) Maturity futures price S(t) Value of the indexequivalent portfolio at t (which is formed as the same value as of the spot value I(t)) I(T) Value of the spot index at time T D Total cash received from dividend payments on the indexequivalent portfolio during the period t to T S(T) Value of the indexequivalent portfolio at T which is formed at t, includes dividend, equal to I(T) + D C , Cost of being long the indexequivalent portfolio Pi C Cost of being short the indexequivalent (which may " include foregone interest on the short sales proceeds) C f Cost of being long the index futures I S C^ Cost of being short the index futures X . Number of endowment units which investor i holds of the S X index equivalent portfolio (This is exogenously given) X Total number of endowment units of the indexequivalent portfolio in the economy X . Number of units of the index portfolio held by the investor i si J X^ Net long number of units of index futures contract held by the investor i C^ Initial cash of investor i b^ Number of units of bond held by the investor i r f Riskfree rate during this period (t,T) Risk tolerance of investor i n Risk tolerance in the economy W . Wealth of the investor i at time t which can be put among bonds, indexequivalent portfolio and the index futures
PAGE 6
w, Ti G i (E i> a i 2) I ,2 ID sF FD S&P 500 VLA NYSE NYFE AMEX OTC CRSP Terminal wealth of investor i at time T Expected value of the terminal wealth of investor i at time T Variance of terminal wealth of investor i at time T Preference function of investor i, which is determined by the expected value and the variance of the terminal wealth Variance of the spot index at time T Variance of the dividend payout which is carried over in the spot indexequivalent portfolio Covariance of the spot index and the random dividend payout over period Variance of the indexequivalent portfolio price Variance of the index futures price Covariance of the index futures price and indexequivalent portfolio price Covariance of the index futures price and random dividend payout Percentage price change of the index futures Rate of return on the indexequivalent portfolio Percentage price change of the index futures in terms of spot index Correlation coefficient of the index futures price and the indexequivalent portfolio Standard and Poor's Composite Average Index Value Line Composite Average Index New York Stock Exchange New York Futures Exchange American Stock Exchange Over the Counter Center of Research in Security Prices, University of Chicago COMPUSTAT Standard & Poor's Compustat Service, Inc.
PAGE 7
FD Futures Discount FCESP Future CertaintyEquivalent Spot Price
PAGE 8
Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy AN EQUILIBRIUM MODEL OF INDEX FUTURES PRICING By Soushan Wu December 1984 Chairman: Robert C. Radcliffe Major Department: Finance, Insurance and Real Estate The objectives of this study, all pertaining to pricing efficiency, are as follows: (1) to develop a theory of cashfutures price relationships for stock index futures which includes dividend risk within the framework of the partial equilibrium approach; (2) to test whether a dividend risk premium has affected cashfutures price relationships; and (3) to provide a framework for evaluating the efficiency and hedging in the pricing of stock index futures. Two types of evidence are employed to attain these objectives. First, the principles of cashfutures price behavior derived from observation of longestablished futures markets serve as points of departure, with substantial consideration given to certain fundamental differences that arise from the specification of stock index futures. Second, empirical tests of the hypothesized equilibrium pricing are evaluated. This study demonstrates that when the dividend risk is taken into consideration, a covariance term between the dividend payout and the indexequivalent portfolio is added to the Perfect Market Model developed by an arbitrage approach. The study also shows that the dividend risk premium viii
PAGE 9
empirically increases with seasonality when time to the maturity increases. Moreover, in the last thirty trading days, the market bears a dividend risk level which is statistically insignificant. ix
PAGE 10
CHAPTER I MOTIVATION OF THE STUDY With the existence of exchange markets, individuals are able to hold their personal endowment in terms of current consumption and investment for the future to consume. The exchange can be improved by enlargening the set of feasible patterns of consumption and investment over time and by sharing of uncertainty associated with consumption and investment in the future. With such a market, satisfaction is increased for the individual and for the society as well. A new financial instrument, the stock index futures contract, has recently been introduced to faciliate intertemporal allocation of resource and risk sharing (since February 1982) . This index futures market, like other existing futures markets, claims to provide two majo advantages to market participants: (1) risk transfer and (2) price discovery (Garbade & Silber, 1983). Both effects allow the capital market to become more complete (Ross, 1976). Risk transfer refers to a hedger' s ability to transfer price risk to another hedger or to a speculator (if no offseting hedger can be found). Specifically, the index futures market can separate market risk from unsystematic risk. Price discovery refers to the information available in explicit prices of the trading transaction. Both the pricing discovery and risk transfer have the ability to extend the range of investment and risk management strategies. These two functions also result in conceptual economic models which relate existing futures prices to existing spot prices.
PAGE 11
2 Conceptual models together with empirical tests which relate spot price and futures are very common in commodity contracts (although results of such models and tests are debatable). However, within the past few years in which trading has begun on stock index futures contracts, little theoretical modeling or empirical testing has been conducted. The studies of stock index futures to date suggest a puzzle: the actual stock futures prices do not conform with existing theoritical pricing models which are based upon a perfect market pricing model under the assumption of a nonstochastic dividend. The perfect market pricing model implies that the index futures price should be equal to the future certainty equivalent of today's spot price (at the maturity date) minus the accumulated dividend payout over this period (Cornell & French, 1983). This dissertation is motivated by the fact that existing theoretical models of stock index futures do not stand up to empirical facts. In the earliest period of their trading (February 1982 to September 1983) , actual stock index futures consistently sold for less than their "theoretically" derived values based on perfect market model (Figlewski, 1983a). A number of possible explanations have been offered by Modest and Sundaresan (1983), Cornell and French (1983), and Figlewski (1983a). Among these studies, Figlewski (1983a) classified the potential reasons as the socalled (market) equilibrium argument and disequilibrium argument. But research on these new financial instruments has just begun and none of the existing studies has been able to adequately explain the differences between actual stock index futures price and "theoretical" price of perfect market model. Researchers reach their own conclusions by examining some specific dates. In the most recent study, Figlewski (1984) employs more complete data to argue that this futures market is in a longrun disequilibrium in
PAGE 12
3 its early stage. Both Modest and Sundaresan's "short sales not fully used" argument (1983) and Cornell and French's "taxrelated timing option" (1983) employ a cashfutures arbitrage approach to obtain their pricing relationships of index futures based on the market equilibrium approach. And in both of these equilibrium models, there is an assumption that dividends to be paid on the spot indexequivalent portfolio are known with certainty. However, if the dividend to be paid on the spot index is unknown in advance, then the strategy of "long indexportfolio and short index futures one unit each" is not sufficient to completely hedge the risk associated with such an unknown dividend payment. If an allowance is made for an unknown dividend payment, it is theoretically plausible that the index futures contract should be priced lower than what the socalled "perfect market" model suggests. This discount occurs because the individual who is long an index futures does not receive dividend payments, but does face a dividend risk. If the dividend payout is stochastic, index futures and the indexequivalent portfolio may not be of the same systematic risk level. Consequently, the arbitrage approach to pricing breaks down. A perfect arbitrage is not possible and index futures pricing models should not be developed around such arbitrage relationships. Instead, a more general equilibrium approach should be employed. The objectives of this study are two: (1) to derive and (2) to test a closed form equilibrium price for a stock index futures contract, which includes the effect of unknown dividends. This closedform equilibrium futures price will also be linked with traditional Capital Assets Pricing Model (CAPM) framework and conventional hedging strategies after some rearrangement . Following a basic review of index futures and a synthesis of past
PAGE 13
4 studies in Chapter II, a basic pricing model with comparative statics analysis and interpretation is presented in Chapter III. Chapter IV provides econometric justification for the form of this predictive model and fits the forecast to actual New York Stock Exchange (NYSE) and Standard and Poor's Compositive (S&P 500) index futures based on a sample taken from New York Stock Futures Exchange and The Chicago Board, starting from the first trading contract to the December 1984 contract. Conclusions of this study are presented in Chapter V.
PAGE 14
CHAPTER II INTRODUCTION TO STOCK INDEX FUTURES In this chapter, the legal characteristics of index futures contract are described in the first section. Following this, the various studies relating to this study are briefly reviewed. Contract Description A stock index futures contract is an obligation to buy or sell a hypothetical portfolio of all the stocks in an index at a stated price at a certain date. It can be liquidated before maturity. At present, there are three main actively traded stock index futures contracts: the Value Line Index traded on the Kansas City Board of Trade, the Standard & Poor's 500 traded on the Index and Option market of the Chicago Mercantile Exchange, and the New York Stock Exchange Composite Index traded on the New York Stock Exchange. In addition, a New York Stock Exchange Financial Index Futures and other contracts, which are on narrower measures of equity market activity such as Utility and Transportation indices, have recently been introduced. Options on these three main indices are also available now. An index futures contract is similar to other kinds of commodity futures contracts paid for on a unique installment plan. The investor who buys a futures contract agrees to buy one unit of a financial product at a specified maturity time at a specified futures price. The index future price is determined when the contract is written and 5
PAGE 15
6 is specified in the contract. The stock index futures price should differ from current or expected future (t, or T) stock index prices for at least two reasons. First, the futures price is chosen so that no payment is made when the contract is written; i.e., at the initiation phase the futures contract has zero market value. But as the contract matures, the investor must make or receive daily installment payments toward the eventual purchase of the financial product. This is referred to as "markingtomarket." The total of the daily installments and the payment at maturity will equal the futures price specified when the contract was initiated. Second, the futures trader does not receive the dividends that are paid to the stockholder, but faces a dividend risk. What makes futures contracts unique among installment plans is that the daily installments are not specified in advance in the contract, but are determined by the daily change in the futures price. If the futures price rises, then the investor, who is long the futures contract, receives a payment from the investor, who is short. The payment is the rise in the futures from the previous day. On the other hand, if the futures price falls, the long holder pays the short holder the change in the daily futures price. The effect of markingtomarket is to rewrite the futures contract each day at the new futures price. Hence the value of the futures contract after the daily settlement will always be zero since the value of a newly written futures contract is zero. When the contract matures, the long investor will have already paid or received the difference between the initial futures price and the futures price at maturity time. With these payments to his credit, he will have a balance due equal to the futures price at maturity. But the value of a futures contract
PAGE 16
7 Therefore, at maturity the futures price must equal the current spot price. Thus the balance due is simply the current spot price at the maturity time. Unlike a forward contract, the value of a futures contract Â— after settlement Â— is always zero. Some special characteristics of index futures contracts are discussed in what follows. The margin on a futures contract represents a "good faith" deposit on the part of the buyer and seller. Margin requirements on stock index futures approximate 510% of the contract's value. This results in a high degree of leverage for the futures trader. Futures margins differ significantly from spot stock margins. They are much lower than stock margins and involve no extension of credit or expense on any unpaid (borrowed) balance. In addition, the futures margin must be restored daily during adverse price movements (marked to the market). Traders in a profitable position may withdraw any excess margin. In certain cases, margin can be posted in Treasury Bills, so interest can be earned while the trading program is in place. The delivery process in the settlement of futures contracts serves a positive economic function in that it assures a certain level of convergence between the cash and futures prices at the expiration date. This price correction is essential to the development of a successful futures contract. Cash/futures convergence in the stock index futures is virtually assured by the "cash settlement" procedure in these contracts. Rather than require the delivery of actual stock, all open positions are "marked to the market" at expiration. In addition, the delivery price is not determined in the futures pit, but by the actual cash market close on the last day of trading (like S&P 500 index futures), or the
PAGE 17
8 average of the closes of the last and next last day (like the NYSE index futures case) . This assures convergence of the futures price with the price for the underlying commodity, the spot index, and prevents any attempts to "squeeze," "corner," or otherwise manipulate the market. On the last trading day of the expiring contract, all three indices futures are traded until 4:00 P.M. (EST) only. Do prices in one market have an impact on prices in the other market? With cash settlement, the arbitrage possibility between the cash market and futures market is encouraged. An arbitrager can profit by going long a unit of index futures, short a unit of the indexequivalent portfolio and investing the balance in risk free bonds simultaneously, if the observed index futures price is lower than the "theoretical" price in mind. Meanwhile, he also can profit by shorting a unit of index futures, borrowing at the riskfree rate, and going long a unit of indexequivalent portfolio simultaneously, if the observed index futures price is higher than the "theoretical" price in mind. This cashfutures arbitrage is often referred to as basis speculation and is best represented as a type of "risk arbitrage." It contributes to the economic activity by both the risk transfer and price discovery functions of futures markets. For taxable investors, under the current tax code, the gain or loss on any futures transaction entered into after June 23, 1981, is be treated as 60 percent longterm capital gains, and 40 percent as shortterm, without regard to the period of time for which the position was held. It means that the tax rate of index futures trade is 32% currently, i.e . , (60% x 20) + (40% x 50) 32%.
PAGE 18
9 Value Line Composite Average (VLA) Published and maintained by Arnold Bernhard and Co., the VLA is an equally weighted geometric average of about 1700 stock prices expressed in index form (June 30, 1961 = 100.0) Nearly 90 percent of the stocks included in the VLA are traded on the New York Stock Exchange. Based on this spot index, a Value Line Index Futures Contract is specified in Table 2.1. TABLE 2.1 Value Line Composite Index Futures Exchange: Kansas City Board of Trade Trading Months: March, June, September, December; 18 months forward Trading Hours: 10:00 A.M. 4:15 P.M., N.Y. time Contract Size: Futures price x $500 Minimum Price Fluctuations: 5 points, equivalent to $25 Daily Price Limits: 500 points ($2,500) Last Trading Day: Last business day of contract month Trading began: Feb. 24, 1982 Standard & Poor's 500 This index, commonly called the S&P 500, is a broadly based arithmetic average, utilizing the share prices of 500 different companies: 400 industrials, 40 utilities, 20 transportation and 40 financial companies. The market value of the index is approximately 80% market value of all stocks on the New York Stock Exchange. Each stock in the index is weighted to reflect the stock's total influence on the index relative to its market value. To determine the weighting, the number of shares outstanding is multiplied by the price per share. Thus, a stock's total market value determines its importance in this index.
PAGE 19
The S&P 500 has been widely accepted as a benchmark for portfolio manager performance as well as a measure of economic activity. It is one of the components of the Index of Leading Economic Indicators. The S&P 500 is based on the average weekly closing value of 19411943 and indexed to a value of 10. The stock index futures contract on this index is specified in Table 2.2. TABLE 2.2 Standarded & Poor's 500 Index Futures Exchange : Trading Months: Trading Hours: Contract Size: Minimum Price Fluctuation: Daily Price Limits: Last Trading Day: Trading began: Index and Options Market March, June September, December, 18 months forward 10:00 A.M. 4:15 P.M., N.Y. time Futures price x $500 5 points, equivalent to $25 500 points ($2500) Third Thursday of contract month April 21, 1982 New York Stock Exchange Composite This index is an arithmetic average consisting of all common stocks listed on the NYSE. As with the S&P 500, each stock in the NYSE index is weighted in proportion to the stock's market value. The index is based on the close price of December 31, 1965, and indexed to a value of 50. The composite futures and financial futures based on this are specified as Table 2.3 and Table 2.4.
PAGE 20
TABLE 2.3 New York Stock Exchange Index Exchange : Trading Hours: Contract Size: Minimum Price Fluctuation: Daily Price Limits: Last Trading Day: Trading began: New York Futures Exchange 10:00 A.M. 4:15 P.M., N.Y. Futures x $500 Time 5 points, equivalent to $25 none Second last business day of contract month May 6, 1982 TABLE 2.4 New York Stock Exchange Financial Index Exchange: New York Futures Exchange Trading Months: March, June, September, December; 18 months forward Trading Hours: 10:00 A.M. 4:15 P.M., N.Y. time Contract Size: Futures price x $1000 Minimum Price Fluctuation: 1 point, equivalent to $5 Daily Price Limits: none Last Trading Day: Second last business day of contract month. However, effective with December 1984 contract and all subsequent contracts, the last business day will be the third friday of the month. Review of Literature To date, three basic arguments have been proposed to explain why actual futures prices are lower than what the perfect market pricing model suggests. These include 1) a taxrelated timing option, 2) no full use of the short sales, and 3) longrun disequilibrium in the index futures market in the early stage. In sum, the taxrelated timing option suggested by Cornell and French (1983) , and the no full use of short sales suggested by Modest
PAGE 21
12 and Sundaresan (1983) have been classified as equilibrium arguments by Figlewski (1983a). Furthermore, he believes that the market is in disequilibrium stage in the early stage. In a market of no transaction cost, no tax, dividend certainty, and no information access cost, a perfect market pricing model is developed as follows: F(t,T)I(t) exp((r f d)) (2.1) where F(t,T) Kt) r f the index futures price at time t, maturity at T. the indexequivalent portfolio at time t. riskfree rate dividend rate (total dividend payout from t to T divided by the current spot index) . Due to possible imperfections of the capital market, as Modest and Sundaresan suggest, empirically observed futures discount may be explained by the fact that short sellers of spot seldom obtain full use of the proceeds of the short sales. They argue that if the proceeds of the short sale of spot are not available to be invested in the money market, the "short spot Â— long futures" arbitrage becomes profitable only when the futures price is below the current spot index by an amount greater than the dividend yield on the index portfolio because a short seller must pay the dividends on the shares he has borrowed. An upper bound and a lower bound thus are developed for index futures pricing. For illustrative purpose Modest and Sundaresan present the bounds and observed futures prices for the June 1982 and December 1982 S&P 500 stock index futures contract. The bounds are presented under the
PAGE 22
alernative assumptions that short sellers have zero, half, and full use of the proceeds, both in the no dividend adjustment case and in the dividend adjustment case under the assumption of nonstochastic dividend. Thus, for each contract, six sets of bounds and prices are presented. For the cases where dividends are adjusted, the bounds are given by the eq(2.2) which can be written as I(t) + C . + C, D pi fs exp(r f ) I(t) " C ps " C fl ~ D > F(t,T) > exp(r f ) (2.2) where Pi C ps C fs I(t) D r f F(t,T) Cost of being long in the indexequivalent portfolio Cost of being short in the spot (which may include foregone interest on the short sales proceeds) Cost of being short in the index futures Price of one unit of indexequivalent portfolio at time t Known dividend with reinvested in riskfree rate Riskfree rate Index futures price at t, maturity at T 3ased on this equation, the figures for December 82 S&P 500 contract are reproduced here as Figure (2.1 2.3) for reference. This argument, according to Figlewski (1983a), only explains why arbitrage might not force the futures discount to disappear once it develops. It gives no insight into why the discount should exist in the first place. This means that the motivation to short the index equivalent portfolio is not clear. Besides, trading index futures involves no investment except some margin requirement which can be met by using securities such
PAGE 23
M 01 0) 2. 14 cn 0) u H U a QJ U S u T3 03 * C X 0 0J Â—1 u CO H w U Â•H a CO Â£ 3 u DO C CU 0) CU a M a 3 H .u > 3 ~3 Â•a U u Â• o 3 CM Ul a X o i0 a c Â— i CD u a; CO 3 n S n c 0> CO o Â• cu a cn cn c CJ CO Â— 1 cu u o fÂ— J Â•H o li u 3 a 3 (J a 03 41 01 c L t u i> 03 a CO a 3 B 3 u O ^ C M 0 cu N C Â» Â— ' 03 o o CO u '.n 3 go a 01 tu 3 T3 MS Q O 03 A X o> 3 =C Â— I 01 CJ 3 0 cn
PAGE 24
15 M cu CJ 2 CD CJ Â•H N fa CO OJ M 3 U 3 fa r t o ; n t Â— * t r 2 r > J > Â• a 3 H CN fa a co 3\ M fH CJ Â• 0 3 CM IM * a cc m o a\ 00 h C fa Â•H M CD u 41 CD U 3 W a 3 e m S Â•I ; CD a a 41 cn a a a Â• CD CD a. a M as W / N co a 1 CO a CD CO Â•H CD 01 .41 U CN tÂ— i Â— 1 0 M u H 3 a "H a 41 M ca a gg s O < re 41 a 3 U cn o 3 ag 3 C h 1 3 u Ul 3 w C H a 0 3 u Â•3 c_> 4J 3 O 3 O H o gg a 41 T3 gg 3 3 CD fa 3 4J a C 3 c a 2 CN CN 3 u 3 00 3 CJ U 3 3 CO 3
PAGE 25
16 M D 1) 3 U H h 00 n CU 3 U 5 n N ""I fc3 t Â« E b F Â• t F 2 t s h Â» t "2 3 r O T3 co C 4J co co Q 3 "H fa TJ Â• O Â• CM <4l CN 00 00 on ao on Â•Â— i c Â— i Â•H k il Â• cu co m 1 3h e n cj co cu CU xs Q Â« S 01 o 00 T3 AJ c cu a CJ CO o U I a CO VM CM 35 O W CU i4 O to h ca 3 a n < 4J rH C 11 U O 3 O U iu >u O TJ O co cu m "O u c Â« Ph 3 O iiJ O "H to j a 01 M 3 H 3 0 1 so 00 a a >> 43 T3 CJ CJ 3 a M a cc en CO c ca 00 C 3 00 TJ C 03 0) 0 u ii 3 0 Cfl 3 3
PAGE 26
as Treasury Bill which would be held regard less of the futures today. Both "long the spot Â— short the futures" and "short the spot Â— long the futures" strategies are different in risk level and investment level. Cornell and French (1983) point to a difference in the way stock and futures returns are taxed. For index futures all paper profits are taxed as if they had been realized by the end of the fiscal year or by the maturity of the index futures, whichever comes first. Returns on a stock portfolio, however, are taxed as short term gains if realized within one year. But if the holding period is extended beyond one year, the tax rate drops to 40% of the short term rate. Therefore, a stock portfolio offers a "taxrelated timing option" that the futures contract does not possess. According to Cornell and French (1983), the timing option is a right to defer to pay the capital gain taxes. Consider an investor who buys an indexequivalent stock portfolio. If its value goes down in a year, he can sell and deduct the loss at short term rates. On the other hand, if it goes up he has the option to extend his holdings period to take advantage of the long term gains rates. Thus, for a taxable investor who knows the dividend payout in advance, Cornell and French derive a functional form for stock index futures to be priced below their theoretical level by an amount equal to the value of the timing option.^" Consequently, the relative timing option is defined as the difference between the "theoretical" price and the observed price relative to the "theoretical" price. The relative timing option of some selected
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18 contract is adapted in Table 2.5 for reference. However, the timing option is difficult, if not impossible, to compute a theoretical value (Wu, 1983) , if one can not specify how long investors are going to hold the indexequivalent portfolio. In addition, the timing option must be positive. There exist, moreover, a huge volume of stocks held by tax exempt investors and by taxable investors whose holding periods are already greater than one year. If the effect of the timing option on futures prices were the main cause of the futures discount, these investors should be writing timing options by selling their stock and buying index futures and riskfree securities to increase their returns. Furthermore, if the dividend to be received on the index is unknown in advance, the perfect arbitrage argument used to derive the model breaks down. Besides these two articles, other researchers which make more comments but offer fewer convincing explanations on the index futures discount can be found in Figlewski (1983a). They are all branches of the market "equilibrium" argument. Figlewski (1983a) presents a permanent market disequilibriun argument. He believes that there exists no temporary short run disequilibrium in the sense that prices do not adjust to equate supply and demand but that there does exist a long run disequilibrium in the index futures market. In other words, actions of investors already in the market create profitable investment opportunities for outside investors who, for one reason or another, are slow to take them up. Reasons for this include an unf amiliarity with the new markets, intertia in developing systems to take advantage of the opportunities they present, legal aspects
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TABLE 2.5 Relative Value of the Timing Option S&P 500 NYSE Contract Days to Relative Value Days to Relative Value Maturity Maturity of Option Maturity of Option 1 June 1982 June 18 1.69% 28 1.95% September 108 2.77% 120 2.97% December 199 3.52% 211 3.20% March 83 290 4.00% 304 3.73% 1 July 1982 September 77 0.01% 90 0.13% December 168 0.95% 181 0.83% March 83 259 2.03% 272 2.06% June 350 2.93% 363 2.97% 2 August 1982 September 45 0.82% 58 0.91% December 136 0.54% 149 0.62% March 83 227 0.14% 240 0.19% June 318 0.72% 331 0.36% 1 September 1982 September 15 1.12% 28 1.07% December 106 1.41% 119 1.39% March 83 197 1.76% 210 1.85% June 288 2.33% 301 2.42% Source: Cornell and French (1983) p. 691, reproduced by permission. ^he relative value of the timing option is expressed as a percentage of the stock price. It is estimated as C dw(/
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20 of trading futures for cautious portfolio managers, the time required to intergrate accounting procedures for futures trading into daily operations, etc. He regresses the discounts on a time trend and the number of days to expiration, to adjust for the fact that the discount must go to zero as a futures approaches maturity. He observes that futures discounts have been decreasing over time, after taking account of the effect of the time to expiration and the high degree of serial correlation in the relationship. Finally, he believes that the futures market is slowly coming into equilibrium. In a more recent article, Figlewski (1984) presents some further results in the hedging performance and basis risk on the stock index futures. This empirical study stands on the ground of a "perfect market" pricing model, which is relevant to our topic. However, he argues that the effects of dividend risk, the length of the holding period, and the time remaining to expiration of the futures contract should be considered. He defines the "return" of index futures in terms of the spot index and takes the variance operator to get the hedge ratio which minimizes risk. Furthermore, he states that the risk minimizing hedge ratio is the portfolio's beta coefficient with respect to the market index if the hedge portfolio is held to the maturity and dividends are not random. He reports the effectiveness of Standard and Poor's 500 index futures in hedging major stock index portfolios over a one week holding period in the sample, July 1, 1982, to September 30, 1983. For a one week holding period, hedging a diversified portfolio weighted toward large capitalization stocks can yield fairly good risk reduction, from about 20 to 30 percent of unhedged portfolio's standard deviation. However, hedging
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21 effectiveness is substantially reduced by the presence of unsystematic risk, even in the amount contained in a broadly diversified portfolio of small stocks like the American Stock Exchange (AMEX) and Over the Counter (OTC) portfolios. He concludes that a short duration hedge for an individual stock or a small portfolio might be quite unsatisfactory. Finally, because of basis risk, the minimum risk hedge ratio was less than the portfolio's beta in every case, with the adverse effects of overhedging being more serious for returns than for risk levels. To explain the basis, Figlewski uses the perfect market pricing model for the futures price to prevent portfolio arbitrage. Price difference between this "equilibrium" price and the actual price is analyzed. The sample is then split into thirds to show the difference in the market's behavior over time. He shows that underpricing of futures was significant in the first third of the sample, but this was not true of the period as a whole for the nearest contracts. In considering the sources of basis risk in a hedge of the S&P 500 portfolio itself, Figlewski believes that dividend risk was not an important factor, while hedge duration and time to expiration of the futures contract were, to some extent. With regard to the pricing of stock index futures, he finds that the significant underpricing that was widely remarked in the very early months of trading seems to have disappeared, and that deviations from the theoretical pricing relation have diminished. This implies that underpricing does not reflect an equilibrium differential, which a factor like the value of the tax timing option would cause.
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22 In this dissertation, a dividend risk is considered to be a potential factor to explain the futures discount of perfect market model. Because the dividend might be uncertain before it is announced, a dividend risk premium might be priced in the market. Suppose the investor holds a hedge portfolio which consists of index futures and indexequivalent portfolio one unit each, the dividend risk should be considered. If the dividend is indeed unknown, a dividend risk premium explicit in an equilibrium model should be developed. The following chapter, Chapter III, presents the theoretical setting in which the effect of an unknown dividend is taken into account so as to determine (1) the price relationship between the spot and the index futures, (2) the systematic risk of index futures contract, and (3) the hedging relationship between the spot and the index futures. All three equations are testable and estimatable. Note [1] The value of the timing options increases monotonically with the variance of return, while return variance generally rises following a stock split. Consequently, Cornell (1984) uses stock splits data to test the theory of the timing option in the index futures market. He finds that the timing option is not empirically important or that the expected increase in variance is the same for all stocks.
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CHAPTER III THE PRICING OF THE INDEX FUTURES In the perfect market pricing model, it is assumed that the dividend for the indexequivalent portfolio underlying the index futures is known in advance. Thus, there is capital gain risk but no dividend risk. The implication of this is that the theoretical equation might systematically overprice the index futures. In this chapter, however, index futures pricing is extended to incorporate stochastic dividend yields under a partial equilibrium approach, assuming that market participants are risk averse. Adopting a set of common assumptions, a closed form solution is found, which can link the pricing relationship with the CAPM and with the hedging strategy as well. We discover that a dividend risk premium is required in evaluating the index futures pricing. To derive these results, a set of notations, definitions, and assumptions are needed. These are presented in the first section of this chapter. In the second section we develop the model necessary for the analysis of futures contracts in the comparative statics analysis of these results. Consequently, several interpretations of this comparative statics analysis are provided in the third section. A summary of this chapter is provided in the forth section. Notation, Assumptions, and Definitions The notation, definitions and assumptions used here is similar to that used to derive the CAPM model as in Sharpe (1964), Lintner (1965), and Mossin (1966). We only add several assumptions to facilitate the
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24 inclusion of an index futures contract, such as cash settlement, and stochastic dividend effect. We will also assume that the investor determines his current consumption level first and then emphasizes on the composition and size of his investment portfolio in these three assets: index futures contract, a spot indexequivalent portfolio and a risk free bond. We assume the investor preference function is of the form, G (E a 2 .), where E and a 2 , represent the singleperiod expected value 1 X f X X X and variance of value of the ith investor's portfolio of risky assets. We further assume that dG i > 0, and 3G i < 0, WT ToT 2 i i that is, greater expected values are preferred and variance of value is not. Notations t Current period T Maturity of the index futures contract F(t,T) Current futures price maturity at T F(T,T) Maturity futures price S(t) Value of the indexequivalent portfolio at t (which is formed as the same as value of I(t)) I(T) Value of the spot index at time T D Total cash received from dividend payment on the indexequivalent portfolio during (t,T) period S(T) Value of the indexequivalent portfolio at T which is formed at t, include dividend, equal to I(T)+D X* Number of endowment units which investor i holds of the S X index equivalent portfolio (This is exogenously given)
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25 Number of units of the index portfolio held by the investor i Net long number of units of index futures contract held by the investor i Initial cash of investor i Number of units of bond held by the investor i Riskfree rate during this period (t,T) Risk tolerance of the investor i Wealth of the investor i at time t which can be put among bonds, indexequivalent portfolio and the index futures Terminal wealth of investor i at time T Expected value of the terminal wealth of investor i at time T Variance of terminal wealth of investor i at time T ,a 2 4 ) Preference functions of investor i, which is determined by the expected value and the variance of the terminal wealth Variance of the spot index at time T Variance of the dividend payout which is carried over in the spot indexequivalent portfolio Covariance of the spot index and the random dividend payout over period Variance of the indexequivalent portfolio price Variance of the index futures price Covariance of the index futures price and indexequivalent portfolio price Covariance of the index futures price and random dividend payout Percentage price change of the index futures Rate of return on the indexequivalent portfolio Percentage price change of the index futures in terms of spot index Correlation coefficient of the index futures price and the indexequivalent portfolio
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26 Note: "Tidle" is used to denote random variable notation. And "bar" is used to denote the mean value of random variable. Assumptions Al: The investor i deals with only three assets (i.e. index equivalent portfolio, index futures and bonds). The market of the specification is as follows: Market Price at t Future Value at T # held Aggregate Spot Market S(t) ?(T) X . X si s Index Fututes 0 F(T,T)F(t ,T) X fi 0 Bond exp(r f ) 1 b^^ B Since this assumption can develop a one period pricing model only, the markedtothemarket effect is thus neglected. A2: Homogeneous belief on the spot price, S(T) dividend payout, E), and index futures price, F(T,T), is over all investors. A3: Every investor maximizes his utility function which is G i (E i ,a i 2 ), where E.^ and a Â± 2 are the expected value and the variance of terminal wealth, respectively. It is at least second dif f erentiable with respect to E.^ and a Â± 2 . The preference function, is also an increasing function of E^^ and decreasing function of a^ 2 . Furthermore, the marginal utility of the expected terminal wealth is decreasing ( 3G, < 0) 3E i A4: The total units of demand in the spot market, X , is set in the s form specified below for all the investors (N) in this market, namely, 2x gi = Â£S gi = X g = X g (stands for gross supply) . A5: The interest rate over the buying date to the maturity is treated as constant. Basically, it follows that the price of the future
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27 is the same as the price of the forward contract (Cox, Ingersoll and Ross, 1981) A6: No taxes, no transaction costs, pricetaking investors, no indivisibility, and costless information are available to everyone. Definitions Dl: S(T) = l"(T) + D The indexequivalent portfolio, S(t), is formed at time t. From time t to t ime T, this portfolio receives a. total dividend payout, D. The value of the indexequivalent portfolio after dividend payout is I(T) at time T. Thus, the index portfolio, S(T), at time T is the sum of I(T) and D. D2: The risk tolerance of investor i is defined by 3G i /3E i " K5 n i ZT1 i ~ 11 3G i /3a 2 i 1 1 where n is the total risk tolerance in economy. D3: The percentage of the futures price change (or the "return") of the index futures in terms of index futures is m F(T,T)F(t,T) r = F F(t,T) D4: The return of the spot indexequivalent protfolio _ SCOS(t) r = s S(t) D5: The systematic risk of the index futures Cov(r , r ) B = 3 F F Var(r) s
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28 D6: The percentage of the futures price change of the index futures in the terms of spot index is ~ F(T,T) F(t,T) R = F I(t) D7: The hedge ratio of the index futures against the spot index is Cov(F(T,T), S(T)) a B = = sF H Var(S"(T)) (T 3 Model This approach is to determine conditions for equilibrium of exchange of the three assets: the indexequilibrium portfolio, the index futures contract underlying this spot portfolio and the bond. Each individual brings to the market his or her present holdings of these three assets and an exchange takes place. This equilibrium approach needs to know what the price must be in order to satisfy demand schedules and also fulfill the condition that supply and demand be equal for all three assets. To answer this question, two requirements must be met. First relationship describing individual demand must be established. Second, these relations of all investors' demand are incorporated into a system to describe general equilibrium. Assuming that there are a large number of risk averse investors labeled i (i=l,2, Â— ,N) , we can consider the behavior of a typical investor. He or she has to form a portfolio by choosing from the following three assets, namely, the indexequivalent portfolio, the index futures, and the riskfree bond. The future value on these three assets is assumed to be a random variable whose distribution is known to
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29 the investor (Assumption Al). Moreover, all investors are assumed to have identical perceptions of these probability distributions (Assumption A2). The future value on a whole portfolio is, of course, a random variable. The portfolio analyses mentioned earlier assume that, in the choice from all the possible combinations, the investor is satisfied to be guided by its expected future value and its variance only (Assumption A3) . It is important to make precise the description of a portfolio in these terms. It is obvious that the holdings of these three assets must be measured in some kind of units. Because the index futures contract is exactly a derivative asset under a particular indexequivalent portfolio, we select one "physical" unit of the spot portfolio as our measure unit and define expected future value and variance of future value relative to this unit. It is convenient to give an intrepretation to the concept of "future (dollar) value" by assuming discrete market dates with intervals of one time unit in an equilibrium perfect market (Assumption A4A6). The future value to be considered on any asset on a given market date may then be thought of as the value per unit that the asset will have at the next market date (including accrued dividends). The main purpose of this dissertation is to compare the relations between the price and the future value of the indexequivalent portfolio and the index futures. To facilitate such comparisions , a risk free bond is used as yardstick. Now, partial equilibrium conditions are capable of determining relative prices: to fix the indexequivalent portfolio's price and express the index futures on it.
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30 The utility function of investor i is denoted as G^(E^,cr^ 2 ). Likewise, the current value of investor i's portfolio, or his budget constraint, is the following eq (3.1). Therefore, the terminal wealth of investor i's portfolio is expressed in eq (3.2). The Budget Constraint: X si E W ti = X si I(t) + b i ex P<" rf > + OXX f . (3.1) The Terminal Wealth: W Ti = X gi 7(T) + X fi [F(T,T) F(t,T)] + b. (3.2) The expected terminal wealth and variance of the terminal wealth are listed as eq (3.3) and eq (3.4), respectively. The Expected Terminal Wealth: E i E E(W Ti } = X si * (T) + X fi [ * (T ' T) " + b Â± (3.3) Variance of Terminal Wealth: a 1 2 = Var (W,^) = X %i Â°s 2 + X %i Â°f 2 + 2X si X fi a sF " X si 2 (a i 2 + 2 Â°ID + V> + X2 fi Â°F 2 + 2X si X fi Â°U + 2X si X fi a DF (3.4) Formally, then, we postulate that each investor i, who behaves as if attempting to maximize the utility function, is subject to the budget constraint, the expected value and variance of his portfolio to form his or her portfolio. Forming the LAGRANGEAN for each investor in eq (3.5) and differentiating with respect to X , X , and b. yields S X 1 1 X the first order conditions in eq (3.6) to eq (3.9).
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31 The Lagrangean Form: L. = G Â± (E Â± , o Â± 2 ) + A. (W ti X si I(t) b. exp(' r f)) (3.5) Maximization: Max Li X .,X Â£ .,b J s.t. (3.3) and (3.4) si fi 1 2 First Order Condition : 3L 3G. 3G. Â— " Â— < r < T > +D > + Â— a t 2X si Â°s 2 + 2X fi a sF ] " X i I(t) " 0 3X . 3E. 3a. si i l 3L = 3G. _ 3G. 1 (F(T,T) F(t,T) + 1 [2X f , a 2 + 2X . a ] = 0 3X^ Â± 3F~ 3o^ fl F 81 sF (3.6) (3.7) 3L 3G _ = JiK exp(" r f) = 0 (3.8) 3b Â± 1 3L = W ti X gl I(t) b Â± exp(" r f) = 0 (3.9) 3A i These firstorder conditions may be aggregated to derive the equilibrium relationship. Substract eq (3.6) from eq (3.7), and rearranging, we obtain eq (3.10). 3G, 1 {[I(T)+D] [F(T,T)F(t,T)]> 3E i 3G Ji 2[(a s 2 a sp ) X si + (a gF a *) X fÂ± ] X s(t) 0 3a Â± (3.10)
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32 Dividing both side of eq (3.10) yields eq (3.11) 3G./3E _ _ _ [KT) + D F(T,T) + F(t,T) ] + 2[X .(a/ a ) 3G./3a 2 31 s sF l i 3G./3E + X (a a *)] _J * exp( r f) I(t) = 0 fl SF F 3G./3a 2 (3.11) To obtain the pricing relationship, we have to aggregate over all investors. To determine general equlibrium, we must also specify equality between demand and supply for each asset. Recall that X,.,, X ,, b. were fi si i defined as the units of (net long) index futures, spot indexequivalent portfolio and the bond held by investor i. These market clearing conditions can be written as the following: EX^^O, EX = X , Eb = B. fi si s i This essentially completes the equations describing a partial equilibrium. Also, in equilibruim, eq (3.11) must hold for all investors and if we assume that all investors have homogeneous expectations regarding and 2 o\ we can sum eq (3.11) over i and define the risk tolerance, n , to 3 arrive at the closedform in eq (3.12). F(t,T) = I(t) e r f + 2/n X (a 2 o ) [D I(T) + F(T,T) ] S S Sr = I(t) e r f + 2/n X (a a _) D s s sF = I(t) e r f + 2/n X g a Dp D (3.12) In eq (3.12), we have a term, 2X (o 2 a )/n, to measure the risk S S Qc premium of the random dividend effect. The sign and value of this premium 3G is thus an empirical question. Dividing eq (3.6) and eq (3.7) by yields eq (3.13) and eq (3.14), respectively. I(t) exp( r f) I(T) D = 2/n X a 2 ' (3.13) s s F(t,T) F(T,T) = 2/n X s a gF (3 .14)
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33 After rearrangement of eq (3.13) and eq (3.14), we have eq(3.15). r Q sF F(T,T) F(t,T) = [I(T) + D I(t) exp( r f)] a 2 s = [S(T) I(t) exp( r f)] ^7 (3.15) Both sides of eq (3.15) divided by F(t,T) yields eq (3.16) o r S(T) I(t) I(t) r v = " I (exp( r f) 1) 77 L Kt) J F(t,T) 9 Cov (r , r ) Var (?) S Â£ * e F (f s " r f } (3 * 16) Equation (3.16) is similar to the conventional CAPM which applies to an asset holding no initial positions. That is, the introduction of index futures market does not change the basic structure of capital asset pricing under conditions of uncertainty. Hereafter, eq (3.16) is referred as LikeCAPM model. COV(? , rl)/ VAR(r' ) is the risk level of the index s r s futures relative to the spot index equivalent portfolio. The covariance term, cov(r g , r p ) , is the key element for the risk premium of the index futures contract. This should come as no surprise, since the index futures contracts do include dividend risk. Furthermore, due to no investment in the index futures at the beginning, we have no intercept term. However, if we take margin into account, we should have the full CAPM. This is to say that the margin is treated as the performance bond.
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34 Now, let us examine the conventional hedging strategy: regress the index futures price change on the spot index change to yield the hedge ratio subject to a holding period as the price change duration. After dividing eq (3.15) by I(t), we arrive at eq (3.17). hlÂ£ CF. r f > o s 6 H (r g r f ) (3.17) With a position of long the spot indexequivalent portfolio and short the index futures at time t, using 1/$ as the hedge ratio, we n would have the riskfree rate of return. In the next section, we intend to demonstrate a comparative statics analysis of this model, to explain who are going to short or long. Comparative Statics Analysis The comparative statics of security risk premium in a meanvariance context has received rigorous treatment in the literature.^ Including the index futures market, the comparative statics analysis of market equilibrium is not examined since the equilibrium price of index futures not yet throughly understood. We are interested in the factors affecting the optimal number of index futures contract. In this section, the types of comparative statics changes considered include the spot index futures price change, spot price change, the interest rate change, the expected dividend yield change, the expected basis change and the relative risk aversion measurement change. Statements thus are made concerning the expected effect on the demand of assets of an increase in these factors.
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35 To facilitate a comparative statics analysis, we need to develop the optimal units of index futures and indexequivalent portfolio held by the investors. 3G i Dividing eq(3.6) and eq(3.7) by , together with the budget constraint, we have the optimal set as eq (3.18). * * "s i 0 s 2 + X fi a sF = V 2 [I(T) + Â° " I(t) ex P (if >l X si* a sF + X fi V = V 2 [ F < T ' T > ~ FCt'T)! b Â± * = e r f ( W t . X gi * s(t))J (3.18) Rearranging, we find the optimal holdings to be eq (3.19) * [I(T) + D I(t) e r f] a 2 [F(T,T) F(t.T)] a X , = se si A [F(T,T) F(t,T) ] a g 2 [I(T) + D I(t) exp( r f)] a p X fi " _ s bi* = e r f(W ti X gi * s(t) (3.19) where Z 5 2/tk (a 2 a 2 Â„ a 2 ) l s F sF ' These are functions of index functions price, spot index price, interest rate, dividend payout, degree of risk aversion, etc. Then we do some partital derivatives to examine factors which might affect the optimal hold units of index futures. For risk adverse investors (which we assume) , the riskaversion measure, is positive. Therefore, Z, defined as 2(o g 2 o^, 2 a gF 2 )/ T l i f is positive also. Thus, we have the following results, eq (3.20) to eq(3.26). 3X o 2 = 8 < 0 ....(3.20) 3F(t,T) Z
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36 3X f I(t)exp( f)o _ = _ > 0 (3.21) 9r f Z 3X exp( r f)a 11 = _ > 0 (3.22) 3s(t) Z 3X fi _0 sF _ = s * < 0 (3.23) 3D Z 3X. a 2 = 1 > 0 (3.24) 3F(T,T) Z 3X fi = [F(T,T) F(t,T)]a g 2 [I(T) + D I(t)exp( r f)] O gF 3n Â± Z > > 0 if numberator = 0 (3.25) < < 3F(t,T) 2 3Â° ^ n J sD i Â— s X > 0 (3.26) A model of demand or supply for the index futures and indexequivalent portfolio in equilibruim stage is shown as eq (3.19). These results are based on previous closedform, pertaining to some restricted assumptions. Here, the optimal number of units of the index futures is derived first, assumed to be net long if positive. Equation (3.20) shows the law of demand in the index futures. The higher the index futures price is, the less one goes long the index futures. The index futures is a normal goods. The impact of the interest rate on the number of units of the index futures is shown in eq(3.21). The higher the interest rate is, the more the long position in the index futures would be. This result means that the index futures contract tends to be a free good if the interest
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37 rate is very high. Furthermore, equation (3.22) demonstrates the relationship between holding the indexequivalent portfolio being long the index futures. The higher the spot index is, the more to long the index futures would be. So, these two financial products are substitute goods. Equation (3.23) shows the impact of dividend payout on the number of units to long index futures. If the expected dividend increases, the demand to long the index futures decreases. In equation (3.24), it is shown that expected maturity futures price affects the number of units to long the index futures. The higher the expected maturity futures price (or the expected profit to long the index futures) is, the more to long the index futures would be, other factors being constant. According to the budget constraint and eq (3.25), individual initial wealth affects the investment decision through the combined effect of the degree of the riskaversion, the expected price and the variance of the index futures price, and the covariance of indexequivalent portfolio price and index futures price. With regard to the performance bond, the performance bond holding is not influenced by the expectation of the index futures contract. It might suggest that bond holding is a supplementary good of the index futures market. In addition, the initial wealth does not directly affect the holding of these assets. 9F(t,T) Comparative statics indicates that > 0. Thus the index sD futures price could increase, according to eq (3.26), when the underlying indexequivalent portfolio dividend yield is stochastic if the covariance of future value on the indexequivalent portfolio and the dividend payout increases. This is due to the covariance of dividend payout and the indexequivalent portfolio being equal to the variance of dividend payout
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38 plus the covariance of dividend payout and the futures price, (i.e., Cov(D, SCO) = Var(D) + Cov(F(T,T) , S(T)). Therefore, if dividend payout increases following the increase of the indexequivalent portfolio, then the index value would also increase following the increase of dividend payout. This means that the index futures price would increase too. Thus, the higher the correlation between the indexequivalent portfolio and dividend payout, the higher the index futures price would be. The index futures price is influenced by the covariance term between dividend payout and indexequivalent portfolio. Intuitively, this has to do with the fact that the index futures price still has to face the dividend risk even if the dividend payout does not benefit the individual who trades in the index futures contracts in long side or in short side. Summary We have presented a closed form model for index futures pricing in a partial equilibrium approach. When we allow for the existence of random dividend yield, this model does bear a risk premium for this random term. A covariance term of dividend payout and indexequivalent portfolio is added to the perfect market model. We can see that to evaluate the index futures contract is the same as a project evaluation. Market equilibrium causes the inflows received from being long a unit of index futures contract, (holding to the maturity i.e., F(T,T)F(t,T) ,) to be inflows from borrowing fund to purchase a unit of indexequivalent portfolio (valued as $500 times spot index) and hold this to the maturity of its corresponding index futures contract, over this period to collect the expected dividend yield and its associated dividend risk premium (i.e., I(T)I(t)e rf + D 2X Cov(D .S^T) ) /n) )
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39 It means that the cash inflow to long an index futures contract is quite similar to implementing a capital investment project with borrowing funds, over the holding period to collect its profit or loss which is marked to the market and including the uncertain dividend payout or earning performance. Meanwhile, with comparative analysis, we know the index futures is a normal good. The demand to be long index futures increases, if the interest rate, the indexequivalent portfolio price, the expectation of the future spot price, or the correlation between indexequivalent portfolio and dividend payout increases. On the contrary, the demand to be long the index futures decreases if expected dividend decreases. Moreover, the more riskadverse the investor is, the less the demand to long the index futures will be. When we express this model in rate of return terms, somewhat surprisingly, the equilibrium market relationship between risk and "market return" on index futures contract is still of the same general linear form as that of the SharpLintnerMossin model. The primary testable implications of the model are that the linearity of the relationship between risk and "expected return," or the "hedge ratio" is greater than one, or the dividend risk premium exists in the market.
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40 Notes If preference function is a quadratic function, then this measure is identical to risk tolerance defined by Pratt (1964). Expected quaratic utility function is only a special function of G i ( E i Â» tJ i 2 ) Â• Without loss of generality, the utility function satisfies some other condition such that second order condition of optimality is satisfied. It is consistent with the Ederington (1979) hedge argument. If dividend payout is a random term indeed, the hedge ratio is not equal to one as the traditionalists argued. Furthermore, assuming that the market setting as follows: Market Price at t Future Value at t+1 I held Aggregate Spot Market S(t) " S(t+1) X . X Index Futures 0 F(t+1 ,T)F(t,T) x Jt 0 s Bond exp(r f ) 1 b" B then we could have the pricing relation as eq (3.12.1) F(t,T) = I(t)e r f + 2 /n X [Var(S(t+l) Cov(S(t+l), F(t+1,T))] s + Â£~Vt+l V t+1 > T > " * t < t+1 >] (3.12.1) where D F t , I are expected value of D, F, I based on the information set at time t. With arrangement, it becomes eq (3.12.2) [F(t,T)F t (t+l,T)] [I(t) I t (t+1,T)] D +I(t)(e r f1) + 2/n X e (Var(S(t+l)) Cov (S(t+1), F(t+1),T))] C,t 1 S (3.12.2) It shows the equilibrium basis relationship. The aggregate basis risk thus is affected by the interest rate, the dividend, the risk aversion in the economy, the market return, the total units of indexequivalent portfolio in the capital market, and the holding period. The analysis of the spot market is reported by others in detail. So, we only analyze the index futures case.
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CHAPTER IV THE EMPIRICAL STUDY In chapter III, the theoretical price of an index futures contract using an equilibrium model was stated in eq(3.12). A major difference between this model and those based upon an arbitrage approach is the potential importance of an unknown dividend. A covariance term between the dividend payout and the indexequivalent portfolio is added to the equilibrium pricing model. In the arbitrage model, the dividend payout is assumed to be known in advance. In this chapter, we investigate whether this covariance term empirically appears to exist in the actual trading market. The dividend risk argument of this dissertation is contingent on the existence of a covariance between dividend payouts and an indexequivalent portfolio. We have also linked the theoretical index price using the equilibrium approach to the conventional CAPM model and a hedging relationship as shown in eq(3.16) and eq(3.17), respectively. If dividend risk is important in index futures pricing, the LikeCAPM model, i.e., eq(3.16), suggests a null intercept as well as a systematic risk level greater than one. Therefore, the LikeCAPM model can serve as a supportive test. If dividend risk is present and priced in the markets, the LikeCAPM model predicts that the systematic risk is greater than one. In addition, the existence of a nonzero intercept in the LikeCAPM model is important in that it is suggestive of another missing variable, such as the margin requirement. If the intercept of the regression is not significantly 41
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42 different from zero, we have no evidence that this is an important factor in pricing index futures. Similarly, a conventional hedge relationship between the spot and the index futures is derived, as shown in eq(3.17). This hedge relationship also serves as a supportive test to the dividend risk argument: the slope of the regression is expected to be greater than one if dividend risk exists in the market. These three equations, eq(3.12), eq(3.16) and eq(3.17), serve as empirical devices to evaluate the impact of dividend risk and to describe the behavior of any such dividend risk premium. The pricing equations are, strictly speaking, correct only in a one period case. This means that the futures contract is being treated as a forward contract.*" The dividend risk argument has two critical considerations which must be understood. First, when a dividend payment is declared by a company, there is no dividend risk present any longer between the date of the announcement and subsequent payment. However, for the indexequivalent portfolio, it is not easy to eliminate total dividend risk at any specified date, since not all companies in the indexequivalent portfolio announce dividend payments at the same date. A varying degree of dividend risk is therefore expected to exist in the market. The nearer the maturity date of the index futures contract is approaching, the less the corresponding dividend risk would probably be. Therefore, the last several trading days might bear insignificant dividend risk. Second, most companies pay dividends quarterly which causes a seasonality in dividend payout behavior. Since the dividends of most companies are declared before paid, the dividend risk premium of the index futures price might be affected by the behavior of the dividend payout pattern of the spot index.
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43 For these two reasons, we report two tables of dividend payout distributions. The first table (Table 4.1) describes the frequency of the number of days between the date of a dividend announcement and the eventual exdividend date for all companies listed in the S&P 500 and the NYSE. This table reveals the degree of certainty about future dividend payments at points in time prior to when the dividend actually is paid. The second table (Table 4.2) describes the monthly payout pattern during the years 19811983. These two tables motivate the basic empirical design of this dissertation. In what follows, we discuss first how the prelimiary results in Table 4.1 and 4.2 were obtained. Following that variable measurement, test methodology, test hypotheses and data set problems are discussed. The main content of this second section is to examine the difficulty of using a direct testing methodology, and, in its place develop a dummy variable approach for indirect testing. In the third section, results of a stricter test methodology as well as tests of the LikeCAPM and the hedging relationship are evaluated. Finally, a brief summary is provided in the fourth section. A Prelimiary Result In this section, two tables are presented to highlight the empirical study: (1) a table regarding the frequency of dividend events and (2) a table regarding the seasonality of monthly dividend payments. Based on the period 19821983, the dividend data on the CRSP tape was used to evalute the degree of dividend certainty over different
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44 lengths of time. This is done by noting the number of days which transpire between the date of a dividend declaration and its eventual payment (actually, the exdividend date.) This was done for (1) all companies included in the S&P 500 and (2) all companies listed on the NYSE. These two indicies were chosen since it is exactly these indices on which the principle stock index futures are written. The results are reported in Table 4.1. Notice that seventyfive percent of the observations had announcement dates within thirty days from the exdividend date. This is true for both the NYSE and S&P 500. It suggests that dividend risk is not present in the period up to thirty days before the exdividend date for seventy five percent of all firms. Thus, trading in the indexequivalent portfolio and the index futures bear substantially less dividend risk within the thirty days before exdividend date. In addition, a monthly dividend payout pattern (from 1981 to 1983) is displayed in Table 4.2. There are four peak dividend payout months. They are February, May, August, and November. These results suggests that the behavior of dividend risk may be characterized by seasonality and by the length of time before the contract matures. Basic Discussion Direct testing of eq(3.12) requires data on futures prices, interest rates, spot price of the index, an expected dividend, and the covariance between dividend payout and the indexequivalent portfolio. Of these, the futures price, spot price, and interest rate are directly observable. The expected dividend could be estimated in a number of
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45 ways. Unfortunately, the covariance term is a cross section term and presents a severe estimation problem. To compute the covariance between the dividend payout and the indexequivalent portfolio requires that we specify the holding period being considered. One can not simply examine the historical covariance between, say, one day spot price changes and dividends paid on that date. First, the dividends would be known on that date since they would have been previously declared. Second, this would focus only on a one day holding period. It would not begin to evaluate the covariance when the holding period is different. And, if we wish to evaluate holding periods longer than one day, it is almost impossible to specify which holding periods should be examined. Directly testing is quite difficult. We turn, instead, to an indirect test which uses a dummy variable approach. But, it is still necessary to estimate the expected dividend payout per unit for the indexequivalent portfolio. Denoting D as the expected dividend, the estimation method used is as follows: Accumulated Dividend of the index from t to T S(t) D = x (4.1) Market Value of the index at t 500 In addition, we set up a dummy variable approach to represent the covariance term over the holding period. This is done by using the additive property of the covariance term to handle the holding period problem. The dummy variable coding is set to a specified value in an attempt to measure the holding period effect of dividend risk. Two approaches are used. First, a twelvecalendarmonth dummy variable approach captures the monthly effect. Second, an approach based on a threemonth payout pattern is used to capture any seasonal effect in the dividend risk premium.
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46 Equipped with an estimate of the expected dividend, (D as defined above) and the number of indexequivalent portfolio, (I(t)) in the market, eq(3.12) can be empirically tested as a regression of F(t,T) (adjusted by the expected dividend of indexequivalent) on the future certaintyequivalent portfolio, and the twelve dummy variables. However, to get a conventional Rsquare, we need to subtract from the right hand side the future certaintyequivalent portfolio. This results in a new dependent variable, which we call the "futures discount." This "futures discount" is regressed on the various dummy variables. Using this model, we might still have two problems. First, a serial correlation might arise. Second, we have to consider the stationarity of the residuals. In order to assure that the regression residuals are stationary, the regression should be performed on a rate of change or on a generalized difference price change. In fact, it is the rate of change in the futures price, not the price level alone, that is the variance of economic interest. Taking generialized first differences of eq(3.12), we solve the serial residual problem and the serial correlation adjustment could provide a more efficient testing. In summation, the generalized difference equation states that the change in the index futures price, adjusted for both the expected dividend and the value of the future certaintyequivalent of spot portfolio, is equal to the dividend risk premium as captured by various dummy variables. Data Sets Following the preliminary result above, five formal tests are motivated: (1) the use of a calendarmonth dummy variable approach (2) the
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47 use of seasonal dividend dummy variable approach (3) a test of the market pricing model, (4) a test of the LikeCAPM relationship, and (5) a test of the hedge relationship. These five tests are performed on a data base of the two most commonly traded indices: the S&P 500, and the NYSE. There are six index futures simultaneously traded in the futures market, since every futures contract lasts eighteen months (except in the case of eariest contracts offered) and new contracts have maturity dates which are three months apart. For all futures contracts, daily closing prices (at 4:15 P.M. Chicago Time) are obtained. For the spots, daily closing prices are also observed (at 4:00 P.M. Chicago Time). The Treasury Bill futures series is used as an interest rate series. The sample period is from the first trading date of every futures to March 8, 1984. In all cases, the percentage price change (or the "return") of the index futures is computed as a oneday holding. In addition, the dividend payout is assumed to be reinvested at the riskfree rate daily during the holding period. Since all subsequent statistical measures are assumed to be serially independent price changes, serial correlation coefficients of order one will be computed for each futures price series in order to adjust to the fact that price tend to move together. Since we use generalized first difference least squares to obtain efficient estimators, all observations which involve Monday or the day after a holiday are excluded in order to avoid a different time span problem. Unexpectedly, the dividend data series is not easily accessible. For New York Index Futures, the daily dividend payout series of the unit indexequivalent portfolio is only available from January 1, 1983.
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48 This was provided by the NYSE statistics department. For the S&P 500, data is aggregated from the CRSP daily tape (from Center Research of Security Prices, University of Chicago) based on the December 1982 COMPUSTAT TAPE (from Standard & Poor's Computing Service, Inc) cusip list. Unfortunately, 20 overthecounter companies are not accessible from the CRSP tape and data from 13 other companies are not fully covered over the sample period. Therefore, 6.6% of the S&P 500 is missing in the data set, meaning that we only create an S&P 467 dividend and market value data set. In terms of both the dividend and market value on December 1982, this S&P 467 dividend payout data series underestimates the total S&P 467 dividend payout data series the total S&P 500 by 6.6%. What we need, however, is the dividend payout per unit of index futures. As a consequence, we divide the total dividend by the market value to get the dividend per unit. This "unit" dividend is still approximately acceptable. The effective sample period is from January 1, 1983 to December 31, 1983 for the NYSE Index Futures and from April 21, 1982 to December 31, 1983 for the S&P 500 Index Futures. The data base is listed in Appendix A. Seven data subsets are named and tested for each contract. They are follows: 1. WHOLE: all observations of the index futures contracts. 2. LE30: all observations of the index futures contracts, excluding the last thirty tradingday observations. 3. LAST30: the last thirty tradingday observations of the index futures contract. 4. LAST90: the last ninety tradingday observations of the index futures contracts.
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49 5. LE90: all observations on the index futures contracts, excluding the last ninety tradingday observations. 6. L120LE30: the last one hundred and twenty observations, excluding the last thirty tradingday observations. 7. LE120: all observations of the index futures contracts, excluding the last one hundred and twenty tradingday observations. The reason for forming these data sets is that they enable us to analyze the dividend risk, in different respects. In the first place, WHOLE is necessary for us to have a whole picture of this "futures discount." Second, we need to know what happens to the nearest (to the maturity) contract. Therefore, LAST90 is useful. Third, the trading in the maturity month should have minimal dividend risk exposure. Thus, LAST30 is needed in order for us to see whether the perfect market model holds. Others are examined to see if there is a change in the pattern of dividend risk over different periods. Test Methodology and Results In the previous section, five tests related to three equations, eq(3.12), eq(3.16), and eq(3.17), were briefly discussed. In this section, the test methodology of each is presented in more detail. Results are reviewed in the subsequent subsections. The first subsection discusses the use of a calendarmonth dummy variable approach. The second subsection discusses the use of a dividendpayoutpattern dummy variable approach. The above two tests methods essentially employ the same concept except that they use a different dummy variable coding to capture the dividend risk premium. Based on initial results,
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50 the last thirty trading days before the index futures mature might not be exposed to dividend risk. If so, the perfect market model should hold in LAST30 data set. Therefore, the test of the perfect market model is performed in the third subsection. Other related issues, such as the test of LikeCAPM and the test of the hedge relationship, are discussed in the fourth subsection as supportive tests of the dividend risk argument. Finally, the fifth subsection summarizes the findings of this chapter. Use of a CalendarMonth Dummy Variable Approach In eq(3.12), the dividend risk adjustment is contingent on the existence of a covariance term between dividend payout and the indexequivalent portfolio. The dividend payout referred to is the accumulated daily dividend payment over the holding period. Assume that the dividend accumulating process is as follows: Tt1 D = ZD 10 t+1 ' t * 1 * 1 (4.3) Since the covariance has an additive property, we have the following equation. Tt1 Cov (D, S(T)) S Cov(D t+i,t+i+l' ?(T)) (4>3) The covariance term which is of interest depends upon two things: (1) individual covariance over shorter period of time (say, monthly or seasonal) and (2) the number of such periods which will occur from the date of pricing to maturity, i.e., time. To capture this covariance relationship, we will first use a monthly dummy variable approach as follows. Define the dummy variable coding, M j_Â» i = lÂ»2 Â— ,12, for the dividend risk premium of each calendar month.
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51 Values of any M. can vary from zero to two, depending on how many times the calendar month is duplicated over the holding period. For example, if an investor buys or sells a unit of index futures and holds from t to T, then the covered calendarmonth dummy variable of the twelve dummy variables is valued as one or two depending on whether the months are duplicated one or two times. Other noncovered calendar month dummy variables are set to be zero. For example, if we buy a December 1984 contract in June 1983, then the dummy variables standing for June through December are two and others are one. Similarly, if it is bought in June 1984, then the dummy variables standing for June through December are one and others are zero. To facilitate the statement of empirically testable null hypotheses, the ex ante equilibrium relation can be restated as an ex post relation assuming that (1) the independent variables contain no measurement errors, (2) that they are stationary, and (3) that individual expectations defined over the variates are unbiased. For OLS estimators to be efficient, it is required that observations be serially uncorrelated. Unfortunately, serial correlation is usually present in most time series regression in the first step of the estimation process. In our test, if serial correlation occurs, the Durbin procedure is used to adjust the ordinary leastsquares regression procedure to obtain efficient parameter estimates. This is referred to here as the second step of the estimation process. This procedure involves the use of generalized differencing to alter the linear model into one in which the errors are independent. Economically, it implies that the prior price information is carried over to today by the degree of serial correlation. Estimation of this generalized differencing results in a new, more efficient, set of parameter estimates.
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52 Since the empirical model consists of the index futures regressed on the future certaintyequivalent portfolio, and the twelve calendarmonth dummy variables without an intercept term, the Rsquare would be almost invariably be one. This is due to the exclusion of the intercept in the regression. Given this, a new dependent variable, which has been labeled futures discount, is created. Futures discounts is defined to be the value of the index futures price after subtracting the future certaintyequivalent spot portfolio and adding back the expected accumulated dividend payout per unit. In addition, the future certaintyequivalent spot portfolio must be an independent variable in the regression. Hereafter, we use FD and FCESP to indicate futures discount and the future certaintyequivalent spot portfolio, respectively. To summarize, a regression of FD on M and FCESP is performed. 2 The results of the first step regression are reported in Appendix B. The results of the second step regression are interpretated after the mathematical details of this test are examined. Test I : Test of eq(3.12) using monthly dummy variable approach Step 1: The first step regression Model: FD(W) o M (W) + 6 FCESP(W) + Â£ (4.4) w where T 1) FD(W) = F(W,T) + Z D(i) exp(r (Ti)/(Tt)) i=W I(W) exp(r f (TW)/(Tt)) 2) M(W) i M. (W) M 12 (W) M i (W) = \ 1 0 1 ( 2 il,2,...,12 3) FCESP = I(W) exp(r f (TW)/(Tt)) 4) e = pe , + I w w1 w
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53 5) t < W < T THE NULL HYPOTHESES: (1) O = 0 (2) 6 = 0 THE ALTERNATIVE HYPOTHESES : (1) o i 0 (2) 8*0 These estimators, however, are not efficient due to serial correlation in the error term. To obtain an efficient estimator, a second step regression using the serial correlation estimate of the first step regression, 'p*', is performed. Step 2: The second step regression Model: A FD(W) = a A M(W) + 6 A FCESP(W) + Â£ (4.5) Â— Â— w where 1) A FD(W) = FD(W) ^ FD(Wl) 2) A M(W) = M(W) ^MCWl) 3) A FCESP(W) = FCESP(W) / p FCESP(Wl) 4) t < W < T THE NULL HYPOTHESES: (1) a = 0 (2) 6=0 THE ALTERNATIVE HYPOTHESES: (1) a * 0 (2) 6*0 Note that this linear transformation does not change the null and alternative hypotheses. Results of step one on the data sets of WHOLE and LE30 are reported
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54 in Appendix B, Table B.1B.2, respectively. Results of step two are in Table 4.3 and Table 4.4 for data sets of WHOLE and LE30. We find that the slope coefficients of the calendarmonth dummy variables are mostly negative with significance at 1% significance level. Not surprisingly, the absolute value of the slopes are larger in the data set LE30 than they are in the data set WHOLE (which includes the last month of trading.) This implies that the earlier the transaction is, the larger the futures discount will be. Intuitively, the accumulated dividend risk (i.e., the sum of the coefficients of several calendarmonth dummy variables) is a potential factor which would cause this discount to decrease over time. It suggests that a transaction in the early trading days of contract is riskier due to the higher uncertainty of total dividend payout. If dividend uncertainty is reduced, such as at the time when many companies submit their 10K form and annual report around February, the dividend risk is not a significant factor in creating futures discounts any more. In some cases, the March dummy variable is positive with significance (at 1% significance level.) This positive effect might be caused by any number of things, a change in the investment opportunity set, for example. Occasionally, multicollinearity does occur in some regressions. This is caused by the fact that the observations all duplicate at the maturity (calendar) month, such as L120Le30 consisting of maturity month observations, or the effective observations of some months are missing due to no active trading in that month. In this case, we simply drop one variable which is collinear to another one. The slope of the remaining variable in the regression would then be the sum of the slopes of the two which are multicollinear before we drop it. At any rate, it does not affect
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results and their implications. There is evidence that dividend risk is priced in the market. Meanwhile, since all observations always duplicate at the maturity month, the calendarmonth dummy variable, which also represents the maturity month, would have a larger coefficient. This larger value actually is mixed by the true value with the intercept. 3 Use of a DividendPayoutPattern Dummy Variable Approach The results of the monthly dummy variable approach discussed above are consistent with the posibility of a dividend risk premium. They are, however, consistent with other explainations. For example, simply time alone (the greater the time to expiration the greater the discount due to whatever reason) could be the cause of these results. The only part of the tests which points strongly towards an exclusion dividend risk impact is the insignificance of February, a month in which corporate reports would tend to reduce risk. Yet the financial literature has not observed a strong February effect on other areas of risk measurement. What is needed is a test which is more closely associated with dividends themselves, a test which confirms or rejects solely the dividend risk argument. That is precisely what the tests associated with seasonal dummies are designed to do. In Table 4.2, a dividend payout pattern was seen with peak months in February, May, August, and November. Three dummy variables are defined in order to investigate the possible seasonality of dividend risk premium. These three dummy variables are called the dividendpayoutpattern dummy variables, S Q , and S 2 Â» Their corresponding values are set as follows. S 0 = M 2 + M 5 + M 8 + M ll
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56 = M, + M, + M., + M 10 M 12 + MÂ„ + M, + M 9 Since a futures contract is eighteen months forward, the value of the dividendpayoutpattern dummy variables can take on values between one and six. The basic idea underlying this approach is to examine whether a seasonality of the futures discount appears corresponding to the dividend payout pattern. In other words, we want to investigate the behavior of the dividend risk premium. In spite of this objective, the behavior of dividend risk might have other complications. The changing pattern along with its magnitude could be further estimated. The test procedure is the same as the first subsection. The mathematical details and hypotheses are as follows. Test II : Test of eq(3.12) in a dividendpayoutpattern dummy variable approach Step 1: The first step regression Model: FD(W) = a S(W) + g FCESP(W) + e, w (4.6) where T 1) FD(W) = F(W,T) + Â£ D(i) exp(r. (Ti)/(Tt)) i=w I(W) exp(r f (TW)/(Tt)) S Â± (W) i = 0,1,2
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57 3) FCESP(W) = I(W) exp (r f (TW)/(Tt) 4) e = p e . + Â£ w w1 w 5) t < w < T THE NULL HYPOTHESES: (1) o = 0 (2) 8 = 0 THE ALTERNATIVE HYPOTHESES: (1) o f 0 (2) 6 J* 0 Step 2: The second step regression Model: A FD(W) a A S(W) + 8 FCESP(W) + Â£ (4.4) Â— w where 1) A FD(W) = FD(W) ^ FD(Wl) 2) A S(W) = S(W) S(Wl) 3) A FCESP(W) = FCESP(W) FCESP(Wl) 4) t < W < T THE NULL HYPOTHESES: (1) a = 0 (2) 80 THE ALTERNATIVE HYPOTHESES: (1) a i 0 (2) 8^0 The results summarized are derived after performing step one and step two on WHOLE, LE30, LAST30, L120LE30, LE90, LE120 data sets, respectively. Some of the results are quite significant. The results of the first step regression are presented in Appendix B, Table B.3B.5, respectively. The results of the second step regression are
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58 reported in Table 4.5, Table 4.6, and Table 4.7. Before examining the results, a review of the test's logic is appropriate. Suppose a quarterly dividend payout is just announced, the dividend risk should be partially or temporarily reduced. Will the announcement affect the structure of the dividend risk premium? This hypothesis expands on the results of Test I (monthly dummies.) The extent to which uncertainty is resovled by the announcement can be understood by examining the trend of the slope coefficient of dividendpayoutpattern dummy variables. If the announcement essentially resolves all uncertainty, the dividend risk premium would be zero. On the other hand, the announcement could represent partial resolution only. In Table 4.5, Table 4.6 and Table 4.7, we find that the risk premium of two months before and one month before are significant over the following data subsets: WHOLE, LE120, LE90. However, the risk premium of the dividendpeakpayout month is not significantly different from zero. Furthermore, the observations of two months before the dividend peak month bear the sum of the dividend risk premium of two months before and one month before dividendpeakmonth, suggesting that dividend risk decreases when the dividend announcement is forthcoming. Finally, the uncertainty is resolved at the announcement. These results confirm the earlier conclusion that the futures discount can be explained by a factor which has seasonal pattern. Furthermore, since we are also interested in the changing pattern of dividend risk of the nearest contract, we apply the regression on the L120LE30 and LAST90 . As shown in Table 4.7, only two months before the matuity is significant (at 1%
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59 significance level). Recall that Table 4.1 shows that 98 percent of dividend events leave no dividend uncertainty within sixty tradingday of the exdividend date. Therefore, the fact that ther is no significance in both one month before and during the dividendpeakmonth dummy variables suggests strongly that the dividend risk is reduced when time to the maturity is within sixty days before dividend announcement. The dividend risk indeed disappears in the market for the very near contract. We thought that perfect market pricing model would hold in LAST30 at least. Two step least square procedure could provide the better efficient estimators. The coefficients of the dummy variables in Test I and Test II, therefore, provide more reliable estimations of the dividend risk premium of the S&P 500 for every futures contract to the corresponding one of the NYSE futures contract. We find that S&P 500' s dividend risk premium (in absolute value) is larger than that of NYSE's. Recall that our model is constructed in terms of a "physical unit." Therefore, the dividend risk premium is measured by a unit. But one should recall that the transaction value of S&P 500 index futures is currently greater than that of the NYSE index futures by approximately fifty percent. Therefore, it is not surprising that the S&P 500 bears larger dividend risk premium. Test of the Perfect Market Model Regressing FD on FCESP using the LAST 30 data set, we expect that both the intercepts and the slopes are not significantly different from zero with low Rsquare. This is due to the lack of any explanatory power of the FCESP variable. It means that the following model can
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60 test the perfect market model. Test III ; Test of eq(3.12) for LAST30 data set Step 1: The first step regression Model: FD(W) = a + 6 FCESP(W) + e (4.8) w where T 1) FD(W) s F(W,T) + Z D(i) exp (r f (Ti)/(Tt) i=w I(W) exp (r f (TW)/(Tt)) 2) FCESP(W) = I(W) exp (r f (TW)/(Tt) 3) e = p e . + C w w1 w 4) t < w < T THE NULL HYPOTHESES: (1) a = 0 (2) 6 = 0 THE ALTERNATIVE HYPOTHESES: (1) a + 0 (2) 6^0 Step 2: The second step regression Model: AFD(W) = a + 8 A FCESP(W) + Â£ (4.9) w where 1) AFD(W) = FD(W) ~'p N FD(Wl) 2) AFCESP(W) = FCESP(W) "p" FCESP (W1) 3) t < w < T THE NULL HYPOTHESES: (1) a = 0 (2) 6=0 THE ALTERNATIVE HYPOTHESES: (1) a jf 0 (2) 6 f 0
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61 The results are shown in Table 4.8. None of the parameters are significant. The perfect market pricing model is confirmed in the LAST30 data set. This is true for all contracts, including the very earlier stage contracts, such as March 1983 contract. This suggests that the index futures market was in equilibrium at an earlier stage than other researchers have suggested. Supportive Tests; LikeCAPM and the Hedge Relation Dusak (1973) starts from the CAPM to explain the "return" of a futures contract in order to study individual commodity futures. The following equation is employed. E(P(i,l))P'(i,0) = Bi (E(R^)r f ) p(i,0) where Rj,: the return of the market portfolio r f : riskfree rate P(i,0): spot price of ith commodity at period 0 P(i,l): spot price of ith commodity at period 1 p'(i,0): futures price of ith commodity at period 0 The above form is similar to our hedging relation. However, the model was obtained from a heuristic discussion as opposed to an equilibrium derivation which was used in this research. The test hypotheses and the regression used to test the LikeCAPM and the hedging relation are follows: Test IV : Test of eq(3.16), the LikeCAPM Model: F p = a + 6 (? r f ) (4.10)
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62 THE NULL HYPOTHESES: H : a = 0, 6 = 1 o THE ALTERNATIVE HYPOTHESIS: a f 0 or B + 1 Test V: Test of eq(3.17), the hedging relation Â— Â— Model: R^, = a + 6(r g r f ) THE NULL HYPOTHESIS: H : a = 0, 6= 1 o THE ALTERNATIVE HYPOTHESIS: H.: a f 0 or B 1* 1 Both regressions' null hypotheses state that the intercepts are zero and the slopes are one. Under the dividend risk argument, we also expect the intercepts of eq(4.7) and eq(4.8) to be zero. But the dividend risk argument suggests that the slope of eq(4.7) and eq(4.8) will be significant and greater than one. As noted before, serial correlation would be adjusted if necessary. However, performing ordinary least square at the rate of return level might adequately avoid serial correlation in general. The above two tests are performed on all of the seven data subsets. The results of the LikeCAPM test are reported in Tables 4.94.12. No intercepts of the regressions are significantly different from zero. The statistical insignificance of the intercept term only indicates that the regression line does go through the origin. This suggests that the margin requirement does not affect the index futures pricing and that no abnormal return is earned corresponding to its systematical risk. The index futures is priced efficiently in general, even in the early stage of the index futures market development. Examining the trend of the slope coefficient over the different data subsets, we find some evidence that
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63 earlier observations have greater slope values. This is consistent with the belief that less dividend risk is priced in the market as time approaches the maturity date of the index futures contract. This result is consistent with Test I suggesting that the effect of dividend risk decrease over time to the maturity. The LikeCAPM test actually is derived from Test I, after we have defined the term "return" of index futures contract. It is therefore not surprising that the results are not changed. In Table 4.12, the slope is not significantly different from one. This would support our conclusion that the perfect market pricing model holds in LAST30. The results of the hedge relationship are reported in Table 4.134.16. Again, the intercepts are not statistically significant over most contracts. However, the main idea of this test is to examine the behavior of hedging overnight. Thus, our main concern is the slope coefficient. The results, however, do not lend support to the claim of the null hypothesis that the slopes, in general, are equal to one. It means that the overnight hedge ratio is less than one since the slope is greater than one. One day perfect hedge of one index futures to one indexequivalent portfolio is impossible. Dividend risk, along with holding period and interest rate change, appear to be important factors for short term hedgers to consider. In sum, the extended analysis suggests that the basic findings is robust for many different definitions of the variables used in the analysis. The results are basically suportive of the dividend risk hypothesis since they adequately explain the futures discount, in the sense that the dividend risk premium is priced in the market and is negative for the investor who shorts the index futures.
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64 However, there is one puzzling aspect of the results in most tables. The slope coefficients of the FCESP are significantly different from zero. This variation in the reported value may be caused by four reasons. In the first place, it might be due to a term structure problem; thus a multiperiod pricing model should be developed to guide the further empirical study. In the second place, it might be due to a measurement error of the interest rate series for its unmatched maturity date to futures price series. In the third place, the approximation technique is not totally accurate regarding the collected dividend payout series. Finally, there might exists the liquidity risk associated with the maturity dynamics of the earier days' trading. Future research should be done in this direction. Summary This chapter presents an empirical test of whether a dividend risk is priced in the index futures market and whethsr the pricing model explains the structure under study. We empirically investigate the closed forms developed in Chapter III. The results support the dividend risk argument. Futures discounts could be explained by a factor which increases with seasonality as the time to maturity increases. Dividend uncertainty is resolved by dividend announcements and as the contract approaches maturity. The unit dividend risk premium is estimated using the pricing model developed in this dissertation. Due to the greater transaction value of S&P 500 index futures, S&P 500' s unit dividend risk premium is priced more in the market than NYSE index futures'. Of course, one has to interpret these comparison with considerable caution. In sum, the results reported in this chapter may be
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65 of use in analyzing historical price change, evaluatin market efficiency, and developing strategies to identify and exploit arbitrage opportunities. However, the use of models of equilibrium pricing under uncertainty in empirical study of futures markets is very limited. As yet, no precise pricing model of index futures contracts has been found which offers completely satisfactory explanatory power, though some progress has been made here.
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66 Notes Cox, Ingersoll, and Ross (1978) examine the theoretical differences between forward and futures prices in a variety of contests. However, Cornell and Reinganum (1981), and Elton, Gruber, and Rentzler (1982) indicate that the difference is economically insignificant. The reason to include FCESP in the right hand side is to make the test completely meet the econometric considerations. See Pindyck and Rubinfeld (1981), p. 82. However, if we include the intercept in these regression a new multicollinerity arises.
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67 TABLE 4.1 Frequency of Days Between the Announcement Date and the Exdate of Dividend Events in NYSE and S & P 500 19821983* Days Accumulated Percentage of NY^F Accumulated Percentage ui oar jvjvj 5 0.06 0.06 i n VJ Â• JH VJ Â• JO U Â• Jl U.JJ n fin VJ . DU n i VJ . DO VJ . DO n 7 s VJ Â• / J n 7 35 0.81 0.81 40 0.85 0.86 45 0.90 0.90 50 0.93 0.93 55 0.95 0.95 60 0.97 0.97 98 1.00 1.00 Note: * Dividend Code of CRSP tape, 1212, 1232, 1239, 3225, 3285, 3723, 2763, 3823, 3825, 3863, 4533, 4822, 5523, 5533, 6521, are excluded. In addition, 20 over the counter companies of S & P 500 are not available. In sum, NYSE has 11992 dividend events and S & P 500 has 4550 dividend events
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68 TABLE 4.2 198183 Monthly Dividend Yields for the NYSE Composite Index// Month 1983 1982 1981 January3.49% 2.16% 2.24% February 13.68% 9.84% 8.39% March 6.69% 4.92% 3.83% April 4.62% 3.60% 2.31% May 14.57% 10.92% 8.15% June 6.38% 4.08% 3.87% July 3.46% 2.76% 2.24% August 13.99% 11.76% 8.90% September 7.70% 4.08% 3.72% October 4.65% 3.36% 4.15% November 14.27% 9.24% 8.68% December 6.50% 3.00% 3.63% Note: // The dividend yields are measured by the difference between the monthly valueweighted return, including dividends, and the valueweighted return, excluding dividends, for the New York Stock Exchange Stocks. The yields have been converted to annnual estimates by multiplying the monthly estimates by 12. The data is from CRSP monthly tape.
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69 CO fa 0) U M fa ca ca CT CO + 4J SI w 3 I a) i! 1 a o CN 00 sO o og CO ^ Â— V 00 Ik C iH 41 4J CJ ca ca M Sl 3 41 41 a M o S cj v Â— * * * * * * * * * * * * * * * * Nf43QO'H^cÂ»ij , iHNcncn Â•j'cooi'iincM^owosoino *H O o o o o o o o I I I I I I co i * * * * * * * *Â« ****** cn co oo c * * O O O Â•Â— ' O Â•Â— i iÂ— i vo O Â•Â— I OO I I I I I I I I ** * ** *** *Â« * ** *** oo(TimcocncÂ»icnocT\coinHr~ Oooo^ Â— tvomr^cNcNr^ooo o o Â•Â— ' O Â— i iÂ— 4 I III o o o o o I I I o o I * * * * * * * O\HO\00COOMflC0cncÂ»1C0NvO MsonorninjsNcoo OOPOOCNÂ— lOOOOOO I I II I I I I * * * * * * * N eo n O h ui i(uinr* CO OjflClClH^ps^oOO cc ^HOCNnOOOOr^O II II I Â« Â« * * co co h n co (J in t H on cn O O CN O r~ a* >Â—h ih On O O rÂ• Â• Â• rÂ— O CN O oo o i io o\ o\ O lO O o rwo cm ^ ON O O O CN O vO S CI 4 ON 00 O CN pi on o en ON ON O sT O i o vO Uliv CO oo oo o h O O o rn so m cn r. O O ON ON O (SI vO H O o 00 CM CN 3 c* p Q. T3 u ca u 4J c o o CN U s s ca 00 41 > ca ca CJ CD CJ H H e E M 3? Â•* 0 41 m OJ OJ 0) .c 41 4J CO u 4J 41 o ca ca A 41 41 41 CO c s a CJ ca ca a f4 CJ CJ ca ca 11 rH 41 144 ijÂ— i CO gg *H Â•H a C SO CU 00 50 O c Â•H Â•H CN 41 CO CO 00 c Â•X X Â•x o
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70 w CO >< 25 PM CO U 0) h co cr CO co. S I52 CO 8  0> Vj II a. a .o\M'j n Â• icno n n i Â— o\ o Â• Â• Â• Â• Â• ON Â•Â•Â•Â•Â•Â•Â• iOOÂ— coir>Â»3"r^cMcoo o I en o o o CM^OOOOOO I I I I I cd c * * * * He Â« Â•H O N 0> CO CO O vO b O CM tÂ— t O O O O < < cm on rr>irÂ» no Â• Â• Â• i Â— / O CM o 00 CO 3ON o o O Â• Â• iÂ—i o CM o ON CO a Â• Â• Â• iÂ— 1 o Â—4 o 4> cu CJ CJ B B (0 CO CJ a Â— 1 o vO Â•H *H ON a o CM VIH Â• Â• Â• Â•H Â•H c CM o B B 00 00 Â•H Â•H CO CO of of H CM CO CM CO 00 o O > > Â• Â• Â• cu cu 4J O IÂ—* o iH rÂ— i Â•H Â•X CO m CU c a) CU 11 H CM CN uo SO 4J Â— ' H NO CN m ai Â• Â• Â• u a o Â— H o CO CO Â•H 41 4J 41 iÂ—i B c 3 CO CO i U CJ Â•H H 41 m Â— 1 O 41 CJ GO c iÂ—4 Â•H Ti cu Â• Â• C B 41 O . Â— i o co CO Â•H T1 CJ CO CO Â« < * M 3 41 OS Q a T3 o 25
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71 TABLE 4.5 Test IV: FD = a S + B FCESP Second Step Least Square Data Set: WHOLE 2 month 1 month during FCESP R 2 DW P df Contract before before S & P 500: 8206 2.9925 0.0801 0.4134 0.4134 0.29 1.69 0.10 28 8209 1.4823** 0.7420** 0.2742 0.0216** 0.60 1.47 0.25 76 8212 1.2417** 0.3397 0.0699 0.0113** 0.72 1.98 0.00 124 8303 0.8653** 0.0594 0.1075 0.0072** 0.75 2.19 0.10 152 8306 0.6663** 0.8150** 0.0479 0.0016 0.91 1.87 0.03 130 8309 0.4722** 0.7632** 0.0782 0.0039 0.89 2.28 0.03 122 8312 0.3879** 0.6263** 0.0942 0.0087** 0.91 2.13 0.09 165 NSYE: 8312 0.2124** 0.2077** 0.0540 0.0063** 0.88 2.30 0.11 180 8402 0.2205** 0.0738 0.0511 0.0182** 0.81 2.49 0.16 180 Note: * Significant at the 5% level of significance ** Significant at the 1% level of significance
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72 TABLE 4.6 Test IV: FD = o S + 6 FCESP Second Step Least Square Data Set: LE30 2 month 1 month during FCESP R 2 DW p df Contract before before S & P 500: 8209 1.3500** 0.6906* 0.3333 0.0231 0.60 1.4140 0.2611 59 8212 0.3498** 0.4474* 0.1540 0.0082 0.72 0.9297 0.0345 108 8303 0.9157* 0.0395 0.0755 0.0056 0.78 2.1905 0.0963 135 8306 0.6659** 0.8098** 0.0433 0.0015 0.92 0.8701 0.0372 119 8309 0.4696* 0.7531** 0.0758 0.0039 0.89 2.2883 0.0291 112 8312 0.3569** 0.6730** 0.1370 0.0099 0.92 2.1349 0.0927 149 NYSE: 8312 0.2293** 0.2387** 0.0674 0.0065* 0.90 2.0932 0.0541 171 8403 0.2389** 0.1097 0.0687 0.0166** 0.87 2.2406 0.0929 180 Note: * Significant at the 5% level of significance ** Significant at the 1% level of significance
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o en QJ U CO 3 cr CO cn cO cu iÂ— I CU oo co u cn T3 C O CJ CU CO 4) CO 3 CT CO cn to , (J H 41 S c o > en Z 00 at cu > > > Â•H Â•H CU CU cn cn rH H II CU U CO cn O CJ cu CU 3 C cn CO CO CO cu CU O CJ X J3 Â•H Â•H 41 4> Â«H
PAGE 84
TABLE 4.9 Test II: r p = a + B F (r g r f ) Data set: WHOLE Market Contract a B v R 2 DW p df S & P 500 NYSE 8209 0.0002 1.21** 0.78 2.28 0.16 78 8212 0.0004 1.20** 0.78 2.58 0.29 126 8303 0.0003 1.20** 0.75 2.83 0.42 154 8306 0.0004 1.02 0.70 2.57 0.27 132 8309 0.0005 1.20** 0.74 2.79 0.33 124 8312 0.0000 1.08 0.79 2.70 0.37 167 8308 0.0008 1.29** 0.80 2.52 0.24 41 8306 0.0002 1.57 0.20 2.57 0.28 81 8309 0.0001 1.02 0.54 3.08 3.08 127 8312 0.0004 1.16** 0.80 2.51 0.23 182 8403 0.0004 1.16** 0.77 2.67 0.23 182 Note: * Significant at the 5% level of significance ** Significant at the 1% level of significance
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TABLE 4.10 Test II: ? F = a + 3 F (r g r f ) Data set: LE30 Market Contract a 8 R 2 DW p df S & P 500 8209 0.0005 1.36** 0.72 2.19 0.13 61 8212 0.0006 1.23** 0.78 2.56 0.28 110 8303 0.0002 1.21** 0.75 2.87 0.44 137 8306 0.0004 1.01** 0.70 2.57 0.27 121 8309 0.0007 1.23** 0.74 2.83 0.35 114 8312 0.0000 1.08** 0.78 2.70 0.37 151 NYSE 8303 0.0008 1.33** 8306 0.0002 1.59 8309 0.0001 1.02 8312 0.0004 1.16** 8403 0.0004 1.16** 0.83 2.79 0.41 28 0.20 2.56 0.28 77 0.54 3.05 0.53 126 0.77 2.45 0.23 173 0.77 2.57 0.23 182 Note: * Significance at the 5% level of significance ** Significance at the 1% level of significance
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77 TABLE 4.11 Test II : r F = a + 8 F (r s ' r f } Contract a R 2 DW P df Data Set S & P 500 8209 0.0003 1 23* 0 81 2.41 0 25 0.0014 1.54** 0.74 2.46 0.24 45 L120LE30 0.0002 1.16 0.69 1.74 0.05 30 LE90 0.0009 0.88 0.84 2.37 0.30 13 LE120 8212 0 0004 1 18 X Â• X U 0 77 Â• 7U _n A 9 0.0015 1.24* 0.78 2.87 0.44 45 L120LE30 0.0005 1.23 0.79 2.28 0.14 78 LE90 0.0000 1.26** 0.78 2.18 0.11 62 LE120 8303 0 0010 1 27** 0 84 ? AP. i. HO 0 9 7 AS T AQTQO 0.0005 1.24** 0.84 2.75 0.40 43 L120LE30 0.0003 1.20** 0.73 2.89 0.44 106 LE90 0.0001 1.23** 0.73 2.85 0.44 91 LE120 8306 0 0009 X Â• & J fl P,A A1 LAO 1 7 U 0.0012 1.23* 0.82 2.14 0.07 45 L120LE30 0.0002 0.98 0.68 2.58 0.27 89 LE90 0.0001 0.97 0.67 2.63 0.29 73 LE120 8309 0.0002 0.96 0.88 2.94 0.47 38 LAST90 0.0002 1.03 0.84 2.64 0.33 44 L120LE30 0.0013 1.38** 0.74 2.93 0.36 83 LE90 0.0015 1.40** 0.73 3.05 0.38 67 LE120 8312 0.0004 0.94 0.86 2.49 0.25 45 LAST90 0.0002 0.93 0.88 2.52 0.27 45 L120LE30 0.0001 1.12* 0.77 2.73 0.39 119 LE90 0.0002 1 . 15 0.76 1.17 0.00 103 LE120 NYSE 8312 0.0004 1.03 0.85 2.19 0.07 37 LAST90 0.0001 0.99 0.85 2.11 0.09 44 L120LE30 0.0005 1.19** 0.79 2.49 0.25 143 LE90 0.0007 1.21 0.80 2.49 0.25 127 LE120 Note: * Significant at the 5% level of significance ** Significant at the 1% level of significance
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78 TABLE 4.12 Test II: r p = a + 8 F (r g r f ) Data Set: LAST30 Market Contract a $ R 2 DW p df S & P 500: 8206 0.0016 1.29 0.73 2.51 0.26 15 8209 0.0018 1.24 0.84 2.44 0.30 14 8212 0.0002 1.08 0.82 2.51 0.30 13 8303 0.0003 1.05 0.81 2.10 0.08 14 8306 0.0000 1.08 0.85 2.42 0.27 13 8309 0.0004 0.92 0.86 2.48 0.28 U 8212 0.0010 1.05 0.79 2.50 0.30 13 NYSE 8312 0.0009 1.01 0.75 1.53 0.05 6 Note: * Significant at the 5% level of significance ** Significant at the 1% level of significance
PAGE 88
79 TABLE 4.13 Test III: Rp = ct + e R (r g r f ) Data Set: WHOLE Market Contract o & R 2 DW p df S & P 500: 8209 0.0002 1.22** 0.78 2.29 0.16 78 8212 0.0004 1.20** 0.78 2.59 0.30 126 8303 0.0003 1.21** 0.75 2.84 0.42 154 8306 0.0004 1.02 0.70 2.57 0.27 132 8309 0.0006 1.22 0.74 2.78 0.33 124 8312 0.0000 1.10* 0.78 2.69 0.37 167 NYSE 8303 0.0008 1.30** 0.80 2.51 0.24 41 8306 0.0002 1.60 0.20 2.64 0.32 81 8309 0.0001 1.04 0.53 3.09 0.54 127 8312 0.0004 1.18** 0.80 2.50 0.23 182 8403 0.0004 1.18** 0.77 2.67 0.24 182 Note: * Significance at the 5% level of significance ** Significance at the 1% level of significance
PAGE 89
TABLE 4.14 Test III: Rp = a + 6 R (r" g r f Data Set: LE30 Market Contract a 8Â„ R 2 DW p df S & P 500 NYSE 8209 0.0005 1.36** 0.72 2.20 0.14 61 8212 0.0006 1.24** 0.78 2.57 0.29 110 8303 0.0003 1.22** 0.75 2.88 0.44 137 8306 0.0004 1.02** 0.70 2.57 0.27 121 8309 0.0007 1.25** 0.74 2.82 0.35 114 8312 0.0000 1.10* 0.78 2.69 0.37 151 8303 0.0009 1.34** 0.83 2.79 0.41 28 8306 0.0002 1.62 0.20 2.64 0.32 77 8309 0.0001 1.04 0.54 3.05 0.52 126 8312 0.0004 1.18** 0.80 2.45 0.23 173 8403 .00004 1.19** 0.77 2.58 0.23 182 Note: * Significance at the 5% level of significance ** Significance at the 1% level of significance
PAGE 90
81 TABLE 4.15 lest ill: a + e H (r s _ r f ) Contract a n R 2 DW P df Data Set O a z JUU a 9Piq Â— u Â• UUU J 1 9 9* pi a i u . 0 1 9 /. 1 Z . H 1 a 9 U . Z j /, R H J T A CTO A LAbiyu 0.0014 1.54** 0.74 2.47 0.24 45 L120LE30 0.0002 1.17 0.70 1.76 0.04 30 LE90 Â— u . UU 1U n an u Â• yu pi a ^ U . Oj 9 T7 Z.J/ PI 9PI Â— U . JU 1 9 1 J T T 1 9 A 89 19 Dili _a nnn/i Â— U Â• UUUh 1 is 1.10 n 77 u . / / 9 z . yy PI CPI Â— U . jU 4j T A CP A A 0.0016 1.24* 0.78 2.88 0.44 45 L120LE30 0.0005 1.24** 0.79 2.29 0.15 78 LE90 Â— u Â• uuuu 1 97** 1 Â• Z 1 pi 78 u . / o 9 18 Z . lo a ii Â— U . 1 1 C 9 OZ T P1 OA LElzU OJUj Â— u . UU1U 1 97** 1 . Z 1 U . O't 9 /. Q Z . 4o PI O 7 U.J./ 4j LAST9U 0.0006 1.25** 0.83 2.76 0.40 43 L120LE30 0.0001 1.20** 0.73 2.90 0.45 106 LE90 Â— u . UUU 1 1 71** n 7 t u . / J 9 8 ^ Z . OJ U . 44 A 1 LElzO O JUO _n pipipiq Â— u Â• uuuy 1 9 9* 1 . ZJ" U . OJ Z Â• J J A OA Â— u . zu 4 1 LAST90 0.0012 1.23* 0.82 2.14 0.07 45 L120LE30 0.0003 1.00 0.68 2.58 0.27 89 LE90 U Â• UUUZ n qq U . J J fl Â£7 U . 0 / Z . 0 J A 90 Â—v . zy 7 O 7 J LE1Z0 n nnri9 u Â• uuuz PI Q7 U . 7 / pi 88 U . OO 9 O 9 z . y j A A 7 U .4/ O Q JO LAST90 0.0002 1.02 0.84 2.64 0.32 44 L120LE30 Â— u . uu it i /, pi** f\ 11. U Â• /H z . yj A 1 Â£ u . 36 83 LE90 0.0016 1.42 0.73 3.05 0.39 67 LE120 891 9 Â— U . UUUf u . yt a 07 U . o / o /.n 0 . 25 LAST90 u Â• uuuz pi 0 1 ; a qq u . oo O CO 0.27 45 L120LE30 _p> nnn? Â— U . UUU j 1.14* a 2. 73 0.38 119 LE90 0.0002 1.17* 0.76 2.27 0.41 103 LE120 NYSE 8312 0.0004 1.04 0.85 2.19 0.07 37 LAST90 0.0001 1.00 0.85 2.11 0.10 44 L120LE30 0.0005 1.21** 0.79 2.49 0.25 143 LE90 0.0007 1.23 0.79 2.49 0.25 127 LE120 Note: * Significance at the 5% level of significance ** Significance at the 1% level of significance
PAGE 91
82 TABLE A. 16 Test III: Rp = a + 3 R (r g r f ) Data Set: LAST30 Market Contract a 8 Rf_ DW p df S & P 500: 8206 0.0015 1.28 0.73 2.51 0.26 15 8209 0.0018 1.24 0.85 2.43 0.30 14 8212 0.0002 1.07 0.82 2.52 0.30 13 8303 0.0003 1.05 0.81 2.08 0.07 14 8306 0.0000 1.08 0.85 2.42 0.27 13 8309 0.0004 0.93 0.86 2.48 0.28 14 8312 0.0010 1.05 0.79 2.50 0.30 13 NYSE 8312 0.0009 1.01 0.75 1.53 0.04 6 Note: * Significant at the 5% level of significance ** Significant at the 1% level of significance
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CHAPTER V CONCLUSION The adjustment for dividend uncertainty, either constant or stochastic, appears to reduce index futures prices. The extent of the difference between the constant and stochatic cases depends on the size of the constant dividend yield, the size of the variance of the dividend process, and the extent correlation between the return on the indexequivalent portfolio and the dividend yield. As time to maturity increases, the difference is magnified. Using a market equilibrium approach, a model which includes a dividend risk premium was developed. This dividend risk premium can not be hedged away, if we buy a classic riskarbritrage portfolio which goes long one unit of the indexequivalent portfolio and short one unit of the index futures simultaneously. In addition, the risk level of the futures contract relative to the indexequivalent portfolio has been examined. In Chapter III, three empirically testable equations were derived. The first specifies the price relationship between the spot and the index futures which includes the effect of a random dividend. The second is a LikeCAPM model which shows the corresponding systematic risk level of index futures contracts. Finally, a hedge relationship was examined. Based on these three equations, five tests were performed on seven data sets to examine the dividend risk behavior empirically. In Chapter IV, the dividend risk argument is examined empirically. The empirical results show that the dividend risk premium with 83
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84 seasonality the closer the contract is to maturity. The total risk premium two months before the dividendpeakmonth is large; one month before the dividendpeakmonth it is somewhat smaller, but still significant; and during the dividendpeakmonth it is not significantly different from zero. As to the magnitutite, S&P 500 has larger market value with a larger associated dividend risk premium in general. In addition, the nearest contract appears to be priced according to the perfect market pricing model, which implies that the dividend risk is not important for the last thirty tradingdays at least. Since we have examined the daily price of index futures in New York Futures Exchange and Chicago Board Futures Exchange over a one and a half year time period, inferences drawn from this research must be tentative. However, we have analyzed in excess of four contracts in each market and have found similarities in the results for index futures on these two markets. Therefore, we consider this research to be one of the most extensive empirical examinations of index futures price to be reported in the literature to date.
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APPENDIX A DATA BASES
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86 TABLE A.l Data Base of Index Futures Data Market Index NYSE*** Contract 8206* 8209 8212 8203 8206 8306 8312 8403 StartTrading Date 820506** 820506 820506 820506 820630 820722 820723 820930 EndTrading Date 820629 820929 821230 830330 830629 830929 831229 840329 Number of Observations 38 102 166 229 253 302 364 379 S&P 500**** 8206 8209 8212 8303 8306 8309 8312 8403 820421 820421 820421 820421 820625 820920 821223 830325 820617 820916 821216 830317 830616 830915 831215 840308 41 104 168 213 202 196 232 200 Note: * 8206 stands for YYMM, meaning index futures contract maturity on June 1982. Others are the same format ** 820506 stands for YYMMDD, meaning the date of May 6, 1982. Others are the same format *** The data source of NYSE is directly provided by statistics department NYSE. Data are in 4:15 P.M. New York time. **** xhe data source of S&P 500 is from a ticker by ticker tape, provided by Chicago Board. The close price at 4:15 P.M. is sorted as the data base.
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87 TABLE A. 2 Data other than Index Futures **** Market Data type StartDate EndDate NYSE Spot 820104 840329 Dividend* 830101 831231 S & P 500 Spot 820421 840308 Dividend** 820104 831231 TreasuryBill*** 820421 840308 Dividend payout of market index is provided by statistics department, NYSE. Dividend payout of S & P 500 is collected from CRSP daily tape, using December 1982 name list in C0MPUSTAT tape. This data series is a approximation value, 6.6% number of companies series is a approximation value, 6.6% number of companies missing or 3.6% dividend payout of total dividends missing, based on the value of December 31, 1982. C0MPUSTAT tape. TreasuryBill Futures data is sorted from a ticker by ticker tape, provided by Chicago Board. Market value of S & P 500 and NYSE are computed from CRSP tape. The market value of S & P 500 underestimates 3.6% value of total value, based on the value of December 31, 1982 COMPUSTAT tape. The names of missing companies are available upon request. Note: * ** *** ****
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APPENDIX B RESULT OF FIRST STEP LEAST SQUARE
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89 C CO s a* CO U + CD Â• . * SI CN O vO (n lA CO CO X Â•X .Â— i O O "H I I X X X m cn in cn o o I I o o I ooiooo\iOvOÂ«tr^rÂ» o o I o o I I He * X X X X * * * * * * on co oo in on o Â•Â— ' o\ O oo r. cn cn o I Â— ' o I o I ^m 1 O I I I * o a\ m sr Â— i O O ~m O Â— i h CO I I I I I H I * * CO 00 CO NO no vO K * * * \0 rÂ» Â•Â— I Â•Â— I on Â•Â— ' H O O Â— i o I I I X X X cti in Â— i oo in co o o I * * o a> n o i OO00OOC0HO I I I Â•X X 00 00 CO a ccj b CO I o o I X X oo m On sr O ON cm sr o sr co on see r. Â—) o Â•r ci h zee I I S C B Â— < CN co 25 2 25 m <Â£ n co o\ 35 35 55 X S * X * cn 00 o ON CN ~m CO sr o ON o in rÂ— t Â• Â• Â—* i i <Â—> 1 /Â— N w J NÂ— ' 1 1 o o On CN CN CN CO \Q 00 m fH O ON CN CO * * Â— n Â« Â— ' 1 1 CN o o CO CN CN in ON o On CN rÂ— n 1 1 I /Â— N Â•Â— ' w Jt T> 1 o cn CO o iH ON CO Â— 1 ,Â— ^ crt W o *^ i d 1 1 ^> 00 (it nJ o m o 00 u 0 cn CO ON o ON o Â— c a VN at nl 0 Q Q 1 1 c i w i 1 1 1 *tH 1 IW lw Â•W Â•H is K s B Ow CN LJ 1 tH *H _ Q ON CN cn IU * * * Â•^ u Â• Â— ' o O 1 1 1 1 I o c tÂ— 1 4C CD cu CÂ— J > > art ON ' w Â» CvJ CO CU CD CO CO ON Â• Â• Â• Â• H O o rÂ— ( o Â— i / Â— N, On i i m cu CD A ^= CO 00 CO o cn 4J o ON m cO Â• Â• Â• 4J u CO CO o O o o CO co b C B 4J u B B CO CO ON CN co a U o NO NO CN Â•H *H CO CO cO Â• Â• Â• Â• <4l o o O o Â•H Â•H (3 c a B C 00 00 Â•H H CO 02 * X X Cm CO cu O . Â— 1 CN a iÂ— i rH > Â— i u N Ul o 25 X 25 b Pi G
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90 oo >> S5 fa co a) fa M CJ Â« fa 3 CO. oo + w x 3 all; II e. 0) 4J .. co M U H 4> fa CO CD O O a, a cn W fa 1) 00 cO C2 CU C u CU U cfl 2 cn o Â«* oc m CO cn m CO o cn 00 vC O cn oo cn o en CO CN CN CO O CN 00 vC o CN CO cj *h at m 41 u C Q o X * * * O O CT\  > St ON I Â— m CN 00 vO O o 0) CD cn cn i Â— l o ON CO CN , Â— 1 , Â— 1 Â• Â• Â• c O cn rÂ— i O o vO m o o o 8ve 1 1 I  Â— Â— Â» 1 U 1 X CU CU Â•X & sr CO Â—i rl . Â— i u N us o X s X X X X X X fa fa G
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91 TABLE B.3 Test IV: FD = a S + 6 FCESP Data set: WHOLE Variable 2 month 1 month during Contract before before S&P 500: 8206 2.103 0.476 0.312 8209 3.052** 1.399** 0.399 8212 2.439** 0.579* 0.161 8303 1.894** 0.124 0.133 8306 0.875** 1.208** 0.057 8309 0.681** 1.492** 0.272 8312 0.860** 1.597** 0.444 FCESP R 2 DW p df 0.012 0.69 0.686 0.62 28 0.022** 0.78 0.98 0.50 76 0.011** 0.86 0.97 0.51 124 0.008** 0.91 0.89 0.55 152 0.001 0.95 1.25 0.33 130 0.003** 0.96 1.17 0.48 122 0.008** 0.97 0.75 0.60 165 NYSE: 8312 0.605** 0.662** 0.279** 0.006*** 0.96 0.67 0.70 180 8403 0.606** 0.724** 0.388** 0.012*** 0.98 0.42 0.81 180 Note: * Significant at the 5% level of significance ** Significant at the 1% level of significance
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92 TABLE B.4 Test IV: FD = a S + 6 FCESP Data set: LE30 Variable 2 month 1 month during FCESP R 2 DW P df Contract before before S&P 500: 8206 9.558 0.342 3.242 0.060 0.88 1.60 0.08 10 8209 3.058** 1.423** 0.356 0.021** 0.79 0.88 0.54 59 8212 2.587** 0.704* 0.382 0.008** 0.87 1.01 0.48 108 8303 1.937** 0.084 0.079 0.006** 0.91 0.94 0.53 135 8306 0.875** 1.190** 0.066 0.001 0.95 1.24 0.33 119 8309 0.681** 1.483** 0.278 1.003** 0.96 1.16 0.49 112 8312 0.793** 1.546** 0.522** 0.009** 0.98 0.76 0.60 149 NYSE: 8312 0.606** 0.664** 0.277** 0.006** 0.98 0.82 0.66 171 8403 0.606** 0.724** 0.388** 0.012** 0.98 0.44 0.77 180 Note: * Significant at the 5% level of significance ** Significant at the 1% level of significance
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TABLE B.5 Test IV: FD = a S + 6 FCESP 2 month 1 month during FCESP R DW p df Data Set before before S&P 500: 8212 1.780** 1 1.528** 1 3.370** 1 3.345 1 .114** 0.052 .519** 0.204 .654** 0.041 .761 0.068 0.011** 0.74 0.011** 0.86 0.017** 0.92 0.014 0.92 2.002 0.063 2.088 0.049 1.082 0.454 1.011 0.486 43 LAST90 43 L120LE3 76 LE90 60 LE120 8303 0.701 1.060 0.629 0.002 0.69 1.912 0.023 43 LAST90 1.327** 0.811* 0.878* 0.015 0.88 1.614 0.184 41 L120LE30 2.085** 0.302 0.295 0.001 0.93 0.993 0.507 104 LE90 2.271** 0.362 0.631 0.003 0.93 1.024 0.487 89 LE120 8306 0.080 1959* 0.052 0.004 0.344 1994** 1.322* 0.010 1.472** 0.848** 0.000 0.007** 1.234** 0673* 0.448 0.008** 0.91 1.460 0.266 39 LAST90 0.94 1.306 0.334 43 L120LE30 0.96 1.476 0.184 87 LE90 0.97 1.655 0.079 71 LE120 8309 0.315 0.340 12.176 0.076 1.163 0.727 0.682 0.008** 0.231 2.000** 0.205 0.006** 0.330 2.054 0.470 0.008 0.86 1.534 0.208 36 LAST90 0.89 1.064 0.457 42 L120LE30 0.97 1.559 0.225 81 LE90 0.97 1.698 0.158 65 LE120 8312 0.673* 0.851** 0.033 0.004 0.77 1.180 0.403 43 LAST90 0.910** 0.006** 1.178 0.015** 0.93 1.083 0.439 43 L120LE30 0.565** 1.730** 0.281 0.003 0.98 0.873 0.535 117 LE90 0.678** _1.701** 0.099 0.002 0.98 0.891 0.534 101 LE120 NYSE: 8312 15.319** 0.103 0.030 0.162** 0.692** 0.237 10.926** 0.109** 0.355** 0.743** 0.326** 0.000 0.451** 0.750** 0.102 0.003 0.85 1.349 1.349 35 LAST90 0.93 0.822 0.822 42 L120LE30 0.98 0.773 0.773 141 LE90 0.98 0.852 0.852 125 LE120 Note: * Significant at the 5% level of significance ** Significant at the 1% level of significance
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LIST OF REFERENCES Anderson, R.A. , and J. Danthine, "Hedging Diversity in Futures Markets: Backwardation and the Coordination of Plans," Research Paper No. 71A, Columbia School of Business, New York (January 1978). Athans, M. , "The Importance of Kalman Filtering Methods for Economic Systems," Annuals of Economic and Social Measurement 3 (January 1974), 4964. Baron, D.P., "Flexible Exchange Rates Forward Markets, and the Level of Trade," American Economic Review 66 (June 1976), 253266. Black, F., "Equilibrium in the Creation of Investment Goods under Uncertainty," In Studies in the Theory of Capital Markets , edited by Michael C. Jensen. Praeger, New York (1972). Black, F., and M. Scholes, "From Theory to a New Financial Product," Journal of Finance 29 (May 1974), 399412 Blume, M. , "On the Assessment of Risk," Journal of Finance 26 (March 1976), 110 3rennan, M.J., "The Supply of Storage," American Economics Review 48 (March 1958). Bodie, Z., and Victor I. Rosansky, "Risk and Return in Commodity Futures," Financial Analysis Journal 36 (MayJune 1980), 2739. Chow, G., "Tests of Equality between Subsets of Coefficients in Two Linear Regression," Econometrica 28 (July 1960), 591605. Constantinidies, G.M. , and M.S. Scholes, "Optimal Liquidation of Assets in the Presence of Personal Taxes: Implications for Asset Pricing," Journal of Finace 35 (May 1980), 439452. Cootner, P.H. , "Return to Speculations: Telser versus Keynes," Journal of Political Economy 48 (August 1960), 396404. Cootner, P.H., Speculation and Hedging Proceedings of a Symposium on Price Effects of Speculation in Organized Commodity Markets, Food Research Institute Studies, Supplement to Vol. VII, 1967. Cornell, B., "Testing the Tax Timing Option Theory: A New Approach," Working Paper #584, Graduate School of Management, University of California, Los Angles (March 1984). Cornell, B., and K.R. French, "Taxes and the Pricing of Stock Index Futures," Journal of Finance 38 (June 1983), 663674. 94
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95 Cornell, B., and M.R. Reinganum, "Forward and Futures Prices: Evidence From the Foreign Exchange Markets," Journal of Finance 36 (1981), 10351045. Cox, J., J. Ingersoll, and S.A. Ross, "A Theory of the Term Structure of Interest Rate." Research Paper No. 468, Graduate School of Business, Stanford University (August 1978). Cox, J., J. Ingersoll, and S.A. Ross, "The Relation between Forward Prices and Futures Prices," Journal of Financial Economics 9 (December 1981), 321346. Dumas, B., "The Theory of the Trading Firm Revisited," Journal of Fiance 33 (June 1978), 10191033. Dusak, K., "Futures Trading and Investor Returns: An Investigation of Commodity Market Risk Premiums," Journal of Political Economy 81 (December 1973), 13871406. Dybvig, P.H., "An Explicit Bound on Individual Assets Deviations form APT Pricing in a Finite Economy," Working Paper, School of Management Yale University, New Haven, 1983. Ederington, L.H., "The Hedging Performance of the New Futures Markets," Journal of Finance 34 (March 1979), 158170. Elton, E. , M. Gruber, and J. Rentzler, "Intraday Tests of the Efficiency of the Treasure Bill Futures Market." unpublished working paper, New York University, New York 1982. Ethier, W. , "International Trade and Forward Exchange Market," American Economic Review 63 (June 1973), 6883. Figlewski, S., "Why Are Prices for Stock Index Futures So Low?" Working Paper No. 138, Graduate School of Business, University of California, Berkeley (1983a). Figlewski, S., "Hedging with Stock Index Futures: Theory and Application in a New Market," Working Paper No. 139, University of California, (1983b). Figlewski, S., "Hedging Performance and Basis Risk in Stock Index Futures," Journal of Finance 39 (July 1984), 657669. Fischer, S., "Call Option Pricing When the Exercise Price is Uncertain, and the Valuation of Index Bond," Journal of Finance 33 (March 1978), 169186. Fisher, L. , "Some New StockMarket Indexes," Journal of Business 196 (January 1966), 191225. French, K.R., "A Comparison of Futures and Forward Price," Journal of Financial Economics 12 (November 1983), 311342.
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96 Garbade, K., and Z. Lieber, "On the Independence of Transactions on the New York Stock Exchange," Journal of Banking and Finance 1 (September 1977), 151172. Garbade, K. , and W. Silber, "Dominant and Satellite Markets: A Study of DuallyTraded Securities," Review of Economics and Statistics 61 (August 1979), 455460. Garbade, K. , and W. Silber, "Price Movements and Price Discovery in Futures and Cash Markets," Review of Economics and Statistics 65 (May 1983), 289297. Grauer, F., and R. Litzenberger , "The Pricing of Commodity Futures Contracts, Nominal Bonds and Other Risky Assets under Commodity Price Uncertainty," Journal of Finance 34 (March 1979), 6983. Henry, A.L., D.L. Tuttle, and W.E. Young, "Market Indexes and their Implications for Portfolio Management," Financial Analysis Journal (September October 1971), 7585. Hirsheifer, J., "Speculation and Equilibrium: Information, Risk and Market," Quarterly Journal of Economics 89 (November 1975), 520542. Jensen, M.C., Financial Analysis Journal 23 (November December 1967), 7785. Johnson, L.L., "The Theory of Hedging and Speculation in Commodity Futures," Review of Economic Studies 28 (June 1960), 139151. Johnston, J. Econometric Methods . McGrawHill Book Inc., New York (1984). Latane, H.A., and W.E. Young, "Test of Portfolio Building Rules," Journal of Finance 24 (September 1969), 595612. Lintner, J., "The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets," Review of Economics and Statistics 47 (February 1965), 1337. Levy, H. , "The CAPM and Beta in an Imperfect Market," Journal of Portfolio Management 6(Winter 1980), 511. Mayers, D., "Nonmarketable Assets and Capital Market Equilibrium under Uncertainty, "in Studies in the Theory of Capital Markets , edited by Michael C. Jensen. Praeger, New York (1972). Modest, D.M., and M. Sundaresan, "The Relationship Between Spot and Futures Prices in Stock Index Futures Markets: Some Presliminary Evidence." Journal of Futures Markets 3 (Spring 1983), 1541. Mossin, J., "Equilibrium in a Capital Asset Market," Econometrica 34 (October 1966), 768783.
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97 Peck, A., "Hedging and Income Stability: Concepts, Implications and Example," American Journal of Agricultural Economics 57 (August 1975). Raynauld, J., and T. Jacques, "Risk Premiums in Futures Markets: An Empirical Investigation," Journal of Futures Markets 4 1984, 189211. Pinduck, R.S., and D.L. Rubinfeld, Econometric Models and Econometric Forecase . McGraw Hill Book Co., New York (1981). Pratt, J.W., "Risk Aversion in the Small and in the Large," Econometrica 32 (JanuaryApril 1964), 122136. Radcliffe, R.C., Investment: Concepts, Analysis and Strategy . Scott, Foresman and Company, Glenview, Illinois (1982) Richard, S.F., and M. Sundaresan, "A Continuous Time Equilibrium Model of Forward Price and Futures Price in a Multigood, Economy," Journal of Financial Economics 9 (1980), 347372. Ross, S.A. , "Options and Efficiency," Quarterly Journal of Economics (February 1976), 7589. Samuelson, P. A., "Proof that Properly Anticipated Price Fluctuatis Randomly," Industrial Management Review (Spring 1965), 4149. Samuelson, P. A., "Is RealWorld Price a Tale Told by the Idiot of Chance?" Review of Economic and Statics 58 (February 1976) , 120123. Sarnat, M. , "Capital of Optimal Portfolios," Journal of Finance 29 (September 1974), 12411253. Sarris, A., "A Bayesian Approach to Estimation of TimeVarying Regression Coefficients," Annuals of Economic and Social Measurement 2 (October 1973), 501523. Schlarbaum, G.G., W. Lewellen, and R.C. Lease, "Realized Returns on Common Stock Investments: The experience of Individual Investors," Journal of Business 51 (April 1978), 299325. Sharpe, W.F., "Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk," Journal of Finance 19 (September 1964), 419442. Simkowitz, M.A., and W.L. Beedles, "Diversification in a ThreeMovement World," Journal of Financial and Quantitative Analysis 13 (December 1978), 924941. Smith, K.V., "Stock Price and Economic Indexes for Generating Efficient Portfolios," Journal of Business 41 (April 1968), 326336. Stein, J.L., "The Simultaneous Determination of Spot and Futures Prices," America Economic Review 51 (December 1961), 10121025.
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98 Stoll, H., "Commodity Futures and Spot Price Determination and Hedging in Capital Market Equilibrium," Journal of Financial and Quantitative Analysis 14 (1979) 873893. Telser, Lester G. , "Safety First and Hedging," Review of Economic Studies 23, (1955), 116. Wu, S., "The Pricing of Index Futures," Term Paper (To Dr. A. A. Heggested) , Department of Finance, University of Florida, Gainesville (May 1983).
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BIOGRAPHICAL SKETCH Soushan Wu was born on June 3, 1950, in Kaoshiung, Taiwan, Republic of China. He received his Bachelor of Commerce (in accounting and statistics) degree in 1972 from National ChungHsin University, Taipei, Taiwan. In 1974, he received his Master of Science degree in management science from National ChiaoTung University, Hsinch, Taiwan. Since then, he continues to be on the faculty of accounting and information systems with National ChiaoTung University. He also served as a consultant to BristolMyers, Taiwan Subsidry, Inc. in 19791980. In 1981, he entered the University of Florida to pursue his graduate study in finance. During his graduate work, he received financial support from the National Science Foundation of Republic of China, as well as student assistantship from the Department of Finance, University of Florida. At the completion of his study, he will return to his country to continue his career in teaching and research. 99
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. / Robert C. Radcliffe, Chairman Associate Professor of Finance, Insurance, and Real Estate. I certifiy that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. S. R. Cosslett Associate Professor of Economics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Raytto^d Chiang Associate Professor of Finance, Insurance, and Real Estat I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a disseration for the degree of Doctor of Philosophy. Roger Huang ~~A Assistant Professor of Finance, Insurance, and Real Estate
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This dissertation was submitted to the Graduate Faculty of the School of Accounting in the College of Business Administration and to the Graduate School, and was accepted as a partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 1984 Dean for the Graduate School and Research

