ON THE PERFORMANCE OF THE
GNMA HEDGE
IN VOLATILE MONEY MARKETS
BY
JAMES WAYNE EARLE
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1982
Copyright 1982
By
James Wayne Earle
To Anne, whose unselfish spirit and
extraordinary resilience give evidence
of how much there is yet to learn.
ACKNOWLEDGMENTS
Acknowledgment and appreciation are extended to the
College of Business Administration at the University of
Florida, where this research originated, and to the
University of Alaska, Anchorage, where the project achieved
maturity and culmination.
Special thanks to Dr. Halbert C. Smith, who served as
chairman of my dissertation committee and provided guidance
throughout my doctoral effort; to Dr. Clayton C. Curtis, who
served as a member of my dissertation committee and first
suggested the subject matter of this research; and to
Dr. Stanley Su, who served as a member of my dissertation
committee and ably advised me in the area of my strong
minor.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS . . . . . . . . . .. iv
LIST OF TABLES . . . . . . . ... .vii
LIST OF FIGURES . . . . . . . . ... . ix
ABSTRACT . . . . . . . . .. . ... .xii
CHAPTER I. BACKGROUND AND HISTORICAL PERSPECTIVE . 1
Basis for a Futures Market and for Futures
Trading in Financial Instruments . . . . 1
Development of an Interest Rate Futures
Market on the Chicago Board of Trade . . . 3
Advantages and Disadvantages of the New
Interest Rate Futures Market Over the
Existing Forward Market . . . . . . 7
CHAPTER II. THE CONCEPT OF HEDGING .. . . . 10
Rationale for Hedging . . . . . . .. 10
Factors Affecting the Performance of the Hedge .17
Theories of Hedging . . . . . . .. .19
CHAPTER III. THE MARKET ENVIRONMENT . . . .. .31
The Potential of the Prime Rate as a Proxy
for Uncertainty in U.S. Money Markets ... .31
The Concepts of Volatility and Stability in
U.S. Money Markets . . . . . . ... 36
CHAPTER IV. METHODOLOGY . . . . . . .. .39
Statement of Objectives . . . . . .. .39
Data Collection and Refinement . . . . .. .40
Critical Parameters . . . . . . ... 43
The Concept of a Moving Performance Parameter .47
Statistical Model . . . . . . . . 50
CHAPTER V. ANALYSIS AND INTERPRETATION . . . .. .62
The Performance of the 28Day Hedge in the
NearTerm Contract . . . . . . .. .62
Page
The Performance of the Hedge Under Selected
Alternative Conditions . . . . . .. .84
CHAPTER VI. CONCLUSIONS AND RECOMMENDATIONS ... .95
APPENDIX A. SELECTED PLOTS DISTANT CONTRACTS . . .103
APPENDIX B. SELECTED STATISTICS DISTANT CONTRACTS . 129
REFERENCES . . . .. . . . . . . 134
SUPPLEMENTARY BIBLIOGRAPHY . . . . . . .. .136
BIOGRAPHICAL SKETCH . . . . . . . . .. .139
LIST OF TABLES
Table Page
1. The Classic Short Hedge . . . . ... .13
2. The Classic Long Hedge . . . . ... 16
3. Summary of Current and Previous Findings
28Day Hedge in the NearTerm Contract . 69
4. Selected Statistics for Multivariate
Regression . . . . . . . ... 75
5. Selected Statistics for Multivariate
Regression . . . . . . . . 76
6. Multiple Coefficient of Determination (r) 77
7. Selected Statistics for Multivatiate
Regression (Autocorrelation Corrected) . 82
8. Selected Statistics for Multivate Regression
(Autocorrelation Corrected) . . . .. ..83
9. Multiple Coefficient of Determination (r2) 83
10. Summary of Selected Hedge Characteristics
from the Viewpoint of the Classic Short
Hedger 28Day Hedge, All Contracts . .. .85
11. Summary of Selected Current Findings 28Day
Hedge in the Distant Contracts . . ... .90
12. Summary of Selected Hedge Characteristics
from the Viewpoint of the Classic Short
Hedger 126Day Hedge, All Contracts .... .93
13. Summary of Selected Current Findings 126Day
Hedge in the Distant Contracts . . ... .94
B.1. Selected Statistics for Multivariate
Regression 28Day Hedge, 6Month Contract
Dependent Variable b* . . . . ... .129
vii
B.2. Selected Statistics for Multivariate
Regression 28Day Hedge, 6Month Contract
Dependent Variable e . . . . . . 129
B.3. Selected Statistics for Multivariate
Regression 28Day Hedge, 9Month Contract
Dependent Variable b* . . . . ... 130
B.4. Selected Statistics for Multivariate
Regression 28Day Hedge, 9Month Contract
Dependent Variable e . . . . .. 130
B.5. Selected Statistics for Multivariate
Regression 28Day Hedge, 12Month Contract
Dependent Variable b* . . . . ... 131
B.6. Selected Statistics for Multivariate
Regression 28Day Hedge, 12Month Contract
Dependent Variable e . . . . ... 131
B.7. Selected Statistics for Multivariate
Regression 126Day Hedge, 9Month Contract
Dependent Variable b* . . . . ... 132
B.8. Selected Statistics for Multivariate
Regression 126Day Hedge, 9Month Contract
Dependent Variable e . . . . ... 132
B.9. Selected Statistics for Multivariate
Regression 126Day Hedge, 12Month Contract
Dependent Variable b* . . . . .. 133
B.10. Selected Statistics for Multivariate
Regression 126Day Hedge, 12Month Contract
Dependent Variable e . . . . ... 133
viii
LIST OF FIGURES
Figure Page
1. Prime Rate Activity . . . . . ... 33
2. Market Period Indices . . . . ... .60
3. Weekly Average Change in Basis GNMA8/CBOT/
4 Week Hedge/3 Month Contract, 19761981 . 64
4. Performance of Hedge Efficiency Index
GNMA8/CBOT/4 Week Hedge/3 Month Contract,
19761981 . . . . . . . ... 71
5. Performance of RiskMinimizing Hedge Ratio
GNMA8/CBOT/4 Week Hedge/3 Month Contract,
19761981 . . . . . . . ... 72
6. Performance of Hedge Efficiency Index
GNMA8/CBOT/4 Week Hedge/3 Month Contract
(Autocorrelation Corrected), 19761981 . 80
7. Performance of RiskMinimizing Hedge Ratio
GNMA8/CBOT/4 Week Hedge/3 Month Contract
(Autocorrelation Corrected), 19761981 . 81
A.1. Weekly Average Change in Basis GNMA8/CBOT/
4 Week Hedge/6 Month Contract, 19761981 . 103
A.2. Performance of Hedge Efficiency Index
GNMA8/CBOT/4 Week Hedge/6 Month Contract,
19761981 . . . . . . . . . 104
A.3. Performance of RiskMinimizing Hedge Ratio
GNMA8/CBOT/4 Week Hedge/6 Month Contract,
19761981 . . . . . . . ... 105
A.4. Performance of Hedge Efficiency Index
GNMA8/CBOT/4 Week Hedge/6 Month Contract
(Autocorrelation Corrected), 19761981 . 106
A.5. Performance of RiskMinimizing Hedge Ratio
GNMA8/CBOT/4 Week Hedge/6 Month Contract
(Autocorrelation Corrected), 19761981 . 107
A.6. Weekly Average Change in Basis GNMA8/CBOT/
4 Week Hedge/9 Month Contract, 19761981 .108
A.7. Performance of Hedge Efficiency Index
GNMA8/CBOT/4 Week Hedge/9 Month Contract,
19761981 . . . . . . . ... 109
A.8. Performance of RiskMinimizing Hedge Ratio
GNMA8/CBOT/4 Week Hedge/9 Month Contract,
19761981 . . . . . . . . . 110
A.9. Performance of Hedge Efficiency Index
GNMA8/CBOT/4 Week Hedge/9 Month Contract
(Autocorrelation Corrected), 19761981 . 111
A.10. Performance of RiskMinimizing Hedge Ratio
GNMA8/CBOT/4 Week Hedge/9 Month Contract
(Autocorrelation Corrected), 19761981 . 112
A.11. Weekly Average Change in Basis GNMA8/CBOT/
4 Week Hedge/12 Month Contract, 19761981 113
A.12. Performance of Hedge Efficiency Index
GNMA8/CBOT/4 Week Hedge/12 Month Contract,
19761980 . . . . . . . . . 114
A.13. Performance of RiskMinimizing Hedge Ratio
GNMA8/CBOT/4 Week Hedge/12 Month Contract,
19761981 . . . . . . . ... 115
A.14. Performance of Hedge Efficiency Index
GNMA8/CBOT/4 Week Hedge/12 Month Contract
(Autocorrelation Corrected), 19761981 . 116
A.15. Performance of RiskMinimizing Hedge Ratio
GNMA8/CBOT/4 Week Hedge/12 Month Contract
(Autocorrelation Corrected), 19761981 . 117
A.16. Weekly Average Change in Basis GNMA8/CBOT/
18 Week Hedge/9 Month Contract, 19761981 118
A.17. Performance of Hedge Efficiency Index
GNMA8/CBOT/18 Week Hedge/9 Month Contract,
19761981 . . . . . . . ... 119
A.18. Performance of RiskMinimizing Hedge Ratio
GNMA8/CBOT/18 Week Hedge/9 Month Contract,
19761981 . . . . . . . ... 120
A.19. Performance of Hedge Efficiency Index
GNMA8/CBOT/18 Week Hedge/9 Month Contract
(Autocorrelation Corrected), 19761981 . 121
A.20. Performance of RiskMinimizing Hedge Ratio
GNMA8/CBOT/18 Week Hedge/9 Month Contract
(Autocorrelation Corrected), 19761981 . 122
A.21. Weekly Average Change in Basis GNMA8/CBOT/
18 Week Hedge/12 Month Contract, 19761981 123
A.22. Performance of Hedge Efficiency Index
GNMA8/CBOT/18 Week Hedge/12 Month Contract,
19761981 . . . . . . . ... 124
A.23. Performance of RiskMinimizing Hedge Ratio
GNMA8/CBOT/18 Week Hedge/12 Month Contract,
19761981 . . . . . . . . . 125
A.24. Performance of Hedge Efficiency Index
GNMA8/CBOT/18 Week Hedge/12 Month Contract
(Autocorrelation Corrected), 19761981 . 126
A.25. Performance of RiskMinimizing Hedge Ratio
GNMA8/CBOT/18 Week Hedge/12 Month Contract
(Autocorrelation Corrected), 19761981 . 127
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
ON THE PERFORMANCE OF THE
GNMA HEDGE
IN VOLATILE MONEY MARKETS
By
James Wayne Earle
August 1982
Chairman: Halbert C. Smith
Major Department: Finance, Insurance and Real Estate
Since the Middle Ages, merchantmen have attempted to
insulate themselves from price change risk, a phenomenon
inherent in business transactions involving passage of time.
In no area of business endeavor has price change risk been
more acute in recent years than in the area of financial
securities.
A common means by which an investor can obtain a degree
of protection from price change risk is by "hedging" a cash
market position in financial futures. The brief existence
of the markets in financial futures has encompassed both
relatively benign and highly volatile money market
conditions. Such differing economic climates provide the
exigency for examining the performance of the hedge over
time.
xii
This research constructs hypothetical sample hedges
from the population of all possible hedges to make
inferences about the performance of the Government National
Mortgage Association (GNMA) hedge over time. Bivariate
regression is used to obtain estimates of two critical
measures of hedge performance, the riskminimizing hedge
ratio and the hedge efficiency index. By computing each
respective measure over time in a manner similar to
computing a moving average, the longterm secular trend in
hedge performance can be observed. Multivariate regression,
utilizing as independent variables a series of descriptive
coefficients developed from the movement of the prime rate,
is employed in an attempt to link the measures of hedge
performance to uncertainty in the money market.
It appears hedge efficiency may be far more volatile
and significantly reduced in magnitude from that implied by
previous research. This is likely a result of uncorrected
positive autocorrelation in the error terms of the early
regression models. The erosion of the positive
changeinbasis characteristic of early markets has resulted
in a generally increased riskminimizing hedge ratio over
that observed in previous research.
Efforts to bind the performance of the hedge to the
uncertainty indicators developed from the movement of the
prime rate prove largely disappointing but do reveal the
considerable masking of weakness in the performance of the
hedge due to violation of the critical nonautoregression
assumption of the regression model.
xiii
CHAPTER I
BACKGROUND AND HISTORICAL PERSPECTIVE
Basis for a Futures Market and for Futures
Trading in Financial Instruments
Origin of a Futures Market
Modern futures trading has it origins in the ancient
trade networks and markets that developed in 12th century
Europe. At that time, merchantmen and farmers began
developing increasingly sophisticated means of reducing
their business risks by transferring those same risks to
other individuals. Advance pricing arrangements, which
would eventually evolve into the widespread use of both
forward and futures contracts that are common today, became
commonplace at the trade fairs and seasonal festivals of
medieval Europe. By 1570, the first organized futures
market in Western civilization was created with the
establishment of the Royal Exchange of London [13]. Almost
simultaneously, organized futures trading was beginning in
feudal Japan, where trading dealt exclusively with rice
contracts. Remarkably, the Imperial government supervised
trading with rules and procedures not unlike our own today.
Futures activity in America was initiated in 1851 when
the first contract was recorded with the Chicago Board of
Trade. The activity was compelled by risk considerations
not unlike those which faced medieval merchantmen in Europe
and Japan centuries before. Goaded by frequent market
crashes that could and often did wipe out their profits in
cotton and grain, merchants began selling their commodities
on a "toarrive" basis. Importers, aided by the formation
of a transAtlantic cable service, began selling their
purchases while the goods were still at sea on slowmoving
surface vessels. This futures activity enabled the transfer
of price change risk to a middleman, the speculator, who
would assume that risk in hopes of commensurate profits.
By 1967 the Chicago Board of Trade had, since its
inauguration over 100 years earlier, developed primarily if
not exclusively as a grain exchange. Futures trading
activity had concentrated in wheat, soybeans, corn, and
oats. The unqualified success of the grain futures markets
understandably resulted in increased interest in the type of
price protection and speculative opportunities offered. In
1968, a new kind of diversification began with the
introduction of trading in iced broilers. Trading in silver
futures and the highly successful plywood contract was
initiated in 1969. By 1974 the Chicago Board of Trade, now
the world's oldest and largest futures market, was
experiencing record volumes. In the six years since the
initial diversification, the number of contracts traded had
increased from 4.7 million to 14.5 million [11].
Significant growth and development of activity on the
Chicago Board of Trade, essential to authentication of the
futures markets as a viable vehicle for minimizing price
change risk, were clearly being demonstrated.
Development of an Interest Rate Futures Market
on the Chicago Board of Trade
Conceptualization. The origin of an interest rate
futures market can be traced back to 1969. A market
phenomenon coined a "credit crunch" had in that year
produced significant interest rate risk for financial
institutions. Mortgage bankers, insurance companies, and
savings and loan associations found themselves faced with
the problem of having to make commitments of funds at
current interest rate levels in an era of increasingly
volatile rate structures. An association of businessmen and
economists, desirous of securing a satisfactory interest
rate on a securities transaction planned for months in the
future, noted the similarity between a businessman's
portfolio of fixedincome securities and a farmer's bins of
wheat. Both the businessman and the farmer could be
adversely affected if prices fell. The farmer could,
however, protect his holdings against price change risk by
selling futures contracts on the Chicago Board of Trade.
The businessman, at least at this point in time, had no such
comparable alternative. Since the organized futures markets
had provided a simple, proven, and popular mechanism for the
farmer or grain dealer to minimize price change risk, it
seemed reasonable to conclude that the same might be
possible for the financial manager. A financial instrument
known as the Government National Mortgage Association (GNMA)
PassThrough Certificate was to provide the vehicle for just
that opportunity.
Method. During the middle 1960's, considerable effort
was invested in finding ways to streamline the mortgage debt
instrument in order to make it a more attractive investment
to a broader base of investors. In particular, ways were
sought to tap the high volume of financial resources in the
bond and securities markets that never found their way into
the mortgage market because of the complex legalities, the
need to be familiar with local housing markets, the need to
underwrite individual mortgages, and the need to service
individual mortgages. The result of these efforts was the
Government National Mortgage AssociationMortgage Backed
Security (GNMAMBS) program with its GNMA PassThrough
Certificate. (Most of the MBS issues are called
"passthroughs" to indicate the issuers "passthrough" to
the investors the principal and interest payments made each
month on the mortgages.) Through the vehicle of the Federal
guaranty, securities backed by mortgages were to become as
safe, as liquid, and as easy to hold as Treasury securities.
5
By 1975, the GNMA Certificate had emerged as an
immensely successful instrument. As one of the most popular
forms of mortgage investment to reach the secondary mortgage
market, the GNMABMS had proven to be very marketable and
highly liquid with its safety virtually unquestioned. The
program represented one of the most significant innovations
in residential finance in the previous 40 years [11].
With the growth of competitive trading of GNMA's in
both primary and secondary markets, the certificate became
competitive with other longterm instruments such as AAA
corporate bonds and longterm government bonds. With GNMA
yields representative of a variety of longterm rates, it
was thought a GNMA futures contract could potentially be
used to minimize interest rate risk throughout the longterm
market. The establishment of an organized futures market in
GNMA securities was the next logical step in the
evolutionary process.
Result. The Commodity Futures Trading Commission Act
of 1975 provided the enabling legislation for trading
mortgage interest rate futures on a commodity exchange. On
May 28, 1975, the Chicago Board of Trade requested
permission from the Commodity Futures Trading Commission in
Washington, D.C., to establish such a market. Trading was
actually begun on October 20, 1975.
By April 20, 1976, the fledgling market's sixmonth
birthday, total trading volume for the GNMA certificate
contract stood at roughly 46,000 contracts. The comparable
volume for the Chicago Board of Trade's previously most
immediately successful contract, plywood, was
11,400 contracts. By the market's first anniversary on
October 20, 1976, about 114,000 GNMA futures contracts had
changed hands, almost triple the firstyear volume in the
successful plywood futures contract [11]. The acceptance
and popularity of financial futures had been solidly
established.
In the brief years that have passed since that
successful inauguration, a veritable trading explosion in
financial futures has spread to other exchanges and a broad
variety of other instruments such as U.S. Treasury bills and
bonds, bank certificate of deposit, foreign currencies, and
even Eurodollars. Chicago Board of Trade and Chicago
Merchantile Exchange trading volumes have increased more
than eightfold in the past decade, with trading in financial
futures now accounting for about 30% of all futures trading
volume, compared to nothing a decade ago. Some believe
trading in financial futures will in the next few years
overtake conventional commodity futures, such as corn,
soybeans, and wheat, in total volume. In 1981, the Chicago
Board of Trade's Treasury bond contract moved to the top of
the list of all futures trading contracts in volume,
dropping corn and soybeans to Nos. 2 and 3.
Today, the big exchanges believe they are on the verge
of a further explosion in trading activity perhaps doubling
the present volume by 1985, and have committed $300 million
for new trading facilities to handle the anticipated load.
Advantages and Disadvantages of the New Interest Rate
Futures Market Over the Existing Forward Market
Forward commitments in financial instruments, like
futures, exist because of the risk of changes in interest
rates over time. Both deal with contracts for future
delivery. However, there are distinguishing characteristics
between the forward market and the futures market that
produce advocates for both contracts and are therefore
important to identify.
Futures contracts specify a standard contract, with a
standard set of terms, which is traded in large volume on an
organized exchange such as the Chicago Board of Trade.
Uniform market quotations are always publically available.
A forward contract is tailored to the particular needs of
the individual buyer and seller by negotiation between those
respective parties. Thus, futures trading creates a
centralized, highly visible market whereas forward
contracts, because they are entered into directly between
individual dealers, lack that attribute. The existence of a
single market, as in the case of futures transactions, can
reduce the search costs of soliciting quotes from numerous
dealers and assures the investor of getting the optimal
price currently available. However, by fixing delivery
date, quantity, coupon, and other factors, futures gain
liquidity at the cost of flexibility. This loss of
flexibility may lead to "basis risk," which will be
addressed in some detail in a later section. With futures,
one is dealing with a stable, recognized, reputable entity
in the Chicago Board of Trade, and there is virtually no
risk that the contract will default and result in
inconvenience and possible litigation as is the case with
forwards.
The futures markets trade in contracts for delivery up
to two and onehalf years into the future, whereas cash
market dealers typically limit their trading to the current
month and perhaps six months thereafter. Therefore, the use
of the futures contract is necessary to deal with risks
associated with changes in interest rates and yields during
periods beyond the coming few months.
The futures trade will require a margin deposit and
maintenance margin costs, referred to as "marking to the
market," when paper losses occur. Forward contracts are
generally made without margin for those institutions which
meet the dealer's credit criteria. One of the difficulties
with the forward market is concerned with the fact that it
has enabled individuals and firms to speculate on interest
rate movements without putting up an investment equivalent
to margin. This may have served to make forward commitment
activity more popular than it might otherwise have been.
This research notes that the relative advantage or
disadvantage of the forward commitment visavis the futures
contract is in some dispute [2,9]. However, the popularity
9
of the new futures markets appears unquestionable, and the
performance of those markets over time provides the focus
for this research.
CHAPTER II
THE CONCEPT OF HEDGING
Rationale for Hedging
Definition
It has been noted that since the middle 1960's,
institutions that borrow and lend large sums of money may
experience a sizeable increase in risk because of interest
rate volatility. These institutions must in the course of
conducting business commit themselves to lend or borrow
money at rates that may become intolerably costly in just
days or weeks. This condition will likely give them an
incentive to utilize the riskminimizing characteristics of
the futures market. While the strategies employed by those
who trade in the futures markets are diverse, often those
who attempt to use the futures as a convenient way to
protect the value of fixedincome trading or investment
portfolios from volatility in interest rates use a technique
called "hedging."
Hedging, at least in its classical sense, is an
activity whereby an individual or firm seeks to minimize
price change risk for a given level of return. The
classical notion of hedging is rooted in the concept of
correlation, the statistical measure that indicates the
relationship, if any, between series of numbers. If two
series move together, they are positively correlated; if the
series are countercyclical, they are negatively correlated.
The idea of simple (two variable) linear correlation
suggests that the best representation of perfect correlation
is a straight (linear) regression line fitted to the
observed data. The equation for such a line is the familiar
Y = a + bX. (1)
In order to diversify and therefore minimize or
eliminate systematic risk, projects best combined or added
to the existing portfolio of projects are those that have a
negative correlation with existing projects. By combining
negatively correlated projects, the overall variability of
returns or risk can be reduced. Thus, the gain (or loss) on
position X is offset by the loss (or gain) on position Y.
Obviously, the strategy works best for perfectly negatively
correlated series.
The Classic Short Hedge
Suppose that on August 24, 1979, a savings and loan
association had expected to close $1 million worth of FHA/VA
loans during the next four weeks and create a GNMA pool.
The institution then intends to sell this pool in the
secondary market. If in the interim interest rates rise,
the pool will have to be sold at a discount thus producing a
loss for the institution. (The fact that interest rates
could fall thus allowing the pool to be sold at a premium
producing a gain over the interim is not the point, at least
for now. Traditionally, price change risk is borne by
speculators and is not the domain of the typical financial
intermediary. To gain the riskreducing advantages of
hedging, the opportunity to profit from favorable interest
rate movements is forfeited.) The risk can be hedged in the
futures market as follows:
On August 24, 1979, the savings and loan association
sells ten GNMA futures contracts for March, 1980, delivery.
This portion will, under the assumptions of the classic
hedge, protect the institution's planned sale in the cash
market. Assume the pool is sold on September 21, 1979. It
is worth noting that on August 24, 1979, the prime rate as
reported by the Federal Reserve Bulletin stood at 12%; on
September 21, 1979, it made its fourth upward move in
28 days and stood at 13%. Although the institution will
have to take a loss in selling the pool at a discount, it
will obtain a hopefully equivalent gain from the futures
position. Since rates had risen between August 24, 1979,
and September 21, 1979, the price of March, 1980, futures
should, according to the assumptions of the classic hedge,
have fallen below the price at which March, 1980, futures
were selling on August 24. A numerical illustration of this
scenario, an example of the classic short hedge, follows:
Date
Aug 24, 1979
Sept 21, 1979
Result
TABLE 1
The Classic Short Hedge
Cash Market Futures Market
Action Action
Current Cash Sell 10 Mar 80 GNMA's
Price: $895,937.50 Price: $863,906.30
Sell $1 Million GNMA Buy 10 Mar 80 GNMA's
Price: $876,875.00 Price: $844,843.80
$19,062.50 $19,062.50
LOSS GAIN
The given example is referred to as a classic "short"
hedge because the institution sold or "shorted" the March,
1980, futures to initiate the hedge and maintained that
short position until the offsetting purchase at the close of
the hedge.
Note that in this example the cash market result, which
consists of the ending or closing cash price less the
beginning or opening cash price, is represented by Y in
Equation (1) and the offsetting, countercyclical result in
the futures market, consisting of the beginning or opening
futures price less the ending or closing futures price, is
represented by X in the same equation. Thus, when loss
(gain) in the cash market is regressed on gain (loss) in the
futures market,
(P2 l) = a + b(P1 P2) (2)
s s f f
2
where Ps = the cash or "spot" price at time t2,
S z
Ps = the cash or "spot" price at time tl,
1
Pf = the futures price at time tl, and
2
P = the
In the case of
b = 1,
futures price at time t2.
perfect negative correlation with a = 0 and
(P P) = (P P
s s f f
and
(P2 pl ( 2 pl) = 0
s s f f
which was in fact the case in the example. Stated
alternatively, when the change in the cash market is
regressed on the change in the futures market,
2 1 1 2
X(P P =P a + bXf(P P )
s s s f f
where Xs = cash or spot market holdings, and
Xf = futures market holdings.
In the case where Xs = Xf, as in the classic hedge,
(P2 P = a + b (P ). (6)
s s f f
Now with perfect positive correlation, a = 0, b = +1 and
Equations (3) and (4) hold.
It is observed that the change in the cash position has
been matched by an equal change in the futures position.
The resulting net change of zero completes the socalled
"perfect" hedge, often depicted in the literature but in
fact an exceedingly rare phenomenon, as shall be later
observed.
The Classic Long Hedge
Similar to the short hedge, a "long" hedge can be
initiated to insure an investment yield sometime in the
future. For example, an institution or investor who can
forecast his cash inflow over the next investment period may
consider current yields attractive and wish to lock into
this yield. If, when the time comes to actually invest in
the cash market, yields have fallen and the price of the
financial instrument has risen, the investor's loss on his
cash market position will hopefully be offset by a gain in
the futures market position.
For example, if on March 18, 1980, the firm or
individual had concluded that in four weeks there would be
$1 million available for investment in financial instruments
and the current yields of 12.65% were considered attractive,
a purchase of ten September GNMA contracts could have been
initiated. By April 15, 1980, the prime rate peaked at 20%
and cash yields, leading the prime rate as they often do,
had declined to 12.05%. The firm or individual then could
buy the instruments) in the cash market as planned but at a
loss from the March 18 rate. However, the futures market
position can now be liquidated with the offsetting gain
predicted by the classic hedge as depicted below:
Date
Mar. 18, 1980
Apr 15, 1980
Result
TABLE 2
The Classic Long
Cash Market
Action
Current Cash
Price: $720,312.50
Buy $1 million GNMA
Price: $749,062.50
$28,750.00
LOSS
Hedge
Futures Market
Action
Buy 10 Sep 80 GNMA's
Price: $697,187.50
Sell 10 Sep 80 GNMA's
Price: $725,937.50
$28,750.00
GAIN
This second example is referred to as a "long" hedge
because the institution bought or went "long" in September,
1980, futures to initiate the hedge and maintained that long
position until the offsetting sale at the close of the
hedge.
It is observed that the mathematical requirements of
the "perfect" hedge are equally applicable for either the
short or the long hedge, i.e., the change in the cash
position has been matched by an equal change in the futures
position resulting in a net change of zero, as suggested by
Equation (4).
It is important to emphasize that the examples
presented thus far are predicated on the assumption that the
objective is price insurance. The Exchange and the contract
specifications are designed principally to facilitate the
hedging function. Fewer than 1% of all futures contracts
are settled by delivery. The rest have been offset before
the delivery month.
Factors Affecting the Performance of the Hedge
The real world seldom sees the perfect hedge so often
depicted. One reason for not having perfect hedges relates
to price movements over time called "convergence."
Convergence, which suggests that as time passes the futures
price will approach the cash price, is a phenomenon
resulting directly from the aspect of futurity inherent in
any futures market and involves the behavior of "spreads"
over time. Price differentials between different futures
market contracts of the same commodity are called "spreads."
Whereas spreads refer to the price difference between
specified months in the same market, "basis" can refer to a
price difference between specified months in different
markets, i.e., the December cash market vs. the December
futures market.
In GNMA markets, spreads are caused by two principal
factors: the first is the profitability of financing
inventories of securities for later sale, a factor referred
to as "warehousing," and the second is the market's
expectation of price changes for GNMA securities. In
positive carry markets where shortterm rates are lower than
longterm rates, the first of these factors is dominant and
acts to offset the convergence phenomenon. Essentially, the
value of warehousing determines the difference between
immediate delivery and futures prices. For example, if one
owns GNMA securities and agrees to sell to a dealer for
delivery one month later, he receives a month's positive
income by financing the securities at a lower rate than the
securities yield. The dealer, in turn, compensates for the
positive carry by bidding a lower price for the one month's
deferred delivery than he would pay for immediate delivery.
Thus while convergence will work consistently against the
short hedge, there is supposedly an equally consistent
offset to the convergence phenomenonthe interest
differential earned by warehousing the closed loans on
GNMA's against which the hedge is placed. If warehousing
profits set the intermonth spreads in both cash and futures
markets, it follows that convergence losses in a short hedge
should equal warehousing profits.
To interpret the results of any hedge, a financial
manager can look at a bottom line consisting not only of the
gain or loss on the sale of the inventory and the futures
market profit or loss, but also interest income and expense,
imputed interest cost on margin funds and all transaction
costs. The inclusion of so many pieces to the short hedge
implies that any individual performance goals placed on a
longterm hedging strategy should be set as part of an
overall marketing goal. However, to seek to recover costs
of transactions from futures activity, one must insert an
element of arbitrage transaction into the equation which
will increase risk from the minimum. Such action is foreign
to the true hedger and not particularly relevant to this
study, for only as basis risk approaches zero is the true
hedger, who has shifted price change risk to basis risk,
fulfilled.
Theories of Hedging
Traditional
The price insurance potential of the futures markets is
the foundation for traditional hedging theory, where those
seeking to minimize price change risk are expected to take
futures market positions equal in magnitude but of opposite
direction or sign to their position in the cash market.
Traditional theory, which continues to underlie many
articles on hedging, is predicated upon the assumption that
cash and futures prices move together in perfect positive
correlation, the difference between the two at any given
time being the "basis." In the case of the socalled
"perfect" hedge, it is observed from Equation (4) that
(P2 P) (P P) = 0
s s f f
or alternatively,
(P2 P) (l Pf) = 0 (7)
s f s f
and the change in the "basis" is zero over the duration of
the hedge. By taking cash and futures market positions of
opposite direction or sign, the desired perfect negative
correlation which eliminates price change risk is supposedly
achieved. What is clear from empirical observation is that
the basis changes so that the great majority of such
"traditional" hedges are not perfect. Many current writers,
while acknowledging this fact, dismiss the question by
making the implicit assumption that such deviations are
distributed around an assumed mean net change of zero.
Therefore, if enough samples are selected from the
distribution of all possible hedges, the distribution of
sample means will be normal and its mean will indeed
approximate zero. According to the Central Limit Theorem,
this will occur only if the true population mean is equal to
zero. It is this critical assumption, in this instance
analogous to a positive slope of 1 for the regression line
2 1
when the change in cash price (P P ) is regressed on the
s s
2 1
change in futures price (P Pf), that has provided one of
several focal points for recent inquiry into the performance
of the GNMA hedge.
Literature Review
In one of the earlier inquires into the theory of
hedging, Working [14] questioned the view that hedgers were
singularly motivated by the price insurance potential of the
futures markets and suggested that most hedging is done in
expectation of a change in spotfutures price relations. He
viewed hedgers as functioning much like speculators who,
seeking to maximize profit, are more concerned with the
relative basis between cash and futures prices than with
absolute price changes. Working reasoned that, depending on
the hedger's future expectations regarding this
relationship, he would be inclined to either hedge
completely his cash position or remain completely unhedged.
However, Ganis [4] observes that trading the cash/futures
basis in an effort to protect an institution's cash position
from loss due to changes in interest rates would be
classical hedging, whereas the very same trades, if part of
an aggressive effort to profit from anticipated changes in
the basis relationship, would be said to constitute
arbitrage transactions. An arbitrage is, of course, a hedge
position taken in anticipation of a change in market
relationships in an effort to realize a gain.
Johnson [6], Stein [12] and more recently Ederington
[1] have used basic portfolio theory to integrate the price
insurance potential of traditional theory with profit
maximization. This application allowed these authors to
explain why hedgers may elect neither the completely hedged
position suggested by traditional theory nor the dichotomus
alternative suggested by Working's hypothesis, but rather a
fractional or partially hedged position. From the
standpoint of this research, Ederington's work is
particularly significant since it is one of the first
attempts to delve into the performance of the fledgling GNMA
hedge. Ederington viewed cash market holdings as fixed and
the decision was how much of the cash market holding to
hedge. There was no presumption as in traditional theory
that Xs = Xf, where Xs represents cash or spot market
holdings and Xf represents futures market holdings.
Continuing to use Ederington's notation for clarity, it is
noted that in a traditional classic hedge with Xs = Xf,
b = +1 where b is the slope of the regression line when the
2 1
change in cash or spot price (P P ) is regressed on the
s s
change in futures price (P2 P ). Going back to
Equation (2)
(P2 ) =) a + b(P1 P )
s s f f
and
2 1 1 2
(X)(P P) = a + b(Xf)(P P )
where a = 0
2 1 1
b = (Xs)(P P) / (X) (P P )
s s s f f f
and, if the hedge is perfect, i.e., (P2 p 1) = _(P1 )
s s f f
then
B = Xs / Xf
and if the hedge is traditional, i.e., Xs = Xf, then
b = +1.
However, as observed later, the classic perfect hedge
is rarely achieved. In the great majority of cases
(P2 P (P2 P)
s s f f
and the traditional assumption that the population mean is
equal to zero has heretofore been an implicit assumption
rather than a proven fact. It follows that the true
regression line, when the change in cash price is regressed
on the change in futures price, will produce the minimum
variance. Therefore, the riskminimizing b, b*, is the
slope of the regression line when the change in cash price
is regressed on the change in futures price. This is as
Ederington found it. Optimal b, b as defined by Ederington
[1], will of necessity be a function of each individual's
riskreturn relationship. Obviously, if b is greater than
one, an individual takes a position in futures greater than
that in the cash market. If b is less than zero, one has
assumed the same position in both the cash and the futures
market, the structure whimsically known as a "Texas" hedge.
Ederington notes that "while the risk reduction
achieved by any one hedger depends on the chosen b, the
futures market's potential for risk reduction can be
measured by comparing the risk on an unhedged portfolio with
the minimum risk that can be obtained on a portfolio
containing both spot (cash) and forward (futures)
securities" [1, p. 163]. This position corresponds to the
variance of return on a portfolio where b equals b*. The
measure of effectiveness he used was, therefore, the percent
reduction of the variance, or
e = 1 [Var(R*) / Var(U)] = R2
where Var(R*) denotes the minimum variance on a portfolio
containing securities futures; R2 is the familiar population
coefficient of determination between the change in cash
price and the change in futures price. Ederington estimated
e using the sample coefficient of determination,
2
r for twoweek and fourweek hedges noting both the
2
observed b and r
We have already observed that traditional theory, while
acknowledging that the hedge will not often be perfect,
implicitly assumes movement or volatility in the change in
basis in any given period is random with zero mean and
standard deviation of o, and is independent of changes in
other periods. As such, traditional theory implies the
widely accepted "random walk" model for the movement of the
basis. However, the generalization of this model known as
the "Martingale" model differs from a random walk in that
volatility may change over time, while it is a fixed
constant for a random walk. Thus, volatility is an ideal
measure of basis variation because it captures the intuitive
notion of stability being related to the magnitude of random
fluctuations. It is this volatility that Ederington
captures in r2 or e.
Ederington hypothesized that expectations of the near
future would be more affected by unexpected changes in the
cash price than expectations of more distant futures and
predicted e would decline as one hedges in more distant
contracts. He further hypothesized that e will be greater
for fourweek hedges than for twoweek hedges because
absolute changes in cash prices should generally be greater
and future prices would have more time to respond over the
longer period.
Ederington confirmed the second hypothesis and
supposedly proved the "lag" in futures, but failed to
conclude, at least for GNMA's, that the near future would be
affected more by unexpected changes in the cash market.
Ederington found that b* was, in most cases, significantly
different from one and in general was less than one.
An important question raised by Ederington is whether
over the long run the expected change in the basis will tend
to be consistently positive or negative. He suggests that
since the basis must be approximately zero at the
delivery date, the expected change in the basis
will generally be positive if the current cash
price exceeds the current futures price and will
generally be negative if the futures price exceeds
the cash price. [1, p. 167]
Ederington felt his results tended to confirm this
hypothesis and, since he had concluded risk reduction to be
approximately the same for nearby and distant contracts, he
suggested that long (short) hedgers would have been well
advised to hedge in the nearby (distant) contracts.
What is perhaps most significant about Ederington's
work, at least from the standpoint of this study, is the
time period observed. His observations were of necessity
confined to the "market experience to date," namely 197677,
the first two years of GNMA futures market existence.
Although turbulent by pre1966 standards, the period may be
characterized as relatively benign when compared with the
years to follow.
Franckle and Wurtzebach [3] refine Ederington's
regression equation
2 1 2 1
(P2 ) = a + b(P2 P )
s s f f
by incorporating E(APs) and E(AP ) where E(APf) = 0,
1 2
i.e., P is an unbiased estimate of P and E(AP )
1 1f
= (P P )[i/(i + j)] where
f s
i = length of hedge
j = time to convergence of spot and futures.
Thus, their equation becomes
2 1 2 1
P2 pl E(AP = a + b[(P P) E(AP)]
S s s f f f
2 1 2 1
P2 Ps E(AP = a + b(P2 P).
s s s f f
They incorporate, using linear interpolation, the
phenomenon of convergence into the equation regressing
unexpected change on unexpected price change in futures
instead of using total change as in previous research.
Unexpected change A is computed under the assumption
that the futures price is an estimate of the spot price at
either the beginning or the end of the delivery month.
Unexpected change B assumes that the futures price is an
estimate of the spot price on the last day of futures
trading.
These varying assumptions all determine the value of j
or the length of time from the closing of the hedge until
price convergence.
Franckle and Wurtzebach observed that estimated e and
the riskminimizing hedge ratio, b*, increased during
197879 over that period observed by Ederington (197677).
This suggests the erosion of the positive change in the
basis characteristic of the earlier period and perhaps the
increased maturity in the markets suspected by Ederington.
They observed that variance of futures price was about twice
as great as variance of cash prices in the less volatile
period of 197677 while there was little difference between
the two in the later period. Because
b* = o / ,
s f
this would account for a lower b* in the earlier period.
Interestingly, they conclude that the futures market is most
effective during periods of volatile interest rates.
Working's hypothesis is not without its present day
advocates. Howard and D'Antonio [5], in a recent and as yet
unpublished work, conclude that while one's cash position
may be only partially hedged, the optimal level of futures
contracts held will vary greatly depending on one's
perception on the relative expected performance of futures
to cash holdings. The authors have thus chosen to
concentrate on optimal b, which Ederington defined as b, as
opposed to the riskminimizing b, which Ederington defined
as b*. The difference between the two has to do with
individual portfolio considerations as contrasted with
market potential for risk reduction. This study is
concerned with the latter.
In a study concentrating on activity throughout the
calendar year 1978, Raleigh [10] took a hypothetical firm
with a cash inventory position through a simulation using
cash forwards four months out and compared results using
futures contracts for four months protection, at all times
being 100% hedged. He then developed a "managed hedge" for
a classic short hedger where futures are used to adjust from
less than 100% coverage in apparent uptrends to more than
100% coverage in apparent downtrends, thus roughly
simulating Ederington's hypothesis that the astute hedger
may not desire a b* = 1 at all times. Trends were
identified through the use of technical analysis. The
arbitrary target was to be 75% covered in uptrends, 100%
covered in sideways markets, and 125% covered in downward
trends. The key, of course, to the success of the managed
hedge was the accurate assessment of the trends. Whereas
Ederington's b* reflected the riskminimizing hedge ratio
over the observed period (in his case, approximately two
years), Raleigh in effect shortened the periods and altered
the hedge ratio over a much shorter term, producing
apparently successful results which confirm Ederington's
conclusions regarding b*.
The matter of the optimal hedge ratio has been refined
further by Kolb, Corgel, and Chaing [8]. The objective is
development of an optimal hedge ratio which will overcome
some of the limitations of the traditional and portfolio
approaches while taking into account differing price
sensitivities for the hedged and hedging instruments.
The authors contend that all hedging in GNMA's is
essentially "crosshedging" since price sensitivites rarely
if ever match. Their suggested technique seeks to control
for the "mismatches" that must of necessity arise.
The approach is applied to hedge a particular risk and
therefore is more sharply focused than either Ederington,
Franckle and Wurtzebach, or the other authors mentioned.
The method is more responsive to individual price
sensitivities than Ederington's universal b* which suggests
a single hedge ratio regardless of the nature of one's cash
position.
Much as with Franckle and Wurtzebach [3] and with
Howard and D'Antonio [5], this approach represents a rather
sharp refinement which, while reflecting the continuing
improvement in technique, does not bear directly on this
study, which is more global in character.
CHAPTER III
THE MARKET ENVIRONMENT (19761980)
The Potential of the Prime Rate as a Proxy
for Uncertainty in U.S. Money Markets
The Prime Rate
The GNMA futures market, in existence for five complete
calendar years at the time of this study, has witnessed some
of the most dramatic volatility in the history of U.S. money
markets. While not the only indication of such volatility
and perhaps not even the best, the movement of the prime
rate is likely the most visible and conspicuous of the
indicators of market volatility and uncertainty.
The concept of a prime rate was born in the Depression
to protect the banks against cutthroat competition. Banks
define the prime rate as their lowest interest rate for
their most creditworthy customers.
In 1974, increases leading to a 12% prime rate ushered
in a massive credit crunch and the worst recession since the
era of the Great Depression. The apex of that historic
credit crunch was reached on July 5, 1974, when the prime
rate reached a relative maximum of 12%. After holding at
that historic level for a period of 94 days, a steady
decline began on October 7, 1974. This decline lasted
284 days and consisted of 20 quarterpoint declines, ending
with a quarterpoint advance on July 18, 1975.
On October 20, 1975, when trading in GNMA futures was
initiated on the Chicago Board of Trade, the prime rate
stood at 8%. By the first of January, 1976, the beginning
of the period reviewed by Ederington [1], the rate had
receded to 7%%. For the next two calendar years, the range
of the observed market period for the Ederington study, the
market would experience only 14 movements of the prime rate.
All would be at quarterpoint intervals with the highest
level of 7 3/4% occurring from October 24, 1977, to
December 31, 1977, and the lowest level of 6% occurring
from December 13, 1976, to May 13, 1977.
A steadily increasing prime rate, actually begun with
the move to 6% on May 13, 1977, was to continue without
interruption throughout 1978. During this calendar year,
the prime rate experienced 15 quarterpoint advances, more
absolute movement than had been experienced in the previous
two years combined. A peak was reached on December 26,
1978, with an 11 3/4% prime rate, only 1/4% below the
historic high set on July 5, 1974 (see Figure 1).
The first half of 1979 witnessed a stable prime rate,
although historically high. On August 16, 1979, the prime
rate moved to 12% to equal the record established five years
earlier. It remained there only 12 days before continuing
upward. During the next 17 months, the prime rate would
19761981
Figure 1. Prime Rate Activity.
move on 52 occasions. Thirteen of these movements would
occur in the remaining months of 1979 and an incredible
39 such movements would occur in calendar 1980. The prime
rate would reach another relative maximum of 20% from
April 2, 1980, to April 18, 1980, plunge to a relative
minimum of 11% on July 25, 1980, and peak again at 21% on
December 19, 1980, moving at times in quantum leaps of a
full percentage point. It is this sharp increase in the
volatility and the level of the prime rate that
characterizes the market period of this study.
It is noted that the rise in interest rates which began
midway through 1979 was pointed, regardless of which
interest rates are considered. The threemonth Treasury
Bill rate rose from 9% at the end of 1978 to over 12% by
the same period one year later. In the capital markets,
1979 bond rates registered record increases. The movement
of the prime rate was equally symptomatic of the behavior of
other indicative interest rates in 1980. Sixmonth Treasury
Bills reached a record of 15.7% on March 24, declined to a
low of 6.6% at the June 16 Treasury auction, then climbed to
a new record of 15.42% on December 15, 1980. The late
Marchearly April period of 1980 registered historically
high interest rates on both short and longterm securities.
While interest rates have no doubt shown increased
volatility in recent years due to factors such as structural
changes in the financial markets, greater emphasis on
monetary policy as a means of implementing a countercyclical
stabilization policy and higher and more variable rates of
inflation, the greater volatility in 1980 than in previous
years is due in large part to change in the Federal
Reserve's operating strategy of focusing on the behavior of
bank reserve positions and, hence, money supply growth, with
less emphasis on interest rates. The new approach, formally
announced on October 6, 1979, coincides with the sharp
fluctuations in market interest rates. The period since
October 6, 1979, differs significantly from earlier periods
of comparable length. Monetary growth, interest rates, and
the economy have been notably more erratic. The interest
rate fluctuations since that new approach was announced are
unprecedented, at least since the War Between the States.
It is noted that while there appears to be positive
correlation between the movement of the prime rate and
longterm rates, the correlation is by no means perfect.
During the second quarter of 1980, shortterm interest rates
plunged 8 to 10 percent, while longterm rates fell 3 to
4 percent. As the prime rate becomes more volatile, it may
be losing its position as the sole base from which other
rates are scaled upward. As recently as a decade ago, rates
seemed to move in tandem with the prime rate. There is some
current evidence that there may be more "sidestepping" of
the prime rate than ever before. The prime rate does appear
to "lag" behind other short term interest rates and, as
already observed, longterm rates are no longer seen as
moving in synchronization with the prime
rate. Admittedly, there is evidence that the linkage
between consumer loans and the prime rate is deteriorating.
Despite these and other recent challenges, the movement
of the prime rate does reflect volatility in the period in
question and, as importantly, does have the characteristic
of high visibility. Therefore, this study will attempt to
utilize characteristics of the movement of the prime rate as
a proxy for uncertainty in financial markets.
The Concepts of Volatility and Stability
in U.S. Money Markets
The concept of absolute stability in money markets
suggests that the regression of interest rates on time will
be linear with slope and variance equal to zero. This
proposition is axiomatic to the discussion and analysis to
follow. Thus, over any given range of observations,
interest rates remain unchanged, each day's quote being
identical to that of the immediately preceding day's quote.
Any curvilinear relationship which would produce a
regression equation that accounted for all variability in
rates would suggest that rates move in a predictable manner,
a condition not supported by logic nor by observation. A
linear regression equation whose variance was indeed zero
but with slope different from zero would likewise imply a
dependable change or movement in rates. It is this
movement, whatever its course, direction, or dependability,
that is antithetical to the concept of absolute stability.
If, therefore, we can infer that absolute stability results
when there is no movement or change in interest rates,
volatility would then infer departure from this norm as
reflected by a variance and/or slope not equal to zero.
Consequently, it is suggested that volatility in money
markets can be characterized over any given range of
observations by departure from the axiom of absolute
stability, i.e., by a slope and/or variance not equal to
zero.
There is considerable intuitive appeal as well as
empirical evidence to support the notion that the
variability of inflation increases when inflation increases.
To the extent that interest rates embody inflation
expectations, higher average levels of interest rates might
also be associated with increased market uncertainty,
however stable the movement of interest rates over the given
range of observations.
Therefore, a third factor which is postulated to
characterize a money market over any given range of
observations is the level of interest rates. An absolutely
stable prime, holding at a historically high level, could
characterize a market possessing a greater degree of
uncertainty than would an absolutely stable prime holding at
some lower level.
Admittedly, many factors may influence a defined
market, making it unreasonable to ascribe all differences in
market behavior between periods to the existence of
uncertainty in the money market. However, to the extent
38
that we are able to define specific market behavior in terms
of uncertainty or volatility in the economy at large, we
will have advanced another step towards the better
understanding of our markets and the complex relationships
that bind them together with macroeconomic forces.
CHAPTER IV
METHODOLOGY
Statement of Objectives
The simplistic and naive assumptions of the classic
hedge as depicted in traditional hedging theory have been
demonstrated earlier. Most notable among these are (1) the
assumption that cash and futures markets are perfectly
positively correlated and (2) the assumption that the astute
hedger will utilize a riskminimizing hedge ratio, b*, of 1.
Recent empirical research (Ederington [1], Franckle and
Wurtzebach [3]) has shown that, over a given market period,
the degree of positive correlation between movement in cash
and futures markets, while strong, is in fact somewhat less
than perfect (b* not equal to 1) and the efficiency of the
hedge, e (estimated by the sample coefficient of
determination), is in fact less than 1.00. What this
previous research has neglected to consider in any detail
thus far is how the riskminimizing hedge ratio, b*, and the
measure of hedge efficiency, e, may be altered in different
market periods, periods which may be segmented by the degree
of uncertainty present in the money market. Franckle and
Wurtzebach [3] only touched on the possibilities. No other
known research has yet addressed this particular aspect of
hedge performance. Therefore, an initial objective of this
research will be to develop a methodology which will allow
observation of Ederington's b* and e over time. Such a
series will provide graphic evidence of any changes in hedge
performance which may be occurring as market conditions
change. By developing a series of coefficients descriptive
of the movement of the prime rate over comparable periods,
this research attempts to draw conclusions about the
performance of the hedge relative to uncertainty in the
money market.
Data Collection and Refinement
Sources
The raw data to construct the hypothetical market
transactions necessary to this study are obtained from data
tapes acquired through the Chicago Board of Trade
Foundation. The foundation was able to supply three
separate Chicago Board of Trade Commodity Futures Data Tapes
containing in the aggregate over 243,000 data lines of
information concerning daily open, high, low, and closing
futures prices, as well as other pertinent market
information. A subset of this master database, consisting
of all activity in the original GNMA 8 contract dating from
October 20, 1975, to December 31, 1980, and containing
approximately 13,000 data lines, was generated to produce
the first of two raw data files necessary to this study.
GNMA cash prices from January 1, 1976, to December 31,
1980, were researched from daily quotations published in The
Wall Street Journal. Bid, asked, and yield information was
recorded on a daily basis, creating a 1,346 line raw data
file from which cash prices could be obtained.
Refinement
It is noted that Ederington, in selecting futures
prices for constructing his hypothetical hedges, used
"weekly closing prices" [1, p. 164], presumably Friday's
closing price. Franckle and Wurtzebach also used a weekly
observation, Thursday's settlement price [3, p. 8]. While
the option to use either the closing price or the settlement
price and confine observations to any given weekday were
similarly available here, there was no perceived necessity
to limit observations to a given weekday nor was there
particular intuitive appeal in selecting either the closing
price or the settlement price. Since in practice a
commodity futures transaction can and likely would be made
at various and random times throughout the trading day and
week, this study has utilized the arithmetic mean between
high and low quotations as the hypothetical futures price
and has constructed observations for every day within the
selected range and not just a selected weekday.
Although Ederington presumably dropped paired
observations with missing data and Franckle and Wurtzebach
do not make clear their handling of missing data, this study
utilizes the following criteria: If a futures quotation for
a given day is necessary to either open a hedge or to close
a hedge and the quotation for that day is missing, as would
occur if that day were a holiday, then an appropriate
quotation is simulated by echoing the data from the
immediately preceding trading day. The only exception to
this rule occurs when a closing futures quotation is
required, the data are missing, and the immediately
preceding quote is of a contract not compatible with the
opening hedge data. In that rare instance, the closing
futures quotation is simulated by mirroring the immediately
following trading date.
These criteria for handling missing data points in the
futures quotes will occasionally result in sample hedges of
slightly shorter or longer duration than that originally
specified. The result is considered a more reasonable
reflection of what might actually occur in market trading
and in any case should have negligible impact on the
results.
In selecting cash prices for hypothetical hedges,
Ederington [1], Franckle and Wurtzebach [3], and Figlewski
[2] all utilize prices as quoted in The Wall Street Journal,
as does this study. Ederington apparently used the bid
price for GNMA 8's. The 8% coupon is normally the delivery
security for the futures contract, but different coupon
GNMA's can be substituted according to a set formula.
Figlewski and Franckle and Wurtzebach observed that during
the periods considered in their studies, the structure of
GNMA prices was such that it was nearly always more
advantageous to deliver the 9% coupon. Thus, Franckle and
Wurtzebach used the converted asked price of the GNMA 9%
Certificate. Figlewski used an average price. It is noted
that if bid prices are used for "sale" of inventory and
asked prices are used for "purchase" of inventory, there
would be a change in the price of the security without any
real change in the market rates. This study has elected to
use the arithmetic average of bid and asked in order to
obtain a cash quote.
The criterion for handling missing data in the cash
database is similar to that previously described for the
futures database. There were isolated instances when cash
quotes as reported in The Wall Street Journal were either
unreadable or differed so significantly from surrounding
figures that they were presumed to be misquotes. In those
instances as well as on holidays, the cash quote is
simulated using the immediately preceding day's quote.
Critical Parameters
Hedge Length
Initially Ederington and later Franckle and Wurtzebach
used only two and fourweek hedges when constructing their
sample observations. Ederington's choice was admittedly
arbitrary [1, p. 164]. Presumably, Franckle and Wurtzebach
selected similar lengths in order to compare their findings
more directly with Ederington's initial study. As a means
of more direct comparison, this research has likewise
developed hypothetical fourweek (28day) hedges, although
that is where the similarity ends. It has already been
noted how the handling of missing data and the selection of
both futures and closing quotations differ slightly from
earlier studies. It must also be noted that earlier studies
allowed futures trading to continue into the spot month,
which the researcher was reluctant to do. These subtle
differences in methodology between this and previous
research work to increase the possibility of divergent
results, although perhaps not significantly.
Following the logic of Raleigh [10] and others who have
constructed hypothetical hedges for the purpose of
demonstrating characteristics of the hedge, this study also
develops an 18week (126day) hedge.
Hedge Distance
With the fourweek (28day) hedge, this study will make
observations, as did the previous researchers, using futures
contracts three, six, nine, and twelve months distant.
Because there are only four annual delivery months available
to the hedger, it is obvious that not all hedges can be
initiated in contracts exactly x months distant. Therefore,
this research has constructed hypothetical hedges according
to the following criteria: If a 28day hedge in the nearby
or threemonth contract is originated in November, December,
or January, the March contract is utilized. Similarly, if a
hypothetical 28day hedge in the nearby or threemonth
contract is originated in February, March, or April, the
June contract is utilized. The logic is similar for all 28
and 126day hedges constructed in either the three, six,
nine, or twelvemonth contracts. In the extreme case, a
28day hedge in the threemonth or nearby contract initiated
on July 31 would close out on August 27, still prior to the
September spot or delivery month.
In the case of the 18week hedge, the three and six
months contracts are ignored and only the nine and
twelvemonth contracts are examined.
Range of Observations
The selection of the range of observations is one of
the more critical decisions in the study. Ederington chose
a twoyear frame [1, p. 164] to examine his b* and e,
presumably because this was the extent of "market experience
to date" [1, p. 163] and there was no compelling reason to
attempt to segment the market period. Franckle and
Wurtzebach also used market experience to date (197679) to
come up with their initial results but then, to compare more
directly with Ederington's effort, divided their
observations into two identical twoyear periods possessing
obviously differing characteristics with respect to interest
rate volatility. Once again, market experience to date
appeared to be the primary deciding factor for aggregating
1976 through 1979 data, and Ederington's twoyear
observations fit neatly into Franckle and Wurtzebach's
fouryear frame, providing two equal observation periods.
This study does not rely completely on the precedents
established by Ederington and by Franckle and Wurtzebach,
partly because the selection appears more expedient than
wellconceived.
A glance at money market activity over the first five
year operating history of the Chicago Board of Trade GNMA
futures market would indicate major market movements can be
roughly segmented into 18month intervals when a new degree
of uncertainty was actively influencing the market. For
instance, the steady rise from the market low of a 6%% prime
rate began on May 13, 1977, and reached an obviously stable
plateau on December 26, 1978; the severe dislocations in
late 1979 and 1980 can be traced to a beginning on June 19,
1979, when prime first moved following a sixmonth lethargy,
and a peak on December 19, 1980. The objective, then, is to
choose a range that will be short enough to capture the full
essence of welldefined market periods without the diluting
or excessive smoothing that would result from distant
historical observations and yet long enough to avoid
distorting aberrations and extremes. An 18month range is
considered adequate to accomplish these goals and yet remain
reasonable consistent with earlier efforts.
Measures of Hedging Performance
This study utilizes Ederington's b* and e to assess
hedge performance where b* is in effect the slope of the
linear regression line when the change in cash or spot price
is regressed on the change in futures price and e is the
sample coefficient of determination. The propriety of this
choice is dictated by the nature of this study. Admittedly,
more refined measures such as those suggested by Kolb,
Corgel and Chaing [8] or by Howard and D'Antonio [5] could
likely be incorporated into future, more sharply focused
efforts. For measuring the potential of these markets to
minimize price change risk, Ederington's measures have
considerable practical, intuitive appeal.
The Concept of a Moving Performance Parameter
It is noted that Ederington, in using "market
experience to date," selected hypothetical hedges from
within that defined period and drew conclusions about market
performance applicable to that period. The effect was to
stop or freeze attention on a single frame in time. If, for
instance, days were numbered beginning at some arbitrary
starting point, say, July 1, 1974, and Ederington's
observations and conclusions relate to a single twoyear
period ending with activity on December 31, 1977, he in
effect froze attention at frame number 1280. Rather than
allowing the range of observations to constantly expand to
"market experience to date," he could have on the following
day deleted an observation from January 1, 1976, and
simultaneously added an observation from January 1, 1978,
effectively freezing attention on frame number 1281.
Effectively, this is what Franckle and Wurtzebach do when
they discard the notion of "market experience to date" and
segment their observations into two welldefined market
periods, 197677 and 197879, thus using a twoyear range of
observations. Franckle and Wurtzebach effectively froze
attention at frames 1280 (December 31, 1977) and 2010
(December 31, 1979). What this study does is construct a
series of "moving" performance indicators, namely bt and et,
with each indicator computed in a manner not dissimilar to a
moving average.
As the name implies, a moving average is a series of
averages computed from the terms in a series by successively
dropping one old term and adding one new term for each
average. Moving averages tend to smooth or eliminate
erratic movements from the series, the degree of smoothing
being a function of the number of terms used in computing
the moving average. The primary use of such a smoothing
technique is to provide a pictorial description of the time
series secular trend. In this case, the selection of an
18month observation range determines the computed value for
b* and et similar to the way the twoyear observation range
did for Ederington and for Franckle and Wurtzebach. The
difference is that instead of selecting one frame for
viewing as did Ederington or even two frames as did Franckle
and Wurtzebach, this study computes b* and et for all such
"frames" beginning with frame number 1096, which is
descriptive of the period from January 1, 1976, through
June 30, 1977, and ending with frame number 2376,
descriptive of the period from July 1, 1979, through
December 31, 1980. The actual equations, then, for
computing the respective series are as follows:
T T T
[ mE (X X) (Y Y) E (X X) E (Y Y) ]
j=tm+l j=tm+l j=tm+l
t T T T T
[ m (X.X) ( (X.X))2] [ mE (Y.Y) ( E (Y i ))2]
j=tm+l j=tm+l j=tm+l j=tm+l
T
Z (X.X) (YY)
[ E (X.X))
j=tm+l
where M = range of observations,
t = 1097, 1098, . ., T
The observed values of the chosen performance indicators, b*
t
and et, are connected using a HewlettPackard Model 7221C
Digital Plotter to depict graphically the movement or
pattern of changes in the respective performance indicators
which, in the case of Franckle and Wurtzebach's
observations, showed significant change between market
periods.
Statistical Model
Bivariate Regression
The Model. The basis for the statistical model which
is used to examine characteristics of the performance of the
GNMA hedge over volatile money market eras is found in
bivariate regression, a straightforward technique which
involves fitting a line to a scatter of points. The
simplest such relationship between an independent variable
and a dependent variable is the straight line expressed by
equation (1). Because relationships in the social sciences
are almost always inexact, the simple equation for a linear
relationship is more realistically written as
Y. = a + bX. + e. (8)
1 1 1
where e represents the error term.
The familiar least square principle, in which the
optimal line is the one which minimizes the sum of the
squares of the error terms, is appropriate to apply in this
research. Lacking the resources to review all possible
hedges which could have been constructed during the period
from 1975 through 1980, a total that would surely be
astronomical, a stratified, systematic sample was selected
from the enormous, although finite, population of all
possible hedges. The sample is systematic in that the
hypothetical hedges constructed are assumed to be executed
at the midpoint of the range of each day's quotations. It
is stratified in that the population of all possible
hypothetical hedges is divided into strata consisting of
either 28day hedges or 126day hedges in contracts ranging
from three to twelve months distant at three month
intervals.
The statistic b in this model, the estimate of the
slope of the true population regression line, indicates the
estimated average change in the dependent variable Y
associated with a unit change in the independent variable X.
(A causal process is not established, however, by the
regression of Y on X and is more appropriately determined
outside the estimation procedure. Often, it is based on
theoretical considerations, good judgment, and past
research.)
The statistic a, the intercept, is of lesser interest
to this study. In the application this research is to make
of bivariate regression, this statistic should, as
traditional hedging theory suggests, be close to zero,
except for the effect of the convergence phenomenon.
Generally, social scientists stress explanation rather
than prediction. The research question at this point
suggests emphasis on the former. To examine how powerful an
explanation the bivariate regression model provides, a
formal measure, the coefficient of determination (R2), is
essential. This population coefficient of determination
(R2) is estimated by the sample coefficient of determination
(r2), the same measure referred to by Ederington as e and
assumed by traditional hedging theory to equal 1.00.
Assumptions and inferences. The basic assumptions of
the classical linear regression model, never properly
ignored or overlooked, suggest
1. no specification error (linearity): the
relationship between Y. and X. is linear;
1 1
2. no measurement error (nonstochastic X and Y): the
variables X. and Y. are accurately measured;
1 1
3. zero mean: the error term, ei, has the
characteristic that the sum of all the error terms
from 1 to n is equal to zero;
4. constant variance (homoscedasticity): the variance
of the error term is constant for all values of X.;
and
5. no autocorrelation: the error terms are
uncorrelated.
By adding the following assumption we have the
classical normal linear regression model.
6. Normality: the error term is normally distributed.
The assumption of linearity is common in postulating
relationships among social science variables. Although not
always correct, its adoption is at least a starting point
that might be justified on several grounds. First,
empirical research has found numerous such relationships
among social science variables to indeed be linear.
Secondly, theory has not suggested thus far what the
nonlinear specification would be. Most importantly,
inspection of the data in this case fails to suggest a clear
alternative to the linear model. While alert to the
possibility that our bivariate relationship may actually be
nonlinear, the assumption of linearity expressed in
Equation (8) is entirely plausable.
The importance of the second assumption is
selfevident. The researcher is satisfied that the data
supplied by the Chicago Board of Trade Foundation and
researched from the library microfilm are as accurate as can
be reasonably expected. Therefore, the possibility of
measurement error can, as a practical matter, be ruled out.
The remaining assumptions involve the error term. The
initial one, a zero mean, is of little concern because the
least square estimate of the slope will be unchanged
regardless. It is true that, if this assumption is not met,
the intercept estimate will be biased. However, since the
intercept estimate is of secondary interest in this
research, this potential source of bias is relatively
unimportant.
Violation of the assumption of homoscedasticity is more
serious. While the least squares estimates remain unbiased,
significance tests would be inaccurate. Diagnosis of
heteroscedasticity depends upon observation and analysis of
residuals. After visual analysis of scattergrams and review
of previous research, there is no reason to suspect a
condition of heteroscedasticity.
The assumption of no autocorrelation suggests that the
error corresponding to an observation is not correlated with
any of the errors for the other observations. When
autocorrelation is present, the least square parameters
estimates are still unbiased; however, the formulas used by
most regression computer programs are no longer appropriate.
Where there is evidence of positive autocorrelation, as is
often the case in political and economic data observed over
time, this means the estimated variances of a and b will be
seriously underestimated and the sample coefficient of
determination, critical to this and to previous research in
measuring the effectiveness of the GNMA hedge, will be
overstated. It is critically important to check for
violation of this assumption when using economic data
collected over time. Observation and tests conducted during
this research suggest strong positive autocorrelation
requiring corrective action.
The normality assumption, necessary only for the
statistical testing of the model, is not crucial as long as
one is willing to believe that individual errors due to
measurement and to omission are small and independent of
each other. Furthermore, the normality assumption can be
ignored when the sample size is large enough, for then the
Central Limit Theorem can be invoked.
Significance of violation of the autocorrelation
assumption. There is some disagreement in the statistical
literature over how serious the violations of the regression
assumptions actually are. Often, the question turns on
whether one's perspective on regression analysis is "robust"
or "fragile." The "robust" perspective on regression
suggests the parameter estimates are not significantly
influenced by violations of the regression assumptions. At
the other extreme, the "fragile" view of regression analysis
suggests violations of the assumption render the regression
results almost useless. Clearly, some of the assumptions
are more likely to be critical than others. The
consequences of violation of the nonautocorrelation
assumption are so serious in this case due to the impact
upon the estimate of the population coefficient of
determination (the measure of hedge effectiveness in this
and previous research), its possibility cannot be ignored.
Previous research (Ederington [1], Franckle and Wurtzebach
[3]) did not mention the possibility of autocorrelation of
the error terms in their regression equations, despite the
fact that they did indeed observe their economic data over
time, as does the research. Their failure to mention the
possibility and the likely effect on their findings may have
resulted from a "robust" perspective of regression in
general or from other considerations not readily apparent.
Nevertheless, evidence of violation of that assumption
prompts consideration of modification of the simple
bivariate equation.
Correction for autocorrelation. The focus on the
nonautocorrelation assumption is appropriate because of the
timeseries context implied by the research question. The
problem arises in a timeseries context because the
disturbances, which are a summary of a large number of
theoretically random factors that enter into the
relationship under study, are likely to carry over into
subsequent time periods.
Virtually all work in regression of this type assumes
that a first order autoregressive process generates the
disturbance. While not the only process possible, the first
order autoregressive process is often studied because of its
statistical tractibility and because it yields a crude
approximation to the process in which we are interested.
The equation for a first order autoregressive process is
et = Pet + vt
St1
where P is the coefficient of correlation between et and
etl and is the basic indicator of whether the
nonautocorrelation assumption is violated.
In correcting for evidence of positive autocorrelation,
one of several approaches available to the researcher, an
alternative identified as the CochraneOrcutt method [7],
was used. The procedure begins with the familiar bivariate
regression of Equation (8) where previous research has
apparently left off. The estimated residuals are used to
obtain a "first round" estimate of P which is defined as
^ ^ ^ ^ 2
P = Zee /Zet t=2,3,....,T
Transformed data points are constructed yielding "second
round" residuals which are then used to obtain a new
estimate of P. This iterative procedure is continued until
the values of the estimators converge. The resulting
transformed data points are free from the effects of
autocorrelation as is reflected by a P of approximately
zero.
The Multivariate Regression Model
The basis for the statistical model which is used to
examine the relationships between the characteristics of
performance of the hedge, bt and et, and the chosen
indicator of uncertainty in money markets, the prime rate,
is multiple regression. The elementary fourvariable case
which this research uses is written
Y = a0 + blX1 + b2X2 + b3X3 + e (9)
where
Y = a performance indicator, either b* or e
X1 = an uncertainty indicator
X = an uncertainty indicator
X = an uncertainty indicator
The bivariate regression assumptions are carried over
to the multivariate case. However, for multiple regression
to produce the best linear unbiased estimates, it must meet
one additional assumptionthe absence of perfect
multicollinearity. Observation of the plot of the
uncertainty indicators, to be developed shortly, does not
suggest multicollinearity in the independent variables.
Using the same range of observations thought
justifiable for examining the performance of the GNMA hedge,
performance characteristics have been developed for the
movement of the prime rate, the "proxy" for uncertainty in
today's money markets. A parameter called prime rate trend
(PRMTND) is, in fact, a series of coefficients which reflect
the slope of the regression line of prime rate on time for
each "frame" from 1097 to 2376. It is important to realize
that these coefficients reflect changes in the slope of the
regression line of the prime rate on time and not in the
prime rate itself. Otherwise, misinterpretation of the
resulting curve might occur. For instance, in early 1979 a
plot of these coefficients reveals the prime rate trend
decreasing when in fact the prime rate had only stabilized
after approximately 18 months of uninterrupted upward
movement. Thus, the rate of increase of the prime rate was
decreasing although the prime rate itself was not.
A parameter called prime rate volatility (PRMVOL) is a
series of variances associated with the mentioned regression
lines. Thus, prime rate could be rising steadily,
uniformly, and predictably, and prime rate volatility could
be approaching zero, for it takes uncertainty reflected by
undulating patterns to result in higher levels of
volatility.
Finally, a parameter called prime rate average (PRMAVE)
is a series of weighted averages which allow the insertion
of the general level of the prime rate into Equation (9).
In this case the relationships are not so obvious as in
the bivariate case. Therefore, we confront the null
hypothesis which states there is NO linear relationship
between the parameter and the performance of b* and e, as
opposed to the alternative hypothesis which states there IS
a linear relationship.
The main focus in the use of this technique as a
descriptive tool is the evaluation of measurement of overall
dependence of a variable, namely, b* or e, on a set of other
Figure 2.
Market Period Indices.
Coefficients Developed
Prime Rate, 19761981.
25
I
20 N
T
E
R
E
15 S
T
R
T
10 E
S
A Plot of Descriptive
from the Movement of the
61
variables, the uncertainty indicators of PRMTND, PRMVOL, and
PRMAVE.
CHAPTER V
ANALYSIS AND INTERPRETATION
The Performance of the 28Day Hedge
in the NearTerm Contract
Explanatory Results
From the 12,848 line database containing a daily record
of market activity of all GNMA futures contracts for
197680, inclusive, a total of 1,284 hypothetical beginning
hedge positions were observed according to the previously
defined selection criteria. This includes a hedge for every
business day during the period. Of this total,
approximately 3% are simulated. These hypothetical
beginning hedge positions provide the first of four input
1
variables, specifically Pf, necessary to construct the
hypothetical hedges requisite to this empirical observation.
Using the same 12,848 line master database of all GNMA
futures activity and the 1,284 line data base just created
containing beginning hedge positions, a matching 1,284 line
database containing hypothetical ending hedge positions was
2
spawned. The observations, corresponding to Pf, provide a
second input variable for the hypothetical hedge.
Similarly, approximately 3% of these observations were
simulated.
To construct the hypothetical beginning and ending cash
1 2
positions, P and P the database containing cash activity
s s
as reported daily in The Wall Street Journal was utilized
according to predefined selection criteria. Two 1,284 line
files were generated completing the selection of the
possible parameters of 1,284 hypothetical 28day hedges in
the nearterm contract.
A summary of these hypothetical hedge transactions was
then created by merging the previous four data files. This
summary file was embellished by the addition of the observed
basis change, expressed both fractionally, as the numerator
of a 32nd, and decimally. This latter observation
corresponds to the error, or residual, when the observed
changeinbasis differed from the predicted changeinbasis,
namely zero, where the regression coefficient is +1. A plot
of these residuals on time, smoothed by reducing them to
weekly averages, is reflected in Figure 3. Before
commenting on that display, some characteristics of this
particular series of hypothetical hedges are worthy of
comment.
Of the 1,284 hypothetical hedges observed,
approximately 6% contain simulated data points. These
observations, while admittedly differing slightly from the
established selection criteria, do closely approximate the
others.
250
200
150
100
50
P 0
0
I
N 50
T
S100
150
200
Figure 3.
Weekly Average Change in Basis GNIA8/CBOT/4 Week
Hedge/3 Month Contract, 19761981.
There were only 15 observed instances of the socalled
perfect hedge, slightly over 1% of the sample total.
There were surprisingly 127 observed instances, nearly
10% of the total, when cash and futures price movement
exhibited negative correlation rather than the expected
positive correlation. In these cases, the short (long)
hedger would have gained (lost) on both cash and futures
transactions on 29 occasions while losing (gaining) in both
markets on 98 occasions.
There were 316 observed instances, nearly 25% of the
sample total, when the short (long) hedger's cash position
improved (deteriorated) but the futures position
deteriorated (improved) more rapidly, producing a net loss
(gain); there were 375 observed instances, nearly 30% of the
sample total, when the short (long) hedger's cash position
deteriorated (improved) while the futures position improved
(deteriorated) less rapidly, thus failing (succeeding) in
covering (preserving) the hedger's cash market loss (gain).
Lastly, there were 168 observed instances, over 13% of
the sample total, where the short (long) hedger's cash
position improved (deteriorated) with the futures position
deteriorating (improving) less rapidly, thus preserving
(covering) the hedger's cash market gain (loss); there were
283 observed instances, over 22% of the sample total, where
the short (long) hedger's cash position deteriorated
(improved) with futures positions improving (deteriorating)
more rapidly, thus producing a net gain (loss).
Admittedly, any one hedger's net results from employing
a series of traditional hedges would be a function of the
magnitude of each transaction and the degree of the change.
Nevertheless, these observations provide insight into the
workings of the market over the 197680 period. From the
viewpoint of the classic short hedger, there were 1,269
opportunities (99% of the sample total) to hedge one's cash
inventory position and experience a disconcerting basis
change. On 702 occasions, over 55% of the sample total, the
futures position added to the yield on an unhedged position
or perfectly hedged the position. On 582 occasions, about
45% of the sample total, the futures position subtracted
from the yield on an unhedged position. The consideration
of transaction costs could be expected to narrow the
apparent margin favoring the hedge.
What is noteworthy is that the expected positive change
in the basis, observed by Ederington [1] to generally
characterize the market during the 197677 period and
attributed to the convergence phenomenon, has not subtracted
from the short hedger's performance opportunities with the
frequency that might have been expected.
Close examination of Figure 3, the plot of residuals on
time, suggests the dominance of the positive change in the
basis, clearly observed for 197677, may have been broken in
197880. Obvious from the plot is the increased volatility
in the performance of the classic traditional hedge in the
197880 period and particularly from October, 1979, through
the end of 1980. If all hedges were perfect, as traditional
theory suggests, the plot of residuals would be the
reference line with a slope of +1. But Figure 3 reveals
such is not the case nor can the variance or volatility over
time be considered constant. (This should not be construed
as suggesting the presence of heteroscedasticity, which is
best observed by a scattergram and not by a plot of
residuals over time. Increase in volatility in the
performance of the hedge may be caused by any combination of
relative change in cash or futures and is not necessarily
suggestive of large changes in futures or cash being
associated with large residuals. The large residual could
result when either the change in cash or futures is quite
small.)
Having constructed a sample of 1,284 hypothetical
hedges in the nearterm contract and examined some of the
characteristics of the performance of the hedge assuming
traditional hedging theory, it is appropriate to examine
the performance of the hedge given the remaining assumptions
of this research. This requires construction of a moving
546day observation range which begins at frame 1097 and
observes 390 hypothetical hedges that could have occurred in
the previous 546day period. After observing
characteristics of performance during that frame (namely, bt
and et), the moving observation range is stepped forward
one frame to repeat the process. This continues until
frame 2349, the last possible observation point for
28day hedges during the 197680 period. The result is 895
observation platforms on which to stand and examine hedge
characteristics over the immediately previous 18month
period.
A summary of these observations was produced utilizing
the data files generated earlier. With each of the 895
frames containing 390 overlapping observations, the summary
can be embellished with such descriptive statistics as the
sample regression coefficient (Ederington's b*) and the
sample coefficient of determination (Ederington's e).
Frame 1097 corresponds, if somewhat roughly, to
Ederington's observations for the 197677 period. His
observed measure of hedging efficiency for this period was
.785, whereas this study produced an et of .812. Franckle
and Wurtzebach [3], in examining the 197677 period,
produced an e of .871, explaining the discrepancy between
their findings and Ederington's results for the same period
by noting differences in data collection, a factor mentioned
earlier. This study's findings of .812 are not inconsistent
with earlier findings, given the varying techniques of data
collection and the other assumptions previously mentioned.
Ederington noted b* at .848 for this period, whereas
Franckle and Wurtzebach found .697, again attributing the
difference to data collection. Observation of the slope at
.907 in this research is compatible with Ederington's
findings.
Franckle and Wurtzebach provide one more opportunity
for comparison. In dividing their sample into two equal
periods, 197677 and 197879, they provide a second frame or
platform from which to observe. This platform corresponds
roughly to frame 2010. Franckle and Wurtzebach observed b*
and e at .936 and .898 for third period, whereas this
research found b* and e at frame 2010 to be .948 and .937.
t t
The last such frame observed in the case of the 28day hedge
is frame 2349.
TABLE 3
Summary of Current and Previous Findings
28Day Hedge in the NearTerm Contract
197677
Estimated Estimated
b* e
Ederington .848 .785
Franckle and Wurtzebach .697 .871
Earle (t=1097) .907 (.835) .812 (.614)
197879
Franckle and Wurtzebach .936 .898
Earle (t2010) .948 (.888) .937 (.802)
1980
Earle (t=2349) 1.032 (.860) .891 (.680)
Having made 893 additional observations of b* and e to
t t
supplement the two observations noted by Franckle and
Wurtzebach, they can now be plotted on a graph with time as
the xaxis and the coefficient of determination or the
regression coefficient on the Yaxis. The results are
depicted in Figures 4 and 5.
The plots in these figures reveal a pattern of
performance that can be attributed to the hedge given the
assumptions. Note the efficiency of the hedge (e) was
improving steadily at the time of the initial observation
(frame 1097) until the end of calendar 1977. Ederington [1]
suggested that continuing maturity of these fledgling
markets could result in increased efficiency. Note that
this period of increasing e corresponds to a period of
decreasing b* at a time when the market was exhibiting
consistent positive change in the basis. Therefore, as b*
was continually being lowered in a positive change
environment, e was predictably increasing. The
stabilization of hedge efficiency from early 1978 until mid
to late1979 may have reflected near optimal efficiency
under the given conditions. (The Federal Reserve action in
October, 1979, corresponds suspiciously with the dramatic
increase in efficiency indicated by the data in mid to
late1979, but this author discounts any significance in
this apparent relationship. More likely, the increased
efficiency results from a combination of a relatively low b*
and the elimination of the last vestiges of the positive
change in the basis characteristic which occurred about
early 1978. Thereafter, a new, higher level of market
1.1
1
D 1.0
j 
S0.9
I
I w
0.8
0.7
Figure 4.
I I I I I
DEC DEC DEC DEC DEC
Performance of Hedge Efficiency Index
GNMA8/CBOT/4 Week Hedge/3 Month Contract,
19761981.
DEC
1.50
H 1.25
E
D
G
E1.00
R
A 
T
I
0 0.75
0.50
DEC DEC DEC DEC DEC DEC
Figure 5. Performance of RiskMinimizing Hedge Ratio
GNMA8/CBOT/4 Week Hedge/3 Month Contract,
19761981.
efficiency dominated until the effects of the turbulent 1980
produced a sharp decline in that year.
These findings confirm Franckle and Wurtzebach's
conclusion for 197879 visavis 197677. The markets
apparently were more efficient in times of increasingly
higher interest rates (although causation is not necessarily
established). However, those conclusions may not be safely
projected beyond October, 1979, when volatility was combined
with overall high levels of interest rates.
The series of riskminimizing hedge ratios (b*)
exhibits a pattern of generally increasing values,
suggesting that the riskminimizing proportion hedged may,
over time, be approaching that suggested by traditional
theory. Indeed, as positive changeinbasis markets are
balanced by negative changeinbasis markets over long
periods of time, this can be anticipated. Observe the
correlation between the performance of the riskminimizing
hedge ratio and weekly average changeinbasis. Severe
negative changeinbasis (indicating futures are moving
slower than cash) is matched by increases in the
riskminimizing hedge ratio. In 1980, with the prime rate
falling during the first half of the year, futures prices
rose slower than cash prices resulting in a negative
changeinbasis for the first half of 1980. However, in the
second half of 1980 with the prime rate rising, futures
prices fell faster than cash resulting in a positive
changeinbasis for the second half of 1980. This explains
the leap in b* at about February, 1980, and the violent
negative spike in the weekly average changeinbasis at
about the same time.
The violent positive spike in the weekly average
changeinbasis occurring in mid1980 leveled off b* (not
reduced it) since it was having to "offset" the effects of
the immediately earlier opposite spike. Efficiency (e),
having leaped to a new level in October, holds until
mid1980 when return of a strong positive changeinbasis
condition (suggesting an immediately lower b*) catches b* at
high levels (because of the 18month range). The result is
a dramatic decrease in et.
Franckle and Wurtzebach [3] observed a sizeable
increase in b* between the 197677 period and the 197879
period. A declining b* as experienced from mid1977 to
mid1978 would suggest a more rapid change in futures prices
than in cash prices, consistent with the generally positive
changeinbasis characteristic of this period. However,
with erosion of the dominance of the generally positive
changeinbasis over 197880, it is noted that the b* has
indeed approached or even exceeded that stipulated by
traditional hedging theory.
Inferential Results
Multiple regression, using PRMVOL, PRMTND, and PRMAVE
as independent variables to predict the value of both b* and
e, is a technique apparently unexplored by previous
research. The results of an SPSS run to evaluate and
measure the overall dependence of b* on these three
variables is summarized in the following table:
TABLE 4
Selected Statistics for Multivariate Regression
Multiple R 0.92433 ANOVA DF SS F
R Square 0.85439 Regression 3 6.30608 1742.68940*
Std Error 0.03473 Residual 891 1.07472
Variable B Beta Std Error B F
PRMVOL 0.0272943 0.53319 0.00190 206.083*
PRMAVE 0.0600282 1.75613 0.00168 1273.127*
PRMTND 16.5203899 0.76201 0.73349 507.291*
Constant 0.4534595
*Significant at the .05 level.
Since the computed F value is larger than the
statistical tables value for a level of significance of .05
for 3 and 891 degrees of freedom, the observed linear
association is statistically significant, although not
necessarily causal. Similarly, each of the independent
variables exhibits a coefficient significantly different
from zero at the .05 level, implying a linear relationship
does exist between each of these independent variables and
the dependent variable, b*. Reference to the multiple r
leads to the initial conclusion that a significant
proportion (85.439%) of the variation in b* can be explained
by the independent variables.
Because of the different units of measure used for the
independent variables, reference to the beta values provides
a more sensible way to compare the relative effect on the
dependent variable by the independent variables. Other
things being equal, the betas indicate that one standard
deviation per unit change of PRMAVE would produce the
greatest change in b* and one unit change of PRMVOL would
produce the least.
The results of the SPSS run to evaluate and measure the
overall dependence of e on the three independent variables
are summarized in the following table:
TABLE 5
Selected Statistics for Multivariate Regression
Multiple R 0.80119 ANOVA DF SS F
R Square 0.64190 Regression 3 0.54765 532.38332*
Std Error 0.01852 Residual 891 0.30552
Variable B Beta Std Error B F
PRMVOL 0.0021678 0.12456 0.00101 4.573*
PRMAVE 0.0081887 0.70461 0.00090 83.339*
PRMTND 1.6598608 0.22519 0.39208 507.014*
Constant 0.8106573
*Significant at the .05 level.
The observed linear association is statistically
significant at the .05 level as are each of the partial
coefficients. The strength of the relationship as evidenced
by a multiple r2 of 64.190% is not nearly so strong as was
by a multiple r of 64.190% is not nearly so strong as was
the case with b*. Again the observed beta weights suggest
PRMVOL could be dropped as a predictor variable with little
2
impact on the multiple r
For purposes of comparing different market
environments, the 895 item data set was subdivided and
examined in separate parts. The first data set contained
the first 598 observations, roughly spanning the period from
1976 to 1978, whereas the second data set contained the
remaining 297 observations roughly corresponding to the
relatively volatile 197980 period. Selected results
follow:
TABLE 6
Multiple Coefficient of Determination (r2)
Dependent Variable b* e
Combined Subsets 0.85439 0.64190
Subset 1 (197678) 0.72711 0.33360
Subset 2 (197980) 0.89796 0.83220
What is notable is the significantly increased strength
of the relationships during the relatively volatile 197980
period as contrasted with the relatively benign 197678
period.
Detection of and Correction for Positive Serial Correlation
in the Data Observations
Detection. Spurred by the intuitive notion that the
data points collected and observed in previous research and
thus far in this study could likely be affected by serial
correlation, the researcher generated a plot of residuals by
an SPSS run on selected samples of hedges. The null
hypothesis was NO evidence of serial correlation (P=0)
whereas the alternative hypothesis was positive serial
correlation (P > 0). The decision rule was to reject the
null if the DurbinWatson dstatistic was less than 1.52.
The computed dstatistic of .48981 was indeed less than 1.52
suggesting significance at the .01 level. Therefore, the
null was rejected and the conclusion was evidence of
positive serial correlation. Initial P on selected samples
ranged from .45 to .80, further confirming the evidence of
strong positive autocorrelation in the data points.
Correction. Using the CochraneOrcutt technique [7]
through three iterations to eliminate positive
autocorrelation in the residuals produced transformed data
points that consistently reflected 0.03 < P < 0.03 where
any 0.20 < P < 0.20 is considered relatively free of serial
correlation in the error terms. (It should be noted that
from this point only every third data point was observed,
reducing the sample from 895 observations to 299
observations.)
The b* and e are recalculated using the transformed
data points thus obtained. The results are predictable and
are depicted in Figures 6 and 7.
We expect the intercept and slope to remain relatively
unchanged after the removal of positive autocorrelation in
the residuals. However, the r2's are predictably reduced
and greater exaggeration is obtained in the plot of the
efficiency index.
Note from Table 3 on page 69 the efficiency index of
.812 reported at t=1097 is reduced to .614. In fact,
Figure 6 indicates the entire family of efficiency indices
is noticeably modified, including the observations at t=2010
and t=2349 reported in Table 3. While the general
characteristics of the performance of the efficiency index
are recognizable, the efficiency index now operates over a
greater range with increased volatility and substantially
reduced magnitude.
1.0
0.9
08
0.7
O.R
0.5
I I I I I
DEC DEC DEC DEC DEC
DEC
Figure 6. Performance of Hedge Efficiency Index
GNMA8/CBOT/4 Week Hedge/3 Month Contract
(Autocorrelation Corrected), 19761981.
I,. *
.wI

I
1.50
H 1.25
E
D
G
E
1.00
I 
0 0.75
0.50
DEC DEC DEC DEC DEC
Figure 7. Performance of RiskMinimizing Hedge Ratio
GNMA8/CBOT/4 Week Hedge/3 Month Contract
(Autocorrelation Corrected), 19761981.
New multiple regressions utilizing the revised estimate
of b* are summarized below:

TABLE 7
Selected Statistics for Multivariate Regression
(Autocorrelation Corrected)
Multiple R 0.84184 ANOVA DF SS F
R Square 0.70869 Regression 3 0.69748 239.21789*
Std Error 0.03118 Residual 295 0.28671
Variable B Beta Std Error B F
PRMVOL 0.0017121 0.53151 0.00030 33.461*
PRMAVE 0.0342306 1.58710 0.00262 170.947*
PRMTND 0.0081647 0.59582 0.00114 51.162*
Constant 0.5893686
*Significant at the .05 level.
The table compares directly with Table 4 on page 75.
Note the strength of the relationship is weakened by the
effect of the autocorrelation in the original data.
The revised multiple regression run utilizing e as the
dependent variable produces the following:
TABLE 8
Selected Statistics for Multivariate Regression
(Autocorrelation Corrected)
Multiple R 0.39091 ANOVA DF SS F
R Square 0.15281 Regression 3 0.14354 17.73652*
Std. Error 0.05194 Residual 295 0.79583
Variable B Beta Std Error B F
PRMVOL 0.0009501 0.30192 0.00049 3.713
PRMAVE 0.0047759 0.22665 0.00436 1.199
PRMTND 0.0066992 0.50040 0.00190 12.409*
Constant 0.7475657
*Significant at the .05 level.
*Significant at the .05 level.
The table compares directly with Table 5 on page 76.
There is considerable loss in goodnessoffit as reflected
by a reduced r2 from 0.64190 to 0.15281. Although a linear
relationship may be said to exist, it is weak.
Selected summary statistics are as follows:
TABLE 9
Multiple Coefficient of Determination (r2)
Dependent Variable b* e
Combined subsets 0.70869 0.15281
Subset 1 (197678) 0.48698 0.46434
Subset 2 (197980) 0.88447 0.73295
This table may be compared directly with Table 6 on
page 77. The pattern is similar to the one obtained with
the original data points. The relationship between b* and e
and the three independent variables, although weaker,
appears much stronger in the latter, more volatile period.
The Performance of the Hedge Under
Selected Alternative Conditions
The Performance of the 28Day Hedge
in the Distant Contracts
As did Ederington [1], this effort has constructed and
observed hypothetical hedges in the socalled distant
contracts, those six, nine, and twelve months removed from
the originating date of the hedge. The criterion for
selecting the hedge parameters is identical to that observed
for the nearterm or threemonth contract.
In the case of the 28day hedge, a total of 1,284
hypothetical transactions were constructed in each of the
respective contracts. The measures of hedge performance, b*
t
and et, are developed similar to the previous case.
Table 10 summarizes from the viewpoint of the classic
short hedger certain characteristics of the performance of
the 28day hedge in the various contracts, including the
nearterm contract.
There is an apparent increasing incidence of unexpected
negative correlation between the movement of cash and
futures prices as one hedges in increasingly distant
TABLE 10
Summary of Selected Hedge Characteristics from the
Viewpoint of the Classic Short Hedger
28Day Hedge, All Contracts
Category Description 28Day 28Day 28Day 28Day
(+) Adds to the Yield Hedge Hedge Hedge Hedge
of an Unhedged Position 3Mo 6Mo 9Mo 12Mo
() Subtracts from the Yield Cntrct Cntrct Cntrct Cntrct
of an Unhedged Position
Observed Instances of Both
Cash and Futures Positions
Improving (+) 29 36 46 48
Observed Instances of Both
Cash and Futures Positions
Deteriorating () 98 86 92 108
Observed Instances of the
Perfect Hedge 15 11 11 10
Observed Instances of Cash
Position Improving with
Futures Position
Deteriorating Faster () 316 305 272 245
Observed Instances of Cash
Position Deteriorating
with Futures Position
Improving Slower (+) 375 366 358 360
Observed Instances of Cash
Position Improving with
Futures Position
Deteriorating Slower () 168 176 200 224
Observed Instances of Cash
Position Deteriorating
with Futures Position
Improving Faster (+) 283 304 305 289
contracts. As observed earlier, there were 127 such
occurrences in the nearterm contract, comprising
approximately 9.9% of the sample total. This instance
increases in the ninemonth and twelvemonth contracts to
138 (10.7%) and 156 (12.2%), respectively. A confidence
interval on the first proportion suggests the statement that
the true proportion of negatively correlated hedges is
between 8.3% and 11.5% is correct with a probability of
.95. Thus, there appears to be a significant increase in
the proportion of negatively correlated hedges in the most
distant contract. The puzzling dominance of instances in
this set where cash prices are decreasing while futures
prices are increasing appears uniform across all contracts.
The observed instance of the socalled "perfect" hedge
appears consistently in the range of 0.8% to 1.2% of the
sample total.
Note the decreasing incidence of futures market
deterioration (increasing futures prices) being sufficient
to override cash market improvement (increasing cash prices)
as one hedges in the more distant contracts. It was
observed in the analysis of the positive/negative
changeinbasis characteristics of the market that when cash
prices rise, as they may be expected to do when interest
rates are falling, futures prices may rise more slowly
suggesting longterm pessimism on the part of the market and
generating negative changeinbasis characteristics. For a
classic short hedger, increasing cash prices would reflect
improvement in the cash position, while increasing futures
prices would reflect deterioration in the futures position.
Thus, on the basis of trend alone, this decreasing incidence
of futures market deterioration being sufficient to override
cash market improvement as one hedges in increasingly
distant contracts may suggest the short hedger would be well
advised to hedge in the distant contracts.
As observed earlier, there were 316 such occurrences in
the nearterm contract, comprising approximately 24.6% of
the sample total. This instance decreases in the sixmonth,
ninemonth, and twelvemonth contracts to 305 (23.8%), 277
(21.2%), and 245 (19.1%), respectively. A confidence
interval on the first proportion suggests the statement that
the true proportion of such hedges is between 22.3% and
27.0% is correct with a probability of .95. There appears,
then, to be a significant difference in the proportions
experienced in the more distant contracts.
This result is not inconsistent with the conclusion
that might be drawn from examination of the other
significant set where the short hedger's yields are reduced
by futures market participation. Note the increasing
incidence of futures market deterioration (increasing
futures prices) being insufficient to override cash market
improvement (increasing cash prices) as one hedges in the
more distant contracts. As observed earlier, there were 168
such occurrences in the nearterm contract, comprising
