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Quantifying the probability of default as assessed by the bond market

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Quantifying the probability of default as assessed by the bond market an analysis of default risk measures
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Bond market
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Broske, Mary Stearns, 1944-
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Bond rating ( jstor )
Default risk ( jstor )
Financial bonds ( jstor )
Financial investments ( jstor )
Financial portfolios ( jstor )
Government bonds ( jstor )
Investment horizon ( jstor )
Investment risks ( jstor )
Investors ( jstor )
Ratings ( jstor )
Bonds ( lcsh )
Default (Finance) ( lcsh )
Risk ( lcsh )
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bibliography ( marcgt )
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non-fiction ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1982.
Bibliography:
Includes bibliographical references (leaves 115-120).
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Typescript.
General Note:
Vita.
Statement of Responsibility:
by Mary Stearns Broske.

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QUANTIFYING THE PROBABILITY OF DEFAULT
AS ASSESSED BY THE BOND MARKET;
AN ANALYSIS OF DEFAULT RISK MEASURES











BY

MARY STEARNS BROKE


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















To my parents, Mr. and Mrs. George L, Stearns, Sr., in

gratitude for their unfailing support, confidence and

encouragement throughout my life, and to my husband,

Ernest, and daughter, Elizabeth, in gratitude for their

support and many sacrifices throughout the course of

this study.















ACKNOWLEDGMENTS


The guidance and direction of Professor Haim Levy,

chairman of my dissertation committee, and the many

helpful comments of the other members, Professor Moshe

Ben-Horim and Professor Rashad Abdel-khalik, are grate-

fully acknowledged. In addition, financial support for

this research from the Center for Econometrics and

Decision Sciences, University of Florida, is gratefully

acknowledged.


iii

















TABLE OF CONTENTS


PAGE


ACKNOWLEDGMENTS ... .

LIST OF TABLES ..

LIST OF FIGURES. .

ABSTRACT .

CHAPTER

I INTRODUCTION. .

II REVIEW OF THE LITERATURE. .

Decomposition of Bond Yields and
of Bond Prices .
Bond Risk and Return: CAPM and
Bonds .
Bond Ratings and Bond Risk .
Prediction of Bankruptcy and of
Default. .
Conventional Ranking Techniques:
A Review .

III PRESENTATION OF THE MODEL .


Introduction .
The Definition of Default .
Measuring the Probability of
Default .
An Algorithm for Calculating the
Probability of Default When There
Are N Intersections Between the
Cumulative Probability Distributions
Numerical Examples .. .

IV METHODOLOGY AND RESULTS .. .

Introduction. .
Methodology .
Description of Data and Presentation
of Empirical Results .
Analysis of Results. .


. iii

........ vi

. viii


. 6

S 10
S 15

S 23

S 26

S 32


. 32
S. 36

S. 37




. 43
. 47

. 50

. 50
* 52

* 56
* 62











CHAPTER


V SUMMARY AND CONCLUSIONS .

Introduction. .
Summary of Results. .
Proposals for Future Research .
Portfolio Implications Research.
Efficient Set Research .
Bond Market Efficiency Research.
Early Warning System Research. .
Bond Rating Accuracy Research. .
Other Researcn .
Conclusions .

APPENDICES

A A COMPARISON OF MOODY'S AND S&P'S
CORPORATE BOND RATING DESCRIPTIONS


B A COMPARISON OF MOODY'S AND S&P'S
CORPORATE BOND RATING DETERMINATIONS 100

C REVIEW OF STOCHASTIC DOMINANCE RULES. 104

D SIX MONTH TOTAL HOLDING PERIOD RETURNS
CALCULATED FOR PORTFOLIOS 108

E CHANGES IN UNADJUSTED GNP USED TO CONVERT
NOMINAL SIX MONTH TOTAL HOLDING PERIOD
RETURNS TO REAL SIX MONTH TOTAL HOLDING
PERIOD RETURNS ... 113

REFERENCES .. 115

BIOGRAPHICAL SKETCH. ... 121


PAGE















LIST OF TABLES


TABLE PAGE

1 Bond Ratings and Default Experience:
1900-1943. .... 25

2 Bond Rating and Default Experience
by Decade; 1920-1939. .. 25

3 Relative Probabilities of Default of
Corporate Bonds When the Investment
Horizon is Six Months (nominal data). ... 60

4 Relative Probabilities of Default of
Corporate Bonds When the Investment
Horizon is Six Months (real data). .. 60

5 Relative Probabilities of Default of
Corporate Bonds and Government Bonds
When the Investment Horizon is Ten
Years. .. 61

6 Relative Probabilities of Default of
Corporate Bonds When the Investment
Horizon is Twenty Years (1971-1980) 63

7 Relative Probabilities of Default of
Corporate Bonds When the Investment
Horizon is Twenty Years (1969-1980) 63

8 Additivity of Relative Probabilities
of Default as Demonstrated by Table
5 Data .. 68

9 The Impact of the Investment Horizon
on the Magnitude of the Relative
Probability of Default of Baa
Corporate Bonds Over Aaa Corporate
Bonds. .. 72

10 Analyze Yield-to-Maturity Data for
Sensitivity of Delta to Change
in D1 . 76











11 Analysis of the Variable D1 for Data
Used in Calculating Delta as
Presented in Table 7 and in Table 10 78

12 Statistical Analysis of the Distributions
of D1 in Order to Determine Whether Table
11(a) is Not Significantly Different
From Table 11(b) . 80

13 Values of Statistical Tests for
Equality of Means and of Variances
Within Bond Rating Categories When
D is Calculated for Bonds Grouped
in Portfolios and for Individual
Bonds. . .. ... 80


vii


TABLE


PAGE
















LIST OF FIGURES


FIGURE

1 A Comparison of Two Securities;
The Treynor Index. .

2 A Comparison of Two Securities:
The Sharpe Index .

3 Shift in Aaa Distribution
(Case i: Default Results in
Zero Return) .

4 Shift in Aaa Distribution
(Case 2: Default Results in
Return Less Than Promised Return).


5 Cumulative Probability Distributions
for Example 2 Alternatives C and D
Demonstrating Shift in D to D' When
Implied Delta is Included in D .


viii


PAGE


* 9 .


........ .....


. 49















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


QUANTIFYING THE PROBABILITY OF DEFAULT
AS ASSESSED BY THE BOND MARKET:
AN ANALYSIS OF DEFAULT RISK MEASURES

By

Mary Stearns Broske

August, 1982

Chairman: Professor Haim Levy
Major Department: Finance, Insurance and Real Estate

Bond rating agencies classify corporate bonds and state

and municipal bonds into different categories according to

the agencies' assessment that the issuer will default on

the bond prior to maturity. Bond ratings rank bonds according

to the agencies' assessment of the bonds' relative proba-

bilities of default. If the market is efficient and is in

equilibrium, the market price of a bond is a function of the

risk incurred by ownership of that bond. There are two types

of risk inherent in bonds: (a) interest rate risk (vari-

ability risk) and (b) default risk.

This study proposes a technique for quantifying default

risk (type (b) risk) as estimated by the bond market. The

technique developed involves comparing cumulative probability

distributions of rates of return (or of yields-to-maturity)

of bonds which differ only in default risk once interest










rate risk is neutralized. The empirical observation that

corporate bonds Csay Aaa) dominate government bonds by First

Degree or by Second Degree Stochastic Dominance cannot hold

in equilibrium, for Aaa bonds possess default risk and

government bonds do not. The technique proposed in this

paper involves systematically changing the Aaa bond dis-

tribution until the dominance disappears. This inclusion

of the implied probability of default results in quantifying

the probability of default as assessed by the bond market.

The probabilities of default as presented in this study

were consistently in the direction predicted by economic

theory, as follows: (1) the magnitude of default probability

varied inversely with the quality of the bond (indicated by

the bond rating), (2) the magnitude of default probability

for a given rating category was consistently larger in

periods of economic contraction than in periods of economic

expansion, and (3) the magnitude of default probability

strictly increased with the length of the investment

horizon.














CHAPTER I
INTRODUCTION



The required return on any risky asset contains a

premium to compensate the holder of that asset for the

risk incurred by the ownership of the asset. The esti-

mation of the risk premium has been a subject of research

in both economics and finance, and is of interest in theory

and in practice.

There exist several commonly accepted and widely used

models of the relationship between the expected return (in

equilibrium, this is identical with the required return) and

the risk inherent in the ownership of risky assets. This

body of literature evolved from the study of stocks and how

they are priced. By extension, existing valuation theory is

assumed to apply to all risky assets. The Capital Asset

Pricing Model, or CAPM (Sharpe, 1964; Lintner, 1965; Mossin,

1966), suggests that the relationship between risk and

expected return is linear, and that only systematic risk,

and not total risk (variance), is relevant to investors.

Assets are priced so as to compensate investors for this

systematic risk. Since the remaining portion of total risk

is diversifiable risk, and the marginal investor is assumed

to hold a diversified portfolio, the expected return contains










only a premium for the non-diversifiable risk. The CAPM

assumes that investors have von Neumann-Morgenstern utility

functions, and are concerned with the first two moments only

and that the distributions are normal. Empirical tests of

the CAPM have produced mixed results. In general, when the

CAPM is tested on stocks, the relationship is linear but the

intercept is higher and the slope is less than theory predicts.

The Arbitrage Pricing Model, or APM (Ross, 1976), repre-

sents an attempt to avoid Roll's (1977) criticisms of the

CAPM in that the market portfolio plays no part in deter-

mining the expected return on a given risky asset. Rather,

the expected return is a linear function of economy-wide

factors, the levels of which are estimated by well-diversified

investors, times a beta-factor for the sensitivity of the

security to that factor. Ross himself suggests that valida-

tion of the APM is empirically intractable since it is not

possible to identify all the factors.

Stock valuation models are even less accurate when used

to evaluate risky bonds. This suggests that risk for bonds

may be multi-dimensional rather than linear in nature. The

measures of bond risk suggested in the literature are uni-

dimensional in that they are assumed to be of equal signifi-

cance for all risky bonds. The horizon (the holding period)

is of significance in bond valuation and implies that any

attempt to develop one risk measure (or one group or cluster

of risk measures) appropriate for bonds of different









maturities held for different holding periods may be inap-

propriate. There are two types of risk inherent in bonds:

(a) variability risk and (b) default risk.

Bond rating agencies provide an estimate of the quality

of most large, publicly held corporate and municipal and

state governmental bond issues by assigning ratings to these

issues, and by revising these ratings when the agency's

estimate of the quality changes. Bond ratings are assigned

primarily on the basis of the agency's assessment of the

probability that the firm will default, and are intended to

rank bond issues according to their relative probabilities

of default. Rating agencies state that there is no set

formula for determining the rating, but rather that all

available information about the firm and the issue are

considered. The relationship of the bond rating to the

probability of default has been analyzed in the literature.

In general, the quality of bonds as indicated by bond ratings

has been determined to be inversely related to the proba-

bility of default as indicated by the frequency of default.

The purpose of this paper is to estimate the probability

of default as assessed by the bond market. In order to

quantify the relative probabilities of default of bond

rating categories, a stochastic dominance measure of the

probability of default is derived. The measure is applied

to distributions of yields-to-maturity and of holding period

returns for investment grade corporate bonds and for govern-

ment bonds, and the resulting relative probabilities of










default are analyzed. The stochastic dominance measure

developed in this study is proposed as a technique for

analyzing the manner in which the bond market estimates

relative probabilities of default.

Chapter II presents an overview of the literature

concerned with bond risk and its measurement. Both those

studies which developed predictors of default and those

which analyzed bond ratings as predictors of default are

reviewed in this section.

Chapter III includes the development of the model, some

simple examples to illustrate its application, and the

derivation of an algorithm for implementing the model in

the general case.

Chapter IV presents the methodology employed and the

empirical results. It includes a description of the data

followed by an analysis of the data for the full period of

the study and for sub-periods of economic expansion and of

economic contraction.

Chapter V presents the summary of the study and the

conclusions to be drawn from it. In general, the results of

the empirical work indicate that the stochastic dominance

technique developed in this study has economic validity.

The resulting estimates of the relative probabilities of

default were consistently in the right direction in that the

estimates were correctly related to the level of economic

activity. The estimates were in general larger in periods






5


of economic expansion than they were in periods of economic

contraction. In addition, the magnitude of the estimates

increased with the length of the investment horizon, as one

would expect.














CHAPTER II
REVIEW OF THE LITERATURE



Decomposition of Bond Yields
and of Bond Prices


Economic theory suggests that the yield on a bond, as

on any risky asset, should consist of the real rate of

interest, a premium to compensate for expected inflation,

and a risk premium to compensate for any other non-

diversifiable risk inherent in ownership of the asset (Hicks,

1939; Keynes, 1930; Lutz, 1940). In general, there are four

suggested components of the yield on a bond: (1) the pure,

or certain, rate of interest which reflects the underlying

dynamics of the economy; (2) a premium for credit risk,

which is the risk of defaulting on either the payment of

interest or principal; (3) a premium for purchasing power

risk, which is the risk of a decline in the purchasing power

of the interest and principal payments; and (4) a premium

for interest rate risk, which is the risk of an increase in

the market rate while the bond is held (Levy and Sarnat,

1972, pp. 104-114; Foster, 1978, pp. 444-445). These are

not derived from a developed theory of bond pricing, but

rather they have been developed to explain empirically

observed differences in yields of different bonds at a










point in time, or in yields of the same bond at different

points in time (Foster, 1978, pp. 444-445).

The classic empirical work on the determinants of the

bond risk premium is that of Fisher (1959). He defined the

risk premium as the difference in yield between a corporate

bond and a government security of the same term to maturity.

The risk premium was hypothesized to have two components--a

default premium and a marketability premium. The risk of

default was associated with three variables; (1) the coef-

ficient of variation of the firm's earnings (net income)

over the last nine years; (2) the length of time that the

firm has operated without creditors having suffered a loss;

and (3) the ratio of the market value of equity to the par

value of debt. The marketability of the bond was estimated

by the market value of all the publicly traded bonds that

the firm had outstanding. This was assumed to proxy

transaction frequency.

Fisher found that the logarithms of the four variables

accounted for approximately 75 percent of the variance in

the logarithm of the risk premium. Unlike others, he found

that the logarithm regression coefficients were relatively

stable over time. As is often the case with earlier empir-

icists, the model that Fisher developed was ad hoc in

nature. The variables were selected based on Fisher's

hypothesis that they had explanatory power, rather than

selected according to any existing theory of the components

of the risk premium.









Hastie (1972) employed Fisher's methodology in studying

municipal bond yields. He also suggested the existence of a

default premium and a marketability premium, but he esti-

mated them differently. The default premium was estimated

by (1) the ratio of overall debt to true property values,

(2) default history, (3) economic diversification, and

(4) college students as a percentage of the issuer's popula-

tion. The marketability premium was estimated by (1) the

size of the block offered, (2) the net debt of the issuer,

and (3) the past population growth. He found that the

relative significance of the variables depended on whether

commercial banks or individuals dominated the market. His

regression coefficients explained about 86 percent of the

variability in the risk premium, but the coefficients were

not stable over time and were sample-sensitive at a given

point in time.

Hastie's model is also ad hoc in nature. Although he

suggests that his inclusion of a default premium and a

marketability premium is theoretically based, a closer

analysis reveals that his work is based on his hypotheses

about what the rational investor would like or dislike.

There is no explanation offered as to why some variables are

selected and other possibilities are omitted.

An alternative approach to bond valuation is exempli-

fied by Silvers (1973) who investigated the determinants of

the bond price, rather than the risk premium. He tested the

relationship between bond price and the following independent









variables: (1) a vector of coupon certainty equivalent

coefficients, (2) a call variable, and (3) a marketability

measure. He sorted his sample by bond rating category, and

concluded that all the independent variables except market-

ability were significant.

A current study by Boardman and McEnally (1981) read-

dresses the issue of the bond yield (or premium) which

concerned Fisher and Hastie, and the issue of bond price

determinants which was addressed by Silvers. The study

decomposes corporate bond prices into components repre-

senting (1) the pure price of time, (2) the default risk of

the bond's agency rating class, and (3) bond-unique risk.

The authors, however, in analyzing bond price determinants,

do so by adding variables so that up to 45 variables enter

into each of the 16 versions of the price equation. This

study is also ad hoc in that variables are added because

they are thought to be of relevance in predicting bond price.

The study is neither based on an existing theory of bond

valuation nor does it attempt to develop such a theory.

Rather, in the absence of theory, it attempts to analyze the

significance of a large number of variables which were

included either in order to replicate earlier studies or

in order to add variables the authors thought missing in

previous work.

Thus, attempts to identify the components of the risk

premium or the components of bond prices in general seem to

involve researchers selecting a set of variables which are









thought to be significant, then testing to see if they are

indeed significant. All such studies are subject to the

criticism that they are not based on theory. Even though

it is common for an author to state economic interest rate

theory as the basis for his research, a closer analysis

indicates that the selection of the variables to be used is

ad hoc.



Bond Risk and Return: CAPM and Bonds


There is not a well-developed theory of bond pricing

under uncertainty which can be said to correspond to the

CAPM. In theory, since the CAPM is an expression of the

risk and return relationship for any risky asset, it is appli-

cable to bonds although it is more commonly applied to stocks.

The linear relationship between risk and return which

is expressed by the CAPM has been tested both on stocks (for

example, Douglas, 1969; Lintner, 1965; Miller and Scholes,

1972; and Black, Jensen and Scholes, 1972) and on bonds

(Percival [1974] did the initial work, followed by Reilly

and Joehnk [1976], and Friend, Westerfield and Granito

[1978]).

In general, the empirical tests using stock data show a

linear relationship, although the intercept is higher than

the observed risk-free rate and tne slope is less than theory

would predict. In addition, there is some evidence that the

standard deviation in some cases, or the residual variance










in others, is significant. Thus, in general, the theoreti-

cal model does not explain the empirical evidence in a

satisfactory manner.

When the CAPM is applied to bonds, the results are even

weaker, as a survey of the literature indicates. The initial

attempt to apply the CAPM to bonds was done by Percival

(1974) for the period 1953-1967. Betas were estimated over

the full period, and realized returns were calculated as the

geometric mean holding-period yields. When the regression


Ri = a + bBi + ei


was run, the regression results were as follows: R2 = .15,

a = .0389, b = -.00931. The t-value was -5.629. Percival

explained the negative sign as resulting from generally

rising interest rates during the period (Percival, 1974,

p. 464). He then added dummy variables for industry and for

bond rating. The regression results then were R2 = .4680,

a = .0370, and b = .00983. The b-coefficient and the bond

rating coefficients were all significant at the a > .05

level. Finally, he analyzed beta as a function of the

industry, the rating, the coupon rate, and the term to

maturity, getting R2 = .3345, with intercept = 1.19605.

The significant variables at a > .05 level were railroad

industry, coupon rate, and maturity (Percival, 1974, p. 464).

Percival concluded that bond betas are measures of

interest rate risk, but that they must be combined with a









non-diversifiable default-risk measure in order to explain

realized bond returns (Percival, 1974, pp. 464-468).

Reilly and Joehnk (1976) assumed that a market-derived

beta for a bond should be an appropriate risk measure for

bonds, and that this beta should be inversely related to bond

ratings. Since they assumed that the CAPM should relate to

bonds, they did not actually test it. Their resulting study

of bond ratings and risk measures, however, provides infor-

mation about the use of the CAPM on corporate bonds. The

CAPM assumes that beta risk is the relevant risk, whereas

Reilly and Joehnk's results indicate that relevant risk for

bonds is total risk (ai2). They found that the bond rating

contained useful information for the pricing of bonds, and

as bond rating was a better indicator of total risk than of

market risk, then by extension, total risk has importance

for bond pricing.

Yawitz and Marshall (1977) applied the CAPM to the

government bond market. They used the excess return form of

the CAPM, and found in every case that the intercept was

positive and statistically significant, and that both beta

and sigma had about equal explanatory power. (They were not

included in the same regression at the same time.) R2 ranged

from about 63 and 77 when beta was used to about 70 and 76

when sigma was used as the risk measure (Yawitz and Marshall,

1977, p. 20). Their finding of the equivalent strength of

market risk and total dispersion of return (a) as a risk

measure indicates that the CAPM is not satisfactory for bonds.









Finally, Friend, Westerfield and Granito (1978) tested

the CAPM on corporate bond returns as part of a comprehen-

sive re-testing of the CAPM by incorporating bonds into the

market index. The Sharpe-Lintner version of the CAPM yielded

the following regression results for corporate bonds:

R2 = -.001, a = 1.016, b = .001. When a second regression

was run, including both beta and the standard deviation of

the residual term, the results were as follows: R2 = .013,

a = 1.017, b = .001. None of the risk measures was statis-

tically significant for individual bonds (Friend, Westerfield

and Granito, 1978, p. 910). It should be noted that they did

not use bond ratings as an explanatory variable. Hence, in

conclusion, none of the tests of the CAPM using bond data

has validated the use of that model in explaining the rele-

vant risk for bonds. Rather, the evidence seems to indicate

that market risk is not the only relevant risk for investors

in bonds. The relationship of risk to return in bonds does

not seem to be linear. Historical studies of bonds have

always indicated a non-linear relation. McCallum's (1975)

study of Canadian government bonds (1948-1968) indicates

that the total holding period return was a non-linear func-

tion of maturity, standard deviation, and beta (McCallum,

1975, Tables 1, 2, 3).

Further work needs to be done on the nature of risk for

bonds, especially for bonds of different quality. None of

the above studies separated bonds by rating, for instance,










before applying the CAPM. Such a procedure would indicate

whether the intercept, slope(s), significance, and R2's are

different for different grades of bonds. This would indi-

cate whether the nature of risk is different for different

bonds.

Recent studies suggest that the estimates of the CAPM's

beta depend on the assumed investment horizon, as systematic

risk is a function of the length of the horizon for stocks

(Levy, 1981), and for stocks and bonds (Kaufman, 1980).

These authors conclude that the failure of prior tests of

the CAPM on stock data and on bonds may be explained by the

omission of the investment horizon. Kaufman offers this

as the explanation of why the CAPM is more successful in

pricing equities than it is in pricing bonds (Kaufman, 1980,

p. 1). Levhari and Levy's (1978) finding that the systemat-

ic risk of aggressive stocks (0j > 1) increases with the

investment horizon and the systematic risk of defensive

stocks (3j < 1) decreases with the horizon led Levy to

re-examine the results of the classic empirical tests of the

CAPM in order to see whether the effect of the assumed invest-

ment horizon could explain the documented poor results (Levy,

1981, p. 37). The results were not much different from those

of the earlier tests, but the empirical results varied with

the assumed horizon (Levy, 1981, p. 38). These studies

cited suggest that it is necessary to consider the investment

horizon when analyzing risk and return for stocks and,

perhaps even more importantly, for bonds.










As it is the purpose of this paper to analyze existing

risk measures and to apply Stochastic Dominance Criteria in

quantifying default risk, then it is appropriate to consider

bond ratings as they relate to bond risk.



Bond Ratings and Bond Risk


Bond rating agencies provide an estimate of the quality

of most large, publicly held corporate and municipal and

state governmental bond issues by assigning ratings to

these issues, and by revising the assigned ratings when

the agency's estimate of the quality changes. The two pri-

mary rating agencies are (1) Moody's and (2) Standard and

Poor's. Their rating assignments and descriptions of the

rating categories are available by subscription, and are

published weekly and monthly. A description of Moody's

bond ratings and a comparable description of Standard and

Poor's bond ratings are displayed in Appendix A. In addi-

tion, bond rating assignments and changes in ratings are

reported in The Wall Street Journal.

Bond rating services, such as Moody's and Standard and

Poor's, assign ratings to bond issues primarily on the basis

of the agency's assessment of the probability that the firm

will default. A partial description of the considerations

which enter into the bond rating is presented in Appendix B.

Bond ratings rank issues in order of the probability of

default. Thus, a rating of Aaa (Moody) or AAA (Standard









and Poor) is assigned to bonds having a negligible proba-

bility of default. The second category is Aa or AA (Moody

and S&P, respectively), followed by A (for both) and Baa

or BBB (Moody and S&P, respectively). Bonds rated below

this last category are considered speculative, and are not

considered to be investment quality. Only issues of the

Federal government are assumed to have no risk of default,

as the Congress has the authority to issue money to settle

its debt.

Bond rating agencies state that there is no set formula

for determining the bond rating, but rather that all avail-

able information about the firm is considered. However,

there seem to be criteria in common use by all bond analysts

in setting the rating (Cohen, Zinbarg and Zeikel, 1977,

p. 388). These criteria are as follows (pp. 389-390):


1. Protective provisions of the issue

2. Ratio analyses

a. Fixed charge coverage
b. Long-term debt to equity
c. Liquidity position, both current and projected

3. Other considerations

a. Size and economic significance of the firm
b. Economic significance of the industry


Perhaps one-half of all bonds are rated identically by

different agencies. Where there are differences, they are

usually not greater than one category (Cohen, Zinbarg and

Zeikel, 1977, p. 385).









The function of bond rating agencies in an efficient

market is not clear, for there are many empirical studies

which succeed in predicting bond ratings on the basis of

publicly available information. Kaplan and Urwitz (1979)

provide a thorough survey of statistical models of bond

ratings. They then develop and test a linear model using

a dummy variable for subordination and variables for total

assets, long-term debt to total assets ratio, and the common

stock's beta and standard error of residuals. Their model

correctly predicted the rating (Moody's) for about two-

thirds of the new issues studied, and the errors in predic-

tion were no more than one rating category away from the

actual rating category assigned by the bond rating agency.

In addition, their results led them to conclude that their

model may be predicting the actual riskiness of bonds (based

on calculated market yield of these bonds) better than the

bond rating prediction ot this risk in the case of some of

the misclassified bonds (Kaplan and Urwitz, 1979, p. 256).

Their finding that market risk (0) is significant in pre-

dicting ratings on new issues, whereas unsystematic risk as

measured by the estimated standard error of residuals in the

market model used to estimate beta is significant in pre-

dicting ratings on seasoned issues (issues which have been

traded for awhile in the secondary market), lends logical

support to an hypothesis that the bond rating itself may be

contributing to the magnitude of the risk premium. Perhaps










this is an indication of bond market inefficiency. There

is no economic reason that on the average new bond issues

should contain a premium for systematic risk, but not for

non-systematic risk, whereas seasoned issues should contain

a premium for non-systematic risk. Of course, it should be

noted that the systematic risk measure is the stock beta

and the non-systematic risk measure is the estimated standard

error of residuals resulting from estimating the stock beta.

Nevertheless, these variables were determined to be signif-

icant in predicting bond ratings, with the stock beta having

significance for new issues and the estimated standard error

or residuals for seasoned issues.

Another approach to the study of bond ratings and bond

market efficiency is exemplified by research analyzing bond

price changes around the date of bond rating changes.

Weinstein (1977), Katz (1974), Grier and Katz (1976), and

Hettenhouse and Sartoris (1976) represent this body of

literature. Weinstein is the only one of these authors to

find some evidence of price changes anticipating rating

changes. He found some indication of price changes occurring

from eighteen months to seven months prior to the announce-

ment of the rating change. He found no evidence of price

change during the six months prior to the rating change,

and little or no price reaction during the month of the

change or for six months thereafter (Weinstein, 1977, p. 342).

The earlier studies cited above found no evidence that the










market anticipates bond ratings changes, and instead report

evidence indicating that bond prices adjust to the announce-

ment of a change in rating. Katz (1974), for example,

reports that there is on average a lag of six to ten weeks

in the price adjustment process (Katz, 1974, p. 558).

Weinstein concludes that his findings indicate that the

bond market is semi-strong efficiently and that the price

change that he found prior to the rating change results from

information that eventually leads to a change in the rating.

In conclusion, the question of whether the bond market is

efficient is not conclusively answered in the literature.

Also, the question of whether the bond rating has value in

an efficient market, given that bond rating can be predicted

to a certain degree, needs further research.

The relationship between bond rating and bond risk has

been studied empirically. The initial investigation of the

relationship of bond rating to market-determined bond risk

was that of Reilly and Joehnk (1976). They estimated several

market-derived betas by using various proxies for the market,

and selected Moody's Average Corporate Bond Yield Series as

the best for their purposes. Capital market theory influ-

enced their hypothesis that the market-derived beta for a

bond should be an appropriate risk measure for bonds, and



1Semi-strong market efficiency implies that no investor
can earn excess returns by using trading rules based on
publicly available information. The definition appears in
most basic finance textbooks, and credit is given to Eugene
Fama (1970) for developing and operationalizing the idea.









that this beta should be inversely related to bond ratings

(Reilly and Joehnk, 1976, p. 1389). They refer to the

empirical relationship between accounting-determined risk

measures, based on internal corporate variables, and the

market-determined risk measure, 3. They cite studies by

Beaver, Kettler and Scholes (1970), Logue and Merville

(1972), Breen and Lerner (1973), and Gonedes (1973) as

documenting the relationship (Reilly and Joehnk, 1976,

p. 1388). They reasoned that it bond ratings are related

to some of the same corporate variables as are bond betas,

then there should be an inverse relationship between bond

betas and bond ratings.

They performed several tests of the association between

bond betas and bond ratings and hypothesized a significant

difference in the betas for different rating classes. Their

results did not consistently support the hypothesized signif-

icant and negative relationships between bond betas and

bond ratings. In general, bond betas for adjacent rating

classes (i.e., Aaa versus Aa) were not significantly dif-

ferent. In addition, even when the hypothesized differences

were significant, the differences were in the wrong direc-

tion. These results remained as stated above even when the

rating categories were widely separated (Aaa versus Baa).

They concluded that bond betas were not consistently related

to bond rating. They did not investigate the possibility

that the strength of the relationship between bond betas and










bond ratings may differ from one bond rating category to

another.

Reilly and Joehnk in the same paper also investigated

the relationship between bond rating and total risk (defined

as the standard deviation of monthly percent price changes).

The total risk measures for industrial bonds were all in the

predicted order. The Aaa measure was the smallest, and each

successive rating class average was larger, as predicted.

The risk measures for the top three classes were signifi-

cantly lower than for the lowest class (Baa). Thus, there

is an inverse relationship between bond ratings and total

risk, as defined by Reilly and Joehnk. However, they did

not address the question of whether the strength of this

relationship varies with the bond rating category.

Another study which contains information about the

relationship between bond ratings and bond risk, as repre-

sented by B and by a, is Friend, Westerfield and Granito

(1978). In testing the CAPM based on bond returns, it was

necessary for them to estimate beta and sigma so as to rank

the bonds by beta (Bi) decile and then by residual standard

deviation (ori) decile within each beta decile. The expected

values of Bi and Cri were estimated from regressions of these

measures of risk on the bond's S&P quality rating, its

maturity, and its coupon rate. Although the study did not

address the question of the relationship between bond rating

and bond risk measures, the results of the regressions










contain useful information about the question. The regres-

sion results were as follows:


B. = .28Q(1) + .23Q(2) + .23Q(3) + .19Q(4) + .30Q(5)
S (11.1) (10.5) (10.6) (6.4) (7.1)

+ .23Q(6) + .47Q(7) .06C + .01M
(4.8) (11.8) (-.38) (7.6)

R2 = .13

Ori = .03Q(1) + .03Q(2) + .03Q(3) + .05Q(4) + .06Q(5)
(11.1) (13.5) (14.3) (15.3) (12.6)

+ .09Q(6) + .08Q(7) .12C + .0004M
(17.2) (19.4) (-7.2) (5.4)

R2 = .33


where Q(1) is the highest and Q(7) the lowest S&P quality

rating, C is the coupon rate, and M is the years to maturity;

the numbers in parentheses are the t-statistics of the

regression coefficients, and R2 is the coefficient of deter-

mination adjusted for degrees of freedom (Friend, Westerfield

and Granito, 1978, p. 912).

The coefficients for the bond rating dummy variables

are all statistically significant in both regressions. The

interesting question is whether they are significantly dif-

ferent from one another in their contributions to the level

of the dependent variables. There is an Analysis of Covar-

iance technique discussed in McNeil, Kelly and McNeil (1975,

pp. 237-239) for determining whether the group membership

(in this case, bond rating category) has additional explana-

tory power on the level of the dependent variable over and










above any effects attributable to initial differences in

the covariables (in this case, coupon rate and years to

maturity). This technique is applicable to the above two

regressions and would require only that they be re-run with-

out any of the dummy variables being included. The pro-

cedure can also be done after removing the dummy variables

one at a time in order to isolate the relative contribution

of each rating to the dependent variable, holding constant

the effect of coupon rate and time to maturity. Such an

analysis should indicate the type of risk which is reflected

in the different bond rating categories. In this study,

sigma is not total risk; rather, sigma is the standard

deviation of the residual.

The literature reviewed suggests that the relevant risk

for high-grade bonds may differ from the relevant risk for

low-grade bonds. Further study is needed to ascertain

whether total risk or systematic risk is relevant for bonds,

and whether the nature of the relevant risk varies with the

bond rating. As this paper is concerned with default risk,

it is appropriate to consider the relationship between bond

ratings and bankruptcy and between bond ratings and default

risk as presented in the literature.



Prediction of Bankruptcy and of Default


The prediction of bankruptcy from publicly available

information has also been documented, and is primarily










associated with Altman (1968, 1971 and 1977). Multiple

discriminant analysis is utilized to develop a bankruptcy

classification model which uses financial statement data and

market data. The most recent paper (1977), which is repre-

sented as demonstrating significant improvement over earlier

models, presents the ZETA bankruptcy identification model.

The ZETA model had a range of prediction accuracy from 96

percent one period prior to bankruptcy to 70 percent five

annual reporting periods earlier (Altman, Haldeman and

Narayan, 1977, p. 50).

The relationship of bond rating to the probability of

default has also been analyzed empirically. The classic

study of the relationship of bond rating to the frequency

of default is that of Hickman (1958). He collected exten-

sive data on default experience for all large outstanding

issues during the period 1900 to 1943. His results are

summarized in Table 1, and indicate that in the period

studied, the probability of default (as indicated by the

occurrence of default) is inversely related to the quality

of the bond as reflected in the bond rating. Hickman (1960)

reported the number of defaults by bond rating broken down

by decade for the twenties and thirties and for the two

decades together. Table 2 presents Hickman's results as

reported by Pye (1974). Pye concluded that Hickman's data

indicates that almost all the spread between Aaa and Baa

bonds in the twenties and thirties would have to be due to

a default premium. He defined the default premium as the












Table 1. Bond Ratings and Default Experience: 1900-1943

Bond Comparable Moody's % Defaulting Prior
Rating Rating to Maturity

I Aaa 5.9%

II Aa 6.0

III A 13.4

IV Baa 19.1

V-IX Below Baa 42.4

Source: W. Braddock Hickman, Corporate Bond Quality and
Investor Experience, Princeton, N.J.: Princeton
University Press, 1958, Table 1, p. 10.


Table 2. Bond Rating and Default Experience by Decade:
1920-1939

Bond Period
Rating 1920-1929 1930-1939 1920-1939

I .12% .42% .3%

II .17 .44 .3

III .20 1.94 1.1

IV .80 3.78 2.3

Source: W. Braddock Hickman, Statistical Measures of
Corporate Bond Financing Since 1900 (1960) cited
in Pye (1974).









difference between the yield (the coupon rate, or promised

return) and the expected return (the yield-to-maturity)

(Pye, 1974, p. 49). Pye analyzed default occurrences in

the fifties and sixties and found that default experience

on investment grade bonds was quite different than for the

earlier period studied by Hickman. Pye found virtually no

incidence of default in the fifties ana sixties for bonds

rated Baa or better. Incidentally, he noted that Lockheed

was rated Baa in 1960, but did not default because Congress

intervened (Pye, 1974, p. 52). Pye concluded that for the

post-war period, the probability of default is so small as

to be insignificant; thus the premium on low grade bonds

when compared with high grade bonds is a risk premium. It

is Pye's opinion that the post-war default experience should

continue into the future (Pye, 1974, p. 52). This is an

empirical question and remains to be answered in relation to

the observed differences in bond ratings.



Conventional Ranking Techniques: A Review


Because bond valuation theory is not as well developed

as is stock valuation theory, the bond investor has fewer

and less sophisticated methods available for making the

investment decision than does an individual wishing to

invest in stocks. Models for the independent estimation of

what the equilibrium price on a stock should be are more

accurate than are techniques for the equivalent estimation









of the equilibrium price on a bond. This may be due to the

relative lack of quantifiable measures of risk for bonds.

As a result, there have evolved techniques which may be

used for ranking bonds, thus enabling the bond investor to

select the preferred bond, or bonds, from the set under

consideration.

The reward-to-risk measures of Jensen, of Treynor, and

of Sharpe can be used to rank bonds, thus enabling the

investor to select the preferred bond, or bonds, from the

set under consideration. These measures are similar, and

have in common the goal of reducing the risk-return evalua-

tion of investment performance to a single measure.2

Treynor (1965) suggests that the relationship of excess

return to nondiversifiable risk is an indication of the

performance of either a security or a portfolio of secur-

ities. The Treynor Index is stated as follows:

Exi RF

i=

where

Exi = the expected return on the security or portfolio

Si = the systematic risk of the security or portfolio

RF = the risk-free rate



2Levy and Sarnat (1972, p. 480) cite Friend and Blume
(1970) as providing the formal relationship of the three
performance measures we shall consider (Jensen, Treynor,
and Sharpe).









This index relates the excess return above the risk-free

rate earned by the security or portfolio to its systematic

risk, and thereby assumes that the market prices only system-

atic risk. If one is comparing securities characterized

by the risk premium being a function of the systematic risk

only, then the Treynor Index has value as a tool for ranking

securities.

An alternative to Treynor's index is Sharpe's (1966)

reward-to-variability measure, which is as follows:


Exi RF
I =
"i

where

ai = the standard deviation of returns on the security.


Sharpe's index was intended to be used as an indicator

of the performance of portfolios in general and of mutual

funds in particular. Since capital market theory presumes

that in equilibrium risky assets are priced so as to com-

pensate investors for the asset's systematic risk, but not

for its non-systematic risk, Sharpe's index normally is not

applied to individual securities or to inefficient port-

folios. The denominator, ai, is the square root of the

variance of returns, or the total risk of the asset. One

may justify applying sharpe's index to individual bonds on

the basis that (1) the precise nature of bond risk is not

uniformly agreed upon in the literature and (2) it may be










that systematic risk is the only relevant risk for some

bonds, whereas non-systematic risk, or perhaps total risk,

is significant in the pricing of other bonds.

The Treynor index and the Sharpe index are the slopes

of transformation lines connecting the risk-free rate and a

point representing the risk-return characteristics of the

given security. A graphical demonstration ot the use of the

Treynor index in comparing two securities, A and B, is pre-

sented in Figure 1. A comparable demonstration for the

Sharpe index is shown in Figure 2. The greater is the

magnitude of the index calculated for a given security or

portfolio, the greater is the level of expected utility

attainable by a risk-averse investor in that security. This

interpretation is presented by Levy and Sarnat (1972, p. 482)

for Sharpe's reward-to-variability ratio. We apply it to

Treynor's index also, as the indexes differ only in the

risk measure used in the denominator.

Whereas Treynor and Sharpe devised ratios to indicate

the excess return to non-diversifiable risk and the excess

return to total risk respectively, Jensen's (1968) Abnormal

Performance Index is based on the excess return, and is not

a ratio. It is expressed as follows:


ai = (Ri Rf) Bi(Rm Rf)

where


R = the return on the market.
m









E(Ri)


E(xA)



E(x,) -
E(xB)


RF


Ex, R F
aA =
A' &


Ex. RF
8 = BP


Figure 1.


A Comparison of Two Securities:


Tne Treynor Index


E(Ri)

E(xA) -

E(x,) -


R


Ex RF
A OA


Ex. -R
aB


Figure 2. A Comparison of Two Securities: The Sharpe Index









Jensen's index is applied to either individual secur-

ities or to portfolios of mutual funds, and is based on the

Capital Asset Pricing Model. A positive value for ai

indicates that after adjusting for risk, and for movements

in the market index, the abnormal performance of the

security or portfolio is also positive.

Although these measures are commonly used to rank bonds

or portfolios, their dependence on the assumptions of and

the validity of the underlying model (the Capital Asset

Pricing Model) precludes the inclusion of them as methods

of ranking bonds. In addition, the well-documented problem

of beta instability, especially as it related to bonds

(Weinstein, 1981), adds further validity to the decision not

to include these measures as ranking techniques. The tech-

niques examined in this study, bond ratings and the Stochastic

Dominance method, are free from dependence on a model, and

directly concern themselves with the probability of default.

Only the prevalence of the Sharpe, Treynor, and Jensen

indexes in practice justifies their inclusion in this survey

of ranking techniques.














CHAPTER III
PRESENTATION OF THE MODEL



Introduction


As stated, the purpose of this study is to estimate the

probability of default as assessed by the bond market. In

order to achieve this goal, the following assumptions are

made. Investors, who are assumed to be risk-averters, are

faced with the choice of investing SI either in Government

bonds or in corporate bonds. For purpose of analysis,

assume these corporate bonds are rated Aaa by the bond

rating agency. Denote the cumulative probability distri-

bution of the rates of return on an investment in the Govern-

ment bonds by FG(x) and the cumulative probability distribution

of the rates ot return on an investment in the Aaa bonds by

FAaa(X), where x is the rate of return. The expected return

on the investment of $I is a function of the risk to which

the investor is exposed. As the investor is a risk-averter,

he/she will require a higher rate of return for exposure to

a higher level of risk. For any given holding period (e.g.

one month, one year, etc.), there are two main components

included in the risk involved in each investment: (a) risk

arising from possible changes in interest rates over the

holding period, and (b) risk arising from the probability









that the issuer may default on the bond prior to the end of

the holding period. As the purpose of this study is to

quantify only default risk, it is necessary to neutralize

type (a) risK. This is accomplished by holding maturity

(or, ideally duration) constant when comparing the cumula-

tive probability distributions of rates of return or the

two types of bonds. This point is discussed further in the

presentation of the methodology in Chapter IV.

If type (a) risk is held constant, the risk-averse

investor will pay a higher price for the default-free Govern-

ment bond than tor the Aaa corporate bond. The lower price

for the Aaa corporate bond implies that the holding period

rates of return on this investment will be higher than will

be the comparable holding period rates of return on an

investment in the Government bond.

On an ex-ante basis, neither FG(x) nor FAaa(X) should

be expected to dominate the other by Second Degree Stochastic

Dominance (SSD) once the default risk is compensated for in

the required rate of return. (Stochastic Dominance rules

are reviewed in Appendix C.) However, the data consist of

ex-post rates of return, and include only bonds of firms

which did not default. As a result, on an ex-post basis,

FAaa(X) is expected to dominate FG(x) by SSD, for an invest-

ment in Aaa bonds exposes the investor to default risk,

whereas investment in Government bonds does not. There are

two possible states (6) which accompany an investment in

corporate bonds:










81 no default, the state wherein the investor obtains

an observation drawn from FAaa(x) as observed in

the past.

82 default, the state wherein the investor receives

either a zero return or some compensation, the

magnitude of which depends on the severity of

the default.


As this study utilizes ex-post data, it includes only

firms which did not default during the period covered by the

data. The ex-post data, as they include only firms which

did not default, consider only state 81, and state 82 is not

represented explicitly in the data. As stated, ex-post data

are expected to reflect tne SSD dominance of FAaa(x) over

FG(x). As a result, it is possible to derive trom distri-

bution FAaa(x) a new distribution FAaa(x) which assigns some

probability to the occurrence of state 92. We can change

the magnitude of this probability until neither FG(x) nor

FAaa(x) dominates the other by SSD. This probability of

state 02 which results in neither distribution dominating

the other by SSD is the risk of default of the Aaa bond as

assessed by the bond market. Assuming that the market is

efficient and is in equilibrium, neither FG(x) nor F' (x)
G Aaa
is expected to dominate the other by SSD.

In this study, ex-post data will be used to estimate

the premium required by bondholders in the past for the

probability of default in the future. As a result, the









conclusions will pertain to the probability of default which

was assessed by the market in the past. Of course, if addi-

tional information regarding the future becomes available,

the probability of default in the future may change. Thus,

the assumption is not made that ex-post distributions are

necessarily stable or that they represent ex-ante distribu-

tions. Rather, ex-post data are assumed to incorporate the

market's assessment at a given point in time of the possi-

bility of default in the future.

It is reasonable to assume that the probability of

default as assessed by the market may vary from year to year.

For practical reasons (and due to statistical limitations)

default risk as attributed to each and every year will not

be measured. Rather, the study distinguishes between years

of economic recession and prosperity, with the expectation

that the derived probability of default will be larger in

years of economic recession or contraction than in years of

economic prosperity or stability.

Finally, before moving to the derivation of and illustra-

tions of the use of this technique, it should be noted that

this analysis can logically be extended to the portfolio

setting under quite general conditions (refer to Kroll, 1981,

for the necessary framework). Obviously in practice the

investor may diversify either bond with other assets rather

than be limited to the assumption that he/she buys either

the Aaa corporate bond or the government bond.









The Definition of Default


In a one-period setting with no taxes, the holder of a

bond which matures at the end of that period is promised a

return of (1 + C), where C is the coupon rate. Risk for

the bondholder is the probability of realizing a return less

than the promised return as a result of the firm defaulting

on the bond agreement. Default occurs when the firm has

generated earnings before interest and taxes (EBIT) less

than the principal and interest legally owed to the bond-

holder at maturity. There exists a distribution of possible

levels of EBIT, only one of which will be the outcome at the

end of the period. Thus, there are two equivalent statements

of the default risk inherent in the ownership of a bond:


Default risk = Pr[(l+r) < (1+C)] (1)


where r is the realized rate of return on the bond and

alternatively,


Default risk = Pr[EBIT < (1 + C)B] (2)


where B is the face value of the bond.

The differences among bonds in the probability of

default should be reflected in the market value of the

bonds and hence in the distributions of returns (and dis-

tributions of EBIT) when bonds of different rating categories

are compared.









Measuring the Probability of Default


Define two firms, F and G, which differ in their distri-

butions of EBIT such that firm G has the greater probability

of default. Let 6 represent the greater probability of

default inherent in firm G's EBIT distribution. If the

bonds of F and of G are matched in all respects except for

the probability of default, it is possible to derive that

value of 6 such that when it is incorporated in G's EBIT

distribution, an investor (with a utility function of a

given class of utility functions) will derive greater or

equal expected utility from investing in F's bonds when

compared with G's bonds. In the marginal case, the investor

would be exactly indifferent between selecting F and selecting

G as an investment.

The Stochastic Dominance Criteria when applied to two

distributions insure that if F dominates G by the given

degree of stochastic dominance (first, second or third

degree), then all investors with a utility function which

is a member of the associated set of utility functions

(first degree, u' > 0; second degree, u' > 0, u' < 0; third

degree, u' > 0, u'' < 0, u''' > 0 will gain greater or equal

expected utility from investing in F. The Stochastic

Dominance Criteria are based on the von Neumann-Morgenstern

axioms. If these axioms hold, and as a result F is preferred

to G, then it follows that the expected utility of F is

greater than the expected utility of G. Thus, we can state









the relationship between expected utility and stochastic

dominance as follows:


EFU(x) > EGU(x) <==> F dominates G (or FDG)


As a result of this analysis, we justify applying stochastic

dominance as a technique for quantifying the probability of

default.

In deriving 6, we shall consider two cases:


Case 1:

Case 2:


The return to

follows:


Default results in zero return.

Default results in a return greater
than zero but less than the promised
return of (1 + C).


the bondholder in Case 1, (1 + r), is as


(1 + r) =


0

(1 + C)


The return to the bondholder in Case

EBIT is defined as Y, is as follows:


if default

if no default


2, (1 + r)B, where


(1 + r)B =


0
(0) / Yf(Y)dY +
-CO


(1+C)B

0


(1 + C)B


if
Yf(Y)dY default

if no
default


As we have defined 6 as the probability of default which

when incorporated in the risky distribution would make the

marginal risk-averse investor indifferent between F and G,

we can make equivalent the expected outcome from investing









in F and from investing in G. For this derivation, we shall

use Aaa bonds as representing the investment with default risk

and government bonds (GOVT) as. the default-free investment.

As this study considers only bonds which did not default

we expect FAaa to dominate FG and that FAaa will thus be pre-

ferred by all risk-averters (and maybe by all investors).

As a result, we expect to find

x
I [FG(t) FAaa(t)]dt > 0


for all values of x. Such a result implies that all risk-

averse investors would be better off by investing in FAaa

rather than in FG. However, this conclusion is incorrect, as

corporate bonds are exposed to default risk whereas government

bonds are not. Thus, we change FAaa by incorporating the

market estimate of default until

x
I [FG(t) FA'(t)]dt

is negative for at least one value of x, that is, until the

dominance exactly disappears (where Fa is the FAaa distri-

bution revised into incorporate default risk). That value of

6 which causes the dominance to disappear is the market esti-

mate of the risk of default.

In many cases, we observe empirically that the two dis-

tributions FG and FAaa intersect only once (or do not inter-

sect at all), FAaa intersects FG from below, and the mean

return of FAaa is greater than the mean return of FG. In this

specific case, every risk-averse investor would prefer FAaa









over FG if and only if EAaa(x) > EG(x). So, we change the dis-

tribution FAaa and hence the mean of FAaa until the dominance

disappears.

To be more precise, in case I above, the expected return

on the investment in Aaa bonds, once the probability of default

is incorporated, is

6(0) + (l-6)EAaa(x)

where EAaa(x) is the expected value of the distribution of

rates of return on the Aaa bond. We set this equal to the ex-

pected return on the government bond [EGOVT(x)], then solve

for 6:

6(0) + (l-6)EAaa(x) = EGOVT(x)
SEGOVT(x)
EAaa(x)
A graphical illustration of the Aaa' distribution which re-

sults from including 6 is displayed in Figure 3. The inclu-

sion of 6 probability of default in the Aaa distribution has

resulted in a shift in Aaa's cumulative probability distri-

bution such that the revised Aaa distribution and the GOVT

distribution intersect at 6. As a result, whereas initially

Aaa D GOVT by FSD, the inclusion of 6 has caused dominance to

reverse so that GOVT D Aaa' by SSD. The cumulative difference-

in the areas under the two curves is exactly zero, and there

is no value of x for which it is negative. Thus, GOVT exactly

dominates Aaa' by SSD.

In Case 2, the expected value of an investment in Aaa

bonds, once the probability of default is incorporated, is









Cum P(x)



GOVT Ac / aac








(I+C)B- x

Figure 3. Shift in Aaa Distribution
(Case 1: Default Results in Zero Return)


Cum P(x)



GOVT AaA aa







(I+C)B-
Figure 4. Shift in Aaa Distribution
(Case 2: Default Results in Return
Less Than Promisea Return)






42

0 (1+C)B-
6[(0) f xf(x)dx + / xf(x)dx] + (1 6)EAaa(x)
--0 0


We set this equal to the expected value of the government

bond, then solve for 6.

0 (l+C)B~
6[(0) / xf(x)dx + / xf(x)dx] + (1-6)EAaa(x) = EGOVT(x)
-0 0

EGOVT(x) EAaa(x)
S 0 (1+C)B (4)
(0) f xf(x)dx + / xf(x)dx EAaa(x)
-00 0


A graphical illustration of the shift in the Aaa distribu-

tion which results from including 6 is displayed in Figure 4.

In this case also, the inclusion of 6 causes the Aaa cumula-

tive probability distribution to shitt so that it intersects

with the GOVT distribution at 6. Again, the cumulative

difference in the two distributions is zero, and there is

no value of x for which the cumulative difference is nega-

tive. Whereas initially Aaa dominated GOVT by FSD, domi-

nance has reversed by including 6 so that GOVT exactly

dominates Aaa' by SSD. The manner in which we derived 6

assures that SSD dominance will result, that dominance will

have reversed, and that the cumulative difference will be

exactly zero. These results follow as a mathematical

necessity from our solving for the precise value of 5 that

would, when included in the risky distribution, result in

the expected outcomes being exactly identical.

To this point, the graphical analysis has been limited

to those cases where there is no more than one point of









intersection between the two cumulative distributions. It

is necessary to extend the work to include the derivation

of 6 in the general case of n-points of intersection. The

derivation of an algorithm for the general case is presented

below, and is based on Levy and Kroll's (1979) observation

that in the case where the cumulative distributions are

discrete it is only necessary to check the points of inter-

section when testing for stochastic dominance (Levy and

Kroll, 1979, p. 126).



An Algorithm for Calculating the Probability of Default
When There Are N Intersections Between the
Cumulative Probability Distributions


For discrete distributions, as noted above, it is only

necessary to check the points of intersection. This is the

basis for developing the algorithm. For purpose of analysis,

assume two discrete cumulative probability distributions of

a given variable, which we call FX (x) and Fy(y) where X'

has default risk and Y does not, and proceed as follows:


1. Order (rank) the observations of the given
variable for X' and then for Y from the
smallest value to the largest.

2. As X' is defined as having some probability
of default, we know that it contains some
implied level of delta. Since we have dis-
crete distributions, the cumulative proba-
bility at any point can be expressed as an
interval as follows:









0 0 < p <

xi 6 < p < [(1-6)(-)+6

x [(1-6) ()+6] < p < [2(1-6) (

x' [2(1-6) (1)+6]



Sn n
x'





x' [(n-1)(1-6) ()+6 < p l

where n is the number of observations, and all observations
1
are assigned the same probability i.

1

Y () < p I ()

Y3 (2) < P < (3)






n-l
Yn Z( ) < p 1

3. Calculate (y x') for each change in
probability. Call these areas, 1 2n.
Formulas for each area (y x') follow:

Area y x'

1. (y1-0)6

2. (yl-xi) ( 6)

3. (y2-xi) [(( ( )(- )+6)-() I

4. (y2-x2) [(R) ((n) (1-6)+6)]















2n. (Yn_-x) (n-_(n-l) (1-6)+6])
n n

4. Begin with a large value for 6 (so that YDX' by
SSD) and reduce it until the cumulative difference
is no longer greater than zero (i.e. until exact
SSD results and any further reduction in 6 would
result in X'DY).


In order to present the algorithm, it is necessary to define

the following variables:

i = (Yi+l x)[6(1 i)], i=0, 1, .., n-l (5)

and a = 0
n
i = (y x ) [[ + 6(i-1 1)], i=l, ., n (6)

and 80 = 0

S = + i = 0, 1, n (7)


The following two rules must hold for the 6 which results

in precise SSD:

I
Rule 1: E > 0 for all i (8)
i=0

I
Rule 2: Z ai a> 0 for all i (9)
i=l


The algorithm is based on the assumption that default

results in a zero return to the bondholder (Case 1). As a

result, the application of the algorithm to the data will

yield market estimates of the probability of the worst

possible outcome occurring. A comparison of the probability










of getting zero (5 for Case 1) with the probability of getting

some outcome less than the promised outcome (6 for Case 2)

yields the result that the latter 6 is larger. A comparison

of equation (3) with equation (4) yields the following

observations:

(1) The numerators are identical.

(2) The denominator in equation (4) is smaller by
the amount,

0 (l+C)B-
(0) 1 xf(x)dx + f xf(x)dx
--0 0

(3) As a result, the 6 for Case 2 is larger than
the 6 for Case 1.


The economic interpretation of this result is straightforward.

The probability of getting something, even though smaller

than promised, is greater than the probability ot getting

zero in the event the firm defaults. The mathematical

implication is that a larger value of 6 will be required to

be incorporated in the distribution with default risk (Aaa

in this example) so that it will precisely dominate the

default-free option (GOVT in this example) by SSD. Because

the algorithm is developed for Case 1, the resulting values

of the relative probabilities of default when the empirical

data are analyzed represent the minimum values of this

measure as assessed by the market.









Numerical Examples


Example 1: (G is the default-free distribution, A is the

distribution having default risk, A' is the A distribution

after the inclusion of the implied 6.)


A G A'
x P(x) x P(x) x P(x)
1.08 0.25 1.08 0.25 0 6
1.10 0.25 1.10 0.25 1.08 (1- ) (0.25)
1.18 0.25 1.12 0.25 1.10 (1-6)(0.25)
1.20 0.25 1.20 0.25 1.18 (1-6) (0.25)
EA(x) =1.14 EG(x) =1.125 1.20 (1-6) (0.25)

EG(x)
S= 1 E(x) = 0.01315789
A


We interpret this as meaning that the investment in option A

exposes the investor to a 1.32% greater probability of default

than would an investment in option G. Note that this formu-

lation of 6 calculates the probability of getting a zero

return. The probability of getting a return less than the

promised return, but greater than or equal to zero, of

course, will be larger than 1.32%. This example yields a 6

value in the range of those found in the empirical analysis.

However, the magnitude of the delta of 1.32% prohibits a

graphical illustration. 1.32% when graphed is so small that

the resulting shift in the A distribution cannot readily be

seen.

In order to graphically demonstrate the estimation of

delta, the following example is one for which there is no

economic definition of the two distributions. Rather, this









is a purely mathematical example. Define two alternatives

which differ only according to a given attribute. Assume

that alternative C possesses the attribute and that alterna-

tive D lacks it. We can then solve for the magnitude of

this difference. For consistency, we shall again call this

difference delta, but without defining delta as the relative

probability of default. This example illustrates the more

general use of this technique.


Example 2: (D possesses the given attribute, C does not

possess it, and D' is the D distribution after the inclusion

of the implied 6).


D C D'
x P(x) _x P(x) x P(x)
10 0.25 8 0.25 0 6
20 0.50 10 0.50 10 (1-6)(0.25)
30 0.25 12 0.25 20 (1- ) (0.50)
ED(x)=20 EC(x)=10 30 (1-6) (0.25)

EC(x)
6 = 1 = 0.50
ED(x)


The selection of alternative D is accompanied by a 50%

greater probability of incurring the given attribute than

if alternative C had been chosen. Figure 5 provides a

graphical demonstration of this example, and illustrates

the shift in the D distribution when the implied delta is

included.


















(D

, .p0 0
O_ o P or4

OC
(r 0
n" ri" I






PI-.
07
rt(D





H- 0
1:' CI-













Om t
r HO
Ort 0





F-C "
(DO



0 0








(D 0

SH-
0 0







. *
*
0 0
ac X -*^
r *~L
03 -
n>O_
il--
rr(T
CD


C)




3



0














CHAPTER IV
METHODOLOGY AND RESULTS



Introduction


The significance of this application of stochastic

dominance is in its potential for quantifying relationships.

The technique developed in this paper has many potential

applications, for it can be used to calculate the magnitude

of differences in cumulative probability distributions.

Thus, it can be used to quantify the magnitude of the

remaining dimension along which distributions differ once

all other relevant differences are neutralized.

The stochastic dominance technique developed in this

paper is applied to data for groups of investment grade

corporate bonds and government bonds which are comparable

along all feasible dimensions except for bond rating. As

the bond rating is assigned primarily on the basis of the

bond rating agency's assessment of the relative probability

of default prior to maturity, the bond rating serves as a

proxy for the relative probability of default. The data are

stratified according to bond rating category in order that

the magnitude of the relative probabilities of default may be

quantified by the stochastic dominance technique developed

in this study. Market price data and corporate bond coupon

50









rates are used to calculate observations of holding period

returns which are then analyzed for the relative probabili-

ties of default faced by the investor with a short horizon

(6 months in this study). Both real and nominal holding

period return series are analyzed. For the investor with a

longer horizon (in this study both 10 and 20 year data are

analyzed) statistical limitations required the use of yield-

to=maturity data rather than holding period return data.

The yield-to-maturity series are relevant for the institu-

tional investor who plans to hold the bond in one case for

10 years and in the other case for 20 years, for his/her

investment horizon is then identical with the bond's term to

maturity. When the investment horizon is not equal to the

10 or 20 years (or any other given term to maturity), it is

necessary to analyze either holding period return series or

yield-to-maturity series when the term to maturity is equal

to the given investor's investment horizon. Failure to per-

form this matching of time to maturity with the length of

the investor's horizon results in an incomplete neutraliza-

tion of interest rate risk (type (a) risk in this model) and

in a distortion of the relative probabilities of default

calculated from such data. The likelihood of default by the

firm increases in times of economic contraction and decreases

in times of economic expansion. As a result, it is neces-

sary to analyze the probability of default as a function of

the state of the economy by considering both periods of

economic contraction and of economic expansion.









Methodology


There are many characteristics of bonds which distin-

guish one from another. In a study of this nature, it is

necessary to examine these distinguishing characteristics in

order to identify those which contribute to the expected

return on investment in them. The coupon rate reflects the

magnitude of the promised cash flows over the remaining life

of the bond, and the face value indicates the promised cash

flow at maturity. Of course, when there is risk of default,

the promised cash flows are not equal to the expected cash

flows. Rather, in the case of bonds with default risk, the

expected cash flows will be consistently less than the

promised cash flows. The term to maturity indicates the

length of time over which the investor is exposed to the

risk of ownership (provided the bond is held to maturity).

Callability is another source of risk to the bond holder.

Its' impact on expected cash flows is the same as is the

impact of default risk in that a call feature increases the

probability that the actual return on the investment will

be less than the promised return on the investment (the yield).

Bonds differ also in sinking fund provisions (if any) and in

status in the event of bankruptcy. The presence of a

sinking fund and the presence of a collateral securing the

debt (i.e. the bond) in the event of bankruptcy both serve

to reduce the likelihood that the investor will experience

a loss in the event the firm defaults. They do not reduce










the probability that the firm will default, but rather reduce

the magnitude of the loss to the bondholder if default is not

identical with bankruptcy. The legal definitions of the

two conditions differ. Default exists when the firm is

unable to meet a scheduled interest payment or is unable to

repay the principal at maturity. At either such point, the

firm is technically in default on the bond. The state of

default exists when the firm is unable to meet the terms of

the bond indenture. The state of bankruptcy exists when

the firm has been judicially declared subject to having its'

assets administered under the bankruptcy laws for the benefit

of its creditors. Finally, bonds differ in the tax status

of income to the investor. Interest income from corporate

bonds and from U.S. government bonds is subject to federal

income tax. However, interest income from U.S. government

bonds is exempt from state and local tax levies (Reilly,

1982, p. 373). The treatment of capital gains income from

U.S. government bonds by state and local taxing authorities

varies. Ideally, one would analyze data on bonds matched

for all relevant characteristics except bond rating in order

to obtain accurate measures of the relative probabilities

of default.

The goal, as stated in the introduction to the model

(Section III), is to neutralize interest rate risk so as









to focus on default risk. We know that the duration1 of a

bond takes into account both the time over which the bond

provides payments and the pattern of those payments over

time, whereas the maturity of a bond considers only the

first of these. As a result, it is suggested that duration

measures on bond's characteristics more accurately (Sharpe,

1981, p. 88). For bondholders, interest rate risk consists

of both price risk and coupon reinvestment risk. Bierwag

and Kaufman (1977) have mathematically verified that both

price risk and coupon reinvestment risk can be neutralized

completely only when the duration of the bond equals the

investor's holding period.2 For these reasons, matching

bonds by duration rather than by maturity is theoretically

correct in order to neutralize interest rate risk. When

bonds are subject to different degrees of default risk,

however, duration is not an unambiguous measure of effective

maturity because of the problem of expected cash flows being



1There are several definitions of duration, for dura-
tion is an index, and is simply the weighted average of the
time periods at which the cash flows are received. The
definitions of duration differ according to their assump-
tions about the stochastic process generating unexpected
interest rate changes over the course of the investor's
planning period.

2Kaufman (1980, p. 3) points out that immunization
against interest rate risk is perfect only if the investor
has selected the formulation of duration which is based on
the stochastic process which actually generated the unex-
pected interest rate changes over the period the bond was
held. This, of course, cannot be determined in advance.









less than promised cash flows (Weinstein, 1981, pp. 258-259).

The calculation of duration uses the yield to maturity as

the discount rate, and the promised coupon payments and

maturity value as cash flows. It is possible for two bonds

of differing degrees of default risk to have the same dura-

tion if they have the same term-to-maturity, the same

coupon rate and the same yield-to-maturity. Duration

measure D1 (Macaulay, 1938) is used to illustrate this point,

where

n Ct(t)

t=l (l+YTM)t (1)
D1 = (1)
n Ct
t=l (l+YTM)t

where

t = the period in which the coupon and/or
principal payment occurs

Ct = the interest and/or principal payment
occurring in period t

YTM = the market yield-to-maturity on the bond


The formulation of duration measure Dl does not consider the

possibility that the expected cash flows might be less than

the promised cash flows (as is the case when the probability

of default exists). If the expected cash flows are less

than the promised cash flows, then the use of the risk-

adjusted discount rate (the yield-to-maturity) is theoreti-

cally incorrect, as it results in the same type of distortion

of Dl as was noted by Robichek and Myers (1966) in their









work on the calculation of net present value. As the

calculation of duration is biased in the case where bonds

are identical except for differences in default risk, we

conclude that the matching of bonds of differing degrees

of default risk for duration will not result in neutralizing

interest rate risk. In addition, there are practical prob-

lems involved in identifying bond's durations in order to

select them for inclusion in this study. As duration is

not published along with the other items of publicly avail-

able bond data (as in Moody's,Standard and Poor's, and The

Wall Street Journal's information), it would be necessary

to calculate each bond's duration, then select those with

the desired duration as a sample. Both the theoretical and

practical problems identified led the researcher to decide

against matching bonds for duration.



Description of Data and Presentation of
Empirical Results


Data were collected and analyzed for investment grade

corporate bonds and for government bonds. Both holding

period return data and yield-to-maturity data were included

in the study. The assumed investment horizons ranged from

six months for the holding period return data to ten to

twenty years for yield-to-maturity data. The sources of

data were various issues of the Federal Reserve Bulletins

for the yield-to-maturity data for a ten-year investment

horizon and Moody's Bond Record for the six month holding









period return data assuming a six-month investment horizon

and also for the yield-to-maturity data assuming a twenty

year investment horizon. The influence of the level of

economic activity on the magnitude of the relative proba-

bilities of default was addressed by analyzing data for

years of economic expansion and for years of economic con-

traction separately. The cyclical behavior of yield dif-

ferentials between long-term Treasury bonas and Aaa and Baa

corporate bonds is well documented, and is presented in

many textbooks (Brigham, 1979, as an example). Histori-

cally, the yield differentials widen in periods of economic

contraction and narrow in periods of economic expansion.

In order to account for the impact of the level of economic

activity on the magnitude of the relative probabilities of

default, data were sorted into sub-sets consisting of periods

of economic expansion and of economic contraction as iden-

tified and defined by the National Bureau of Economic

Research, Inc. (1981, p. 21). Expansionary periods were

identified as 1969, 1971 through 1973, and 1976 through

1979. Periods of economic contraction were identified as

1970, 1974 through 1975, and also 1980. Both nominal and

real data were studied for the six-month holding period

return analysis. Nominal data were converted to real data

by using the percent change in the Consumer Price Index (CPI)

unadjusted for seasonal differences, for all urban consumers

for each six month period. The relationship between the

nominal and the real holding period return is as follows:










(1 + RN) = (1 + RR) (1 + h) (2)


where RN is the nominal holding period return, RR is the

real holding period return, and h represents the realized

rate of inflation for the period. The value of h used to

convert nominal data to real data was the percent change

in the CPI from the beginning to the end of the six month

period. The total nominal six month holding period return

was calculated as follows:


(1 + RN) = P1 + C/2 (3)
P0


where Po and P1 are the market price of the bond in dollars

at the beginning and at the end of the period, and C is the

annual coupon expressed in dollars.

The relative probabilities of default displayed in

Tables 3 through 7 are the result of applying the algorithm

derived in this paper (Chapter III) for the general case

of n-points of intersection between cumulative probability

distributions. The computer program written to do this

reiteratively tests until it has found the last value of

delta for which both rules of the algorithm hold. This

value of delta is the smallest value of delta for which the

cumulative difference in the cumulative probability dis-

tributions is greater than or equal to zero. A reduction

in the value of delta by 0.001 would result in the cumula-

tive difference being strictly less than zero. The values










of delta in the tables are to be read as percent, that is,

0.013 is read as 1.3 percent.

The first data analyzed consisted of nominal six month

holding period returns. The sample consisted of 465 bonds

matched for maturity (20 years) and for the presence of a

sinking fund, for callability, and for the lack of sub-

ordination. The bonds were all investment grade corporate

bonds with the above listed characteristics, The sample

was stratified, where the strata were the four investment

grade bond rating categories (Aaa, Aa, A, and Baa). The

bonds for a given rating category were combined in port-

folios of either 10 or the population, whichever was smaller.

The source of the data was Moody's Bond Record. For each

portfolio, the semi-annual holding period return was cal-

culated. As the period covered was 1969 through 1980, there

were 24 observations of semi-annual holding period returns

for each bond rating category for the total period, 16 for

periods of economic expansion and 8 for periods of economic

contraction. Tables 3 and 4 present the relative probabili-

ties of default calculated by the algorithm presented in

this paper as it is applied to distributions of six-month

nominal and six-month real holding period returns respectively.

There were an insufficient number of comparable government

bonds (20 years to maturity) to permit their inclusion.

Appendix D presents the values for the six-month total

holding period returns both nominal and real, which were








60








m co m r 0n m
) 0 0 u
Q O O 0 0 0

U o o o o o o






0 0 00 0 00

0o 0 0 0 I o oo
a) a)







oa cn > > a >









0 0 O O 000 n1
c > o o o > o o














S-H 0 0 E 1 0 0
O Oo









0 0 0 ra 0
H = n Oa) r.
m0 O (d 0 ZH -40









r-44-J0 0 Q*H) 0 4
0.0 0 0 C CD C) > 0 < 00
*U O o3T S O 01 *




O 0 00 O3 0 00
aui om a 0













4-1 4- :4
H g e r rH 5
C) CO C* 0 HO M C 0 HO
aQ o CrH U 0 r- rl Q o I -I o1 0 r- 1









NCD I > 0 00 N I > 0 0 0
IcQ .( Ui > -r4 3 cn '









O00 | 0 0 O 0 0










0 4 0
Q) +J 1- Cu -4 cJ (u CH













r ro m ( a -4 ) -,

a) 0a 00 a) 0*d 00
4J 0 ro 0 -W -, 1 0 ( 0

*H ( H O 4 *H w E O O
- H O( O U) r- I 1 (3 H1 U O- r Q4 0









Sa)4 3 03 o o o 0 u f o o
- (d O 0 0 0 Ot 0 0 0









E-i W a HE-i H






61

used to calculate the values of delta. Appendix E presents

the values of (l+h) used to convert nominal data to real data.

Various issues of the Federal Reserve Bulletin were the

source of yield-to-maturity data on Aaa and Baa corporate

bonds and on government bonds. These data were used to

calculate the relative probabilities of default of corporate

bonds and of government bonds when the investment horizon is

ten years or more. These relative probabilities of default

are displayed in Table 5, both for the total period of 1971



Table 5. Relative Probabilities of Default of
Corporate Bonds and Government Bonds When
the Investment Horizon is Ten Years
(Based on 1971-1980 Federal Reserve Data)

Aaa vs. Govt. Baa vs. Govt. Baa vs. Aaa

Total Period 0.013 0.023 0.011
Economic
Expansion 0.012 0.021 0.010
Economic
Contraction 0.014 0.027 0.013


through 1980 and for sub-periods of economic expansion and

of economic contraction. The yield-to-maturity data pub-

lished in the Federal Reserve Bulletin for corporate bonds

are based on averages of daily figures from Moody's Investor's

Service, and are for seasoned issues. The government bond

data as published by the Federal Reserve are yields-to-

maturity for all government bonds neither due nor callable

in less than ten years. The relative probabilities of

default presented in Table 5 are calculated by the model









developed in this paper and are based on monthly observa-

tions of yields-to-maturity. As a result, there are 120

observations for the total period, 84 for periods of econom-

ic expansion and 36 for periods of economic contraction.

Tables 6 and 7 present relative probabilities of default

for corporate bonds when the investment horizon is twenty

years. These calculations are based on the same set of

data from Moody's Bond Record which was used to calculate

the relative probabilities of default when the investment

horizon is six months (presented in Tables 3 and 4). Table

6 is based on data for 1971 through 1980, while Table 7 is

based on data for 1969 through 1980. As a result, the

number of observations for the total period are 20 for

Table 6 and 24 for Table 7. For periods of economic expan-

sion, the number of observations for Tables 6 and 7 are 14

and 16 respectively, and for periods of economic contraction,

the numbers of observations are 6 and 8 also respectively.



Analysis of Results


There are two aspects to analyzing empirical results,

one of which is peculiar to research projects in economics,

and the other of which is applicable to all empirical

research. As this study addresses an economic question,

i.e., what is the market assessment of the relative proba-

bilities of default, it is necessary to provide an economic

interpretation of the results presented in the study. In














































E-

( 0
O0

0
No
NO


0

H-




u)0
Sa)
>o





z
0 r
a i


Ul
r1

a)
4-








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0


0
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4a)
mO





a)
z
C







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00
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O
O







0





































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O














O































CD4
O













































0
o
o
o
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O
r-
O











O








ON
o
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y


CO
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H
0
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order to do this, three aspects of the economic interpreta-

tion of relative probabilities of default are analyzed:

(1) the additivity characteristics of deltas; (2) the rela-

tionship of delta to the level of economic activity,

including both the relative magnitudes of deltas and the

relative sensitivities of deltas to a change in the level of

economic activity; and (3) the relationship of delta to the

length of the investment horizon. Secondly, it is necessary

to examine the validity of the techniques employed in the

research, for the results are only as valid as are the

techniques employed in obtaining them. In addressing this

question, the sensitivity of the results to the data grouping

technique employed is tested. In addition, it is necessary

to compare the relative probabilities of default generated

in this study with the relative incidences of default reported

in the literature. As a result, both the internal and the

external validity of the research will be examined. As the

deltas presented in this study are relative probabilities

of default, they should be amenable to being represented

graphically as are risk premia. When historical risk premia

are plotted, one observes that the relative magnitudes of

the risk premia are consistently positively correlated with

the quality of the bond as indicated by the bond rating.

The premia of Aaa corporate bonds over Treasury (government)

bonds, of Baa corporate bonds over Aaa corporate bonds, and

of Baa over government bonds are consistently positive.









In addition, when periods of economic contraction occur,

the magnitudes of these relative risk premia increase.

There are, of course, economic rationales for these obser-

vations which should also apply to the results of this study.

The relative probability of default of high grade corporate

bonds (say Aaa) over the government bonds is expected to be

less than the relative probability of default of low grade

corporate bonds (say Baa) over the same government bonds,

as the quality of the bond as indicated by the rating is

inversely related to the probability of default. The rela-

tive riskiness of bonds is reflected in historical differ-

ences in risk premia. It is necessary to determine if

relative probabilities of default are additive, as are risk

premia. In order to do this, it is first necessary to assume

that bond rating assignments correctly rank bonds according

to their relative probabilities of default. If one does not

make this assumption, then the question is actually a joint

hypothesis: (1) assigned bond ratings indeed rank bonds

accurately by relative probability of default prior to

maturity; and (2) the technique developed in this paper

actually calculates relative probabilities of default which

are economically valid. The literature, reviewed above,

suggests that bond ratings do correctly rank bonds according

to their relative probabilities of default as indicated by

the historic incidence of default. It is necessary to

determine the mathematical relationship between relative









probabilities of default, and then to test the deltas pre-

sented in Tables 3-6 for this relationship.

Consider three investment alternatives, B, A,. and G and

interpret them such that B is more likely to default than

is A, and A is more likely to default than is G. Assume

that the only dimension along which they differ is the

probability of default. Define 61 as the relative proba-

bility of default of B over G, 62 as the relative proba-

bility of default of A over G, and 63 as the relative

probability of default of B over A. If the deltas are

strictly additive, then,


61 =2 + 63 (4)

We know that in the simple case of no more than one inter-

section between the cumulative probability distributions

and when default results in a return of zero,


(1 1)EB(x) = EG(x) (5)


(1 62)EA(x) = EG(x) (6)


(1 63)EB(x) = EA(x) (7)


Subtracting (6) from (5) yields,

(1 61)EB(x) (1 62)EA(x) = 0


Rearrange terms, so that,

(1 61)EB(x) = (1 62)EA(x)









Substitute (7) for EA(x), so that


(1 (6)EB(X) = (1 62) (1 63)EB(x)


Divide both sides by EB(x), so that


(1 61) = (1 2)(1 63


Expand right hand side


(1 6 ) = (1 2 63 + 623)


Multiply both sides by (-1) and solve for 68


61 = 62 + 63 6203 (8)


So, in this case, the deltas should be additive, but less a

cross product term. Note that we assume that G was default-

risk free. Equation 8 is applied to the data in Table 5

to produce Table 8 which demonstrates the nature of the

additivity of deltas. An examination of Table 8 leads to

the conclusion that relative default probabilities are

additive, as the differences in the calculated total range

of delta and the observed range of delta are virtually zero.

As the relationships were derived for a pairwise comparison

in which not all members possessed default risk, it was not

correct to demonstrate additivity for any other sets of

deltas. Thus, it seems that the data confirm the hypothesis

that the relative probabilities of default as calculated by

this technique are additive as are risk premia. The mathe-

matical relationship existing within the structure of deltas








68






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has been determined to exist in deltas which are calculated

from empirical data. Thus, there exists a default risk

structure composed of relative probabilities of default.

In this paper, we have derived this structure for three

different investment horizons, for the default risk struc-

ture relevant for a given individual is a function of his/her

investment horizon. The default risk structure for a given

investment horizon, of course, is based on the market's

assessment of the relative probabilities of default for the

different bond rating categories.

The default risk structure is also a function of the

level of economic activity. As the probability of default is

related to the state of the economy, the magnitudes of the

relative probabilities of default are expected to be greater

in periods of economic contraction than in periods of econom-

ic expansion. An examination of Tables 5 through 7 leads to

the conclusion that the relative probabilities of default

for every pairwise comparison except one (A vs. Aa is

unchanged) are larger in periods of economic contraction

than they are in periods of economic expansion. The fact

that the A vs. Aa comparison is unchanged is not significant,

as this comparison is not present in Table 5, and Tables 6

and 7 are based essentially on the same data (only the number

of years covered varies). Essentially, then this A vs. Aa

observation, as the only one not as predicted, represents

one calculation out of nine (three pairwise comparisons in









Table 5, and six in Tables 6 and 7 together). These results

are exactly as we would hope they would be if indeed the

stochastic dominance measure of the relative probability of

default has economic validity.

Before moving to an analysis of the impact of the length

of the investment horizon on the magnitudes of the relative

probabilities of default, it is necessary to assign a more

precise interpretation to the values of delta as presented

in Tables 5 and 6. There are two distinctive features of the

data used to calculate Table 5 deltas which need elaboration

and which affect the comparison of these results with those

of Table 6. First, as was noted earlier, Table 5 data are

for government bonds which are neither due nor callable in

less than ten years and for corporate bonds with ten years

to maturity whereas the data for Table 6 are for callable

bonds with twenty years to maturity. Thus, the ten-year (or

more) horizon deltas and the twenty-year horizon deltas differ

in the degree which maturity has been held constant. As a

result, the Table 5 deltas are biased when compared with

Table 6 deltas, but there is no reason to assume bias within

the Table 5 results when corporate bonds are compared with

one another. As corporate bonds in general possess a call

feature, no bias is expected to exist either in Table 5 Baa

versus Aaa deltas or in any pairwise comparisons in Table 6

(or Table 7). Rather, a call feature is assumed for all

corporate bonds with no difference between bond rating









categories as to its presence or lack of presence. There is

a difference in callability as noted when government bonds

and corporate bonds are compared, as in Table 5. The dif-

ference in the degree to which maturity is held constant in

the 10-year versus the 20-year data and the difference in

callability of government bonds and corporate bonds impinge

on the strict interpretation of the calculated relative

probabilities of default for the 10-year investment horizon

when compared with the 20-year investment horizon.

As the corporate bond data for deltas presented in

Table 5 and in Table 6 are characterized by maturity matching

of 10 and of 20 year respectively, and by no bias expected

in the treatment of callability, it is valid to analyze deltas

calculated for the pairwise comparison of Baa corporate bonds

versus Aaa corporate bonds. Such a comparison is presented

in Table 9. It should be noted that the relative probability

of default consistently increases with the length of the

investment horizon. The relative probability of default of

Baa over Aaa corporate bonds is consistently greater for the

20 year investment horizon than for the 10 year investment

horizon. It should be noted that the magnitude of the dif-

ference is quite small, but the fact that the value of (b)-

(a) is consistently 0.003 and the value of (1)-(2) is con-

sistently 0.001 lends credibility to tne suggestion that

these results are not a mere statistical fluke. This

conclusion holds for the total period covered by the data









Table 9. The Impact of the Investment Horizon
on the Magnitude of the Relative Probability
of Default of Baa Corporate Bonds Over
Aaa Corporate Bonds


20 Year Horizon 10 Year Horizon Difference
Baa vs. Aaa Baa vs. Aaa in Delta
(1) (i2) (1)-(2)
Total Period 0.012 0.011 0.001

Economic
Expansion (a) 0.011 0.010 0.001

Economic
Contraction (b) 0.014 0.013 0.001

Difference in
Delta (b)-(a) 0.003 0.003 0.000


(1971 through 1980) as well as for sub-periods of economic

expansion and of economic contraction. It is interesting to

note that the differences in the relative probabilities of

default (reflected in column 3) are consistently of the same

magnitude, and do not vary with the definition of the period

(total, economic expansion, or economic contraction). It

also should be noted that the difference in the magnitude of

delta when periods of economic expansion are compared with

periods of economic contraction is invariant to the length

of the investment horizon. For the 10-year investment

horizon, the difference in the sub-period's deltas is 0.003

and for the 20-year investment horizon, the difference in the

sub-period's deltas is the same 0.003. Thus, one can conclude

from this that the increases in deltas when the 20-year

investment horizon is compared with the 10-year investment









horizon are a function of the difference in the horizon. In

this case, the difference in the relative probability of

default is not affected by the state of economic activity,

but rather only by the length of the investment horizon. In

other words, the level of economic activity has no more effect

on the default risk faced by an investor with a 20-year

investment horizon than it does on an investor with a 10-year

investment horizon. The difference in the length of the

investment horizon is the crucial factor in the amount of

default risk assumed by investment in a given rating cate-

gory of corporate bonds. This issue is explored further

in Chapter V in the discussion of future research.

Before moving to addressing the validity of this research,

it is necessary to examine the relative probabilities of

default when the investment horizon is short (six months in

this study). These data are presented in Tables 3 and 4, but

have not been analyzed to this point. The six month holding

period return data which are the basis for these tables are

nominal returns in Table 3 and real returns in Table 4 as

noted earlier when these tables were initially presented.

The results from using nominal versus real data are not

significantly different from one another. Neither presen-

tation of the relative probabilities of default when the

holding period (the investment horizon) is six months has

the economic validity observed in comparable presentations

for the 10-year and 20-year investment horizons. With two









exceptions (A vs. Aa increases and Baa vs. A is unchanged)

the magnitudes of the relative probabilities of default are

smaller in periods of economic contraction than they are in

periods of economic expansion. We expect the reverse to be

true, as was observed in the 10-year and in the 20-year

investment horizon series. In addition, the relationships

predicted by the concept of additivity hold only for the

sub-periods of economic expansion. It should be noted that

the data set used for calculating six-month holding period

returns (Tables 3 and 4) is the same data set which is the

basis for the 20-year investment horizon analysis (Table 7).

They are based on the same set of bonds. The only difference

is that observations of market prices and coupon rates were

used for the six-month holding period return calculations

(Tables 3 and 4), and observations of yield-to-maturity were

used for the 20-year investment horizon calculations (Table

7 is for the same period of 1969 through 1980, and Table 6

is for a sub-set of 1971 through 1980). The explanation for

the inconsistent results does not seem to lie in the data

themselves, but rather in either the choice of the horizon

(6 months) or the use of holding period return data rather

than yield-to-maturity data to calculate relative probabili-

ties of default. As stated earlier, bond ratings are

assigned on the basis of the bond rating agency's assessment

of the relative probability that the bond issuer will default

on the bond prior to maturity. We can interpret this as









meaning that the relative probabilities of default as reflected

in the market's assessment of default probability (in market

data) are based on the probability of default occurring prior

to maturity. This is evidence that the relative probabili-

ties of default as calculated by the technique developed in

this study are to be interpreted as the probability of default

before maturity. The market seems to be assessing the rela-

tive likelihood of default prior to maturity. This inter-

pretation lends support to the use of yield-to-maturity

series where the time to maturity is equal to the investor's

investment horizon as a basis for calculating delta. This

is a reasonable conclusion if the market is indeed inter-

preting bond ratings as ranking bonds according to the prob-

ability of default prior to maturity, as rating agencies

intend.

Although the results of the analysis of the yield-to-

maturity series have been demonstrated to have economic

validity, the possibility remains that the results were

biased due to the theoretical and practical reasons leading

to the preclusion of duration-matched data. The impact of

duration on the calculation of delta is examined by analyzing

the sensitivity of delta to duration measure D1 and by testing

the impact of the portfolio grouping of data on the sensi-

tivity of delta to duration.

Table 10 analyzes the 20-year investment horizon yield-

to-maturity data for sensitivity of the value of delta to a






















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change in the range of duration measure Di. In order to

produce Table 10, the range of D1 was increased from a range

of (10.00 to 10.99) to a range of (8.50 to 12.49) by incre-

ments of 1.00.

Table 10 indicates that delta has some sensitivity to

a change in duration measure DI. It is possible that the

grouping procedure used to prepare the bond data (i.e., bonds

combined in portfolios, then delta calculated for portfolio

data) introduced a bias which is mathematical in nature. In

order to examine the extent to which the calculations of

delta based on data combined into portfolios was influenced

(biased) by differences in the magnitude of Dl, Table 11

presents moments of the distributions of D1 calculated both



Table 11. Analysis of the Variable D1 for Data
Used in Calculating Delta as Presented
in Table 7 and in Table 10

n x a2
(a) 465 Bonds Individually
Aaa 114 11.3370 1.2951
Aa 120 10.7913 1.3925
A 119 10.3954 1.6455
Baa 112 10.0333 1.1982
Total Bonds 465
(b) 465 Bonds in Portfolios
Aaa 12 11.2018 0.9811
Aa 12 10.7238 1.1480
A 12 10.4583 1.2098
Baa 12 10.2808 1.6595
Total Bonds 48










for individual bonds (as presented in Table 10) and for

portfolios of bonds (as presented in Table 7). If the data

in Table lla are not significantly different from the data

in llb, then the relative probabilities of default are not

significantly biased by the methodology which grouped the

bond data into portfolios. It was first necessary to deter-

mine if the values of Di underlying (a) and (b) can be assumed

to be a random sample from a normal distribution. The SAS

computer software package was used to address this question.

Table 12 presents the results. For Table 11(a), the modi-

fied Kolmogorov-Smirnov D-statistic was calculated and

tested with the Null hypothesis being that the data are a

random sample drawn from a normal distribution. For Table

11(b), the Shapiro-Wilk W-statistic was calculated and

tested with the Null hypothesis again being that the data

are a random sample drawn from a normal distribution. The

statistics indicate that the null hypothesis cannot be

rejected. Finally, it is necessary to see if the mean is the

same and the variance is the same within each rating category.

A Z-statistic was calculated for each rating category to see

if the means are not statistically different from one another,

and a F-statistic was calculated for each rating category

to see if the variances are not statistically different from

one another. These values are presented in Table 13. Table

13 results taken with Table 12 results indicate that the

grouping of bonds into portfolios has not introduced a







80


Table 12. Statistical Analysis of the Distributions of
D1 in Order to Determine Whether Table 11(a) is Not
Significantly Different From Table 11(b)


Rating D Prob>D t(mean=0) Prob
(a) Analysis of Table 11(a)
(modified Kolmogorov-Smirnov D-statistic)
Aaa 0.07380 0.129 106.4630 0.0


<0.010

<0.010

>0.150


(b) Analysis of Table 11(b)
(Shapiro-Wilk W-statistic)
Aaa 0.98140 0.964


0.656

0.714

0.929


100.1750

88.4034

97.0042



39.1768

34.6720

32.9386

27.6456


>Itl


0.0001

0.0001

0.0001



0.0001

0.0001

0.0001

0.0001


n


114

120

119

112



12


Table 13. Values of Statistical Tests for Equality
of Means and of Variances Within Bond Rating
Categories when D1 is Calculated for Bonds
Grouped in Portfolios and for
Individual Bonds

Hypothesis Test Statistic Aaa Aa A Baa
H0: X(a) = X(b) Z 0.443 0.206 (0.186) (0.641)

H0: a2 (a) = (b) F 1.320 1.213 1.360 0.722


0.12419

0.10402

0.06580


Baa


0.95470

0.95910

0.97650


Baa


001









significant bias due to the impact of duration (DI) on the

values of the relative probabilities of default, If anything,

the fact that the variance of D was reduced for each bond

rating category when bonds are combined into portfolios

would serve to reduce what impact not matching for duration

might have had on the results.

Finally, although historical data on default experience

is presented (Tables 1 and 2) one cannot compare these prob-

abilities of default with the ones in this paper. The

main factor which prevents such a comparison is that Table 1

and Table 2 data are not controlled for maturity, but rather

are for all bonds of a given rating regardless of maturity.

Thus, although the incidence of default can be interpreted

as the probability of default over a given time period,

Hickman's and Pye's data do not consider the investment

horizon, and cannot be interpreted as the relevant proba-

bility of default for any given investor. In addition, of

course, Hickman's and Pye's data cover a different time

period than does this study.















CHAPTER V
SUMMARY AND CONCLUSIONS



Introduction


Bond rating agencies classify corporate bonds and state

and municipal bonds into different categories according to

the agencies' assessment that the issuer will default on

the bond prior to maturity. The rank (bond rating category)

assigned to a given bond is based primarily on publicly

available information taken from the issuer's financial

statements. Bond ratings rank bonds according to their

relative probabilities of default as assessed by the bond

rating agencies. Once one assumes that the bond market is

efficient and is in equilibrium, then it follows that the

market price of any given bond is a function of the risk

incurred by ownership of that bond. There are two types of

risk inherent in bonds: (a) interest rate risk (i.e.,

variability risk) and (b) default risk.

In this paper, a technique is proposed for quantifying

default risk (type (b) risk). As a result, the probability

of default as estimated by the bond market can be measured.

The technique developed involves comparing the cumulative

probability distributions of rates of return (or of yields-

to-maturity) of bonds which differ only in default risk

82










once we have neutralized interest rate risk, A comparison

of the cumulative probability distributions of say Aaa bonds

with government Donds leads to the observation tnat the Aaa

bonds dominate the government bonds by First Degree or by

Second Degree Stochastic Dominance. In equilibrium, this

observation cannot hold, for the Aaa bonds possess default

risk and the government bonds do not. It follows then that

the magnitude of the probability of default of the Aaa bonds

over the government bonds can be quantified. The technique

for quantifying this probability of default involves changing

the distribution of rates of return (or yields-to-maturity)

of the Aaa bond in a systematic manner until the dominance

disappears. This manner of revising the Aaa distribution

in order to include the implied probability of default results

in the quantifying of the probability of default as assessed

by the bond market.



Summary of Results


The significance of the technique developed in this

paper is in its potential as a tool for quantifying

relationships. It can be used to quantity the remaining

dimension along which distributions differ after all other

relevant differences are neutralized.

An algorithm for the stochastic dominance technique

developed in this paper is applied to aata for groups of










investment grade corporate bonds and government bonas which

are comparable to the extent empirically feasible except for

bond rating assignment. The bond rating serves as a proxy

for the relative probability of default. Market price data

and corporate bond coupon rates are used to calculate holding

period returns which are then analyzed for the relative

probabilities of default faced by an investor with a six-

month investment horizon. Both real and nominal holding

period return series were analyzed. Statistical limitations

required tne use or yield-to-maturity data rather than holding

period return data for the investor with a longer horizon

(both 10 and 20 year data were analyzed). In addition, as

the magnitude of the probability of default is also a function

of the state of the economy, it was necessary to calculate

relative probabilities of default for botn periods of economic

contraction and of economic expansion.

Data were collected and analyzed for investment grade

corporate bonds and for government bonas for a variety of

investment horizons. both holding period return data and

yield-to-maturity data were studied. The sources of data

were various issues of the Federal Reserve Bulletin and

Moody's Bond Record. Definitions of periods of economic

expansion and of economic contraction were provided Dy the

National Bureau of Economic Research, Inc. (1981, p, 21).

Nominal data were converted to real data by using the

percent change in the Consumer Price Index.










The probabilities of default as calculated by the

technique proposed in this paper were consistently in the

direction predicted by economic theory. The magnitude of

the probability of default was found to vary inversely with

the quality of the nona as indicated by the bond rating.

Lower grade bonds consistently exhibited a greater

probability of default than did higher grade bonds. Also,

the magnitude or default probability for a given bond rating

category consistently was larger in periods of economic

contraction than in periods of economic expansion. This

result is as expected, since the likelihood of default for a

given firm is expected to be related to the state of the

economy. Finally, the probability of default was found to be

positively related to the length of the investment horizon.

It was consistently observed that the magnitude of the prob-

ability of default increased with tne length or the investment

horizon. This result held for all categories of bonds.

This calculation of the analysis of probabilities of

default as assessed by the bond market is only one application

of the technique developed in this paper. Other applications

and extensions are discussed later in this chapter. The

application or this technique to the quantifying of bond

default probability, as assessed by the market, is charac-

terized by limitations, as is the case when any such technique

is applied in research. It is necessary to address

limitations in the research methodology in order that the










analysis may be completed. The discussion of potential

shortcomings of this research will be followed by suggestions

for future research which is proposed in order to address

these perceived limitations. It is suggested that the

acknowledgment of research limitations followed by proposals

for extensions of this present study in order to address

these limitations should serve to bound this research without

suggesting that it is complete in itself.

As the suggested major contribution of this paper is in

its presentation of a technique for quantifying differences

in cumulative probability distributions, the fact that the

algorithm derived herein is for tne case when default results

in zero rather tnan in the more realistic result of a return

less than expected, is a potential limitation. This is

addressed by acknowledging that the empirical results must be

interpreted as the minimum value of the implied probability

of default. The results in this paper are interpreted as the

market estimate of the likelinooa of the worst possible

outcome, and thus represent a lower bound on the estimated

probability of default. In other words, the probability of

getting zero in the event of default is less than the prob-

ability of default resulting in a return less than the

expected return. The outcome in the event of default is

represented by a continuum stated mathematically as,


(1+C)B
/ xf(x)dx
0










It is possible to extend this research to estimate empiri-

cally the magnitude of the implied probability of default

when default results in the return as defined above. The

derivation of the value of the expression for tne magnitude

of default probability in this case is presented in Equation

(4) on page 42 of this study. It remains to test this

equation empirically.

A second potential limitation of this research is that

there is no statistical analysis of tne results in order to

determine whether the resulting relative probabilities of

default are significantly different from one another. As the

analysis of bond data was performed in order to demonstrate

the use of the technique developed and presented in this

study, rather than the focus of the paper being on the

probabilities of default themselves, it is sufficient that

the results are as economic theory would predict. Future

research is proposed to analyze the measure presented in this

paper as an alternative to bond ratings. This suggested

extension actually consists of a series of related topics.

when addressing these related topics, it is appropriate to

consider the question of whether the results are significantly

different from one another, or indeed whether they are

significantly different from zero. Further elaboration on

this is to be found in tne next section of this chapter,

where brief proposals for future research are outlined.










Another possible limitation of the research is seen in

the use of data on portfolios of bonds rather than on

individual bonds. The magnitudes of probabilities of

default as calculated on portfolio data may not be relevant

for individual bond selection. A degree of bias may be

introduced by using portfolio averages of yields-to-maturity

as the basis for calculating tne relative probability of

default. It is possible to address this question by

calculating the probability of default for individual bonds

of a given category, say Aaa, then determining if tne average

of these calculations is significantly different from the

relative probability of default of Aaa bonds in general. The

addressing of this question would fall logically under the

suggested broad area of research on this measure of the

relative probability of default as an alternative to bond

ratings.

In addition, it may be that this research suffers from

the limitation of being based on Moody's bond rating assign-

ments only. This may be perceived as a narrowness of scope.

As noted earlier, however, the suggested contribution of this

paper is in the development and presentation of a technique

which can be used to address many questions which are

economic in nature. Tne selection of this particular question

as a means to demonstrate the use of this technique is

secondary to the development of the technique itself. The

comparison of the probabilities of default for data stratified










by Moody's nond rating as compared with data stratified by

rating assigned by an alternative bond rating agency

(Standard and Poor, or maybe Fitch) remains to be done, and

would follow logically either in the proposed study of bond

market efficiency or in the proposed testing of this measure

as an alternative to bond ratings.

A final suggested limitation of this research is in the

selection of maturity and coupon rather than of duration for

matching data. The theoretical justification for this

decision is discussed in detail in Chapter IV, and is best

summarized by the statement that duration is not always an

unambiguous measure of effective maturity for bonas which

differ in default risk. In addition, there exists a major

practical barrier to studying large numbers of bonds matched

for duration, as duration is not published, but rather must

be calculated. Botn theoretical reasons and practical reasons

led the researcher to decide against matching bonds for

duration. In addition, the impact of this decision on the

results was statistically analyzed with the conclusion that

no significant bias was found to have been introduced.

The discussion of possible limitations to the research

leads logically into a discussion of proposals for future

research. Tne following section provides brief outlines of

further theoretical extensions of this technique and of

other applications of this research.










Proposals for Future Research


Portfolio Implications Research


This study analyzed the risk of default incurred by

holding a given bond in isolation. Of course, in reality

bonds may be held in portfolios. They may be diversified

with each other or they may be diversified with the market

portfolio.

If one considers years (or periods) as states, we can

apply State Contingent Stochastic Dominance (Kroll, 1981) as

a technique for quantifying the probability of default when

bonds can be diversified with each other or witn other assets.

The dominance by Second Degree State Contingent Stochastic

Dominance tSSD) of one option over another in the pairwise

comparison guarantees the existence of dominance in the

portfolio context (Kroll, 1981, pp. 10-11). In order to

develop an SSD measure of default probability for bonds held

in portfolios, it is crucial to justify viewing years (or

periods) as states of the world. It is an empirical fact

that bond returns (or yields) move together. When the return

(or yield) on Aaa bonds has increased, one expects to observe

that the return on say Baa bonds has also increased. The

fact that bond returns move together suggests that years (or

periods) may be treated as states of the world so that an

SSD measure of the probability of default may be calculated

for the bond data used in this study as well as for other




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