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
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S-H 0 0 E 1 0 0
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m0 O (d 0 ZH -40
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Sa)4 3 03 o o o 0 u f o o
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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-
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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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(,Q0 0 4 n 0 0
C> o o o
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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