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QUANTIFYING THE PROBABILITY OF DEFAULT AS ASSESSED BY THE BOND MARKET; AN ANALYSIS OF DEFAULT RISK MEASURES BY MARY STEARNS BROKE A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1982 To my parents, Mr. and Mrs. George L, Stearns, Sr., in gratitude for their unfailing support, confidence and encouragement throughout my life, and to my husband, Ernest, and daughter, Elizabeth, in gratitude for their support and many sacrifices throughout the course of this study. ACKNOWLEDGMENTS The guidance and direction of Professor Haim Levy, chairman of my dissertation committee, and the many helpful comments of the other members, Professor Moshe BenHorim and Professor Rashad Abdelkhalik, 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: 19001943. . .... 25 2 Bond Rating and Default Experience by Decade; 19201939. . .. 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 (19711980) 63 7 Relative Probabilities of Default of Corporate Bonds When the Investment Horizon is Twenty Years (19691980) 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 YieldtoMaturity 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 yieldstomaturity) 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 nondiversifiable risk. The CAPM assumes that investors have von NeumannMorgenstern 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 economywide factors, the levels of which are estimated by welldiversified investors, times a betafactor 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 multidimensional 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 yieldstomaturity 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 subperiods 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. 104114; Foster, 1978, pp. 444445). 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. 444445). 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 componentsa 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 samplesensitive 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) bondunique 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 welldeveloped 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 riskfree 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 19531967. Betas were estimated over the full period, and realized returns were calculated as the geometric mean holdingperiod yields. When the regression Ri = a + bBi + ei was run, the regression results were as follows: R2 = .15, a = .0389, b = .00931. The tvalue 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 bcoefficient 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 nondiversifiable defaultrisk measure in order to explain realized bond returns (Percival, 1974, pp. 464468). Reilly and Joehnk (1976) assumed that a marketderived 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 retesting of the CAPM by incorporating bonds into the market index. The SharpeLintner 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 nonlinear relation. McCallum's (1975) study of Canadian government bonds (19481968) indicates that the total holding period return was a nonlinear 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 reexamine 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. 389390): 1. Protective provisions of the issue 2. Ratio analyses a. Fixed charge coverage b. Longterm 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 onehalf 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, longterm 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 nonsystematic risk, whereas seasoned issues should contain a premium for nonsystematic risk. Of course, it should be noted that the systematic risk measure is the stock beta and the nonsystematic 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 semistrong 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 marketdetermined bond risk was that of Reilly and Joehnk (1976). They estimated several marketderived 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 marketderived beta for a bond should be an appropriate risk measure for bonds, and 1Semistrong 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 accountingdetermined risk measures, based on internal corporate variables, and the marketdetermined 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 tstatistics 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. 237239) 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 rerun 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 highgrade bonds may differ from the relevant risk for lowgrade 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: 19001943 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 VIX 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: 19201939 Bond Period Rating 19201929 19301939 19201939 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 yieldtomaturity) (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 postwar 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 postwar 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 rewardtorisk 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 riskreturn 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 riskfree 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 riskfree 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) rewardtovariability 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 nonsystematic 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 nonsystematic 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 riskfree rate and a point representing the riskreturn 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 riskaverse investor in that security. This interpretation is presented by Levy and Sarnat (1972, p. 482) for Sharpe's rewardtovariability 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 nondiversifiable 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 welldocumented 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 riskaverters, 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 riskaverter, 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 riskaverse investor will pay a higher price for the defaultfree 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 exante 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 expost rates of return, and include only bonds of firms which did not default. As a result, on an expost 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 expost data, it includes only firms which did not default during the period covered by the data. The expost 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, expost 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, expost 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 expost distributions are necessarily stable or that they represent exante distribu tions. Rather, expost 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 oneperiod 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 NeumannMorgenstern 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 riskaverse 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 defaultfree 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 riskaverters (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 riskaverse 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) + (l6)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) + (l6)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] + (16)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 npoints 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 < [(16)()+6 x [(16) ()+6] < p < [2(16) ( x' [2(16) (1)+6]
