Estimating Default Probabilities of Corporate Bonds over Various Investment Horizons

Estimating Default Probabilities of Corporate Bonds over Various Investment Horizons Edward I. Altman Max L. Heine Professor of Finance NYU Stern School of Business New York City In advance of forthcoming regulatory changes, commercial banks have developed sophisticated internal credit models that consider the risk of their counterparties. Even though these models suffer from high Type II errors false forecasts of default they tend to be quite accurate in bankruptcy prediction. Although external credit rating agencies excel at assessing the initial rating of a corporate bond, they are slow to change their ratings in light of new information. The flexibility and ease of use of a well-known internal credit model, first developed in the 1960s, and the extensive default histories of the external rating agencies can be combined to predict the probability of default over investment horizons stretching from 1 year to 10 years. D espite low long-term interest rates, near-recordlow numbers of bankruptcies and defaults (especially in the high-yield bond market), and narrow credit spreads on corporate debt, investors and analysts should not be complacent about credit risk. Even in today s benign investment environment, credit risk is as important as it has been in more turbulent times for two good reasons. First, credit quality of new corporate bond issuers has deteriorated recently. The proportion of new issuers with an initial credit rating of B or lower from one of the established rating agencies reached nearly 43 percent in 2004, an all-time high. And the probability of default of a bond initially rated B or lower is extremely high, particularly in the second and third years after issuance, as my analysis will demonstrate. So, investors can expect a spike in credit risk and widening of spreads if the future unfolds like the past. Second, credit ratings have become more prominent in the banking industry because of anticipated regulatory changes. The new capital adequacy framework, known as Basel II, was agreed upon in 2004 by the heads of bank supervisory authorities of This presentation comes from the 2005 Financial Analysts Seminar held in Evanston, Illinois, on 17 22 July 2005. the G 10 countries. Basel II will change the way large banks calculate minimum capital requirements to cushion against unexpected losses in their credit and loan portfolios. The current regulatory framework allows banks to set aside 8 percent of capital, irrespective of counterparty risk. Capital requirements under Basel II, however, will be risk based: the higher the counterparty risk, the more capital the bank must set aside for losses. In anticipation of Basel II, banks have been developing sophisticated internal credit rating models that estimate each counterparty s credit rating, probability of default, and probability of loss given default. Obviously, these internal models will not always give the same credit rating as the external agencies do. Moody s Investors Service, Standard & Poor s, and Fitch Ratings are the most familiar rating agencies, and their rating methodologies are consistent across industries, time, and countries. Criticism, however, has been leveled against how slowly they change a rating in light of new information. And when they change a rating, the change is not as great as it should be. In other words, a bias exists in ratings changes. For example, after an initial downgrade, the odds for the next change being a downgrade are between 2:1 and 3:1 for industrial companies; for firms in the financial 2006, CFA Institute cfapubs.org MARCH 2006 65

CFA Institute Conference Proceedings Quarterly industry, the odds are closer to 5:1. Thus, researchers have observed positive serial autocorrelation of downgrades a series of biased rating changes in the same direction. Alternatively, after an upgrade, the likelihood of an upgrade or downgrade is close to random, or even odds. The agencies are cautious by design because they have an objective, which is high stability of ratings. 1 The agencies have a migration policy (sometimes explicit, sometimes implicit) that says to make a change only if it will be enduring. As a result, they are slow to change a rating, and when they make a change, it is only partial. Still, the agencies excel at providing accurate ratings for new issues. And they offer a rich default history for any given initial rating. Therefore, one can combine the predictive power of an internal rating system with the default history of the agencies in a three-step, forward-looking modeling process: 1. Construct an internal credit scoring system based on relevant samples of defaulted and nondefaulted firms, 2. Convert the resulting internal credit score into external bond rating equivalents (BREs), and 3. Calculate cumulative and marginal probabilities of default over a chosen investment horizon. For example, if a new issue has a BRE of B, an investor might wish to know the probability of default for any investment horizon up to 10 years after issuance. Internal Credit Model: Altman s My first work with internal credit models was in 1968 when I built the first-generation multivariate model known as Z-score. 2 It is a tool, one of many now available commercially, that can be used for any investment, although I demonstrate it here with corporate bonds. The Z-score model uses discriminant analysis, a statistical technique not to be confused with regression analysis. Discriminant analysis is similar to regression analysis in that there are many independent variables that the researcher believes can explain the behavior of a quantitative dependent variable. The dependent variable in the Z-score model, however, can take on only qualitative categories. So, I 1 See Edward I. Altman and Herbert A. Rijken, How Rating Agencies Achieve Rating Stability, Journal of Banking & Finance (November 2004):2679 2714. Alternatively, this and other papers can be downloaded from the author s own website at http:// pages.stern.nyu.edu/~ealtman/. 2 Edward I. Altman, Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy, Journal of Finance (September 1968):589 609. explored how the interaction of various independent variables can explain the external credit rating, or prediction of corporate distress. Discriminant analysis also calculates the weight of each independent variable. In the Z-score model, the independent or discriminant variables are familiar accounting ratios that predict credit quality and, ultimately, bankruptcy and default of a corporate issuer. Figure 1 shows the results of a discriminant analysis. Using a sample of bankrupt and nonbankrupt companies, I plotted the ratio of equity to debt along the x-axis and earnings before interest and taxes (EBIT) to total assets along the y-axis (the relevance of these variables in the Z-score model is explained later). The circles represent a sample of bankrupt companies, and each square represents a sample of nonbankrupt companies. The discriminant model maximizes the squared distance between the group means and minimizes the squared distance within groups (another statistical technique called analysis of variance or ANOVA). In this way, the coefficients or weights of each independent variable are established. Every set of independent variables has a unique set of weights, and when a variable is added or removed, all the weights change. Discriminant analysis does not, however, optimize the coefficients: It does not indicate which model is best. Figure 1. EBIT/Total Assets Forecasting Distress with Discriminant Analysis Equity/Debt Exhibit 1 shows the Z-score weights of the five financial ratios that serve as the independent discriminant variables: 1. Working capital to total assets. Working capital is the difference between current assets and current liabilities, and this ratio measures a company s internal liquidity relative to its total capitalization. (For this ratio, and throughout this analysis, total assets exclude intangible assets.) 66 MARCH 2006 2006, CFA Institute cfapubs.org

Estimating Default Probabilities of Corporate Bonds over Various Investment Horizons Exhibit 1. Component Definitions Variable Definition Weighting Factor X 1 Working capital/total assets 1.2 X 2 Retained earnings/total assets 1.4 X 3 EBIT/total assets 3.3 X 4 Market value of equity/book value of total liabilities 0.6 X 5 Sales/total assets 1.0 2. Retained earnings to total assets. Retained earnings represent the cumulative profitability of the company, less any dividend payments, and are a proxy for the age of the company. The ratio is also a backdoor measure of leverage. A company that has high retained earnings to total assets has financed its growth primarily with internally generated profits, and a company with a low ratio relies on external equity or debt. 3. EBIT to total assets. This ratio is return on total assets, which measures the productivity of the company s assets. EBIT is independent of tax and leverage, and the earning power of assets can indicate whether a company has enough resources to satisfy bondholders claims against it. 4. Market value of equity to book value of total liabilities. The numerator uses forward-looking market values rather than historical accounting values. Liabilities include non-interest-bearing debt, such as trade payables, and interest-bearing obligations, such as long-term and short-term debt. Market value of equity for publicly traded companies indicates the credit quality of the company indirectly through its cost of equity. It also measures the company s ability to raise equity at a reasonable price and the market s consensus of the future growth of the company. 5. Sales to total assets. This ratio measures the company s ability to compete with other companies in the same industry. Interestingly, this ratio is the least significant variable by itself, but it is significant when combined with the other four variables. For all five discriminant variables, one would expect that the higher the ratio, the stronger the company and the higher its Z-score. Furthermore, the signs of the coefficient weights are positive, as expected. The constant term of the model has been suppressed. In general, this Z-score model should be used only for manufacturing companies; other model specifications exist for companies operating in other industries. Linking s with External Bond Ratings The original 1968 model placed the Z-score for every company in a sample into one of three regions: Scores above 2.99 were in the safe zone, meaning that the probability of default was relatively small; scores between 1.80 and 2.99 were in the grey zone, with a slightly higher probability of default; and scores below 1.80 were in the distress zone, with a clear warning of impending default. In those days, rarely did a company survive when its Z-score was below 1.80. I do not use these three zones as much anymore. Instead, Z-scores are mapped to bond rating equivalents. Table 1 shows the mapping of the Z-scores of companies in the S&P 500 Index to the initial bond rating provided by Standard & Poor s for the 1996 2001 period. Interestingly, in 1968, a company with a Z-score of 1.80 would have been labeled distressed ; but today, that same Z-score denotes a company with a BRE of B. Over the entire sample period, Z-scores at or below 0.20 correspond to a BRE of D, which would be a distressed company. Obviously, like any other statistical investigation, the Z-score and BRE will be sensitive to the start and end dates of the sample period, which are chosen by the analyst. Table 1. Average by S&P Bond Rating, 1996 2001 Rating No. of Firms Average Standard Deviation AAA 66 6.20 3.06 AA 194 4.73 2.36 A 519 3.74 2.29 BBB 530 2.81 1.48 BB 538 2.38 1.85 B 390 1.80 1.91 CCC + CC 9 0.33 1.16 D 150 0.20 Probability of Default over an Investment Horizon Given a BRE, the lengthy default history provided by the rating agencies can be exploited. One way to do so is to calculate mortality rates by bond rating for companies over a period of time. Think of mortality rates of bonds like an insurance actuary thinks of mortality rates of people. It is one thing to say that a company has a certain default rate on average, but investors may want to know what the default rate will be over an investment horizon say one year after purchase or two years after purchase. Note that this 2006, CFA Institute cfapubs.org MARCH 2006 67

CFA Institute Conference Proceedings Quarterly analysis ignores subsequent rating changes and instead (as investors do) focuses on the initial rating at the time the investment decision is made. Thus, the probability of default at any time for one year ahead, known as the marginal mortality rate (MMR) of default, can be computed as MMR () t Total value of defaulting debt in year t = Total value of population at start of year t, and the cumulative mortality rate (CMR) over the entire investment horizon can then be computed if all the prior MMRs are known and converted into survival rates (SRs): CMR() t = 1 SR() t, t = 1, where CMR (t) = cumulative mortality rate in t SR (t) = survival rate in t, 1 MMR (t) When calculating MMRs and CMRs, the population of bonds is reduced in dollar terms for the next period if the issuer defaults, if the issue is called, or if the principal outstanding is reduced through a sinking fund provision. For example, $1.5 billion of bonds are issued in a particular year and are rated BB. In the first year, $50 million of bonds default, and therefore, the MMR is 3.3 percent $50 million divided by $1,500 million. Also in the first year, another $125 million of bonds goes out of existence for other noncredit reasons, such as calls and sinking funds. So, the original cohort loses $175 million of the original $1,500 million. Therefore, at the beginning of the second year, the value in the denominator of the MMR calculation is $1,325 million, not $1,500 million. If in the second year $100 million of bonds default, then the MMR is 7.5 percent $100 million divided by the reduced cohort of $1,325 million. In the first year, the SR is 1 minus the MMR or 1 0.033 (i.e., 0.967 or 96.7 percent), and the SR in the second year is 1 0.075 (i.e., 0.925 or 92.5 percent). Therefore, the CMR in the second year equals 1 minus the product of the MMRs or 1 (0.967 0.925), which equals 0.1055 or 10.55 percent. Table 2 shows MMRs and CMRs for initial ratings of corporate bonds over the sample period from 1971 to 2004. MMRs are low in the first year because companies retain some cash from the issue to make their first few interest payments. In the earlier years, one generally observes increasing MMRs, and the pattern for B issues demonstrates this tendency. But notice the MMR pattern for BBB issues: 0.36 percent one year after issuance, a spike to 3.22 percent two years after issuance, then falling to 1.43 percent three years after issuance. Clearly, something happened to a large BBB rated bond issue two years after issuance: In 2000, WorldCom issued several billion dollars of BBB rated bonds, and the company defaulted in 2002. This default highlights that the weights of issues in this analysis are dollar weighted. So, all else being equal, a $30 billion default will have 10 times the influence on MMR as a $300 million default will. Applications The July 1986 bankruptcy of steel producer LTV Corporation is shown in Figure 2. 3 According to the original Z-score model (the one that used the three zones), the company at the start of the 1980s was in the safe zone (or the top of the grey zone); it then fell into the distress zone and stayed there four years until the company defaulted. Although the Z-score prediction was correct, the timing was wrong. Still, one day before its bankruptcy announcement, LTV s bonds were selling at $83 and a Wall Street investment bank issued a buy recommendation for LTV s bonds. On the next day, the bond price fell to $35. Those investors who bought the bonds on the basis of that recommendation lost 50 percent of their money within 24 hours. The model s output could have been helpful to challenge the analyst s recommendation. Figure 3 shows IBM Corporation, which did not go bankrupt. IBM was rated AAA, and it deserved to be AAA based on its Z-score in 1983. The external rating remained at AAA until 1993, when the company was downgraded to AA and then to A. At the time, the Z-score was deteriorating with BREs falling below investment grade until the company was restructured under its new CEO, Louis ( Lou ) Gerstner. Since then, the company has displayed a dramatic turnaround with commensurate improvements in its Z-score. This analysis can be used not only for bankruptcy prediction but also for the tactical investment strategy of buying quality junk and selling junk quality. In other words, buy the junk bonds that look like investment grade and sell the investment grades that look like junk. Conclusion The Z-score model is 85 90 percent accurate. It is easy to use, and it is available at a low cost. The model is not perfect because it has high Type II errors. A Type II error occurs when the model says a company is in distress but the company does not go bankrupt. Still, Type II errors are cheaper than Type I errors, in which an investor buys a bond because of a favorable Z-score and the bond eventually defaults. That is an expensive error. Type II errors, in which the investor does not 3 LTV at the time was one of the longest (1985 1993) nonfinancial, nonrailroad bankruptcies. The company again went to bankruptcy in December 2000. 68 MARCH 2006 2006, CFA Institute cfapubs.org

Estimating Default Probabilities of Corporate Bonds over Various Investment Horizons buy the bond and it does not default, cost the investor only the opportunity cost on the lost yield. Therefore, with a Type II error, all the investor has lost is the little bit of time it took to try these models. Even though the timing of an eventual bankruptcy can be wrong, a low BRE can challenge an analyst who recommends a buy of a stock or bond when the model indicates the reverse. Although any model is but one of many tools used in making credit decisions, with the model I have outlined, investors can scrutinize an issuer whose fundamentals appear sound but whose Z-score and BRE indicate distress. This article qualifies for 0.5 PD credits. Table 2. Mortality Rates by Original Rating for All Rated Corporate Bonds by Years after Issuance, 1971 2004 (percent) Rating/ Mortality Rates 1 Year 2 Years 3 Years 4 Years 5 Years 6 Years 7 Years 8 Years 9 Years 10 Years AAA MMR 0.00 0.00 0.00 0.00 0.03 0.00 0.00 0.00 0.00 0.00 CMR 0.00 0.00 0.00 0.00 0.03 0.03 0.03 0.03 0.03 0.03 AA MMR 0.00 0.00 0.32 0.16 0.03 0.03 0.00 0.00 0.03 0.02 CMR 0.00 0.00 0.32 0.48 0.51 0.54 0.54 0.59 0.57 0.59 A MMR 0.01 0.10 0.02 0.09 0.06 0.11 0.06 0.21 0.11 0.06 CMR 0.01 0.11 0.13 0.22 0.28 0.39 0.45 0.65 0.76 0.82 BBB MMR 0.36 3.22 1.43 1.28 0.77 0.45 0.20 0.20 0.14 0.40 CMR 0.36 3.56 4.49 6.16 6.89 7.31 7.50 7.68 7.87 8.18 BB MMR 1.19 2.48 4.40 2.01 2.51 1.16 1.60 0.88 1.70 3.60 CMR 1.19 3.64 7.88 9.74 12.00 12.93 14.36 15.07 16.52 19.60 B MMR 2.85 6.85 7.40 8.55 6.00 4.16 3.72 2.28 1.96 0.86 CMR 2.85 9.51 16.20 23.37 27.94 30.96 33.46 34.97 36.25 36.80 CCC MMR 7.98 15.57 19.55 12.10 4.26 9.45 5.60 3.15 0.00 4.28 CMR 7.98 22.31 37.50 45.06 47.37 52.35 55.01 56.43 56.43 58.30 Note: Based on 1,796 issues rated by Standard & Poor s at issuance. Source: Standard & Poor s (New York) and author s compilation. 2006, CFA Institute cfapubs.org MARCH 2006 69

CFA Institute Conference Proceedings Quarterly Figure 2. : LTV Corporation, 1980 1987 3.5 Safe Zone 3.0 2.5 Grey Zone 2.0 1.5 Distress Zone 1.0 0.5 0 0.5 1.0 1.5 80 81 82 83 84 85 86 87 Note: LTV filed for bankruptcy in July 1986. Figure 3. : IBM Corporation, 1980 2002 6.0 5.5 5.0 4.5 4.0 Safe Zone 3.5 3.0 2.5 2.0 Grey Zone 1.5 Distress Zone 1.0 0.5 0 80 82 84 86 88 90 92 94 96 98 00 02 Notes: In January 1993, IBM was downgraded from AAA to AA. In July 1993, it was further downgraded to A. 70 MARCH 2006 2006, CFA Institute cfapubs.org

Q&A: Altman Question and Answer Session Edward I. Altman Question: How do you correct accounting variables that are subject to manipulation by the company s management? Altman: I am more concerned with selecting the right variable than with fine-tuning it by making ad hoc adjustments for different companies. Still, adjustments can be made to companies across an entire industry. For example, for department stores, I would capitalize all operating leases and add them to the company s debt, which will affect relevant accounting ratios. Question: If an issue s credit has migrated, say from BBB to BB then to B, have you then calculated MMR and CMR conditional on subsequent movements? Altman: No, that is much more complex than the analysis I have presented here. And although that analysis can be conducted, the point that I stress is that investors are concerned about initial ratings at the beginning of their investment horizon. Moreover, that s when I believe the rating agencies do their best job at accurately ascribing a rating to an issue. The agencies aren t great at re-ratings, so even if those conditional probabilities were available, they probably wouldn t be as accurate as the MMRs and CMRs that are based on an initial rating. Question: How do you define default in your default rate analysis? Altman: There are three definitions: bankruptcy; missed interest payments that remain unpaid over the grace period; and distressed restructuring, where the bondholders agree to a lower-priority security, such as equity in place of the debt, an extension of the time to pay, or a reduction in the interest rate. Incidentally, about 40 percent of the time default occurs before a company files for Chapter 11 bankruptcy protection. Question: Are recovery rates stable across time? Altman: As a rule of thumb, bonds that default sell for around 40 cents on the dollar just after default. But, of course, that figure varies with time. In 2001, the average price of a defaulted bond was 25 cents on the dollar just after default; today, less than four years later, it is 60 cents on the dollar. Generally, bondholders whose issues go into default do not usually lose everything because a significant chance of recovery exists. Question: How do you compute a Z-score for a private company? Altman: Use the Z-double-prime model, which replaces the market value of the company s equity with book value in the fourth discriminant independent variable. This model can also be applied to nonmanufacturers, and we have found high accuracy for it when applied to emerging market debt in countries such as Argentina, Brazil, and Mexico. Question: What is the latest trend in corporate bond investing? Altman: Distressed debt investing, or vulture investing, is hot right now in investment banking. I define distressed bonds as those selling at more than 1,000 bps over comparable U.S. Treasuries. This niche sector today is worth about $720 billion in face value and about $540 billion in market value, down slightly from a few years ago. Distressed debt is a hybrid of fixedincome and equity analysis because the value of the debt going through bankruptcy is based on the value of equity coming out of bankruptcy. Skills that are important here, such as legal and negotiating prowess, are more complex than those needed for considering either debt or equity alone. Question: How can you perform discriminant analysis? Altman: Plenty of low-cost solutions can be launched within familiar spreadsheet packages, such as Microsoft Excel. Many statistical and econometric software packages will include a module for discriminant analysis. Question: Under Basel II, how might banks alter their lending terms with small- and mediumsized enterprises (SMEs)? Altman: We have conducted research on this topic (available on my website), and we believe that, although the lending rate to SMEs could increase, these enterprises shouldn t be hurt by the proposed risk-adjusted capital requirements. Indeed, on average, they should benefit because of lower capital requirements on SME lending, especially if the credit can be considered a retail-related one. 2006, CFA Institute cfapubs.org MARCH 2006 71