CREDIT RISK: QUANTITATIVE APPROACHES

Transcription

1 sau71566_ch13.qxd 9/5/00 2:48 PM Page 317 CHAPTER 13 CREDIT RISK: QUANTITATIVE APPROACHES Learning Objectives This chapter will formalize the concepts of credit risk measurement in quantitative terms. It will introduce you to several models of loan pricing. We will discuss Pricing of a loan in a risk neutral setting The interaction of probability of default and loss given default in loan pricing The portfolio effect in a fixed income context Two full models of credit risk monitoring Creditmetrics Credit Risk Introduction Chapter 12 introduced credit risk measurement as an art. Although we referred to several quantitative techniques in the chapter, we made no attempt to unify credit risk measurement into a single model that the FI could implement. Our emphasis was on the diversity of the problem. Yet as FIs become increasingly large and complex, they seek stable, reliable quantitative models that can be implemented with the aid of computers and information technology regardless of the artistic talents of employees. 1 Throughout this chapter, we have recourse to three concepts, which define the credit risk problem. As we will rediscover in Chapter 16, Insurance Risk, these three concepts are analogous to those confronted by the insurance underwriter. The concepts are probability of default (called in insurance loss frequency), loss given default (called in insurance loss severity) and the correlations of default on individual credits (or the distribution of the loss function). Probability of default is simply the chance that the borrower will not make payment of interest and principal when due. In Chapter 12, we usually approached the question from the point of view of the credit officer who set up policies to minimize the possibility of default on any given credit. In this chapter, we invert the question and say, Given the suc- 1 For a discussion of these methods, see Anthony Saunders, Credit Risk Measurement, John Wiley & Sons, New York,

2 sau71566_ch13.qxd 9/5/00 2:48 PM Page Part II Measuring Risk cessful implementation of our credit system, there will still exist a probability that the borrower will default. What is that probability, and how can we use it to price loans and mitigate the inevitable credit risk associated with it? Loss given default takes the question one step further. Again, in Chapter 12, we were concerned that loans be structured with appropriate collateral so that they would, even in default, involve as little loss as possible for the lender. Assuming the successful implementation of those structures, however, defaults will still occur. Following such defaults, the FI will implement its policies (pursuing the borrower taking appropriate measures such as rescheduling, realization of collateral, use of collection agencies, bankruptcy proceedings, etc.) to realize some recovery on each of those defaulted loans. That recovery will, at most, be 100 percent of the principal and accrued interest lent and, at least, be zero. Loss given default is one minus the percentage recovery. Correlations among defaults and losses given default. As Figure 12 1 demonstrates, credit portfolio values are pro-cyclical. A rigorous model of credit risk must consider not only how each credit individually performs. It must also consider how default rates and losses given default will, because of industry, country-wide or world-wide correlated economic factors, move together across loans. Clearly, there are very high analytical and data demands in order to describe realistically these three parameters. In this chapter, we will first look at the components of the promised return on a single loan; it is a combination of promised interest rate, implicit and explicit fees and reserve requirements. Then we will examine the pricing of a single loan. Although simple, the model suggests that the manager is able to price default risk. This is true, but since loan risk pricing exists together with credit rationing in the market, we examine the reason for credit rationing. The simple model also assumes we already know loan loss frequency and severity and we ignore loan portfolio effects. In reality, of course, the FI must estimate those parameters. We then turn to models that estimate the probabilities of default and data provided by bond credit rating agencies to build on several pricing models of debt. Following that, we will apply a simple binomial model to illustrate portfolio effects. Finally, the chapter describes two models of credit risk measurement. The approach we prescribe in this chapter is rigorously quantitative. Just because these models are numerical and complex, however, does not mean that they are always right or indeed always better than the generic approaches described in Chapter 12. As the Professional Perspectives box counsels, the analyst should be skeptical of the promises of black box models: critical examination is necessary. The Return on a Loan The Contractually Promised Return on a Loan A number of factors affect the promised gross return an FI achieves on any given loan. 2 These factors include: 1. The interest rate on the loan. 2. Any fees relating to the loan. 3. The credit risk of the loan. 2 To calculate the net return, you must also account for the cost to the FI of funding the loan and the noninterest expenses of booking and servicing it.

3 sau71566_ch13.qxd 9/5/00 2:48 PM Page 319 Chapter 13 Credit Risk: Quantitative Approaches 319 Professional Perspectives Model Behaviour Banks credit-risk models are mind-bogglingly complex. But the question they try to answer is actually quite simple: how much of a bank s lending might plausibly turn bad? Armed with the answer, banks can set aside enough capital to make sure they stay solvent should the worst happen. No model, of course, can take account of every possibility. Credit-risk models try to put a value on how much a bank should realistically expect to lose in the 99.9% or so of the time that passes for normality. This requires estimating three different things: the likelihood that any given borrower will default; the amount that might be recoverable if that happened; and the likelihood that the borrower will default at the same time others are doing so. This last factor is crucial. In effect, it will decide whether some unforeseen event is likely to wreck the bank. Broadly speaking, the less likely it is that many loans will go bad at the same time that is, the lower the correlation of the individual risks the lower the risk will be of a big loss from bad loans. None of this is easy to do. Many of the banking industry s brightest rocket scientists have been given over to the task. Credit Suisse Financial Products has launched Credit- Risk, which attempts to provide an actuarial model of the likelihood that a loan will turn bad, much as an insurance firm would produce a forecast of likely claims. McKinsey, a consultancy, has a model that links default probabilities to macroeconomic variables, such as interest rates and growth in GDP. J. P. Morgan s Credit-Metrics applies a theoretical model of when borrowers default, using credit ratings for bonds and drawing on another model developed by KMV, a Californian firm, which calculates the risk that a firm will default by looking at changes in the price of its shares. With the help of Taylor-series expansions, Gamma integrals, negative binomial distributions and so forth (we ll spare you the details), the models go from calculating the probability that any one borrower will default, to estimating the chances that Wal-Mart, say, will default at the same time as Woolworth or that loans to French property developers will go bad at the same time as loans to Air France. This leads to a series of loss probabilities for the bank s entire portfolio of loans. This will indicate the maximum loss that the bank needs to prepare for by setting aside capital. Last year s model Credit-risk models have evolved from value-at-risk models, which were developed to estimate how much of a bank s trading portfolio foreign exchange, cash, securities and derivatives it could lose in a single day because of adverse movements in financial prices. These models have been criticised for assuming that past correlations in the prices of different assets will hold in future and for making simplistic assumptions about the range of possible price changes. They also fail when prices for the underlying assets become unavailable when a stockmarket suspends trading, for example. These criticisms apply just as well to credit-risk models. Value-at-risk models have one big advantage over creditrisk models, however. They generally deal with assets that are publicly traded, so there is a vast amount of data for the models to crunch. It is far harder to come up with data on the market value of loans or on how much of the value of bad loans banks eventually recover. That leaves it uncertain whether the results cranked out by credit-risk models are statistically valid. The models are clever, all right. But how much relation they bear to reality may not be clear until after the next recession. Source: The Economist, February 28, 1998, p The Economist Newspaper Group, Inc. Reprinted with permission. Further reproduction prohibited. Compensating Balances A proportion of a loan that a borrower is required to hold on deposit at the lending institution. 4. The collateral backing of the loan. 5. Other nonprice terms (especially compensating balances and reserve requirements). Compensating balances are a proportion of a loan that a borrower cannot actively use for expenditures. Instead, these balances have to be kept on deposit at the FI. For example, a borrower facing a 10-percent compensating balance requirement on a $100 loan would have to place $10 on deposit (traditionally on demand deposit) with the FI and could use only $90 of the $100 borrowed. This requirement raises the effective cost of loans for the borrower, since the deposit rate earned on compensating balances is less than the borrowing

4 sau71566_ch13.qxd 9/5/00 2:48 PM Page Part II Measuring Risk rate and may, in the case of a demand deposit, be zero. Thus, compensating balance requirements act as an additional source of return on lending for an FI. 3 Consequently, while credit risk may be the most important factor ultimately affecting the return on a loan, FI managers should not ignore these other factors when evaluating loan profitability and risk. Indeed, FIs can compensate for high credit risk in a number of ways other than charging a higher explicit interest rate or risk premium on a loan or restricting the amount of credit available. In particular, higher fees, high compensating balances, and increased collateral backing all offer implicit and indirect methods of compensating an FI for lending risk. Next we look at an example of how an FI can vary the promised return on a commercial loan by choosing specific noninterest rate terms. Suppose an FI makes a spot one-year, $1 million loan. The loan rate is set at Base lending rate 12% L Spread 2% m 14% L m The base lending rate (L) is either the market-determined marginal cost of funds for the FI or an FI-determined basic lending rate that is, prime. In Canada, the BA rate is used to proxy the market-determined marginal cost of funds for banks. In other markets, an interbank deposit rate is used. For example, the offshore markets frequently use the London Interbank Offered Rate (LIBOR). Alternatively, the base lending rate used could be the prime lending rate announced by the FI itself. Traditionally, the prime rate is the rate charged to the FI s lowest-risk customers. In the United States now, however, it is more of a base rate from which positive or negative risk premiums can be added. There, the best and largest borrowers are now commonly required to pay below U.S. prime so as to be competitive with the commercial paper market. 4 In Canada, subprime loans are not explicitly offered, but loans priced against market benchmarks (such as the BA rate), rather than prime, can achieve the same effect. In Figure 13 1, you can see the extent to which the prime rate exceeds an FI s cost of funds: 1.4 percent on average over the last 15 years. Prime represents the cost of funds for the FI plus a spread representing the minimal credit risk of a prime corporate borrower plus the noninterest expenses incurred by the FI in obtaining and servicing plus a profit margin. 5 Suppose the FI also 1. Charges a 1 8-percent loan origination fee (f) to the borrower. 2. Imposes a 10-percent compensating balance requirement (b) to be held as noninterest-bearing demand deposits. 3. Pays reserve requirements (R) of 10 percent imposed by the Federal Reserve on the FI s demand deposits, including any compensating balances. 3 They also create a more stable supply of deposits, thus mitigating liquidity problems. Note that if the FI wishes to require compensating balances, this requirement must be set out with other loan terms and conditions in the loan agreement signed by the borrower and the FI. 4 For more information on the prime rate, see P. Nabar, S. Park, and A. Saunders, Prime Rate Changes: Is There an Advantage in Being First? Journal of Business 66, 1993, pp ; and L. Mester and A. Saunders, When Does the Prime Rate Change? Journal of Banking and Finance 19, The reader will note from Figure 13 1 that prime changes infrequently relative to the market rates of interest and that prime has fallen below the banks short-term cost of funds at several times in the last decade. For example, in the October 1992 week following the Charlottetown Accord s electoral defeat (which coincided with Clinton s election in the U.S.), short-term interest rates rapidly increased by approximately 2 percent before dropping back to former levels, while prime stayed relatively stable.

5 sau71566_ch13.qxd 9/5/00 2:48 PM Page 321 Chapter 13 Credit Risk: Quantitative Approaches 321 FIGURE 13 1 Canadian prime s spread over bank cost of funds Percent Jan-85 Jan-87 Jan-89 Jan-91 Jan-93 Jan-95 Jan-97 Jan-99 Jan-86 Jan-88 Jan-90 Jan-92 Jan-94 Jan-96 Jan-98 Jan-2000 Year Prime Interbank funds Spread Then the contractually promised gross return on the loan, k, per dollar lent would equal: 6 f (L m) 1 k 1 1 [b(1 R)] This formula may need some explanation. The numerator is the promised gross cash inflow to the FI per dollar, reflecting fees plus interest. In the denominator, for every $1 in loans the FI lends, it retains b as noninterest-bearing compensating balances. Thus 1 b is the cash outflow from the FI (ignoring reserve requirements). However, since b (compensating balances) are held by the borrower at the FI as demand deposits, the FI holds noninterest-bearing reserves at the rate R against these compensating balances. Thus, the net benefit from compensating balances has to take into account noninterest-bearing reserve requirements. The net outflow by the FI per $1 of loans is 1 [b(1 R)], or 1 minus the reserve-adjusted compensating balance requirement. 7 6 This formula ignores present-value aspects that could easily be incorporated. For example, fees are earned in upfront undiscounted dollars, while interest payments and risk premiums are normally paid on loan maturity and thus should be discounted by the FI s cost of funds. 7 Note that downward adjustment of compensating balances to account for noninterest-bearing reserves held for liquidity reasons is excessive if the reserves are interest-paying secondary or buffer reserve assets (see Chapter 7). If the reserves are interest-bearing, compensating balances should be reduced only to the extent that they earn an expected risk-adjusted rate of interest lower than that of the FI s other assets. The specification above is most accurate in jurisdictions (such as the U.S.) where reserves are required and must be primary (i.e., noninterest-bearing). Note that, although provincially regulated credit unions have reserve requirements, those reserves may be kept in interest-bearing securities. Where risk-free reserves bear interest at the risk-free market rate (e.g., T-bills), the alternative specification (1 k) 1 [f (L m)]/(1 b) is appropriate.

6 sau71566_ch13.qxd 9/5/00 2:48 PM Page Part II Measuring Risk Plugging in the numbers from our example into this formula, we have ( ) 1 k 1 1 [(0.10)(0.9)] k k , or k 15.52% This is, of course, greater than the simple promised interest return on the loan, L m 14 percent. In the special case where fees are zero (f 0) and the compensating balance is zero (b 0), the contractually promised return formula reduces to 1 k 1 (L m) That is, the spread (m) is the fundamental factor driving the promised return on a loan once the base rate on the loan is set. Note that as credit markets become more competitive, both origination fees (f) and compensating balances (b) are becoming less important. For example, where compensating balances are still charged, the FI may now require them to be held as interest-earning time deposits. As a result, borrowers opportunity losses from compensating balances have been reduced to the difference between the loan rate and the compensating balance timedeposit rate. Further, in nondomestic dollar loans made offshore, compensating balance requirements are very rare. 9 The Expected Return on the Loan The promised return on the loan (1 k) that the borrower and lender contractually agreed upon includes both interest rate and noninterest rate features such as fees. Therefore, it may well differ from the expected and, indeed, actual return on a loan. Default risk is the risk of the borrower being unable or unwilling to fulfill the terms promised under the loan contract. It is usually present to some degree in all loans. Thus, at the time the loan is made, the expected return (E(r)) per dollar loaned is related to the promised return by E(r) p (1 k) where p is the probability of repayment of the loan. 10 This specification assumes only two possible futures: on the day of loan maturity, either the borrower defaults and pays nothing 8 If we take into account the compounding effects on the fees and if we assume that FI s discount rate (d) was percent, then the f term needs to be compounded by 1 d In this case, k is percent. 9 For a number of interesting examples using similar formulas, see John R. Brick, Commercial Banking: Text and Readings (Haslett, Mich.: Systems Publication Inc., 1984), ch. 4. If compensating balances held as deposits paid interest at 8 percent (r d 8%) then the numerator (cash flow) of the FI in the example would be reduced by b r d, where r d 0.08 and b 0.1. In this case, k percent. This assumes that the reserve requirement on compensating balances held as time deposits (R) is 10 percent. However, while currently in the U.S. reserve requirements on demand deposits are 10 percent, the US reserve requirement on time deposits is 0 percent (zero). Recalculating, but assuming R 0 and interest of 8 percent on compensating balances, we find k percent. 10 This specification is the simplest usable model of expected return. A slightly more complex model is S E[R] p s (1 k s ) s 0 where there are S possible outcomes, each yielding a 1 k, return where k s 1. Here partial repayment is possible. One could further complicate the picture by modelling payments occurring at different times in the future or positing continuous states of the world in continuous time (necessitating integration over states and times).

7 sau71566_ch13.qxd 9/5/00 2:48 PM Page 323 Chapter 13 Credit Risk: Quantitative Approaches 323 Binomial Distribution A statistical distribution of random, independent events with only two possible outcomes, success or failure, where the probability of success in each event is p and the probability of failure is (1 p). or the borrower repays totally the principal and accrued interest (and fees if charged). This means that the performance of loans in a portfolio can be described as a binomial distribution, a statistical model to which we will return later in this chapter. The probability of no loss is p. The probability of loss on the loan is (1 p). In this simplest case, we assume that the loss given default (i.e., the severity) is 100 percent of the principal plus accrued interest. To the extent that p is less than 1, default risk is present. This means the FI manager must (1) set the spread (m) sufficiently high to compensate for this expectation and (2) recognize that setting high spreads as well as high fees and base rates may actually reduce the probability of repayment (p). That is, k and p are not independent. Indeed, over some range, they may be negatively related. As a result, FIs usually have to control for credit risk along two dimensions: the price or promised return dimension (1 k) and the quality or credit availability dimension. In general, the quantity dimension controls credit risk differences on retail loans more than the price dimension when compared to wholesale loans. We discuss the reasons for this in the next section. The section after it evaluates different ways in which FI managers can assess the appropriate size of m, the spread on a loan. This is the key to pricing wholesale loan and debt risk exposures correctly. Concept Questions 1. Calculate the promised return (k) on a loan if the base rate is 13 percent, the spread is 2 percent, the compensating balance required is 5 percent, fees are 1 2 percent, and noninterest-bearing reserves on compensating balances are 10 percent. 2. What is the expected return on this loan if the probability of default is 5 percent? Credit Rationing In the simple world discussed above, if an FI manager knew the expected probability of repayment and loss given default (above, we assumed it was complete loss), she could adjust the default risk premium she charges to achieve the expected return she requires on a loan. In practice, however, the manager is limited in the extent to which she can alter the promised rates of return. These limits differ among different classes of loans. Retail Because of their small dollar size in the context of an FI s overall investment portfolio and the higher costs of collecting information on household borrowers, loan decisions made for many types of retail loans are reject or accept decisions. All borrowers who are accepted are often charged the same rate of interest and by implication the same risk premium. For example, a wealthy individual borrowing from a bank, trust, or finance company to purchase a Cadillac is likely to be charged the same auto loan rate as a less wealthy individual borrowing to finance the purchase of a Geo. In the terminology of finance, retail customers are more likely to be sorted or rationed by loan quantity restricts rather than by price or interest rate differences The wealthy individual, however, may forgo the standardized auto loan in favour of a cheaper personal loan from an FI that holds sufficient liquid assets of the borrower to render the loan virtually riskless. Moreover, the volume of business that individual brings to her main FI may allow her to negotiate a considerable reduction in loan fees, further reducing total borrowing costs.

8 sau71566_ch13.qxd 9/5/00 2:48 PM Page Part II Measuring Risk Residential mortgage loans provide another good example. While two borrowers may be accepted for mortgage loans, an FI discriminates between them according to the loan/price ratio the amount it is willing to lend relative to the market value of the house being acquired rather than by setting a different mortgage rate. 12 Credit Rationing Restrictions on the quantity of loans made available to an individual borrower or to a category or class of borrowers. Wholesale Generally, at the retail level, an FI controls its credit risks by credit rationing rather than by using a range of interest rates or prices. At the wholesale level, FIs use both interest rates and credit quantity to control credit risk. Thus banks quote a prime lending rate (L) to certain high-quality borrowers, while more risky borrowers are charged prime plus a spread, or a default risk premium (m), to compensate the FI for the additional risk involved. FIs may be willing to lend funds to lower-quality wholesale borrowers as long as they are compensated by high enough interest rates. But too-high lending rates can backfire on the FI, as discussed earlier. For example, a borrower charged 15 percent for a loan a prime rate of 10 percent plus a spread of 5 percent may be able to repay only by using the funds to invest in highly risky investments with some small chance of a big payoff. However, by definition, many high-risk projects fail to pay off and the borrower may default. In an extreme case, the FI receives neither the promised interest nor the original principal lent. This suggests that very high contractual interest rate charges on loans may actually reduce an FI s expected return on loans, because high interest rates induce the borrower to invest in risky projects. 13 Alternatively, only borrowers with risky projects might be interested in borrowing at high interest rates; low-risk borrowers might drop out of the potential borrowing pool at high-rate levels. This lowers the average quality of the pool of potential borrowers. We show these effects in Figure At very low contractual interest rates (k), borrowers do not need to take high risks in their use of funds and those with relatively safe investment projects use bank financing. As interest rates increase, borrowers with fairly safe low-return projects no longer think it is profitable to borrow from FIs. Hence we observe that firms typically borrow less as interest rates rise. This is the reason behind the inverse relationship between real interest rates and economic growth. Alternatively, however, a borrower may switch to higher-risk investment projects to have a chance of being able to pay off the loan. In terms of Figure 13 2, when interest rates rise above k*, the additional expected return earned by the FI through higher interest rates (k) is increasingly offset by an increase in the expected default risk on the loan (1 p). In other words, an FI charging wholesale borrowers loan rates in the 9- to 15-percent region can earn a lower expected return than an FI charging 8 percent. This relationship between interest rates and the expected returns on loans suggests that beyond some interest rate level it may be best for the FI to credit ration its wholesale loans; that is, not to make loans or to make fewer loans. Rather than seeking to ration by price by charging higher and higher risk premiums to borrowers, the FI can establish an upper ceiling on the amounts it is willing to lend to maximize its expected returns on lending. In the context of Figure 13 2, borrowers may be charged interest rates up to 8 percent, with the 12 However, as the cost of information falls and comprehensive databases on individual households creditworthiness are developed and as FIs compute with increasing accuracy individual account RAROCs, the size of loan for which a borrower-specific interest rate becomes optimal will shrink. 13 In the context of the previous section, a high k on the loan, reflecting a high base rate (L) and risk premium (m), can lead to a lower probability of repayment (p) and thus a lower E(r) on the loan, where E(r) p(1 k). 14 See also J. Stiglitz and A. Weiss, Credit Rationing in Markets with Imperfect Information, American Economic Review 71, 1981, pp

9 sau71566_ch13.qxd 9/5/00 2:48 PM Page 325 Chapter 13 Credit Risk: Quantitative Approaches 325 FIGURE 13 2 The relationship between the promised loan rate and expected return on the loan Expected return on loan ( p(1 + k)) percent k* Promised loan rate (k) percent most risky borrowers also facing more restrictive limits or ceilings on the amounts they can borrow at any given interest rate. In the context of Canadian banking practice, Hatch and Wynant report that only 5 percent of loans carry spreads in excess of 3 percent over prime clear evidence of credit rationing by banks to limit risky lending. 15 Concept Questions 1. Can a bank s return on its loan portfolio increase if it cuts its loan rates? 2. What might happen to the expected return on a wholesale loan if a bank eliminates its fees and compensating balances in a low-interest-rate environment? Estimating the Probability of Default Economists, bankers, and analysts have employed many different models to assess the default risk on loans and bonds. These vary from the relatively qualitative to the highly quantitative. These models are not mutually exclusive; an FI manager may use more than one to reach a credit pricing or loan quantity rationing decision. We analyze a number of these models in the next sections, starting with the most simple. Credit Scoring Models In Chapter 12, we noted that credit scoring models are frequently used by FIs evaluating applicants for mortgage and credit card loans. Credit scoring models use historical data on observed borrower characteristics to calculate the probability of default or to sort borrowers 15 See James E. Hatch and Larry Wynant, Canadian Commercial Lending (Toronto: Carswell, 1995), pp. 2,

10 sau71566_ch13.qxd 9/5/00 2:48 PM Page Part II Measuring Risk into different default risk classes. By selecting and combining different economic and financial borrower characteristics, an FI manager may be able to: 1. Numerically establish which factors are important in explaining default risk. 2. Evaluate the relative degree or importance of these factors. 3. Improve the pricing of default risk. 4. Be better able to screen out bad loan applications. 5. Be in a better position to calculate provisions needed to meet expected future loan losses. To employ credit scoring models in this manner, the manager must identify objective economic and financial measures of risk for any particular class of borrower. For consumer debt, the objective characteristics in a credit scoring model might include income, assets, age, occupation, length of employment, home ownership, length of residence, debt-service ratios, number of finance company loans, number of credit cards, credit card balances, and marital status, among other factors. Additional inputs to models include frequency of credit inquiries and uses of cash advantages, variables whose high values may indicate excessive credit seeking behaviour. While all of the large credit-granting FIs in Canada currently use credit scoring models to assess consumer credit risk, many use them to assess small- and mid-market corporate risk as well. Credit scoring models for corporate customers typically include financial ratios, type and size of business, length of time in business, assessment of managerial abilities, and other criteria. Credit scoring models have for corporate credit risk become increasingly popular since Edward Altman first published his classic Z-score linear discriminant model of bankruptcy prediction in Using data from U.S. publicly traded companies, Altman estimated a function to discriminate between 33 firms that went bankrupt and 33 firms that remained healthy, where each of the healthy firms was matched with a bankrupt firm of the same size and industry. The independent variables used to discriminate between the two samples were five ratios for each firm taken from the year before the failed firm s bankruptcy. The model he estimated was 16 See Edward I. Altman, Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy, Journal of Finance 23:4 (September 1968), pp For an application to commercial lending, see Altman, Managing the Commercial Lending Process in Handbook of Banking Strategy, eds. R. C. Aspinwall and R. A. Eisenbeis (New York: John Wiley, 1985), pp The discriminant model is a highly effective classification tool often used in the natural sciences to discriminate between two populations based on observed characteristics. Since Altman published his work, however, discriminant analysis has shown to be less useful than logit analysis in causal models such as the above (e.g., poor earnings cause bankruptcy). See Daniel McFadden, A Comment on Discriminant Analysis versus Logit Analysis, Annuals of Econometric and Social Measurement 5:4 (1976), pp Logit analysis constrains the cumulative probability of default on a loan to lie between zero and one by assuming the probability of default to be of the logistic functional form: 1 P 1 e BX where P is the probability of defaults, B is a 1 by n vector of coefficients, and X is an n by 1 vector of independent variables. An alternative to the logit specification is a probit analysis, which assumes that the probability of default is a cumulative normal distribution. The parameters are solved for by the method of maximum likelihood. See William Greene, Econometric Analysis (New York: Macmillan, 1990), ch. 20 for a very readable econometric discussion of logit and probit models. Greene explains how using an ordinary leastsquares linear regression of independent variables on a dependent variable that takes the value of zero or one can lead to a heteroscedastic error. For a more detailed treatment, see G. S. Madalla, Limited Dependent and Quantitative Variables in Econometrics (Cambridge: Cambridge University Press, 1983).

11 sau71566_ch13.qxd 9/5/00 2:48 PM Page 327 Chapter 13 Credit Risk: Quantitative Approaches 327 Out-of-Sample Performance Ability of a model to predict accurately an outcome from new data rather than from the data used in model estimation. Z 1.2X 1 1.4X 2 3.3X 3 0.6X 4 1.0X 5 where X 1 Working capital/total assets ratio X 2 Retained earnings/total assets ratio X 3 Earnings before interest and tax/total assets ratio X 4 Market value of equity/book value of debt ratio X 5 Sales/total assets ratio The higher the value of Z, the lower the default risk classification of the borrower. Suppose that the financial ratios of a potential firm took the following values: X X X X X The ratio of X 2 is zero and X 3 is negative, indicating that the firm has had negative earnings (losses) in recent periods. Also, X 4 indicates that the borrower is highly leveraged. However, the working capital ratio (X 1 ) and the sales asset ratio (X 5 ) indicate that the firm is reasonably liquid and is maintaining its sales volume. The Z score provides an overall indication of the borrower s credit risk, combining these five factors as follows: Z 1.2(0.2) 1.4(0.0) 3.3(.02) 0.6(0.1) 1.0(2.0) 1.64 According to Altman, any firm with a Z score of less than 1.81 should be placed in the high default region. The FI should not make a loan to this borrower until it improves it earnings. There are a number of problems with credit scoring models like the discriminant analysis model shown above. The first problem concerns its predictive ability. A model should be judged not just on its ability to predict the bankruptcy on the basis of the observations that were used to estimate the parameters of the model (i.e., the model s fit ) but also, and more practically, on predicting bankruptcies using new data. Altman s original discriminant analysis model s credibility rested in part on its out-of-sample performance, as documented in his original paper. An FI manager implementing a model today should always test it by analyzing its ability to predict new bankruptcies. The second problem concerns the applicability of the model. There is no obvious economic reason, for example, to expect that the independent variables Edward Altman selected that allowed him to distinguish between healthy an bankrupt publicly listed firms in the United States during the years 1946 and 1965 would be useful for distinguishing between healthy and problematic borrowers in Canada in the 21st century. Hence, it is somewhat surprising that his model was found to have considerable out-of-sample predictive ability using Canadian data from the 1970s. 17 Altman s results, then, provide a good start- 17 See Earl G. Sands and Gordon L. V. Springate, Predicting Business Failures: A Canadian Approach, CGA Magazine, May 1993, pp Altman and Lavallée conducted discriminant analysis on 21 Canadian firms that went bankrupt from 1970 to Although X 3 and X 5 were the same as in Altman s 1968 study, they used the current ratio instead of net working capital/assets; net profit/total debt instead of EBIT/assets; and growth rate of equity growth rate of assets instead of retained earnings/total assets. Coefficients differed considerably from those of the original Altman model, but the model had good predictive ability. See Un

12 sau71566_ch13.qxd 9/5/00 2:48 PM Page Part II Measuring Risk Watch-listed Account An account that is identified by the lending FI as having a high risk of future debt-service problems and therefore deserving of special monitoring. ing point for someone building a model today. Yet an FI manager should not just adopt his model but build her own tailored to the type of firm (or customer) and availability of information. Weights in a credit scoring model may not be constant over time, so any model should be regularly updated. A third problem with the models is that they discriminate only between two extreme values of borrower behaviour: default and no default. In the real world there are various gradations of borrower difficulty. An otherwise healthy account may be watch-listed by the lender because the company faces difficulties. The loan may experience minor delays in payment. The borrower may renegotiate repayment because of difficulties. The loan may be classified as nonperforming or the borrower may outright default on its obligations. This suggests that a more accurate, more finely calibrated sorting among borrowers may require defining more classes in the model. The fourth problem is that these models ignore important hard-to-quantify factors that may play a crucial role in the default/no default decision. For example, the reputation of the borrower and the implicit contractual nature of long-term borrower-lender relationships could be important characteristics, as could macroeconomic factors such as the phase of the business cycle. Although credit scoring models usually ignore these variables, they can be fruitfully employed by FI managers in conjunction with the results of models. Moreover, credit scoring models rarely use publicly available information, such as the prices in asset markets in which the outstanding debt and equity of the borrower are already traded. 18 A fifth problem relates to the functional form of the model. Inconveniently, the output of the model is not a probability. It is a score that must be transformed into a probability of default. Other models (see footnote 16) give a probability as their output. Usually models are designed as if the X j variables did not interact among each other. Recent work in nonlinear discriminant analysis has sought to relax this assumption. Although nonlinear relationships among variables can be explicitly laid out by the model builder, there may be some true interrelationships that are lost by not being modelled. Neural networks (computer algorithms seeking complex links between X j variables to improve classification power) show considerable promise. There is a cost, however: while neural networks improve predictive ability, they do not give the analyst meaningful parameters that allow satisfactory interpretation of the model. 19 A final problem relates to default records kept by FIs. There is currently no centralized database on defaulted loans, for proprietary and other reasons. While some task forces of consortiums of commercial banks and consulting firms in the United States are in the process of constructing such databases (e.g., the Corporate Credit Database project of the Commercial Bank Consortium), it may be years before they develop a sufficient data series. In Canada, we know of no such inter-fi initiative. This constrains the ability of FIs (especially smaller ones) to use traditional credit scoring models only insofar as the sizes of their internal databases permit. For consumer loans, their use is well established; for larger business loans, they are less useful. modèle discriminant de prédiction des faillites au Canada, Finance 1:1 (1980), pp The work, however, was criticized on econometric grounds by Riding and Job, The Prediction of Corporate Bankruptcy: A Canadian Context, Finance 5:1 (1984), pp For example, S.C. Gilson, K. John, and L. Lang show that three years of low or negative stock returns can usefully predict bankruptcy probabilities. In fact, this market-based approach supplements the market-based information models discussed in later sections. See An Empirical Study of Private Reorganization of Firms in Default, Journal of Financial Economics, 1990, pp See P. K. Coats and L. F. Fant, Recognizing Financial Distress Patterns: Using a Neural Network Tool, Working Paper, Department of Finance, Florida State University, September 1992, and New Tools for Routine Jobs, Financial Times, September 24, 1994.

13 sau71566_ch13.qxd 9/5/00 2:48 PM Page 329 Chapter 13 Credit Risk: Quantitative Approaches 329 FIGURE 13 3A U.S. Yield Spreads Over Aaa Corporate Debt Percent Year Aa A Baa Average corporate Note: Graphs show each long-term annual average Moody s corporate bond yield average minus the Moody s Aaa corporate annual average yield. Source: Bond Market Association Web site These criticism, however, denigrate neither quantitative techniques in general nor Altman in particular. Altman started (and is still active in) a whole bankruptcy predicting industry. We chose to present his 1960s default model because of its simplicity. Today, the analyst can pick from models of great complexity in a very large and growing field of bankruptcy prediction. It is incumbent on each FI that accumulates proprietary default data to analyze that data to help it improve pricing decisions and credit risk management. In the next section, we turn to using publicly available data to obtain estimates of borrower probabilities of default. Concept Question 1. Suppose that X in the preceding discriminant model example. Show how this would change the default risk classification of the borrower. Term Structure Derivation of Credit Risk Bond credit rating agencies, as we described in Chapter 12, are playing an increasing role in providing publicly available credit risk analysis, thereby reducing the informational asymmetries associated with credit risk. Their bond ratings, when combined with market rates of return of differently rated bonds, provide powerful tools for estimating probabilities of default of borrowers. The probabilities can be estimated by analyzing the term structure of credit risk. Look at these spreads for coupon-bearing bonds in Figure 13 3A, which shows the different yields of each risk class of U.S. corporate bond compared to Aaa-rated bonds. Figure 13 3B shows three risk classes of Canadian bonds: investment-grade corporate bonds,

14 sau71566_ch13.qxd 9/5/00 2:48 PM Page Part II Measuring Risk FIGURE 13 3B Long-term corporate and provincial bonds spreads over Canada bonds Percent Jan-80 Jan-82 Jan-84 Jan-86 Jan-88 Jan-90 Jan-92 Jan-94 Jan-96 Jan-98 Jan-00 Jan-81 Jan-83 Jan-85 Jan-87 Jan-89 Jan-91 Jan-93 Jan-95 Jan-97 Jan-99 Date Provincial spread Corporate spread Canada bond yield Canada Bond Strips and Zero-coupon Corporate Bonds Bonds that are created or issued bearing no coupons and only a face value to be paid on maturity. As such, they are issued at a large discount from face value. bonds issued by provincial governments, and Canada bonds (i.e., bonds issued by the federal government of Canada). Only the yield on Canada bonds is plotted, but the graph also shows the yield spreads of corporates and provincial bonds over Canada bonds. Those spreads have averaged 1 percent (for provincials) and 1.2 percent (for corporates) over the last 20 years. Similar yield spreads might also be constructed for Canada bond strips bonds issued by large provinces and public utilities (like Ontario Hydro) and zero-coupon corporate bonds if sufficient issues are available. Because strips and zero-coupon corporates are single-payment discount bonds, it is possible to extract required credit risk premiums and implied probabilities of default from actual market data on interest rates. That is, the spreads between risk-free Canada bond strips issued by the government and zero-coupon bonds issued by corporate borrowers of differing quality may reflect perceived credit risk exposures of corporate borrowers for single payments at different times in the future. Next, we look at the simplest case of extracting an implied probability of default for an FI considering buying one-year bonds from, or making one-year loans to, a risky borrower. Then we consider multiyear loans and bonds. In each case, we show that we can extract a market view of the credit risk the probability of default of an individual borrower. Probability of Default on a One-period Debt Instrument. Assume that the FI requires an expected return on a one-year corporate debt security equal to the risk-free return on Canada bonds of one year s maturity. This implies that the FI manager is not risk averse;

15 sau71566_ch13.qxd 9/5/00 2:48 PM Page 331 Chapter 13 Credit Risk: Quantitative Approaches 331 that is, applies risk-neutral valuation methods with no risk premium. Let p be the probability that the corporate debt, both principal and interest, will be repaid in full; therefore, 1 p is the probability of default. If the borrower defaults, the FI is assumed to get nothing. Once again, we assume that the loss given default is 100 percent. in a moment, we will relax this assumption. By denoting the promised return on the one-year corporate security as 1 k and on the credit-risk-free government security as 1 i, the FI manager would be indifferent between corporate and government securities when p(1 k) 1 i or the expected return on corporate securities is equal to the risk-free rate. Suppose that i 10% and: k 15.8% This implies that the probability of repayment on the security, as perceived by the market, is: 1 i p k If the probability of repayment is 0.95, this implies a probability of default (1 p) equal to Thus, in this simple one-period framework, a probability of default of 5 percent on the corporate bond (loan) requires the FI to set a spread of 5.8 percent: 20 k i 5.8% Clearly, as the probability of repayment (p) falls and the probability of default (1 p) increase, the required spread, between k and i, increases. This analysis can easily be extended to the more realistic case where the FI does not expect to lose all interest and all principal if the corporate borrower defaults. 21 Realistically, the FI lender can expect to receive some partial repayment even if the borrower goes into bankruptcy. For example, using U.S. data, Altman has estimated that when firms default on their junk bonds, the investor receives, on average, around 40 cents on the dollars. 22 As discussed in Chapter 12, many loans and bonds are secured or collateralized by first liens on various pieces of property or real assets should a borrower default. Let be the proportion of the loan s principal and interest that is collectable on default, where in general is positive. For example, in the junk-bond case, is approximately 0.4. The FI manager would set the expected return on the loan to equal the risk-free rate in the following manner: (1 k) (1 p) p(1 k) 1 i The new term here is (1 k) (1 p); this is the payoff the FI expects to get if the borrower defaults. 20 In the real world, an FI could partially capture this required spread in higher fees and compensating balances rather than only in the risk premium. In this simple example, we are assuming away compensating balances and fees, however, they could easily be built into the model. 21 See J.B. Yawitz, Risk Premia on Municipal Bonds, Journal of Financial and Quantitative Analysis 13, 1977, pp , and J.B. Yawitz, An Analytical Model of Interest Rate Differentials and Different Default Recoveries, Journal of Financial and Quantitative Analysis 13, 1977, pp E.I. Altman, Measuring Corporate Bond Mortality and Performance, Journal of Finance 44, 1989, pp

16 sau71566_ch13.qxd 9/5/00 2:48 PM Page Part II Measuring Risk As might be expected, if the loan has collateral backing such that 0, the required spread on the loan will be less for any given default risk probability (1 p). Collateral requirements are a method of controlling default risk; they act as a direct substitute for spreads in setting required loan rates. To see this, solve for the spread <special character> between k (the required yield on risky corporate debt) and i (the risk-free rate of interest): (1 i) k i (1 i) ( p p ) If i 10 percent and p 0.95 as before, but the FI can expect to collect 64 percent of the promised proceeds if default occurs ( 0.64), then the required spread 2.0 percent. 23 Interestingly, in this simple framework, ( and p are perfect substitutes for each other. That is, a bond or loan with collateral backing of 0.7 and p 0.8 would have the same required risk premium as one with 0.8 and p 0.7. An increase in collateral is a direct substitute for an increase in default risk (i.e., a decline in p). Probability of Default on a Multiperiod Debt Instrument. We can extend this analysis to derive the credit risk of default probabilities occurring in the market for longer-term loans or bonds. For the simple one-period loan or bond, the probability of default (1 p) was 1 i 1 p 1 [ 1 k] Marginal Default Probability The probability that a borrower will default in any given year. or: p 1 [ ] p 0.05 Suppose that the FI manager wanted to find out the probability of default on a twoyear bonds. 24 To do this, the manager must estimate the probability that the bond would default in the second year conditional on the probability that it does not default in the first year. The probability that a bond will default in any one year is clearly conditional on the fact the default hasn t occurred earlier. The probability that a bond will default in any one year is the marginal default probability for that year. For the one-year loan, 1 p is the marginal and total or cumulative probability (Cp) of default in year one. However, for the two-year loan, the marginal probability of default in the second year (1 p 2 ) can differ from the marginal probability of default in the first year (1 p 1 ). Later we discuss ways to estimate p 2, but for the moment suppose that 1 p Then, 1 p probability of default in year 1. 1 p probability of default in year According to the CBA, Canadian banks realizing their security on commercial loans have a recovery rate of around 64 cents on the dollar. Helen Sinclair, CBA president, Canadian Bankers Association Submission to the Standing Committee on Industry: A Review of Access by Small and Medium-Sized Businesses to Traditional and New Sources of Financing, April 19, 1994, p For more details of this approach, see R. Litterman and T. Iben, Corporate Bond Valuation and the Term Structure of Credit Spreads, Journal of Portfolio Management, 1989, pp , and Jerome S. Fons, Using Default Rates to Mode the Term Structure of Credit Risk, Financial Analysts Journal, September/October 1994, pp