Measuring Credit Risk

1. Measuring Credit Risk FIN 653 Lecture Notes From Saunders and Cornett Ch. 11

2. I. Credit Scoring Models Credit Scoring Models use data on observed borrower characteristics either to calculate the probability of default or to sort borrowers into different default risk classes.

3. I. Credit Scoring Models By selecting and combining different economic and financial borrower characteristics, a financial institution 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; intelligent ; 3. Improve the pricing of default risk; 4. Be better able to screen out bad loan applicants; 5. Be in a better position to calculate any reserves needed to meet expected future loan losses.

4. I. Credit Scoring Models 1. Linear Probability Model: Uses past data as input into a model to explain repayment experience on old loans. The relative importance of the factors used in explaining past repayment performance then forecasts repayment probability on new loans. Zi = ??j Xi,j + ?i Then take these estimated ?js and multiply with the observed Xij for a prospective borrower, we can derive an expected value of Zi for the prospective borrower. That value can be interpreted as the expected probability of default. E(Zi) = 1 - Pi.

5. I. Credit Scoring Models E.g., there were two factors influencing the past default: the leverage or debt-equity ratio (D/E) and the sales-asset ratio (S/A). Based on the past default experience, the linear probability model is estimated as Zi = .5(D/Ei) + .1(S/Ai). � Assume a prospective borrower has a D/E = .3 and an S/A = 2.0. Its expected probability of default is then Zi = .5(.3) + .1(2.0) = .35.

6. I. Credit Scoring Models The major weakness of Credit Scoring Models is that the estimated probabilities of default can often lie outside the interval (0 to 1). The logit and probit models overcome this weakness by restricting the estimated range of default probability to lie between 0 to 1.

7. I. Credit Scoring Models 2. The Logit Model: The logit model constrains the cumulative probability of default on a loan to lie between 0 and 1 and assumes the probability of default to be logistically distributed according to the functional form: 1 F (Zi) = ------------------ 1 + e -Zi where F (Zi) = the cumulative probability of default on the loan. � Zi = estimated by regression in a similar fashion to the linear probability model.

8. I. Credit Scoring Models It's major weakness is the assumption that the cumulative probability of default takes on a particular functional form that reflects a logistic function.�

9. I. Credit Scoring Models 3. The Probit Model: The probit model also constrains the projected probability of default to lie between 0 and 1, but differs from the logic model in assuming that the probability of default has a (cumulative) normal distribution rather than the logistic function. However, when multiplied by a fixed factor, logit estimates may produce appropriately correct probit values.

10. I. Credit Scoring Models 4. Linear Discriminant Models: Discriminant models divide borrowers into high or low default risk classes contingent on their observed characteristics (Xj). The discriminant model by E.I. Altman: Z = 1.2X1 + 1.4 X2 + 3.3 X3 + 0.6 X4 + 1.0 X5 Where Z = an overall measure of the default risk X1 = Working Capital/Total Assets X2 = Retained Earnings/Total Assets X3 = Earnings before Interest & Taxes/Total Assets X4 = Market Value of Equity/Book Value of Long-term Debt X5 = Sales/Total Assets

11. I. Credit Scoring Models According to Altman's credit scoring model: Z > 1.81 ? Low default risk class � Z < 1.81 ? High default risk class Ex: Suppose that the financial ratios for a potential borrowers: X1 = .20 X2 = .0 X3 = -.20 X4 = .10 X5 = -2.0 Then Z score = 1.64 The FI should not make a loan to this borrower until it improves its earnings.

12. I. Credit Scoring Models Problems with the discriminant analysis: 1. This model usually discriminates only between two extreme cases of borrowers behavior, default and no default. 2. No obvious economic reason to expect the weights in the discriminant function to be constant over short periods. The same concern also applies to the variables (Xj). 3. These models ignore important hard-to-quantify factors that may play a crucial role in the default or no default decision.

13. II. Term Structure Derivation of Credit Risk One market-based method of assessing credit risk exposure and default probabilities is to analyze the risk premiums inherent in the current structure of yields on corporate debt or loans to similar risk-rated borrowers. The spreads between risk free deep-discount bonds issued by the Treasury & deep-discount bonds issued by corporate borrowers of different quality may reflect perceived credit risk exposure of corporate borrowers for single payments at different times in the future.

14. II. Term Structure Derivation of Credit Risk 1. Probability of Default on a One-Period Debt Instrument Assume that the Financial Institution requires an expected return on a one year corporate debt at least equal to the risk free return on T-bonds of one year's maturity. Let P be the probability that the debt will be repaid, then (1-P) is the probability of default. The FI would be indifferent when P (1 + k) + (1-P) * 0 = 1 + i

15. II. Term Structure Derivation of Credit Risk Suppose, i =10%, k = 15.8% Then the probability of repayment as perceived by the market is 1 + i 1+10% P = --------- = --------------- = 0.95 1 + k 1+15.8% => Probability of default (l-P) = 5%. And a probability of default of 5% on the corporate bond requires the FI to set a risk premium of 5.8%: ? = k - i = 5.8%

16. II. Term Structure Derivation of Credit Risk Let ? be the proportion of the loan's principal and interest that is collectable on default. The FI manager would set the expected return on the loan to equal the risk free rate: ? (1+k) (1-P) + P(l+k) = 1 + i The the probability of default is: (k-i) 1- p = ------------------- (1-?) (1+k) So the larger the ?, the higher the probability of default.

17. II. Term Structure Derivation of Credit Risk Collateral requests are a method of controlling default risk; they act as direct substitute for risk premiums in setting required loan rates:� (1 + i) k - i = ? = -------------------- - (1 + i) (? + p - p? ) �As ? goes up, (k-i) goes down.

18. II. Term Structure Derivation of Credit Risk If i = 10, p = 0.95 as before, but the FI can expect to collect 90% of the promised proceeds if default occurs, then the required risk premium: � ? = 0.6% and the required rate of return on the corporate bond would be: k = 10.6%

19. II. Term Structure Derivation of Credit Risk 2. Probability of Default on a Multiperiod Debt Instrument: What is the probability of default on a two-year bond? We must estimate the probability that the bond could default in the second year conditional on the probability that it does not default in the first year. The probability that a bond would default in any one year is the marginal default probability for that year.

20. II. Term Structure Derivation of Credit Risk Suppose: 1-P1 = 0.05 = prob. of default in yr. 1 1-P2 = 0.07 = prob. of default in yr. 2 The probability of the borrower surviving - not defaulting at any time between now (t=0) and the end of period 2 is: � P1 * P2 = (.95)(.93) = .8835 �The cumulative default probability (CP) is therefore � CP = 1 - [P1*P2] = 1 - [(.95)*(.93)] = .1165

21. II. Term Structure Derivation of Credit Risk Given the presence of both one- and two-year discount bonds for Treasury issues and corporate issues of a particular risk classes, we can derive P2 from the term structure of interest rates. � Maturity 1 yr. 2 yrs ________________________________________ T-Bond 10% 11% Corporate Bond 15.8% 18% ________________________________________ �

22. II. Term Structure Derivation of Credit Risk From the T-Bond Yield Cure: (1+i2)2 = (1+i1)(1+f1) when f1 = the expected one-year forward rate (1+f1) = (1+i2)2/(1+i1) = 1.12 Current One-Year Rate Expected one-Year Rate ________________________________________________________ Treasury 10.0% (i1) 12.0% (f1) Corporate 15.8% (k1) 20.2% (c1) Spread 5.8% 8.2% ________________________________________________________

23. II. Term Structure Derivation of Credit Risk The expected rates on one -year bonds can generate an estimate of the expected probability of repayment on one-year corporate bonds in one-year's time, or � P2 = (1+f1)/(1+c1) = .9318 where f1 and c1 are expected one-year rate on Treasury bill and corporate bond, respectively. Thus the expected probability of default in year 2: � (1 - P2) = 6.82%. �

24. II. Term Structure Derivation of Credit Risk In a similar fashion, the one-year rates expected in two-year's time can be derived from the Treasury & corporate term structures. The probability of repayment on one-year loans originates in two-year's time is � P3 = (1+f2)/(1+c2) � Thus we can derive a whole term structure of expected future one-year default probabilities for grade B bonds.

25. II. Term Structure Derivation of Credit Risk The cumulative probability of default would tell the FI the probability of a loan or bond investment defaulting over a particular time period. In the example, the corporate bond would default over the next two year is � CP = 1 - [(P1)(P2)] = 1 - [(.95) (.9318)] = 11.479%

26. II. Term Structure Derivation of Credit Risk Advantages: 1. It is forward looking and based on market expectations; 2. If there are liquid markets for Treasury & corporate discount bonds - such as T-strips & corporate zeros-, then we can easily estimate expected future default rates. Disadvantage: 1. While the market for T-strips is now quite deep, the market for corporate discount bonds is still small. It might not be able to extract default risk premium for corporate discount bonds.

27. III. Mortality Rate Derivation of Credit Risk The FI manager may analyze the historic or past default risk experience, the mortality rates, of bonds or loans of a similar quality. Let� P1 = the prob. of a grade B bond or loan surviving the first year of its issue; 1 - P1 = the marginal mortality rate or the prob. of the bond or loan dying or defaulting in the first year of issue. For each grade of corporate borrower quality, a marginal mortality rates (MMR) curve can show the historical default rate experience of bonds in any specific quality class in each year after issue on the bond or loan.

28. III. Mortality Rate Derivation of Credit Risk Total value of grade B defaulting in year 1 of issue MMR1 = ----------------------------------------------------------------------- Total value of grade B bonds outstanding in year 1 of issue � � Total value of grade B bonds defaulting in year 2 of issue MMR2 = ----------------------------------------------------------------------- Total value of grade B bonds outstanding in year 2 of issue adjusted for defaults, calls, sinking fund and redemption, and maturities in the prior year.

29. III. Mortality Rate Derivation of Credit Risk TABLE 8-9 Adjusted Mortality Rates by Original Standard & Poor's Bond Rating (Defaults and Issues, 1971-88) � � Years after Issuance (percentage) �_____________________________________________________________________________________________ Original Rating 1 2 3 4 5 6 7 8 9 10 � AAA Yearly 0.00 0.00 0.00 0.00 0.00 0.15% 0.05% 0.00 0.00 0.00 Cumulative 0.00 0.00 0.00 0.00 0.00 0.15 0.21 0.21% 0.21% 0.21% � AA Yearly 0.00 0.00 1.39% 0.33% 0.20% 0.00 0.27 0.00 0.11 0.13 Cumulative 0.00 0.00 1.39 1.72 1.92 1.92 2.18 2.18 2.29 2.42 A Yearly 0.00 0.39% 0.32 0.00 0.00 0.11 0.11 0.07 0.13 0.00 Cumulative 0.00 0.39 0.71 0.71 0.71 0.82 0.93 1.00 1.13 1.13

30. III. Mortality Rate Derivation of Credit Risk Years after Issuance (percentage) �_____________________________________________________________________________________________ Original Rating 1 2 3 4 5 6 7 8 9 10 � BBB Yearly 0.03 0.20 0.12 0.26 0.39 0.00 0.14 0.00 0.21 0.80 Cumulative 0.03 0.23 0.35 0.61 1.00 1.00 1.14 1.14 1.34 2.13 � BB Yearly 0.00 0.5 0 0.57 0.26 0.53 2.79 3.03 0.00 0.00 3.48 Cumulative 0.00 0.50 1.07 1.34 1.86 4.59 7.48 7.48 7.48 10.70 B Yearly 1.40 0.65 2.73 3.70 3.59 3.86 6.30 3.31 6.84 3.70 Cumulative 1.40 2.04 4.72 8.24 11.54 14.95 20.31 22.95 28.22 30.88 CCC Yearly 1.97 1.88 4.37 16.35 2.06 0.00 0.00 0.00 0.00 0.00 Cumulative 1.97 3.81 8.01 23.05 24.64 24.64 24.64 24.64 24.64 24.64 _____________________________________________________________________________________________

31. III. Mortality Rate Derivation of Credit Risk Problems with the Mortality Rate Approach: 1.���Like the credit scoring model, it produces historic or backward looking measures. 2.���The estimates of default rates and, therefore implied future default probability tend to be highly sensitive to the period over which the FI manager calculates the MMRs. 3.���The estimates tend to be sensitive to the number of issues and the relative size of issues in each investment grade.

32. IV. Option Models of Default Risk 1. Theoretical Framework When a firm raises funds either by issuing bonds or increasing its bank loans, it holds a very valuable default or repayment option. That is, if a borrower cannot repay the bond holder or the bank, it has the option of defaulting on its debt repayment and turning any remaining assets over to the debt holders. On the other hand, if things go well, the borrower can keep most of the upside returns on asset investments after the promised principal and interest on the debt have been paid.

33. IV. Option Models of Default Risk 2. The Borrower's Payoff from Loans (Loan as a call option) If the investments turn out badly, the stockholder - owners of the firm would default on the firms debt, turn its assets (Al) over to the debt holders, and lose only their initial stake in the firm(s). By contrast, if the firm does well and the assets of the firm are valued highly (A2), the firm's stockholders would payoff the firm's debt (OB) and keep the difference (A2 - B).

34. IV. Option Models of Default Risk 3. The Debtholder's Payoff from Loans The maximum amount the bank or bondholder can get back is B, the promised payment. However, the borrower who possesses the default or repayment option would only rationally repay the loan if A > B. A borrower whose asset value falls below B would default and turn over any remaining assets to the debtholders. Thus the value of the loan from the perspective of the lender is always the minimum of B or A, or min [B,A]. That is, the payoff function to the debtholder is similar to writing a put option on the value of the borrowers' assets with B, the face-value of debt, as the exercise price.

35. IV. Option Models of Default Risk 4. Applying the Option Valuation Model to the Calculation of Default Risk Premiums: The market value of a risky loan made by a lender to a borrower can be expressed as F(?) = Be-? i [(1/ d)N(h1 ) + N(h2)] Where ? = the length of time remaining to loan maturity D = the borrower�s leverage ratio measured as Be-? i /A; N(h) = the probability that a deviation exceeding the calculated value of h will occur. h1 = -[1/2 ?2 ? - ln(d)]/ ? ?? ; h2 = -[1/2 ? 2 ? + ln(d)]/ ? ?? .

36. IV. Option Models of Default Risk 4. Applying the Option Valuation Model to the Calculation of Default Risk Premiums: Written in terms of a yield spread, this equation reflects an equilibrium default risk premium that the borrower should be charged: k(? )- i = (-1/? ) ln [N(h2) + (1/2)N (h1)] where k(? ) = Require yield on risky debt i = Risk-free rate on debt of equivalent maturity

37. IV. Option Models of Default Risk Thus Merton has show that the lender should adjust the required risk premium as d and ?2 change , i.e., as leverage and asset risk change. Specifically, � ?[k(?)-i] ---------- > 0 ? d � ?[h(?)-i] ---------- > 0 ? ?

38. IV. Option Models of Default Risk Problems with the Option Pricing Models 1.������� The assumption of continuously traded claims on the assets of the borrower. Since many loans are never, or at best, infrequently traded, this assumption is difficult to accept in many real-world applications. The value of ?2 the volatility of the underlying assets of the borrower - plays a crucial role in setting the equilibrium risk premium. The value of option-based premium is extremely sensitive to errors made in measuring ?2. Moreover, volatility itself is variable over time.

39. IV. Option Models of Default Risk An Option Model Application� B = $100,000 ? = 1 year i = 5 % d = 90% or .9 ?2 = 12 % Substituting these values into the equations for h1 and h2 and solving for the areas under the standardized normal distributions, we find: N(hl) = .174120, and N(h2) = .793323

40. IV. Option Models of Default Risk An Option Model Application� where: -[1/2(.12) 2 - ln (.9)] h1 = ------------------------ = -.938 .12 and -[1/2(.12) 2 + ln (.9)] h2 = ------------------------- = + .818 .12

41. IV. Option Models of Default Risk The current market value of the loan is: L (t) = Be?i [(1/ d)N(h1) + N(h2) ] $ 100,000 = ----------- [.793323+(1.1111)(.17412)] 1.05127 $100,000 = ------------ [.986788] = $93,866.18 1.05127

42. IV. Option Models of Default Risk and the required risk spread or premium is: � k(? )- i = (-1/? ) ln [N(h2) + (1/2)N (h1)] = (-1)ln [.986788] = 1.33% Thus, the risky loan rate k(? ) should be set at 6.33 percent when the risk-free rate (i) is 5 percent.

43. V. RAROC Models RAROC (Risk Adjusted Return On Capital) was pioneered by Bankers Trust and has now been adopted by virtually all the large banks, although with some proprietary differences between them. The essential idea behind RAROC is to balances expected loan income against the loan's risk. Thus, rather than dividing loan income by assets, it is divided by some measure of asset (loan) risk: One year income on a loan RAROC = --------------------------------------- Loan (asset) risk or Risk capital

44. V. RAROC Models A loan is approved only if RAROC is sufficiently high relative to a benchmark cost of capital for the bank. Alternatively, if the RAROC on an existing loan falls below a bank's RAROC benchmark, the lending officer should seek to adjust the loan's terms to make it "profitable" again. One problem in estimating RAROC is the measurement of loan risk. Duration showed that the percentage change in the market value of an asset such as a loan (?L/L) is related to the duration of the loan and the size of the interest rate shock (?R/1+R): ? L = - DL ?R L 1+R

45. V. RAROC Models The same concept is applied here, except that interest late shocks are replaced by credit quality shocks: � ? L = - DL * L * (?R/1+R) (dollar capital (duration of (risk amount (expected risk change of risk exposure the loan) or size amount) in credit factor on or loss amount) loan)

46. V. RAROC Models While the loan's duration (2.7 years) and the loan amount ($1 million) are easily estimated, it is more difficult to estimate the maximum change in the credit risk premium on the loan. Since publicly available data on loan risk premium are scarce, we turn to publicly available corporate bond market data to estimate premium. First, an S&P credit rating (AAA,AA, A, and so on) is assigned to borrower. Thereafter, the risk premium changes of all the bonds traded in that particular rating class over the last year are analyzed.

47. V. RAROC Models The ?R in the RAROC equation equals: � ?R = Max. [?(RI - RG) > 0 ] � Where ?(RI - RG ) is the change in the yield spread between corporate bonds of credit rating class I (RI) and matched duration treasury bonds (RG) over the last year. In order to consider only the worst-case scenario, the maximum change in yield spread is chosen, as opposed to the average change.

48. V. RAROC Models Example: To evaluate the credit risk of a loan to a AAA borrower. Assume there are currently 400 publicly traded bonds in that class. The first step is to evaluate the actual changes in the credit risk premium (RI - RG) on each if these bonds for the past year. The range from a fall in the risk premiums of negative 2 percent to an increase of 3.5 percent. Since the largest increase maybe a very extreme (unrepresentative) number, the 99 percent worst-case scenario is chosen (i.e., only 4 bonds out 400 have risk premium increases exceeding the 99 percent worst case).

49. V. RAROC Models The estimate of loan(or capital) risk, assuming that the current average level of rates ( R )on AAA bond is 10 percent, is � ? L = DL * L * ( ?R/ 1+R) = -(2.7) ($1 million) (.011/1.1) = -$27,000 Thus, while the face value of the loan amount is $1 million, the risk amount or change in the loan's market value due to a decline in its credit quality is $ 27,000.

50. V. RAROC Models To determine whether the loan is worth making, the estimated loan risk is compared to the loan income (spread over the FI's cost of funds plus fees on the loan). Suppose the projected spread plus fees is as follows: � Spread = 0.2% * $ 1 million = $2.00 Fees = 0.1 % * $ 1 million = $1,000 Total Loan Income = $3,000

51. V. RAROC Models The loan's RAROC is One-year income on loan RAROC = ----------------------------------------- Loan risk (or Capital risk) (?L) = $3,000/ $27,000 = 11.1% If the 11.1 percent exceeds the bank's internal RAROC benchmark (based on its cost of funds), the loan will be approved. If it less, the loan will either be rejected outright or the borrower will be asked to pay higher fees and/or a higher spread to increase the RAROC to acceptable levels.