1. Loan Portfolio Selection and Risk Measurement Chapters 10 and 11

2. Saunders & Allen Chapters 10 & 112 The Paradox of Credit Lending is not a “buy and hold”process. To move to the efficient frontier, maximize return for any given level of risk or equivalently, minimize risk for any given level of return. This may entail the selling of loans from the portfolio. “Paradox of Credit” – Fig. 10.1.

3. Saunders & Allen Chapters 10 & 113

4. 4 Managing the Loan Portfolio According to the Tenets of Modern Portfolio Theory Improve the risk-return tradeoff by: –Calculating default correlations across assets. –Trade the loans in the portfolio (as conditions change) rather than hold the loans to maturity. –This requires the existence of a low transaction cost, liquid loan market. –Inputs to MPT model: Expected return, Risk (standard deviation) and correlations

5. Saunders & Allen Chapters 10 & 115 The Optimum Risky Loan Portfolio – Fig. 10.2 Choose the point on the efficient frontier with the highest Sharpe ratio: –The Sharpe ratio is the excess return to risk ratio calculated as:

6. Saunders & Allen Chapters 10 & 116

7. 7 Problems in Applying MPT to Untraded Loan Portfolios Mean-variance world only relevant if security returns are normal or if investors have quadratic utility functions. –Need 3 rd moment (skewness) and 4 th moment (kurtosis) to represent loan return distributions. Unobservable returns –No historical price data. Unobservable correlations

8. Saunders & Allen Chapters 10 & 118 KMV’s Portfolio Manager Returns for each loan I: –R it = Spread i + Fees i – (EDF i x LGD i ) – r f Loan Risks=variability around EL=EGF x LGD = UL –LGD assumed fixed: UL i = –LGD variable, but independent across borrowers: UL i = –VOL is the standard deviation of LGD. VVOL is valuation volatility of loan value under MTM model. –MTM model with variable, indep LGD (mean LGD): UL i =

9. Saunders & Allen Chapters 10 & 119 Valuation Under KMV PM Depends on the relationship between the loan’s maturity and the credit horizon date: Figure 11.1: DM if loan’s maturity is less than or equal to the credit horizon date (maturities M 1 or M 2 ). MTM if loan’s maturity is greater than credit horizon date (maturity M 3 ). See Appendix 11.1 for valuation.

10. Saunders & Allen Chapters 10 & 1110

11. Saunders & Allen Chapters 10 & 1111 Correlations Figure 11.2 – joint PD is the shaded area.  GF =  GF /  G  F  GF = Correlations higher (lower) if isocircles are more elliptical (circular). If JDF GF = EDF G EDF F then correlation=0.

12. Saunders & Allen Chapters 10 & 1112

13. Saunders & Allen Chapters 10 & 1113 Role of Correlations Barnhill & Maxwell (2001): diversification can reduce bond portfolio’s standard deviation from $23,433 to $8,102. KMV diversifies 54% of risk using 5 different BBB rated bonds. KMV uses asset (de-levered equity) correlations, CreditMetrics uses equity correlations. Correlation ranges: –KMV:.002 to.15 –Credit Risk Plus:.01 to.05 –CreditMetrics:.0013 to.033

14. Saunders & Allen Chapters 10 & 1114 Calculating Correlations using KMV PM Construct asset returns using OPM. Estimate 3-level multifactor model. Estimate coefficients and then evaluated asset variance and correlation coefficients using: First level decomposition: –Single index model – composite market factor constructed for each firm. Second level decomposition: –Two factors: country and industry indices. Third level decomposition: –Three sets of factors: (1) 2 global factors (market-weighted index of returns for all firms and return index weighted by the log of MV); (2) 5 regional factors (Europe, No. America, Japan, SE Asia, Australia/NZ); (3) 7 sector factors (interest sensitive, extraction, consumer durables, consumer nondurables, technology, medical services, other).

15. Saunders & Allen Chapters 10 & 1115 CreditMetrics Portfolio VAR Two approaches: –Assuming normally distributed asset values. –Using actual (fat-tailed and negatively skewed) asset distributions. For the 2 Loan Case, Calculate: –Joint migration probabilities –Joint payoffs or loan values –To obtain portfolio value distribution.

16. Saunders & Allen Chapters 10 & 1116 The 2-Loan Case Under the Normal Distribution Joint Migration Probabilities = the product of each loan’s migration probability only if the correlation coefficient=0. –From Table 10.1, the probability that obligor 1 retains its BBB rating and obligor 2 retains it’s a rating would be 0.8693 x 0.9105 = 79.15% if the loans were uncorrelated. The entry of 79.69% suggests a positive correlation of 0.3.

17. Saunders & Allen Chapters 10 & 1117 Mapping Ratings Transitions to Asset Value Distributions Assume that assets are normally distributed. Compute historic transition matrix. Figure 11.3 uses the matrix for a BB rated loan. Suppose that historically, there is a 1.06% probability of transition to default. This corresponds to 2.3 standard deviations below the mean on the standard normal distribution. Similarly, if there is a 8.84% probability of downgrade from BB to B, this corresponds to 1.23 standard deviations below the mean.

18. Saunders & Allen Chapters 10 & 1118 Joint Transition Matrix Can draw a figure like Fig. 11.3 for the A rated obligor. There is a 0.06% PD, corresponding to 3.24 standard deviations below the mean; a 5.52% probability of downgrade from A to BBB, corresponding to 1.51 std dev below the mean. The joint probability of both borrowers retaining their BBB and A ratings is: the probability that obligor 1’s assets fluctuate between –1.23  to +1.37  and obligor 2’s assets between –1.51  to +1.98  with a correlation coefficient=0.2. Calculated to equal 73.65%.

19. Saunders & Allen Chapters 10 & 1119

20. Saunders & Allen Chapters 10 & 1120 Calculating Correlation Coefficients Estimate systematic risk of each loan – the relationship between equity returns and returns on market/industry indices. Estimate the correlation between each pair of market/industry indices. Calculate the correlation coefficient as the weighted average of the systematic risk factors x the index correlations.

21. Saunders & Allen Chapters 10 & 1121 Two Loan Example of Correlation Calculation Estimate the systematic risk of each company by regressing the stock returns for each company on the relevant market/industry indices. R A =.9R CHEM + U A R Z =.74R INS +.15R BANK + U Z  A,Z =(.9)(.74)  CHEM,INS + (.9)(.15)  CHEM,BANK Estimate the correlation between the indices. If  CHEM,INS =.16 and  CHEM,BANK =.08, then  AZ =0.1174.

22. Saunders & Allen Chapters 10 & 1122 Joint Loan Values Table 11.1 shows the joint migration probabilities. Calculate the portfolio’s value under each of the 64 possible credit migration possibilities (using methodology in Chap.6) to obtain the values in Table 11.3. Can draw the portfolio value distribution using the probabilities in Table 11.1 and the values in Table 11.3.

23. Saunders & Allen Chapters 10 & 1123 Credit VAR Measures Calculate the mean using the values in Table 11.3 and the probabilities in Tab 11.1. –Mean = –Variance = –Mean=$213.63 million –Standard deviation= $3.35 million

24. Saunders & Allen Chapters 10 & 1124 Calculating the 99 th percentile credit VAR under normal distribution 2.33 x $3.35 = $7.81 million Benefits of diversification. The BBB loan’s credit VAR (alone) was $6.97million. Combining 2 loans with correlations=0.3, reduces portfolio risk considerably.

25. Saunders & Allen Chapters 10 & 1125 Calculating the Credit VAR Under the Actual Distribution Adding up the probabilities (from Table 11.1) in the lowest valuation region in Table 11.3, the 99 th percentile credit VAR using the actual (not normal) distribution is $204.4 million. Unexpected Losses=$213.63m - $204.4m = $9.23 million (>$7.81m). If the current value of the portfolio = $215m, then Expected Losses=$215m - $213.63m = $1.37m.

26. Saunders & Allen Chapters 10 & 1126 CreditMetrics with More Than 2 Loans in the Portfolio Cannot calculate joint transition matrices for more than 2 loans because of computational difficulties: A 5 loan portfolio has over 32,000 joint transitions. Instead, calculate risk of each pair of loans, as well as standalone risk of each loan. Use Monte Carlo simulation to obtain 20,000 (or more) possible asset values.

27. Saunders & Allen Chapters 10 & 1127 Monte Carlo Simulation First obtain correlation matrix (for each pair of loans) using the systematic risk component of equity prices. Table 11.5 Randomly draw a rating for each loan from that loan’s distribution (historic rating migration) using the asset correlations. Value the portfolio for each draw. Repeat 20,000 times! New algorithms reduce some of the computational requirements. The 99 th % VAR based on the actual distribution is the 200 th worst value out of the 20,000 portfolio values.

28. Saunders & Allen Chapters 10 & 1128 MPT Using CreditMetrics Calculate each loan’s marginal risk contribution = the change in the portfolio’s standard deviation due to the addition of the asset into the portfolio. Table 11.6 shows the marginal risk contribution of 20 loans – quite different from standalone risk. Calculate the total risk of a loan using the marginal contribution to risk = Marginal standard deviation x Credit Exposure. Shown in column (5) of Table 11.6.

29. Saunders & Allen Chapters 10 & 1129 Figure 11.4 Plot total risk exposure using marginal risk contributions (column 6 of Table 11.6) against the credit exposure (column 5 of Table 11.4). Draw total risk isoquants using column 5 of Table 11.6. Find risk outliers such as asset 15 which have too much portfolio risk ($270,000) for the loan’s size ($3.3 million). This analysis is not a risk-return tradeoff. No returns.

30. Saunders & Allen Chapters 10 & 1130

31. Saunders & Allen Chapters 10 & 1131 Default Correlations Using Reduced Form Models Events induce simultaneous jumps in default intensities. Duffie & Singleton (1998): Mean reverting correlated Poisson arrivals of randomly sized jumps in default intensities. Each asset’s conditional PD is a function of 4 parameters: h (intensity of default process); (constant arrival prob.); k (mean reversion rate);  (steady state constant default intensity). The jumps in intensity follow an exponential distribution with mean size of jump=J. So: probability of survival from time t to s:

32. Saunders & Allen Chapters 10 & 1132 Numerical Example Suppose that =.002, k=.5,  =.001, J=5, h(0)=.001 (corresponds to an initial rating of AA). Correlations across loan default probabilities: V c =common factor; V=idiosyncratic factor. As v  0, corr  0 As v  1, corr  1. If v=.02, V=.001, V c =.05: the probability that loan i intensity jumps given that loan j has experienced a jump is = vV c /(V c +V) = 2%. If v=.05 (instead of.02), then the probability increases to 5%. Figure 11.5 shows correlated jumps in default intensities. Figure 11.6 shows the impact of correlations on the portfolio’s risk.

33. Saunders & Allen Chapters 10 & 1133

34. Saunders & Allen Chapters 10 & 1134

35. Saunders & Allen Chapters 10 & 1135 Appendix 11.1: Valuing a Loan that Matures after the Credit Horizon – KMV PM Maturity=M 3 in Figure 11.1. Use MTM to value loans. Four Step Process: –1. Valuation of an individual firm’s assets using random sampling of risk factors. –2. Loan valuation based on the EDFs implied by the firm’s asset valuation. –3. Aggregation of individual loan values to construct portfolio value. –4. Calculation of excess returns and losses for portfolio. Yields a single estimate for expected returns (losses) for each loan in the portfolio. Use Monte Carlo simulation (repeated 50,000 to 200,000 times) to trace out distribution

36. Saunders & Allen Chapters 10 & 1136 Step 1: Valuation of Firm Assets at 3 Time Horizons – Fig. 11.7 A 0, A H, A M valuations. Stochastic process generating A H, A M : The random component  = systematic portion f + firm-specific portion u. Each simulation draws another risk factor. Using A H and A M can calculate EDF H and EDF M

37. Saunders & Allen Chapters 10 & 1137 Step 2: Loan Valuation Using Term Structure of EDFs Convert EDF into QDF by removing risk-adjusted ROR. Also value loan as of credit horizon date H:

38. Saunders & Allen Chapters 10 & 1138 Step 3: Aggregation to Construct Portfolio Sum the expected values V H for all loans in the portfolio.

39. Saunders & Allen Chapters 10 & 1139 Step 4: Calculation of Excess Returns/Losses Excess Returns on the Portfolio: Expected Loss on the Portfolio: Repeat steps 1 through 4 from 50,000 to 200,000 times.

40. Saunders & Allen Chapters 10 & 1140 A Case Study: KMV PM valuation of 5 yr maturity $1 loan paying a fixed rate of 10% p.a. Using Table 11.8:

41. Saunders & Allen Chapters 10 & 1141 Valuing the Loan at the Credit Horizon Date =1 Using Table 11.9:

42. Saunders & Allen Chapters 10 & 1142 KMV’s Private Firm Model Calculate EBITDA for private firm j in industry j. Calculate the average equity mulitple for industry i by dividing the industry average MV of equity by the industry average EBITDA. Obtain an estimate of the MV of equity for firm j by multiplying the industry equity multiple by firm j’s EBITDA. Firm j’s assets = MV of equity + BV of debt Then use valuation steps as in public firm model.

43. Saunders & Allen Chapters 10 & 1143 Credit Risk Plus Model 2 - Incorporating Systematic Linkages in Mean Default rates Mean default rate is a function of factor sensitivities to different independent sectors (industries or countries). Table 11.7 shows as example of 2 loans sensitive to a single factor (parameters reflect US national default rates). As credit quality declines (m gets larger), correlations get larger.