chap11_12

A bank is a place that will lend you money if you can prove that you dont need it.Bob Hope

Saunders & Cornett, Financial Institutions Management, 4th ed.

Why New Approaches to Credit Risk Measurement and Management?

Why Now?


Structural Increase in BankruptcyIncrease in probability of defaultHigh yield default rates: 5.1% (2000), 4.3% (1999, 1.9% (1998). Source: Fitch 3/19/01Historical Default Rates: 6.92% (3Q2001), 5.065% (2000), 4.147% (1999), 1998 (1.603%), 1997 (1.252%), 10.273% (1991), 10.14% (1990). Source: AltmanIncrease in Loss Given Default (LGD)First half of 2001 defaulted telecom junk bonds recovered average 12 cents per $1 ($0.25 in 1999-2000)Only 9 AAA Firms in US: Merck, Bristol-Myers, Squibb, GE, Exxon Mobil, Berkshire Hathaway, AIG, J&J, Pfizer, UPS. Late 70s: 58 firms. Early 90s: 22 firms.


DisintermediationDirect Access to Credit Markets20,000 US companies have access to US commercial paper market.Junk Bonds, Private Placements.Winners Curse Banks make loans to borrowers without access to credit markets.


More Competitive MarginsWorsening of the risk-return tradeoffInterest Margins (Spreads) have declinedEx: Secondary Loan Market: Largest mutual funds investing in bank loans (Eaton Vance Prime Rate Reserves, Van Kampen Prime Rate Income, Franklin Floating Rate, MSDW Prime Income Trust): 5-year average returns 5.45% and 6/30/00-6/30/01 returns of only 2.67%Average Quality of Loans have deterioratedThe loan mutual funds have written down loan value


The Growth of Off-Balance Sheet DerivativesTotal on-balance sheet assets for all US banks = $5 trillion (Dec. 2000) and for all Euro banks = $13 trillion.Value of non-government debt & bond markets worldwide = $12 trillion.Global Derivatives Markets > $84 trillion.All derivatives have credit exposure.Credit Derivatives.


Declining and Volatile Values of CollateralWorldwide deflation in real asset prices.Ex: Japan and SwitzerlandLending based on intangibles ex. Enron.


TechnologyComputer Information TechnologyModels use Monte Carlo Simulations that are computationally intensiveDatabasesCommercial Databases such as Loan Pricing CorporationISDA/IIF Survey: internal databases exist to measure credit risk on commercial, retail, mortgage loans. Not emerging market debt.


BIS Risk-Based Capital RequirementsBIS I: Introduced risk-based capital using 8% one size fits all capital charge.Market Risk Amendment: Allowed internal models to measure VAR for tradable instruments & portfolio correlations the 1 bad day in 100 standard.Proposed New Capital Accord BIS II Links capital charges to external credit ratings or internal model of credit risk. To be implemented in 2005.


Traditional Approaches to Credit Risk Measurement20 years of modeling history


Expert Systems The 5 CsCharacter reputation, repayment historyCapital equity contribution, leverage.Capacity Earnings volatility.Collateral Seniority, market value & volatility of MV of collateral.Cycle Economic conditions. 1990-91 recession default rates >10%, 1992-1999: < 3% p.a. Altman & Saunders (2001)Non-monotonic relationship between interest rates & excess returns. Stiglitz-Weiss adverse selection & risk shifting.


Problems with Expert SystemsConsistencyAcross borrower. Good customers are likely to be treated more leniently. A rolling loan gathers no loss.Across expert loan officer. Loan review committees try to set standards, but still may vary.Dispersion in accuracy across 43 loan officers evaluating 60 loans: accuracy rate ranged from 27-50. Libby (1975), Libby, Trotman & Zimmer (1987).SubjectivityWhat are the optimal weights to assign to each factor?


Credit Scoring ModelsLinear Probability ModelLogit ModelProbit ModelDiscriminant Analysis Model97% of banks use to approve credit card applications, 70% for small business lending, but only 8% of small banks (

Linear Discriminant Analysis The Altman Z-Score ModelZ-score (probability of default) is a function of:Working capital/total assets ratio (1.2)Retained earnings/assets (1.4)EBIT/Assets ratio (3.3)Market Value of Equity/Book Value of Debt (0.6)Sales/Total Assets (1.0)Critical Value: 1.81


Problems with Credit ScoringAssumes linearity.Based on historical accounting ratios, not market values (with exception of market to book ratio).Not responsive to changing market conditions.56% of the 33 banks that used credit scoring for credit card applications failed to predict loan quality problems. Mester (1998).Lack of grounding in economic theory.


The Option Theoretic Model of Credit Risk MeasurementBased on Merton (1974)KMV Proprietary Model


The Link Between Loans and Optionality: Merton (1974)Figure 4.1: Payoff on pure discount bank loan with face value=0B secured by firm asset value.Firm owners repay loan if asset value (upon loan maturity) exceeds 0B (eg., 0A2). Bank receives full principal + interest payment.If asset value < 0B then default. Bank receives assets.


Using Option Valuation Models to Value LoansFigure 4.1 loan payoff = Figure 4.2 payoff to the writer of a put option on a stock.Value of put option on stock = equation (4.1) = f(S, X, r, , ) whereS=stock price, X=exercise price, r=risk-free rate, =equity volatility,=time to maturity. Value of default option on risky loan = equation (4.2) = f(A, B, r, A, ) whereA=market value of assets, B=face value of debt, r=risk-free rate, A=asset volatility,=time to debt maturity.


Problem with Equation (4.2)A and A are not observable.Model equity as a call option on a firm. (Figure 4.3)Equity valuation = equation (4.3) =E = h(A, A, B, r, ) Need another equation to solve for A and A:E = g(A) Equation (4.4)Can solve for A and A with equations (4.3) and (4.4) to obtain a Distance to Default = (A-B)/ A Figure 4.4


Mertons Theoretical PDAssumes assets are normally distributed.Example: Assets=$100m, Debt=$80m, A=$10mDistance to Default = (100-80)/10 = 2 std. dev.There is a 2.5% probability that normally distributed assets increase (fall) by more than 2 standard deviations from mean. So theoretical PD = 2.5%.But, asset values are not normally distributed. Fat tails and skewed distribution (limited upside gain).


Mertons Bond Valuation ModelB=$100,000, =1 year, =12%, r=5%, leverage ratio (d)=90%Substituting in Mertons option valuation expression: The current market value of the risky loan is $93,866.18The required risk premium = 1.33%


KMVs Empirical EDFUtilize database of historical defaults to calculate empirical PD (called EDF):

Fig. 4.5


Number of firms that defaulted within

a year with asset values of 2( from

Empirical EDF = B at the beginning of the year

Total population of firms with

asset values of 2( from B

at the beginning of the year

50 Defaults

Empirical EDF = Firm population of 1, 000 = 5 percent

Accuracy of KMV EDFsComparison to External Credit RatingsEnron (Figure 4.8)Comdisco (Figure 4.6)USG Corp. (Figure 4.7)Power Curve (Figure 4.9): Deny credit to the bottom 20% of all rankings: Type 1 error on KMV EDF = 16%; Type 1 error on S&P/Moodys obligor-level ratings=22%; Type 1 error on issue-specific rating=35%.


Monthly EDF credit measureAgency Rating


Problems with KMV EDFNot risk-neutral PD: Understates PD since includes an asset expected return > risk-free rate.Use CAPM to remove risk-adjusted rate of return. Derives risk-neutral EDF (denoted QDF). Bohn (2000).Static model assumes that leverage is unchanged. Mueller (2000) and Collin-Dufresne and Goldstein (2001) model leverage changes.Does not distinguish between different types of debt seniority, collateral, covenants, convertibility. Leland (1994), Anderson, Sundaresan and Tychon (1996) and Mella-Barral and Perraudin (1997) consider debt renegotiations and other frictions.Suggests that credit spreads should tend to zero as time to maturity approaches zero. Duffie and Lando (2001) incomplete information model. Zhou (2001) jump diffusion model.


Term Structure Derivation of Credit Risk MeasuresReduced Form Models: KPMGs Loan Analysis System and Kamakuras Risk Manager


Estimating PD: An Alternative ApproachMertons OPM took a structural approach to modeling default: default occurs when the market value of assets fall below debt valueReduced form models: Decompose risky debt prices to estimate the stochastic default intensity function. No structural explanation of why default occurs.


A Discrete Example:Deriving Risk-Neutral Probabilities of DefaultB rated $100 face value, zero-coupon debt security with 1 year until maturity and fixed LGD=100%. Risk-free spot rate = 8% p.a.Security P = 87.96 = [100(1-PD)]/1.08 Solving (5.1), PD=5% p.a.Alternatively, 87.96 = 100/(1+y) where y is the risk-adjusted rate of return. Solving (5.2), y=13.69% p.a.(1+r) = (1-PD)(1+y) or 1.08=(1-.05)(1.1369)


Multiyear PD Using Forward RatesUsing the expectations hypothesis, the yield curves in Figure 5.1 can be decomposed:(1+0y2)2 = (1+0y1)(1+1y1) or 1.162=1.1369(1+1y1) 1y1=18.36% p.a.(1+0r2)2 = (1+0r1)(1+1r1) or 1.102=1.08(1+1r1) 1r1=12.04% p.a.One year forward PD=5.34% p.a. from: (1+r) = (1- PD)(1+y) 1.1204=1.1836(1 PD) Cumulative PD = 1 [(1 - PD1)(1 PD2)] = 1 [(1-.05)(1-.0534)] = 10.07%


The Loss Intensity ProcessExpected Losses (EL) = PD x LGDIf LGD is not fixed at 100% then:(1 + r) = [1 - (PDxLGD)](1 + y)Identification problem: cannot disentangle PD from LGD.


Disentangling PD from LGDIntensity-based models specify stochastic functional form for PD.Jarrow & Turnbull (1995): Fixed LGD, exponentially distributed default process.Das & Tufano (1995): LGD proportional to bond values.Jarrow, Lando & Turnbull (1997): LGD proportional to debt obligations.Duffie & Singleton (1999): LGD and PD functions of economic conditionsUnal, Madan & Guntay (2001): LGD a function of debt seniority.Jarrow (2001): LGD determined using equity prices.


KPMGs Loan Analysis SystemUses risk-neutral pricing grid to mark-to-marketBackward recursive iterative solution Figure 5.2.Example: Consider a $100 2 year zero coupon loan with LGD=100% and yield curves from Figure 5.1.Year 1 Node (Figure 5.3):Valuation at B rating = $84.79 =.94(100/1.1204) + .01(100/1.1204) + .05(0)Valuation at A rating = $88.95 = .94(100/1.1204) +.0566(100/1.1204) + .0034(0)Year 0 Node = $74.62 = .94(84.79/1.08) + .01(88.95/1.08) Calculating a credit spread:74.62 = 100/[(1.08+CS)(1.1204+CS)] to get CS=5.8% p.a.


Noisy Risky Debt PricesUS corporate bond market is much larger than equity market, but less transparentInterdealer market not competitive large spreads and infrequent trading: Saunders, Srinivasan & Walter (2002)Noisy prices: Hancock & Kwast (2001)More noise in senior than subordinated issues: Bohn (1999)In addition to credit spreads, bond yields include:Liquidity premiumEmbedded optionsTax considerations and administrative costs of holding risky debt


Mortality Rate Derivation of Credit Risk MeasuresThe Insurance Approach:Mortality Models and the CSFP Credit Risk Plus Model


Mortality AnalysisMarginal Mortality Rates = (total value of B-rated bonds defaulting in yr 1 of issue)/(total value of B-rated bonds in yr 1 of issue).Do for each year of issue.Weighted Average MMR = MMRi = tMMRt x w where w is the size weight for each year t.


Mortality Rates - Table 11.10Cumulative Mortality Rates (CMR) are calculated as:MMRi = 1 SRi where SRi is the survival rate defined as 1-MMRi in ith year of issue.CMRT = 1 (SR1 x SR2 xx SRT) over the T years of calculation.Standard deviation = [MMRi(1-MMRi)/n] As the number of bonds in the sample n increases, the standard error falls. Can calculate the number of observations needed to reduce error rate to say std. dev.= .001No. of obs. = MMRi(1-MMRi)/2 = (.01)(.99)/(.001)2 = 9,900


CSFP Credit Risk Plus Appendix 11BDefault mode modelCreditMetrics: default probability is discrete (from transition matrix). In CreditRisk +, default is a continuous variable with a probability distribution.Default probabilities are independent across loans.Loan portfolios default probability follows a Poisson distribution. See Fig.8.1.Variance of PD = mean default rate. Loss severity (LGD) is also stochastic in Credit Risk +.


Distribution of LossesCombine default frequency and loss severity to obtain a loss distribution. Figure 8.3.Loss distribution is close to normal, but with fatter tails.Mean default rate of loan portfolio equals its variance. (property of Poisson distrib.)


Pros and ConsPro: Simplicity and low data requirements just need mean loss rates and loss severities.Con: Inaccuracy if distributional assumptions are violated.


Divide Loan Portfolio Into Exposure BandsIn $20,000 increments.Group all loans that have $20,000 of exposure (PDxLGD), $40,000 of exposure, etc.Say 100 loans have $20,000 of exposure.Historical default rate for this exposure class = 3%, distributed according to Poisson distrib.


Properties of Poisson DistributionProb.(n defaults in $20,000 severity band) = (e-mmn)/n! Where: m=mean number of defaults. So: if m=3, then prob(3defaults) = 22.4% and prob(8 defaults)=0.8%.Table 8.2 shows the cumulative probability of defaults for different values of n.Fig. 8.5 shows the distribution of the default probability for the $20,000 band.


Loss Probabilities for $20,000 Severity Band


Table 8.2 Calculation of the Probability of Default, Using the Poisson Distribution

N

Probability Cumulative Probability

0

0.0497870.049789

1

0.1493610.199148

2

0.2240420.42319

3

0.2240420.647232

.

7

0.021604 0.988095

8

0.0081020.996197

Economic Capital CalculationsExpected losses in the $20,000 band are $60,000 (=3x$20,000)Consider the 99.6% VaR: The probability that losses exceed this VaR = 0.4%. That is the probability that 8 loans or more default in the $20,000 band. VaR is the minimum loss in the 0.4% region = 8 x $20,000 = $160,000.Unexpected Losses = $160,000 60,000 = $100,000 = economic capital.


Calculating the Loss Distribution of a Portfolio Consisting of 2 Bands:$20,000 and $40,000 Loss Severity


Aggregate Portfolio (Loss on v = 1, Loss on v = 2)

Loss ($)in $20,000 units Probability

0 (0,0) (.0497 x .0497)

20,000 (1,0) (.1493 x .0497)

40,000 [(2, 0) (0,1)] [(.224 x .0497) + (.0497 x.1493)]

60,000 [(3, 0) (1, 1)] [(.224 x .0497) + (.1493)2]

80,000 [(4, 0) (2,l) (0, 2)] [(.168 x.0497) + (.224 x.1493) +

(.0497x.224)]

Add Another Severity BandAssume average loss exposure of $40,000100 loans in the $40,000 bandAssume a historic default rate of 3%Combining the $20,000 and the $40,000 loss severity bands makes the loss distribution more normal. Fig. 8.8.


OversimplificationsThe mean default rate was assumed constant in each severity band. Should be a function of macroeconomic conditions.Ignores default correlations particularly during business cycles.


Loan Portfolio Selection and Risk MeasurementChapter 12


The Paradox of CreditLending is not a buy and holdprocess.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.


Managing the Loan Portfolio According to the Tenets of Modern Portfolio TheoryImprove 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


The Optimum Risky Loan Portfolio Fig. 10.2Choose the point on the efficient frontier with the highest Sharpe ratio:The Sharpe ratio is the excess return to risk ratio calculated as:


Problems in Applying MPT to Untraded Loan PortfoliosMean-variance world only relevant if security returns are normal or if investors have quadratic utility functions.Need 3rd moment (skewness) and 4th moment (kurtosis) to represent loan return distributions.Unobservable returnsNo historical price data.Unobservable correlations


KMVs Portfolio ManagerReturns for each loan I:Rit = Spreadi + Feesi (EDFi x LGDi) rf Loan Risks=variability around EL=EGF x LGD = ULLGD assumed fixed: ULi = LGD variable, but independent across borrowers: ULi =

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): ULi =


CorrelationsFigure 11.2 joint PD is the shaded area.GF = GF/GFGF =

Correlations higher (lower) if isocircles are more elliptical (circular). If JDFGF = EDFGEDFF then correlation=0.


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