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### “A bank is a place that will lend you money if you can prove that you don’t need it.”

### Why New Approaches to Credit Risk Measurement and Management?

### Traditional Approaches to Credit Risk Measurement

### The Option Theoretic Model of Credit Risk Measurement

### Term Structure Derivation of Credit Risk Measures

### Mortality Rate Derivation of Credit Risk Measures

### Loss Probabilities for $20,000 Severity Band

### Loan Portfolio Selection and Risk Measurement

Bob Hope

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

Why Now?

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

Structural Increase in Bankruptcy

- Increase in probability of default
- High yield default rates: 5.1% (2000), 4.3% (1999, 1.9% (1998). Source: Fitch 3/19/01
- Historical Default Rates: 6.92% (3Q2001), 5.065% (2000), 4.147% (1999), 1998 (1.603%), 1997 (1.252%), 10.273% (1991), 10.14% (1990). Source: Altman
- Increase 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.

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

Disintermediation

- Direct Access to Credit Markets
- 20,000 US companies have access to US commercial paper market.
- Junk Bonds, Private Placements.
- “Winner’s Curse” – Banks make loans to borrowers without access to credit markets.

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

More Competitive Margins

- Worsening of the risk-return tradeoff
- Interest Margins (Spreads) have declined
- Ex: 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 deteriorated
- The loan mutual funds have written down loan value

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

The Growth of Off-Balance Sheet Derivatives

- Total 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.

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

Declining and Volatile Values of Collateral

- Worldwide deflation in real asset prices.
- Ex: Japan and Switzerland
- Lending based on intangibles – ex. Enron.

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

Technology

- Computer Information Technology
- Models use Monte Carlo Simulations that are computationally intensive
- Databases
- Commercial Databases such as Loan Pricing Corporation
- ISDA/IIF Survey: internal databases exist to measure credit risk on commercial, retail, mortgage loans. Not emerging market debt.

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

BIS Risk-Based Capital Requirements

- BIS 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.

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

20 years of modeling history

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

Expert Systems – The 5 Cs

- Character – reputation, repayment history
- Capital – 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.

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

Problems with Expert Systems

- Consistency
- Across 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).
- Subjectivity
- What are the optimal weights to assign to each factor?

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

Credit Scoring Models

- Linear Probability Model
- Logit Model
- Probit Model
- Discriminant Analysis Model
- 97% of banks use to approve credit card applications, 70% for small business lending, but only 8% of small banks (<$5 billion in assets) use for small business loans. Mester (1997).

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

Linear Discriminant Analysis – The Altman Z-Score Model

- Z-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

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

Problems with Credit Scoring

- Assumes 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.

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

Based on Merton (1974)

KMV Proprietary Model

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

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.

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

Using Option Valuation Models to Value Loans

- Figure 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, , ) where

S=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, ) where

A=market value of assets, B=face value of debt, r=risk-free rate, A=asset volatility,=time to debt maturity.

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

Problem with Equation (4.2)

- A andAare 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 andA:

E = g(A) Equation (4.4)

Can solve for A andA with equations (4.3) and (4.4) to obtain a Distance to Default = (A-B)/A Figure 4.4

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

Merton’s Theoretical PD

- Assumes assets are normally distributed.
- Example: Assets=$100m, Debt=$80m, A=$10m
- Distance 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).

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

Bond Valuation Model

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

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

KMV’s Empirical EDF

- Utilize database of historical defaults to calculate empirical PD (called EDF):
- Fig. 4.5

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

Accuracy of KMV EDFsComparison to External Credit Ratings

- Enron (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/Moody’s obligor-level ratings=22%; Type 1 error on issue-specific rating=35%.

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

Agency Rating

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

Problems with KMV EDF

- Not 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.

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

Reduced Form Models: KPMG’s Loan Analysis System and Kamakura’s Risk Manager

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

Estimating PD: An Alternative Approach

- Merton’s OPM took a structural approach to modeling default: default occurs when the market value of assets fall below debt value
- Reduced form models: Decompose risky debt prices to estimate the stochastic default intensity function. No structural explanation of why default occurs.

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

A Discrete Example:Deriving Risk-Neutral Probabilities of Default

- B 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)

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

Multiyear PD Using Forward Rates

- Using 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%

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

The Loss Intensity Process

- Expected Losses (EL) = PD x LGD
- If LGD is not fixed at 100% then:

(1 + r) = [1 - (PDxLGD)](1 + y)

Identification problem: cannot disentangle PD from LGD.

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

Disentangling PD from LGD

- Intensity-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 conditions
- Unal, Madan & Guntay (2001): LGD a function of debt seniority.
- Jarrow (2001): LGD determined using equity prices.

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

KPMG’s Loan Analysis System

- Uses risk-neutral pricing grid to mark-to-market
- Backward 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.

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

Noisy Risky Debt Prices

- US corporate bond market is much larger than equity market, but less transparent
- Interdealer 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 premium
- Embedded options
- Tax considerations and administrative costs of holding risky debt

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

The Insurance Approach:

Mortality Models and the CSFP Credit Risk Plus Model

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

Mortality Analysis

- Marginal 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.

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

Mortality Rates - Table 11.10

- Cumulative 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 x…x 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.= .001
- No. of obs. = MMRi(1-MMRi)/2 = (.01)(.99)/(.001)2 = 9,900

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

CSFP Credit Risk Plus Appendix 11B

- Default mode model
- CreditMetrics: 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 portfolio’s 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 +.

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

Distribution of Losses

- Combine 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.)

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

Pros and Cons

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

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

Divide Loan Portfolio Into Exposure Bands

- In $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.

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

Properties of Poisson Distribution

- Prob.(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.

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

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

Economic Capital Calculations

- Expected 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.

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

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

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

Add Another Severity Band

- Assume average loss exposure of $40,000
- 100 loans in the $40,000 band
- Assume 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.

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

Oversimplifications

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

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

Chapter 12

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

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.

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

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

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

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:

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

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 3rd moment (skewness) and 4th moment (kurtosis) to represent loan return distributions.
- Unobservable returns
- No historical price data.
- Unobservable correlations

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

KMV’s Portfolio Manager

- Returns for each loan I:
- Rit = Spreadi + Feesi – (EDFi x LGDi) – rf
- Loan Risks=variability around EL=EGF x LGD = UL
- LGD 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 =

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

Correlations

- Figure 11.2 – joint PD is the shaded area.
- GF = GF/GF
- GF =
- Correlations higher (lower) if isocircles are more elliptical (circular).
- If JDFGF = EDFGEDFF then correlation=0.

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

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