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Saunders & Cornett, Financial Institutions Management, 4th ed.
1
“A bank is a place that will lend you money if you can prove that
you don’t need it.”Bob Hope
Saunders & Cornett, Financial Institutions Management, 4th ed.
2
Why New Approaches to Credit Risk Measurement and
Management?
Why Now?
Saunders & Cornett, Financial Institutions Management, 4th ed.
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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.
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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.
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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.
6
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.
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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.
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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.
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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.
10
Traditional Approaches to Credit Risk Measurement
20 years of modeling history
Saunders & Cornett, Financial Institutions Management, 4th ed.
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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.
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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.
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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.
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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
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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.
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The Option Theoretic Model of Credit Risk Measurement
Based on Merton (1974)
KMV Proprietary Model
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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.
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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, , ) 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.
Saunders & Cornett, Financial Institutions Management, 4th ed.
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$ Payoff
Assets0 A1 B A2
Figure 4.1 The payoff to a bank lender
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$ Payoff
Stock Price (S)0
X
Figure 4.2 The payoff to the writer of a put option on a stock.
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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
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Value ofAssets (A)
Value ofEquity (E)
($)
B
L
A1 A20
Figure 4.3 Equity as a call option on a firm.
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B$80m
A$100m
t0 t1 Time(t)
Default Region
A
A
Figure 4.4 Calculating the theoretical EDF
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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).
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Merton’s 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%
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KMV’s Empirical EDF
• Utilize 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
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5%
EmpiricalEDF
Figure 4.5default (DD): A hypothetical example.Empirical EDF and the distance to
0 Distanceto Default
(DD)
ProprieteryTrade-Off
2
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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%.
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12/96 06/97 12/97 06/98 12/98 06/99 12/99 06/00 12/00 06/01
20 CCCCC
B
KMV EDF Credit Measure
Source: KMV.
Agency Rating
BB
BBB
A
AA
AAA
151075
2
1.0
.5
.20
.15
.10
.05
.02
Figure 4.6 KMV expected default frequency TM and agency rating for Comdisco Inc.
12/96 06/97 12/97 06/98 12/98 06/99 12/99 06/00 12/00 06/01
20 CCCCC
B
KMV EDF Credit Measure Agency Rating
BB
BBB
A
AA
AAA
151075
2
1.0
.5
.20
.15
.10
.05
.02
Source: KMV.
Figure 4.7 KMV expected default frequency TM and agency rating for USG Corp.
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Monthly EDF™ credit measure
Agency Rating
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1009080706050
Percent of Population Excluded
40302010
100
90
80
70
60
50
40
30
20
10
00
Figure 4.8
Source: Kealhofer (2000).
agency ratings (1990-1999) for rated U.S. companies.KMV EDF Credit Measure vs.
EDF Power
S&P Company Power
S&P Implied Power
Moodys Implied Power
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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.
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Term Structure Derivation of Credit Risk Measures
Reduced Form Models: KPMG’s Loan Analysis System and Kamakura’s Risk Manager
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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.
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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)
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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%
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16%
14%
10%
8%
1 Yr. 2 Yr. Time to Maturity
SpotYield
Zero-CouponTreasury Bond
A Rated Zero-Coupon Bond
B Rated Zero-Coupon Bond
11.5%
13.69%
Figure 5.1 Yield curves.
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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.
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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.
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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.
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0 1 2 3
Time
4D
C
B
B RiskGrade
A
Figure 5.2 The multiperiod loan migrates overmany periods.
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Period 1Period 0
Figure 5.3 Risky debt pricing.
Period 2
$100 A Rating
$100 B Rating
$85.43
$67.14$80.28
$0 Default
5%5%
0.34%
94%
94%
1%5.66%
94%
1%
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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
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Mortality Rate Derivation of Credit Risk Measures
The Insurance Approach:
Mortality Models and the CSFP Credit Risk Plus Model
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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.
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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
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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 +.
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Default Rate
BBB Loan
Credit Risk Plus
CreditMetrics
Possible Path of Default Rate
Time Horizon
Default Rate
BBB Loan
Possible Pathof Default Rate D
BBB
AAA
Time Horizon
Figure 8.1Comparison of credit risk plusand CreditMetrics.
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Frequencyof Defaults
Distribution ofDefault Losses
Severityof Losses
Figure 8.2 The CSFP credit risk plus model.
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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.)
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Probability
Model 1
ActualDistributionof Losses
Losses
Figure 8.3 Distribution of losses with defaultrate uncertainty and severity uncertainty.
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Probability
ExpectedLoss
EconomicCapital
99thPercentileLoss Level
Loss0
Figure 8.4 Capital requirement under the CSFPcredit risk plus model.
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Pros and Cons
• Pro: Simplicity and low data requirements – just need mean loss rates and loss severities.
• Con: Inaccuracy if distributional assumptions are violated.
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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.
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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.
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.008
.05
.168
.224
Defaults
843210
Figure 8.5 Distribution of defaults: Band 1.
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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 0.049787 0.049789 1 0.149361 0.199148 2 0.224042 0.42319 3 0.224042 0.647232 . 7 0.021604 0.988095 8 0.008102 0.996197
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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.
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0
0.25
0.15
0.05
0.1
0.2
0
Amount of Loss in $
ExpectedLoss
EconomicCapital
UnexpectedLoss
350,000400,000250,000300,000160,000200,00060,000 100,000
Figure 8.6 Loss distribution for single loan portfolio —severity rate = $20,000 per $100,000 loan.
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0
0.25
0.15
0.05
0.1
0.2
0
Amount of Loss in $
350,000400,000250,000300,000150,000200,00050,000 100,000
Figure 8.7 Single loan portfolio — severity rate = $40,000per $100,000 loan.
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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)]
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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.
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0
0.120
0.060
0.020
0.040
0.080
0.100
0.000
Amount of Loss in $
350,000400,000250,000300,000150,000200,00050,000 100,000
Figure 8.8 Loss distribution for two loan portfolios withseverity rates of $20,000 and $40,000.
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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.
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Loan Portfolio Selection and Risk Measurement
Chapter 12
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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.
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Return
The EfficientFrontier
A
B
C
Risk0
Figure 10.1 The paradox of credit.
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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
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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:
p
fp rR
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Return (Rp)rf
A
BD
C
Risk (p)
Figure 10.2 The optimum risky loan portfolio
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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
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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 =
)1( EDFEDF
22)1( ii EDFiVOLLGDEDFiEDFi
222 )1()1( iii VVOLEDFiEDFiVVOLLGDEDFiEDFi
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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.
)1()1(
)(
FFGG
FGGF
EDFEDFEDFEDF
EDFEDFJDF
Saunders & Cornett, Financial Institutions Management, 4th ed.
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Firm F
Firm G
Firm F’sDebt Payoff
100
100(1-LGD)
Market Valueof Assets - Firm G
Market Valueof Assets - Firm F
Face Value of Debt
Figure 11.2 Value correlation.