Credit Risk Management

Credit Risk Management

Credit Risk Management VAR

Philippe Jorion University of California at Irvine July 2004 © 2004 P.Jorion E-mail: [email protected]

Please do not reproduce without author’s permission \var\wbank\4credit.ppt

Credit Risk Management: Plan (1) The nature of credit risk (2) Credit risk measurement (3) The Basel capital charges (4) Procyclicality of credit risk charges (4) Conclusions

Philippe Jorion

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Credit Risk Management (1) The Nature of Credit Risk

Credit Risk: Definition Risk of economic loss from the failure of a counterparty to fulfill its contractual obligations ! Risk can be defined either: !

» as default mode (only if actual default occurs) » as mark-to-market (accounting for intermediate changes in value due to changes in default probability)

Risk Management - Philippe Jorion

Philippe Jorion

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Drivers of Credit Risk Default event: discrete variable for default binomial event b = 1 (default) or 0 (no default), with probability p = E(b) ! Credit exposure: amount exposed at default EAD = exposure at default ! Loss given default: fraction of exposure lost LGD = 1 – fractional recovery = 1 – f ! Correlations across variables !


Distribution of Credit Losses Define N as the number of counterparties ! Default-mode credit loss is !

CL =

∑

N i =1

bi × E A D i × LG D i

This is a complex random variable, which depends on the distribution of the risk drivers and all correlations thereof ! This also involves market risk for EAD !


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Credit Risk: Probability of Default !

Traditional approaches: Based on » accounting data (Altman’s Z-score) » historical default rates across credit rating

Reduced-form models: Estimate PD from credit spreads ! Structural models: Estimate PD from equity prices and economic explanation of default process !


Credit Risk: Exposure at Default Bonds, loans, receivables, guarantees: exposure is close to the notional amount ! Performance letters of credit, commitments have lower exposures ! Swaps and forward contracts: initial exposure is zero but then takes on positive value— potential exposure is stochastic ! Long options: exposure is positive ! Short options: exposure is zero !


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Credit Risk: Loss Given Default Primary determinant is seniority ! Collateral is relevant ! Stochastic and can be correlated with probability of default !


Credit Risk: Correlations Across Defaults Structural model is required because default correlations cannot be measured directly for obligors ! Approaches: (1) Fixed correlations across all industries (2) Correlations measured from equity prices (3) Correlations from country/industry indices and mapping of obligor on indices !


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Measuring Credit Risk: Example 1 credit of $100 million

100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% -$100

N=1, E(Loss)=$1 million, SD=$10 million

-$90

-$80

-$70

-$60

-$50

-$40

-$30

-$20

-$10

$0

10 independent credits of $10 million 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% -$100


-$90

-$80

-$70

-$60

-$50

-$40

-$30

-$20

-$10

$0


cr2a.wmf at 80%

Measuring Credit Risk: Example 100 independent credits of $1 million

100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% -$100


-$90

-$80

-$70

-$60

-$50

-$40

-$30

-$20

-$10

$0

1000 independent credits of $100,000 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% -$100

N=1000, E(Loss)=$1 million, SD=$0.3 million

-$90


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-$80

-$70

-$60

-$50

-$40

-$30

-$20

-$10

$0 cr2b.wmf at 80%

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Credit Risk Management (2) Credit Risk Measurement

Credit Risk Models: Taxonomy

Default mode: considers default only

Bottom-up: models each credit individually (e.g. corporate, sovereign) Mark-to-market: considers changes in market values

Conditional model: accounts for changing parameters (PD), function of macro conditions Structural correlation model: explains correlations by joint asset/stock prices

Unconditional models: uses fixed parameters (but can be changed) Reduced-form correlation model: relates default to common macro factors

Top-down: aggregates risk (e.g. retail)


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Measuring Credit Risk: CreditMetrics Bottom-up, mark-to-market, unconditional, structural model of portfolio credit risk ! Credit risk driver is migration of credit rating, from transition matrix (unconditional model) ! Correlations in credit rating changes inferred from equity market correlations (structural model) ! Bond/loan values based on credit ratings and credit spreads ! Expected exposures from market risk model; no integration of market and credit risk !


Exposures User portfolio

Market volatilities

Expected exposure

Credit VAR Credit rating

Credit spreads

Correlations

Equities Seniority correlations

Bond Recovery Correlation Rating model rate migration valuation

Distribution of values for a single credit

Joint rating changes

Portfolio Value at Risk due to credit Source: CreditMetrics

Philippe Jorion

crm.wmf

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Distribution of Losses

Frequency

Unexpected loss at 99% level

Expected loss

Loss Risk Management - Philippe Jorion

Credit Loss Distribution Skewed to the left ! Similar to a short position in an option !

» Merton model decomposes a position in a credit -sensitive bond as a long position in a risk-free bond plus a short position in a put on the firm value


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Option Approach to Corporate Debt Value of the firm

Equity

Face value K of debt

Debt

0

K

Value of the firm Risk Management - Philippe Jorion

merton.wmf at 80%

Option Approach to Corporate Debt Equity

0 Debt

K

0

K Value of the firm


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merton2.wmf at 80%

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Option Distribution Option value

Distribution of option values

Spot price

Distribution of spot prices Risk Management - Philippe Jorion

dist-put.wmf at 80%

Option Distribution


Philippe Jorion

ERM-risk.swf

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Using the Loss Distribution (1): Expected Losses Pricing of products: bid-ask spreads, or corporate bond yields should be sufficient to cover expected credit losses (plus remuneration of economic capital) ! Provisions: charge-offs against income to account for expected losses !


Expected Credit Loss !

Over the target horizon (say 1 year), this is

ECL = ∫ (b × EAD × LGD)f (b,EAD,LGD)db dEAD dLGD !

When the variables are independent, this is

ECL = [∫ bf (b)db][∫ EADf (EAD)dEAD][∫ LGDf (LGD)dLGD]

ECL = Pr ob[default] × E[EAD] × E[LGD] Risk Management - Philippe Jorion

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Expected Credit Loss: Example !

Compute the ECL on a BBB-rated $100 million 5-year bond » the expected cumulative default rate is 2.28% » the exposure is assumed constant » the expected recovery rate is 47%

ECL = Pr ob[default] × E[EAD] × (1 − f ) ECL = 2.28%× $100m × (1 − 47%) = $1.2m !

Note that this does not depend on correlations


Using the Loss Distribution (2): Unexpected Losses Covered by equity capital: institution should set aside enough capital to cover worst loss ! Can be lowered by diversification or credit derivatives ! Remuneration of economic capital should also go into the pricing of the product !


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Comparison of Capital Charges Portfolio A

Portfolio B

Portfolio C

High credit quality, diversified (500)

High credit quality, concentrated (100)

Low credit quality, diversified (500)

Assuming Zero Correlations CreditMetrics

777

2,093

1,989

CreditRisk+

789

2,020

2,074

Internal Model 1

767

1,967

1,907

Internal Model 2

724

1,906

1,756

Basel Rules

5,304

5,304

5,304

CreditMetrics

2,264

2,941

11,436

CreditRisk+

1,638

2,574

10,000

Internal Model 1

1,373

2,366

9,654

Basel Rules

5,304

5,304

5,304

Assessing Correlations

ISDA-Portfolio of $66.3 billion notional, 99%1-year VAR Risk Management - Philippe Jorion

Using the Loss Distribution (3): Marginal Contribution to Risk Individual credits should be evaluated not only on the basis of their stand-alone risk, but also their contribution to portfolio risk ! Economic capital should be allocated to credit positions in relation to their contribution to risk: risk premium ! Banks that use economic capital efficiently, or diversify, can charge lower risk premiums, and be more competitive !


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Portfolio Credit Risk Models: Issues Ideally, internal credit risk models give best estimate of economic capital, like internal model approach for market risk ! In practice, models need to be validated ! Validation difficult: !

» credit events more rare, probabilities difficult to estimate » validation needs to be performed over many years, across economic cycles » correlation across borrowers is an important driver of credit risk and is difficult to measure Risk Management - Philippe Jorion

Risk Management (3) The Basel Capital Charges

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Basel Capital Adequacy: Credit Risk Charges !

Standardized Approach: » assign external ratings into new risk weights

!

Foundation Internal Ratings Based Approach: » map internal ratings into probability of default (PD), risk weights; supervisors supply other inputs

!

Advanced Internal Ratings Based Approach: » banks supply internal ratings, loss given default (LGD), exposure at default (EAD), maturity (M) CRC =

∑

i

K i × EAD i = 8%

(∑

i

RWi × EAD i

)


Standardized Approach New Basel Risk Weights: Standardized Approach Credit Rating Claim

AAA/AA-

A+/A-

BBB+/BBB-

BB+/B-

Below B-

Unrated

Sovereign

0%

20%

50%

100%

150%

100%

Banks-option 1

20%

50%

100%

100%

150%

100%

Banks-option 2

20%

50%

50%

100%

150%

50%

Short-term

20%

20%

20%

50%

150%

20%

Claim Corporates

AAA/AA20%

A+/A50%

BBB+/BB100%

Below BB-

Unrated

150%

100%

Notes: • Under option 1, the bank rating is based on the sovereign country in which it is incorporated • Under option 2, the bank rating is based on an external credit assessment • Short-term is below three month Risk Management - Philippe Jorion

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Basel Committee Goals Maintain the 8% minimum ratio for the banking system as a whole ! Establish risk-sensitive risk charge to minimize regulatory arbitrage ! Provide incentives for further development of good risk management systems ! Calibrate capital charge to unexpected loss over one year horizon at 99.9% confidence level !


Basel Committee Philosophy “[T]he new framework is intended to align regulatory capital requirements more closely with underlying risks, and to provide banks and their supervisors with several options for the assessment of capital adequacy.” -- William McDonough, January 2001


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Internal Rating Based (IRB) Approach !

The capital multiplier for credit i is

 N−1 ( PD) + ρ( PD) ⋅ N−1 ( 0.999)  1+( M−2.5) ⋅b( PD)  −PD⋅ LGD]⋅ K =[LGD⋅ N  1−1.5⋅b( PD)    1 ρ PD − ( )    

N(.) is the normal cumulative distribution fct., M is the effective maturity, ρ(PD) a correlation function, b(PD) a maturity function ! The total capital charge is CRC =

∑

i

K i × EAD i


IRB Approach Regulatory capital adds up individual charges ! Economic capital charge, however, should reflect the contribution of obligor i to the portfolio risk; generally depends on portfolio ! This is “portfolio-invariant” with a one-factor model and sufficiently diversified portfolio ! The capital charge should be proportional to the probability that the obligor return is less than a cutoff, conditional on common factor m !

Pr ( Ri ≤ Z i | m ) Risk Management - Philippe Jorion

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IRB Approach: Rationale (1) Obligor i has return Ri on its asset value and defaults if return less than threshold Zi ! Assume Ri∼N(0,1) ! Since Prob(Ri≤Zi)=PDi, we have Zi=N-1(PDi) ! Decompose the return into a market return, with correlation √ρ, and idiosyncratic return !

ρ i ⋅ M + 1 − ρi ⋅ ε i

Ri =

assuming M and ε∼N(0,1) Risk Management - Philippe Jorion

IRB Approach: Rationale (2) Pr ( Ri ≤ Zi | m ) = Pr

(

ρ i ⋅ M + 1 − ρi ⋅ ε i ≤ N −1 ( PDi ) | m

)

 N −1 ( PDi ) + ρ i ⋅ (− M )  Pr ( Ri ≤ Zi | m ) = Pr  ε i ≤ | m   1 ρ − i  

Taking the extreme value such Pr(-M≤m) = 0.999,  N −1 ( PDi ) + ρ i ⋅ N −1 (0.999)  Pr ( Ri ≤ Zi | m ) = N     − 1 ρ i  

With an adjustment for expected loss, maturity, and LGD, this is the Basel IRB model Risk Management - Philippe Jorion

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IRB Approach: Correlation Function Correlation Function

!

0.25

Corporate Mortgages Other Retail

0.20

!

0.15

0.10

! !

0.05

0.00 0%

2%

4%

6%

8% 10% 12% 14% 16% 18% 20% Probability of Default

Parameter ρ measures sensitivity of obligor to business cycle Values around 0.1-0.2 for rated corporates (but imprecisely estimated) Values lower for retail Also, lower values for smaller obligors, proxied by higher PD Risk Management - Philippe Jorion

IRB Approach: Maturity Adjustment Maturity Function for Corporates

!

3.00

PD= 1.30% PD= 0.25% PD=10.00%

2.50

!

2.00

!

1.50

! 1.00

Shorter maturities give banks more control over deteriorating loan Baseline for 1 year Increase function for longer maturities Effect is weaker for lower credits, however; maturity function depends on PD

0.50 0

1

2

3

4

5

6

Maturity (years)

Philippe Jorion

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8

9

10 Risk Management - Philippe Jorion

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IRB Capital Requirements 20% 18% 16%

Corporates Mortgages Other Retail

14% 12% 10% 8% 6% 4% 2% 0% 0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

Probability of Default Risk Management - Philippe Jorion

IRB Approach: Diversification Issues Basel capital charges add up credit risk charges for each individual obligor ! Some allowance is made for diversification effects by having different asset classes ! Within each class, “portfolio invariance” is achieved with a single-factor model and perfect diversification; capital has been calibrated to “typical” portfolio correlations ! However, this penalizes banks with greater than average diversification !


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Risk Management (4) Procyclicality of Credit Risk Charges

Issues Concern is that new capital standards will exacerbate business cycles ! In a downturn, !

» bank capital is eroded due to increased losses, » existing obligors are downgraded; higher PD leads to greater capital charge » raising capital may be more difficult !

This could lead banks to tighten lending activity during a downturn, making it worse


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Example A bank starts with 100 loans and $8 capital ! 10 loans default, with LGD=45% (loss=$4.5) ! Under Basel I, capital drops from $8 to $3.5 and remaining charge is 90x8%=$7.2: shortfall is 7.2-3.5=$3.7 ! Under Basel II, remaining 90 loans become more risky with charge of 90x10%=$9.0: shortfall is 9.0-3.5=$5.5 ! Capital charge would have increased by 25% =(10-8)/8 !


Numerical Exercise Consider typical bank portfolio over 1998-02 ! Compute new capital charge under IRB ! Using S&P ratings, charge would have increased by 30-45%: ratings are more stable and supposed to be “through the cycle” ! Using KMV's EDFs based on stock prices, charge would have increased by 3-83%: EDFs are “point-in-time” and more responsive !

Source: Kashyap and Stein (2004) Risk Management - Philippe Jorion

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Solutions Bank regulators can force capital to be higher than Basel minimum: not optimal ! Procyclical effects can be dampened by: (1) smoothing PD, (2) flattening the IRB curve, or (3) changing the aggregate capital charge (e.g. !

using a moving average or a countercyclical index)

!

The latter solution is preferred because it maintains informativeness in reported capital charges, does not invite regulatory arbitrage


Risk Management (5) Conclusions

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Conclusions (1) Credit risk models are much more complex than market risk models but are still based on Value-at-Risk methodology ! Like VAR models, diversification effects and correlations are essential ! Credit risk models are still difficult to validate and not trusted by bank regulators !

» no backtesting framework » validation needs to economic cycles Risk Management - Philippe Jorion

Conclusions (2) The Basel 2 charges represent a considerable improvement over the Basel 1 rules, introducing risk-sensitive capital charges ! The charges are calibrated to a one-factor model with perfectly diversified remaining risk, so that total capital is still about 8% ! Risk-sensitive credit charges, however, may introduce procyclicality, inducing banks to tighten lending during a recession ! Basel 2 will spread risk management methods !


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Bibliography !

!

Basel Committee on Banking Supervision, 2004, International Convergence of Capital Measurement and Capital Standards: A Revised Framework Gordy, Michael, 2003, “A Risk-Factor Model Foundation for Ratings-Based Bank Capital Rules,” Journal of Financial Intermediation 12 (July) » mgordy.tripod.com/research.html#pubs

!

Anil K Kashyap and Jeremy C. Stein, 2004, “Cyclical implications of the Basel II capital standards,” FRB Chicago Economic Perspectives 28 (1st quarter) » www.chicagofed.org/publications/economicperspectives/ ep_1qtr2004_part2_kashyap_stein.pdf

References !

!

!

!

Philippe Jorion is Professor of Finance at the Graduate School of Management at the University of California at Irvine Author of “Value at Risk,” published by McGraw-Hill in 1997, which has become an “industry standard,” translated into 7 other languages; revised in 2000 Author of the “Financial Risk Manager Handbook,” published by Wiley and exclusive text for the FRM exam; revised in 2003 Editor of the “Journal of Risk”

Phone: (949) 824-5245 FAX: (949) 824-8469

Philippe Jorion

E-Mail: [email protected] Web: www.gsm.uci.edu/~jorion

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Philippe Jorion

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