Chapter 6

The VAR Approach: CreditMetrics and Other Models

2

The Concept of VAR

• Example of VAR applied to market risk.• Market price of equity = $80 with estimated

daily standard deviation = $10.• If tomorrow is a “bad day” (1 in 100 worst)

then what is the value at risk?• If normally distributed, then the cutoff is

2.33 below the mean = $80 – 2.33(10) = $56.70. 99% VAR = $23.30 Figure 6.1.

3

Time1 Day(Tomorrow)

P $80

P $56.7

2.33 $23.3

0(Today)

Figure 6.1The VAR of traded equity.

4

CreditMetrics

• What is VAR is next year is a “bad” year?

• Since most loans are not publicly traded, then we do not observe the price or the standard deviation.

• Consider VAR for an individual loan. Portfolios are covered in Chapter 11.

5

Rating Migration

Table 6.1 One-Year Transition Probabilities for BBB-Rated Borrower

____________________________________________________________________ AAA 0.02% AA 0.33 A 5.95 BBB 86.93 <------------------------------Most likely to stay BB 5.30 in the same class B 1.17 CCC 0.12 Default 0.18 _____________________________________________________________________ Source: Gupton, et. al., CreditMetrics-Technical Document, J.P. Morgan, April 2,1997, p. 11.

6

Valuation

• Consider BBB rated $100 million face value loan with fixed 6% annual coupon and 5 years until maturity. Cash flow diagram – Figure 6.2.

P = 6 + 6 + 6 + 6 + 106 (6.1)

(1+r1+s1) (1+r2+s2)2 (1+r3+s3)

3 (1+r4+s4)4

7

0

Today(Loan

Origination)

(Credit Events are: upgrades, downgrades, or defaults.)

LoanMaturity

1

$6m $6m $6m $6m

$106m

CreditEventOccurs

2 3 4 5

Figure 6.2 Cash flows on the five-year BBB loan.

8

Valuation at the End of the Credit Horizon (in 1 year)

• Calculate one year forward zero yield curves plus credit spread (see Appendix 6.1)

Table 6.2 One Year Forward Zero Curves Plus Credit Spreads

By Credit Rating Category (%) Category Year 1 Year 2 Year 3 Year 4 AAA 3.60 4.17 4.73 5.12 AA 3.65 4.22 4.78 5.17 A 3.72 4.32 4.93 5.32 BBB 4.10 4.67 5.25 5.63 BB 5.55 6.02 6.78 7.27 B 6.05 7.02 8.03 8.52 CCC 15.05 15.02 14.03 13.52 Source: Gupton, et. al., CreditMetrics-Technical Document, J.P. Morgan, April 2,1997, p. 27.

9

Using the Forward Yield Curves to Value the Risky Loan

• Under the credit event that the loan’s rating improves to A, the value at the end of yr 1:

• Must repeat 8 times for each possible credit migration.

P = 6 + 6 + 6 + 6 + 106 = $108.66 (1.0372) (1.0432)2 (1.0493)3 (1.0532)4

10

Table 6.3Value of the Loan at the End of Year 1

Under Different Ratings

Year-End Rating Value (millions) AAA $109.37 AA 109.19 A 108.66 BBB 107.55 BB 102.02 B 98.10 CCC 83.64 Default 51.13 Source: Gupton, et. al., CreditMetrics-Technical Document, J.P. Morgan, April 2,1997, p. 10.

11

Calculation of VARNormal vs. Actual Distribution

Figure 6.3Table 6.4 VAR Calculations for the BBB Loan

(Benchmark Is Mean Value of Loan) ______________________________________________________________________ New Loan Difference Value Plus Probability of Value Probability Year-End Probability Coupon Weighted from Weighted Rating of State (%) (millions) Value ($) Mean ($) Difference Squared ___________________________________________________________________________________ AAA 0.02 $109.37 0.02 2.28 0.0010 AA 0.33 109.19 0.36 2.10 0.0146 A 5.95 108.66 6.47 1.57 0.1474 BBB 86.93 107.55 93.49 0.46 0.1853 BB 5.30 102.02 5.41 (5.07) 1.3592 B 1.17 98.10 1.15 (8.99) 0.9446 CCC 0.12 83.64 1.10 (23.45) 0.6598 Default 0.18 51.13 0.09 (55.96) 5.6358 $107.09 8.9477 = variance mean value of value = Standard deviation $2.99 Assuming 5 percent VAR = 1.65 X = $4.93. normal 1 percent VAR = 2.33 x = $6.97. distribution Assuming 6.77 percent VAR = 93.23 percent of =$107.09 - $102.02 = $5.07. actual actual distribution distribution* 1.47 percent VAR = 98.53 percent of = $107.09 - $98.10 = $8.99. actual distribution 1 percent VAR = 99 percent of = $107.09 - $92.29 = $14.80. actual distribution ______________________________________________________________________________________ *Note: Calculation of 6.77% VAR (i.e., 5.3%+1.17%+0.12%+0.18%) and 1.47% VAR (i.e., 1. 17% + 0.12% + 0.18%). The 1% VAR is interpolated from the actual distribution of the loan’s values under different rating migrations. Source: Gupton, et. al., CreditMetrics-Technical Document, April 2,1997, p. 28.

12

51.13

1%

86.93

100.12 107.55

Value of Loan if RemainingBBB Rated throughout ItsRemaining Life

107.09 Mean

ExpectedLoss

UnexpectedLoss

Probability%

ReservesEconomicCapital

$0.46$6.97

109.37

Figure 6.3 Actual distribution of loan values onfive year BBB loan at the end of year 1(Including first year coupon payment).

13

Technical Problems

• Rating Migration– Assumes stable Markov process. Nickell, Perraudin &

Varotto (2001) find autocorrelation (2nd order process or higher)

– Cyclical impact on transition matrix. Bangia, Diebold & Schuermann (2000). Figure 6.5 shows CreditMetrics Z macro shift factor: Finger (1999), Kim (1999).

– Impact of bond “aging” on transition probabilities: Altman & Kishore (1997). Non-homogeneity within ratings classes: Kealhofer, Kwok, & Weng (1998).

– Bond transition matrices not applicable to loans.

14

3 2 1

0.5

0.4

0.3

0.2

0.1

10 2 3

Market Factor

Figure 6.5

Source: Finger (1999), p.16.

Unconditional asset distribution andconditional distributions with positive andnegative Z.

UnconditionalConditional, Z2Conditional, Z2

15

Technical Problems (cont.)

• Valuation– Non-stochastic interest rates and credit spreads.– Fixed LGD. Volatility of LGD adds to VAR.

• MTM vs. DM Models– VAR is less under DM than for MTM because

DM does not consider loss of upside gain potential (if credit upgrades).

16

Appendix 6.1Calculating the Forward Zero Yield Curve for

Valuation

• Three steps:– Decompose current spot yield curve on risk-

free (US Treasury) coupon bearing instruments into zero coupon spot risk-free yield curve.

– Calculate one year forward risk-free yield curve.

– Add on fixed credit spreads for each maturity and for each credit rating.

17

Step 1: Calculation of the Spot Zero Coupon Risk-free Yield Curve Using a No Arbitrage Method

• Figure 6.6 shows spot yield curve for coupon bearing US Treasury securities.

• Assuming par value coupon securities:

• Figure 6.7 shows the zero coupon spot yield curve.

Six Month Zero: 100 = C+F = C+F = 100+(5.322/2) 1+0r1 1+0z1 1 + (.05322/2) Therefore, the six month zero riskfree rate is: 0z1 = 5.322 percent per annum One Year Zero: 100 = C + C+F = C + C+F 1+0r2 (1+0r2)

2 1+0z1 (1+0z2)2

100 = (5.511/2) + 100+(5.511/2) = (5.511/2) + 100+(5.511/2) 1+(.05511/2) (1+.05511/2)2 1+(.05322/2) (1+.055136/2)2

18

6.47%

Yield toMaturity

p.a.

6Mos.

1Yr.

2Yr.

3Yr.

Maturity

CYC RF

2.5Yr.

1.5Yr.

6.25%

6.09%

5.98%

5.511%

5.322%

Figure 6.6

Maturity

Yield toMaturity p.a.

Figure 6.7

6 Mos 1 Yr 1.5 Yrs 2 Yrs 2.5 Yrs 3 Yrs

CYC RF

ZYCRF

5.511%

5.98%6.09%

6.25%

0.0647%

5.322%

5.5136%

5.9353%

6.1079%

6.2755%

7.6006%

ZYCRF

Maturity

Yield toMaturity p.a.

Figure 6.8

6 Mos 1 Yr 1.5 Yrs 2 Yrs 2.5 Yrs 3 Yrs

5.322% 5.5136%

5.9353%6.1079%6.7813%

6.6264%

6.9475%

14.3551%

7.2813%

7.1264%

7.4475%

14.8551%

6.2755%

7.6006%

1 Year Forward

1 Year ForwardFYC RF

FYC R

19

Step 2: Calculating the Forward Yields• Use the expectations hypothesis to calculate

6 month maturity forward yields:(1 + 0z2)

2 = (1 + 0z1)(1 + 1z1) (1+(.055136/2)2 = (1+.05322/2)(1+1z1) Therefore, the rate for six months forward delivery of 6-month maturity US Treasury securities is expected to be: 1z1 = 5.7054 percent p.a. (1 + 0z3)

3 = (1 + 0z2)2(1 + 2z1)

(1+(.059961/2)3 = (1+.055136/2)2(1+2z1) Therefore, the rate for one year forward delivery of 6-month maturity US Treasury securities is expected to be: 2z1 = 6.9645 percent p.a.

20

Use the 6 month maturity forward yields to calculate the 1 year forward risk-free yield curve

Figure 6.8

(1 + 2z2)2 = (1 + 2z1)(1 + 3z1)

Therefore, the rate for 1 year maturity US Treasury securities to be delivered in 1 year is: 2z2 = 6.703 percent p.a. (1 + 2z3)

3 = (1 + 2z1)(1 + 3z1)(1 + 4z1) Therefore, the rate for 18-month maturity US Treasury securities to be delivered in 1 year is: 2z3 = 6.7148 percent p.a. (1 + 2z4)

4 = (1 + 2z1)(1 + 3z1)(1 + 4z1)(1 + 5z1) Therefore, the rate for 2 year maturity US Treasury securities to be delivered in 1 year is: 2z4 = 6.7135 percent p.a.

21

Step 3: Add on Credit Spreads to Obtain the Risky 1 Year Forward Zero Yield Curve

• Add on credit spreads (eg., from Bridge Information Systems) to obtain FYCR in Figure 6.8.

Table 6.8 - Credit Spreads For Aaa Bonds Maturity (in years, compounded annually) Credit Spread, si

2 0.007071 3 0.008660 5 0.011180

10 0.015811 15 0.019365 20 0.022361

22

Appendix 6.2Estimating Unexpected Losses Using

Extreme Value Theory (EVT)• VAR measures the minimum loss that will occur with

a certain probability.• EVT examines the size of the loss that exceeds VAR –

the tail distribution.• Figure 6.4: Loss distribution is normal up until the

95%tile, shown to be a loss of $4.93 million. After this, the distribution has the fat tails of a Generalized Pareto Distribution (GPD).

• If used normal distribution then 99% VAR would be $6.97 million. But under GPD, the 99% VAR is $22.23 million.

23

0Mean

Distributionof

UnexpectedLosses

Probability

$4.93 $6.97 $22.23 $53.53

GPD

NormalDistribribution

95% 99% 99%

ES

Mean ofExtremeLosses

Beyond the99th percentile

VAR underthe GPD

VAR VAR VARNormalNormalDist. Dist.

GPD

Figure 6.4(ES = the expected shortfall assuming a generalized Pareto Distribution (GPD) with fat tails.)

Estimating Unexpected Losses Using Extreme Value Theory.

24

Calculating the EVT VAR and the Expected Shortfall (ES)

• The GPD is a 2 parameter skewed distribution:

• With 10,000 observations, the 95% threshold is set by worst 500 observations.

VARq = u + (/)[(n(1 - q)/Nu)- - 1] VAR.99, = 22.23 = $4.93 + (7/.5)[(10,000(1-.99)/500)

-.5 – 1]

G, (x) = 1 – (1 + x/)-1/ if 0, = 1 – exp(-x/) if = 0

The two parameters that describe the GPD are (the shape parameter) and (the scaling parameter). If > 0, then the GPD is characterized by fat tails

25

Expected Shortfall = Mean of the excess distribution of unexpected losses beyond the threshold VAR

• McNeil (1999) shows that:

• Thus: 2.4 times more capital would be needed to secure the bank against catastrophic credit losses compared to unexpected losses occurring up to the 99th percentile level, even using the “fat tails” VAR measure. This may be excessive: Cruz, Coleman & Salkin (1998).

ESq = (VARq/(1 - )) + ( - u)/(1 - ) ES.99 = ($22.23)/.5) + (7 - .5(4.93))/.5 = $53.53

ES.99 /VAR.99 = $53.53 / $22.23 = 2.4