Chapter 6
The VAR Approach: CreditMetrics
and Other Models
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.
2
Figure 6.1 The VAR of traded equity.
P  $80
2.33
 $23.3
P  $56.7
0
(Today)
1 Day
(Tomorrow )
Time
3
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.
4
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.
5
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
+
(1+r1+s1)
6
(1+r2+s2)2
+
6
(1+r3+s3)3
+
106
(6.1)
(1+r4+s4)4
6
Figure 6.2 Cash flow s on the five-year BBB loan.
(Credit Ev ents are: upgrades, downgrades, or def aults.)
$106m
Credit
Event
Occurs
0
Today
(Loan
Origination)
$6m
$6m
$6m
$6m
1
2
3
4
5
Loan
Maturity
7
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
AA
A
BBB
BB
B
CCC
3.60
3.65
3.72
4.10
5.55
6.05
15.05
4.17
4.22
4.32
4.67
6.02
7.02
15.02
4.73
4.78
4.93
5.25
6.78
8.03
14.03
5.12
5.17
5.32
5.63
7.27
8.52
13.52
Source: Gupton, et. al., CreditMetrics-Technical Document, J.P. Morgan, April 2,1997, p.
27.
8
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:
P=6+
6
+
6
+
6
+
106 = $108.66
2
3
(1.0372)
(1.0432)
(1.0493)
(1.0532)4
• Must repeat 8 times for each possible credit
migration.
9
Table 6.3
Value of the Loan at the End of Year 1
Under Different Ratings
Year-End Rating
AAA
AA
A
BBB
BB
B
CCC
Default
Value (millions)
$109.37
109.19
108.66
107.55
102.02
98.10
83.64
51.13
Source: Gupton, et. al., CreditMetrics-Technical Document, J.P. Morgan, April 2,1997, p.
10.
10
Calculation of VAR
Normal vs. Actual Distribution
Figure 6.3
Table 6.4 VAR Calculations for the BBB Loan
(Benchmark Is Mean Value of Loan)
__________________________________________________________________
____
New Loan
Difference
Value Plus
Probability
of Value
Probability
Year-End
Probability
Coupon
Weighted
from
Weighted
Rating
of State (%)
(millions)
Value ($)
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
normal
distribution
5 percent VAR =
1 percent VAR =
Assuming
actual
distribution*
$8.99.
6.77 percent VAR =
1.65 X  = $4.93.
2.33 x  = $6.97.
1.47 percent VAR =
93.23 percent of
actual distribution
98.53 percent of
1 percent VAR =
actual distribution
99 percent of
=$107.09 - $102.02 =
$5.07.
= $107.09 - $98.10 =
= $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.
11
Figure 6.3 Actual distribution of loan values on
five year BBB loan at the end of year 1
(Including first year coupon payment).
Probability
%
86.93
1%
51.13
100.12
Unexpected
Loss
Expected
Loss
Economic
Capital
$6.97
Reserv es
$0.46
107.09 107.55 109.37
 Mean
Value of Loan if Remaining
BBB Rated throughout Its
Remaining Life
12
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.
13
Figure 6.5 Unconditional asset distribution and
conditional distributions w ith positive and
negative Z. Source: Finger (1999), p.16.
0.5
Unconditional
Conditional, Z 2
Conditional, Z  2
0.4
0.3
0.2
0.1
3
2
1
0
1
2
3
Market Factor
14
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).
15
Appendix 6.1
Calculating the Forward Zero Yield Curve for
Valuation
• Three steps:
– Decompose current spot yield curve on riskfree (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.
16
Step 1: Calculation of the Spot Zero Coupon Riskfree 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:
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
• Figure 6.7 shows the zero coupon spot yield curve.
17
Figure 6.7
Figure 6.6
Yield to
Maturity p.a.
Yield to
Maturity
p.a.
7.6006%
CYCRF
ZYCRF
6.47%
6.25%
6.09%
5.98%
6.2755%
6.1079%
5.9353%
5.511%
CYCRF
0.0647%
5.322%
6.25%
5.5136%
6 1 1.5 2 2.5 3
Mos. Yr. Yr. Yr. Yr. Yr.
Maturity
5.98%
6.09%
5.511%
5.322%
6 Mos
1 Yr 1.5 Yrs 2 Yrs 2.5 Yrs 3 Yrs
Maturity
Figure 6.8
Yield to
Maturity p.a.
FYCR 1 Year Forw ard
FYCRF 1 Year Forw ard
14.8551%
14.3551%
ZYCRF
7.4475%
7.1264%
7.6006%
6.9475%
7.2813%
6.6264%
6.7813%
5.322%
6 Mos
5.9353%
6.1079%
6.2755%
5.5136%
1 Yr 1.5 Yrs 2 Yrs 2.5 Yrs 3 Yrs
Maturity
18
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.
19
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.
20
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)
2
3
5
10
15
20
Credit Spread, si
0.007071
0.008660
0.011180
0.015811
0.019365
0.022361
21
Appendix 6.2
Estimating 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.
22
Figure 6.4
Estimating Unexpected Losses Using Extreme Value Theory.
(ES = the expected shortf all assuming a generalized Pareto Distribution (GPD)
with f at tails.)
Distribution
of
Unexpected
Losses
Probability
GPD
Normal
Distribribution
0
Mean
$4.93
$6.97 $22.23 $53.53
ES
95%
99% 99%
VAR
VAR VAR Mean of
Normal Normal GPD Extreme
Dist.
Dist.
Losses
Bey ond the
99th percentile
VAR under
the GPD
23
Calculating the EVT VAR and
the Expected Shortfall (ES)
• The GPD is a 2 parameter skewed
distribution:
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
• With 10,000 observations, the 95%
threshold is set by worst 500 observations.
-
VAR q = u + (/)[(n(1 - q)/Nu) - 1]
-.5
VAR .99, = 22.23 = $4.93 + (7/.5)[(10,000(1-.99)/500) – 1]
24
Expected Shortfall = Mean of the excess distribution
of unexpected losses beyond the threshold VAR
• McNeil (1999) shows that:
ES q = (VAR q/(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
• 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).
25