Value At Risk Models

(Example from Anthony Saunders, "The VAR Approach",

in Credit Risk Measurement, New York: John Wiley and Sons, 1999, pp. 37-57)
©2002
Value at Risk (VAR) models measure the maximum loss (in value) on a given asset or liability over a given
time period at a given confidence level (e.g., 95%, 97.5%, 99%). Most VAR models, like those based on
earnings at risk, rely on an underlying distribution of returns to generate explicit potential loss values.
Our initial focus is on single asset VAR models.
Example: What is the value at risk(VAR) of a 1.00% percent loss in equity? First, derive the correspondin
normal standard deviation associated with a given probability. Then multiply the corresponding
normalized standard deviation times the standard deviation of an asset and subtract this product
from the current asset value (e.g., current equity value).
$80.00 Current equity value 1.00% Query Standard Probability (QSP)
$10.00 Standard Deviation 2.33 Query Standard Deviation (QSD)
Adjusted Value = $56.74 = ($80.00 -(2.33)x ($10.00)
VAR = $23.26 = ($80.00 -$56.74)
The lower the query standard probability, the higher will be the query standard deviation, and the higher the VAR.
Table 1
Query Standard Probability, Query Standard Deviation, and Value at Risk
QSP QSD VAR
1.00% 2.33 $23.26
2.00% 2.05 $20.54
3.00% 1.88 $18.81
4.00% 1.75 $17.51
5.00% 1.64 $16.45
6.00% 1.55 $15.55
7.00% 1.48 $14.76
8.00% 1.41 $14.05
9.00% 1.34 $13.41
10.00% 1.28 $12.82
QSP = Query Standard Probability
QSD = Query Standad Deviation
VAR = Value at Risk

Value At Risk Under Alternative Levels of Risk

25.00

20.00

15.00

10.00

5.00

0.00
1.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% 10.00%
Query Standard Deviation Value At Risk

Assessing VAR models:

1 The current market value of a loan is not directly observable because most loans are not traded.
Possible solution: create a secondary market in loans.

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 2 If the current market value of a loan is not directly observable, there is no time series on which
to calculate the standard deviation of its value.
3 As with other models, the normality assumption of returns creates potential bias in small samples.
Loans tend to have highly truncated upside returns and long downside risks., i.e., they are asymmetric.

Existing approaches involve compiling data on a borrower's credit rating (credit scoring), on the
probability that a rating will change in the next year (a rating transition probability matrix), recovery
rates on defaulted loans, and credit spreads and yields in the bond (or loan) market.
The CreditMetrics Approach to Loan Valuation Revisions
CreditMetrics, a public research arm of J.P. Morgan, has developed an approach to VAR
in debt and equity markets. We consider here a debt example.
Table 2
Bond Categories, Interest Rates, and Ratings Transition Probabilities
for a Hypothetical Benchmark BBB Bond
Rating Interest Rate Trans.Prob.
AAA 5.00% 0.20%
AA 5.25% 0.40%
A 5.50% 5.00%
BBB 6.00% 85.00%
BB 6.50% 6.00%
B 7.00% 1.50%
CCC 8.00% 1.20%
Default 9.50% 0.70%
Total 100.00%
Source: Adapted from CreditMetrics-Technical Document, J.P. Morgan, April 2, 1997, p. 11
Transition probabilities illustrate the likelihood of a benchmark bond moving from its base level to either
an upgrade or a downgrade. These transition probabilities are derived by industry analysts based on
credit scoring systems, the level and volatility of earnings over time, and, where available, observed
changes in volatility of a firm's equity capital share prices. Changes in ratings translate into the required credit
risk spreads or premiums on a loan's remaining cash flows, and thus, on the implied market (or present)
value of a loan. If a loan is downgraded, the required credit spread premium should rise, while an upgrade
produces the opposite effect.

We now illustrate how bond ratings are used in conjunction with the term structure of interest rates to
derive the corresponding VAR. Consider a relatively riskless asset such as a T-bond. In the absence of
risk, for various time horizons, the yield curve portrays the underlying rate of discount. Usually the yield
curve will be upward sloping, although inverted yield curves can occur in the presence of expected declines
in interest rates that are reflective of current and evolving economic conditions.

Once one introduces risky assets such as bank loans and commercial bonds, one expects that for any
given time horizon, the corresponding yield will be higher. For our present purposes, we will assume
a monotonic relationship, I.e., for a given time horizon and a given bond category, the term interest rate
will be proportionately higher by the ratio of the corresponding T-bond rate to the period 0 rate.

Consider the term structure of interest rates in Table 3. If T-Bonds have a current rate of 3 percent and
a year-1 rate of 3.72 percent, we derive the year-1 rate for AAA bonds as 1.0891. We do this as follows:
1.0891 = (1.0372÷ 1.0000)x 1.0500 Of course, for actual grades
of bonds, the spread of yields will vary according to market conditions and perceived levels of risk.
Table 3
The Term Structure of Interest Rates

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 Rate Category 0 1 2 3
0.0300 T-Bonds 1.0000 1.0372 1.0432 1.0493
0.0500 AAA Bonds 1.0000 1.0891 1.0954 1.1018
0.0525 AA Bonds 1.0000 1.0917 1.0980 1.1044
0.0550 A Bonds 1.0000 1.0942 1.1006 1.1070
0.0600 BBB Bonds 1.0000 1.0994 1.1058 1.1123
0.0650 BB Bonds 1.0000 1.1046 1.1110 1.1175
0.0700 B Bonds 1.0000 1.1098 1.1162 1.1228
0.0800 CCC Bonds 1.0000 1.1202 1.1267 1.1332
0.0950 Default Bonds 1.0000 1.1357 1.1423 1.1490

The Term Structure of Interest Rates

1.2000

1.1500

1.1000

1.0500

1.0000

0.9500

0.9000
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
T-Bonds AAA Bonds AA Bonds A Bonds BBB Bonds
BB Bonds B Bonds CCC Bonds Default Bonds

Let us now examine the effect of different bond ratings on the present value of a remaining loan.
Suppose now that we have the following information on a benchmark bond whose current value, coupon rate,
and interest rate are given below. The coupon payment is the same, whose stream and end-term principal
we discount using the term structure of interest rates as present worth factors.
Example:
B= $100.00 Existing value of a loan
i= 0.06 Current interest rate
C= $6.00 Current coupon payment
t= 5 Remaining years on loan
Using the term structure of interest profile in Table 3 above, we derive the corresponding present value of
bond assets using the term structure of rates as present worth factors (PWF):

Table 4
Present Value of a T-Bond
0 1 2 3
Coupon (+ end Principal) $6.00 $6.00 $6.00 $6.00
PWF 1.0000 1.0372 1.0432 1.0493
Annual Present Values $6.00 $5.78 $5.75 $5.72
Asset Present Value (PV) = $123.90

From Table 3, we now make the same calculations for each category of commercial debt.

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 Table 5
Present Values of Various Grades of Benchmark Commercial Debt
0 1 2 3 4
T-Bonds $6.00 $5.78 $5.75 $5.72 $100.65
AAA $6.00 $5.51 $5.48 $5.45 $95.85
AA $6.00 $5.50 $5.46 $5.43 $95.63
A $6.00 $5.48 $5.45 $5.42 $95.40
BBB $6.00 $5.46 $5.43 $5.39 $94.95
BB $6.00 $5.43 $5.40 $5.37 $94.50
B $6.00 $5.41 $5.38 $5.34 $94.06
CCC $6.00 $5.36 $5.33 $5.29 $93.19
Default $6.00 $5.28 $5.25 $5.22 $91.91
The present values (PV) in the right-hand column reflect the effect of differential interest rates of various
bond rating categories.

Let us now integrate the role of interest rates and bond ratings within a VAR framework. To do so, we
generate a probability distribution of present values, whose standard deviation is then used to calculate
the corresponding VAR.
Table 6
VAR Calculations for a Benchmark Loan
A B C D E
= AxB =B-Mean PV =(D)^2
Rating Trans.Prob. PV Loan Prob.PV Difference Difference^2
AAA 0.20% $118.29 $0.24 $1.12 $1.26
AA 0.40% $118.02 $0.47 $0.86 $0.74
A 5.00% $117.75 $5.89 $0.59 $0.35
BBB 85.00% $117.23 $99.64 $0.06 $0.00
BB 6.00% $116.70 $7.00 -$0.46 $0.21
B 1.50% $116.19 $1.74 -$0.97 $0.95
CCC 1.20% $115.17 $1.38 -$1.99 $3.98
Default 0.70% $113.67 $0.80 -$3.49 $12.18
Weighted Mean PV = $117.16 Variance = $19.67
St.Deviation = $4.44
VAR: 5.00% 1.00%
Standard Normal Distribution $7.30 $10.32
Actual Distribution $5.84a $8.09b
a. Actual distribution 5% level approximated by 9.4%=6%+1.5%+1.2%+.7%
b. Actual distribution 1% level by 3.4%=1.5%+1.2%+.70%

Distribution of Transition
$113.67 0.70% Probability Present Values
$115.17 1.20%
90.00%
$116.19 1.50%
80.00%
$116.70 6.00%
70.00%
$117.23 85.00%
60.00%
$117.75 5.00%
50.00%
$118.02 0.40%
40.00%
$118.29 0.20%
30.00%
20.00%
10.00%
0.00% -4-
$113.00 $114.00 $115.00 $116.00 $117.00 $118.00 $119.00
 50.00%
40.00%
30.00%
20.00%
10.00%
0.00%
$113.00 $114.00 $115.00 $116.00 $117.00 $118.00 $119.00

Differences in the transition probability distribution create a bias in terms of the actual VAR,
as is shown in the above figure.
What implications derive from the VAR calculations? First is that a VAR estimate for an asset
provides a benchmark standard for the level of required capital, or reserves. This may be quite
different from a standardized capital reserve requirement as set by a central bank or by
an international standard such as the tier reserve requirements of the Basle Accords of 1988.
While VAR models represent an important step from simple capital reserve models, it should
be kept in mind that VAR estimates depend on the intertemporal stability of transition
probabilities as well as on the normality of the underlying probability distribution. VAR may
not capture the effects of asymmetric information arising the presence of moral hazard or
adverse selection.

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 P. LeBel
over a given
ased on
alues.

derive the corresponding

corresponding
act this product

ard Probability (QSP)

ard Deviation (QSD)

nd the higher the VAR.

9.00% 10.00%

are not traded.

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ries on which

n small samples.
they are asymmetric.

oring), on the
y matrix), recovery

vel to either
based on
observed
the required credit
(or present)
hile an upgrade

erest rates to
absence of
ally the yield
pected declines

s that for any

assume
interest rate

3 percent and
s as follows:
r actual grades
of risk.

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 4
1.0532
1.1059
1.1085
1.1111
1.1164
1.1217
1.1269
1.1375
1.1533

4 4.5

ng loan.
alue, coupon rate,
d-term principal

esent value of

4
$106.00
1.0532
$100.65

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 PV
$123.90
$118.29
$118.02
$117.75
$117.23
$116.70
$116.19
$115.17
$113.67
rates of various

To do so, we
to calculate

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sset

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