Portfolio Models of Credit Risk, Merton Model, and KMV
1. A risk analyst is trying to estimate the Credit VaR for a risky bond. The Credit VaR is defined
here as the maximum unexpected loss at a confidence level of 99.9% over a one-month horizon.
Assume that the bond is valued at $1,000,000 one-month forward, and the one-year cumulative
default probability is 2% for this bond, what is the best estimate of the Credit VaR for this bond
assuming no recovery?
a) 20,000
b) 1,682
c) 998,318
d) 0
Solution: c
First, transform the annual default probability into a monthly probability. Using (1-
2%)=(1-d)^12, find d=0.00168, which assumes a constant probability of default during the year.
The expected credit loss is d*$1M = $1,682. Finally, we calculate the WCL at the 99.9%
confidence level, which is the lowest number CL
i
such that
P
(CL
≤
CL
i
)
≥
99
.
9%. We have
P
(CL = 0) = 99
.
83%;
P
(CL
≤
1
,
000
,
000) = 100
.
00%. Therefore, the WCL is $1,000,000, and the
CVAR is $1
,
000
,
000
−
$1
,
682 = $998
,
318
2. A senior unsecured BB rated bond matures exactly in 5 years, and is
paying an annual coupon of 6%. The 1-year forward price of the bond, if the
obligor stays BB is __________
Solution: 102.0063
3. Following is a set of identical transactions. Assuming all counterparties
have the same credit rating, which transaction should preferably be executed?
a.
Buying gas from a trading firm
b.
Buying gas from a gas producer
c.
Buying gas from a distributor
d.
Indifferent between a), b), and c).
Solution: b. This is an example of right-way trade. To have lower credit risk,
it would be preferable to engage in a trade where there is a lower probability
of a default by the counterparty when the contract is in-the-money. This will
happen if the counterparty enters a transaction to hedge an operating
exposure. For instance, a gas producer has a natural operating exposure to