22-Credit Risk Model Derivs Page 715 Wednesday, February 4, 2004 1:12 PM
Credit Risk Modeling and Credit Default Swaps 715
These survival probabilities can then be used to compute forward
default probabilities defined in equation (22.8):
p 1
Q( 00 , )– Q( 01 , ) 1 – 99.00%
()= ---------------------------------------------- = ------------------------------ = 1.00%
Q( 00 , ) 1
and
p 2
Q( 01 , )– Q( 02 , ) 99.00% – 97.63%
()= ---------------------------------------------- = ------------------------------------------------ = 1.39%
Q( 01 , ) 99.00%
Since we assume a 5% flat risk-free rate for two years, the risk-free dis-
count factors are
P( 01 ) = e
- 5%
,
P( 02 , ) = e - 5% × 2
for one and two years, respectively. Assuming a 20% recovery ratio, we
can then calculate, using equation (22.11), what the total protection
value (V) of the default swap contract is providing
V = e –5%( 1 – 0.99)( 1 – 0.2)+ e –5% ×^2 (0.99 – 0.9763)( 1 – 0.2)
= 0.00761 + 0.010134
= 0.017744 = 177.44 basis points
As mentioned, the default swap premium is not paid in full at the
inception of the swap but paid in a form of spread until either default or
maturity, whichever is earlier. From equation (22.12), we can compute
the spread of the default swap as follows:
0.017744
s = ---------------------------------------------------------------------------------------------------------------------
0.99 ×exp(–0.05)+ 0.9763 ×exp(–0.05 × 2 )
0.017744
= -------------------------= 0.009724
1.824838
which is 9.724 basis points for each period, provided that default does not
occur. This is a payment in arrears. That is, if default occurs in the first
period, no payment is necessary. If default occurs in the second period,
there is one payment; if default never occurs, there are two payments.