The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 692 Wednesday, February 4, 2004 1:12 PM

692 The Mathematics of Financial Modeling and Investment Management

If we calculate the yield as before, we find that the spread to the risk-

free rate is 1.5 basis points. If the recovery is the asset value, then we do

need to follow equation (22.5) and the debt value is

ln200 – ln100 + (0.05 – 0.2^2 ) × 1

d 2 = --------------------------------------------------------------------------------------= 3.6157

0.2 1

d 1 = 3.6157 + 0.2 = 3.8157

Et()= 200 ×N(3.8157)– e –0.05 × 100 ×N(3.6157)

= 104.877

DtT( , 1 )= 200 – 104.8777 = 95.1223

The small difference in the two results is because the default probability

is really small (only 0.015%). When the default probability gets bigger,

the debt value difference will get larger.

The second bond is more complex to evaluate. It can be defaulted in

t= 1 when the first debt is defaulted or t= 2 when only itself is defaulted.

The retiring of the first debt can be viewed as the dividend of the stock.

Under the lognormal model described above, we can write the firm

value at the end of the two-year period as

( , 2 )= [AtT

(r–σ^2 ⁄ 2 )(T 1 – t)+ σWT 1

AtT ( , 1 )– K 1 ]e

()

= At

(r–σ^2 ⁄ 2 )(T 2 – t)+σWT 2

()e

()

• K 1 e

(r–σ^2 ⁄ 2 )(T 1 – t)+ σWT() 1

where K 1 is the face value of the 1-year debt and

Wt()= ∫t 0 d Wu()du

The default probability of the second debt is the sum of the first year

default probability and the second year default probability as follows:

Pr[AT( 1 ) <K 1 ]+Pr[A(T 1 ) >K 1 and (AT( 2 ) <K 2 ) ]