Question

Exercise 2: The one-year risk-free rate is i1 = 5% and the two-year risk-free rate is i2 = 5%. The probability of default for the first year is (1 − p1) = 10%. The marginal probability of default for the second year is (1 − p2) = 20%.

a) Calculate the two-year cumulative probability of default (1 − Cp2).
b) Calculate the one year corporate interest rate k1 and the corporate forward rate for the second year c2.
c) Calculate the annual interest rate for a two-year loan k2.

Answer 1

1 - p1 = 10%

Hence, p1 = 1 - 10% = 90% = 0.90

1 - p2 = 20%

hence, p2 = 1 - 20% = 80% = 0.80

a) Calculate the two-year cumulative probability of default (1 − Cp2).

The two-year cumulative probability of default = 1 - p1 x p2 = 1 - 0.9 x 0.8 = 1 - 0.72 = 0.28 = 28%

b) Calculate the one year corporate interest rate k1 and the corporate forward rate for the second year c2.

p1 = (1 + f1) / (1 + c1); for the first year, risk free forward rate = f1 = same as risk free spot rate = i1 and corporate forward rate c1 = corporate bond rate k1

Hence, p1 = 0.90 = (1 + i1) / (1 + k1) = (1 + 5%) / (1 + k1)

Hence, k1 = 1.05/0.9 - 1 =  0.1667 = 16.67%

p2 = (1 + f2) / (1 + c2)

i1 = i2 = 5%

(1 + i1) x (1 + f2) = (1 + i2)²

Hence, (1 + 5%) x (1 + f2) = (1 + 5%)²

Hence, f2 = (1 + 5%)² / (1 + 5%) - 1 = 5%

p2 = 0.8 = (1 + 5%) / (1 + c2)

Hence, c2 = 1.05 / 0.8 -1 =  0.3125 = 31.25%

c) Calculate the annual interest rate for a two-year loan k2

(1 + k2)² = (1 + k1) x (1 + c2) = (1 + 16.67%) x (1 + 31.25%) =  1.53125

Hence, k2 =  1.53125^1/2 - 1 =  0.2374 = 23.74%