Question

Suppose the risk-free rate yield curve is flat at 1.5% with annual compounding. 1-year, 2-year, 3-year, and 4-year bonds yield are 2.643%, 3.063%, 3.329%, and 3.529% respectively with annual compounding. All of the bonds pay 5% annual coupons. Assume that in case of default, the recovery rate is 30% of principal, with no payment of accrued interest. Find the risk-neutral probability of default during each year.

Answer 1

Risk neutral default probability is given by =

(y-r) / ( 1-R)

here

y = Bond's Yield

r = yield on a risk-free bond promising the same cash flows as the bond

R = Recovery Rate

For 1 year:-

Risk-neutral probability of default = (0.02643 - 0.015) / ( 1-0.3)

Risk-neutral probability of default = 0.014 / 0.7

Risk-neutral probability of default = 0.016329

For 2 year:-

Risk-neutral probability of default = (0.03063 - 0.015) / ( 1-0.3)

Risk-neutral probability of default = 0.0156 / 0.7

Risk-neutral probability of default = 0.022329

For 3 year:-

Risk-neutral probability of default = (0.03329 - 0.015) / ( 1-0.3)

Risk-neutral probability of default = 0.0183 / 0.7

Risk-neutral probability of default = 0.026129

For 4 year:-

Risk-neutral probability of default = (0.03529 - 0.015) / ( 1-0.3)

Risk-neutral probability of default = 0.0203 / 0.7

Risk-neutral probability of default = 0.028986

Thank You!!