Question

An annual coupon bond has 2 years remaining until maturity. The bond has a 6% expected default rate on each coupon date, as long as the default has not yet occurred. If default occurs, the bond will pay no coupon for that period, but will pay 30% of the redemption amount from the sale of bond collateral.  

The bond has a coupon rate of 10%, and a face amount of 100,000.

Using expected present value, what price should a bond investor pay based on a desired annual yield of 8%?

Answer 1

let's assume coupons are paid semi annually, so there are two payments per period.

FV = Face Value = 100,000

C =coupon per period = Annual coupon rate / 2 x Face Value = 10% / 2 x 100,000 = 5,000

P_D = probability of default = 6%

P_S = probability of survival = 1- P_D = 1- 6% = 94%

R = Recovery on default = 30% of Face Value = 30% x 100,000 = 30,000

In the interim periods 1 to 3, if bond survives it pays the coupon C otherwise it pays the recovery amount R.

At the end of period 1:

Expected payout = P_S x C + P_D x R = 94% x 5,000 + 6% x 30,000 = 6,500

At the end of period 2:

The bond has survived period 1

Expected payout = P_S x P_S x C + P_S x P_D x R = 94% x 94% x 5,000 + 94% x 6% x 30,000 = 6,110

At the end of period 3:

The bond has survived period 1 & 2

Expected payout = P_S x P_S x P_S x C + P_S x P_S x P_D x R = 94%³ x 5,000 + 94%² x 6% x 30,000 =   5,743

At the end of period 4:

The bond has survived period 1, 2 & 3

Expected payout = P_S³ x P_S x (C + FV) + P_S³ x P_D x R= 94%³ x 94% x (5,000 + 100,000) + 94%³ x 6% x 30,000 =   83,474

hence Price of the bond = PV of all the expected payouts

Annualised yield = 8%

Yield per period = 8% / 2 = 4%

Hence, price of the bond

$=\frac{6500}{1.04} + \frac{6110}{1.04^2}+\frac{5743}{1.04^3}+ \frac{83474}{1.04^4}$

= $ 88,358.56