Question

Assume a corporate bond with a $1000 face value matures 3 years and 5 months from today and has an annual coupon rate of 5% paid semiannually. There is a 20% chance that the issuer will default at maturity. If the firm defaults, it will pay 80% of what is promised (final coupon + face value) at maturity. Treasuries with the same maturity earn a yield to maturity of 3% and investors in these corporate bonds demand a 5% risk premium over the current rate on Treasuries (thus requiring an expected return of 8%) to compensate for the risk they face (All rates are APRs with semiannual compounding).
Calculate the clean price of the bond.

Answer 1

Expected rate return per annum= 8%

ie. 8%/2= 4% per semi-annual compounding period

Semi-annual Coupon amt.=1000*5%/2=25

r= Expected semi-annual rate of return , found out above-- 4% per s/a period

No.of semi-annual coupons still to maturity=As it is 1 month from the previous coupon--(3yrs.*2 s/a periods )=6 annuity --coupon pmts,

final coupon is combined with amount to be received at maturity ---for ease of calculating probable payments to be received, so, it is 6-1 =5 s/a coupons

Amt. to be received on maturity---(20%*80%*(1000+25))+(80%*(1000+25))= (20%*80%*1025)+(80%*1025)=984

This final pmt. Is at end   of s/a period, 6

so, now using the formula to find the clean price of a bond,

Price=(Pmt.*(1-(1+r)^-n)/r)+(Amt. at maturity/(1+r)^n)

ie.(25*(1-1.04^-5)/0.04)+(984/1.04^6)=

888.97

So, the answer is:

the clean price of the bond= $ 888.97