Question

A $5000 semi-annual coupon bond paying interest at 12.4%/year compounded semi-annually is redeemable at par in 16 years. It is callable at 115 at the end of 8 years and at 111 at the end of 11 years. Determine the price to guarantee a yield rate of 13.4%/year compounded semi-annually.

Answer 1

From the bond-holders perspective, there are 3 scenarios-

Scenario 1: Hold the bond for 16 years and redeem at par($100 per bond) at the end of 16th year
Scenario 2: Hold the bond for 8 years, then bond-issuer exercises the call option and bond holders receive $115 per bond at the end of 8th year.
Scenario 3: Hold the bond for 11 years, then bond-issuer exercises the call option and the bond holder receives $111 per bond at the end of 11th year.

Face Value = $5000 (50 bonds of Face Value 100 each)
Interest = 12.4%/year
Coupon = Semi-Annual
Coupon Payment Frequency = 2 (times in an year)
Compounding Frequency = Semi-Annual
YTM = 13.4%/ year compounded semi-annually

Coupon Amount = Face-Value * Interest Rate/Coupon Payment Frequency = 5000*12.4%/2 = 310

Value of Bond in Scenario 1:

"PV" formula of the excel can be used to find the value. "PV" takes following inputs -
rate = applicable discount rate = YTM/2 = 13.4%/2 = 6.7%
nper = No. of payments = 16 years * 2 = 32
payment = coupon payment = 310
FV = Final redemption value = 50*100 = 5000

Applying the above formula -

Value of the Bond = PV(6.7%, 32, 310, 5000) = -4673.70

Value of Bond in Scenario 2:

"PV" formula of the excel can be used to find the value. "PV" takes following inputs -
rate = applicable discount rate = YTM/2 = 13.4%/2 = 6.7%
nper = No. of payments = 8 years * 2 = 16
payment = coupon payment = 310
FV = Final redemption value = 50*115 = 5750

Applying the above formula -

Value of the Bond = PV(6.7%, 16, 310, 5750) = -5024.79

Value of Bond in Scenario 3:

"PV" formula of the excel can be used to find the value. "PV" takes following inputs -
rate = applicable discount rate = YTM/2 = 13.4%/2 = 6.7%
nper = No. of payments = 11 years * 2 = 16
payment = coupon payment = 310
FV = Final redemption value = 50*111 = 5550

Applying the above formula -

Value of the Bond = PV(6.7%, 22, 310, 5550) = -4848.51

Calculations in excel will look like below -

Formulae -

Values:

Hence, out of all 3 scenarios, minimum price of the bond is coming from scenario 1, i.e. $4,673.70

So, price of the bond should not be more than $4,673.70 to guarantee a yield of 13.4%/year compounded semi-annually.

Answer: $4,673.70