Question

You are examining three bonds with a par value of ​$1000 ​(you receive ​$1 comma 000 at​ maturity) and are concerned with what would happen to their market value if interest rates​ (or the market discount​ rate) changed. The three bonds are

Bond A-bond with 6 years left to maturity that has an annual coupon interest rate of 12 ​percent, but the interest is paid semiannually. Bond B-bond with 9 years left to maturity that has an annual coupon interest rate of 12 ​percent, but the interest is paid semiannually. Bond C-bond with 15 years left to maturity that has an annual coupon interest rate of 12 ​percent, but the interest is paid semiannually. What would be the value of these bonds if the market discount rate were

a. 12 percent per year compounded​ semiannually?

b. 7 percent per year compounded​ semiannually?

c. 16 percent per year compounded​ semiannually? d. What observations can you make about these​ results?

Answer 1

Price of bond is calculated as present value of coupon payment plus present value of face value of bond.
Price of bond	Coupon amount*PVAF + Face value*PVF
Formula to calculate Present value of annuity
PVAF	(1-((1+r)-n))/r
r represents interest rate and n represents no of payments
PVF	1/((1+r)^n)
Semi-annual coupon amount for bond A	$60.00	1000*(12%/2)
No of payments under bond A	12.00
Semi-annual coupon amount for bond B	$60.00	1000*(12%/2)
No of payments under bond B	18.00
Semi-annual coupon amount for bond C	$60.00	1000*(12%/2)
No of payments under bond C	30.00
a.
Calculation of price of each bond if market rate is 12% (Semi-annual 6%)
Price of bond A	(60*(1-(1.06^-12))/0.06) + 1000*(1/(1.06^12))
Price of bond A	60*8.383844 + 1000*0.496969
Price of bond A	$1,000
Price of bond B	(60*(1-(1.06^-18))/0.06) + 1000*(1/(1.06^18))
Price of bond B	60*10.8276 + 1000*0.350344
Price of bond B	$1,000
Price of bond B	(60*(1-(1.06^-30))/0.06) + 1000*(1/(1.06^30))
Price of bond B	60*13.76483 + 1000*0.17411
Price of bond B	$1,000
b.
Calculation of price of each bond if market rate is 7% (Semi-annual 3.5%)
Price of bond A	(60*(1-(1.035^-12))/0.035) + 1000*(1/(1.035^12))
Price of bond A	60*9.663334 + 1000*0.661783
Price of bond A	$1,242
Price of bond B	(60*(1-(1.035^-18))/0.035) + 1000*(1/(1.035^18))
Price of bond B	60*13.818968 + 1000*0.538361
Price of bond B	$1,330
Price of bond B	(60*(1-(1.035^-30))/0.035) + 1000*(1/(1.035^30))
Price of bond B	60*18.39205 + 1000*0.70138
Price of bond B	$1,805
c.
Calculation of price of each bond if market rate is 16% (Semi-annual 8%)
Price of bond A	(60*(1-(1.08^-12))/0.08) + 1000*(1/(1.08^12))
Price of bond A	60*7.536078 + 1000*0.397114
Price of bond A	$849
Price of bond B	(60*(1-(1.08^-18))/0.08) + 1000*(1/(1.08^18))
Price of bond B	60*9.371887 + 1000*0.250249
Price of bond B	$813
Price of bond B	(60*(1-(1.08^-30))/0.08) + 1000*(1/(1.08^30))
Price of bond B	60*11.25778 + 1000*0.099377
Price of bond B	$775
d.
Thus, we can see that when market interest is equal to coupon rate then all the three bonds would sell at par which is $1,000, irrespective of the time to maturity.
When market interest rate is lower than coupon rate then all three bonds would sell at premium (price higher than par value)
When market interest rate is higher than coupon rate then all three bonds would sell at discount (price lower than par value)