Question

Bond A is a 12-year 7% annual coupon bond. Bond B is a 12-year 9% annual coupon bond. Bond C is a 12-year 11% annual coupon bond. Each of these three bonds has a yield to maturity (YTM) of 9%. Assume the market rate of interest does not change over time. Specify the bond that sells at premium, sells at discount, and sells at par. What is value of each bond at this moment (t=0)? Specify the inputs for your financial calculator. What would be the price of each bond 1 year from now? Is the total return earned on Bond A the same as the total return earned on Bond C? Explain. If the capital gains yield (CGY) earned on Bond A greater than the CGY on Bond C? Explain. Is the interest yield (IY) on Bond A for year 2 greater than the IY on Bond C? Explain.

Answer 1

The solution to part 1

In terms of Bond Pricing, the price of the bond is inversely proportionate to the market interest rates.

Hence, Bond A is selling at a discount as the coupon rate (7%) is less than the market interest rate (9%)

Bond B is selling at par as coupon rate (9%) is equal to market interest rate (9%)

and Bond C is selling at a premium as the coupon rate (11%) is higher than the market interest rate (9%)

The solution to part 2

Assuming the Par value of all the bond $ 100, the Value of Bonds at t=0 will be:

The price of a bond is the sum total of present values of all coupon payments and the present value of maturity value.

Bond A price at t₀ = PV of all coupon payments + PV of maturity value

=[ (coupon payments) * cumulative discount factor for 1-12 years discounted at the YTM ]+ [Face value * discount factor for 12th-year discounted at the YTM

= [($ 100.00 * 7% ) * ( (1-1/(1+YTM)^N) / YTM )] + [ $ 100 * (1/(1+YTM)^N)

= $ 7 * ((1- 1/(1+9%)¹²)/9%) + $ 100 * (1/(1+9%)¹²)

= $ 7 * 7.1607 + $ 100 * 3.555

= $ 85.68

Likewise,

Bond B price at t₀ = PV of all coupon payments + PV of maturity value

=[ (coupon payments) * cumulative discount factor for 1-12 years discounted at the YTM ]+ [Face value * discount factor for 12th-year discounted at the YTM

= [($ 100.00 * 9% ) * ( (1-1/(1+YTM)^N) / YTM )] + [ $ 100 * (1/(1+YTM)^N)

= $ 9 * ((1- 1/(1+9%)¹²)/9%) + $ 100 * (1/(1+9%)¹²)

= $ 9 * 7.1607 + $ 100 * 3.555

= $ 100

Bond C price at t₀ = PV of all coupon payments + PV of maturity value

=[ (coupon payments) * cumulative discount factor for 1-12 years discounted at the YTM ]+ [Face value * discount factor for 12th-year discounted at the YTM

= [($ 100.00 * 11% ) * ( (1-1/(1+YTM)^N) / YTM )] + [ $ 100 * (1/(1+YTM)^N)

= $ 11 * ((1- 1/(1+9%)¹²)/9%) + $ 100 * (1/(1+9%)¹²)

= $ 11 * 7.1607 + $ 100 * 3.555

= $ 114.32

The solution to part 3

In the case of one year from now (t=1), the remaining life of all the bonds will be 12 - 1 = 11 years.

Hence, the price of all the bonds at t=1 will be:

Bond A price at t₁ = PV of all coupon payments + PV of maturity value

=[ (coupon payments) * cumulative discount factor for 1-11 years discounted at the YTM ]+ [Face value * discount factor for 11th-year discounted at the YTM

= [($ 100.00 * 7% ) * ( (1-1/(1+YTM)^N) / YTM )] + [ $ 100 * (1/(1+YTM)^N)

= $ 7 * ((1- 1/(1+9%)¹¹)/9%) + $ 100 * (1/(1+9%)¹¹)

= $ 7 * 6.8052 + $ 100 * 3.875

= $ 86.39

Likewise,

Bond B price at t₁ = PV of all coupon payments + PV of maturity value

=[ (coupon payments) * cumulative discount factor for 1-11 years discounted at the YTM ]+ [Face value * discount factor for 11th-year discounted at the YTM

= [($ 100.00 * 9% ) * ( (1-1/(1+YTM)^N) / YTM )] + [ $ 100 * (1/(1+YTM)^N)

= $ 9 * ((1- 1/(1+9%)¹¹)/9%) + $ 100 * (1/(1+9%)¹¹)

= $ 9 * 6.8052 + $ 100 * 3.875

= $ 100

Bond C price at t₁ = PV of all coupon payments + PV of maturity value

=[ (coupon payments) * cumulative discount factor for 1-12 years discounted at the YTM ]+ [Face value * discount factor for 12th-year discounted at the YTM

= [($ 100.00 * 11% ) * ( (1-1/(1+YTM)^N) / YTM )] + [ $ 100 * (1/(1+YTM)^N)

= $ 11 * ((1- 1/(1+9%)¹¹)/9%) + $ 100 * (1/(1+9%)¹¹)

= $ 11 * 6.8052 + $ 100 * 3.875

= $ 113.61

The solution to part 4

Total return earned on Bond A will be = Interest Yield + Capital Gain Yield

= 7% + [{$86.39-$85.68)/85.68*100]

= 7% + 0.83%

= 7.83%

Total return earned on Bond C will be = Interest Yield + Capital Gain Yield

= 11% + [{$113.61-$114.32)/114.32*100]

= 11% - 0.62%

= 10.38%

The total return yield is higher in the case of Bond C as compared to Bond A because Bond C is having higher interest yield and is selling at a premium whereas the Bond A is selling at a discount.

The solution to part 5

The capital gain yield is higher in the case of Bond A in comparison of Bond C because Bond A is selling at discount, Hence in order to provide capital appreciation, the investor will get the amount equal to par value at the time of maturity whereas while buying the Bond A he has only paid $ 85.68, thus the capital gain yield of Bond A will be higher.

The solution to part 6

The interest yield is higher in the case of Bond C in comparison of Bond A because Bond C is selling at a premium that is the coupon rate offered by Bond C is higher than the market interest rate on a similar bond,