Question

Bond X is noncallable and has 20 years to maturity, a 10% annual coupon, and a $1,000 par value. Your required return on Bond X is 12%; if you buy it, you plan to hold it for 5 years. You (and the market) have expectations that in 5 years, the yield to maturity on a 15-year bond with similar risk will be 11%. How much should you be willing to pay for Bond X today? (Hint: You will need to know how much the bond will be worth at the end of 5 years.) Do not round intermediate calculations. Round your answer to the nearest cent.

Answer 1

We need to calculate the bond value at the end of 5 years. We will use yield to maturity as 11% (as it is given that the yield to maturity on a 15-year bond with similar risk will be 11%) and at that point, the given bond will have 15 years to maturity.
Annual coupon rate=10%
Face value=$1,000
Yield to maturity=11%
Annual coupon payment=Face value*Annual coupon rate=$1,000*10%=$100
Time period we will use for our calculation=15 years

Using excel, we calculated the present value of the bond (at the end of 5 years) as $928.09.

Note: The present value is shown with a negative sign because it is a cash outflow.
Now, if we buy the bond today, we will get annual coupon payments of $100 for 5 years and also selling price of the bond for an amount of $928.09

Given that, the required return on Bond X is 12%. We will use the discount rate as 12% to calculate the present value of the cash flows.
Present value=Cash flow in year 1/(1+discount rate)^1+Cash flow in year 2/(1+discount rate)^2+Cash flow in year 3/(1+discount rate)^3+Cash flow in year 4/(1+discount rate)^4+Cash flow in year 5/(1+discount rate)^5 + Sale price of the bond/(1+discount rate)^5

Here, the cash flows refers to annual coupon payments=$100
Discount rate=12%
Sale price of the bond (present value of the bond at the end of 5 years)=$928.09
=$100/(1+12%)^1+$100/(1+12%)^2+$100/(1+12%)^3+$100/(1+12%)^4+$100/(1+12%)^5+$928.09/(1+12%)^5
=$100/(1.12)^1+$100/(1.12)^2+$100/(1.12)^3+$100/(1.12)^4+$100/(1.12)^5+$928.09/(1.12)^5
=$100/1.12+$100/1.2544+$100/1.404928+$100/1.57351936+$100/1.762341683+$928.09/1.762341683
=$89.28571429+$79.71938776+$71.17802478+$63.55180784+$56.74268558+$526.6231906
=$887.1008109 or $887.10 (Rounded up to 2 decimal places)