Calculate the prices for both bonds.

What are the prices for the bonds next year, if everything remains the same?

What happens to the prices of these bonds if the YTM increases to 7% in the next year, everything else being the same? (Hint: calculate the price for next year with YTM = 7%)

What happens to the prices of these bonds if the YTM decreases to 5% in the next year, everything else being the same? (Hint: calculate the price for next year with YTM = 5%)

Which of the two bonds, based on your previous answers, is the most sensitive to a change in the interest rate (YTM), or, in other words, which of the two has the highest interest rate risk?

Calculate the duration measure for both bonds as of now.

Without calculations, what will happen to the duration for these bonds next year and why?

SOLUTION (Assuming a Face Value of $1,000)

Price of Bond A = 250 / 0.06  * ( 1  - ( 1 / 1.06¹⁵) + 1,000 / 1.06¹⁵ = $2,845.33 (Premium Bond)

Price of Bond B =  50 / 0.06  * ( 1  - ( 1 / 1.06⁵) + 1,000 / 1.06¹⁵ = $957.88 (Discount Bond)

Price of Bond A = 250 / 0.06  * ( 1  - ( 1 / 1.06¹⁴) + 1,000 / 1.06¹⁴ = $2,766.05 (Premium Bond)

Price of Bond B =  50 / 0.06  * ( 1  - ( 1 / 1.06⁴) + 1,000 / 1.06⁴ = $965.35 (Discount Bond)

Without calculations: When the YTM increases, the price of the bond decreases.
With calculations, use the above formula with YTM=7%. Assuming we are calculating the price for year 1:

Price of Bond A = 250 / 0.07  * ( 1  - ( 1 / 1.07¹⁴) + 1,000 / 1.07¹⁴ = $2,574.18 (Premium Bond)

Price of Bond B =  50 / 0.07  * ( 1  - ( 1 / 1.07⁴) + 1,000 / 1.07⁴ = $932.26 (Discount Bond)

Without calculations: When the YTM decreases, the price of the bond increases.
With calculations, use the above formula with YTM=7%. Assuming we are calculating the price for year 1:

Price of Bond A = 250 / 0.05  * ( 1  - ( 1 / 1.05¹⁴) + 1,000 / 1.05¹⁴ = $2,979.73 (Premium Bond)

Price of Bond B =  50 / 0.05  * ( 1  - ( 1 / 1.05⁴) + 1,000 / 1.05⁴ = $1,000 (Par Bond)
(Note that you don't need calculations for this price, because the YTM is equal to the coupon rate).

Without calculations: a longer time to maturity and a lower coupon rate make a bond more sensitive
to a change in the interest rate (YTM).

Given a 1% increase in the YTM, Bond A decreases (2,766.05 - 2,574.18) / 2,766.05  = 6.94%

Given a 1% increase in the YTM, Bond B decreases (965.35 - 932.26) / 965.35  = 3.43%

or

Given a 1% decrease in the YTM, Bond A increases (2,979.73 - 2,766.05) / 2,766.05  = 7.73%

Given a 1% decrease in the YTM, Bond B increases (1,000 - 965.35) / 965.35  = 3.59%

Clearly, Bond A has a higher interest rate sensitivity, or higher interest rate risk than Bond B.

Duration measures the average maturity of a bond. Higher duration means higher interest rate sensitivity. Calculating the duration for a 15-year bond is tedious, and would not be asked on an exam. Using a spreadsheet, financial calculator, or patience, will yield the following duration for year 0:

D_{Bond A} = { [( 250 / 1.06 ) * 1] + [(250 / 1.06²) * 2] + . . . . . +  [(1,250 / 1.06¹⁵) * 15] } / 2,845.33 = 8.11
 

D_{Bond B} = { [( 50 / 1.06 ) * 1] + [(50 / 1.06²) * 2] + . . . . . +  [(1,050 / 1.06⁵) * 5] } / 957.88 = 4.53

Again, Bond A has a higher interest rate risk, because of a higher duration.

If all else remains the same, then the duration must decrease. We know that a longer time to maturity makes a bond more interest rate sensitive. If a year goes by, the time to maturity has decreased, therefor the sensitivity has gone down, which should be reflected in a lower duration.