Question

Interest Rate Risk [LO2] Both Bond Sam and Bond Dave have 9 percent coupons, make semiannual payments, and are priced at par value. Bond Sam has 3 years to maturity, whereas Bond Dave has 20 years to maturity. If interest rates suddenly rise by 2 percent, what is the percentage change in the price of Bond Sam? Of Bond Dave? If rates were to suddenly fall by 2 percent instead, what would the percentage change in the price of Bond Sam be then? Of Bond Dave? Illustrate your answers by graphing bond prices versus YTM. What does this problem tell you about the interest rate risk of longer-term bonds?

Answer 1

As the bond is selling at par that meant interest rate or yield to maturity (YTM) is equal to its coupon rate of 9%

Assume that interest rate is increased by 2%; Bond’s price can be calculated the help of following formula

Bond price P = C* [1- 1/ (1+i) ^n] /i + M / (1+i) ^n

Where,

Assume that par value or face value of bond M = $100

Market price of the bond P =?

C = coupon payment or annual interest payment = 9% per annum = $9, but it makes coupon payments on a semi-annual basis therefore coupon payment = 9%/2 of $100 = $4.5

n = number of payments = 6 (3*2 for semiannual payments of up to maturity) for Bond Sam

i = yield to maturity or priced to yield (YTM) = 11% per annum or 11%/2 = 5.5% semiannual

Therefore,

P (Bond Sam) = $4.5 * [1 – 1 / (1+5.5%) ^6] /5.5% + $100 / (1+5.5%) ^6

= $22.48 + $72.52

= $95.00

n = number of payments = 40 (20*2 for semiannual payments of up to maturity) for Bond Dave

Therefore,

P (Bond Dave) = $4.5 * [1 – 1 / (1+5.5%) ^40] /5.5% + $100 / (1+5.5%) ^40

= $72.21 + $11.75

= $83.95

If the interest rate is decreased by 2%;

i = yield to maturity or priced to yield (YTM) = 7% per annum or 7%/2 = 3.5% semiannual

Therefore,

P (Bond Sam) = $4.5 * [1 – 1 / (1+3.5%) ^6] /3.5% + $100 / (1+3.5%) ^6

= $23.98 + $81.35

= $105.33

And

P (Bond Dave) = $4.5 * [1 – 1 / (1+3.5%) ^40] /3.5% + $100 / (1+3.5%) ^40

= $96.10 + $25.26

= $121.36

From above calculations we can make out that if the maturity is for longer term then the impact of increase/decrease in interest rate is more on the price of the bond