Question

Construct some simple examples to illustrate your answers to the following:

a. If interest rates fall, do bond prices rise or fall?

b. If the bond yield to maturity is less than the coupon, is the price of the bond greater or less than the par value?

c. If the price of a bond is below the par value, is the yield to maturity greater or less than the coupon?

d. Do low-coupon bonds sell at higher or lower prices than high-coupon bonds?

e. If interest rates change, do the prices of high-coupon bonds change proportionately more than that of low-coupon bonds?

Answer 1

a. If interest rates fall, do bond prices rise or fall?

Bond price is inversely related to interest rates (or YTM). Hence, if interest rate falls, prices of the bond rises.

Example: Assume a zero coupon bond of face value $ 1,000 and with 5 years maturity and yield of 8%. Price = Face value / (1 + y)ⁿ = 1000 / (1 + 8%)⁵ =  $ 680.58. Let's say interest rate falls by 1%. That is YTM = y = 8% - 1% = 7%. HEnce, Price now = 1000 / (1 + 7%)⁵ = $  712.99

Thus price of the bond rises when interest rate falls.

b. If the bond yield to maturity is less than the coupon, is the price of the bond greater or less than the par value?

If the bond yield to maturity is less than the coupon, the bond is trading at premium and hence the price of the bond will be greater than the par value.

Example: Consider an 8% annual coupon paying bond with maturity of 5 years and yield to maturity of 7%. Assume its par value to be $ 1,000.

It's price = - PV (Rate, period, PMT, FV) = - PV (7%, 5, 8% x 1000, 1000) = $ 1,041 which is greater than its par value of $ 1,000.

c. If the price of a bond is below the par value, is the yield to maturity greater or less than the coupon?

If the price of a bond is below the par value, the bond is trading at discount and hence its yield to maturity greater than the coupon.

Example: Consider an 8% annual coupon paying bond with maturity of 5 years and trading at $ 950. Assume its par value to be $ 1,000

It's YTM = Rate (Period, PMT, PV, FV) = Rate (5, 8% x 1000, 950, -1000) = 9.30% > 8%, the coupon.

d. Do low-coupon bonds sell at higher or lower prices than high-coupon bonds?

Price of a bond is PV of all the future coupon payments and repayments. Hence, all things being same, a low coupon bond will sell at lower price than high coupon bond.

Example: Determine the prices of Bond A & B that have annual coupon of 5% and 7%. Both of them have maturity of 5 years and carry a yield of 8%.

Price of bond A = - PV (Rate, Period, PMT, FV) = - PV (8%, 5, 5% x 1000, 1000) = $ 880.22

Price of bond B = - PV (Rate, Period, PMT, FV) = - PV (8%, 5, 7% x 1000, 1000) = $ 960.07

Bond A is lower coupon bond and hence it price is lower than that of B

e. If interest rates change, do the prices of high-coupon bonds change proportionately more than that of low-coupon bonds?

A high coupon bond has lower duration and hence lower modified duration and hence lower sensitivity to interest rate changes. Hence, If interest rates change, the prices of high-coupon bonds change proportionately less than that of low-coupon bonds?

Consider the example form part (d)

Example: Determine the prices of Bond A & B that have annual coupon of 5% and 7%. Both of them have maturity of 5 years and carry a yield of 8%. Calculate the %age changes in the bond prices of each of them if interest rate changes to 9%.

Current Price of bond A = - PV (Rate, Period, PMT, FV) = - PV (8%, 5, 5% x 1000, 1000) = $ 880.22

Current Price of bond B = - PV (Rate, Period, PMT, FV) = - PV (8%, 5, 7% x 1000, 1000) = $ 960.07

When interest rate changes to 9%,

Price of bond A will be = - PV (Rate, Period, PMT, FV) = - PV (9%, 5, 5% x 1000, 1000) = $ 844.41

Price of bond B wil be = - PV (Rate, Period, PMT, FV) = - PV (9%, 5, 7% x 1000, 1000) = $ 922.21

%age change in price of bond A = 844.41 / 880.22 - 1 = - 4.07%

%age change in price of bond B = 922.21 / 960.07 - 1 = - 3.94%

Bond B has higher coupon, but %age change in the price on change in interest rate is lower than that in case of Bond A that has higher coupon.