Questions and Problems on Duration and Interest rate risk.

Questions:

1. What is the reinvestment risk?

(Reinvestment risk is the risk that as investors receive interest and principal payments on their bonds they may have to reinvest these cash flows at a rate lower than they currently receive on the existing bond.)

2. How does duration differ from maturity?

(Duration differs from maturity as a measure of interest rate sensitivity because duration takes into account the time of arrival and the rate of reinvestment of all cash flows during the assets life).

3. What are the different general interpretations of duration?

1) Technically, duration is the weighted-average time to maturity using the relative present values of the cash flows as the weights.

2) Duration has economic meaning as the interest sensitivity (or interest elasticity) of an asset’s value to changes in the interest rate.

3) Duration shows how much time do we need to recover our initial investments.

4. If there is a decline in interest rates, which bond you rather be holding, long term bonds or short-term bonds? Why? Which type of bond has the greater interest rate risk?

(You would rather be holding long-term bonds because their price would increase more than the price of the short-term bonds, giving them a higher return).

Problems:

1. Calculate the duration of a $1, 000 6% coupon bond with three years to maturity. Assume that all market interest rates are 7%.

Solution:

This bond has a duration of 2.83 years. Note that the current price of the bond is $973.76, which is the sum of the individual “PV of payments.”

2. Two bonds are available for purchase in the financial markets. The first bond is a 2-year, $1, 000 bond that pays an annual coupon of 10 %. The second bond is a 2-year, $1, 000, zero-coupon bond.

a. What is the duration of the coupon bond if the current yield-to-maturity (YTM) is 8 %?

Duration = 1.910 years

b. Suppose the YTM changes from 8% to 10%. How does this change affect the duration of the coupon bond? (Please answer whether duration increases or decreases?)

Increasing the yield-to-maturity decreases the duration of the bond.

c. Calculate the duration of the zero-coupon bond with an YTM of 8 %.

Duration of the zero-coupon bond is equal to its maturity.

Duration = 2 years

d. If the YTM changes from 8% to10%, how does the change affect the duration of the zero-coupon bond?

Changing the yield-to-maturity does not affect the duration of the zero coupon bond. Duration = Maturity

e. Why does the change in the YTM affect the coupon bond differently than the zero-coupon bond?

Increasing the YTM on the coupon bond allows for a higher reinvestment income that more quickly recovers the initial investment. The zero-coupon bond has no cash flow until maturity.

f. Which bond is subject to the larger interest rate risk and why? Calculate the approximate percentage change in the price of two bonds if the interest rate is expected to increase from 8% to 10%.

1. (%) Δ P/P» - DUR x Δ i/(1+i) = - 0.0354

(%) Δ P/P= -1.910* (10%-8%)/(1+8%)

(%) Δ P/P» - 0.037

2. (%) Δ P/P= -2 * (10%-8%)/(1+8%) = -0.037

($) Δ P» - DUR x Δ i/(1+i) x P

(-4.54*(4.5%-6%)/(1+6%) *957.87 = +61.54