Question

a. A 10-year 8% coupon bond has a yield of 9%.  Using annual compounding, what would the duration of the bond equal?

b. If interest rates were to decrease by 30 basis points, what percentage change in price would you expect for the bond?

c. Find the new price.

d. What does it mean to immunize yourself from interest rate risk using duration?  How would you do it with a coupon bond?  Zero-coupon bond?

Answer 1

The formula for Duration is

Duration =Summation (n x PVn)/Summation(PVn) = (1 x 8/1.09^1 + 2 x 8/1.09^2 + 3 x 8/1.09^3 + 4 x 8/1.09^4 + 5 x 8/1.09^5 + 6 x 8/1.09^6 + 7 x 8/1.09^7 + 8 x 8/1.09^8 + 9 x 8/1.09^9 + 10 x 108/1.09^10)/( 8/1.09^1 + 8/1.09^2 + 8/1.09^3 + 8/1.09^4 + 8/1.09^5 + 8/1.09^6 + 8/1.09^7 + 8/1.09^8 + 8/1.09^9 + 108/1.09^10)

Duration = 7.1459.

b. To calculate this, we find the effective duration given as Duration/(1+yield) = 7.146/1.09 = 6.556. Hence a decrease in interest rates by 30 bps, will cause an increase in the value of bond by 30 x 6.556 = 196.68 bps or 1.96%.

c. The new price will be 1.96% more than the previous price. Previous price = ( 8/1.09^1 + 8/1.09^2 + 8/1.09^3 + 8/1.09^4 + 8/1.09^5 + 8/1.09^6 + 8/1.09^7 + 8/1.09^8 + 8/1.09^9 + 108/1.09^10) = 93.582.

Hence, new price = (1+0.0196) x 93.582 = $95.416.

d. Immunizing against interest risk duration means to have total duration of the portfolio zero. We need to balance the asset and liability durations so that the net duration is 0. Doing this will not cause any movement (or cause negligible movement in the price of bond) with a change in interest rates.

We would find a coupon bond with an effective duration = 6.556 and short the bond.

With a zero-coupon bond, the duration is equal to the maturity of the bond. Hence, we would short a zero coupon bond with an effective duration of 7 i.e. a ZCB maturaing in 7 years.