Question

You are a bond portfolio manager at XYZ and the investment committee has asked you to buy a bond with price B1 and sell short a certain quantity N of a second bond with price B2:

Bond with price B1 is a 1-year zero coupon bond with a yield-to-maturity of 1%

Bond with price B2 is a 2-year zero coupon bond with a yield-to-maturity of 2%

The resulting portfolio value is Π = B1- NB2

Questions:

1. How would you choose N to optimally hedge the interest rate exposure of the portfolio Π and thus minimize its sensitivity to interest rate change? Find a numerical value for N

2. Using the value of N that you found in part 1), what is your portfolio's profit or loss if both of the yield-to-maturities of bonds B1 and B2 suddenly decrease by 1%?

Answer 1

1.

For zero coupon bond, duration is same as year to maturity

Bond B1 is a 1-year zero coupon bond, hence duration for bonds B1 is 1 year

Similarly duration for 2 year zero coupon Bond B2 is 2 years

We are buying one bond B1 and selling N number of bonds B2

Portfolio value =B1 - N * B2

As Bond B2 has twice the duration than B1 and hence Bond B2 is 2 times more sensitive than B1 for the change in the interest rates.

Change in portfolio value should be 0 for any change in interest rates, assuming 1% increase interest rates

change in portfolio value = change in value of B1 - N * change in value of B2

0 = -1 * 1% - N * (-2 * 1%)

0 = -0.01 + N * 0.02

N = 0.01 / 0.2

N = 0.5

Hence in order to minimize the interest rate risk, we will have to purchase 0.5 amount of Bond B2

Hence N value in this case = 0.5

2)

For 1% decrease in yield to maturity of Bonds B1 and B2

Change in Portfolio value = -1*-1% - 0.5 * (-2 * -1%)

Portfolio profit / loss = 0