# Bond Duration and Portfolio

3.2. Suppose you own a portfolio of two zero-coupon bonds, one maturing in three years and one maturing in five years. Both have a face value of 100 euro. The three year rate is currently 3% and the five year rate 4%. What is the value of your portfolio? What is its modified duration? What is the sensitivity of the portfolio value to one basis point increase in each of the time buckets? What is the present value of a basis point? After some up-beat economic news, the three years rate moves up to 3.17% and the five years rate to 4.40%. What is your loss, if you calculate the loss by re-pricing the portfolio precisely? What loss do you obtain if you use the deltas?

Solution

The value of the portfolio is:

Bond 1: 91.5142

Bond 2: 82.1927

Total value of the portfolio = 173.707

The Macaulay duration of a zero-coupon bond is its maturity. The modified duration of a zero coupon bond is therefore:

We therefore obtain the following two modified durations for our two bonds:

Bond 1:

Bond 2:

The modified duration of a bond portfolio is equal to the weighted average modified duration of the individual bonds, the weights being the present values of the bonds. Therefore, the modified duration of this bond portfolio is:

D = (91.5142 * 2.9126 + 82.1927 * 4.80769) / 173.707 = 3.8093

Alternatively, we also know that:

For calculating , we have to use the rule :

For Bond 1:

(same for Bond 2)

The sensitivities of the portfolio value to the two rates is equal to the two modified duration of the exposures to the two rates. The sensitivity of the portfolio to shifts in the entire yield curve is equal to the modified duration of the portfolio.

The present value of a basis point of the portfolio is calculated as follows, as the difference between the portfolio value at the initial zero coupon rates and the zero coupon rates shifted upwards by one basis point:

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