Investment Science: Duration of a bond

Sunday, September 5, 2010


In this post, we will look into the formula for the duration of a bond, in simple terms, the duration of a bond reflects the sensitiveness of the price of the bond with respect to changes in interest rates or duration measures (dP/dλ) (Percentage change in price over changes in yield) of the bond, duration is mentioned in years and in general bonds with longer duration are more sensitive to changes in yield than bonds with shorter durations.

In this article, we will look into the formula for the Macaulay duration of a bond with a simple example on computing the same to get a quick understanding of the duration of a bond, in further posts, we will explore more on the duration of a bond and how it plays an important role when considering a bond portfolio to meet a given obligation.

The Macaulay duration of a bond is given by

Where

m = Number of coupon periods per year
c = Coupon rate per period
y = Yield per period
n = Number of coupon periods left (with m coupon periods per year)

To compute the Macaulay duration of a bond, we will solve a problem from
Investment Science by Luenberger (Problem 3.10)

Problem 3.10 Find the price and duration of a 10-year, 8% bond that is trading at a yield of 10%.

Here

m = 2 (By default, assume that coupon payments happen every 6 months or 2 periods per year)
C = Coupon payment = 8% of 100 = $8 (By default, assume the face value of the bond F = 100)
Coupon rate per period c = 8%/2 = 4% = 0.04
Yield per period y = 10%/2 = 5% = 0.05
Number of coupon periods remaining, n = 10, n * m = 20

The price of the bond is given by (Refer the article on the price and yield of a bond)

P = $87.537

Based on the above formula, the duration of the bond is given by



D = 6.84


Duration of a zero coupon bond:
What will be the duration of a 10-year, zero coupon bond that is trading at a yield of 10%? It is 10 years (Check the formula for additional proof).

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