Understanding the duration of a bond

Duration measures the sensitivity of a bond price to changes in interest rates. Given a 1% change in interest rates, the price of a bond will change by its duration. For example, if we have a par bond ($100) with a duration of five years and interest rates increase by 1%, the price of the bond will fall five points to $95 ($100 minus the five duration). 

Bond yields and prices are inversely correlated. When yields go up, prices go down. When rates fall, prices go up. If rates drop by 1%, the new price of the bond is $105 ($100 plus the five duration). Duration is a measure of risk.

The price of a bond is the sum of the present value of its cash flows. A zero-coupon bond is purchased at a discount and matures at par. There is only one cash flow. The duration of a zero-coupon bond is equal to its maturity date. 

A five-year bond that pays annual interest has five cash flows. As an example, let us go back to the “good old days” and pretend interest rates are 5%. One year from now, the investor receives $5 in interest for every $100 invested. Assuming interest rates remain at 5%, what is that cash flow worth today? Discounted at 5%, it is worth $4.76 ($5 divided by 1.05). 

Continuing with the example, two years from now, the investor receives another $5. What is that cash flow worth today? The answer: $4.53 ($5 divided by 1.05^2). At maturity, the investor receives $105, the final coupon and (hopefully) the return of the investment. That future cash flow is worth $82.27 today ($105 divided by 1.05^5).   

Visually, bond duration is like a seesaw. It is the point of balance between the present values of the coupon payments and the final cash flow. In the above example, the duration is 4.55 years. Low interest rates (approaching zero) lengthen duration, whereas high rates shorten duration.   

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