Analysis of Duration

The previous discussion describes the intuition that the duration on a bond is a measure of this interest rate sensitivity. I develop this relationship more rigorously in this section and end it with a rule of thumb based on modified duration, which is used by bond managers to assess interest rate risk on their portfolios. Duration is actually a weighted average of the cash flow delivery dates. To see this, consider again the standard present valuation of a cash flow stream on an n period coupon-bearing bond. We are interested in the interest rate sensitivity of the bond's price. This sensitivity is what we call the bond's delta, or ΔP/Δr. To see this, take the derivative with respect to the interest rate. (The derivative is a simple application of the exponent rule). Note that .

Now consider the terms and recognize that these are weights on the timing of the cash flows. Rewriting the equality, we get:

The right-hand side is a weighted average of the times with the weights being the discounted cash flows.

If we then divide by the price of the bond P, we get the familiar form of an elasticity; in this case, divided by . This quotient is called Macaulay duration. It is an elasticity, but the best way to think of duration is that it is the weighted average of times until delivery of cash flows. Collecting terms, Macaulay duration is written as:

Go to the companion website for more details (see the Duration spreadsheet).

This is a complicated formula. You will find the Duration.xlsx spreadsheet much more intuitive since it breaks this computation down into separable and additive parts.

Now, to get back to the issue of sensitivity of the bond's price to changes in yield, understand that Macaulay duration D can be written as:

which, upon multiplying both sides by P and dividing by , gets us:

The term is modified duration and it measures the rate sensitivity of the bond's price. Given a bond's modified duration, we can then approximate its instantaneous risk by substituting in the current bond price P and its yield r.

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Example 2.5

The data for a 10-year bond that pays a 10 percent coupon and has a 5 percent yield is available on the companion website. This bond is priced at $138.97 (confirm this for yourselves) and has Macaulay duration of 7.11. The modified duration is and therefore, the price sensitivity dP. For a 1 percent change in yield, is equal to .

Go to the companion website for more details.

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Example 2.6

In 2009, the central bank targeted effective rates at close to zero. A bond manager estimates his portfolio as having duration 5. If the central bank were to tighten interest rates by 100 bps (1 percent), then his portfolio would lose approximately 5 percent of its value, that is, .

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