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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 ...