three bonds. Modified duration measures the size of the interest rate sensitivity. Sometimes we can be misled into thinking that it measures which part of the yield curve the instrument is sensitive to. After all, the modified duration (% change in price) is almost the same number as the Macaulay duration (a kind of weighted average years to maturity). For example, the annuity above has a Macaulay duration of 4.8 years, and we might think that it is sensitive to the 5-year yield. But it has cash flows out to 10 years and thus will be sensitive to 10-year yields. If we want to measure sensitivity to parts of the yield curve, we need to consider key rate durations. For bonds with fixed cash flows a price change can come from two sources: The passage of time (convergence towards par). This is of course totally predictable, and hence not a risk. A change in the yield. This can be due to a change in the benchmark yield, and/or change in the yield spread. The yield-price relationship is inverse, and the modified duration provides a very useful measure of the price sensitivity to yields. As a first derivative it provides a linear approximation. For large yield changes, convexity can be added to provide a quadratic or second-order approximation. Alternatively, and often more usefully, convexity can be used to measure how the modified duration changes as yields change. Similar risk measures (first and second order) used in the options markets are the delta and gamma. Modified duration and DV01 as measures of interest rate sensitivity are also useful because they can be applied to instruments and securities with varying or contingent cash flows, such as options. Embedded options and effective duration For bonds that have embedded options, such as putable and callable bonds, modified duration will not correctly approximate the price move for a change in yield to maturity. Consider a bond with an embedded put option. As an example, a $1,000 bond that can be redeemed by the holder at par at any time before the bond's maturity (i.e. an American put option). No matter how high interest rates become, the price of the bond will never go below $1,000 (ignoring counterparty risk). This bond's price sensitivity to interest rate changes is different from a non-puttable bond with otherwise identical cash flows. To price such bonds, one must use option pricing