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Convexity is also positively correlated with duration. 17) shows that convexity is the rate at which price sensitivity to yield changes as yield changes. That is, it describes how much a bond’s modified duration changes in response to changes in yield. 18) expresses this relationship formally. The convexity term can be seen as an “adjustment” for the error made by duration in approximating the priceyield curve. 18), is measured is the number of interest periods. 19) can be used to convert the convexity measure from interest periods to years.

It is clearly unlikely that all the coupons of any but the shortest-maturity bond will be reinvested at the same rate. As noted earlier, market interest rates are in a state of constant flux, and this would affect money reinvestment rates. Therefore, although yield to maturity is the main market measure of bond levels, it is not a true interest rate. This is an important point. Chapter 2 will explore the concept of a true interest rate. Another problem with YTM is that it discounts a bond’s coupons at the yield specific to that bond.

This measure is obtained by applying a mathematical property known as a Taylor expansion to the basic equation. The relationship between price volatility and duration can be made clearer if the bond price equation, viewed as a function of r, is expanded as a Taylor series (see Butler, pp. 112–114 for an accessible explanation of Taylor expansions). 13). 13) where r = the yield to maturity of an annual-coupon-paying bond As stated above, Macaulay duration equals modified duration multiplied by (1+r).

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