Convexity measure

We will discuss two approaches for assessing the interest rate risk exposure of a bond or a portfolio. The first approach is the full valuation approach that involves selecting possible interest rate scenarios for how interest rates and yield spreads may change and revaluing the bond position. The second approach entails the computation of measures that approximate how a bond s price or the portfolio s value will change when interest rates change. The most commonly used measures are duration and convexity. We will discuss duration/convexity measures for bonds and bond portfolios. Finally, we discuss measures of yield curve risk. 

For a discussion, see Steven V. Mann and Pradipkumar Ramanlal, Duration and Convexity Measures When the Yield Curve Changes Shape, Journal of Financial Engineering (March 1998), pp. 35-58. 

The reason for this result is that duration is in fact a first (linear) approximation for a small change in yield.The approximation can be improved by using a second approximation. This approximation is referred to as convexity. The use of this term in the industry is unfortunate since the term convexity is also used to describe the shape or curvature of the price/yield relationship. The convexity measure of a security can be used to approximate the change in price that is not explained by duration. 

The convexity measure of a bond is approximated using the following... 

Substituting these values into the convexity measure given by equation (4.4) ... 

Mathematically, any function can be estimated by a series of approximations referred to a Taylor series expansion. Each approximation or term of the Taylor series is based on a corresponding derivative. For a bond, duration is the first-term approximation of the price change and is related to the first derivative of the bond s price with respect to a change in the required yield. The convexity measure is the second approximation and related to the second derivative of the bond s price. 

We ll see how to use this convexity measure shortly. Before doing so, there are three points that should be noted. First, there is no simple interpretation of the convexity measure as there is for duration. Second, it is more common for market participants to refer to the value computed in equation (4.4) as the convexity of a bond rather than the convexity measure of a bond. Finally, the convexity measure reported by dealers and vendors will differ for an option-free bond. The reason is that the value obtained from equation (4.4) is often scaled for the reason explained after we demonstrate how to use the convexity measure. 

Given the convexity measure, the approximate percentage price change adjustment due to the bond s convexity (i.e., the percentage price... 

For example, in some books the convexity measure is defined as follows ... 

Equation (4.6) differs from equation (4.4) since it does not include 2 in the denominator. Thus, the convexity measure computed using equation (4.6) will be double the convexity measure using equation (4.4). So, for our earlier illustration, since the convexity measure using equation (4.4) is 165.35, the convexity measure using equation (4.6) would be 330.68. 

Which is correct, 165.35 or 330.68 The answer is both. The reason is that the corresponding equation for computing the convexity adjustment would not be given by equation (4.5) if the convexity measure is obtained from equation (4.6). Instead, the corresponding convexity adjustment formula would be... 

Equation (4.7) differs from equation (4.5) in that the convexity measure is divided by 2. Thus, the convexity adjustment will be the same whether one uses equation (4.4) to get the convexity measure and equation (4.5) to get the convexity adjustment or one uses equation (4.6) to compute the convexity measure and equation (4.7) to determine the convexity adjnstment. 

Equation (4.8) differs from equation (4.4) by the inclusion of 100 in the denominator. In our illustration, the convexity measure would be 1.6535 rather than 165.35 nsing equation (4.4). The convexity adjustment formula corresponding to the convexity measure given by equation (4.8) is then... 

Similarly, one can express the convexity measure as shown in equation (4.9) ... 

For the UK gilt principal strip we have been using in our illustrations, the convexity measure is 3.3068. 

Convexity adjustment to percentage price change = (Convexity measure/2) x (Ay) x 10,000... 

Consequently, the convexity measure (or just simply convexity as it is referred to by some market participants) that could be reported for this UK strip are 165.35, 330.68, 1.6535, or 3.3068. All of these values are correct, but they mean nothing in isolation. To use them to obtain the convexity adjustment to the price change estimated by duration requires knowing how they are computed so that the correct convexity adjustment formula is used. It is the convexity adjustment that is important—not the convexity measure in isolation. 

As with duration, there is little difference between modified convexity and effective convexity for option-free bonds. However, for bonds with embedded options there can be quite a difference between the calculated modified convexity and effective convexity measures. In fact, for all option-free bonds, either convexity measure will have a positive value. For bonds with embedded options, the calculated effective convexity measure can be negative when the calculated modified convexity measure is positive. 

The unit in which convexity, as defined by (2.18), is measured is the number of interest periods. For annual-coupon bonds, this is equal to the number of years for bonds with different coupon-payment schedules, formula (2.19) can be used to convert the convexity measure from interest periods to years. 

The convexity measure increases with the square of maturity it decreases as both coupon and yield rise. It is a function of modified duration, so index-linked bonds, which have greater duration than conventional bonds of similar maturities, also have greater convexity. For a conventional vanilla bond, convexity is almost always positive. Negative convexity occurs most frequently with callable bonds. 

If we use an HP calculator to find the price of the bond at the new yield of 7.50%, we see that It Is 82.83980, a change In price of 13.92%. The convexity measure of 1.4756% Is an approximation of the error we would make when using the modified duration value to estimate the price of the bond following the 200 basis point rise in yield. 

---