Convexity

Study guide: Exam FM-2

Convexity measures the curvature of the price–yield relationship for a bond or portfolio. While Modified Duration gives the first-order approximation of price sensitivity to yield changes, convexity captures the second-order effect. On Exam FM, "convexity" means modified convexity unless a question says otherwise:

$C_{Mod} = \frac{1}{P} \cdot \frac{d^2P}{dj^2}$

$C_{Mod} = \frac{\displaystyle\sum_{t=1}^{n} t(t+1) \cdot C_t \cdot v^{t+2}}{P}$

Macaulay convexity is the present-value-weighted average of the squared times of the cash flows, and the two convexities are linked through the Macaulay Duration $D_{Mac}$ :

$C_{Mac} = \frac{\displaystyle\sum_{t=1}^{n} t^2 \cdot C_t \cdot v^{t}}{P}$

$C_{Mod} = \frac{C_{Mac} + D_{Mac}}{(1+j)^2}$

The second-order approximation of the price change uses the modified measures:

$\frac{\Delta P}{P} \approx -D_{Mod} \cdot \Delta j + \tfrac{1}{2} \cdot C_{Mod} \cdot (\Delta j)^2$

In the formulas, $C_t$ is the cash flow at time $t$ , $j$ the effective yield per period, $v = 1/(1+j)$ , and $P = \sum C_t v^t$ is the price
Modified convexity weights each cash flow by $t(t+1)v^{t+2}$ ; Macaulay convexity by $t^2 v^t$ . For a single cash flow at time $t$ , $C_{Mac} = t^2$ and $C_{Mod} = t(t+1)/(1+j)^2$
Higher convexity is beneficial: to second order, for the same price and modified duration, a more convex bond rises more when yields fall and falls less when yields rise
Redington Immunization requires the convexity of the assets to exceed the convexity of the liabilities

Convexity Price Approximation A bond has modified duration $D_{Mod} = 7.5$ years and convexity $C_{Mod} = 68$ . If yields rise by $0.5\%$ ( $\Delta j = 0.005$ ), estimate the percentage price change.
Answer
 $\begin{align*} \frac{\Delta P}{P} &\approx -D_{Mod} \cdot \Delta j + \tfrac{1}{2} \cdot C_{Mod} \cdot (\Delta j)^2 \\ &= -(7.5)(0.005) + \tfrac{1}{2}(68)(0.005)^2 \\ &= -0.0375 + 0.00085 \\ &= -0.03665 \end{align*}$ 
The bond price decreases by approximately $3.67\%$ . Without the convexity term the estimate would be $-3.75\%$ ; convexity reduces the actual price decline slightly.

Macaulay and Modified Convexity of an Annuity A 10-year annuity-immediate pays $1{,}000$ at the end of each year. At $j = 7\%$ its price is $P = 7{,}023.5815$ and its Macaulay duration is $D_{Mac} = 4.9460710$ . Find its Macaulay and modified convexity.
Answer
 $\begin{align*} C_{Mac} &= \frac{\sum_{t=1}^{10} 1{,}000\, t^2 (1.07)^{-t}}{7{,}023.5815} \\ &= \frac{228{,}451.20}{7{,}023.5815} \\ &= 32.526311 \end{align*}$ 
 $\begin{align*} C_{Mod} &= \frac{C_{Mac} + D_{Mac}}{(1.07)^2} \\ &= \frac{32.526311 + 4.946071}{1.1449} \\ &= 32.729830 \end{align*}$ 
Summing $t(t+1)(1{,}000)(1.07)^{-(t+2)}$ directly and dividing by $P$ gives the same $32.729830$ .