Duration and Convexity: Interest Rate Risk for Exam FM

Macaulay Duration

Macaulay duration D is the weighted average time of cash flows, where each cash flow is weighted by its present value: D = (sum of t * PV(CF_t)) / P, where P = sum of PV(CF_t). For a zero-coupon bond maturing at time n, D = n. For a coupon bond, D < n because earlier cash flows pull the average time forward.

For an n-period annuity-immediate: D = (Ia)-angle-n / a-angle-n. For a bond: D = (Fr * (Ia)-angle-n + n * C * v^n) / P.

Modified Duration

Modified duration D* = D / (1+i) measures the percentage price sensitivity to a small change in yield: dP/P is approximately -D* * di. This first-order approximation works well for small yield changes. On Exam FM, you may be asked to estimate the price change for a given yield shift using modified duration.

Example: A bond has modified duration 7.5 and price $1000. If the yield increases by 0.5%, the estimated price change is -7.5 * 0.005 * 1000 = -$37.50.

Convexity

Convexity C = (sum of t*(t+1) * PV(CF_t)) / (P * (1+i)^2) measures the curvature of the price-yield relationship. The second-order price approximation is: dP/P is approximately -D* * di + (1/2) * C * (di)^2. Including convexity improves the estimate for larger yield changes. Convexity is always positive for standard bonds (no embedded options), meaning duration alone overestimates price decreases and underestimates price increases. Exam FM tests both the formulas and the interpretation of duration and convexity.