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. Duration is measured in the same units as the payment periods (years, half-years, etc.).

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. The increasing annuity (Ia)-angle-n = (a-double-dot-n - n*v^n)/i plays a central role in duration calculations. Memorizing this formula saves significant time on exam day.

Example: A 5-year zero-coupon bond has Macaulay duration equal to 5, regardless of the yield. A 5-year bond with annual coupons of $60 and redemption value $1,000 at yield 8% has D = (60*(Ia)-angle-5 + 5*1000*v^5) / P. Computing (Ia)-angle-5 at 8% = (a-double-dot-5 - 5*v^5)/0.08 = (4.3121 - 3.4029)/0.08 = 11.365, the numerator becomes 60*11.365 + 5*1000*0.6806 = 681.9 + 3403.0 = 4084.9. With P = 60*3.9927 + 1000*0.6806 = 920.16, D = 4084.9/920.16 = 4.44 years.

Properties of Duration

Several properties of Macaulay duration appear on Exam FM. For a coupon bond, duration increases with maturity but at a decreasing rate. For a par bond, the limiting duration as n approaches infinity is (1+i)/i, which is the duration of a level perpetuity. Duration decreases as the coupon rate increases because higher coupons place more weight on earlier cash flows.

The portfolio duration is the weighted average of the component durations, with weights equal to the proportion of total present value contributed by each component. If a portfolio consists of bonds with present values P_1 and P_2 and durations D_1 and D_2, the portfolio duration is (P_1*D_1 + P_2*D_2)/(P_1 + P_2).

Duration also has a useful interpretation as the percentage price sensitivity. Specifically, D = -(1+i) * (dP/di) / P. This connects the weighted-average-time interpretation to the risk measure interpretation, showing they are equivalent.

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. Modified duration answers the practical question: by what percentage does the price change for a 1% shift in yield?

Example: A bond has modified duration 7.5 and price $1,000. If the yield increases by 0.5%, the estimated price change is -7.5 * 0.005 * 1000 = -$37.50. The new estimated price is $962.50. If instead the yield decreases by 0.5%, the estimated price increase is +$37.50, giving $1,037.50.

For a zero-coupon bond maturing in n periods, D* = n/(1+i). For an annuity-immediate, D* = ((Ia)-angle-n / a-angle-n) / (1+i). Modified duration is always less than Macaulay duration (since 1+i > 1). When working with nominal rates compounded m times per year, use the effective rate per period for the (1+i) divisor.

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. This is because the actual price-yield relationship is convex (curves upward). An investor benefits from convexity: for a given duration, higher convexity means the bond gains more when rates fall than it loses when rates rise by the same amount.

Example: A bond has modified duration 8, convexity 75, and price $1,000. For a yield increase of 1% (di = 0.01): first-order estimate = -8 * 0.01 * 1000 = -$80.00. Second-order estimate = -80.00 + 0.5 * 75 * (0.01)^2 * 1000 = -80.00 + 3.75 = -$76.25. The convexity correction of +$3.75 partially offsets the negative duration effect.

Dollar Duration and Dollar Convexity

Dollar duration (also called DV01 or price value of a basis point) measures the absolute dollar change per unit change in yield: Dollar Duration = D* * P. DV01 specifically measures the dollar change for a one-basis-point (0.01%) change in yield: DV01 = D* * P * 0.0001. For a bond with modified duration 7.5 and price $1,000, DV01 = 7.5 * 1000 * 0.0001 = $0.75 per basis point.

Dollar convexity = C * P measures the absolute curvature. These dollar measures are useful for hedging because they express sensitivity in dollar terms that can be matched across assets and liabilities. When constructing immunization strategies, matching dollar durations (rather than Macaulay durations) ensures the surplus is protected in dollar terms.

Duration of Specific Cash Flow Patterns

For a level annuity-immediate paying $1 per period for n periods at rate i: D = (Ia)-angle-n / a-angle-n. As n approaches infinity (perpetuity-immediate), D = (1+i)/i. For a level annuity-due: D = (Ia-double-dot)-angle-n / a-double-dot-n. For a perpetuity-due: D = 1/d = (1+i)/i, the same as the perpetuity-immediate because the timing shift does not change the weighted average when taken to infinity.

For a bond with semiannual coupons, compute duration in half-year units first, then divide by 2 to convert to years if the problem requests annual duration. Be consistent with units: if yields are given as nominal semiannual rates, use the per-period rate throughout.

Study Tips and Common Pitfalls

The most common mistake is confusing Macaulay and modified duration. Macaulay duration is a weighted average time; modified duration is a price sensitivity measure. They are related by D* = D/(1+i). On Exam FM, read carefully to determine which one the question asks for.

Another frequent error is incorrect unit conversion for semiannual bonds. If the coupon period is 6 months and you compute D = 9 half-year periods, the Macaulay duration in years is 4.5, not 9. Similarly, modified duration computed per period must be adjusted if the question asks for annual modified duration.

Practice computing duration and convexity for bonds, annuities, and simple portfolios. Be able to use the first-order and second-order approximations to estimate price changes. At least 6 practice problems covering different cash flow patterns will build the fluency needed for exam day.