Structural Limitations of Nominal Maturity and Duration Principles

Evaluating bond price sensitivity solely through terminal contractual maturity misses the timing of coupon payments. Two distinct bonds possessing identical ten-year stated maturities display different market risks depending on their coupon schedules. An instrument bearing high periodic coupons returns significant capital cash flows early, shortening the investor's effective waiting timeline. Formulated by Frederick Macaulay, Macaulay duration quantifies the cash-flow-weighted average term-to-maturity, calculating the temporal center of gravity for all discounted future cash flows.

Modified duration is a local approximation

For annualized YTM y compounded m times a year, Dmod = Dmac / (1 + y/m), and ΔP/P ≈ −Dmod × Δy. Enter the yield change as a decimal: 100 basis points is 0.01, a percentage-point change rather than a 1% relative increase in yield. A modified duration of eight implies approximate price changes of −8% and +8% for ±100 bp. It does not imply symmetric actual returns. CFA's bond valuation explanation prices promised cash flows using a discount rate.

Reprice one bond to see the approximation error

Illustrative assumptions: an option-free eight-year zero-coupon bond pays 100 units at maturity; YTM is 5%, compounded annually. P = 100/(1.05)^8 = 67.6839. Dmod = 8/1.05 = 7.6190 and C = 8×9/(1.05)^2 = 65.3061. C is the second derivative of price with respect to yield divided by price, using the same annual-compounding convention.

If YTM rises to 6%, duration alone gives −7.6190%. Adding convexity, −Dmod×0.01 + 0.5×C×0.01², gives −7.2925%. Direct repricing gives 100/(1.06)^8 = 62.7412, a −7.3026% change from the original price. At 4% YTM, the exact change is +7.9562%, rather than the linear +7.6190%. The second-order correction reduces error in this example; it does not guarantee accuracy in extreme moves.

Options and curve changes need different treatment

The public summary of Dunetz and Mahoney's 1988 research distinguishes local duration estimates from callable-bond behavior. Positive convexity for an option-free bond with fixed positive cash flows is not a rule for every bond. A call option can change cash flows and produce negative convexity.

Check units, compounding frequency and whether a quoted measure is modified or effective duration before applying a small shock. For larger shocks, reprice the cash flows. This single-YTM illustration does not measure nonparallel curve shifts, changing credit spreads, liquidity risk or foreign-exchange risk.