Modified Duration

Macaulay duration is a weighted average of the times to a bond's cash flows, with weights equal to each cash flow's present value as a share of the bond's full price. That figure is expressed in years, but it is not directly a price-sensitivity statistic. Dividing it by one plus the yield per period converts it into modified duration, which is read as a percentage price response per unit of yield.

Periodicity matters. If Macaulay duration is measured in semiannual periods and the bond is priced off a semiannual yield, the division uses the semiannual yield, and the result must then be halved-in-reverse, that is, divided by the number of periods per year, to be quoted on an annual basis. Because the denominator is greater than one, modified duration is always slightly smaller than Macaulay duration, and the gap widens as yields rise.

In practice analysts rarely compute duration from the weighted-average formula. Instead they bump and reprice: raise the yield by a small amount, lower it by the same amount, take the difference between the two full prices, and divide by twice the initial full price times the yield change expressed in decimal form. This approximate modified duration converges on the analytical value as the bump shrinks, and it works on any pricing model, including ones without a closed-form solution.

When a bond's cash flows are not fixed, yield duration is meaningless. Callable bonds, putable bonds and mortgage-backed securities require effective duration, which shifts the entire benchmark yield curve rather than a single yield to maturity and reprices through an option-adjusted model. Floating-rate notes have very low effective duration to the reference rate because coupons reset, though they retain credit spread sensitivity.

Modified duration is a percentage measure, so it must be scaled to be useful in currency terms. Money duration is modified duration multiplied by the full price of the position, giving the expected value change for a one hundred basis point yield move. Divide by ten thousand and you obtain the price value of a basis point, the desk convention that traders and risk systems quote as PVBP or DV01.

Portfolio duration is normally computed as the market-value-weighted average of the constituent durations, which implicitly assumes all yields move in parallel. This is the number that drives asset-liability immunization, where duration of assets is matched to the duration of the liability, and it is the basis for hedging with futures: the hedge ratio compares portfolio money duration to the money duration of the cheapest-to-deliver contract.

Candidates should be able to move fluently in both directions. Given Macaulay duration and a yield, produce modified duration; given a yield change in basis points, produce the estimated percentage price change by multiplying modified duration by the negative of the yield change. Then extend to the second-order term: half of annual convexity multiplied by the squared yield change, always added, which corrects the systematic bias of the linear estimate.

Two details separate correct answers from near-misses. Duration statistics apply to the full price, including accrued interest, not the quoted flat price. And a single duration number describes only parallel shifts in the curve; steepening, flattening and twists require key rate durations, which measure sensitivity to individual maturity segments while holding the rest of the curve fixed.