Macaulay and Modified Duration

Distinguish a final date from weighted timing

Two bonds can mature on the same date while paying very different proportions of their total value beforehand. Macaulay duration captures that distinction by weighting each payment's time by its share of the bond's present value. It is not the calendar date when an investor has recovered the purchase price through undiscounted receipts, and it is not an additional contractual maturity.

FINRA's duration explanation distinguishes timing-oriented and sensitivity-oriented measures. That distinction is essential whenever a report gives a number labeled only duration. Ask whether the number is Macaulay, modified or another measure, which rate it varies, and whether the payments are held fixed. A familiar unit such as years does not make differently defined duration statistics directly interchangeable in a price-change formula.

Construct weights from present values

For the regular coupon-date model, calculate PV_k = CF_k/(1+y/m)^k and total price P. Each weight is w_k = PV_k/P, and its time in years is t_k = k/m. Macaulay duration is the sum of t_k*w_k. With positive receipts and a positive compounding base, the weights are positive and sum to one, apart from numerical rounding.

This construction supplies useful bounds. Weighted time cannot precede the earliest included payment or exceed the latest one. A zero-coupon structure places all weight at maturity. Adding positive earlier coupons distributes some weight to earlier dates. These checks concern the stated fixed cash-flow model; they do not license using the same bounds for signed derivative cash flows or instruments with uncertain exercise-dependent payments.

Work through two discounted receipts

Assume a hypothetical USD 100 face amount, a 10% annual coupon, annual payments, two whole years remaining and a 10% annually compounded yield. Valuation is immediately after a coupon payment, with no default, calls, accrued interest, taxes or costs. The remaining receipts are USD 10 in one year and USD 110 in two years. Their present values are USD 9.090909 and USD 90.909091.

Total price is USD 100. The corresponding weights are approximately 0.090909 and 0.909091. Macaulay duration is 1*0.090909 + 2*0.909091 = approximately 1.909091 years, using unrounded weights in the calculation. The result is below the two-year maturity because part of the value arrives after one year. It is not a prediction of an actual sale date or holding-period return.

Translate weighted time into price sensitivity

CFA Institute distinguishes yield-based duration measures, including modified duration. For annual nominal yield y compounded m times, Dmod = Dmac/(1+y/m). In the example, 1.909090909/1.10 = approximately 1.735537 years. More precisely, modified duration equals -(1/P)*dP/dy, where differentiation is with respect to the annual nominal yield expressed as a decimal, while cash-flow amounts and dates remain fixed.

The local estimate is change in price divided by starting price approximately equal to -Dmod*change in y. A one-percentage-point yield increase is 0.01, not 1. Multiplying the example's modified duration by -0.01 gives approximately -0.01735537, or -1.735537%. The minus sign is essential: raising the discount rate lowers the present value of these fixed positive receipts.

Compare the tangent estimate with repricing

For the same instantaneous increase from 10% to 11%, exact cash-flow repricing gives 10/1.11 + 110/(1.11)^2 = USD 98.287477. The duration-only estimate is USD 98.264463. Both describe a decline from USD 100, but they are not identical. The difference reflects curvature in the price-yield relationship rather than a change in the bond's contractual payments.

This comparison also explains why a duration statement should name the shock size. A local slope summarizes behavior near the starting yield; using it farther away accumulates approximation error. The direction of the effect remains useful, but a rounded duration multiplied by a large shock is not a substitute for discounting the unchanged receipts at the complete stressed yield.

Keep yield sensitivity distinct from curve exposure

Modified duration here varies one bond-specific yield. A real term structure has several maturities that can move differently. Translating a single-yield result into a statement about a benchmark curve requires an additional mapping between that curve and the bond's valuation. A maturity-specific curve shock can affect early coupons differently from final principal, which one scalar yield-duration number cannot describe independently.

Duration also does not measure every source of loss. A small yield sensitivity says little about default, transaction costs or the availability of a buyer. FINRA explicitly cautions that low duration is not equivalent to low overall risk. Keep those categories separate when explaining a result so that a narrow mathematical statistic does not become a general safety rating for an instrument.

Review the units and the supported structure

The parent model covers regular fixed-coupon noncallable bullets at a coupon date, with one, two or four coupon periods per year and integer maturities from one to thirty. A period-weighted result must be converted to years before presentation. Changing frequency without adjusting that conversion can produce a duration twice or four times the intended time scale even when the discounted price is correct.

For a reproducible review, retain the cash-flow weights, yield convention, starting price and exact shock used in any comparison. Check the analytic sensitivity against a small symmetric repricing calculation without confusing that numerical check with an independent investment-validation sample. The purpose is to verify the arithmetic contract, not to establish that a duration-based strategy will succeed or that a particular holding suits a reader.

Primary references

1. CFA Institute: Yield-Based Bond Duration Measures and Properties
2. FINRA: Duration and Interest Rate Risk