Bond Duration: Macaulay, Modified and Effective

Who this page is for — Anyone separating final maturity, the economic timing of cash flows and the sensitivity of a security or portfolio to yield changes.

Duration is not one universal quantity. In fixed income, the term covers related but different measures:

Macaulay duration is the present-value-weighted average time to cash flows;
modified duration approximates the percentage price change for a small change in yield to maturity;
effective duration estimates sensitivity by revaluing the instrument after curve shocks, especially when cash flows can change.

Duration supplies the local slope; convexity describes how that slope changes.

Macaulay duration: economic timing of cash flows

For cash flows CFₜ, times t, present values PV(CFₜ) and full price P:

D_Mac = Σ[t × PV(CFₜ)] / P

The unit is time, usually years. A non-callable zero-coupon bond has Macaulay duration equal to maturity. A coupon bond delivers some present value earlier and normally has duration below final maturity. All else equal, a higher coupon shifts economic weight earlier and tends to reduce duration; longer maturity tends to increase it.

These relationships are conditional on cash flows and rates. Duration is not the period an investor must hold the bond and not the date on which capital is certain to be recovered. Default, calls, prepayment and sale change the outcome.

Modified duration: yield sensitivity

For a fixed-cash-flow security with nominal annual yield y compounded m times per year under the standard convention:

D_mod = D_Mac / (1 + y/m)

The first-order relationship is:

ΔP / P ≈ −D_mod × Δy

If D_mod = 6 and yield increases by 0.001, or 10 basis points, the approximation gives a price change near −0.6% before the convexity correction. The minus sign describes the ordinary local price-yield relationship of a fixed-rate security without dominant optionality.

Yield must be decimal: 1% is 0.01, not 1. The formula also depends on the compounding convention. Duration calculated from a semi-annual yield must not be applied to an incompatible shock definition.

Effective duration and embedded options

When calls, prepayments or other options change cash flows as rates move, a fixed-cash-flow formula may be inadequate. Effective duration uses model- revalued prices:

D_eff = (P₋ − P₊) / (2 × P₀ × Δy)

P₋ is price after a downward yield shock, P₊ after an upward shock, P₀ the initial price and Δy the positive shock size. The report must state which curve moves, what remains fixed and how exercise, prepayment and spreads are modelled.

For mortgage-backed securities or callable bonds, falling rates can accelerate prepayment or calls and shorten cash flows. Duration can move substantially and specialised structures may exhibit unusual signs. The number is inseparable from its valuation model.

Parallel shifts, curves and spreads

One duration compresses yield-curve risk into one shock. If two-, five- and ten-year rates move differently, portfolios with the same total duration can produce different P&L. Key-rate duration or DV01 by node distributes sensitivity across maturities.

The shocked factor also needs to be separated:

risk-free or government curve;
swap curve;
one security's yield to maturity;
credit spread or option-adjusted spread;
real rate or inflation expectation.

Rate duration and spread duration are not interchangeable. Hedging one factor can leave basis, credit and liquidity risk.

From percentage to monetary value

Modified duration gives an approximate relative change. Monetary value per basis point is expressed by DV01:

DV01 magnitude ≈ full position value × D_mod × 0.0001

Full value must represent the position in reporting currency, not necessarily the quoted price per 100 face. Quantity, accrued interest, face value and FX conversion need to be coherent.

Aggregating duration as a weighted average and aggregating signed DV01 answer different questions. A portfolio with near-zero net DV01 may retain large opposing gross exposures at different curve nodes.

Limits and controls

duration is local and weakens for larger shocks;
convexity corrects part of nonlinearity, not every risk;
uncertain cash flows require behavioural and model assumptions;
spread, credit, inflation, currency and liquidity remain separate;
measures built from different curves or compounding are not directly comparable;
duration changes with price, yield, time and composition.

Scenario revaluation should test parallel and non-parallel curve moves, and optional instruments should be tested under alternative behavioural assumptions. The sensitivity is useful precisely when its boundary is visible.

Common mistake — Saying duration 7 means “seven-year maturity” or a certain 7% loss for any rate rise. It is a local sensitivity under a defined shock and convention.

Checklist

Is the measure Macaulay, modified, effective, spread or key-rate duration?
Which curve, yield, compounding and day count are used?
Is price clean or full, and in which currency?
Are cash flows fixed or dependent on options and behaviour?
What are the direction and size of the shock?
Are convexity and curve nodes material?
Which credit, basis and liquidity risks remain outside it?

Sources

Frederick R. Macaulay, Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the United States since 1856, NBER, 1938.
Federal Reserve Board, The U.S. Treasury Yield Curve: 1961 to the Present — Macaulay duration, modified duration and convexity formulas.
FINRA, Brush Up on Bonds: Interest Rate Changes and Duration.
Basel Committee on Banking Supervision, SRP98 — Application guidance on interest rate risk in the banking book — duration profiles, PV01 and limitations of simple shocks.

Bond duration