Bond Duration

What bond duration measures

Duration is a family of measures used to describe a bond's cash-flow timing and sensitivity to yield changes. It is expressed in time units for Macaulay duration and as a price-sensitivity measure for modified duration. Investors use duration to compare interest-rate exposure and estimate how a bond's market price may respond when its yield changes.

Duration is not the same as maturity. Maturity is the date when principal is contractually due, assuming the issuer pays. Duration reflects the timing and present value of all cash flows, including coupons. A bond with a long maturity can have shorter duration than another bond of similar maturity if its coupons are larger or its yield differs.

Macaulay duration is the present-value-weighted average time until a bond's cash flows are received. Each cash flow's weight is its present value divided by the bond's price. A larger share of value arriving sooner produces lower Macaulay duration; greater weight on distant payments raises it.

Modified duration translates Macaulay duration into an estimate of a bond's percentage price sensitivity to a change in yield to maturity. For small parallel yield changes, the approximation is: percentage price change ≈ −modified duration × change in yield. The minus sign reflects the general inverse relationship between bond prices and yields.

Duration is useful for comparing rate risk, but it is not a guarantee. The estimate assumes a simplified relationship between price and yield, and can be less accurate for large yield changes, bonds with options, or changing cash-flow assumptions. It does not measure issuer credit risk or liquidity by itself.

Macaulay duration and cash-flow timing

To calculate Macaulay duration, discount each scheduled cash flow to present value using the bond's yield and period convention. Multiply each present value by the time period in which it arrives, sum those products, and divide by the bond's total present value. The resulting weighted average indicates when the investor receives value, not when the bond legally matures.

A zero-coupon bond has one cash flow at maturity, so its Macaulay duration equals its time to maturity under the standard convention. A coupon bond returns some value earlier through coupon payments, so its Macaulay duration is generally shorter than maturity. Larger coupons shift more present value toward earlier dates and usually reduce duration.

Holding coupon and yield constant, a longer maturity generally increases duration because more value depends on distant payments. Holding other features constant, a lower coupon or lower yield generally increases duration. These are directional relationships, not substitutes for calculation when the bond includes call or prepayment features.

Macaulay duration can help relate an asset's cash-flow timing to a future liability. If a bond portfolio's duration is matched to a liability horizon under suitable assumptions, price and reinvestment effects can partly offset. A yield-curve shift, changing cash flows, credit spreads, or a sale before the horizon can disrupt the match.

Modified duration and price estimates

A bond with modified duration of 5 has an estimated price sensitivity of about 5% for a 1 percentage-point change in yield, with direction opposite the yield movement. If yield rises by 1 percentage point, estimated price change is approximately -5%; if yield falls by 1 percentage point, the estimate is about +5%, before convexity and other effects.

The yield change must be expressed in decimal form in the multiplication. One percentage point is 0.01; 25 basis points is 0.0025. For modified duration of 5 and a 25-basis-point increase, estimated price change is -5 × 0.0025 = -0.0125, or about -1.25%.

If a bond is priced at $1,040 and has modified duration of 4.2, a yield increase of 0.50 percentage point gives an estimated percentage change of -4.2 × 0.005 = -2.1%. The approximate new price is $1,040 × (1 − 0.021) = $1,018.16. This is a first-order estimate, not an exact quote.

For a portfolio, modified duration can be approximated by a market-value-weighted average of constituent durations when measures are compatible. A portfolio worth $60,000 in duration-5 bonds and $40,000 in duration-2 bonds has approximate duration 0.60 × 5 + 0.40 × 2 = 3.8. This estimates overall parallel-rate sensitivity, not issuer or spread exposure.

Money duration converts percentage sensitivity into a currency amount by combining modified duration with market value. Price value of a basis point estimates the dollar change for a one-basis-point yield move. These measures make large and small positions easier to compare, but the same assumptions about cash flows and yield changes still apply.

Why duration changes

Duration changes as time passes, yields move, and cash flows change. A plain bond approaches maturity, so its remaining payment schedule shortens. Its duration generally declines over time, though the path depends on coupon and price. Recalculate when comparing a bond portfolio at different dates.

Coupon rate affects how much value arrives before maturity. A higher coupon returns more cash earlier, generally reducing duration relative to a lower-coupon bond with similar maturity and yield. A zero-coupon bond concentrates all cash flow at maturity and therefore has duration equal to its term.

Yield level also affects present-value weights. When yield changes, later cash flows are discounted differently, changing duration. A duration number quoted without price date, yield, and cash-flow assumptions is incomplete. For a bond fund, duration can change as holdings, cash, and portfolio positioning change.

Embedded options change expected cash flows. A callable bond may be redeemed early when rates fall, limiting price appreciation and shortening expected life. A mortgage security may prepay faster or extend based on rates. Option-adjusted or effective duration measures may be used, but definitions and models vary.

Floating-rate securities reset coupons periodically, so their price sensitivity to benchmark rates may be lower between reset dates than that of similar fixed-rate bonds. They still carry credit, spread, liquidity, and reset risk. Duration should be interpreted in light of reset terms and the rate to which payments adjust.

Duration and convexity

The price-yield relationship for a plain bond is curved rather than perfectly linear. Modified duration estimates the slope at the current yield. Convexity describes how that slope changes as yields move. For small changes, duration may provide a useful approximation; for larger changes, convexity can make the estimate materially inaccurate.

A conventional option-free bond generally has positive convexity: price gains when yields fall tend to exceed price losses when yields rise by the same amount, all else equal. Callable and mortgage securities can have negative convexity in some rate ranges because falling rates encourage early repayment and limit price gains.

A second-order estimate can add a convexity term: percentage price change ≈ −Dmod × Δy + 0.5 × convexity × (Δy)^2, under the convention used. This refinement still depends on a model and does not capture credit spread, liquidity, or changing cash-flow behavior unless those are modeled.

Do not present duration as a precise forecast for a large market move. Yield curves can shift unevenly across maturities, while simple duration often assumes a parallel change. A steepening, flattening, or twist can affect securities with different cash-flow dates differently.

Worked case: shorter and longer bonds

A client compares two hypothetical bonds with similar credit quality and yield. Bond A has modified duration 2.5; Bond B has modified duration 7.0. If market yields rise by 0.75 percentage point, first-order estimates are about -1.875% for A and -5.25% for B. The longer-duration bond has greater estimated price sensitivity.

If the client plans to sell both after one year, the estimates highlight possible market-value differences but do not determine total return. Coupon income, reinvestment, spread changes, and actual sale prices matter. If yields fall instead, the longer-duration bond may gain more under the same simplified approximation, while call features or convexity can alter that relationship.

Bond A might be more appropriate for a client who is especially concerned about price sensitivity, but a lower duration does not make it automatically suitable. Its credit quality, income, maturity, liquidity, and role in the portfolio still matter. Duration is a risk measure, not a product recommendation.

Worked case: a bond liability

A hypothetical nonprofit expects to pay $500,000 in four years for a building project. It invests in fixed-income securities whose Macaulay duration is approximately four years. Under simplified assumptions, the portfolio's price sensitivity and reinvestment effects may partly offset over the horizon if yields shift in a parallel way and the securities are held as planned.

The match can fail if the yield curve twists, credit spreads change, the nonprofit withdraws funds early, cash flows are uncertain, or a bond is called. A duration match is not the same as having $500,000 guaranteed on the due date. Liability management requires attention to amount, timing, credit quality, liquidity, and reinvestment.

If the nonprofit invests in a long corporate bond to match duration, issuer default could prevent the needed payment. A Treasury portfolio may have different credit characteristics but still faces price and reinvestment changes before the target date. Duration should be combined with credit analysis and cash-flow planning.

Duration for bonds with options

Callable bonds have cash flows that can change when the issuer exercises its option. As rates fall, an issuer may call a high-coupon bond, returning principal earlier and limiting the bond's price increase. A fixed cash-flow duration estimate can overstate the bond's price gain in that environment.

Mortgage-backed securities can prepay when homeowners refinance or sell homes. When rates rise, prepayments may slow and expected life may extend; when rates fall, principal may return faster. Effective duration models price changes after recalculating cash flows under rate scenarios, but the result depends on prepayment assumptions.

Putable bonds give investors a right to sell the bond back under specified conditions. That option can shorten expected cash flows when rates or credit conditions change. A plain modified duration may not capture the option's effect. Review the instrument's options and the duration measure provider uses.

What duration leaves out

Duration is primarily an interest-rate sensitivity measure. It does not by itself measure whether the issuer will pay, how easy a bond is to sell, whether currency moves, or whether inflation erodes purchasing power. A bond with low duration can still default or become illiquid.

Credit spreads can change independently of benchmark yields. If Treasury yields fall but a corporate spread widens substantially, the corporate bond price may still decline. Spread duration or scenario analysis may supplement interest-rate duration for credit instruments.

Duration may assume a parallel yield-curve shift. Actual changes can be concentrated at one maturity or involve twists. Key-rate duration measures sensitivity to particular points on the curve, but requires more data and should be interpreted under the model used.

A bond portfolio's duration can change as prices, holdings, cash, and fund flows change. A fund's published duration is a snapshot and may not match a future date. Check the calculation date, source, and whether it accounts for embedded options.

A practical duration review

Confirm whether the figure is Macaulay, modified, effective, spread, or portfolio duration. Record its date, yield convention, currency, and treatment of options. A bare number without the measure type can be misleading.

Use modified duration for a first-order estimate of percentage price change for a small yield move. Convert basis points to decimals, apply the negative sign, and multiply by market value if a dollar estimate is needed. State that the result is approximate.

Compare duration with the client's horizon and cash needs. Review convexity, credit spread, liquidity, call or prepayment features, inflation, and currency exposure. If the goal is a known liability, model full cash flows and downside scenarios rather than relying on duration alone.

For a fund, understand how holdings and duration may change. A low duration can reduce rate sensitivity but may require reinvestment sooner or expose the investor to credit risk. A high duration may support a liability hedge but can create larger mark-to-market movements.

How to answer Series 65 questions

Macaulay duration is a present-value-weighted average timing of cash flows. Modified duration estimates percentage price sensitivity to yield changes. Neither is identical to legal maturity.

For a small yield increase, estimated price change is negative duration times the yield change. For a decrease, the first-order estimate is positive. Convert basis points correctly and remember that the estimate is approximate.

Longer maturity, lower coupon, and lower yield generally increase duration, all else equal. Embedded options, credit spreads, yield-curve shape, and liquidity can change actual price response. Duration does not measure default risk or guarantee a sale value.

Sources

CFA Institute 2026 Refresher Reading, Yield-Based Bond Duration Measures and Properties; CFA Institute Research Foundation, Closed-Form Formulas for Macaulay Duration; NASAA Series 65 Test Specifications, Section I. Price examples are hypothetical.