Identifier: C2T4L1 | Revision: 2026-10-26

Decision and objectives

A five-year bond need not have duration five. Duration summarizes when its value arrives and, after a convention adjustment, how its price reacts locally to yield. Learn the cash-flow weights before using a screen number. The objective is a reproducible estimate using arithmetic, not a calculus derivation.

Objectives: Calculate present-value cash-flow weights; Calculate Macaulay and modified duration; Distinguish maturity from rate sensitivity.

Prerequisites: C2T3L5.

Mechanics

Macaulay duration is the average payment time weighted by each payment's share of the bond's present value. A coupon returns some value before maturity, so its weighted-average time is shorter than final maturity. A zero-coupon bond puts all weight at redemption and therefore has Macaulay duration equal to remaining maturity under the stated convention.

The weights are present-value shares, not nominal coupon shares. Each must use the same price, yield convention and cash schedule. For ordinary positive cash flows they sum to one. Using face value rather than price as the denominator breaks the calculation when a bond trades away from par.

Price and return drivers

For annual effective yield y: weight at t = [CFt / (1+y)^t] / price. Macaulay duration Dmac = sum(t × weight). Modified duration Dmod = Dmac / (1+y). With nominal annual yield and m compounding periods, Dmod = Dmac / (1+y/m), using matching period timing.

Modified duration is a local estimate: percentage price change ≈ −Dmod × change in yield expressed as a decimal. The minus sign reflects the inverse price-yield relationship. Duration in years is a timing measure; the price sensitivity uses a decimal rate move, not an additional year of holding.

Worked case

Hypothetical two-year annual bond pays 10 in year one and 110 in year two per 100 face. Annual effective yield is 10%, valuation just after coupon, no options, tax or costs. Present values are 10/1.10 = 9.090909 and 110/1.10² = 90.909091; price is 100. Weights are 0.090909 and 0.909091. Weighted time = 1 × 0.090909 + 2 × 0.909091 = 1.909091 years. Modified duration = 1.909091 / 1.10 = 1.735537.

For a 0.50 percentage-point rise, use Δy = 0.005. Estimated change = −1.735537 × 0.005 = −0.867769%, or −0.867769 price units per 100. A two-year zero at the same yield has Macaulay duration 2 and modified duration 2/1.10 = 1.818182: later concentrated cash is more sensitive.

Adverse case

Reprice the coupon bond at 10.5%: 10/1.105 + 110/1.105² = 99.138019. Actual loss is 0.861981%, slightly smaller than the duration estimate. The approximation holds the cash schedule fixed and treats the curve move as a single matching yield change. If the issuer calls, defaults, or changes a floating coupon, the relevant cash flows may change and the fixed-schedule estimate can fail substantially.

Practical use

Request the duration definition, date, yield basis, clean/dirty basis and treatment of options before comparing reports. A portfolio's weighted duration normally uses market-value weights, not face weights. Dollar or TRY sensitivity additionally requires the position's market value. A bond fund's duration changes with holdings and market moves; buying a fund does not lock a personal maturity date. Costs and tax are excluded from the local model and still affect realized sale cash.

Decision boundaries

Use duration to compare small rate exposures under consistent conventions. It does not measure default loss, currency risk, liquidity or the exact time at which principal returns. Do not say duration 1.9 guarantees recovery of the purchase price in 1.9 years. For a hard cash deadline, inspect the actual schedule as well as the sensitivity number. Decline a precise hedge claim based on an unexplained duration label.

Practice and references

Exercise: hypothetical three-year zero, annual effective yield 8%, no option or default. Answer: Dmac = 3 years; Dmod = 3/1.08 = 2.777778. A 25-basis-point rise, Δy = 0.0025, implies approximately −0.694444% price change. A basis point is 0.01 percentage point, so 25 basis points is 0.25 percentage point.

Takeaway: cash-flow timing creates the weights; modified duration translates those weights into a local yield sensitivity.

Reference guide: domestic-government-bond, macaulay-duration, modified-duration, Duration, Macaulay duration, Modified duration.

Next: C2T4L2. Topic test: C2T4.

Primary sources (checked 2026-10-26; changing terms require a publication recheck): SEC Investor Bulletin: What Are Corporate Bonds? — Interest rate risk, credit risk, call risk and seniority.

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