Bond duration describes the timing of a bond’s payments and its price sensitivity to a small change in yield. Macaulay duration averages payment times using their present values as weights. Modified duration translates that timing measure into an approximate percentage price change.
Maturity is when the final principal repayment is due. Face value, also called principal, is the amount promised back at maturity. A coupon is a scheduled interest payment. A coupon bond therefore pays some money before its maturity date.
Present value is the value today of a future payment after discounting it by an interest rate. Yield to maturity is the single discount rate that makes the present values of all promised payments add up to the bond’s price. It is not a guaranteed realized return. Here yields use annual compounding, so a payment CF at t years has present value PV = CF ÷ (1 + y)^t, where y is the yield in decimal form.
Macaulay duration Dmac = Σ[t × PV(CF at t)] ÷ price. The Σ symbol means add across all payments. Each present-value weight is PV ÷ price, and the weights sum to 1. With time measured in years, Macaulay duration is in years. A positive-coupon bond with positive promised payments has Macaulay duration below maturity; a bond paying only at maturity has Macaulay duration equal to maturity.
For annual compounding, modified duration Dmod = Dmac ÷ (1 + y). The local approximation is ΔP ÷ P ≈ −Dmod × Δy. P is the starting price; ΔP is its change. Δy is the change in annual yield in decimal form. A move from 5% to 6% is one percentage point, or Δy = 0.01, not 1. Modified duration is conventionally quoted in years under this annual-yield convention.
Illustrative example: a two-year bond with face value 100, annual coupon 5 and annual yield 5% has price 100, Macaulay duration 1.95238 years and modified duration 1.85941 years. A one-percentage-point yield rise therefore implies an approximate price fall of 1.85941%. The two-year maturity alone does not supply that price estimate.
The minus sign captures the opposite movement of price and yield when promised cash flows stay fixed. Duration does not predict whether yields will rise or fall. Its straight-line approximation is local: exact repricing can differ, especially for larger changes, because the price-yield relationship curves. This curvature is called convexity.
The example assumes unchanged promised cash flows, no default and no options to repay early or change payments. It applies one yield to all payment dates, equivalent here to a flat interest-rate curve moving in parallel, with all its rates changing by the same amount. Different changes at different maturities require richer analysis. Default risk, inflation, trading costs and reinvestment of coupons remain separate questions.
No. Maturity gives the final payment date; duration is a weighted timing or sensitivity measure. It does not prescribe a holding period or guarantee recovery of the purchase price.
No. Check whether the number is Macaulay or modified duration, its time units and the yield’s compounding frequency. The annual formula here must be adjusted for other payment and yield conventions.
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Educational research only — not investment advice.