By the end of this lesson, you can discount a two-year bond’s payments, calculate Macaulay and modified duration, and compare a duration price estimate with exact repricing. This helps you read interest-rate exposure without treating a sensitivity calculation as a forecast.
Face value, or principal, is the amount promised back at maturity, the final repayment date. A coupon is a scheduled interest payment. Present value discounts a future payment into a value today: with annual compounding, PV = CF ÷ (1 + y)^t. CF is the payment, t is years until payment and y is an annual yield written as a decimal. For example, 5% = 0.05.
Yield to maturity is the single discount rate that equates the sum of promised payment present values to the bond’s price. It is not the coupon rate or a promised realized return. We value the bond immediately after a coupon date, with no accrued interest, and examine an immediate yield change before any time passes. All amounts are illustrative currency units per 100 of face value.
Assume two years remain, face value is 100, the coupon is 5 paid once per year, and yield is 5% with annual compounding. Assume no default, no early repayment option and unchanged promised payments. The year-one payment is 5; the year-two payment is 105, comprising the final coupon and principal.
The year-one present value is 5 ÷ 1.05 = 4.76190. The year-two present value is 105 ÷ 1.05² = 95.23810. Their sum is price P = 100. A higher discount rate would lower these present values even though the promised payments themselves did not change. Carry unrounded values through the calculations; displayed figures are rounded.
Divide each present value by price to obtain its weight: 4.76190 ÷ 100 ≈ 0.047619 and 95.23810 ÷ 100 ≈ 0.952381. These weights sum to 1. Macaulay duration is the weighted average payment time: Dmac = 1 × 0.047619 + 2 × 0.952381 ≈ 1.95238 years.
Maturity is two years, but some value arrives after one year, bringing the weighted time below two. For annual compounding, modified duration Dmod = Dmac ÷ (1 + y) = 1.95238095 ÷ 1.05 ≈ 1.85941 years. Modified duration supplies the local price sensitivity; inserting maturity or Macaulay duration directly into that estimate would give a different answer.
Now raise yield from 5% to 6%. This is an increase of one percentage point: Δy = 0.06 − 0.05 = 0.01. The relative price change is approximately ΔP ÷ P ≈ −Dmod × Δy = −1.85941043 × 0.01 = −0.01859410, or −1.85941%.
Multiply that relative change by starting price 100: ΔP ≈ −1.85941 currency units. The estimated new price is 100 − 1.85941 = 98.14059. A percentage price change and a currency price change are different quantities; their numerical sizes coincide here only because the starting price is 100.
At 6%, exact repricing of the same promised payments gives Pnew = 5 ÷ 1.06 + 105 ÷ 1.06² = 98.16661. The exact fall is 1.83339%, compared with the duration estimate of 1.85941%. Exact price exceeds the straight-line estimate by about 0.02602 currency units.
The price-yield relationship for this fixed-payment bond bends upward rather than following a straight line. Convexity means that curvature. Duration uses the slope at the starting yield, so it misses the bend as yield moves away. This bond’s positive curvature makes the exact price higher than its tangent-line estimate. We checked the gap by exact repricing; we have not added a convexity correction or claimed the estimate is exact.
Keep the original bond and starting yield of 5%. For an immediate rise to 5.5%, what is Δy, and what price does the original modified duration estimate? Should that estimate equal exact repricing?
Answer: the rise is 0.5 percentage points, so Δy = 0.005. The approximate relative price change is −1.85941043 × 0.005 = −0.00929705, or −0.92971%. Estimated price is 100 × (1 − 0.00929705) = 99.07029. Exact repricing is 5 ÷ 1.055 + 105 ÷ 1.055² ≈ 99.07684. The smaller gap, about 0.00655, illustrates why a local straight-line estimate works better for a smaller move; curvature still prevents equality.
In this exercise one yield discounts every payment date. Interpreted as a change in an initially flat interest-rate curve, the shock is parallel: all rates rise by the same amount. Real curves can twist, with short and long rates moving differently. A single duration then cannot capture each payment’s exposure. Other payment frequencies also require matching yield conventions, not uncritical reuse of the annual formula.
Duration describes an immediate price response under specified assumptions, not total return over a holding period. Coupons, their reinvestment, passage of time, inflation, default, liquidity and costs can all matter. Bonds with early repayment rights can have changing cash flows and different curvature. Comparing exposure is useful when assessing how a possible sale before maturity could be affected, but the calculation neither predicts rates nor identifies a suitable allocation.
General education, not investment advice. All bond amounts and yield changes below are illustrative assumptions, not market observations, forecasts or Quantustik performance claims. Duration measures one source of price sensitivity and cannot establish a suitable holding or prevent losses.
No. Fixed promised payments can be discounted at a different market yield. Their present value, and therefore the sale price in this example, changes.
Modified duration adjusts for the yield used to discount the cash flows. The coupon rate determines payment amounts. They both happen to be 5% in the starting example but need not match.
Investor education only — not investment advice, and never a promise of profit. Every investment can lose value.