Bond prices and yields move in opposite directions, and duration measures how far. Price a bond from its yield or solve for yield to maturity from its price, then see its duration, convexity and sensitivity to interest-rate changes.
How it works
Price a fixed-rate bond from its yield, or solve the yield to maturity from its price, and measure current yield, Macaulay and modified duration, convexity and price sensitivity.
Price = Σ C ÷ (1 + y/m)^t + Face ÷ (1 + y/m)^n, C = Face × Coupon ÷ m
Macaulay duration = Σ t × PV(CF) ÷ Price ÷ m
Modified duration = Macaulay duration ÷ (1 + y/m)
Convexity = Σ t(t + 1) × PV(CF) ÷ [Price × (1 + y/m)² × m²]
ΔPrice ≈ −D(mod) × Δy × Price + ½ × Convexity × Δy² × PriceCommon uses
- Fixed-income investment analysis
- Evaluating debt issuance and refinancing
- Finance coursework and CFA exam preparation
Worked example
The calculator above opens with this example. Change any input to see the results update instantly, or select Reset to return to these figures.
| Solve for | Price (from yield) |
| Face (par) value | $1,000 |
| Annual coupon rate | 5% |
| Years to maturity | 10 |
| Coupon frequency | Semiannual |
| Yield to maturity (annual) | 6% |
| Bond price | $925.61 |
| Yield to maturity | 6.00% |
| Current yield | 5.40% |
| Macaulay duration | 7.89 years |
| Modified duration | 7.67 |
| Convexity | 71.79 |
| Price change if yield +100 bp | ($67.63) |
| Price change if yield −100 bp | $74.27 |
Prices are clean prices on a coupon date; accrued interest between coupon dates is not included. Rate-change estimates use duration plus convexity and are most accurate for moderate moves.
Interpreting the results
- A bond trades at a premium when its coupon exceeds the market yield and at a discount when it falls below.
- Modified duration approximates the percentage price change for a 1% change in yield.
- Longer maturities and lower coupons increase duration and therefore interest-rate risk.
Frequently asked questions
What is yield to maturity?
Yield to maturity is the annual return an investor earns by buying a bond at its current price and holding it to maturity, assuming all coupons are paid and reinvested at the same rate. It is the discount rate that makes the present value of the bond’s cash flows equal its price.
Why do bond prices fall when interest rates rise?
A bond’s fixed coupons become less attractive when new bonds pay higher rates, so its price must fall until its yield matches the market. The longer the bond’s duration, the larger the price change.
What is the difference between Macaulay and modified duration?
Macaulay duration is the weighted-average time, in years, until the bond’s cash flows are received. Modified duration adjusts it for the yield and approximates the percentage price change for a one-percentage-point change in yield.
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