Price/yield see-saw
A 1% yield move has a bigger price impact on long, low-coupon bonds than on short, high-coupon ones—duration in action.
Duration
Bond Calculator — Price, YTM, Coupon, Face Value
Inputs & Solvers
Assumption: clean price (no accrued interest). Coupon dates are simplified; for professional use, verify day-count and accrual conventions.
Negative yields allowed (e.g., -0.25)
Hint: 5 on $1,000 face = $50/year.
Enter exactly four inputs and leave the fifth blank.
Result
Outputs update after you click a “Solve” button.
Result:
Analytics:
Price vs Yield chart
Based on current Face, Coupon, Maturity and Frequency.
Cash-flow table
Future coupon amounts are shown per period; discounting uses the current YTM.
| Period | Cash Flow | Discount Factor | Present Value | Cumulative PV |
|---|
Bond Pricing & YTM — Educational Guide
Summary (voice-friendly): A bond’s price is the present value of all future coupons plus the face value, discounted by the yield per period. YTM is the annualized rate that makes those discounted cash flows equal today’s price.
What is a bond?
A bond is a loan to a borrower (issuer). The investor receives periodic interest payments (coupons) and the face value at maturity. Prices move inversely with yields: when yields rise, present values fall, so prices drop (and vice versa).
Clean vs. dirty price
Clean price excludes accrued interest; dirty price adds accrued interest between coupon dates. This tool uses clean price. For settlement-day accuracy, add accrued interest based on the chosen day-count convention (e.g., 30/360, Actual/Actual).
Coupon frequency & yield conventions
- Frequency: annual, semi-annual, quarterly, or monthly. The discount rate per period is \( r_p = y / f \), where \( y \) is annualized YTM and \( f \) is payments per year.
- Quoted vs. effective yield: Many markets quote a nominal annual yield (APR) with a compounding frequency. Effective annual yield is \( (1 + y/f)^f - 1 \).
Pricing formula (clean)
\( P = \sum_{t=1}^{n} \frac{C}{(1 + r_p)^t} + \frac{F}{(1 + r_p)^n} \), where \( C \) = coupon per period, \( F \) = face value, \( r_p = y/f \), \( n = f \cdot T \) periods.
Worked example
Face \( F = \$1{,}000 \), coupon rate \( 5\% \) (so \( C = \$50 \) per year), semi-annual coupons (\( f=2 \Rightarrow C/f = \$25 \) per period), maturity \( T = 10 \) years (\( n = 20 \) periods), YTM \( y = 6\% \Rightarrow r_p = 0.06/2 = 0.03 \).
Coupon PV \( = 25 \times \frac{1 - (1+0.03)^{-20}}{0.03} \approx \$372.45 \).
Face PV \( = 1000 \times (1+0.03)^{-20} \approx \$553.68 \).
Price \( P \approx \$926.13 \) (discount bond since YTM > coupon rate).
Duration & convexity (intuition)
- Macaulay duration (years): cash-flow weighted average time; a measure of timing.
- Modified duration: approximate % price change for a 1% change in yield.
- Convexity: curvature; improves the accuracy of duration-based price change estimates for larger yield moves.
Common mistakes
- Confusing coupon rate (%) with coupon amount ($). Use the toggle to select your input style.
- Using annual yield but forgetting to divide by frequency for the per-period discount rate.
- Comparing clean prices to dirty prices without adjusting for accrued interest.
People also ask
Why does bond price fall when yield rises?
Higher yields increase the discount rate, which lowers the present value of future cash flows—so price falls.
Is YTM the same as total return?
YTM equals the internal rate of return if you hold to maturity and reinvest coupons at the same yield. Realized return may differ if you sell early or reinvest at different rates.
What affects duration?
Longer maturities and lower coupons generally increase duration (more rate sensitivity). Higher coupons and shorter maturities reduce it.
Glossary
- Face (Par): Principal repaid at maturity.
- Coupon: Interest paid to investors (rate % × face, or $ amount per year).
- YTM: Annualized discount rate equating price to PV of cash flows.
- Duration / Convexity: First/second-order measures of price sensitivity to yield changes.
Note: This page simplifies calendars and accrual. Professional workflows may require day-count conventions, holiday calendars, settlement lags, and callable/putable features.
5 Fun Facts about Bonds
Dirty vs clean
Quotes are usually “clean” (no accrued interest). The cash you pay is “dirty”: clean price + accrued coupon since last pay date.
Accrued
Zero coupons zig hardest
Zeros have the highest duration for a given maturity because every pound comes at the end—great hedge, big rate sensitivity.
Zeros
Yield ≠ total return
YTM assumes reinvesting coupons at the same yield. If reinvestment rates drop, realized return can lag even if held to maturity.
Reinvestment
Convexity is a curve bonus
Positive convexity means price falls less than duration predicts on a rate rise and rises more on a rate drop—nice asymmetry.
Convexity