1-year zero at 5% semi-annual
Face 1,000, 5% discount rate, 1 year, semi-annual compounding — the default inputs.
- Price
- 951.81
- YTM
- 5.062%
- Macaulay / modified
- 1.000 / 0.975
Zero Coupon Bond Calculator
Price a zero-coupon bond and review YTM together with Macaulay and modified duration.
A zero pays no coupons — only face value at maturity — so the price is a single discount. Choose compounding frequency (semi-annual is the default). Figures are theoretical and for information only.
Results
- Price
- —
- YTM (%)
- —
- Macaulay duration
- —
- Modified duration
- —
How the zero is priced
With no coupons, the only cash flow is face value at maturity. The calculator discounts that payment on the compounding grid you select, then converts the price into an annual effective yield.
Price = F / (1 + r/p)t × p
- F — face value paid at maturity
- r — market discount rate in decimal form
- p — compounding periods per year
- t — years to maturity
YTM is (F / price)1/t − 1, shown in percent to three decimals. Macaulay duration equals t. Modified duration is t / (1 + YTM / p). Price rounds to two decimals, matching the results panel. Semi-annual compounding is the default, which is why a 5% one-year discount rate produces a 5.062% YTM rather than 5% exactly.
Worked examples
These cases use the same compounding and rounding as the calculator. Type the inputs in and you should get the same price.
10-year zero at 3%
Face 1,000, 3% discount rate, 10 years, semi-annual. Duration stays equal to maturity because there are still no interim coupons.
- Price
- 742.47
- YTM
- 3.022%
- Macaulay / modified
- 10.000 / 9.851
5-year zero, annual compounding
Face 10,000, 4% discount rate, 5 years, annual compounding. With one compounding per year, YTM matches the 4% discount rate exactly.
- Price
- 8,219.27
- YTM
- 4.000%
- Macaulay / modified
- 5.000 / 4.808
Related calculators on this site
Zeros isolate discounting. Coupon bonds add an annuity, a mortgage amortizes principal, and VaR uses a yield-like volatility number rather than a bond price.
Zero-coupon bond FAQ
There are no coupons, so the price is face value discounted at the market rate for the number of compounding periods until maturity: F / (1 + r/p)years × p. Price is rounded to two decimals.
The price compounds at the chosen frequency, but YTM is reported as an annual effective rate: (face / price)1 / years minus one. With semi-annual compounding those two numbers differ slightly — 5% discounting for one year becomes a 5.062% YTM. Annual compounding makes them match, as in the third example.
A zero pays only at maturity, so Macaulay duration equals the term in years. Modified duration is that term divided by one plus YTM per compounding period.
Semi-annual is the usual market convention for many government zeros and is the default here. Annual, quarterly, and monthly are available if your quote uses a different grid.
Use the standard bond calculator for coupon bonds. That tool adds a coupon annuity, accrued interest on a 30/360 basis, and an approximate YTM. For a fully amortizing loan payment, see the mortgage calculator.
Street quotes may use a different compounding convention, day-count, or a discount yield rather than a true price from (1 + r/p)n. Fees and settlement lags are not modelled. Figures are theoretical and for information only.
Disclaimer: the contents of this website are for informational purposes only and do not constitute any investment recommendation. The visitor acts at his own risk.