How to use the zero-coupon bond calculator
The page has two tools.
- Price from yield. Enter the face value, the yield to maturity and the years to maturity, and choose how the yield is compounded. US zero-coupon bonds, including Treasury STRIPS, are normally quoted with semiannual compounding.
- Yield from price. Enter the face value, the price you would pay and the years to maturity to find the yield you lock in by holding to maturity.
Both tools show the duration, the effective annual yield and a table of the interest that accrues each year.
The zero-coupon bond formula
A zero pays one amount at one date, so its price is a single discounting step:
Continuous compounding: Price = F × e−y × t
Yield = m × [(F ÷ Price)1 ÷ (m × t) − 1]F is the face value, y the annual yield, m the compounding periods per year (1 annual, 2 semiannual) and t the years to maturity. With continuous compounding the yield is ln(F ÷ Price) ÷ t.
Modified duration = t ÷ (1 + y ÷ m)
Imputed interest in year k = Accreted value at the end of year k − Accreted value at the startThe accreted value after s years is Price × (1 + y ÷ m)m × s. It starts at the price and reaches the face value at maturity.
A worked example
A $1,000 zero-coupon bond matures in 10 years. At a 4.5% yield compounded semiannually, it costs $640.82. The $359.18 difference is all the interest you earn, paid in one go at maturity, for a money multiple of 1.56×.
The bond’s value accretes toward face value at the yield. In the first year it grows by $29.16, and by the final year by $43.53, because each year’s growth compounds on a larger base. The table under the tool shows every year.
Because the only payment comes at the end, the Macaulay duration is 10.00 years and the modified duration 9.780: the price changes by about that many percent for each percentage point of yield.
In reverse, buying a $1,000 zero for $610 with 12 years to go locks in a yield of 4.16%.
Compounding conventions change the quote, not the bond
The same yield gives a different price depending on how it is compounded:
| Compounding of the 4.5% yield | Price of the 10-year zero |
|---|---|
| Annual | $643.93 |
| Semiannual | $640.82 |
| Continuous | $637.63 |
More frequent compounding means a higher effective rate, and so a lower price. When you compare zeros quoted by different sources, convert both to the same convention, or compare effective annual yields. Continuous compounding is common in models and textbooks; bond markets quote semiannual yields.
Phantom income and taxes
A zero pays no cash for years, but in the US the IRS treats the discount as interest that is earned gradually. For a taxable zero-coupon bond, you generally report the year’s original issue discount (OID) as interest income, and the bond’s tax basis rises by the same amount, so you are not taxed on it again at maturity. Brokers report it on Form 1099-OID.
The table on this page shows the yearly accrual under the constant-yield method, the way OID is spread over a bond’s life. The amounts on your tax form can differ: they are based on the bond’s issue price and accrual periods, and bonds bought after issue at a different price follow separate rules for market discount and acquisition premium.
Because you owe tax without receiving cash, taxable zeros are often held in IRAs and other tax-advantaged accounts. Interest on most municipal zero-coupon bonds is exempt from federal income tax, so the phantom income question largely disappears there.
Why zeros swing more in price
A 30-year zero at 4.5% costs $263.15. If its yield rises one point to 5.5%, the price drops to $196.38. With a duration equal to its full maturity, a long zero is one of the most rate-sensitive bonds you can own. That cuts both ways: it is why investors use zeros to match a known future expense, such as tuition in a given year, and why the price of a long zero can fall sharply before maturity.
What this calculator does not include
- Credit risk. Treasury STRIPS carry the US government’s credit; corporate zeros do not, and the yield must reflect that.
- Exact tax figures. Imputed interest is illustrated with the constant-yield method from today’s price. Use your broker’s 1099-OID for tax returns.
- Trading costs. Dealer markups on zeros can be significant, especially for small or long-dated purchases.
- Day counts and settlement dates. Maturity is entered in years, not as dates.