Zero Coupon Bond Calculator

What this zero coupon bond calculator does

A zero coupon bond pays you nothing at all until the day it matures, and then hands over its full face value. You make your money by buying it for less than that face value, and the gap between the two is the entire return.

So the only question worth asking is what that gap should be. Give this calculator the face value, the yield you require, and the years to maturity, and it returns what the bond is worth today.

The arithmetic is a single discounting step, which is why zeros are the cleanest instrument in fixed income to price. What is not clean is how they behave, and there are two things about them that catch people out: they swing in price more violently than any other bond, and in a taxable account they can generate a tax bill on income you have not received. Both have sections below.

No currency is assumed. Everything runs in your browser and nothing typed here is stored.

How to use it

1. Face value of bond. What you get at maturity. Usually 1,000 for a corporate or municipal zero, or 100 if you are working in the percentage-of-par convention.
2. Rate or yield. Your required annual yield as a percentage. If you are pricing an existing bond, this is the market yield for that maturity and credit quality.
3. Time to maturity. In years. Decimals are accepted, so eighteen months is 1.5.

Press Calculate. Press Reset to clear it.

The result is the present value, meaning the price at which buying the bond would deliver exactly the yield you entered. Pay less than that and you earn more than your target. Pay more and you earn less.

One convention note before you compare against a broker quote. This discounts once a year. Most bond markets, including US Treasuries, work on a semi-annual basis, and the difference is set out below.

The formula

Price = face value ÷ (1 + yield)^years

That is present value in its purest form. There are no coupons to discount separately, no reinvestment assumptions, and no schedule of payments. One future amount, one discount rate, one exponent.

It is worth appreciating what that removes. With an ordinary coupon bond, the quoted yield to maturity assumes you reinvest every coupon at that same yield, which almost never happens in practice, so the return you actually realise differs from the yield you were quoted. A zero has no coupons to reinvest, so if you hold it to maturity your realised return is exactly the yield you bought at. No assumption, no drift.

That certainty is the main reason zeros exist. If you need a known amount on a known date, a college fee or a pension liability, a zero matches it precisely in a way that a coupon bond cannot.

A worked example

A bond paying 1,000 in ten years, at a required yield of 5 percent.

1,000 ÷ 1.05¹⁰ = 1,000 ÷ 1.62889 = 613.91

So you pay 613.91 today and receive 1,000 in a decade. The 386.09 difference is your entire return, and it works out to 5 percent a year compounding, which is what you asked for.

A few more, to show how sharply time and yield change the price:

The 30 year row is the one worth sitting with. At a 6 percent yield you pay 174 for something worth 1,000 three decades out. That is the power of discounting over long horizons, and it is also a warning about how much of that price is riding on the yield assumption.

Annual and semi-annual, and why your broker's number differs

This calculator compounds once a year. The bond market conventionally works semi-annually, because that is how coupon bonds pay and yields need to be comparable across instruments.

Discounting twice a year at half the rate gives a slightly lower price than discounting once a year at the full rate, because the compounding bites sooner.

So on a 1,000 bond the gap is around four currency units, which is under one percent of the price. Small, and enough to explain a discrepancy if you are checking against a dealer quote for a Treasury STRIP.

If you need the semi-annual figure, halve the yield and double the years, then use the formula yourself. A 5 percent yield over 20 years becomes 2.5 percent over 40 periods.

The tool prices slightly high on the market convention, which means it will suggest paying marginally more than a semi-annual calculation would. Worth knowing on a large purchase and immaterial on a small one.

Zeros move more than any other bond

This is the characteristic that matters most and the one people underestimate.

A coupon bond returns money to you steadily along the way, so its effective average maturity is shorter than its stated one. A zero returns everything at the very end, so its duration equals its full maturity. Longer duration means bigger price swings for a given change in yield, and a zero is the most extreme case there is.

Here is a 20 year zero on a 1,000 face value, priced across yields:

From 2 percent to 8 percent the price falls by more than two thirds. And a single point, from 5 to 6 percent, takes 17.3 percent off the value.

How that scales with maturity, for the same one point rise from 5 to 6 percent:

A quarter of the value of a 30 year zero, gone, on a one point move in yields. That is equity-like volatility from an instrument most people file under safe.

The important caveat is what "safe" actually means here. If you hold to maturity you get the face value regardless of what happened to prices in between, and your return is exactly the yield you bought at. The volatility only becomes real money if you have to sell early. So a zero matched to a date you actually need the money is low risk, and the same zero held loosely is not.

It cuts both ways, of course. When yields fall, long zeros gain more than anything else in fixed income, which is why they are a favourite instrument for anyone taking a view on falling rates.

Tax on money you have not received

This is the practical trap with zeros, and it surprises people every year.

In the United States the tax authorities treat the gradual increase in a zero's value as interest income, earned each year, even though you receive no cash until maturity. It is called original issue discount, and the amount you must include in income each year is set out on a Form 1099-OID that your broker sends you.

Informally the industry calls it phantom income: taxable income with no cash attached.

So on a 613.91 purchase that grows toward 1,000 over ten years, you will owe tax each year on that year's accretion, taxed as ordinary income rather than at capital gains rates, while receiving nothing to pay it with. For a large holding in a high bracket, that is a real cash flow problem.

There are three ordinary responses.

Hold zeros in a tax sheltered account. Inside a retirement account the accretion is not currently taxable, which removes the problem entirely. This is the standard advice and it is why zeros are so common in retirement portfolios.

Use municipal zeros. If the accreted interest is exempt from federal income tax, there is no phantom income to worry about at the federal level. Our tax equivalent yield calculator compares those against taxable alternatives.

Budget for it. If you hold taxable zeros in a taxable account deliberately, plan for the annual liability from other income.

None of this applies outside the United States in the same form, though several countries have comparable accrual rules for discount instruments. Check your own before assuming the gain is only taxed at maturity.

Working out the yield from a price

Often you have the price rather than the yield, because a bond is being offered to you at a number and you want to know what return it implies. Rearranging the formula gives it directly:

Yield = (face value ÷ price)^1/years − 1

Pay 614.46 for a 1,000 bond maturing in ten years and the implied yield is (1000 ÷ 614.46)^0.1 − 1 = 4.99 percent.

Pay 376.89 for a 1,000 bond maturing in twenty years and it is 5.00 percent.

This is the version to use when comparing offers. Two zeros at different prices and different maturities are impossible to rank by price alone, and trivially comparable once both are expressed as yields.

It is the same arithmetic our savings interest rate calculator uses to solve for a required rate, which is worth knowing if you would rather not do the exponent by hand.

Questions people ask

How do I price a zero coupon bond?

Divide the face value by one plus the yield, raised to the number of years. A 1,000 bond at 5 percent over ten years prices at 613.91.

What are STRIPS?

Treasury securities with the coupons separated from the principal, each traded as its own zero coupon instrument. They carry no credit risk beyond the government's and cannot be called.

Do I pay tax before the bond matures?

In a taxable US account, yes. The annual accretion is taxable as original issue discount even though no cash arrives. See the section above.

Are zero coupon bonds risky?

Held to maturity you receive the face value and earn exactly your purchase yield. Sold early they are the most price-volatile bonds available, with a 30 year zero losing about a quarter of its value on a one point rise in yields.

Why does my broker quote a different price?

Probably semi-annual compounding, which is the market convention and gives a slightly lower price. On a 1,000 bond the gap is around four units.

I know the price. How do I get the yield?

Divide face value by price, raise to the power of one over the years, subtract one. See the section above.

Why buy a bond that pays nothing until the end?

Certainty. No coupons means no reinvestment risk, so if you hold to maturity your realised return equals the yield you bought at exactly. That makes zeros ideal for funding a known cost on a known date.

References

A note on sourcing. The pricing formula is standard present value arithmetic. The tax treatment described is that of the United States, where accreted discount on a zero coupon bond is taxable annually as original issue discount rather than at maturity; the Internal Revenue Service sets out the rules and reporting requirements in Publication 1212, and FINRA covers the practical implications for investors.

1. Financial Industry Regulatory Authority, The One-Minute Guide to Zero Coupon Bonds. https://www.finra.org/investors/insights/zero-coupon-bonds
2. Internal Revenue Service, Publication 1212, Guide to Original Issue Discount (OID) Instruments. https://www.irs.gov/publications/p1212
3. U.S. Securities and Exchange Commission, Bonds, Investor.gov. https://www.investor.gov/introduction-investing/investing-basics/investment-products/bonds-or-fixed-income-products/bonds
4. OpenStax, Principles of Finance, Section 7.2, Time Value of Money Basics. https://openstax.org/books/principles-finance/pages/7-2-time-value-of-money-tvm-basics