Zero Coupon Bond Calculator (Price, Duration & Rate Sensitivity)

A zero coupon bond pays nothing until maturity. The entire return comes from buying below par and being repaid in full, so its price is simply the present value of one future payment.

That single cash flow has an important consequence: a zero's Macaulay duration equals its maturity exactly. Every other bond has duration shorter than its maturity, because coupons return capital earlier. A zero has none, so nothing pulls its duration in.

This makes long zeros the most interest-rate-sensitive instruments in the bond market. A 30-year zero falls over 25% on a one-point rise in yields. That is either the reason to own one or the reason not to, depending on your view, but it should never be a surprise.

where F is face value, y the annual yield, n years to maturity and m the compounding frequency. US Treasury convention is semiannual, so m = 2.

Macaulay duration equals n, the maturity, by definition. Modified duration is:

and approximates the percentage price change for a one-point move in yields.

$1,000 face, 4.5% yield, ten years, semiannual:

• Periods: 20. Rate per period: 2.25%
• Price: $1,000 ÷ 1.0225²⁰ = $640.82
• Discount: $359.18, which is the entire return
• Total return to maturity: 56.05%
• Macaulay duration: 10.00 years. Modified duration: 9.78
• A 100bp rise in yields costs $59.57, or 9.30%

A 30-year zero at the same yield: priced at $263.15, with duration of 30 years. A 100bp rise costs 25.37% of its value.

A 2-year zero: $914.84, and a 100bp rise costs only 1.93%.

The three together are the clearest illustration available of what duration means: same issuer, same yield, wildly different risk.

• Phantom income. In a taxable US account the annual accretion of a zero is taxed as interest each year even though no cash is received. This is why zeros are usually held in tax-deferred accounts, and the calculator does not model it.
• Credit risk. A corporate zero can default. Treasury STRIPS carry no credit risk but every other zero does.
• Reinvestment. A zero has no coupons to reinvest, which eliminates reinvestment risk entirely. That is a genuine advantage this page does not quantify.
• Convexity. Modified duration is a linear approximation. For large yield moves the true price change differs, and for a long zero the convexity effect is substantial.
• Liquidity and bid-offer spreads, which are wider on STRIPS than on coupon Treasuries.
• Callability, on the small number of callable zeros.
• Inflation. These are nominal figures. A 30-year nominal zero carries enormous inflation risk.
• Accrued interest conventions and day counts between coupon dates, which do not apply to a zero but do affect comparison with coupon bonds.

• Underestimating long-duration risk. A 30-year zero is not a conservative holding. It loses a quarter of its value on a one-point yield move, which is equity-like volatility from a government bond.
• Holding zeros in a taxable account. The accretion is taxed annually without any cash arriving to pay the tax. Zeros belong in tax-deferred accounts in most cases.
• Comparing a zero's yield with a coupon bond's without adjusting for compounding. Quote conventions differ, and a semiannual yield is not directly comparable to an annual one.
• Assuming duration equals maturity for all bonds. It only does for zeros. Coupon bonds always have shorter duration than maturity.
• Relying on modified duration for large moves. It is a first-order approximation. Convexity matters, and for long zeros it matters a lot.
• Forgetting that the return is fixed only if held to maturity. Sell early and you get the market price, which can be far below your purchase price.

• Standard fixed income mathematics. Zero coupon pricing, Macaulay duration and modified duration are closed-form results with no jurisdictional content.
• The observation that Macaulay duration equals maturity for a zero follows directly from the definition of duration as the weighted average time to cash flows.