Zero-Coupon Bond Calculator: Price, Yield, Phantom Income and Duration

How it works

Price = Face ÷ (1 + y/m)^(n·m) (continuous: Face × e^(−y·n)); yield = m × [(Face ÷ Price)^(1/(n·m)) − 1]; face from a budget = Price × (1 + y/m)^(n·m); effective annual yield = (1 + y/m)^m − 1; OID for a year = value at the end − value at the start, with value at t = Price × (1 + y/m)^(m·t); Macaulay duration = n; modified duration = n ÷ (1 + y/m); convexity = n(n + 1/m) ÷ (1 + y/m)²

A zero-coupon bond pays nothing until maturity, so its whole return is the discount at which you buy it. A 10-year zero at 4.5% (semi-annual) sells for about 64 cents on the dollar and grows to par. That makes it the simplest way to lock in a sum for a known date, such as a tuition bill, and the most rate-sensitive bond of its maturity, because its Macaulay duration equals its full term: a one-point rise in yields cuts a 10-year zero’s price by about 9%. Price, yield and the face value a budget buys are three ways of solving the same compound-interest equation; the compounding choice matters because US Treasury STRIPS and most zero quotes are semi-annual, while many other markets quote annual yields. In a taxable US account the yearly accretion is original issue discount (OID). IRS Publication 1212 has holders accrue it with the constant-yield method: each period’s OID is the adjusted value at the start times the yield per period. It is taxed every year even though no cash arrives, which is why the schedule here shows the tax on that “phantom income” year by year. Holding zeros in an IRA or 401(k), or buying tax-exempt municipal zeros, avoids it. The calculator also shows the price at other yields, convexity, what you would get by selling before maturity at a different market yield, and the face value in today’s money after inflation. It ignores bid-ask spreads, commissions and state tax.

Worked example

A $10,000 zero maturing in 10 years at a 4.5% yield with semi-annual compounding: price = 10,000 ÷ (1 + 0.045 ÷ 2)^20 = 10,000 ÷ 1.56051 = $6,408.16, or 64.08 per 100 of face. The total return is $3,591.84 (56.1%). The effective annual yield is 4.551%. In the first year the bond accretes $6,408.16 × (1.0225² − 1) = $291.61 of imputed interest, which at a 24% tax rate costs about $69.99 though no cash is paid; over ten years the phantom-income tax adds up to $862.04. Modified duration is 10 ÷ 1.0225 = 9.78 years, so if yields rise one point the price falls to about $5,813 (−9.3%). At 2.5% inflation the $10,000 is worth about $7,812 in today’s money.

Sources: IRS Publication 1212: Guide to Original Issue Discount (OID) Instruments · Investor.gov: Zero coupon bond (phantom interest and taxes) · Calculator.net: bond calculator (price, yield, frequency)

Frequently asked questions