Deriving the Zero-Coupon Bond Price Formula

Why a Zero-Coupon Bond Has a Simple Price Formula

A zero-coupon bond makes no scheduled interest payments before maturity. Its holder receives one promised payment at maturity, so its price today is the present value of that single future cash flow. This structure makes it possible to derive a compact pricing formula directly from the present-value equation.

The formula is simple, but using it correctly depends on matching the yield to the time periods in the calculation. A rate per six-month period cannot be used with a count of years, for example. The number of periods and the rate for each period must describe the same timeline.

Definition: A zero-coupon bond is a bond that makes no coupon payments before maturity and promises one payment at maturity. Let $P_0$ be its price at the valuation date, $F$ its promised maturity payment, $y$ the required yield per compounding period, and $n$ the number of periods until the payment. Under periodic compounding, its price is the present value of that one payment.

Deriving the Formula from Present Value

Assume the bond makes one payment $F$ at the end of period $n$, the required yield $y$ is constant for each period, and interest compounds once per period. In the basic model, the payment is certain: there is no default, and no tax, liquidity, or embedded-option adjustment is being modelled. More generally, the discount rate must be appropriate for valuing the promised payment and its risks.

The present-value equation asks what amount invested today grows to $F$ after $n$ periods at yield $y$. If that amount is $P_0$, periodic accumulation gives $P_0(1+y)^n=F$. Solving for $P_0$ gives the price formula.

P_0(1+y)^n=F \quad\Longrightarrow\quad P_0=\frac{F}{(1+y)^n}

This equation can also be read as discounting once per period. Each period’s discount factor is $1/(1+y)$; applying it $n$ times gives $1/(1+y)^n$. Thus, the formula is not a separate rule for zero-coupon bonds: it follows from applying the present-value equation to the bond’s single cash flow.

Identify the payment date.
Count the compounding periods from the valuation date to the bond’s single maturity payment. This count is $n$.

Find the yield per period.
Use the yield that corresponds to the period counted in Step 1. If the yield is quoted in another form, convert it before substituting.

Discount the payment.
Calculate $F/(1+y)^n$. Keep full precision during the calculation and round the final price to the requested unit.

Check the result.
Accumulate the calculated price forward for $n$ periods. It should reproduce the promised maturity payment, apart from any final rounding.

Annual and Periodic Yield Conventions

When the yield is an effective annual rate $i$, and maturity is an integer number $n$ of years away, the annual-period formula is $P_0=F/(1+i)^n$. In actuarial notation, the annual discount factor is $v=1/(1+i)$, so the same result is $P_0=Fv^n$. The effective rate $i$ describes growth over one year; $n$ must therefore count years.

A bond yield may instead be quoted as a nominal annual rate $i^{(m)}$, convertible $m$ times per year. The rate per conversion period is $i^{(m)}/m$. If the bond matures after $t$ years and the maturity date falls on a conversion date, there are $mt$ periods. The price is then calculated using the periodic rate and period count together.

P_0 = \frac{F}{\left(1+\dfrac{i^{(m)}}{m}\right)^{mt}}

This nominal-yield expression depends on its stated compounding convention. It should not be substituted for an effective annual yield without conversion. If the date does not fall on a period boundary, or a different day-count or compounding convention applies, that convention must be specified before a price can be calculated.

Worked Example: Annual Yield Over Three Years

A hypothetical zero-coupon bond promises $ \$1{,}000 $ in three years. The required effective annual yield is $5\%$, with annual compounding. Find the price today and verify it by accumulating the price to maturity.

Method. There is one payment and three annual periods, so use $P_0=F/(1+i)^n$, with $F=\$1{,}000$, $i=0.05$, and $n=3$. The common mistake to avoid is treating the rate as a total three-year discount rather than applying it in each of the three periods.

First calculate the accumulation factor: $1.05^2=1.1025$, and $1.05^3=1.1025(1.05)=1.157625$.

Then discount the payment: $P_0=\$1{,}000/1.157625=\$863.8376\ldots$. Rounded to the nearest cent, the price is $\$863.84$.

Check. Accumulating the unrounded price gives $(\$1{,}000/1.157625)(1.157625)=\$1{,}000$. The rounded price gives approximately $\$1{,}000.00$ at maturity, with only a negligible rounding difference.

Worked Example: Quarterly Yield and a Two-Year Maturity

A hypothetical zero-coupon bond promises $\$5{,}000$ in two years. Its required yield is $1.5\%$ per quarter, compounded quarterly. Calculate the current price, rounded to the nearest cent, and verify the result.

Method. Use the stated yield per quarter directly, but count quarters rather than years. Two years contain $n=2(4)=8$ quarterly periods. The common mistake to avoid is discounting for only two periods because the maturity is two years away.

The quarterly accumulation factor is $1.015^2=1.030225$, $1.015^4=1.030225^2=1.061363550625$, and $1.015^8=1.061363550625^2\approx1.126492586$.

Therefore, $P_0=\$5{,}000/1.126492586\approx\$4{,}438.56$, rounded to the nearest cent.

Check. The unrounded price multiplied by $1.015^8$ is $\$5{,}000$. The calculation uses eight quarterly accumulations, consistent with the two-year maturity.

Worked Example: Converting a Nominal Semiannual Yield

A hypothetical zero-coupon bond promises $\$2{,}000$ in four years. Its quoted nominal annual yield is $6\%$, convertible semiannually. Find the price to the nearest cent and verify it.

Method. A nominal rate convertible twice yearly gives a semiannual yield of $i^{(2)}/2=0.06/2=0.03$, or $3\%$ per half-year. Four years contain $4(2)=8$ half-year periods. The common mistake to avoid is using $6\%$ as the rate for each half-year.

Calculate the accumulation factor: $1.03^2=1.0609$, $1.03^4=1.0609^2=1.12550881$, and $1.03^8=1.12550881^2=1.266770081$ (to nine decimal places).

The price is $P_0=\$2{,}000/1.266770081\approx\$1{,}578.82$, rounded to the nearest cent.

Check. Multiplying the unrounded price by $1.266770081$ gives $\$2{,}000$, subject to the displayed factor’s rounding. The periodic rate and the eight-period count follow from the semiannual conversion convention.

What the Formula Does—and Does Not—Tell You

The formula provides a present value when the payment, timing, yield, and compounding convention are specified. It does not establish that the promised payment will actually be made. In the basic model, payment is certain. If the issuer may default, a valuation requires an appropriate treatment of that risk; simply treating an uncertain payment as certain can overstate its value.

It is also important to distinguish a bond’s promised maturity payment from the price paid today. The formula discounts the maturity payment; it does not add coupon payments, because a zero-coupon bond has none. A bond that makes interim payments requires the present values of those additional cash flows as well, which is a different cash-flow structure.

Key takeaway: For one payment $F$ due after $n$ compounding periods at a yield $y$ per period, the zero-coupon price is $P_0=F/(1+y)^n$. The rate and period count must use the same time unit.

A useful diagnostic is to compare the price with the maturity payment under a positive yield. If $y>0$, then $(1+y)^n>1$, so $P_0<F$. A zero yield gives $P_0=F$. A negative yield can produce $P_0>F$ under the same mathematical model. These relationships are checks on the calculation, not substitutes for matching the yield convention to the payment schedule.

Common Mistakes

Using the wrong number of periods. Convert years to quarters or half-years when the yield is stated per quarter or half-year.
Using a nominal annual rate as a periodic rate. For a nominal rate convertible $m$ times per year, divide by $m$ to obtain the rate per conversion period.
Discounting a single payment as though it were an annuity. A zero-coupon bond has one maturity payment, not a stream of coupons.
Rounding too early. Retain calculator precision through the discounting and round the final price only.
Ignoring the pricing assumptions. The basic formula values a specified payment at a specified yield; it does not itself model default or other features that change cash flows or required returns.

Check Your Understanding

For each problem, match the rate to the compounding period, show the discount factor, and round money to the nearest cent unless stated otherwise.

A hypothetical zero-coupon bond pays $\$800$ in two years. The effective annual yield is $4\%$. Find the price and verify it by accumulating the price for two years. (Answer: $\$739.64$)
A bond pays $\$3{,}000$ in three years and earns a required yield of $2\%$ per quarter. Find its price using the number of quarterly periods, then state the maturity value obtained by accumulating the unrounded price. (Answer: price $\$2{,}365.48$; accumulated value $\$3{,}000$)
A hypothetical bond promises $\$1{,}200$ in five years and has an effective annual yield of $7\%$. Calculate the price and identify whether it is below or above the promised payment. (Answer: $\$855.58$, below the payment)
A zero-coupon bond promises $\$2{,}500$ in 18 months. Its nominal annual yield is $8\%$, convertible quarterly. Convert the yield to a quarterly rate, count the periods, and calculate the price. (Answer: quarterly yield $2\%$, 6 periods, price $\$2{,}219.93$)
A bond with a $\$1{,}000$ maturity payment is priced at $\$900$ exactly two years before maturity. Assuming annual compounding and a constant annual yield, solve the zero-coupon price equation for the annual yield. (Answer: $\sqrt{1000/900}-1\approx5.409\%$ per year)

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