Yield to Maturity for Zero-Coupon Bonds

Why a Zero-Coupon Bond Has a Direct Yield Formula

A zero-coupon bond makes no interim coupon payments. Its only promised payment is its face value at maturity. With only one cash flow to discount, its yield can be calculated directly from the price, face value, and time to maturity. This makes a zero-coupon bond a useful setting for checking whether a yield quote is consistent with a bond’s cash flows.

The yield is meaningful only in relation to a specified compounding convention. An annual effective yield, a nominal annual yield compounded quarterly, and a continuously compounded yield can describe the same price and maturity payment, but their numerical rates differ. Always identify the convention before calculating or comparing a yield.

Definition: A zero-coupon bond is a bond with no scheduled coupon payments before maturity. Let $P$ be its current price, $F$ its face value paid at maturity, and $n$ its term to maturity in years. The yield to maturity under a stated compounding convention is the rate that makes the present value of the maturity payment equal to $P$, assuming the bond is held to maturity and the promised payment is made.

For the formulas below, assume $P>0$, $F>0$, and a fixed maturity payment at time $n$. The bond has no other cash flows, and the yield is constant over the term in the pricing equation. These formulas determine the yield implied by the stated price; they do not guarantee that the issuer will make the promised payment.

Deriving the Yield Formulas

Under an annual effective yield $i$, one dollar invested for one year grows to $1+i$ dollars. After $n$ years, the accumulation factor is $(1+i)^n$. Discounting the face value by this factor gives the bond price:

P=\frac{F}{(1+i)^n}.

To solve for $i$, multiply both sides by $(1+i)^n$, divide by $P$, and take the $n$-th root. This gives:

(1+i)^n=\frac{F}{P}, \qquad i=\left(\frac{F}{P}\right)^{1/n}-1.

The $n$-th root is necessary because the price reflects growth or discounting over the full $n$-year term, not just one year. The result is an annual effective rate, even when $n$ is not a whole number, provided the maturity is stated in years and the annual effective model is appropriate.

Formula: annual effective yield. For a zero-coupon bond paying $F$ in $n$ years at price $P$, the annual effective yield is $i=(F/P)^{1/n}-1$. It is valid when the cash flow is a single maturity payment and the yield is compounded annually on an effective basis.

If the quoted yield is a nominal annual rate $j^{(m)}$, convertible $m$ times per year, then each period has rate $j^{(m)}/m$. There are $mn$ periods in $n$ years. The price equation and its solution are:

P=\frac{F}{\left(1+\frac{j^{(m)}}{m}\right)^{mn}}, \qquad j^{(m)}=m\left[\left(\frac{F}{P}\right)^{1/(mn)}-1\right].

This expression requires $1+j^{(m)}/m>0$, so each period’s accumulation factor is positive. Under continuous compounding at force of interest $\delta$, the corresponding price equation is $P=Fe^{-\delta n}$, giving $\delta=\ln(F/P)/n$. These are different ways of quoting the rate; use the equation that matches the stated convention.

Reading the Price and Yield Together

For a positive price and face value under annual effective compounding, the formula immediately gives a useful consistency check. If $P=F$, then $i=0$. If $P<F$, then $F/P>1$, so $i>0$. If $P>F$, then $F/P<1$, so $i<0$. These conclusions follow from the price-yield equation; they do not depend on a coupon comparison because this bond has no coupons.

A second check is to substitute the calculated yield back into the pricing equation. A yield is correct only if discounting the promised face value at that yield reproduces the original price, subject to any stated rounding. The same check works for nominal periodic and continuous conventions by using their respective price equations.

Worked Example: Annual Effective Yield Over Three Years

A hypothetical zero-coupon bond will pay $1,331 in three years. Its current price is $1,000. Calculate its annual effective yield and check the implied price. Use annual effective compounding.

Method. There is one cash flow, so use the annual effective zero-coupon formula. The common mistake to avoid is dividing the total percentage increase by three; the annual yield is a compound rate, so the cube root is required.

Step 1: Identify the inputs. Here $F=\$1{,}331$, $P=\$1{,}000$, and $n=3$.

Step 2: Calculate the yield.

i=\left(\frac{1{,}331}{1{,}000}\right)^{1/3}-1 =(1.331)^{1/3}-1 =1.10-1 =0.10.

The annual effective yield is 10.00% per year. The cube root is exact because $1.10^3=1.331$.

Step 3: Check by repricing.

P=\frac{1{,}331}{(1.10)^3} =\frac{1{,}331}{1.331} =\$1{,}000.

The calculated yield reproduces the stated price exactly, so the result is consistent.

Worked Example: Quarterly Compounding and a Rounded Price

A hypothetical zero-coupon bond pays $10,000 in two years. Its price is $9,234.83. Find the nominal annual yield compounded quarterly. Report the rate to four significant figures and check the price using the rounded rate.

Method. There are four compounding periods per year and eight periods over two years. Use the nominal periodic formula. The common mistake to avoid is taking a second-year or annual root instead of an eighth root.

Step 1: Identify the inputs. The face value is $F=\$10{,}000$, the price is $P=\$9{,}234.83$, the maturity is $n=2$ years, and the compounding frequency is $m=4$. Thus $mn=4(2)=8$ periods.

Step 2: Solve for the quarterly rate and nominal annual yield.

\frac{j^{(4)}}{4} =\left(\frac{10{,}000}{9{,}234.83}\right)^{1/8}-1 \approx 0.010000, \qquad j^{(4)}\approx 4(0.010000)=0.04000.

The nominal annual yield compounded quarterly is 4.000%. The price is rounded to cents; the unrounded price generated by an exactly 4.000% nominal yield is $10{,}000/(1.01)^8$, which rounds to $9,234.83.

Step 3: Check by repricing. Using the rounded nominal rate, the quarterly rate is $0.04000/4=0.01000$. Therefore:

P=\frac{10{,}000}{(1+0.01000)^8} =\frac{10{,}000}{1.082856706} \approx \$9{,}234.83.

This agrees with the stated price to the nearest cent. Keeping the period count at eight is essential: the bond has two years of quarterly periods, not two periods.

Worked Example: A Price Above Face Value and Two Yield Conventions

A hypothetical zero-coupon bond has a face value of $1,000, a price of $1,200, and four years to maturity. Calculate its annual effective yield and its continuously compounded yield. Check each result by substituting it into the appropriate pricing equation. Report rates to four decimal places as percentages.

Method. Calculate each rate from its own price equation. The common mistake to avoid is treating annual effective and continuously compounded rates as identical, or assuming that a price above face value must produce a positive yield.

Step 1: Annual effective yield. With $F=\$1{,}000$, $P=\$1{,}200$, and $n=4$:

i=\left(\frac{1{,}000}{1{,}200}\right)^{1/4}-1 =\left(\frac{5}{6}\right)^{1/4}-1 \approx -0.0445572.

Thus the annual effective yield is −4.4557% per year. It is negative because the investor pays more than the bond’s sole promised payment.

Step 2: Continuously compounded yield. Use the continuous-compounding equation:

\delta=\frac{\ln(P/F)}{-n} =\frac{\ln(F/P)}{n} =\frac{\ln(1{,}000/1{,}200)}{4} \approx -0.0455804.

The continuously compounded yield is −4.5580% per year. It differs from the annual effective yield because the compounding convention differs.

Step 3: Check both prices. For the annual effective yield, $1+i\approx0.9554428$, so:

\frac{1{,}000}{(1+i)^4} \approx\frac{1{,}000}{0.8333333} =\$1{,}200.

For continuous compounding, $e^{-\delta(4)}=1.2$, so $1{,}000e^{-\delta(4)}=\$1{,}200$. Both conventions reproduce the same observed price, while giving different rate quotes.

Common Errors and Practical Checks

A zero-coupon bond’s one-payment structure makes the algebra short, but several input and interpretation errors can still change the answer:

Using the wrong ratio. The annual effective formula uses $F/P$, not $P/F$. Reversing the ratio reverses the direction of the yield.
Ignoring maturity. The exponent is $1/n$ for an annual effective yield and $1/(mn)$ for a nominal yield compounded $m$ times per year. A multi-year price change is not itself an annual rate.
Confusing a period rate with a nominal annual rate. Under $m$-thly compounding, first solve for the per-period rate, then multiply it by $m$ to obtain $j^{(m)}$.
Assuming every yield is positive. If the purchase price exceeds the sole maturity payment, the yield is negative under these standard models.
Rounding too early. Keep intermediate values unrounded where possible, then report the final rate at the requested precision. Repricing with a rounded yield should agree within the corresponding rounding tolerance.

In practice, the direct formula is useful both for valuing a zero-coupon instrument at a stated yield and for interpreting the rate implied by a quoted price. A sound calculation records the face value, price, maturity, and compounding basis before applying the formula. It then checks the result by discounting the maturity payment back to the original price.

Key takeaway: A zero-coupon bond’s yield is the rate that discounts its single face-value payment to its current price. Use the formula for the stated compounding convention, keep the full maturity period count, and verify the result by repricing.

Check Your Understanding

For each problem, identify the compounding basis before calculating. Answers are provided for checking.

A zero-coupon bond costs $2,000 and pays $2,500 in five years. Find its annual effective yield. (Answer: approximately 4.5638% per year)
A zero-coupon bond priced at $6,000 pays $6,000 in three years. Find its annual effective yield and state the price-to-face-value check. (Answer: 0.0000% per year; price equals face value)
A bond with a price of $7,000 pays $8,000 in four years. Find its annual effective yield. (Answer: approximately 3.3946% per year)
A zero-coupon bond costs $5,000 and pays $5,000 in three years. Find the nominal annual yield compounded semiannually. (Answer: 0.0000% nominal annual yield)
A bond costs $12,000 and pays $15,000 in three years. Find the continuously compounded yield and state whether it is positive or negative. (Answer: approximately 7.4381% per year; positive)

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