Effective Annual Yield and Zero-Coupon Prices

From Effective Annual Yield to Price

A zero-coupon bond makes no periodic interest payments. Its investor pays a price at the valuation date and receives a single redemption payment at maturity. The effective annual yield connects these two amounts: it is the annual compound rate that makes the price grow to the redemption value over the bond’s term.

The essential pricing task is therefore to discount one known future payment over a specified number of years. This tutorial uses a whole-number term in years and an annual effective yield. It does not use a nominal compounding quote or a continuous yield.

Definition: A zero-coupon bond is a bond with no payments before maturity and one payment at maturity. Let $R$ be its redemption value, $n$ its term in whole years from the valuation date to maturity, and $i$ its effective annual yield. Its price $P$ is the amount at the valuation date that, invested at effective annual rate $i$, accumulates to $R$ at the end of year $n$.

The redemption value is the amount actually paid at maturity; it may equal the bond’s stated face amount, but the two terms need not be treated as interchangeable unless the bond terms say so. The price is the value at the valuation date, not a payment made at maturity.

Deriving the Price Formula

Under an effective annual rate $i$, one dollar invested at the valuation date accumulates to $1+i$ dollars after one year. With interest compounded annually at that rate, it accumulates to $(1+i)^n$ dollars after $n$ years. Thus an investment of $P$ grows to $P(1+i)^n$. To fund the bond’s maturity payment, that accumulated amount must equal $R$.

$$ P(1+i)^n=R $$

Solving this accumulation equation for the initial amount gives the price formula. Equivalently, using the discount factor $v=1/(1+i)$, the price is the redemption value multiplied by the discount factor for $n$ years.

P=\frac{R}{(1+i)^n}=Rv^n \qquad\text{where}\qquad v=\frac{1}{1+i}

Theorem: If a zero-coupon bond pays $R$ at the end of $n$ whole years, and the annual effective yield for that term is $i>-1$, its price at the valuation date is $R/(1+i)^n$. This formula assumes one maturity payment, no interim cash flows, annual effective compounding, and no adjustments for default, taxes, or transaction costs.

For a positive effective yield, the discount factor $v$ is less than one, so the price is below the redemption value. If the yield is zero, the formula gives $P=R$. The formula uses the yield appropriate to the bond’s stated term and valuation date; a yield quoted using a different compounding convention must not be substituted as though it were effective annual.

Pricing One Bond and Stating a Quote

Worked Example: Price and Quote for a Two-Year Bond

A hypothetical zero-coupon bond will redeem for $1,210 exactly two years from today. Its annual effective yield is 10%. Find its price today and its price per $100 of redemption value. Use annual compounding and assume no default or transaction costs.

Method. Discount the single maturity payment for two years at the effective annual yield. The common mistake to avoid is treating the 10% yield as simple interest over the full term.

First, convert the rate to decimal form: $i=0.10$. The two-year accumulation factor is $1.10^2=1.10\times1.10=1.21$.

P=\frac{1{,}210}{1.21}=1{,}000

The price is $1,000. The quote per $100 of redemption value is found by dividing the price by the redemption value and multiplying by 100: $100\times 1{,}000/1{,}210=82.6446\ldots$. To the nearest cent, the quote is $82.64 per $100 of redemption value.

Check. Accumulating the price for two years gives $$1{,}000\times1.21=$1{,}210$, exactly the stated redemption payment. The quote is below $100 because the price is below the maturity payment.

A price quote per $100 is a scaled representation of the price, not a different valuation rule. If the redemption value is $R$, a price of $P$ corresponds to a quote of $100P/R$. When converting a quote back to a total price, multiply the quote per $100 by $R/100$.

Scaling the Formula and Pricing a Portfolio

The same discount factor applies to every dollar of redemption value at the same term and yield. Consequently, doubling the redemption value doubles the price. For several zero-coupon bonds valued on the same date, calculate the present value of each maturity payment using that bond’s term and yield, then add the individual prices. Payments at different dates must not be treated as though they all occur at the portfolio’s latest or earliest maturity.

Worked Example: Scaling a Price to Multiple Bonds

A hypothetical zero-coupon issue has a redemption value of $14,641 per bond, a term of four years, and an annual effective yield of 10%. An investor buys three identical bonds. Find the price of one bond, the total purchase price, and the price quote per $100 of redemption value. Assume the bonds have no interim payments and no transaction costs.

Method. Find the accumulation factor for one bond, discount its redemption value, then scale the result by the number of bonds. This avoids the mistake of applying the per-$100 quote to the number of bonds without first accounting for each bond’s redemption value.

The four-year accumulation factor is $1.10^4=1.10\times1.10\times1.10\times1.10=1.4641$. Therefore, the price of one bond is $$14{,}641/1.4641=$10{,}000$.

\text{Total price}=3(10{,}000)=\$30{,}000

The quote per $100 of redemption value is $100\times10{,}000/14{,}641=68.3016\ldots$, or $68.30 to the nearest cent.

Check. Each bond’s $10,000 price accumulates by a factor of 1.4641 to $14,641. For three bonds, the prices total $30,000 and the maturity payments total $43,923; multiplying $30,000 by 1.4641 also gives $43,923.

For a portfolio with different maturity dates, the same principle applies bond by bond. Each payment is discounted for its own number of years. The prices can then be added because they are all values at the same valuation date.

Worked Example: Adding Prices at Two Maturity Dates

An investor considers two hypothetical zero-coupon bonds, both valued today at an annual effective yield of 10%. Bond A redeems for $1,331 in three years. Bond B redeems for $2,420 in two years. Find the combined price. Assume both have no interim payments and no transaction costs.

Method. Discount each redemption payment over its own term, then add the two prices. The common mistake to avoid is discounting both payments for the same number of years merely because the bonds are being considered together.

For Bond A, the three-year accumulation factor is $1.10^3=1.331$, so its price is $$1{,}331/1.331=$1{,}000$.

For Bond B, the two-year accumulation factor is $1.10^2=1.21$, so its price is $$2{,}420/1.21=$2{,}000$.

\text{Combined price}=\$1{,}000+\$2{,}000=\$3{,}000

Check. The first price accumulates for three years to $1,331, and the second accumulates for two years to $2,420. Each discounted cash flow therefore matches its own maturity payment under the stated yield.

Common Mistakes and Exam Traps

The formula is short, but its inputs must match the cash-flow timing and yield convention. A term of two years means two annual accumulation periods in this setting; it does not mean that simple interest is added once using $2i$. Likewise, a quoted rate must be identified as effective annual before it is used in $(1+i)^n$.

Common mistake: using simple interest instead of compound accumulation. For the two-year bond in the first example, the incorrect calculation $1{,}210/(1+2(0.10))=1{,}008.33$ uses a simple-interest accumulation factor of 1.20. The correct annual compound factor is $1.10^2=1.21$, so the price is $1{,}210/1.21=\$1{,}000$. Checking by accumulation exposes the error: $1,008.33 multiplied by 1.21 does not equal $1,210.

Use the redemption payment, not an assumed face amount. Discount the amount specified as payable at maturity.
Match the term to the valuation date. Count the whole years from today to the stated redemption date.
Use the stated rate convention. This formula uses an effective annual yield, not a nominal or continuous quote.
Keep quote units clear. A quote per $100 is not the total dollar price unless the redemption value is exactly $100.
For multiple bonds, discount each cash flow to the same date. Use each payment’s own term before adding present values.

Zero-coupon pricing is useful because one future payment makes the relationship between yield, time, redemption value, and price transparent. It also provides a clean way to check whether a calculated price is consistent: accumulating the price at the stated effective annual yield for the stated term must reproduce the redemption value.

Check Your Understanding

For each problem, assume annual effective compounding, no interim bond payments, no default, and no transaction costs. Round dollar prices to the nearest cent where needed.

A hypothetical zero-coupon bond redeems for $1,331 in three years. Its effective annual yield is 10%. Find its price today. (Answer: $1,000.00)
A bond redeems for $14,641 in four years at an effective annual yield of 10%. Find its price and its quote per $100 of redemption value, to the nearest cent. (Answer: $10,000.00; $68.30 per $100)
A zero-coupon bond will pay $2,420 in two years. At a 10% effective annual yield, find its price. (Answer: $2,000.00)
At a common valuation date and 10% effective annual yield, one zero-coupon bond pays $1,331 in three years and another pays $2,420 in two years. Find the combined price. (Answer: $3,000.00)
A hypothetical bond redeems for $2,500 in three years at an effective annual yield of 8%. Find its price and its quote per $100 of redemption value, to the nearest cent. (Answer: $1,984.58; $79.38 per $100)

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