From Price to Yield
A zero-coupon bond has one payment: its redemption value at maturity. Its market price is the amount paid now for that future payment. When the price, redemption value, and time to maturity are known, the yield is the rate that makes the present value of the redemption payment equal to the price. The calculation depends on the compounding convention used for the yield quote.
For this tutorial, assume the bond has no interim payments, its redemption value is certain, and its term is a positive whole number of years. The price and redemption value are positive amounts in the same currency. Ignore taxes, transaction costs, and any risk of nonpayment. A yield calculated from these assumptions is the rate implied by the stated cash flows; it is not a guarantee of the investor’s realized return if the bond is sold early or does not pay as promised.
Effective Annual Yield
An effective annual yield \(i\) measures the growth over one year, including the effect of annual compounding. With \(n\) whole years until redemption, a dollar invested at this rate accumulates to \((1+i)^n\) dollars. Therefore, the present value of \(R\) is \(R/(1+i)^n\).
The equation is solved by multiplying by \((1+i)^n\), dividing by \(P\), taking the positive \(n\)th root, and subtracting 1. Since \(P\) and \(R\) are positive, the positive root gives the unique real effective annual rate consistent with the model. If \(P<R\), the yield is positive; if \(P=R\), it is zero; and if \(P>R\), it is negative.
Worked Example: Effective Annual Yield from a Discount Price
A hypothetical zero-coupon bond will redeem for $1,440 in 2 years. Its market price is $1,000. Find its effective annual yield and check the result.
Method. Use the present-value equation for one payment at the end of 2 years. The price is below the redemption value, so a positive yield is expected.
Substitute the values and isolate \(i\):
The accumulation factor over 2 years is \(1{,}440/1{,}000=1.44\). Taking the square root gives the annual accumulation factor:
The effective annual yield is 20.00%. To check, $1,000 accumulates for 2 years to \(1{,}000(1.20)^2=1{,}000(1.44)=\$1{,}440\), exactly the redemption value. The method avoids treating the total 2-year increase of 44% as an annual rate.
Nominal Yield Convertible \(m\) Times a Year
A nominal annual rate \(i^{(m)}\) convertible \(m\) times per year applies a periodic rate of \(i^{(m)}/m\) for each conversion period. Over \(n\) years there are \(mn\) periods. Accordingly, the redemption value is discounted using \(mn\) applications of the periodic accumulation factor.
The exponent \(mn\) counts conversion periods, not years. The root first finds the accumulation factor for one conversion period; multiplying the periodic rate by \(m\) then gives the nominal annual quote. This formula assumes the stated conversion frequency applies throughout the term and that \(mn\) is the number of complete conversion periods.
Worked Example: Nominal Yield Convertible Quarterly
A hypothetical zero-coupon bond costs $800 and redeems for $1,000 in 2 years. Find the nominal annual yield convertible quarterly.
Method. There are \(m=4\) conversion periods per year and \(n=2\) years, so the total number of periods is \(mn=8\). First solve for the quarterly accumulation factor:
Take the eighth root, then convert the periodic rate to a nominal annual rate:
Thus the quarterly rate is approximately \(1.028285-1=0.028285\), or 2.8285%. Multiply by 4:
The nominal annual yield convertible quarterly is 11.314%. To check, using the unrounded periodic rate gives \(800(1.25)=\$1{,}000\) after eight quarters. The method avoids taking a 2-year root or multiplying the quarterly rate by the number of years instead of by four.
Continuously Compounded Yield
For a continuously compounded yield \(\delta\), an amount invested for \(n\) years accumulates by the factor \(e^{\delta n}\). The bond price is therefore the redemption value discounted by this factor. Taking natural logarithms isolates the yield.
The logarithm is applied to the ratio \(R/P\), which is dimensionless. If \(R>P\), the logarithm is positive and the continuously compounded yield is positive; if \(R<P\), it is negative. Do not interpret \(\delta\) as an effective annual rate: they are different rate conventions.
Worked Example: Continuous Yield from Price and Redemption Value
A hypothetical zero-coupon bond has a market price of $600 and will redeem for $900 in 2 years. Find its continuously compounded annual yield and verify it.
Method. Use the continuous-compounding price equation and take the natural logarithm of the redemption-to-price ratio:
Using \(\ln(1.5)\approx0.4054651\):
The continuously compounded annual yield is 20.273% (rounded to three decimal places as a percentage). For the check, \(600e^{(0.2027326)(2)}\approx600e^{0.4054652}\approx600(1.5)=\$900\), allowing for rounding. The method avoids using an effective-rate formula on a continuous yield quote.
Choosing and Checking the Formula
The cash flows alone do not determine which yield formula to use; the quote convention does. Before calculating, identify whether the requested rate is effective annually, nominal with a specified conversion frequency, or continuous. Then count the full term in the units required by that convention. The following table summarizes the corresponding equations.
| Yield convention | Price equation | Yield solved from price |
|---|---|---|
| Effective annual | \(P=R(1+i)^{-n}\) | \(i=(R/P)^{1/n}-1\) |
| Nominal, convertible \(m\) times per year | \(P=R(1+i^{(m)}/m)^{-mn}\) | \(i^{(m)}=m[(R/P)^{1/(mn)}-1]\) |
| Continuous | \(P=Re^{-\delta n}\) | \(\delta=\ln(R/P)/n\) |
A useful check is to substitute the solved rate back into the price equation under the same convention. The result should reproduce the original price before rounding. If it does not, review the term, the number of conversion periods, and whether the rate is quoted nominally or effectively. Keep extra digits during intermediate calculations and round the final rate only.
Common Yield-Solving Traps
- Dividing the total percentage increase by the term. The total increase is not generally the annual effective yield. For an effective annual rate, take the \(n\)th root of \(R/P\), then subtract 1.
- Using years instead of conversion periods. For a nominal rate convertible \(m\) times per year, use \(mn\) periods in the exponent and multiply the periodic rate by \(m\).
- Mixing rate conventions. The effective annual, nominal, and continuous formulas are not interchangeable. Apply the convention requested in the question.
- Reversing the ratio. In these yield formulas, the accumulation ratio is \(R/P\), not \(P/R\). A bond priced below redemption should have a positive yield under these assumptions.
For example, suppose a bond priced at $800 redeems for $1,000 in 2 years, with an effective annual yield requested. The incorrect calculation \( (1{,}000-800)/(800\times2)=12.5\% \) divides the total gain evenly across years and ignores compounding. The correct calculation is \( (1{,}000/800)^{1/2}-1=\sqrt{1.25}-1\approx11.803\% \). Substitution confirms that \(800(1.11803)^2\approx\$1{,}000\).
Check Your Understanding
For each problem, identify the rate convention before solving. Answers are rounded to three decimal places where needed.
- A hypothetical bond costs $729 and redeems for $1,000 in 3 years. Find its effective annual yield. (Answer: 11.111%)
- A hypothetical bond costs $10,000 and redeems for $14,641 in 4 years. Find its effective annual yield and verify the accumulation factor. (Answer: 10.000%; accumulation factor \(1.4641\))
- A hypothetical bond costs $800 and redeems for $1,000 in 2 years. Find the nominal annual yield convertible quarterly. (Answer: 11.314%)
- A hypothetical bond costs $600 and redeems for $900 in 2 years. Find its continuously compounded annual yield. (Answer: 20.273%)
- A hypothetical bond costs $850 and redeems for $1,000 in 2 years. Find its nominal annual yield convertible monthly. (Answer: approximately 8.154%)