Continuous Yield Quotes for Zero-Coupon Bonds

Why Continuous Yield Quotes Need Careful Conversion

A zero-coupon bond makes one payment: its stated redemption value at maturity. When its yield is quoted as a constant force of interest, the bond’s price is found by continuously discounting that future payment. The calculation is compact, but one distinction matters throughout: a force of interest of 5% is not the same rate as an effective annual yield of 5%.

This tutorial develops the continuous-yield price formula, explains why it holds, and shows how to convert between a force of interest and an equivalent effective annual yield. The examples assume annual time units and a known redemption date and amount. No interim coupon payments are included.

Definition: A zero-coupon bond with redemption value $F$ at time $t$ pays $F$ at maturity and no other payments. If its yield is a constant force of interest $\delta$ per year, its price at time 0 is $P=Fe^{-\delta t}$. The quoted force is constant over the term, and the formula assumes the stated redemption payment is made in full at time $t$.

Deriving the Continuous-Yield Price

The force of interest describes the instantaneous rate of growth of an accumulated value. If $A(s)$ is the amount at time $s$ and the force is constant at $\delta$, the defining relationship is

\frac{dA(s)}{ds}=\delta A(s).

Separating variables and integrating from time 0 to time $t$ gives the accumulation factor. A present amount $P$ therefore grows to $Pe^{\delta t}$. For a zero-coupon bond, that accumulated value must equal its redemption payment $F$.

\int_{P}^{F}\frac{dA}{A}=\int_0^t\delta\,ds \quad\Longrightarrow\quad \ln\!\left(\frac{F}{P}\right)=\delta t \quad\Longrightarrow\quad F=Pe^{\delta t}.

Solving for the present price produces the continuous-discounting formula. This derivation also gives a useful check: accumulating the calculated price at the quoted force for the full term must recover the redemption value.

P=Fe^{-\delta t}.

Application condition: Use $P=Fe^{-\delta t}$ when the yield is a constant force $\delta$ per year over the entire period from purchase to redemption, and $t$ is measured in years. The formula does not describe a bond with coupon payments unless each payment is separately discounted, nor does it apply unchanged when the force varies over time.

Converting Continuous Yield to Effective Annual Yield

The effective annual rate $i$ is the rate that produces the same one-year accumulation as the quoted force. Under a constant force, one year of accumulation is $e^\delta$. Under an effective annual rate, it is $1+i$. Equating those factors gives the conversion.

1+i=e^\delta \quad\Longleftrightarrow\quad i=e^\delta-1, \qquad \delta=\ln(1+i).

Consequently, the continuous price formula can also be written using the equivalent effective annual rate. Since $e^{\delta t}=(e^\delta)^t=(1+i)^t$, the two expressions agree:

P=Fe^{-\delta t}=\frac{F}{(1+i)^t}.

This equality applies when $i$ is the effective annual rate equivalent to the constant force $\delta$, not when the same numerical percentage is simply inserted in place of $\delta$. For positive rates, $e^\delta-1$ is greater than $\delta$. For example, a force of 5% corresponds to an effective annual rate of about 5.127%, not 5%.

Identify the quoted rate.
Confirm that the quote is a constant force of interest per year, rather than an effective annual rate or a nominal rate convertible $m$-thly.

Set the term and redemption value.
Write the maturity time $t$ in years and identify the single payment $F$.

Discount or convert.
Calculate $P=Fe^{-\delta t}$. If an effective annual quote is also needed, calculate $i=e^\delta-1$.

Check the accumulation.
Verify that $Pe^{\delta t}=F$, or equivalently that $P(1+i)^t=F$.

Worked Applications

Worked Example: Pricing a Bond and Converting Its Yield

A hypothetical zero-coupon bond will redeem for $15,000 in 4 years. Its quoted yield is a constant force of 4.5% per year. Find the price and the equivalent effective annual yield. Report the price to the nearest cent and the rate to 4 significant figures.

Method. Discount the redemption payment under the constant force, then equate the one-year continuous and effective accumulation factors. This avoids treating the 4.5% force as though it were an effective annual rate.

The data are $F=\$15{,}000$, $t=4$, and $\delta=0.045$. First calculate the exponent:

\delta t=(0.045)(4)=0.18.

The price is

P=15{,}000e^{-0.18} =15{,}000(0.8352702) =\$12{,}529.053.

Thus the price, rounded to the nearest cent, is $12,529.05. The equivalent effective annual rate is

i=e^{0.045}-1 =1.0460279-1 =0.0460279.

As a percentage to 4 significant figures, $i=4.603\%$. To check the price, use the continuous accumulation factor:

12{,}529.053e^{(0.045)(4)} =12{,}529.053e^{0.18} \approx \$15{,}000.00.

The price therefore accumulates to the stated redemption value. The result is consistent with the equivalent effective annual rate because $e^{0.045}=1.0460279=1+i$.

Worked Example: Distinguishing a Force Quote from an Effective Quote

A hypothetical zero-coupon bond pays $9,000 in 3 years. One quote is a constant force of 6% per year. A separate quote is an effective annual yield of 6%. Calculate the price under each quote and identify which price is lower. Report prices to the nearest cent.

Method. Price the bond separately under each stated rate convention. The common mistake is to use $1.06$ as the annual accumulation factor for the 6% force quote; that factor belongs to a 6% effective annual rate.

Under the force quote, $\delta=0.06$, so

P_{\text{force}}=9{,}000e^{-(0.06)(3)} =9{,}000e^{-0.18} =9{,}000(0.8352702) =\$7{,}517.43.

Under the effective annual quote $i=0.06$, discount by $(1+i)^3$:

P_{\text{effective}} =\frac{9{,}000}{(1.06)^3} =\frac{9{,}000}{1.191016} =\$7{,}556.57.

The force quote is equivalent to an effective annual rate of

e^{0.06}-1=0.0618365,

or approximately 6.184%, which is higher than the separate 6% effective quote. A higher yield gives the lower price for this single positive payment, so the force-quote price of $7,517.43 is lower than $7,556.57. As a check, $7{,}517.43e^{0.18}$ is approximately $9,000, while $7{,}556.57(1.06)^3$ is approximately $9,000. Small differences in these checks arise from rounding the prices to cents.

Worked Example: Using a Continuous Quote That Represents a Familiar Effective Rate

A hypothetical zero-coupon bond pays $12,000 in 3 years. Its quoted constant force is $\delta=\ln(1.04)$ per year. Find the equivalent effective annual yield and the price. Report the price to the nearest cent and the rate as a percentage to 4 significant figures.

Method. Convert the force before calculating the price. The logarithm in the quote is chosen so the equivalent effective annual accumulation factor is exactly 1.04; this provides a useful check without changing the stated rate convention.

The effective annual rate is

i=e^\delta-1 =e^{\ln(1.04)}-1 =1.04-1 =0.04.

Therefore the equivalent effective annual yield is 4.000%. For the price, either use the force formula or the equivalent effective-rate formula. Using the conversion,

P=\frac{12{,}000}{(1.04)^3} =\frac{12{,}000}{1.124864} =\$10{,}667.96.

The same result follows directly because $e^{-\delta t}=e^{-3\ln(1.04)}=(1.04)^{-3}$. For a final check, accumulating the unrounded price for 3 years at the equivalent effective rate gives

10{,}667.9563(1.04)^3 =10{,}667.9563(1.124864) =\$12{,}000.00.

The calculation uses the specified force, converts it to an effective annual rate, and checks that the accumulated price equals the redemption payment.

Common Mistakes and Checks

The most frequent error is confusing the number in a force quote with the effective annual rate. If $\delta=0.05$, the one-year accumulation factor is $e^{0.05}$, not $1.05$. Using the wrong factor changes the price:

\text{Incorrect for a 5% force over }t\text{ years: } \frac{F}{(1.05)^t} \qquad\text{Correct:}\qquad Fe^{-0.05t}.

The incorrect expression would be appropriate only if the quote were a 5% effective annual yield. Another error is to use $e^{\delta t}$ as a discount factor. That is the accumulation factor: it grows a present value to a future value. A future redemption payment is brought back to time 0 using its reciprocal $e^{-\delta t}$.

Before finalizing a calculation, check that:

$\delta$ is expressed per year and $t$ is in years;
the exponent is the product $\delta t$, with a negative sign in the price formula;
the effective annual conversion is $i=e^\delta-1$, not $i=\delta$;
accumulating the price over the term returns the stated redemption amount.

Continuous-yield pricing is useful whenever a zero-coupon bond is quoted using a force of interest or when quotes in different rate conventions must be compared. The conversion makes those comparisons consistent: first express each quote as an accumulation factor, then apply the matching discount factor. The formula here is for a single known payment at maturity; valuing a bond with multiple cash flows requires discounting each payment at its appropriate time.

Check Your Understanding

For each problem, identify the rate convention before applying a price formula. Prices are rounded to the nearest cent and rates to 4 significant figures where requested.

A zero-coupon bond pays $5,000 in 2 years and has a constant force of 3% per year. Find its price and equivalent effective annual yield. (Answer: $4,708.82; 3.045%)
A zero-coupon bond redeems for $8,000 in 4 years at a constant force of 4% per year. Find the price and verify it using the equivalent effective annual rate. (Answer: $6,817.15; equivalent effective annual yield 4.081%)
A bond redeems for $10,000 in 5 years and its constant force is $\ln(1.06)$ per year. Find the equivalent effective annual yield and price. (Answer: 6.000%; $7,472.58)
A bond pays $10,000 in 5 years. Compare its price at a constant force of 5% per year with its price at an effective annual yield of 5%. State which price is lower. (Answer: force-quote price $7,788.01; effective-quote price $7,835.26; the force-quote price is lower)
A zero-coupon bond pays $25,000 in 8 years at a constant force of 2.5% per year. Find the price and equivalent effective annual yield, then state the accumulation check. (Answer: $20,468.27; 2.532%; accumulating the unrounded price by $e^{(0.025)(8)}$ gives $25,000.00)

What You'll Learn