Pricing Level-Coupon Bonds with Annuity Factors

Pricing a Level-Coupon Bond

A level-coupon bond has two kinds of promised cash flows: equal coupon payments during its term and a redemption payment at maturity. Its price is the value, at the stated yield, of those future payments. An annuity factor makes the coupon part efficient to calculate; a single-payment discount factor handles redemption.

This tutorial considers a conventional bond valued on a coupon date, immediately after any coupon due on that date has been paid. The next coupon is one payment period away. The yield supplied is the effective yield per payment period, so no conversion from an annual rate is needed. Assume the bond makes all promised payments, has no embedded options or transaction costs, and is redeemed at its face amount. Pricing between coupon dates and redemption at a different amount require additional treatment.

Definition: Let $C$ be the level coupon paid at the end of each payment period, $n$ the number of remaining coupon periods, $R$ the face amount paid at maturity, and $i$ the effective yield per payment period. With $v=1/(1+i)$, the bond’s price $P$ is the present value of the $n$ coupons and the redemption payment, all measured at the valuation date.

The Pricing Formula

The coupons arrive at times $1,2,\ldots,n$, so they form an annuity-immediate. The redemption amount arrives at time $n$, on the same date as the final coupon. Therefore the final period’s cash flow includes both $C$ and $R$, but it is convenient to value the coupon and redemption streams separately.

P=C a_{\overline{n}|}+Rv^n

Here $a_{\overline{n}|}$ is the present-value annuity factor at effective rate $i$, and $v^n$ is the present-value factor for one amount due in $n$ periods. For $i\ne 0$,

a_{\overline{n}|}=\frac{1-v^n}{i} \qquad\text{and}\qquad v^n=(1+i)^{-n}.

The rate $i$ must match the length of one payment period. For example, if coupons are paid twice per year, $i$ in this formula is an effective yield for one half-year, and $n$ counts the remaining half-year periods. If the question provides a yield for some other interval, it must first be expressed as an effective yield per coupon period; do not insert a rate with a mismatched time unit.

Important: The coupon amount $C$ is the cash payment made each period, not automatically the coupon rate written as a percentage. If a coupon rate is stated, first apply it to the face amount under the terms given to find the coupon payment.

Why the Formula Holds

Discount each coupon back to the valuation date. The coupon at time $k$ has present value $Cv^k$, so the coupon stream has value $C(v+v^2+\cdots+v^n)$. This is a finite geometric series. Its sum is the annuity-immediate factor, which gives the coupon component of the price.

v+v^2+\cdots+v^n =\frac{v(1-v^n)}{1-v} =\frac{1-v^n}{i} =a_{\overline{n}|}, \qquad\text{since }1-v=\frac{i}{1+i}=iv.

The redemption amount is a single payment due at time $n$, so its present value is $Rv^n$. Adding that value to the coupon present value gives $P=C a_{\overline{n}|}+Rv^n$. This derivation also explains why the last coupon is not omitted: the annuity factor includes all $n$ coupons, including the one due with redemption.

A Useful Price Check for Par Redemption

Because this tutorial assumes redemption at face amount $R$, we can compare the price with $R$ using the coupon amount and the yield. The annuity identity $i a_{\overline{n}|}+v^n=1$ implies:

\begin{aligned} P-R &=C a_{\overline{n}|}+Rv^n-R\\ &=C a_{\overline{n}|}-Ri a_{\overline{n}|}\\ &=(C-iR)a_{\overline{n}|}. \end{aligned}

Since $a_{\overline{n}|}>0$ when $i>0$, this is a useful check: if $C>iR$, the price is above face value; if $C<iR$, it is below face value; and if $C=iR$, it equals face value. This comparison is a check on the calculation, not a substitute for discounting the cash flows when an actual price is required.

Worked Example: Annual Coupons at a Yield Below the Coupon Rate

A hypothetical bond has a face and redemption amount of $1,000, pays a $60 coupon at the end of each year for five years, and is valued just after a coupon date. Its effective yield is 4% per year. Calculate its price to the nearest cent. Name the method and a common mistake to avoid.

Method. Value the five equal annual coupons with an annuity-immediate factor and the $1,000 redemption amount with a single-payment factor. The stated annual yield matches the annual coupon period.

First calculate the discount factor:

v=\frac{1}{1.04}=0.96153846,\qquad v^5=(1.04)^{-5}=0.82192711.

Then calculate the annuity factor:

a_{\overline{5}|} =\frac{1-0.82192711}{0.04} =4.45182225.

The present value of the coupons is $60(4.45182225)=267.109335$, and the present value of redemption is $1{,}000(0.82192711)=821.92711$. Thus,

$$ P=267.109335+821.92711=1{,}089.036445. $$

Rounded only at the final step, the price is $1,089.04. As a check, $C-iR=60-0.04(1{,}000)=20>0$, so the price should exceed face value. Indeed, $P-R=20(4.45182225)=89.036445$.

Common mistake avoided: applying the yield to the coupon rate as if that percentage were itself the coupon cash flow. The cash flow is $60 each year; it is that amount that is multiplied by the annuity factor.

Worked Example: A Bond Priced Below Face Value

A hypothetical bond will pay $70 at the end of each of three annual periods and $1,000 at the end of the third period. The effective yield is 8% per year, and the valuation date is just after a coupon date. Find the price to the nearest cent and check its relationship to face value.

Method. Use the annuity-immediate factor for the three coupons and discount the redemption payment for three periods. The annual period of the yield agrees with the annual payment period.

Calculate the discount factors:

v=\frac{1}{1.08}=0.92592593,\quad v^2=0.85733882,\quad v^3=0.79383224.

The annuity factor is:

a_{\overline{3}|} =\frac{1-0.79383224}{0.08} =2.57709700.

The coupon present value is $70(2.57709700)=180.396790$. The redemption present value is $1{,}000(0.79383224)=793.832240$. Therefore,

$$ P=180.396790+793.832240=974.229030. $$

The price, rounded to the nearest cent, is $974.23. The check gives $C-iR=70-0.08(1{,}000)=-10$, so $P-R=-10(2.57709700)=-25.770970$, consistent with a price below $1,000.

Common mistake avoided: treating the $70 coupon as if it were paid only at maturity. It is paid at each of three dates, so its present value is an annuity, not one discounted lump sum.

Worked Example: Semiannual Payments with a Per-Period Yield

A hypothetical bond has a face amount of $5,000 and makes six remaining semiannual coupon payments of $120 each. It also pays $5,000 at the final coupon date. Its effective yield is 2.5% per half-year. Find its price to the nearest cent.

Method. Count six payment periods, use the supplied half-year yield directly, value the coupons as an annuity-immediate, and discount redemption for six periods. No conversion to an annual rate is part of the data or needed for this calculation.

The single-period discount factor and the six-period factor are:

v=\frac{1}{1.025}=0.97560976,\qquad v^6=(1.025)^{-6}=0.86229687.

The annuity factor is:

a_{\overline{6}|} =\frac{1-0.86229687}{0.025} =5.50812520.

The coupons are worth $120(5.50812520)=660.975024$, and redemption is worth $5{,}000(0.86229687)=4{,}311.484350$. Thus,

$$ P=660.975024+4{,}311.484350=4{,}972.459374. $$

The price is $4,972.46 to the nearest cent. For a check, the per-period yield amount on face value is $0.025(5{,}000)=125$, which exceeds the $120 coupon. The price should therefore be below face value. In fact, $P-R=(120-125)(5.50812520)=-27.540626$.

Common mistake avoided: using six as the number of years or applying an annual yield to six half-year periods. There are six payment periods, and each is discounted at the stated 2.5% effective half-year yield.

Common Errors and a Reliable Procedure

A bond price is sensitive to both cash-flow timing and the rate used for discounting. Before calculating, identify the valuation date and list which payments remain. Under the assumptions here, the first remaining coupon is one period away, the last coupon and redemption coincide at time $n$, and the yield is measured per payment period.

One tempting but incorrect calculation is to add all coupons to redemption and discount the total as though it were paid at maturity. For the semiannual example above, this would give $(6(120)+5{,}000)(0.86229687)=5{,}720(0.86229687)=4{,}932.34$, rather than the correct $4,972.46. The error occurs because coupons before maturity are paid earlier and are not discounted for all six periods.

A dependable calculation sequence is:

Identify the inputs.
Record the coupon cash amount $C$, the redemption amount $R$, the number of remaining coupon periods $n$, and the effective yield $i$ per period.

Match the period units.
Confirm that one period in $i$ is the interval between payments and that $n$ counts those intervals.

Value each stream.
Compute $C a_{\overline{n}|}$ for the coupons and $Rv^n$ for redemption.

Add and check.
Add the two present values, round the price as requested, and compare $C$ with $iR$ to check whether the result should be above or below face value.

This method applies to a level-coupon bond valued on a coupon date under the stated assumptions. It does not by itself handle a bond valued between coupon dates, a different redemption amount, or a yield quoted for a different compounding interval. Those cases change the cash-flow timing, inputs, or discount factors and should not be forced into this setup without adjustment.

Key takeaway: For a level-coupon bond with redemption at face value and valuation just after a coupon date, price the coupons as an annuity-immediate and redemption as a single payment: $P=C a_{\overline{n}|}+Rv^n$. Use the effective yield per payment period, and verify the result against the coupon-versus-yield check.

Check Your Understanding

For each problem, assume the bond is valued just after a coupon date, makes all stated payments, and is redeemed at its face amount. Use the stated effective yield per payment period and round prices to the nearest cent.

A bond has a $1,000 face amount, pays $50 annually for two years, and yields 5% per year. Find its price. (Answer: $1,000.00)
A bond pays $75 annually for four years and redeems for $1,000. Its effective annual yield is 6%. Find its price. (Answer: $1,051.98)
A bond has a $2,500 face amount and pays $55 every half-year for eight remaining periods. The effective yield is 2% per half-year. Find its price. (Answer: $2,536.63)
A bond has a $1,500 face amount, pays a 7% annual coupon for six years, and has an effective annual yield of 8%. First find the annual coupon cash amount, then calculate the price. (Answer: annual coupon $105; price $1,430.66)

What You'll Learn