Finding a Payment from a Target Present Value

From a Present-Value Target to a Payment

An annuity problem often gives the value of a payment stream and asks for the payment amount. The central step is to express the present value as a factor times the level payment, then divide by the factor. The factor depends on when the payments occur: a payment at the end of the first period is discounted once, while a payment at the beginning of the first period is not discounted.

The rate must also match the payment period. In this tutorial, $i$ is the effective interest rate per payment period, and $n$ is the number of level payments. For example, if payments are quarterly, $i$ must be an effective quarterly rate. The formulas below do not by themselves convert an annual rate into a rate for another payment period.

Definition: Let $R$ be a level payment, $n$ the number of payments, and $i$ the effective interest rate per payment period. For payments at times $1,2,\ldots,n$, the present value is $R a_{\overline{n}|}$. For payments at times $0,1,\ldots,n-1$, the present value is $R\ddot{a}_{\overline{n}|}$. Here $v=1/(1+i)$, $a_{\overline{n}|}=(1-v^n)/i$, and $\ddot{a}_{\overline{n}|}=(1+i)a_{\overline{n}|}$, when $i\ne 0$.

These equations assume a constant effective rate per period, equal payments, and regular payment dates exactly one period apart. They value only the stated payment stream at time 0; any fees, down payment, or other cash flow must be included separately in the value equation. If $i=0$, each payment has a present-value factor of 1, so the present value is $nR$ for either timing.

Select the Factor Before Solving

The timing determines which factor belongs in the equation. An annuity-immediate has payments at the ends of periods $1$ through $n$, so its terms are $Rv,Rv^2,\ldots,Rv^n$. An annuity-due has payments at the beginnings of periods $0$ through $n-1$, so its terms are $R,Rv,\ldots,Rv^{n-1}$. In both cases, the payment $R$ is multiplied by the sum of the discount factors for its actual payment dates.

Payment timing	Payment dates	Present-value equation
Annuity-immediate	1, 2, ..., n	$PV=R a_{\overline{n}\|}$
Annuity-due	0, 1, ..., n-1	$PV=R\ddot{a}_{\overline{n}\|}$

Locate the first payment.
A first payment one full period from the valuation date indicates an annuity-immediate. A first payment at the valuation date indicates an annuity-due.

Count the payments and identify the rate period.
Use $n$ for the number of payments, not the number of intervals between the first and last payments. Confirm that $i$ is effective for one payment period.

Write the present-value equation.
Set the target present value equal to $R$ times the factor for the identified timing.

Solve and check.
Divide by the factor, retain full precision during the calculation, and substitute the rounded payment into the original equation as a reasonableness check.

Why the Payment Formulas Work

The present value of a cash-flow stream is the sum of each payment discounted from its payment date to time 0. This gives a direct derivation of the payment equation; it is not necessary to memorize which power of $v$ starts each series.

For an annuity-immediate, the first payment is at time 1. Therefore, the payment at time $k$ has present value $Rv^k$, for $k=1,\ldots,n$. Summing those values gives a geometric series:

PV=R(v+v^2+\cdots+v^n) =R\frac{v(1-v^n)}{1-v} =R\frac{1-v^n}{i} =R a_{\overline{n}|}.

In the third expression, $1-v=i/(1+i)=iv$, so $v/(1-v)=1/i$. Thus, when the target present value is known, rearranging gives the level payment:

R=\frac{PV}{a_{\overline{n}|}} =\frac{PV\,i}{1-v^n}.

For an annuity-due, the payment dates are one period earlier: $0,1,\ldots,n-1$. Its present value is $R(1+v+\cdots+v^{n-1})$. This is the annuity-immediate series shifted one period earlier, so its factor is $(1+i)a_{\overline{n}|}$. Consequently, the payment is $PV/\ddot{a}_{\overline{n}|}$, not $PV/a_{\overline{n}|}$.

Worked Examples

Worked Example: A Three-Payment Annuity-Immediate

A hypothetical equipment lease is represented by three equal payments at the ends of years 1, 2, and 3. The present value of those payments is $10,000, using an effective annual interest rate of 10%. Find the payment. Ignore fees and other cash flows.

Method. The first payment is at time 1, so this is an annuity-immediate. There are $n=3$ payments and $i=0.10$ per payment period.

First calculate the discount factor and annuity-immediate factor:

v=\frac{1}{1.10}=0.9090909091, \qquad v^2=0.8264462810, \qquad v^3=0.7513148009.

a_{\overline{3}|} =v+v^2+v^3 =0.9090909091+0.8264462810+0.7513148009 =2.486851991.

Set the target present value equal to the payment times this factor and solve:

10{,}000=R(2.486851991), \qquad R=\frac{10{,}000}{2.486851991}=4{,}021.148\ldots

Rounded to the nearest cent, the payment is $4,021.15. As a check, $4{,}021.148\ldots\times2.486851991=10{,}000$ before rounding. The method avoids discounting the first end-of-year payment zero times: it occurs at time 1 and must be discounted once.

Worked Example: An Annuity-Due With a Clean Factor

A hypothetical service contract is valued at $7,380 at time 0. It requires four equal payments at times 0, 1, 2, and 3. The effective interest rate per payment period is 25%. Find the payment, ignoring any other cash flows.

Method. Since a payment is made immediately at time 0, this is an annuity-due. Here $n=4$, $i=0.25$, and $v=1/1.25=0.8$. The discount factors for the four payment dates are:

1,\quad v=0.8,\quad v^2=0.64,\quad v^3=0.512.

Their sum is the annuity-due factor:

\ddot{a}_{\overline{4}|}=1+0.8+0.64+0.512=2.952.

Solve the present-value equation and check:

7{,}380=R(2.952), \qquad R=\frac{7{,}380}{2.952}=2{,}500.

Each payment is $2,500. Substitution verifies the target exactly: $2{,}500\times2.952=7{,}380$. The payment at time 0 contributes the full $2,500 to the present value; it is not discounted.

Worked Example: Distinguishing Four Payments at Time 0 Through 3

A hypothetical grant is to be paid in four equal installments at times 0, 1, 2, and 3. Its present value is $8,000, and the effective interest rate per payment period is 20%. Find each installment. Ignore taxes and fees.

Method. The first payment occurs at time 0, so the stream is an annuity-due with $n=4$ and $i=0.20$. The per-period discount factor is $v=1/1.20=0.8333333333$. The successive powers required are:

v^2=0.6944444444,\qquad v^3=0.5787037037.

The present-value factor includes the payment at time 0:

\ddot{a}_{\overline{4}|} =1+v+v^2+v^3 =1+0.8333333333+0.6944444444+0.5787037037 =3.106481481.

Therefore:

R=\frac{8{,}000}{3.106481481}=2{,}575.2608\ldots

Rounded to the nearest cent, each installment is $2,575.26. Using the unrounded payment, $2{,}575.2608\ldots\times3.106481481\approx8{,}000$, confirming the calculation. A common error is to use the annuity-immediate factor, which discounts all four payments, including the one at time 0. That incorrect factor is $v+v^2+v^3+v^4=2.588734568$, giving $8{,}000/2.588734568\approx\$3{,}090.31$, not the payment for the stated schedule.

Common Timing and Calculation Traps

The most important exam check is to translate the words into payment dates before selecting a factor. “At the end of each period” normally places the first payment at time 1. “Immediately,” “today,” or “at the beginning of each period” places the first payment at time 0. The number of payments is not changed just because one occurs at time 0: payments at times 0 through 3 are four payments, not three.

Using the wrong timing factor. An annuity-due contains an undiscounted payment at time 0; an annuity-immediate does not. The difference is substantive, not a rounding adjustment.
Using the wrong rate period. A nominal annual rate or an effective annual rate is not automatically the effective rate per payment period. Convert the rate only when the problem supplies enough information to do so.
Rounding too early. Keep the discount factors and payment unrounded until the final currency answer. Rounding a factor first can shift the result by cents or more.
Forgetting additional cash flows. If a stated amount is a total purchase price that includes a down payment, do not equate the full price to the present value of installments without accounting for that payment separately.

A useful final check is to multiply the calculated payment by the exact factor used in the equation. The product should reproduce the target present value, up to any difference caused by displaying a rounded payment. It is also sensible to check the direction: for a fixed target, a larger present-value factor means a smaller payment.

Key takeaway: First identify the payment dates, then use the matching present-value factor, and finally solve $R=PV/\text{factor}$. A correct calculation with the wrong timing factor answers a different cash-flow problem.

Check Your Understanding

For each problem, identify the payment timing before calculating. Round monetary answers to the nearest cent unless the answer is exact.

A three-payment annuity-immediate has present value $5,000 at an effective rate of 10% per payment period. Find the level payment. (Answer: $2,010.57)
Four equal payments occur at times 0, 1, 2, and 3. Their present value is $7,380 at an effective rate of 25% per period. Find each payment. (Answer: $2,500)
A four-payment annuity-due has present value $8,000 at an effective rate of 20% per payment period. Find the level payment. (Answer: $2,575.26)
A two-payment annuity-immediate has a present value of $10,000 at an effective rate of 10% per period. Find the level payment. (Answer: $5,761.90)
Seven equal payments are made at times 1 through 7, with no interest. The present value is $14,700. Find each payment and state why immediate-versus-due timing does not change the factor when the rate is zero. (Answer: $2,100; each payment has discount factor 1 at zero interest)

Payment timing	Payment dates	Present-value equation
Annuity-immediate	1, 2, ..., n	\(PV=R a_{\overline{n}\|}\)
Annuity-due	0, 1, ..., n-1	\(PV=R\ddot{a}_{\overline{n}\|}\)

What You'll Learn