Solving for a Deferred Annuity Payment

Why the Payment Date Matters

When a deferred annuity’s present value and interest rate are known, the unknown is often the level payment. The central task is to value the payment stream at the correct dates, then divide the given present value by the value of one dollar of payment. A one-period timing shift changes every discount factor, so identifying the first payment date is essential.

This tutorial treats a finite stream of equal payments under a fixed interest rate. Unless stated otherwise, payments are made at the ends of regular periods, the present value is measured at time 0, and there are no fees or other cash flows. The method is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: A deferred annuity-immediate with $n$ payments and a deferral of $d$ payment periods makes payments at times $d+1,d+2,\ldots,d+n$. The first payment is one period after the deferral boundary. Under effective interest rate $i$ per payment period, let $v=1/(1+i)$. If each payment is $R$, its present value at time 0 is $R v^d a_{\overline{n}|}$, where $a_{\overline{n}|}=(1-v^n)/i$.

Solving for the Level Payment

The present-value factor $v^d a_{\overline{n}|}$ is the value at time 0 of payments of one unit each. Therefore, if the given present value is $P$, the level payment is $P$ divided by that factor. This formula assumes the interest rate and payment period are expressed in compatible units and that the payments occur at the stated period ends.

P=R v^d a_{\overline{n}|} \qquad\Longrightarrow\qquad R=\frac{P}{v^d a_{\overline{n}|}}

For an annuity-immediate, the first payment is at time $d+1$, not at time $d$. Its direct-sum present value is $R(v^{d+1}+v^{d+2}+\cdots+v^{d+n})$. Factoring out $R v^d$ leaves the ordinary annuity-immediate factor $a_{\overline{n}|}=v+v^2+\cdots+v^n$.

Fix the timeline.
Write the deferral length, the first payment date, and the last payment date. Count the actual payments rather than inferring the count from the last date alone.

Match the rate period.
Use the effective rate per payment period. If the stated rate is nominal, convert it according to its compounding frequency.

Find the unit-payment factor.
For payments at $d+1$ through $d+n$, calculate $v^d a_{\overline{n}|}$.

Divide and check.
Divide the given present value by the factor to find $R$. Substitute the result back into the present-value equation or discount the payments individually.

Worked Applications

Worked Example: Payments at the Ends of Years 3, 4, and 5

A fund has a present value of $2,500 and earns an effective annual rate of 6%. It will make three equal payments at the ends of years 3, 4, and 5. Find each payment. Use the deferred annuity-immediate method and round only the final payment to the nearest cent.

Timeline and method. The first payment is at year 3, so the deferral is $d=2$ years and the number of payments is $n=3$. The payments are at years $d+1$ through $d+n$, as required for an annuity-immediate.

The effective rate per annual payment period is $i=0.06$, so $v=1/1.06$. The present-value factor can be calculated using the annuity formula or by summing the payment-date discount factors:

v^2a_{\overline{3}|}=v^3+v^4+v^5 =(1.06)^{-3}+(1.06)^{-4}+(1.06)^{-5}

The individual discount factors are approximately $0.839619283$, $0.792093663$, and $0.747258173$. Their sum is $2.378971119$. Thus the payment is

R=\frac{2500}{2.378971119} \approx 1050.87446

Rounded to the nearest cent, each payment is $1,050.87. As a check, multiplying the unrounded payment by the factor gives $1050.87446(2.378971119)\approx 2500$. A common mistake is to use $v^2+v^3+v^4$, treating the first payment as occurring at year 2. That shifts every payment one year early and overstates the payment-date discount factors.

Worked Example: Solving with an Annual Effective Rate

A deferred benefit fund has a present value of $7,859.17. It earns an effective annual rate of 10% and will pay four equal annual amounts at the ends of years 3, 4, 5, and 6. Find the annual payment to the nearest cent.

Timeline and method. There are four payments, beginning at year 3. Therefore $d=2$, $n=4$, and the value at time 0 of one dollar per payment is $v^2a_{\overline{4}|}$.

With $i=0.10$, the required factors are

v^2=(1.10)^{-2}=0.826446281, \qquad v^4=(1.10)^{-4}=0.683013455

Then

a_{\overline{4}|} =\frac{1-v^4}{0.10} =\frac{1-0.683013455}{0.10} =3.169865450

The deferred unit-payment factor is

v^2a_{\overline{4}|} =(0.826446281)(3.169865450) \approx 2.619723512

The payment is consequently

R=\frac{7859.17}{2.619723512} \approx 2999.9998

To the nearest cent, the annual payment is $3,000.00. Substitution checks the result: $3000(2.619723512)\approx \$7,859.17$, to the nearest cent. The common mistake avoided here is using only the four-payment factor $a_{\overline{4}|}$; that factor values the stream one period before its first payment, not at time 0. The deferral discount factor $v^2$ is also required.

Worked Example: Nominal Rate with Quarterly Payments

A reserve has a present value of $3,659.87. Interest is credited at a nominal annual rate of $8\%$ convertible quarterly. The reserve is to make four equal quarterly payments, with the first payment at the end of the third quarter. Find the payment to the nearest cent.

Timeline and method. The payment period is one quarter. The first payment is at quarter 3, so there are two full quarters before the first payment: $d=2$. There are $n=4$ payments, at quarters 3, 4, 5, and 6.

A nominal rate of $8\%$ convertible quarterly gives an effective quarterly rate of

i_{\text{quarter}}=\frac{0.08}{4}=0.02, \qquad v=\frac{1}{1.02}

The value at time 0 of one dollar at each payment date is

v^2a_{\overline{4}|} =v^3+v^4+v^5+v^6 =v^2\left(\frac{1-v^4}{0.02}\right) \approx 3.659869955

Hence

R=\frac{3659.87}{3.659869955} \approx 999.99999

The quarterly payment is $1,000.00. The check is $1000(3.659869955)\approx \$3,659.87$, to the nearest cent. The key trap is using $0.08$ as the rate for each quarter. The stated rate is nominal and convertible quarterly, so the rate for each payment period is $0.08/4=0.02$.

Payment Timing at the Deferral Boundary

The formula above is for an annuity-immediate: the first payment is one full payment period after the deferral boundary. If instead the first payment occurs exactly at the boundary, the payment dates are $d,d+1,\ldots,d+n-1$. This is a deferred annuity-due, not a deferred annuity-immediate. Its present value is $R v^d\ddot{a}_{\overline{n}|}$, where $\ddot{a}_{\overline{n}|}=1+v+\cdots+v^{n-1}$. Do not use the immediate formula without first checking the first payment date.

A quick direct-sum check is especially useful when the wording is ambiguous. Write down the times at which money is actually paid, attach one discount factor to each payment, and verify that the factor sum contains exactly $n$ terms. This check catches both a shifted first payment and a missing or extra payment.

Common Mistakes and Checks

Counting the deferral boundary as a payment. For an annuity-immediate, a deferral of $d$ periods means the first payment is at $d+1$, not $d$.
Using the wrong number of payments. Count the dates in the stated stream. For example, payments at years 3 through 6 inclusive are four payments.
Mixing rate and payment periods. Convert a nominal rate to the effective rate per payment period before calculating $v$ or an annuity factor.
Forgetting to discount the annuity factor to time 0. The factor $a_{\overline{n}|}$ is valued one period before the first payment. For a deferred stream, include $v^d$.
Rounding too soon. Keep several digits in discount factors and the annuity factor; round the payment only at the end.

Key takeaway: For $n$ equal payments at the ends of periods $d+1$ through $d+n$, the present value is $R v^d a_{\overline{n}|}$. Thus, divide the given present value by $v^d a_{\overline{n}|}$ to find the payment, after matching the interest rate to the payment period.

Check Your Understanding

For each problem, identify the payment dates before applying a factor. Amounts are rounded to the nearest cent.

A fund worth $2,000 earns an effective annual rate of 10% and pays equal amounts at the ends of years 2 and 3. Find each payment. (Answer: $1,267.62)
At an effective annual rate of 6%, a present value of $3,083.93 funds four equal payments at the ends of years 3, 4, 5, and 6. Find each payment. (Answer: $1,000.00)
A present value of $2,261.87 earns interest at a nominal annual rate of 8% convertible quarterly. It funds three equal quarterly payments at the ends of quarters 2, 3, and 4. Find each payment. (Answer: $800.00)
A present value of $2,668.36 earns an effective annual rate of 4% and funds three equal annual payments at the ends of years 2, 3, and 4. Find each payment. (Answer: $1,000.00)

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