Deferred Annuity: Present Value and Future Value Made Simple

When is the first payment in a deferred annuity?

An annuity contains equal payments at equal intervals. The word deferred adds a delay before that sequence starts. It does not change the number of payments promised.

A few terms before we go on. An ordinary annuity pays at the end of each period. An annuity due pays at the beginning of each period. An annuity factor is the number you multiply by one payment to value the whole series. And “time 3” simply means the end of Year 3, while time 0 means today.

An ordinary-annuity factor places the value one period before the first payment. Therefore, if annual payments first arrive at the end of Year 4, the factor produces a value at the end of Year 3. Three additional years of discounting bring that value to time 0.

For the last row, another valid method is to value an annuity due at time 3 and discount that value three periods. Both methods describe the same cash dates. Draw those dates before deciding how many periods to discount.

How do you find the present value of a deferred annuity?

Let R be each payment, i the interest rate per payment period, n the number of payments, and d the number of periods before the ordinary-annuity valuation date. The first payment occurs at time d + 1. So for an ordinary annuity deferred 3 years, d is 3. For an annuity due deferred 3 years, the first payment is at time 3, so d is 2.

The bracketed expression divided by i is the present value of an ordinary annuity of 1. Multiply by R to value the payment series, then apply the single-sum factor for d periods.

The subtraction method uses two ordinary-annuity factors instead. Value a hypothetical series covering all d + n periods, then remove the payments that do not actually occur during the first d periods.

The Mathematics of Finance discussion of deferred annuities provides an additional educational reference for locating the valuation date.

Deferred annuity example: four delayed annual receipts

Hypothetical example. Juniper Studio will receive $6,000 at the end of Years 4, 5, 6, and 7. Assume a 6% annual discount rate. What is the value today?

Here is the timeline.

Using subtraction, the seven-period factor is 5.58238144 and the three-period factor is 2.67301195. Their difference, multiplied by $6,000, also gives $17,456.22. Retain unrounded values until the final answer; printed tables can produce small rounding differences.

Want to see that rounding in action? With five-decimal table factors, $6,000 × 3.46511 × 0.83962 gives $17,456.25, a few cents away.

You can also check the answer in Excel with =PV(6%,4,-6000)/(1+6%)^3, which returns $17,456.22.

The $24,000 total receipts are not their present value. Each receipt arrives later than today. Also, $20,790.63 is an intermediate value at time 3, not the final answer at time 0.

What if the payments come at the beginning of each year?

Now say Juniper Studio receives the same four $6,000 payments at the beginning of Years 4, 5, 6, and 7. That means times 3, 4, 5, and 6.

Method 1: The ordinary-annuity factor now places the value at time 2. So $20,790.63 ÷ 1.06^2 = $18,503.59.

Method 2: Value an annuity due at time 3, which is $20,790.63 × 1.06 = $22,038.07. Then discount three periods: $22,038.07 ÷ 1.06^3 = $18,503.59.

Both methods match. The value is $1,047.37 higher than $17,456.22 because every payment arrives one year sooner.

What if you need to solve for the payment?

Exam questions sometimes flip the problem. Say you invest $50,000 today at 5% annually and want five equal withdrawals, with the first at the end of Year 4.

Step 1: Grow the money to time 3: $50,000 × 1.05^3 = $57,881.25.

Step 2: Find the five-period ordinary-annuity factor at 5%, which is 4.329477.

Step 3: Divide: $57,881.25 ÷ 4.329477 = $13,369.11 per year.

How do you find the future value of a deferred annuity?

If no balance exists before the first deposit, the empty waiting periods earn nothing. Future value measured immediately after the final deposit depends on the actual deposits and the interest earned between them.

Hypothetical example. Deposit $2,000 at the end of Years 3, 4, and 5 at 4% annually. At the end of Year 5, the first deposit has earned interest for two years, the second for one year, and the third for zero years.

This is the future value of a three-payment ordinary annuity. Do not use five payments merely because the final date is Year 5. If valuation occurs after the final payment, compound the accumulated balance for that extra interval.

How does a deferred annuity show up in accounting?

Here is a common case. Harbor Supply buys equipment today and signs a note promising $6,000 at the end of Years 4, 5, 6, and 7. The note states no interest, and we’ll assume 6% is the appropriate market rate. These are the same cash flows as our earlier example, so the note’s present value is $17,456.22.

The entries below track Notes Payable at its net carrying amount. The note’s face amount is $24,000. At issuance, the $6,543.78 discount reduces that amount to the $17,456.22 initial carrying amount.

Purchase date:
Debit Equipment $17,456.22
Credit Notes Payable $17,456.22

End of Year 1:
Debit Interest Expense $1,047.37
Credit Notes Payable $1,047.37

Interest builds the balance until payments begin. FASB ASC 835-30 describes the interest method, which applies a constant rate to the amount outstanding at the start of each period.

The balance lands exactly on zero, reconciling the amortization schedule. Total interest is $6,543.78, the same as $24,000 minus $17,456.22. Interest is rounded to the cent each year, so the Year 3 balance is one cent above the present value rounded to $20,790.63.

End of Year 4:
Debit Interest Expense $1,247.44
Debit Notes Payable $4,752.56
Credit Cash $6,000.00

Want more practice with this kind of entry? Explore the Intermediate Accounting course.

Common deferred annuity mistakes to avoid

• Mixing up payments and waiting periods. Count them separately. A 3-year deferral followed by four year-end payments ends at Year 7, not Year 4.
• Using the wrong rate. Match the rate to the payment interval. A 6% annual rate compounded quarterly is 1.5% per quarter.
• Stopping too early. Label the date on every value. $20,790.63 is a time 3 value, not today’s value.
• Ignoring payment timing. Check whether payments come at the beginning or end of each period, because that changes d.
• Counting empty years in future value. Only actual deposits go in the calculation.

A deferred annuity and bond pricing both combine annuity and single-sum calculations, but for different reasons: one shifts a payment series through time; the other adds coupons and maturity principal. If amounts are uncertain, the expected cash flow approach addresses that additional issue.

Three practice questions with explained answers

1. Count the deferral periods

Five equal annual receipts begin at the end of Year 6. After using a five-period ordinary-annuity factor, how many additional periods should you discount?

2. Complete the present-value calculation

A hypothetical deferred payment series is worth $11,025 at time 2. At 5% annually, what is its value at time 0?

3. Interpret the waiting period

Two $3,000 deposits occur at the end of Years 4 and 5. At 10%, what is the balance immediately after the second deposit?

Frequently asked questions

Does “deferred 3 years” always mean the first payment is at time 4?

No. That’s true for an ordinary annuity. A deferred annuity due can begin at time 3, because its payments come at the start of each year. When the wording is unclear, go by the actual payment dates.

Can I discount every payment separately?

Yes. Discount each cash flow from its own date and add the results. This is a useful independent check on the annuity methods.

Does a longer delay reduce present value?

For the same positive payments and positive discount rate, yes. Moving every receipt farther away increases the discounting.

Is a deferred annuity the same as an annuity due?

No. “Deferred” tells you when the payments start. “Due” tells you payments come at the beginning of each period. An annuity can be both.

What is the formula for the present value of a deferred annuity?

PV = R × [1 − (1 + i)^(−n)] ÷ i × (1 + i)^(−d). R is the payment, i is the rate per period, n is the number of payments, and d is one period less than the time of the first payment.

Does the waiting period change future value?

Not at the last payment date, as long as nothing is invested before the first deposit. If you value the balance later, compound it for that extra time.

Is this the same as a deferred annuity from an insurance company?

Not quite. Both involve payments that start later, but the insurance product is a contract you buy. This lesson covers the time value of money calculation.

About the author

Mansour Farhat, CPA, CIA, DBA, is the instructor behind Farhat Lectures. Read his full bio at https://farhatlectures.com/about-me/.