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Exam FM · Deferred Annuities and Perpetuities · Tutorial 167 of 998

Deferred Annuity and Perpetuity Problem Workshop

Learn to translate deferred cash-flow timelines into present values, solve for unknown amounts, and check payment timing—including fractional deferrals.

Beginner 10 min read

What You'll Learn

  • Identify the first payment date and deferral period for immediate and due payments
  • Value deferred finite annuities and perpetuities using a dated cash-flow timeline
  • Combine a finite annuity with a perpetuity and solve for an unknown payment
  • Apply fractional deferrals when annual payments begin at a noninteger time
  • Detect timing errors by comparing formulas with the actual payment dates

Start With the Payment Dates

Deferred annuity problems often look like formula-selection problems, but the essential first step is to identify when each payment occurs. A one-period shift changes present value, and the word “deferred” by itself does not say whether the first payment is at the end of the deferral period or at its beginning. Write down the dates before choosing a factor.

Throughout this tutorial, assume a constant effective annual interest rate \(i>0\), with \(v=1/(1+i)\). Payments are guaranteed, level, and annual unless stated otherwise; there is no mortality contingency. Fractional times use the accumulation model \((1+i)^t\), so discounting by \(t\) years uses \(v^t\). A finite annuity has a positive integer number of payments, even when its deferral is fractional.

Definition: A deferred annuity-immediate with deferral \(d\), payment \(R\), and \(n\) payments has payments at times \(d+1,d+2,\ldots,d+n\). A deferred annuity-due with the same parameters has payments at times \(d,d+1,\ldots,d+n-1\). The deferral \(d\) is the time from the valuation date to the point one period before the first payment of an immediate annuity, or to the first payment of a due annuity.

For integer or fractional \(d\), the present values at time 0 are

$$ R v^d a_{\overline{n}|} \qquad\text{and}\qquad R v^d \ddot{a}_{\overline{n}|}, $$

respectively. Here \(a_{\overline{n}|}\) values payments one period apart at times 1 through \(n\), while \(\ddot{a}_{\overline{n}|}\) values payments at times 0 through \(n-1\).

For a perpetuity-immediate deferred by \(d\), the first payment is at \(d+1\), and its present value is \(R v^d/i\). For a perpetuity-due deferred by \(d\), the first payment is at \(d\), and its present value is \(R v^d\ddot{a}_{\overline{\infty}|}=R v^d(1+i)/i\). These formulas require a positive effective rate and the stated annual payment timing.

Why Deferral Multiplies the Annuity Value by \(v^d\)

The deferral factor is a consequence of discounting every payment date by the same additional interval. For an annuity-immediate, the payment dates are \(d+1\) through \(d+n\). Its present value is the sum of each payment discounted from its own date:

$$ R\sum_{k=1}^{n}v^{d+k} =Rv^d\sum_{k=1}^{n}v^k =Rv^d a_{\overline{n}|}. $$

The same argument applies to a due annuity, whose dates are \(d\) through \(d+n-1\): factoring \(v^d\) out of the discounted sum leaves \(\ddot{a}_{\overline{n}|}\). For a perpetuity, the corresponding infinite sum converges when \(i>0\). With fractional \(d\), the factoring step remains valid under the stated fractional-time accumulation model.

1
Mark the valuation date.
Set time 0 at the date for which present value is required.
2
Write the first payment date.
For an immediate annuity, it is \(d+1\); for a due annuity, it is \(d\). Continue the dates at the stated payment interval.
3
Value each stream at the right date.
Use the finite annuity or perpetuity factor, then discount back to time 0.
4
Check against the timeline.
Confirm that the formula includes exactly the payments shown and that no payment has been shifted by one period.

Combining a Finite Annuity and a Perpetuity

When two streams are both valued at time 0, add their present values. The streams may have different starting dates and different payment amounts; each must be valued using its own timing. The following example uses the total price to solve for an unknown finite-annuity payment.

Worked Example: Solve for a Payment in a Two-Stream Contract

A contract is purchased for $3,000 at time 0. It pays an unknown annual amount \(R\) at times 3, 4, 5, and 6, and a separate $600 perpetuity-immediate beginning at time 8. The effective annual interest rate is 25%. Find \(R\). Assume all payments are certain.

Method. Value each stream at time 0, using its own first-payment date. The four finite payments are an annuity-immediate deferred by 2 years. The perpetuity’s first payment at time 8 means its deferral is 7 years.

Since \(v=1/1.25=0.8\), the finite annuity factor before deferral is

$$ a_{\overline{4}|} =\frac{1-v^4}{i} =\frac{1-0.8^4}{0.25} =\frac{1-0.4096}{0.25} =2.3616. $$

Thus the present-value factor for the four payments is

$$ v^2a_{\overline{4}|} =0.8^2(2.3616) =0.64(2.3616) =1.511424. $$

The perpetuity-immediate has present value

$$ \frac{600}{i}v^7 =\frac{600}{0.25}(0.8^7) =2{,}400(0.2097152) =503.31648. $$

Equating the sum of the two present values to the $3,000 price gives

$$ 1.511424R+503.31648=3{,}000, \qquad R=\frac{3{,}000-503.31648}{1.511424} =1{,}651.875. $$

Rounded to the nearest cent, \(R=\$1{,}651.88\). As a check, using the unrounded amount gives \(1.511424(1{,}651.875)=2{,}496.68352\); adding \(503.31648\) returns exactly $3,000. The key timing check is that the perpetuity is discounted by \(v^7\), not \(v^8\): its first payment is at time 8, but the perpetuity-immediate factor \(600/i\) is located one period before that payment.

Due Payments and the Deferral Boundary

For a due annuity, the first payment occurs at the boundary date \(d\). A payment exactly at that date is not discounted for a further year. In contrast, an immediate annuity with deferral \(d\) begins one payment period later. This distinction is especially useful when a problem gives payment dates directly rather than labeling the annuity “due” or “immediate.”

Worked Example: Check a Deferred Annuity-Due Against an Immediate Annuity

At an effective annual rate of 25%, compare (a) three payments of $500 at times 2, 3, and 4 with (b) three payments of $500 at times 3, 4, and 5. Find the present value of each stream and their difference. Payments are certain.

Method. The first stream is a due annuity deferred by 2 years. The second is an immediate annuity deferred by 2 years. Since \(v=0.8\), the due-annuity factor is

$$ \ddot{a}_{\overline{3}|} =1+v+v^2 =1+0.8+0.64 =2.44. $$

Therefore the first present value is

$$ 500v^2\ddot{a}_{\overline{3}|} =500(0.64)(2.44) =500(1.5616) =\$780.80. $$

For the second stream,

$$ a_{\overline{3}|} =v+v^2+v^3 =0.8+0.64+0.512 =1.952, $$ $$ 500v^2a_{\overline{3}|} =500(0.64)(1.952) =500(1.24928) =\$624.64. $$

The difference is \(\$780.80-\$624.64=\$156.16\). This equals the present value of the one extra earlier payment in the first stream: \(500v^2=500(0.64)=\$320\)? That is not the difference because the streams have three payments each and the first stream is shifted one period earlier throughout. To check correctly, subtract corresponding cash flows: the difference in present values is

$$ 500[(v^2+v^3+v^4)-(v^3+v^4+v^5)] =500(v^2-v^5) =500(0.64-0.32768) =\$156.16. $$

This check confirms that the entire three-payment stream has shifted one year earlier; the difference is not simply the value of one payment.

Fractional Deferrals

A fractional deferral does not mean that the annual payment interval has changed. It means the whole annual payment pattern is shifted by a fraction of a year. For example, a perpetuity-immediate deferred by \(1.5\) years has payments at \(2.5,3.5,4.5,\ldots\). Under the assumed fractional-time accumulation model, discount by \(v^{1.5}\). Do not replace a fractional deferral with a rounded integer unless the problem explicitly defines such a convention.

Worked Example: Value a Fractionally Deferred Perpetuity

A certain perpetuity-immediate pays $500 annually, with its first payment at time 2.5 and each later payment one year afterward. The effective annual interest rate is 25%. Find its present value at time 0.

Method. Since the payment dates are \(2.5,3.5,4.5,\ldots\), the deferral is \(d=1.5\): the first payment is one year after the date \(1.5\). The perpetuity value one payment period before its first payment is \(500/i\), so

$$ \text{PV}_0 =\frac{500}{0.25}v^{1.5} =2{,}000(0.8)^{1.5} =2{,}000(0.715541753) =\$1{,}431.083506. $$

Rounded to the nearest cent, the present value is $1,431.08. A timeline check places the value \(500/i=\$2,000\) at time \(1.5\), one year before the first payment at time \(2.5\). Discounting that amount \(1.5\) years to time 0 gives the same result. Rounding the deferral to 1 or 2 years would value a different payment schedule.

Common Timing Errors

A frequent mistake is to use the first payment date itself as the deferral in an immediate-perpetuity formula. If a $500 perpetuity-immediate begins at time 3 and \(i=25\%\), its deferral is 2, not 3. The correct calculation is

$$ \text{Correct PV}_0 =\frac{500}{0.25}v^2 =2{,}000(0.64) =\$1{,}280. $$

Using \(v^3\) instead gives \(2{,}000(0.512)=\$1{,}024\), which is wrong because it discounts the value one extra year. The perpetuity factor \(R/i\) is located at time 2, immediately before the first payment at time 3.

Before accepting an answer, check the cash-flow dates against the formula. In particular:

  • For an immediate annuity, the first payment is one payment period after its deferral date.
  • For a due annuity, the first payment is on its deferral date.
  • For an immediate perpetuity, the value \(R/i\) is one period before its first payment.
  • For a due perpetuity, the first payment is at the date where the due stream begins.
  • For fractional deferrals, preserve the stated payment dates and use the matching fractional discount factor.
Key takeaway: A deferred value is reliable only when its deferral matches the actual payment timeline. Write the dates, identify whether the first payment is immediate or due, and then apply the annuity or perpetuity factor at the correct point before discounting to time 0.

Check Your Understanding

For each problem, assume certain annual payments and a constant effective annual rate. Check the payment dates before selecting a factor.

  1. At \(i=20\%\), an annuity-immediate pays $700 at times 4, 5, and 6. Find its present value at time 0. (Answer: $853.32)
  2. At \(i=10\%\), a deferred annuity-due pays $250 at times 1, 2, 3, and 4. Find its present value at time 0. (Answer: $792.47)
  3. At \(i=5\%\), a perpetuity-due pays $300 annually, with its first payment at time 2. Find its present value at time 0. (Answer: $5,714.29)
  4. At \(i=16\%\), a certain annuity-immediate pays $900 at times 1.5, 2.5, and 3.5. Find its present value at time 0 using fractional discounting. (Answer: $1,876.73)
  5. At \(i=25\%\), a contract has total present value $3,000. It pays an unknown amount \(R\) at times 2, 3, 4, and 5, plus a $400 perpetuity-immediate beginning at time 8. Find \(R\). (Answer: $1,410.30)

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