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Exam FM · Deferred Annuities and Perpetuities · Tutorial 157 of 998

Deferred Perpetuity-Due Values

Use the first payment date—not just the word “deferred”—to select and apply the correct perpetuity value.

Beginner 9 min read

What You'll Learn

  • Define the deferral date and identify the first payment date on a timeline.
  • Value a deferred perpetuity-due when its first payment is at the deferral date.
  • Distinguish that value from a perpetuity whose first payment is one period later.
  • Derive the deferred values by discounting a perpetuity value from its first-payment date.
  • Check calculations by valuing at the deferral date before discounting.
  • Avoid common errors caused by shifting payments by one period.

Why the First Payment Date Matters

A perpetuity-due makes its first payment at the date its value is being considered. Deferring that stream shifts its payment dates into the future, but the phrase “deferred by three years” can be ambiguous unless the first payment date is also stated. Does the first payment occur exactly three years from now, or one year after that date? Those streams have different present values.

For a deferred perpetuity-due, first locate the deferral date and then locate the first payment. If the first payment falls on the deferral date, include that payment when finding the value there. If the first payment is one period later, the value at the deferral date is instead that of a perpetuity-immediate. The difference is one payment period, so it affects the formula even when every other assumption is unchanged.

The formulas below assume level payments made annually, a constant positive effective annual interest rate, and a certain stream that continues indefinitely. Let \(R\) be each payment, \(i\) the effective annual interest rate, and \(v=1/(1+i)\) the annual discount factor. Define \(d=1-v=i/(1+i)\). A payment at time \(t\) is discounted to an earlier focal date by the appropriate power of \(v\).

Definition: A level perpetuity-due pays \(R\) at its first payment date and then pays \(R\) at every subsequent annual date forever. If the first payment occurs \(m\) periods after the focal date, its value at the focal date is \(R v^m/d\). A perpetuity whose first payment is one period after a specified date has value \(R/i\) at that date.

Two Timing Patterns

Let the focal date be time 0 and let time \(m\) be the deferral date, where \(m\) is a nonnegative integer. The key distinction is whether the payment at time \(m\) is part of the stream.

Payment datesValue at time \(m\)Value at time 0
\(m, m+1, m+2, \ldots\): first payment at the deferral date\(R/d\)\(R v^m/d\)
\(m+1, m+2, m+3, \ldots\): first payment one period after the deferral date\(R/i\)\(R v^m/i\)

In the first row, the stream at time \(m\) is due: it pays immediately at time \(m\). In the second row, there is no payment at time \(m\); the first one arrives one period later, so the stream at time \(m\) is immediate. The identity \(v/d=1/i\) also means \(R v^{m+1}/d=R v^m/i\). This gives a useful equivalent form for the second row, but the timing description should still be checked before choosing a formula.

1
Mark the focal date and deferral date.
Measure the number of interest periods between them. Use consistent periods for payments and the effective rate.
2
Mark the first payment.
If it falls on the deferral date, use the due pattern. If it falls one period later, use the immediate pattern.
3
Value at the deferral date first.
Use \(R/d\) when a payment is due on that date, or \(R/i\) when the first payment is one period later.
4
Discount to the requested focal date.
Multiply the deferral-date value by \(v^m\), where \(m\) is the number of periods from the focal date to the deferral date.

Why the Deferred Perpetuity-Due Formula Holds

At the first payment date, the value of a level perpetuity-due is the sum of the payment made immediately and all later payments discounted to that date. For positive \(i\), the discount factor satisfies \(0<v<1\), so the geometric series converges.

$$ R+Rv+Rv^2+\cdots =R\left(\frac{1}{1-v}\right) =\frac{R}{d}. $$

Thus, if the first payment is at time \(m\), the value at time \(m\) is \(R/d\). Moving that value back \(m\) periods to time 0 multiplies it by \(v^m\), giving \(R v^m/d\). If instead the first payment is at time \(m+1\), then at time \(m\) the stream consists of payments one, two, three, and so on periods away. Its value is

$$ Rv+Rv^2+Rv^3+\cdots =\frac{Rv}{1-v} =\frac{Rv}{d} =\frac{R}{i}. $$

Discounting this value by \(v^m\) gives \(R v^m/i\). The derivation shows why the two timing patterns cannot share one formula: at the deferral date, one includes an immediate payment and the other does not.

Worked Example: First Payment at the Deferral Date

A hypothetical endowment pays $800 annually forever, beginning with a payment at time 3 and continuing at times 4, 5, and so on. The effective annual interest rate is 25%. Find the value at time 0, rounded to the nearest cent.

Method. The first payment is at time 3, exactly the deferral date. Therefore the value at time 3 is a perpetuity-due value. Here \(R=800\), \(i=0.25\), and \(v=1/1.25=0.8\). The discount rate is \(d=1-v=0.2\).

At time 3, the value is

$$ \frac{R}{d}=\frac{800}{0.2}=4{,}000. $$

Time 3 is three annual periods after time 0, so discount by \(v^3\):

$$ PV_0=4{,}000(0.8)^3 =4{,}000(0.512) =2{,}048. $$

The value is $2,048.00. As a check, \(800(0.8)^3/0.2=2{,}048\), the same result. A common mistake is to use the one-period-later formula \(R v^3/i\), which gives \(1{,}638.40\). That formula excludes the payment at time 3, even though the problem includes it.

Worked Example: First Payment One Period After the Deferral Date

A hypothetical trust pays $1,000 annually forever, with its first payment at time 3. The trust is valued at time 0, and the deferral date is time 2. The effective annual interest rate is 25%. Find the value at time 0, rounded to the nearest cent.

Method. Since the first payment at time 3 is one period after the deferral date, there is no payment at time 2. At time 2, the stream has the value of a perpetuity-immediate. With \(R=1{,}000\), \(i=0.25\), and \(v=0.8\), its value at time 2 is

$$ \frac{R}{i}=\frac{1{,}000}{0.25}=4{,}000. $$

The deferral date is two periods after time 0. Discount the value at time 2 by \(v^2\):

$$ PV_0=4{,}000(0.8)^2 =4{,}000(0.64) =2{,}560. $$

The value is $2,560.00. Equivalently, \(R v^2/i=1{,}000(0.64)/0.25=2{,}560\). Using \(R v^2/d\) instead would give $3,200.00 and would incorrectly add a payment at time 2. The payment dates in the problem, not merely the word “deferred,” determine the correct method.

Worked Example: Valuing at a Date Other Than Time 0

A hypothetical scholarship pays $500 annually forever, with its first payment at time 4. The effective annual interest rate is 10%. Find the value at time 1, rounded to the nearest cent.

Method. From the focal date, time 1, to the first payment at time 4 there are three annual periods. The payment at time 4 is the first payment of the perpetuity-due when valued at time 4. Let \(v=1/1.1=10/11\) and \(d=1-v=1/11\). The value at time 4, including the payment due on that date, is

$$ \frac{R}{d} =\frac{500}{1/11} =5{,}500. $$

Discount three periods from time 4 to time 1:

$$ PV_1 =5{,}500\left(\frac{10}{11}\right)^3 =\frac{5{,}500{,}000}{1{,}331} \approx 4{,}132.2314. $$

Rounded to the nearest cent, the value at time 1 is $4,132.23. The check is the same valuation expressed directly from time 1: the first payment is three periods away, so \(500v^3/d=4{,}132.2314\). A common error is to count four periods from the focal date, even though time 1 to time 4 is three periods.

Exam Checks and Common Traps

The most reliable check is to draw a short timeline and write the actual payment dates before substituting into a formula. If the first payment is at the deferral date, the value there includes an immediate payment. If the first payment is one period later, the value there does not. Then check that the discount exponent counts periods from the requested focal date to the date where the perpetuity value was calculated.

  • Confusing deferral date and first payment date. A deferral date at time \(m\) does not, by itself, say whether the stream pays at \(m\) or \(m+1\). Read the payment dates.
  • Using \(d\) and \(i\) interchangeably. They are related but not equal: \(d=i/(1+i)\). The value at a date with an immediate payment is \(R/d\); the value one period before the first payment is \(R/i\).
  • Discounting the wrong number of periods. Count intervals between dates, not the number of dates shown. From time 1 to time 4 is three periods.
  • Mixing annual payments with a rate for another period. The formulas require a rate and a payment period on the same basis. Convert the rate appropriately before using annual payment dates.
  • Applying a finite-annuity formula to an infinite stream. A perpetuity has no final payment, so its value comes from the convergent geometric series, not from a finite number of payments.

These timing checks are important because the difference between the two patterns is not a rounding issue. For fixed \(R\) and \(i>0\), a perpetuity-due at the deferral date is worth more there than a perpetuity-immediate: \(R/d>R/i\), since \(d<i\). That difference is precisely the effect of including one additional payment at the deferral date.

Key takeaway: When the first payment occurs \(m\) periods after the focal date, the value is \(R v^m/d\). When the stream is described by a deferral date \(m\) and the first payment is one period later, the value is \(R v^m/i\). Identify both dates before choosing the formula.

Check Your Understanding

For each problem, identify the payment timing before calculating. Unless stated otherwise, payments are annual and the interest rate is an effective annual rate. Round money to the nearest cent.

  1. A hypothetical grant pays $600 annually forever, beginning with a payment at time 4. The effective annual interest rate is 25%. Find its value at time 0. (Answer: $1,228.80)
  2. A hypothetical fund pays $1,500 annually forever, with its first payment at time 4. Its deferral date is time 3, and the effective annual interest rate is 25%. Find its value at time 0. (Answer: $3,072.00)
  3. A hypothetical scholarship pays $700 annually forever, with its first payment at time 5. Find its value at time 2 if the effective annual interest rate is 10%. (Answer: $5,785.12)
  4. At an effective annual interest rate of 25%, compare the time-0 values of two hypothetical perpetuities that each pay $2,000 annually forever. The first pays at times 5, 6, and onward; the second pays at times 6, 7, and onward. Find the difference between their values. (Answer: $655.36)
  5. A hypothetical perpetuity-due pays annually forever, beginning with its first payment at time 3. The effective annual interest rate is 25%, and its value at time 0 is $4,000. Find the annual payment. (Answer: $1,562.50)

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