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Exam FM · Deferred Annuities and Perpetuities · Tutorial 156 of 998

Deferred Perpetuity-Immediate Values

Separate the value of a perpetuity at the deferral date from the discounting needed to bring it back to today.

Beginner 10 min read

What You'll Learn

  • Define a whole-period deferral for a perpetuity-immediate
  • Identify the first payment date from the deferral length
  • Calculate the perpetuity value at the deferral date
  • Discount that value to the valuation date
  • Verify a deferred value using a geometric series
  • Avoid confusing the effective rate with one minus the discount factor

Introduction

A perpetuity-immediate pays a level amount at the end of each period, forever. If those payments do not begin immediately, its value at an earlier date has two parts: the value of the perpetuity once its payments are about to begin, and the discounting of that value back across the deferral period. Keeping those parts separate makes the first payment date easier to identify and reduces timing errors.

This tutorial considers annual payments and a whole-number deferral measured in years. The annual effective interest rate is constant. The central question is: if the first payment is delayed, how does that delay change the value at the valuation date?

Definition: A level annual perpetuity-immediate pays an amount \(R\) at the end of each year, continuing indefinitely. If there are \(k\) full years with no payments after the valuation date, the first payment is at the end of year \(k+1\). The perpetuity is deferred \(k\) years.

Value at the Deferral Date, Then Discount Back

Assume a positive annual effective interest rate \(i\), the same rate in every year, and annual payments of \(R\) made with certainty. At time \(k\), just before the first payment at time \(k+1\), the perpetuity has the value of an ordinary perpetuity-immediate: \(R/i\). To find its value at time 0, discount that amount for \(k\) years.

Theorem: The present value at time 0 of a level annual perpetuity-immediate deferred \(k\) full years is
$$ PV_0 = \frac{R}{i}v^k = \frac{R}{i(1+i)^k}, \qquad v=\frac{1}{1+i}. $$
Here \(k\) is a nonnegative integer, and the first payment is at time \(k+1\), not time \(k\).

The formula applies when the payments are level, annual, and continue indefinitely, and when the constant annual effective rate satisfies \(i>0\). With \(i=0\), the undiscounted infinite series has no finite present value. If the rate is negative, the values of the positive payments do not form a convergent series, so the formula does not give a finite present value.

1
Locate the first payment.
If the deferral is \(k\) full years, count \(k\) years with no payments. The first payment is at time \(k+1\).
2
Value the perpetuity at time \(k\).
One period before its first payment, its value is \(R/i\).
3
Discount to time 0.
Multiply the time-\(k\) value by \(v^k\). Do not discount for \(k+1\) years: the perpetuity value is already located at time \(k\).

Why the Deferral Factor Is \(v^k\)

The result follows directly from the payment dates. With \(k\) full years of deferral, the payments occur at times \(k+1,k+2,k+3,\ldots\). Discounting each payment to time 0 gives the infinite geometric series below.

$$ PV_0 = Rv^{k+1}+Rv^{k+2}+Rv^{k+3}+\cdots = Rv^{k+1}\left(1+v+v^2+\cdots\right). $$

Because \(i>0\), \(0<v<1\), so the geometric series converges and its sum is \(1/(1-v)\). Also, \(1-v=i/(1+i)=iv\). Substituting these facts shows why the value is \(R/i\) discounted for exactly \(k\) years:

$$ PV_0 = \frac{Rv^{k+1}}{1-v} = \frac{Rv^{k+1}}{iv} = \frac{R}{i}v^k. $$

The same reasoning gives a useful timing check: the first discounted payment has factor \(v^{k+1}\), while the full perpetuity value is \((R/i)v^k\). These expressions agree because the perpetuity includes every payment from time \(k+1\) onward, not just the first one.

Worked Example: Five-Year Deferral at an 8% Effective Rate

A hypothetical fund will pay $2,400 at the end of every year forever, with its first payment at the end of year 6. Find the value at time 0 if the annual effective interest rate is 8%.

Method. Since the first payment is at time 6, there are five full years without payments, so \(k=5\). First value the perpetuity at time 5, then discount five years to time 0.

At time 5, the value is:

$$ \frac{R}{i}=\frac{2{,}400}{0.08}=30{,}000. $$

The discount factor for five years is:

$$ v^5=\frac{1}{(1.08)^5} =\frac{1}{1.4693280768} \approx 0.680583197. $$

Therefore:

$$ PV_0=30{,}000(0.680583197) \approx \$20{,}417.50. $$

Check. The first payment is discounted six years, so the direct series begins \(2{,}400v^6\). Its value is \(2{,}400v^6/(1-v)\). Here \(1-v=1-1/1.08\approx0.0740741\), not \(0.08\). Since \(1-v=0.08v\), the direct-series value simplifies to \(2{,}400v^5/0.08=30{,}000v^5\), agreeing with the result. The common mistake this check avoids is treating \(1-v\) as equal to \(i\).

Read the First Payment Date Carefully

A deferral is a count of complete payment periods with no payments. It is not automatically the time of the first payment. For example, a three-year deferral means no payments at times 1, 2, or 3; the first payment is at time 4. The value \(R/i\) is located at time 3, one year before that first payment. Discounting it back to time 0 uses \(v^3\).

A quick timeline is often safer than relying on a verbal phrase alone. Mark time 0, count the no-payment periods, and then mark the first payment at the next period-end. The distinction between a payment at the deferral boundary and one period after it matters: those are different cash-flow streams and have different present values.

Worked Example: Three-Year Deferral at a 5% Effective Rate

A hypothetical scholarship trust is to pay $1,800 at each year-end forever. The first payment is at the end of year 4. Find the value at time 0 using an annual effective rate of 5%.

Method. The first payment at time 4 means the trust has three full years without payments. Thus \(k=3\), and the perpetuity value is first calculated at time 3.

At time 3, the value is:

$$ \frac{1{,}800}{0.05}=36{,}000. $$

The three-year accumulation factor is:

$$ (1.05)^3=1.157625, \qquad v^3=\frac{1}{1.157625}\approx0.8638375985. $$

Discounting the time-3 value gives:

$$ PV_0=36{,}000v^3 =\frac{36{,}000}{1.157625} \approx \$31{,}098.15. $$

Check. The first payment is $1,800 at time 4, so its present value is \(1{,}800v^4\). The full stream is \(1{,}800v^4/(1-v)\). Since \(1-v=0.05v\), this equals \(1{,}800v^3/0.05=36{,}000v^3\), the same value. The check confirms that the first payment is at year 4, while the perpetuity value is located at year 3.

Worked Example: Using the Deferral Date as an Intermediate Valuation Point

A hypothetical endowment pays $1,000 annually forever, with the first payment at the end of year 4. The annual effective interest rate is 25%. Find the present value at time 0, and verify it from the payment series.

Method. Three complete years have no payments, so \(k=3\). Value the perpetuity at time 3, then discount three years.

The value at time 3 is:

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

At 25%, \(v=1/1.25=0.8\), and \(v^3=(0.8)^3=0.512\). Therefore:

$$ PV_0=4{,}000(0.512)=\$2{,}048.00. $$

Check. The payments begin at time 4, so the direct series is \(1{,}000v^4/(1-v)\). With \(v=0.8\), this is \(1{,}000(0.8)^4/(0.2)=1{,}000(0.4096)/0.2=\$2{,}048.00\). The two methods agree.

Common Exam Traps

  • Discounting one period too many. With \(k\) full years of deferral, the value \(R/i\) is at time \(k\), not time \(k+1\). Apply \(v^k\), even though the first payment occurs at time \(k+1\).
  • Using the first payment date as the deferral length. If the first payment is at time 6, then \(k=5\), not 6.
  • Confusing \(1-v\) with \(i\). Since \(v=1/(1+i)\), \(1-v=i/(1+i)=iv\). For an 8% rate, \(1-v\approx0.0740741\), not \(0.08\).
  • Using a perpetuity formula outside its assumptions. The formula here requires level annual payments, a constant positive annual effective rate, and payments continuing forever. A finite stream or a changing payment amount is a different cash-flow pattern.
Key takeaway: For an annual perpetuity-immediate deferred \(k\) full years, identify the first payment at time \(k+1\), value the perpetuity at time \(k\) as \(R/i\), and discount that value by \(v^k\).

Check Your Understanding

For each problem, assume a constant annual effective interest rate and level annual payments continuing forever. Round dollar values to the nearest cent.

  1. A perpetuity pays $1,500 at each year-end, with its first payment at the end of year 3. The annual effective rate is 5%. Find its value at time 0. (Answer: $27,210.88)
  2. A perpetuity pays $2,400 annually, first at the end of year 6. The annual effective rate is 8%. Find its value at time 0. (Answer: $20,417.50)
  3. A deferred perpetuity pays $900 annually, first at the end of year 4. The annual effective rate is 10%. Find its value at time 0. (Answer: $6,761.08)
  4. A perpetuity pays $1,000 at each year-end, first at the end of year 3. At an annual effective rate of 25%, find the present value and state the number of full years with no payments. (Answer: $2,560.00; 2 full years)
  5. A perpetuity pays $5,000 annually, first at the end of year 2. The annual effective rate is 20%. Find its present value and give the first payment date. (Answer: $20,833.33; time 2)

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