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Exam FM · Interest and Annuity Applications · Tutorial 899 of 998

Deferred Annuities with Intermediate Lump Sums

Learn to value and solve exchanges involving deferred level annuities and intermediate lump sums by moving every cash flow to one focal date.

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What You'll Learn

  • Define the payment dates of a deferred annuity-immediate and choose a useful focal date.
  • Value a deferred level annuity at the end of its deferral period.
  • Accumulate or discount lump sums to the same date as the annuity value.
  • Set up equivalent-value equations with consistent cash-flow signs.
  • Solve for an unknown lump sum or annuity payment and check the result.
  • Avoid off-by-one errors when locating the first annuity payment.

Why a Common Focal Date Matters

An exchange can include a level annuity whose payments begin in the future and one or more lump sums paid before those annuity payments start. The cash flows do not occur at the same time, so they cannot be added directly. The solution is to select a focal date, move every cash flow to that date using the stated interest assumption, and then write an equivalent-value equation.

The focal date is often the end of the deferral period: the deferred annuity can be valued there using the ordinary annuity-immediate factor, while each intermediate lump sum is accumulated or discounted to that same date. Another date, including time 0, is also valid if every cash flow is valued consistently.

Definition: Suppose an annuity-immediate pays \(R\) at the ends of \(n\) consecutive annual periods, with its first payment at time \(m+1\). It is deferred \(m\) years. Under a constant effective annual interest rate \(i\), its value at time \(m\) is \(R a_{\overline{n}|}\), where \(a_{\overline{n}|}=(1-v^n)/i\) and \(v=1/(1+i)\). Its value at time 0 is \(R a_{\overline{n}|}v^m\).

This tutorial assumes annual payments and a constant effective annual rate over the relevant period. A lump sum at time \(t\) is accumulated to a later focal time \(T\) by multiplying it by \((1+i)^{T-t}\), or discounted to an earlier focal time by multiplying it by \(v^{t-T}\). If interest rates vary over time, a single constant-rate annuity factor is not appropriate; the cash flows must instead be valued using the accumulation factors for the applicable periods.

Set Up the Timeline Before the Equation

Use a timeline to distinguish the end of the deferral period from the date of the first payment. If the annuity is deferred \(m\) years, its first payment is at \(m+1\), not at \(m\). The value at time \(m\) is therefore the present value of payments at times \(m+1\) through \(m+n\), exactly as for an ordinary \(n\)-payment annuity viewed from one period before its first payment.

$$ \begin{array}{c|cccccc} \text{Time} & 0 & 1 & \cdots & m & m+1 & \cdots\ m+n\\ \hline \text{Annuity} & 0 & 0 & \cdots & 0 & R & \cdots\ R \end{array} $$

To include intermediate lump sums, assign each cash flow a sign from one consistent perspective. For example, if the question asks for the net value of an investment, receipts may be positive and payments negative. Then move each signed amount to the focal date and add. The equation of value is the statement that the signed equivalent value of the exchange is zero, or that the values of its two sides are equal.

Key result: At the end of a deferral of \(m\) years, the deferred annuity-immediate is worth \(R a_{\overline{n}|}\). A lump sum \(C_t\) paid at time \(t\) has value \(C_t(1+i)^{m-t}\) at that focal date if \(t\le m\). Keep the sign of \(C_t\): a receipt and a payment contribute in opposite directions.

Why the Deferred-Annuity Factor Works

At time \(m\), the first annuity payment is one period away, the second is two periods away, and the final payment is \(n\) periods away. Discounting each payment to time \(m\) gives a geometric series. Summing that series produces the ordinary annuity-immediate factor.

$$ \begin{aligned} V_m &=Rv+Rv^2+\cdots+Rv^n\\ &=R\left(\frac{v(1-v^n)}{1-v}\right)\\ &=R\left(\frac{1-v^n}{i}\right) =R a_{\overline{n}|}, \end{aligned} $$

The last equality uses \(1-v=i/(1+i)=iv\), so \(v/(1-v)=1/i\). To move the value at time \(m\) back to time 0, discount it \(m\) years, giving \(R a_{\overline{n}|}v^m\). The annuity factor does not include the deferral period; the factor \(v^m\) accounts for it when valuing at time 0.

Worked Example: Net Value with an Intermediate Payment

An investor receives \(900\) at the end of each year for four years, with the first receipt at time 3. The investor pays \(500\) at time 1. At an effective annual interest rate of \(5\%\), find the net present value at time 0, from the investor’s perspective. Round the final amount to the nearest cent.

Method. The annuity is deferred two years, so use time 2 as the focal date. The annuity payments occur at times 3, 4, 5, and 6. The time-1 payment is an outflow, so it has a negative sign.

First find the annuity-immediate factor for four payments:

$$ v=\frac{1}{1.05}=0.9523809524,\qquad a_{\overline{4}|}=\frac{1-v^4}{0.05} =\frac{1-0.8227024748}{0.05} =3.545950504. $$

At time 2, the annuity is worth \(900(3.545950504)=3{,}191.355454\). The payment at time 1 accumulates one year to time 2 and is worth \(-500(1.05)=-525\). Therefore, the net value at time 2 is \(3{,}191.355454-525=2{,}666.355454\). Discount two years to time 0:

$$ 2{,}666.355454v^2 =2{,}666.355454(0.9070294785) =2{,}418.462997. $$

The net present value is \(2{,}418.46\). As a check, valuing each cash flow directly at time 0 gives \(900(v^3+v^4+v^5+v^6)-500v\), which is the same equivalent value. The common mistake avoided here is treating the time-1 payment as though it occurred at time 2 without accumulating it first.

Solving for an Intermediate Lump Sum

When an unknown lump sum is paid during the deferral period, the same focal-date method isolates it in the equivalent-value equation. The lump sum must be moved to the focal date before being compared with the deferred annuity. Its timing determines whether it is accumulated or discounted.

Worked Example: Finding a Lump Sum at the Deferral Boundary

A fund receives \(2{,}500\) at time 0 and then pays \(800\) at the end of each year for four years, beginning at time 3. At an effective annual interest rate of \(4\%\), an additional amount \(L\) is received at time 2 so that the fund’s receipts and payments have equal value. Find \(L\). Round to the nearest cent.

Method. Use time 2 as the focal date. Accumulate the initial receipt two years. The annuity’s first payment is one year after the focal date, so its value there is \(800a_{\overline{4}|}\). Since \(L\) is received at the focal date, it is added without accumulation.

$$ \begin{aligned} 2{,}500(1.04)^2+L&=800a_{\overline{4}|},\\ 2{,}500(1.0816)+L &=800\left(\frac{1-(1.04)^{-4}}{0.04}\right). \end{aligned} $$

Calculate the annuity factor:

$$ (1.04)^4=1.16985856,\qquad (1.04)^{-4}=0.854804191, $$ $$ a_{\overline{4}|}=\frac{1-0.854804191}{0.04} =3.629895225. $$

The initial receipt accumulated to time 2 is \(2{,}500(1.0816)=2{,}704\). The annuity value at time 2 is \(800(3.629895225)=2{,}903.916180\). Thus

$$ L=2{,}903.916180-2{,}704=199.916180. $$

The required receipt is \(L=199.92\). Substitution checks the equation: \(2{,}704+199.916180=2{,}903.916180\). The key timing point is that the time-2 lump sum is already at the focal date; discounting it or accumulating it would be an error.

Several Lump Sums and Signed Cash Flows

With multiple intermediate amounts, value each one separately at the chosen focal date. This avoids losing track of different dates and makes the signs explicit. A receipt before the focal date is accumulated; a payment before that date is also accumulated, but remains negative from the chosen perspective.

1
Mark the payment dates.
Write the annuity payment dates and every lump-sum date on a timeline. Identify the first annuity payment, not just the stated deferral length.
2
Choose a focal date.
The end of the deferral period is often convenient because the annuity value there is \(R a_{\overline{n}|}\).
3
Move every amount to that date.
Use the appropriate accumulation or discount factor, retaining the amount’s sign.
4
Write and solve the value equation.
Set the signed equivalent values equal, then check the result by substitution or by using a second focal date.

Worked Example: Annuity Payment with Two Intermediate Lump Sums

At time 0, an account receives \(1{,}500\). It also receives \(200\) at time 1 and pays \(300\) at time 2. The account then pays the same amount \(R\) at times 3, 4, and 5. At an effective annual interest rate of \(8\%\), find \(R\) so that the account’s receipts and payments have equal value. Round the final payment to the nearest cent.

Method. Take time 2 as the focal date. The initial receipt accumulates for two years, the time-1 receipt accumulates for one year, and the time-2 payment is already at the focal date. The annuity has three payments, beginning one year after time 2.

$$ 1{,}500(1.08)^2+200(1.08)-300 =R a_{\overline{3}|}. $$

The left-hand side is \(1{,}500(1.1664)+216-300=1{,}749.60+216-300=1{,}665.60\). Next, calculate the annuity factor:

$$ v=\frac{1}{1.08}=0.925925926,\qquad v^2=0.857338820,\qquad v^3=0.793832240, $$ $$ a_{\overline{3}|}=v+v^2+v^3 =0.925925926+0.857338820+0.793832240 =2.577096986. $$

Therefore,

$$ R=\frac{1{,}665.60}{2.577096986} =646.3086\ldots $$

The annual payment is \(R=646.31\). Checking before rounding gives \(646.3086\ldots(2.577096986)=1{,}665.60\), matching the net value of the earlier receipts and payment at time 2. The \(300\) payment reduces the amount available for the annuity; treating it as a receipt would give the wrong direction of adjustment.

Common Exam Traps

  • Moving the first annuity payment to the wrong date. A two-year deferral of an annuity-immediate means the first payment is at time 3, not time 2. At time 2, use \(a_{\overline{n}|}\), not an annuity-due factor.
  • Applying the deferral twice or not at all. If the focal date is the end of the deferral period, use \(R a_{\overline{n}|}\) there; do not also multiply by \(v^m\). If the focal date is time 0, then \(R a_{\overline{n}|}v^m\) is appropriate.
  • Adding amounts from different dates directly. A lump sum at time \(t\) is not numerically interchangeable with its value at time \(m\). Accumulate or discount it first.
  • Ignoring the cash-flow perspective. Decide whether receipts are positive or negative, then keep that convention throughout. A payment does not become positive merely because it is part of the same transaction.
Key takeaway: A deferred level annuity and intermediate lump sums can be handled in one equivalent-value equation. Define the timeline, choose one focal date, move every signed cash flow to it, and verify that the annuity factor’s first payment is exactly one period after the date at which \(a_{\overline{n}|}\) is applied.

Check Your Understanding

For each problem, use annual effective interest and round any requested dollar amount to the nearest cent.

  1. At \(5\%\), an investor receives \(400\) at each year-end from time 3 through time 5 and pays \(250\) at time 1. Find the net present value at time 0, taking receipts as positive. (Answer: \(749.93\))
  2. At \(5\%\), an account receives \(2{,}000\) at time 0 and an unknown amount \(L\) at time 2, then pays \(600\) at each year-end from time 3 through time 7. Find \(L\) so the receipts and payments are equivalent. (Answer: \(392.69\))
  3. At \(8\%\), a fund receives \(1{,}000\) at time 0 and \(120\) at time 1, pays \(250\) at time 2, and then pays \(R\) at times 3 and 4. Find \(R\) so that receipts and payments are equivalent. (Answer: \(586.56\))
  4. At \(4\%\), an annuity pays \(500\) at times 3, 4, 5, and 6, while a separate payment of \(300\) is made at time 2. Find the net value at time 2, taking the annuity receipts as positive. (Answer: \(1{,}514.95\))

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