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Exam FM · Annuities Immediate and Due · Tutorial 126 of 998

Valuation Immediately Before or After a Payment

Locate the focal instant, decide whether the scheduled payment belongs in the value being calculated, and apply the matching annuity factor.

Beginner 9 min read

What You'll Learn

  • Distinguish a value immediately before a payment from one immediately after it.
  • Identify the payments included in the value of an annuity at a payment date.
  • Use annuity-immediate and annuity-due factors for the remaining payments.
  • Separate the value of future payments from an account balance that includes past deposits.
  • Check a result by comparing the two sides of a payment boundary.

Why the Instant Around a Payment Matters

An annuity can have a different value immediately before and immediately after one of its payments, even though no time has passed between those instants. The difference is not caused by interest: it arises because the payment has not yet occurred just before the payment date, but has occurred just after it. A reliable valuation therefore begins by locating the focal instant precisely and stating what is being valued.

There are two common quantities to keep separate. The value of remaining payments includes only payments that have not yet been made. By contrast, an account balance after a deposit includes that deposit along with earlier deposits and their accumulated interest. The same cash-flow schedule can produce different answers to these questions.

Definition: Immediately before a payment at time \(k\) means the instant \(k^-\), just before the payment is made. Immediately after means \(k^+\), just after it is made. For the value of remaining payments, include a payment due at \(k\) in the value at \(k^-\), and exclude it at \(k^+\). For an account balance, include a deposit in the balance at \(k^+\), but not at \(k^-\).

Assume level payments of amount \(R\) occur at regular intervals, and the effective interest rate per payment interval is \(i\). Let \(v=1/(1+i)\). In this tutorial, \(a_{\overline{n}|}\) denotes the present value of \(n\) payments of 1 at the ends of successive intervals, and \(\ddot{a}_{\overline{n}|}\) denotes the present value of \(n\) payments of 1 at the beginnings of successive intervals. These factors use the same interval as the payment schedule.

Values of Remaining Payments at the Boundary

Consider an annuity-immediate with payments at times \(1,2,\ldots,n\). At time \(k\), where \(1\leq k\leq n\), the payment at time \(k\) is due at the boundary. Just before it is paid, the remaining stream consists of that payment and the payments at times \(k+1,\ldots,n\). Just after it is paid, only the later payments remain.

$$ \begin{aligned} V_{k^-} &= R\ddot{a}_{\overline{n-k+1}|},\\ V_{k^+} &= R a_{\overline{n-k}|}. \end{aligned} $$

The second expression is zero when \(k=n\), because there are no future payments after the final payment. The first expression equals \(R\) in that case: immediately before the final payment, the only remaining payment is the one due at that instant.

For an annuity-due with payments at times \(0,1,\ldots,n-1\), the same boundary logic applies. At a payment time \(k\), where \(0\leq k\leq n-1\), there are \(n-k\) payments from time \(k\) onward. Just before the time-\(k\) payment, the remaining-payment value is \(R\ddot{a}_{\overline{n-k}|}\). Just after it, the remaining future payments have value \(R a_{\overline{n-k-1}|}\), with zero value if no payments remain.

Why the Values Differ by One Payment

At \(k^-\) and \(k^+\), the future payments after time \(k\) are the same. Their values at the boundary are unchanged because no interval of interest accrues between the two instants. The only difference in the remaining-payment stream is the payment of \(R\) due at time \(k\): it is included before payment and excluded after payment.

$$ V_{k^-}-V_{k^+}=R $$

For the annuity-immediate formulas, the same result follows from the factor identity \(\ddot{a}_{\overline{m}|}=1+a_{\overline{m-1}|}\), where \(m=n-k+1\). The \(1\) represents the payment due immediately; the remaining \(m-1\) payments form an annuity-immediate.

$$ R\ddot{a}_{\overline{m}|}-R a_{\overline{m-1}|} =R\left(1+a_{\overline{m-1}|}\right)-R a_{\overline{m-1}|} =R $$

This boundary check is useful, but it does not mean every pair of values called “before” and “after” differ by exactly \(R\). That conclusion applies when comparing the value of remaining payments. An account balance has a different definition: just after a deposit, the deposit is added to the accumulated value of earlier deposits.

Worked Examples

Worked Example: An Annuity-Immediate at an Interior Payment

A contract pays $100 at the end of each year for four years. The effective annual interest rate is 5%. Find the value of the remaining payments immediately before and immediately after the second payment. Give answers to the nearest cent.

Method. Count the payments still included at each instant, then value those payments at time 2. The payment at time 2 is included at \(2^-\), but not at \(2^+\).

Immediately before the second payment, the remaining payments are at times 2, 3, and 4. Their values at time 2 are $100, \(100v\), and \(100v^2\), respectively, where \(v=1/1.05=0.95238095\).

$$ \begin{aligned} V_{2^-} &=100+100(0.95238095)+100(0.95238095)^2\\ &=100+95.238095+90.702948\\ &=285.941043\\ &\approx \$285.94. \end{aligned} $$

Immediately after the second payment, only the payments at times 3 and 4 remain. At time 2, those payments are one and two years away.

$$ \begin{aligned} V_{2^+} &=100v+100v^2\\ &=95.238095+90.702948\\ &=185.941043\\ &\approx \$185.94. \end{aligned} $$

The difference is \(285.941043-185.941043=100\), exactly the payment due at time 2. Common mistake avoided: discounting the payment at time 2 by one year when valuing at \(2^-\). It is due at the focal instant, so its value there is $100, not \(100v\).

Worked Example: An Annuity-Due Immediately After a Payment

A four-payment annuity-due pays $250 at times 0, 1, 2, and 3. The effective annual interest rate is 6%. Find (a) the value of remaining payments immediately before the payment at time 2, and (b) the value of remaining payments immediately after it. Give answers to the nearest cent.

At time \(2^-\), the payments still due are the $250 payment at time 2 and the $250 payment at time 3. The first is due immediately; the second is one year away. With \(v=1/1.06=0.94339623\),

$$ \begin{aligned} V_{2^-} &=250+250v\\ &=250+250(0.94339623)\\ &=250+235.849057\\ &=\$485.85. \end{aligned} $$

At time \(2^+\), the payment at time 2 has been made. Only the time-3 payment remains, one year away:

$$ \begin{aligned} V_{2^+} &=250v\\ &=250(0.94339623)\\ &=\$235.85. \end{aligned} $$

The values differ by $250, the payment at the boundary. Common mistake avoided: treating every annuity-due payment as if it occurred at the end of an interval. Here, the time-2 payment is due immediately at the focal date.

Worked Example: Account Balance After a Deposit

A saver deposits $100 at the end of each year for five years into an account earning an effective annual rate of 4%. Find the account balance immediately before and immediately after the third deposit. Assume deposits are made at the stated times and interest is credited over each complete year. Give answers to the nearest cent.

The deposits are at times 1, 2, 3, 4, and 5. Immediately before the deposit at time 3, only the first two deposits are in the account. The time-1 deposit has accumulated for two years and the time-2 deposit for one year:

$$ \begin{aligned} B_{3^-} &=100(1.04)^2+100(1.04)\\ &=108.16+104.00\\ &=\$212.16. \end{aligned} $$

Immediately after the third deposit, add the $100 just deposited:

$$ \begin{aligned} B_{3^+} &=B_{3^-}+100\\ &=212.16+100\\ &=\$312.16. \end{aligned} $$

This is an account balance, not the value of the contract’s future deposits. The fourth and fifth deposits are not yet in the account. Common mistake avoided: using an annuity-immediate factor for the balance after time 3 without checking which deposits have actually been made. The direct accumulation above identifies each included deposit and its exact number of years of interest.

A Reliable Boundary-Valuation Process

A short timeline check prevents most errors. Use it before substituting into an annuity factor, especially when the wording says “immediately before” or “immediately after.”

1
Mark the focal time.
Write the time \(k\) and distinguish \(k^-\) from \(k^+\). Confirm that the interest rate is effective per payment interval.
2
Name the quantity.
Decide whether the question asks for the value of remaining payments or an account balance. These quantities include different cash flows.
3
List the included payments.
For a remaining-payment value, include the boundary payment at \(k^-\) and exclude it at \(k^+\). For a balance, include a deposit at \(k^+\).
4
Count periods and check the result.
Discount or accumulate each payment by its actual number of intervals. For remaining-payment values across one boundary, check that the difference is \(R\).

Common Timing Traps

  • Counting the boundary payment on both sides. A remaining-payment value includes it at \(k^-\), but excludes it at \(k^+\). Do not include it twice in the after-payment value.
  • Omitting the payment due immediately. At \(k^-\), a payment due at \(k\) has zero time to earn or lose interest. Its value at that instant is \(R\).
  • Confusing a remaining stream with a fund balance. After a deposit, a balance includes that deposit. The value of the remaining contractual payments does not include a payment already made.
  • Using the wrong period count. A payment one interval after the focal date is multiplied by \(v\) for a value at the focal date; a payment due at the focal instant is not discounted.
  • Assuming an interest change at the payment instant. Under the standard level-rate model, no interest accrues in the zero time between \(k^-\) and \(k^+\). The change in the remaining-payment value comes from the payment itself.
Key takeaway: State the focal instant, identify whether the requested value is a remaining-payment value or an account balance, and list the payments included before using an annuity factor. At a payment boundary, the remaining-payment values immediately before and after differ by exactly the payment due there.

Check Your Understanding

For each problem, identify the boundary payment before calculating. Answers are rounded to the nearest cent where money is requested.

  1. An annuity-immediate pays $200 at times 1 through 5. The effective annual rate is 4%. Find the value of remaining payments at time \(3^-\). (Answer: $577.22)
  2. An annuity-due pays $300 at times 0 through 4. The effective annual rate is 5%. Find the value of remaining payments at time \(2^+\). (Answer: $557.82)
  3. Deposits of $150 are made at times 1 through 4 into an account earning 4% effective annually. Find the account balance at time \(3^+\). (Answer: $468.24)
  4. An annuity-immediate pays $800 at times 1 through 6, and the effective annual rate is 3%. Find the values of remaining payments at \(4^-\) and \(4^+\). (Answer: $2,330.78 immediately before; $1,530.78 immediately after)
  5. Deposits of $1,000 are made at times 1 through 5 into an account earning 6% effective annually. Find the account balance immediately before and after the final deposit at time 5. (Answer: $4,637.09 immediately before; $5,637.09 immediately after)

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