Reading Deferred Annuity Timelines

Why a Deferred-Annuity Timeline Needs Clear Boundaries

A deferred annuity has a waiting period between the valuation date and the beginning of payments. The key timeline question is whether the first payment occurs at the end of that waiting period or one payment period later. Confusing those dates shifts every payment and can change a value calculation.

Start by setting the valuation date to time zero. Then label the time unit, the end of the deferral period, the payment dates, and the stated number of payments. Keep the deferral endpoint distinct from the first payment date unless the contract’s timing explicitly places a payment there.

Definition: The valuation date is the date at which a stream is being considered, and is labelled $t=0$. The deferral period is the elapsed time from the valuation date to the stated start of the payment period. A payment date is a date on which a payment is actually made; it must be marked separately from a date that only ends the deferral period.

Two Common Payment Timings

For a level annuity with annual payments, an immediate annuity’s first payment is made at the end of its first payment period. A due annuity’s first payment is made at the beginning of its first payment period. These distinctions still apply when an annuity is deferred.

Timeline convention: If an annuity-immediate is deferred for $m$ annual periods and has $n$ payments, its payment dates are $t=m+1,m+2,\ldots,m+n$. If an annuity-due is deferred for $m$ annual periods, with its first payment made at the end of the deferral period, its payment dates are $t=m,m+1,\ldots,m+n-1$. In both cases, $m$ and $n$ describe different things: $m$ is the stated deferral length, and $n$ is the number of payments.

The endpoint $t=m$ is therefore not automatically a payment date. In the immediate case, it marks the end of the wait, and the first payment is one full payment period later. In the due case described above, the first payment is made at that endpoint. If a problem gives actual payment dates or describes payment timing in words, use that information to establish the timeline rather than assuming the two patterns are interchangeable.

\begin{array}{c|cccccc} \text{Time} & 0 & 1 & \cdots & m & m+1 & \cdots \\ \hline \text{Deferred immediate} & \text{valuation} & & & \text{deferral ends} & \text{payment 1} & \cdots \\ \text{Deferred due} & \text{valuation} & & & \text{payment 1} & \text{payment 2} & \cdots \end{array}

The timeline’s numbers are measured in the payment period used to describe the deferral. If payments are annual, $t=4$ means four years after the valuation date. If payments are made on other regular intervals, label that interval explicitly so a time coordinate cannot be mistaken for a payment count.

Worked Example: Three-Year Deferral With Payments in Arrears

A hypothetical contract is valued at time $t=0$. It makes five annual payments of $800, with the first payment at the end of the first annual payment period after a three-year deferral. Place the deferral endpoint and every payment date on a timeline. Assume annual effective interest $i=5\%$; no mortality model is involved.

Method: identify the payment timing, then label dates. The phrase “at the end of the first annual payment period after a three-year deferral” means the payments are in arrears. The three-year deferral ends at $t=3$, but that is not a payment date.

The first payment occurs one year later:

t_{\text{first}}=3+1=4.

There are five annual payments, so the remaining dates follow by adding one year each time: $t=5$, $t=6$, and $t=7$ for payments 2 through 4, followed by $t=8$ for payment 5. The complete timeline is:

\begin{array}{c|ccccccccc} \text{Time} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \text{Event} & \text{valuation} & & & \text{deferral ends} & \$800\ (1) & \$800\ (2) & \$800\ (3) & \$800\ (4) & \$800\ (5) \end{array}

Check. The first payment is one annual period after the deferral endpoint, and the five payment dates run consecutively from $t=4$ through $t=8$. The annual interest rate is not needed to place these dates. The common mistake this method avoids is putting payment 1 at $t=3$ merely because the deferral ends then.

When the First Payment Is at the Deferral Endpoint

Some payment descriptions place the first payment at the end of the waiting period. That is due timing: the first payment date and the deferral endpoint coincide. The timeline should show this coincidence explicitly rather than shifting the first payment forward an extra period.

Worked Example: Payments Beginning on the Deferral-Completion Date

A hypothetical benefit is valued on January 1, 2026. The contract provides four annual payments of $1,250, beginning on January 1, 2028, exactly when a two-year waiting period ends. Further payments occur on each of the next three January 1 dates. Assume annual effective interest $i=4\%$; no mortality model is involved. Mark the valuation date, deferral endpoint, and payment dates.

Method: translate the dates into years from the valuation date. January 1, 2026 is $t=0$. Two years later, January 1, 2028, is $t=2$. Since the first payment is explicitly made on that date, the deferral endpoint is also payment 1.

The remaining dates are one, two, and three years after the first payment, giving $t=3$, $t=4$, and $t=5$. In calendar dates, the stream is January 1, 2028; January 1, 2029; January 1, 2030; and January 1, 2031.

\begin{array}{c|cccccc} \text{Time} & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \text{Event} & \text{Jan. 1, 2026} & & \text{deferral ends; payment 1} & \text{payment 2} & \text{payment 3} & \text{payment 4} \end{array}

Check. Four payment dates are marked, beginning at $t=2$, so the first payment coincides with the end of the two-year deferral. If the first payment were incorrectly placed at $t=3$, the entire stream would be one year late. The method avoids treating every deferred annuity as an annuity-immediate.

Dates, Payment Counts, and Payment Spacing

A timeline should make three separate pieces of information visible: when payments start, how often they occur, and how many there are. The distance from the first payment to the last is not itself the number of payments. For example, five annual payments have four annual intervals between the first and last payment dates.

If a contract states calendar dates, convert each one to elapsed time from the valuation date. If it gives a payment count and a regular payment interval, mark that many payment points at the stated spacing. In either case, label the payments in sequence so that the last date can be checked against the stated count. Do not infer an unstated payment at the deferral endpoint.

Worked Example: Recovering a Timeline From Calendar Dates

A hypothetical arrangement is valued on December 31, 2027. It pays $900 on each December 31 from 2030 through 2034, inclusive. Assume annual effective interest $i=6\%$; no mortality model is involved. State the deferral endpoint, the first and last payment times, and the payment count.

Method: use elapsed calendar years and mark every payment. The valuation date is December 31, 2027, or $t=0$. The dates December 31, 2028, and December 31, 2029 are one and two years later. The first stated payment date, December 31, 2030, is three years later, so it is at $t=3$.

The subsequent payment dates are December 31, 2031, 2032, 2033, and 2034, at $t=4,5,6,7$. Thus there are five payments, from $t=3$ through $t=7$. The interval between first and last is $7-3=4$ years, which is one fewer than the number of payments because the first date is itself a payment.

\begin{array}{c|cccccccc} \text{Time} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \text{Event} & 2027 & 2028 & 2029 & 2030\ (1) & 2031\ (2) & 2032\ (3) & 2033\ (4) & 2034\ (5) \end{array}

Check. The timeline shows five payment labels and the last date agrees with the stated inclusive range. It also shows that the first payment is exactly three years after the valuation date. The common mistake avoided is counting the four gaps between the first and last payment as if there were only four payments.

Why a One-Period Shift Changes a Value

A timeline is not a value calculation, but it determines which time exponent belongs to each payment. Under a constant effective annual interest rate $i$, a payment $C$ made at time $t$ has value $Cv^t$ at time zero, where $v=1/(1+i)$. This follows by discounting the payment once for each of the $t$ annual periods: after one period its value is $Cv$, after two periods $Cv^2$, and after $t$ periods $Cv^t$. Values of separate payments are then added.

\text{Value at }t=0=\sum_j C_jv^{t_j}, \qquad v=\frac{1}{1+i}.

This relationship assumes annual payment dates expressed in years and a constant annual effective rate over the relevant periods. If the interest rate varies by period, the appropriate period-specific discount factors are needed instead. The point here is that the timeline supplies the $t_j$ values; moving a payment by one period changes its discounting.

Common Timing Trap: Treating the Deferral Endpoint as Payment 1

Suppose a four-payment annuity-immediate is deferred for three years. The correct payment times are $t=4,5,6,7$. At $i=10\%$, the value at time zero is:

1000(v^4+v^5+v^6+v^7) =1000(0.683013+0.620921+0.564474+0.513158) =\$2{,}381.57, \qquad v=\frac{1}{1.10}.

The wrong timeline puts the first payment at $t=3$, giving payment times $t=3,4,5,6$. That produces:

1000(v^3+v^4+v^5+v^6) =1000(0.751315+0.683013+0.620921+0.564474) =\$2{,}619.72.

Both calculations use four payments and the same rate, but the incorrect timeline places each payment one year too early. The corrected timeline is $t=4,5,6,7$ because three complete deferral years end at $t=3$, and an annuity-immediate’s first payment follows one more annual payment period later. Values are rounded to the nearest cent only after summing.

Key takeaway: Label the valuation date, deferral endpoint, first payment date, payment interval, and stated payment count separately. For an immediate annuity the first payment follows the deferral endpoint by one payment period; for a due annuity it may coincide with that endpoint when the contract specifies payment at the end of the wait.

Check Your Understanding

For each situation, mark the relevant times from the stated valuation date and distinguish the end of the deferral from the payment dates.

A valuation is made at $t=0$. A three-year deferral is followed by three annual payments in arrears. Give the deferral endpoint and payment times. (Answer: Deferral ends at $t=3$; payments are at $t=4,5,6$.)
A valuation date is June 30, 2025. A three-year wait ends on June 30, 2028, when the first of three annual payments is made; two further payments follow on each next June 30. Give the payment dates and times. (Answer: June 30, 2028, 2029, and 2030; $t=3,4,5$.)
A stream is valued on December 31, 2024. It pays six annual amounts on December 31, 2029, and each of the next five years. Give the first and last payment times, the number of payments, and the number of annual intervals from first to last. (Answer: First at $t=5$, last at $t=10$; 6 payments and 5 annual intervals.)
A valuation is made at $t=0$. A four-year deferral ends before the first payment period of a five-payment annuity-immediate, with payments every two years. Give the deferral endpoint and payment times. (Answer: Deferral ends at $t=4$; payments are at $t=6,8,10,12,14$.)
A four-payment annuity-due is valued at $t=0$, with a two-year deferral and the first payment made exactly when the deferral ends. Give the payment times and explain where payment 1 falls. (Answer: Payments are at $t=2,3,4,5$; payment 1 coincides with the deferral endpoint.)

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