Why Counting Payments Requires a Timeline
A deferral describes when a payment stream starts; it does not, by itself, tell you whether the first payment occurs at the start or the end of the first payment period. That distinction separates an annuity-due from an annuity-immediate. It also determines whether a payment falls exactly on the date the deferral ends.
For a deferred annuity, first identify the length of one payment period, the number of whole periods deferred, and the number of payments. Then place each payment on a timeline. This tutorial focuses on counting and timing those payments, not on valuing them. No interest rate is needed to determine payment dates.
Use a time unit that matches the payment frequency. For annual payments, one period is one year; for quarterly payments, one period is one quarter. Let \(m\) be the number of whole periods deferred and \(n\) the number of payments, where \(m\) is a nonnegative integer and \(n\) is a positive integer.
The entries are payment dates measured in payment periods from time 0. Each list contains \(n\) dates. The immediate stream’s last payment is at time \(m+n\); the due stream’s last payment is at time \(m+n-1\). In both cases the payments cover \(n\) payment periods in total, but the payment dates sit at different ends of those periods.
What Happens at the Deferral Boundary?
The boundary date is time \(m\): the end of the \(m\)-period deferral and the start of the payment phase. With the standard convention, a due annuity has its first payment exactly at that boundary. An immediate annuity has no payment at the boundary; its first payment is one full payment period later.
A due payment at time \(m\) is not an extra payment in addition to the \(n\) payments. It is the first payment in the stream. Likewise, a payment at the end of the final period is counted if the annuity is immediate, but not if the annuity is due. A timeline prevents both errors.
Translate the stated frequency into one unit, such as one year or one quarter. Express both the deferral and the payment term in that unit.
Place time 0 at the valuation or reference date and mark time \(m\), the end of the deferral.
For an immediate stream, the first payment is one period after the boundary. For a due stream, it is at the boundary.
List the payment dates, count exactly \(n\), and verify that the last date is \(m+n\) for immediate or \(m+n-1\) for due.
Why the Date Rules Hold
An ordinary annuity-immediate with \(n\) payments has payment dates \(1,2,\ldots,n\). Shifting the entire stream forward by \(m\) periods adds \(m\) to each date, giving \(m+1,m+2,\ldots,m+n\). An ordinary annuity-due has dates \(0,1,\ldots,n-1\). Shifting those dates forward by \(m\) gives \(m,m+1,\ldots,m+n-1\).
This shift explains both the payment count and the boundary behavior. Adding the same \(m\) to each date changes when payments occur, but it does not change how many entries are in the list. For a due annuity, the original time-0 payment moves to time \(m\), right at the boundary. For an immediate annuity, the original time-1 payment moves to time \(m+1\), one full period after the boundary.
Worked Example: Six Annual Payments After a Four-Year Deferral
A hypothetical contract makes six equal annual payments after a four-year deferral. In case A, payments are made at the end of each payment year. In case B, payments are made at the beginning of each payment year. Find the payment dates in both cases. Assume the payment interval is one year and dates are measured from time 0. No interest rate or mortality assumption is needed to count dates.
Method: set the boundary, then build each schedule. The deferral is \(m=4\) years, so the boundary is at time 4. The number of payments is \(n=6\).
Case A: annuity-immediate. The first payment occurs one year after the boundary, at time \(4+1=5\). Add one year at a time for six payments:
There are six dates. The last date is \(m+n=4+6=10\), which agrees with the list. These are payments at the ends of years 5 through 10.
Case B: annuity-due. The first payment occurs at the boundary, time 4. The six dates are:
There are six dates. The last date is \(m+n-1=4+6-1=9\), as required. The payment at time 4 is the first payment, not a seventh payment. The common mistake this method avoids is treating the due payment at the boundary as an extra payment before the six-payment sequence begins.
Payment Dates and Payment-Period Endpoints
For an annuity-due, payments occur at the beginnings of the payment periods they relate to. Therefore, the last payment date is one period before the end of the final period covered by the stream. For an annuity-immediate, payments occur at the ends of those periods, so the final payment date coincides with the end of the final period.
This distinction matters when a question describes both a payment term and a final date. For example, a two-year annuity-due with annual payments has payments at the start of year 1 and the start of year 2. Its two-year coverage ends at time 2, even though it has no payment at time 2. A two-year annuity-immediate pays at times 1 and 2.
Worked Example: A Deferred Two-Year Annuity-Due
A hypothetical maintenance plan has a three-year payment-free deferral, followed by a two-year annuity-due with one payment per year. The first payment is made when the payment phase begins. List the payment dates and state when the final covered year ends. Assume annual periods and certain payments; no interest rate is needed.
Method: distinguish the boundary payment from the end of the payment term. Here \(m=3\) and \(n=2\). The deferral boundary is time 3.
Because the stream is due, the first payment occurs at time \(m=3\). There are two payments, one at the beginning of each of the two payment years:
The number of dates is two, as specified. The last payment date is \(m+n-1=3+2-1=4\). That payment is at the beginning of the second covered year, so the coverage ends one year later, at time \(4+1=5\). The payment at time 3 lies on the deferral boundary; it is not inside the three completed deferral years. The common mistake avoided here is calling time 5 the date of the last due payment instead of the end of the final covered period.
Changing the Payment Frequency
The same counting rules apply when payments are more frequent than once a year, provided the deferral and term are converted to the same payment-period unit. For quarterly payments, count quarters rather than years. Keep the time unit consistent in the date list; convert to years only after the payment dates have been established.
For instance, a deferral of 18 months is six quarters, not 18 payment periods, when payments are quarterly. A two-year quarterly term contains eight payment periods and therefore eight payments. The stream type determines whether its first payment is at the six-quarter boundary or one quarter later.
Worked Example: Quarterly Payments After an Eighteen-Month Deferral
A hypothetical benefit makes quarterly payments for two years after an 18-month deferral. Payments are made at the beginning of each payment quarter. Find the number and timing of the payments, in quarters and in years. Assume 12 months per year, four quarters per year, and certain payments. No interest rate is needed.
Method: convert everything to quarters before counting. The deferral is \(18/3=6\) quarters because each quarter is three months. The two-year payment term is \(2\times4=8\) quarters, so \(m=6\) and \(n=8\).
The payments are due at the beginning of each quarter. The first is at time 6 quarters, and the eighth is at time \(6+8-1=13\) quarters. Listing all eight dates gives:
Dividing each date by four gives the dates in years:
There are eight payments, one at the start of each quarter in the two-year payment phase. That phase ends at \(6+8=14\) quarters, or 3.50 years. The final due payment at 13 quarters occurs at the start of the last covered quarter, not at its end. The common mistake this method avoids is counting the 18-month deferral as 18 quarters or placing the first due payment one quarter after the boundary.
Reading Ambiguous Timing Descriptions
Words such as “after,” “at the end of,” and “beginning in year” can be ambiguous if a payment frequency and an exact first date are not stated. Do not resolve ambiguity by relying on the word “deferred” alone. Translate the description into an explicit first payment date, then identify the stream type and check the remaining dates.
- “Deferred \(m\) periods, annuity-immediate” uses the standard shifted schedule, with the first payment at \(m+1\).
- “Deferred \(m\) periods, annuity-due” uses the standard shifted schedule, with the first payment at \(m\).
- “The first payment is at time \(m\)” states the boundary date directly; for a regular stream beginning there, this is due timing.
- “The first payment is one period after time \(m\)” states immediate timing.
If a contract explicitly says that no payment can occur through and including time \(m\), a payment at time \(m\) is excluded. That wording differs from the standard whole-period deferral convention used here, under which a due payment may occur at the boundary. On a problem that specifies an exact first date, follow that date rather than imposing an unstated convention.
Worked Example: The Boundary Date Is Stated Explicitly
A hypothetical annual payment stream lasts three years after a two-year waiting period. The contract says the first payment is made at the end of the first payment year, with later payments one year apart. Determine its type, dates, and payment count. Assume the stated first-payment date controls; no interest rate is needed.
Method: use the exact first date, then count the remaining regular payments. The waiting-period boundary is time 2. The first payment is at the end of the first payment year after that boundary, so its date is time \(2+1=3\). A first payment one period after the boundary is annuity-immediate timing.
The stream lasts three annual payment periods, so the three dates are time 3, time 4, and time 5. There are three payments, and the last date is \(m+n=2+3=5\), confirming the schedule. A common mistake is to see a two-year waiting period and place the first payment at time 2 automatically. That would be appropriate for a due stream under the standard convention, but it contradicts the explicit end-of-first-payment-year wording here.
Common Counting Traps
- Counting periods instead of payments. A stream with \(n\) payment periods has \(n\) payments, not \(n+1\). The boundary date may be included as a due payment, but it is still only one of those \(n\) payments.
- Using the wrong endpoint. The last immediate payment is at \(m+n\); the last due payment is at \(m+n-1\). The due stream’s final covered period ends at \(m+n\).
- Mixing time units. A quarterly schedule must use quarter counts throughout until the dates are deliberately converted to years or months.
- Assuming “deferred” always means the first payment is after the boundary. Under standard convention, a deferred annuity-due pays at the boundary. Read the stream type or the explicitly stated first payment date.
Check Your Understanding
For each problem, assume regular payments, whole payment periods, and the standard deferral convention unless the wording gives an exact first date.
- A hypothetical annuity-immediate is deferred for three years and makes four annual payments. Give the payment dates in years. (Answer: Times 4, 5, 6, and 7; four payments.)
- A hypothetical annuity-due is deferred for six annual periods and makes three payments. Give the first and last payment dates and the total count. (Answer: First at time 6, last at time 8; three payments.)
- A hypothetical annuity-immediate makes monthly payments for one year after a nine-month deferral. Give the payment dates in months. (Answer: Months 10 through 21 inclusive; 12 payments.)
- A hypothetical quarterly annuity-due is deferred for five quarters and lasts three years. Give the first and last payment dates in years, the number of payments, and the end of the final covered period. (Answer: First at 1.25 years, last at 4.00 years, 12 payments; the final covered period ends at 4.25 years.)
- Two hypothetical annual streams are each deferred for two years and each make three payments. One is immediate and the other is due. Compare their payment dates and state which one pays at the boundary. (Answer: Immediate at times 3, 4, and 5; due at times 2, 3, and 4; the due stream pays at the time-2 boundary.)