Identifying Timing in Payment Descriptions

Why Payment Timing Matters

Two payment streams can have the same payment amount and the same number of payments but different values because the payments occur on different dates. A payment today is not discounted; a payment one period from today is discounted once. Correctly interpreting phrases such as “in arrears,” “in advance,” and “first payment today” is therefore part of setting up an annuity calculation, not a minor wording detail.

This tutorial focuses on translating those phrases into payment dates and choosing between an annuity-immediate and an annuity-due. It is aligned with the SOA Exam FM (Financial Mathematics) syllabus. Assume level payments at regular intervals unless the description says otherwise.

Definition: An annuity-immediate with $n$ payments has payments at the ends of $n$ consecutive periods: times $1,2,\ldots,n$, measured from the valuation date. An annuity-due with $n$ payments has payments at the beginnings of those periods: times $0,1,\ldots,n-1$. The first payment of an annuity-due is made at the valuation date.

Translate the Wording into Payment Dates

Start by identifying the valuation date and the length of one payment period. Then write down the date of the first payment and count the remaining payments at the stated regular interval. A timeline is a reliable way to keep the first payment and the total number of payments distinct.

Wording	First payment date	Usual classification
At the end of each period; in arrears	End of period 1	Annuity-immediate
At the beginning of each period; in advance	Beginning of period 1	Annuity-due
First payment today, then once each period	Time 0	Annuity-due
First payment in one period, then once each period	Time 1	Annuity-immediate

“In arrears” means that a payment is made after the period of service, use, or coverage has elapsed. For a regular stream valued at time 0, that places the first payment at time 1. “In advance” means that payment is made at the start of the period, so the first payment is at time 0. The phrase “first payment today” also fixes the first date at time 0, regardless of whether the description uses the word “advance.”

The classification depends on dates, not on the payment’s label or purpose. Rent paid at the beginning of each month and a benefit paid immediately and then monthly have due timing if their first payment is today. A bill paid after each month of service has elapsed has immediate timing if its first payment is one month from today.

Worked Example: Payments Made in Arrears

A hypothetical fund pays $1,000 at the end of each year for two years. The effective annual interest rate is 25%. Find the present value at the start of the first year, and classify the payment stream.

Method: locate the dates before selecting the factor. “At the end of each year” means the payments occur at times 1 and 2. There are two payments, and the stream is an annuity-immediate. The discount factor per year is $v=1/(1+i)=1/1.25=0.8$.

The time-1 payment has present value $1000v=1000(0.8)=\$800$. The time-2 payment has present value $1000v^2=1000(0.8)^2=1000(0.64)=\$640$. Therefore,

\text{Present value}=800+640=\$1{,}440.00.

Equivalently, $a_{\overline{2}|}=(1-v^2)/i=(1-0.64)/0.25=1.44$, so the value is $1000(1.44)=\$1{,}440.00$. The check is that both payments are discounted, once and twice respectively. The common mistake to avoid is treating “end of each year” as a payment today; that would move the first payment one full year too early.

Worked Example: First Payment Today

A hypothetical grant pays $500 immediately and then $500 at the end of each of the next two years. The effective annual rate is 25%. Find the present value and classify the three-payment stream.

Method: count actual payment dates. The payments occur at times 0, 1, and 2. Because the first payment is today, the stream is an annuity-due with three payments. At this rate, $v=1/1.25=0.8$.

The payment at time 0 is worth $500 today. The time-1 payment is worth $500(0.8)=\$400$. The time-2 payment is worth $500(0.8)^2=500(0.64)=\$320$. Adding the three present values gives

\text{Present value}=500+400+320=\$1{,}220.00.

As a factor check, $ \ddot{a}_{\overline{3}|}=1+v+v^2=1+0.8+0.64=2.44$, and $500(2.44)=\$1{,}220.00$. The first payment is not discounted because it occurs at the valuation date. Avoid counting the immediate payment as an extra payment in addition to the three stated payments: it is the first of the three.

Choosing and Justifying the Annuity Factor

Let $R$ be a level payment made once per period, let $i$ be the effective interest rate per payment period, and let $v=1/(1+i)$. The formulas below give present values at time 0. They require regular payment spacing, a fixed payment amount, and the same effective rate for each payment period. If the stated rate is not already effective per payment period, convert it before applying the factor.

Present-value factors: For $n$ payments of $R$, an annuity-immediate has present value $R a_{\overline{n}|}$, where $a_{\overline{n}|}=(1-v^n)/i$. An annuity-due has present value $R\ddot{a}_{\overline{n}|}$, where $\ddot{a}_{\overline{n}|}=1+v+\cdots+v^{n-1}$. These formulas assume $i\ne 0$; when $i=0$, each payment has present value $R$, so either stream has present value $nR$.

The difference between the factors follows directly from the dates. For an immediate annuity, the payment at time $k$ is discounted $k$ periods, so its present value is $Rv^k$. Summing those values for $k=1,\ldots,n$ gives

R(v+v^2+\cdots+v^n) =R\frac{v(1-v^n)}{1-v} =R\frac{1-v^n}{i} =R a_{\overline{n}|},

because $1-v=i/(1+i)=iv$. For an annuity-due, the payment at time $k$ is discounted $k$ periods for $k=0,\ldots,n-1$. Its present value is

R(1+v+\cdots+v^{n-1})=R\ddot{a}_{\overline{n}|}.

The due stream is also the same sequence as an immediate stream shifted one period earlier. Thus its present value is $(1+i)$ times the immediate-annuity value with the same $R$ and $n$:

\ddot{a}_{\overline{n}|}=(1+i)a_{\overline{n}|}.

This identity is a useful check, but it does not decide which factor applies. The payment dates decide that. In particular, do not use the identity to change the number of payments: a due annuity with $n$ payments has dates $0$ through $n-1$, not $0$ through $n$.

Worked Example: “In Advance” with a Monthly Rate

A hypothetical equipment lease requires three payments of $400, one at the beginning of each month, with the first payment today. The nominal annual rate is $i^{(12)}=12\%$, compounded monthly. Find the present value today.

Method: convert to the payment-period rate, then value the dated payments. The monthly effective rate is $i=0.12/12=0.01$, or 1% per month. The three payment dates are month 0, month 1, and month 2. This is a three-payment annuity-due.

The discount factors for months 0, 1, and 2 are $1$, $1/1.01=0.9900990099$, and $1/(1.01)^2=1/1.0201=0.9802960494$, respectively. The present values of the payments are therefore $400.00, approximately $396.039604, and approximately $392.118420. Adding the unrounded values,

400\left(1+0.9900990099+0.9802960494\right) =400(2.9703950593) \approx \$1{,}188.16.

The answer is rounded to the nearest cent. A factor check gives $\ddot{a}_{\overline{3}|}=1+v+v^2=2.9703950593$, consistent with the dated-cash-flow sum. The common mistake to avoid is using an annual rate of 12% for each monthly period; the contract specifies monthly compounding, so the effective monthly rate is 1%.

A Reliable Timing Procedure

When the wording is dense, use the following sequence. It separates the interpretation step from the valuation step, reducing the chance that a familiar formula will lead you to assume the wrong first payment date.

Fix the valuation date.
Identify what “today” or the focal date means. Time 0 is that date.

Identify the payment interval.
Determine whether payments are annual, quarterly, monthly, or at another regular interval.

Translate the first-payment phrase.
“Today” or “at the beginning” places the first payment at time 0. “At the end of the first period” or “in arrears” places it at time 1.

Count the payments and write every date.
For $n$ payments, check that the dates are $1,\ldots,n$ for immediate timing or $0,\ldots,n-1$ for due timing.

Use the matching rate and factor.
Use the effective rate per payment interval, then apply $a_{\overline{n}|}$ or $\ddot{a}_{\overline{n}|}$ according to the dates.

This procedure applies to regular immediate and due annuities. If the dates are irregular, if payments are not level, or if the first payment occurs after a nonstandard gap, do not force the cash flows into one of these two basic timelines without accounting for the difference. First state the actual dates; then value each payment at the appropriate number of periods from the focal date.

Common Timing Mistakes

The most frequent error is to recognize the payment amount and number of payments, then select a factor before checking the first payment date. The resulting calculation may be internally correct for a different timeline, but it does not answer the stated question.

Worked Example: Correcting a One-Period Shift

A hypothetical account pays $700 at the beginning of each year for three years, with the first payment today. The effective annual interest rate is 10%. A candidate treats the stream as an annuity-immediate and calculates $700a_{\overline{3}|}$. Identify the error and find the correct present value.

Incorrect calculation. With $v=1/1.10$, the immediate-annuity calculation discounts payments at times 1, 2, and 3:

700(v+v^2+v^3) =700(0.9090909091+0.8264462810+0.7513148009) \approx \$1{,}740.80.

Correction. The first payment is today, so the actual dates are 0, 1, and 2. The correct classification is annuity-due. Its present value is

700(1+v+v^2) =700(1+0.9090909091+0.8264462810) \approx \$1{,}914.88.

Both calculations use three payments, but the incorrect one shifts every payment one year later. The difference is exactly the effect of that one-period shift: the correct value is $(1.10)$ times the immediate value before rounding.

“Beginning” does not mean the end of the first period. A beginning-of-period payment occurs before that period’s interest has accrued.
“Today” is a payment date, not just a valuation instruction. If the first payment is today, include its full amount at time 0.
Do not confuse the number of periods with the number of payments. Three due payments span the dates 0, 1, and 2; they do not require a payment at time 3.
Do not mix rate units. For monthly payments, use the effective monthly rate, not an annual rate applied once per month.
Do not infer timing from the type of transaction. Read the stated dates; rent, withdrawals, and benefits can each be structured in advance or in arrears.

Key takeaway: Write the payment dates before choosing a factor. Payments at times $1,\ldots,n$ are an annuity-immediate; payments at times $0,\ldots,n-1$ are an annuity-due. The first date determines the classification.

Check Your Understanding

For each problem, identify the payment timing and use the stated rate per payment period to find the present value at time 0. Round money to the nearest cent.

A hypothetical fund pays $300 at the end of each year for two years. The effective annual rate is 25%. State the classification and present value. (Answer: annuity-immediate; $432.00.)
A hypothetical award pays $250 today and at the beginning of each of the next two years, for three payments total. The effective annual rate is 25%. State the classification and present value. (Answer: annuity-due; $610.00.)
A hypothetical trust pays $1,000 at the end of each quarter for two quarters. The nominal annual rate is $i^{(4)}=8\%$, compounded quarterly. State the classification and present value. (Answer: annuity-immediate; $1,941.56.)
A hypothetical scholarship pays $200 at the beginning of each year for five years, with the first payment today. The effective annual rate is 10%. State the classification and present value. (Answer: annuity-due; $833.97.)
A hypothetical service contract requires four payments of $1,000, one at the beginning of each half-year, with the first payment today. The nominal annual rate is $i^{(2)}=10\%$, compounded semiannually. State the classification and present value. (Answer: annuity-due; $3,723.25.)

What You'll Learn