Annuities Due and Payment Timing

Recognizing an Annuity Due

A payment stream’s label depends on when payments occur, not on what they are for or how often the payment amount is quoted. A stream with equal payments at the beginning of successive periods is an annuity due. Moving each payment to the beginning rather than the end of its period changes how long it can earn interest and how many periods it must be discounted.

First mark the valuation date and the payment dates. If the first payment occurs immediately at time zero, the stream is due. If the first payment occurs one full period later, at time one, it is an ordinary annuity. The distinction matters even when both streams have the same payment amount and number of payments.

Definition: An annuity due is a finite series of equal payments made at the beginnings of consecutive, equally spaced periods. For $n$ annual payments, the payment dates are $0,1,\ldots,n-1$. The present value of $n$ unit payments at these dates, using effective annual interest rate $i$, is denoted $\ddot{a}_{\overline{n}|}$. An ordinary annuity instead pays at times $1,2,\ldots,n$.

The time-zero payment is made on the valuation date; it is not discounted. Each later payment is discounted for the number of periods between its payment date and that valuation date. For a positive effective interest rate, an annuity due has a greater present value than an otherwise identical ordinary annuity, because each corresponding payment arrives one period earlier.

Present Value from the Payment Dates

Suppose the valuation date is time zero, the effective annual interest rate is $i$, and $n$ equal payments of $R$ are made annually in advance. Set $v=1/(1+i)$. The payment at time $k$ is discounted by $v^k$, so the present value is the sum of the present values of the individual payments:

\operatorname{PV}_{\text{due}} =R+Rv+Rv^2+\cdots+Rv^{n-1} =R\sum_{k=0}^{n-1}v^k =R\ddot{a}_{\overline{n}|}.

This expression is a direct valuation from the dates. It also explains why the first payment is not multiplied by a discount factor: at time zero, its discount exponent is zero and $v^0=1$. For an ordinary annuity, the payment dates are $1$ through $n$, so each payment has one additional period of discounting.

For accumulated value, choose a future date and compound every payment forward to that date. If the same $n$ payments occur at times $0$ through $n-1$, and the target date is time $n$, then the payment made at time $k$ earns interest for $n-k$ periods:

\operatorname{AV}_{\text{due at }n} =R(1+i)^n+R(1+i)^{n-1}+\cdots+R(1+i) =R\sum_{k=0}^{n-1}(1+i)^{n-k}.

The earliest payment earns interest for the most periods; the last payment, made at time $n-1$, earns interest for one period. This accumulated value is measured at time $n$, one period after the final payment. Always state the target date: the same payment stream can have different accumulated values at different dates.

Worked Example: Present Value of Annual Payments Made in Advance

A hypothetical maintenance contract requires four annual payments of $1,200, with the first payment due immediately and each later payment due one year after the preceding one. Find the present value at time zero. Assume an effective annual interest rate of 5%.

Method. Identify the dates as $0,1,2,3$, then discount each payment from its date to time zero. The rate is annual and the payment spacing is annual, so one period is one year.

The discount factor is $v=1/1.05=0.952381$. The individual discount factors, rounded here to six decimals, are $1.000000$, $0.952381$, $0.907029$, and $0.863838$. Thus:

\begin{aligned} \operatorname{PV} &=1{,}200(1+v+v^2+v^3)\\ &=1{,}200(1+0.952381+0.907029+0.863838)\\ &=1{,}200(3.723248)\\ &=\$4{,}467.90. \end{aligned}

As a check on the timing, if the same four payments instead occurred at times $1,2,3,4$, their present value would be $1{,}200(0.952381+0.907029+0.863838+0.822702)=\$4{,}255.14$. The due stream is worth more at time zero because each payment arrives one year earlier. The common mistake is to discount the first due payment by one year; that incorrectly changes its date from time zero to time one.

Accumulated Value and Timing Comparisons

For a future-value question, the main task is not to discount payments but to carry each payment forward to a common target date. The number of compounding periods is determined separately for every payment. Do not assume that all contributions earn interest for the same duration.

Worked Example: Accumulating Contributions to a Stated Date

A hypothetical equipment reserve receives three annual contributions of $2,000 at times $0,1,2$. Find the accumulated value at time $3$, immediately after one year has passed since the final contribution. Assume an effective annual interest rate of 6%.

Method. Compound each contribution to time $3$. The contribution at time zero earns interest for three years, the contribution at time one for two years, and the contribution at time two for one year.

\begin{aligned} \operatorname{AV}_{3} &=2{,}000\big((1.06)^3+(1.06)^2+(1.06)^1\big)\\ &=2{,}000(1.191016+1.123600+1.060000)\\ &=2{,}000(3.374616)\\ &=\$6{,}749.23. \end{aligned}

As a check, the last contribution has earned exactly one year of interest: $\$2{,}000(1.06)=\$2{,}120$. If instead the target date were time $2$, the final contribution would still be present at that date and would have earned no interest; it would not be correct to report the time-$3$ value. The common mistake is to give the contribution at time two zero years of growth when the question asks for value at time three.

A comparison between payment schedules must put both streams on the same valuation date and use the same rate. Comparing their undiscounted totals alone ignores the value of receiving money earlier. The next example uses present values to compare two schedules with different payment amounts.

Worked Example: Comparing Due and Ordinary Payment Offers

A hypothetical service provider offers either three payments of $2,500 at times $0,1,2$, or three payments of $2,600 at times $1,2,3$. Which schedule has the greater present value at time zero? Assume an effective annual interest rate of 7%.

Method. Value each schedule at time zero using its actual payment dates. For the first schedule, the payments form an annuity due; for the second, they form an ordinary annuity. Set $v=1/1.07=0.934579$, with $v^2=0.873439$ and $v^3=0.816298$.

\begin{aligned} \operatorname{PV}_{\text{due}} &=2{,}500(1+v+v^2)\\ &=2{,}500(1+0.934579+0.873439)\\ &=2{,}500(2.808018)\\ &=\$7{,}020.05, \end{aligned}

\begin{aligned} \operatorname{PV}_{\text{ordinary}} &=2{,}600(v+v^2+v^3)\\ &=2{,}600(0.934579+0.873439+0.816298)\\ &=2{,}600(2.624316)\\ &=\$6{,}823.22. \end{aligned}

The due schedule has the greater present value, by $\$7{,}020.05-\$6{,}823.22=\$196.83$. Although the ordinary schedule has a larger undiscounted total ($\$7{,}800$, compared with $\$7{,}500$), its payments occur later. The common mistake is to select the larger total without accounting for payment timing.

Why the Timing Change Matters

The timing formulas follow from the basic single-payment rule: a dollar at time $k$ has present value $v^k$ dollars at time zero, and a dollar at time $k$ grows to $(1+i)^{n-k}$ dollars at time $n$. Applying those rules separately to every payment gives the annuity expressions above. This is also a useful derivation check: the exponents must match the actual number of periods between the payment date and the valuation date.

For positive interest rates, moving a payment one period earlier increases its present value because it is discounted for one fewer period. It also increases its accumulated value at a fixed later date because it earns interest for one additional period. For example, a payment $R$ due at time $k$ has present value $Rv^k$; moving it to time $k-1$ changes that to $Rv^{k-1}$, which is larger when $i>0$. At a zero rate, timing does not change the dollar value. If the effective rate is negative, the direction of the comparison can reverse, so state the rate assumption before concluding which stream is worth more.

Mark the valuation date.
State the date at which the present or accumulated value is required.

Place each payment on a timeline.
A first payment at time zero indicates a due stream; one at time one indicates an ordinary annuity when periods are annual.

Match the interest period to the payment interval.
Use an effective rate per period consistent with the spacing of the payments.

Value each payment to the common date.
Discount for periods before the valuation date or accumulate for periods after the payment date.

Check the first and last exponents.
For a due present value, the first exponent is zero. For a due accumulation to time $n$, the last payment earns one period.

Annuity-due recognition is especially useful when contracts, contributions, or service fees are described in words rather than with a timeline. Phrases such as “payable immediately and annually thereafter” identify a payment at time zero. Phrases such as “first payment one year from today” indicate an ordinary annuity. Do not infer timing from labels such as “annual premium” alone; the stated due date controls.

Key takeaway: An annuity due has payments at times $0,1,\ldots,n-1$. Value each payment at its actual date: the initial payment is not discounted from time zero, and each payment earns one more period of interest when accumulated to the same future date than its end-of-period counterpart.

Check Your Understanding

For each problem, write down the payment dates before calculating. Round dollar answers to the nearest cent.

A hypothetical subscription requires three annual payments of $400, with the first payment today. Find the present value at 4% effective annual interest. (Answer: $1,154.44)
A reserve receives four contributions of $600 at times $0,1,2,3$. Find its accumulated value at time $4$ at 5% effective annual interest. (Answer: $2,715.38)
Find the present value of three annual payments of $1,000 at times $0,1,2$, using an effective annual interest rate of 8%. (Answer: $2,783.26)
At 6% effective annual interest, compare four payments of $1,500 at times $0,1,2,3$ with four payments of $1,550 at times $1,2,3,4$. Which has the greater present value at time zero, and by how much? (Answer: The due schedule is greater by $138.60; its present value is $5,509.52 versus $5,370.91.)
At 10% effective annual interest, calculate the present values at time zero of two $2,500 payments made at times $0,1$ and two $2,500 payments made at times $1,2$. How much greater is the first value? (Answer: $4,772.73 versus $4,338.84; the first is greater by $433.88.)

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