Valuing Annuities Due With Worked Cash Flows

Reading an Annuity-Due Timeline

A payment made at the beginning of a period is available earlier than a payment made at its end. That timing difference affects its value: at a positive interest rate, a payment received earlier has a greater value at a fixed future date, and a payment made earlier has a greater present value cost. Rent, prepaid service contracts, and periodic deposits can all have this start-of-period pattern.

The first task is not to choose a formula. It is to mark each payment time and the date at which the value is required. In this tutorial, $n$ equal payments of amount $R$ are made at times $0,1,\ldots,n-1$, one at the start of each period. Unless stated otherwise, the effective interest rate $i$ is for one payment period, and payments occur exactly one period apart. Fees, taxes, and payment risk are ignored.

Definition: An annuity due is a finite series of payments made at the beginnings of consecutive periods. For $n$ level payments of $R$, the payment times are $0,1,\ldots,n-1$. Its present value is measured at time $0$; its accumulated value must be tied to a stated future time.

The notation $a_{\overline{n}|}$ denotes the present-value factor for $n$ level payments at the ends of periods, and $s_{\overline{n}|}$ denotes their accumulated-value factor at the end of period $n$. Annuity-due factors are $\ddot{a}_{\overline{n}|}$ and $\ddot{s}_{\overline{n}|}$. The two kinds of timing differ by one period, so the due factor is the ordinary factor multiplied by $1+i$. This adjustment assumes a common effective rate per payment period. If the payment interval changes, the rate must be converted to the effective rate for that interval before using the factors.

Present Value of Level Payments

For a positive effective rate $i$, let $v=1/(1+i)$. The payment at time $k$ is discounted $k$ periods to time $0$. Thus a payment at time 0 is not discounted, the payment at time 1 is discounted once, and so on.

\ddot{a}_{\overline{n}|} =1+v+v^2+\cdots+v^{n-1} =\frac{1-v^n}{1-v} =(1+i)a_{\overline{n}|}, \qquad PV=R\ddot{a}_{\overline{n}|}.

The geometric-series expression is a direct way to check the timeline. If $i=0$, each payment has its original value and $PV=nR$; the quotient expression for the factor is then interpreted by its limit. The formula is not appropriate for irregular payment amounts, unequal time intervals, or an interest rate that is not stated per payment period. In those cases, value each cash flow at its own time.

Worked Example: Present Value of Prepaid Annual Rent

A hypothetical office lease requires four annual rent payments of $2,400, each due at the beginning of the year. The first payment is due immediately. Find the present value at an effective annual interest rate of 6%.

Method and timeline. The payment times are $t=0,1,2,3$, and the requested value is at $t=0$. The annual payment period and the effective annual interest-rate period agree, so $i=0.06$ and $v=1/1.06$.

Value each payment. The time-0 payment is undiscounted. The other payments are discounted one, two, and three years:

PV =2400+\frac{2400}{1.06} +\frac{2400}{1.06^2} +\frac{2400}{1.06^3}.

The individual present values, kept unrounded for the total, are $2,400, $2,264.150943, $2,135.991459, and $2,015.086282. Therefore,

PV=8815.228684\ldots

The annuity-due factor gives the same result: $a_{\overline{4}|}=(1-v^4)/0.06$, and $\ddot{a}_{\overline{4}|}=1.06a_{\overline{4}|}$. Hence $PV=2400\ddot{a}_{\overline{4}|}=8815.228684\ldots$. Rounded to the nearest cent, the present value is $8,815.23.

Check and common mistake avoided. There are four payments, including one at time 0, so the first term must be $2,400, not $2400/1.06$. Treating this as an ordinary annuity would delay every payment by a year and produce a value that is too small by the due-versus-immediate timing adjustment.

Why the Due Adjustment Works

The adjustment follows from the cash-flow times, rather than being an independent rule to memorize. For a level annuity due, the time-0 present value is the sum of $R$ times each payment's discount factor. The ordinary annuity has the same number and amount of payments, but its payment times are $1,2,\ldots,n$. Each of those payments is exactly one period later than its corresponding due payment.

PV_{\text{due}} =R\sum_{k=0}^{n-1}v^k =(1+i)R\sum_{k=1}^{n}v^k =(1+i)PV_{\text{ordinary}},

because $(1+i)v^k=v^{k-1}$. This also explains the factor relationship $\ddot{a}_{\overline{n}|}=(1+i)a_{\overline{n}|}$. The same one-period shift applies to accumulated values, provided both values are measured at dates one period apart.

Accumulated Value and the Required Date

For payments at times $0,1,\ldots,n-1$, there are two common future valuation dates. At time $n$, the payments have accumulated for $n,n-1,\ldots,1$ periods. At time $n-1$, the last payment has not accumulated at all, and the exponents are $n-1,n-2,\ldots,0$. These are different values. The date must be read from the question rather than inferred from the phrase “future value.”

\begin{aligned} AV\text{ at time }n &=R\big((1+i)^n+(1+i)^{n-1}+\cdots+(1+i)\big) =R\ddot{s}_{\overline{n}|} =R(1+i)s_{\overline{n}|},\\ AV\text{ at time }n-1 &=R\big((1+i)^{n-1}+(1+i)^{n-2}+\cdots+1\big) =Rs_{\overline{n}|}. \end{aligned}

These expressions assume that the payments are made as specified and that interest is earned at the same effective rate $i$ for each complete period. To check either result, write one accumulation term for every payment and verify that its exponent equals the number of periods from that payment date to the valuation date.

Worked Example: Deposits Valued at the Last Deposit Date

A hypothetical reserve receives five deposits of $800 at the beginning of each year, starting now. Find its value four years later at an effective annual interest rate of 5%. Also find the value one year after that, to show how the valuation date changes the answer.

Method and timeline. The deposits are made at $t=0,1,2,3,4$. Four years later is $t=4$, which is the time of the fifth and final deposit. The deposits therefore accumulate for $4,3,2,1,0$ years, respectively.

AV_{t=4} =800\big(1.05^4+1.05^3+1.05^2+1.05^1+1.05^0\big).

The accumulation factors are $1.21550625$, $1.157625$, $1.1025$, $1.05$, and $1$. Their sum is $5.52563125$, so the value is $800(5.52563125)=4420.505$. Rounded to the nearest cent, the value at $t=4$ is $4,420.51.

At $t=5$, each deposit earns interest for one additional year. Thus $AV_{t=5}=4420.505(1.05)=4641.53025$, or $4,641.53 to the nearest cent. This agrees with $800\ddot{s}_{\overline{5}|}$, since the five deposits now accumulate for $5,4,3,2,1$ years.

Check and common mistake avoided. The wording asks for the value four years later, so the required date is $t=4$, not $t=5$. Using the annuity-due accumulated value at the end of five periods would answer a different question. The timeline shows that the last deposit has zero years of accumulation at $t=4$.

Irregular Payments Made at Period Beginnings

A cash-flow pattern can have start-of-period timing without having level payments. If payment amounts differ, a single level-annuity factor cannot represent the stream. The reliable procedure is to list each amount, its time, and the number of periods to the valuation date. Then discount or accumulate each cash flow separately and add the results. This method also provides a useful independent check on formula-based calculations.

Mark the payment dates.
Include a payment at time 0 when the first amount is due immediately.

Mark the valuation date.
Do not assume that “after $n$ payments” means the end of period $n$; identify the stated date on the timeline.

Value each cash flow.
Discount a payment by $v^k$ for $k$ periods to the present, or accumulate it by $(1+i)^k$ to a future date.

Add unrounded amounts.
Round the final total to the requested precision, and check that each exponent matches the time between the payment and valuation dates.

Worked Example: Irregular Prepaid Service Payments

A hypothetical three-year equipment service agreement requires payments of $900 at $t=0$, $950 at $t=1$, and $1,050 at $t=2$. Find the present value at an effective annual interest rate of 5%, and the accumulated value at $t=3$.

Method. Because the payment amounts differ, value each payment separately. For the present value at $t=0$, the respective discount periods are 0, 1, and 2:

PV=900+\frac{950}{1.05}+\frac{1050}{1.05^2}.

The three contributions are $900, $904.7619048, and $952.3809524. Their unrounded sum is $2757.1428572\ldots$, so the present value, rounded to the nearest cent, is $2,757.14.

For the accumulated value at $t=3$, the payments accumulate for 3, 2, and 1 years. In particular, the last payment is made at $t=2$, so it earns one year of interest:

AV_{t=3} =900(1.05)^3+950(1.05)^2+1050(1.05).

The accumulated contributions are $1,041.8625, $1,047.375, and $1,102.50. Their sum is $3191.7375$, giving $3,191.74 to the nearest cent. As a check, the present value accumulated for three years is $2757.1428572\ldots(1.05)^3=3191.7375$, the same result.

Common mistake avoided. Applying a level annuity-due factor to these amounts would treat all three payments as equal. Also, accumulating the final payment for zero years would value the stream at $t=2$, not at the requested date $t=3$.

Common Timing and Rate Errors

Annuity-due questions are often short, but a one-period error changes every term in the calculation. The most useful checks are to count payment dates, count periods to the valuation date, and confirm the rate period.

Discounting the immediate payment. A payment at $t=0$ contributes its full amount to present value. Multiplying it by $v$ incorrectly moves it to $t=1$.
Using the wrong future date. For deposits at $t=0,\ldots,n-1$, the value at $t=n-1$ is $Rs_{\overline{n}|}$; at $t=n$, it is $R(1+i)s_{\overline{n}|}$. Do not interchange these expressions.
Using an annual rate for nonannual periods. If payments are quarterly, the calculation needs an effective quarterly rate. A quoted annual rate cannot simply be treated as the quarterly rate unless the problem explicitly defines it that way.
Rounding each contribution too early. Keep unrounded discount or accumulation values through the sum, then round the final monetary answer to cents.
Assuming equal payments from equal timing. Start-of-period cash flows can be irregular. Use annuity factors only when the payments are level and the other factor assumptions are met.

Key takeaway: An annuity-due factor is a compact representation of a specific timeline, not a substitute for one. For a level stream, multiplying the ordinary-annuity factor by $1+i$ accounts for the one-period shift. For any stream, the payment dates and valuation date determine the correct discount or accumulation exponent.

Check Your Understanding

For each problem, identify the payment times and valuation date before calculating. Answers are rounded to the nearest cent where applicable.

A hypothetical lease requires four annual payments of $1,200 at $t=0,1,2,3$. Find the present value at an effective annual rate of 4%. (Answer: $4,530.11)
A reserve receives six deposits of $350 at $t=0,1,2,3,4,5$. Find its value at $t=5$ using an effective annual rate of 3%. (Answer: $2,263.94)
A service contract requires payments of $2,000 at $t=0$, $1,500 at $t=1$, and $1,000 at $t=2$. Find the present value at an effective annual rate of 6%. (Answer: $4,305.09)
A fund receives four quarterly deposits of $500 at the beginning of each quarter, starting now. Find the accumulated value at the end of quarter 4 if the effective quarterly rate is 1.5%. (Answer: $2,076.13)
A buyer can spend at most $5,000 today on two equal annual payments due immediately and one year from now. At an effective annual rate of 4%, find the exact maximum payment amount and the greatest whole-cent payment that does not exceed the budget. (Answer: Exact maximum $5000/(1+1/1.04)=\$2,549.019607\ldots$; greatest whole-cent payment: $2,549.01)

What You'll Learn