Deriving the Equivalent Annual Payment Formula

Why Derive an Equivalent Annual Payment?

An irregular cash-flow stream can be summarized as a level payment over a chosen horizon. This is useful when a decision requires a common annual measure, but the conversion is meaningful only if the payment dates, interest rate, and number of years are specified. The central method is to value the irregular stream at time 0, then set that value equal to the present value of an ordinary annuity.

The method assumes a constant effective annual interest rate and annual comparison payments made at the ends of years 1 through $n$. The irregular cash flows may occur at time 0 or at other specified dates. Their present values must be calculated using the same rate and valuation date as the annual payments. The result is an equivalent amount, not a claim that the original cash flows actually occur evenly.

Definition: For an irregular stream with cash flow $C_j$ at time $t_j$, its equivalent annual payment over $n$ years is the level amount $E$, paid at the end of each year $1,\ldots,n$, whose present value equals the present value of the stream. With a constant effective annual rate $i>0$, $$ \sum_j C_j(1+i)^{-t_j}=E a_{\overline{n}|}, \qquad a_{\overline{n}|}=\frac{1-(1+i)^{-n}}{i}. $$

Use a consistent sign convention: for example, treat receipts as positive and payments as negative. Then $E$ is a net equivalent annual amount. If the question instead asks for the level annual cost, express the costs as positive amounts and calculate their equivalent annual cost. In either convention, retain the signs consistently when forming the present value.

Deriving the Ordinary-Annuity Factor

An ordinary annuity consists of $n$ equal payments $E$ at the ends of years 1 through $n$. Discounting each payment to time 0 gives $E/(1+i)$, $E/(1+i)^2$, and so on. Therefore, the annuity’s present value is a finite geometric series.

\begin{aligned} PV_{\text{annuity}} &=E\left((1+i)^{-1}+(1+i)^{-2}+\cdots+(1+i)^{-n}\right)\\ &=E\frac{1-(1+i)^{-n}}{i}\\ &=E a_{\overline{n}|}. \end{aligned}

To see why the geometric-series expression holds, let $v=(1+i)^{-1}$. The discount-factor sum is $v+v^2+\cdots+v^n$. Multiplying this sum by $1-v$ cancels the middle terms:

(1-v)(v+v^2+\cdots+v^n)=v-v^{n+1}.

Thus the sum is $v(1-v^n)/(1-v)$. Since $1-v=i/(1+i)=iv$, this equals $(1-v^n)/i$, which is the stated annuity factor. Equating the irregular stream’s present value to this annuity present value and dividing by $a_{\overline{n}|}$ gives the equivalent annual payment:

Result: Under a constant effective annual rate $i>0$, if the irregular stream has present value $PV_0$ at time 0, then its equivalent annual payment over $n$ years is $$ E=\frac{PV_0}{a_{\overline{n}|}} =PV_0\frac{i}{1-(1+i)^{-n}}. $$ The comparison payments occur at the ends of years 1 through $n$. Since $a_{\overline{n}|}>0$, there is exactly one amount $E$ with this present value.

If $i=0$, there is no discounting: the ordinary-annuity present value is $nE$, so $E=PV_0/n$. The formula involving division by $i$ should not be used at $i=0$. The rate must also match the payment period: with annual payments, use an effective annual rate. If cash flows are dated at non-integer times, discount them using the stated rate convention and those dates; do not silently treat a partial year as a full year.

A Reliable Calculation Sequence

The algebra is short, but most errors arise before the division: a cash flow is placed at the wrong time, the annual-payment horizon is misread, or the payments are treated as beginning-of-year rather than end-of-year. Use this sequence.

Set the valuation date and sign convention.
Identify the date at which values are to be compared, usually time 0, and mark receipts and payments consistently.

Find the irregular stream’s present value.
Discount each cash flow from its stated date at the applicable rate, then add the signed present values.

Identify the annual-payment pattern.
Confirm the number $n$ of payments and whether they are at the ends or beginnings of periods. For end-of-year payments, use $a_{\overline{n}|}$.

Divide and check.
Set the two present values equal, solve for $E$, then multiply $E$ by the annuity factor to verify that it reproduces the irregular stream’s present value.

Worked Applications

Worked Example: Converting Irregular Receipts to Four Annual Payments

A hypothetical project is expected to generate receipts of $1,200 at the end of year 1, $800 at the end of year 2, and $1,500 at the end of year 4. At an effective annual interest rate of 5%, find the equivalent level receipt at each year-end for four years. Round the final payment to the nearest cent.

Method. First value all receipts at time 0. Then equate that present value to the value of four ordinary-annuity payments. The year-3 cash flow is zero; it does not change the four-payment horizon.

The present value of the irregular stream is

\begin{aligned} PV_0 &=\frac{1200}{1.05}+\frac{800}{1.05^2}+\frac{1500}{1.05^4}\\ &=1142.857143+725.623583+1234.053712\\ &=3102.534438. \end{aligned}

The four-payment annuity factor is

a_{\overline{4}|} =\frac{1-1.05^{-4}}{0.05} =\frac{1-0.822702475}{0.05} =3.545950504.

Equating present values gives

E=\frac{3102.534438}{3.545950504}=874.9514.

Rounded to the nearest cent, the equivalent annual receipt is $874.95 at each year-end for four years. Check: $874.9514(3.545950504)=3102.5344$, matching the irregular stream’s present value before rounding. The common mistake this method avoids is using three payments because there are only three nonzero receipts; the comparison horizon is four years.

Worked Example: Including a Time-0 Amount

A hypothetical equipment upgrade has costs of $1,000 immediately, $1,080 at the end of year 1, and $1,259.712 at the end of year 3. The effective annual interest rate is 8%. Find the equivalent annual cost over three years, with payments at the ends of years 1, 2, and 3. Round to the nearest cent.

Method. Treat each stated cost as a positive amount for the cost calculation. A time-0 cost is not discounted; the other costs are discounted to time 0.

\begin{aligned} PV_0 &=1000+\frac{1080}{1.08}+\frac{1259.712}{1.08^3}\\ &=1000+1000+1000\\ &=3000. \end{aligned}

Here $1.08^3=1.259712$, so the final cost also has present value $1,000. For three end-of-year payments,

a_{\overline{3}|} =\frac{1-1.08^{-3}}{0.08} =2.57709699.

Therefore,

E=\frac{3000}{2.57709699}=1164.1004.

The equivalent annual cost is $1,164.10. Check: $1164.1004(2.57709699)=3000.00$, to the displayed precision. The common mistake this method avoids is discounting the immediate $1,000 cost; it is already at the valuation date.

Worked Example: Finding a Net Annual Equivalent

A hypothetical project requires an immediate outlay of $1,000 and then produces receipts of $550, $605, and $665.50 at the ends of years 1, 2, and 3. If the effective annual interest rate is 10%, find the net equivalent annual amount over those three years. Use positive signs for receipts and a negative sign for the outlay. Round the final amount to the nearest cent.

Method. Calculate the signed present value, then divide by the three-year ordinary-annuity factor. A positive result represents a net annual benefit under this sign convention.

\begin{aligned} PV_0 &=-1000+\frac{550}{1.10}+\frac{605}{1.10^2}+\frac{665.50}{1.10^3}\\ &=-1000+500+500+500\\ &=500. \end{aligned}

The receipts have present values of $500 each because $1.10^2=1.21$ and $1.10^3=1.331$. The annuity factor is

a_{\overline{3}|} =\frac{1-1.10^{-3}}{0.10} =2.486851991.

Thus,

E=\frac{500}{2.486851991}=201.0574.

The net equivalent annual amount is $201.06 of benefit. Check: $201.0574(2.486851991)=500.00$, to the displayed precision. The common mistake this method avoids is converting only the receipts and forgetting the time-0 outlay when finding the net present value.

Timing and Horizon Traps

The factor $a_{\overline{n}|}$ applies to payments at times 1 through $n$, not at times 0 through $n-1$. Payments at the beginnings of periods have a different present-value factor, $\ddot{a}_{\overline{n}|}=(1+i)a_{\overline{n}|}$. Using that factor for end-of-year payments shifts every comparison payment earlier and overstates the present value produced by a given payment.

For example, in the first worked example, incorrectly dividing $3,102.534438 by the annuity-due factor $1.05(3.545950504)=3.723248029$ gives about $833.29. That amount is not the four end-of-year equivalent payment. The correct calculation uses the ordinary-annuity factor and gives $874.95. The smaller amount would be appropriate only for a different timing pattern, with payments at the beginnings of the four years.

The horizon matters just as much. It is the number of level annual payments specified for the comparison, not necessarily the number of nonzero cash flows. A stream may include a time-0 item, a year with no cash flow, or several cash flows in one year; none of those facts by itself changes the chosen number of annual payments. State the payment dates explicitly before selecting the annuity factor.

Key takeaway: The equivalent annual payment is found by matching values at one date: calculate the irregular stream’s present value, calculate the ordinary-annuity factor for the specified payment dates, and divide. The result depends on the interest rate, valuation date, horizon, and payment timing.

Check Your Understanding

For each problem, use end-of-year comparison payments unless the question specifies otherwise. Round monetary answers to the nearest cent.

A stream pays $500 immediately and $540 at the end of year 1. At an effective annual rate of 8%, find its equivalent annual payment over two years. (Answer: $560.77 per year)
A stream has present value $2,400 and the interest rate is zero. Find the equivalent annual payment over six years. (Answer: $400.00 per year)
At an effective annual rate of 10%, receipts of $440 at the end of year 1 and $605 at the end of year 2 are to be replaced by three equal year-end receipts. Find the equivalent annual receipt. (Answer: $361.90 per year)
A project has a time-0 outlay of $1,500 and receipts of $660, $726, and $798.60 at the ends of years 1, 2, and 3. At an effective annual rate of 10%, find the net equivalent annual amount over three years, using positive signs for receipts. (Answer: $120.63 per year of net benefit)
At an effective annual rate of 5%, a stream pays $250 at the end of year 1 and $400 at the end of year 4. Find the equivalent annual payment over four years. (Answer: $159.95 per year)

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