Choosing Projects With the Equivalent Annual Annuity

Why Annualize NPV?

When two mutually exclusive projects have different lives, their total NPVs cover different lengths of time. A longer-lived project may have the higher NPV simply because it generates value over more years. If the projects provide a comparable service and can be undertaken repeatedly under stable conditions, the equivalent annual annuity (EAA) expresses each project’s NPV as an equal annual amount over that project’s own life. The resulting annual values provide a useful basis for ranking the alternatives.

EAA is not a different way to calculate project value: it is an annualized representation of NPV at the same discount rate. The method is appropriate only when the alternatives are comparable and the repeatability assumption is reasonable. If those conditions do not hold, an annualized ranking may not answer the actual decision question. This tutorial is aligned with the SOA VEE Accounting and Finance syllabus.

Definition: For a project with life $n$, net present value $NPV$, and effective annual discount rate $i$, its equivalent annual annuity is the uniform end-of-year amount $EAA$ whose present value over $n$ years equals the project’s NPV. For $i>0$, $$ EAA=\frac{NPV}{a_{\overline{n}|}, \qquad a_{\overline{n}|}=\sum_{t=1}^{n}\frac{1}{(1+i)^t} =\frac{1-(1+i)^{-n}}{i}. $$

The annuity factor $a_{\overline{n}|}$ is the present value of one dollar paid at the end of each year for $n$ years. Use the same discount rate in the annuity factor that was used to calculate the project’s NPV. When $i=0$, the factor is $n$, so $EAA=NPV/n$.

Why the EAA Formula Holds

Suppose an equivalent annual amount $EAA$ is received at each year-end for $n$ years. Its present value is the sum of the discounted annual payments. To be equivalent to the project, this present value must equal the project’s NPV:

NPV=\sum_{t=1}^{n}\frac{EAA}{(1+i)^t} =EAA\sum_{t=1}^{n}\frac{1}{(1+i)^t} =EAA\,a_{\overline{n}|}.

Solving this equality for the annual amount gives $EAA=NPV/a_{\overline{n}|}$. The conversion preserves present value; it does not assert that the project actually pays a level cash flow each year. In particular, EAA is not the project’s accounting profit or its forecast annual cash receipt.

Worked Example: The Higher NPV Has the Lower EAA

Two repeatable projects provide the same service and have the same risk. Use an effective annual discount rate of 10%, with all project cash flows occurring at year-end. Project A costs $100,000 now and pays $40,000 at the end of year 1 and $100,000 at the end of year 2. Project B costs $65,000 now and pays $30,000 at the end of each of years 1 through 4. Calculate both NPVs and EAAs, then rank the projects.

Method. Calculate each project’s NPV at 10%, then divide by the annuity factor for that project’s own life. This comparison avoids treating total NPV over two years as directly comparable with total NPV over four years.

For Project A, the two-year annuity factor is $a_{\overline{2}|}=(1/1.10)+(1/1.10^2)=1.735537$. Its NPV is

\begin{aligned} NPV_A &=-100{,}000+\frac{40{,}000}{1.10}+\frac{100{,}000}{1.10^2}\\ &=-100{,}000+36{,}363.636364+82{,}644.628099\\ &=19{,}008.264463. \end{aligned}

The EAA for Project A is $19{,}008.264463/1.735537190=\$10{,}952.38$ per year, rounded to the nearest cent.

For Project B, the four-year annuity factor is $a_{\overline{4}|}=1/1.10+1/1.10^2+1/1.10^3+1/1.10^4=3.169865$. The present value of its four receipts is $30{,}000(3.169865446)=\$95{,}095.96$. Therefore,

NPV_B=-65{,}000+95{,}095.963380=\$30{,}095.96.

Project B’s EAA is $30{,}095.963380/3.169865446=\$9{,}494.40$ per year, rounded to the nearest cent.

Decision and check. Project B has the larger total NPV, $30,095.96 versus $19,008.26. Project A has the larger EAA, $10,952.38 versus $9,494.40 per year, so choose A on an annual-value basis under the stated repeatability and comparability assumptions. As a check, discounting each EAA over its project’s life recovers its NPV: $10{,}952.380952(1.735537190)=19{,}008.264463$, and $9{,}494.397740(3.169865446)=30{,}095.963380$. The common mistake this example avoids is choosing B from total NPV alone.

Ranking Alternatives and Making an Accept-or-Reject Decision

For mutually exclusive projects that provide comparable service, calculate an EAA for each project using its own life. If each alternative can be repeated on comparable terms and the discount rate is appropriate for both, the alternative with the higher EAA creates more equivalent annual value. If the alternatives are cost-only, express the present value of costs as a positive amount, convert each to an equivalent annual cost, and choose the lower annual cost, assuming the service provided is equivalent.

An optional project presents a different question. When the choice is to undertake the project or do nothing, its NPV and EAA have the same sign because the annuity factor is positive. A positive EAA therefore corresponds to a positive NPV; it does not by itself establish that the project is better than a separate, mutually exclusive investment. In that case, compare the alternatives using a consistent decision method.

Worked Example: Should the Company Undertake the Project?

A company can undertake an optional three-year project or do nothing. The project costs $40,000 now and is expected to generate $20,000 at the end of each of years 1, 2, and 3. Assume an effective annual discount rate of 10%, certain cash flows as stated, and no other project cash flows. Calculate NPV and EAA, then decide whether to proceed.

Method. Discount each year-end receipt at 10% to calculate NPV. Then divide by the three-year annuity factor. All amounts below are in dollars, and rounding is to the nearest cent only at the end.

The three-year factor is $a_{\overline{3}|}=1/1.10+1/1.10^2+1/1.10^3=2.486851991$. Thus,

\begin{aligned} NPV &=-40{,}000+\frac{20{,}000}{1.10} +\frac{20{,}000}{1.10^2} +\frac{20{,}000}{1.10^3}\\ &=-40{,}000+18{,}181.818182 +16{,}528.925620+15{,}026.296019\\ &=\$9{,}737.04. \end{aligned}

The EAA is $9{,}737.039821/2.486851991=\$3{,}915.41$ per year. The project has positive NPV and positive EAA, so the company should proceed if the only alternative is doing nothing. As a check, the annual equivalent can also be found by subtracting the annualized cost from the annual receipts: $20{,}000-40{,}000/2.486851991=\$3{,}915.41$. This confirms the sign and the calculation.

Worked Example: Comparing Two Projects From Their NPVs

Two repeatable alternatives provide the same service, have the same risk, and are evaluated at an effective annual discount rate of 10%. Project C has a three-year life and an NPV of $36,000. Project D has a five-year life and an NPV of $50,000. Which project has the higher annual value?

Method. Use the annuity factor corresponding to each life. The factors are $a_{\overline{3}|}=2.486851991$ and $a_{\overline{5}|}=3.790786769$.

For Project C, $EAA_C=36{,}000/2.486851991=\$14{,}476.13$ per year. For Project D, $EAA_D=50{,}000/3.790786769=\$13{,}189.87$ per year. Thus Project D has the higher total NPV, but Project C has the higher EAA and is preferred on an annual-value basis under the stated assumptions. Rounding is to the nearest cent. The check is that both NPVs are positive, and the EAA calculation preserves each project’s NPV when multiplied by its own annuity factor.

Common Traps and Conditions for Use

A frequent error is to divide NPV by the number of years rather than by the annuity factor. For the two-year Project A above, the incorrect calculation is $19{,}008.26/2=\$9{,}504.13$ per year. That treats the NPV as if it were a sum of undiscounted annual amounts. The correct calculation discounts a level annual amount over both years: $19{,}008.26/1.735537=\$10{,}952.38$ per year.

A second trap is using the wrong project life or discount rate in the annuity factor. The factor must cover the project’s own life and use the same rate used in its NPV calculation. Do not use the longer project’s factor for both alternatives. Also, do not rank projects solely by total NPV when unequal lives mean the totals cover different durations.

EAA is most informative when the alternatives provide equivalent service and can be repeated on comparable terms. If a project cannot be repeated, the alternatives do not provide equivalent service, or the decision is constrained to a specific planning horizon, simply annualizing each NPV may not represent the decision being made. For a cost-only comparison, use equivalent annual costs and minimize them; applying the “choose the highest EAA” rule to negative cost values without first defining the measure can reverse the intended interpretation.

Key takeaway: Calculate each project’s NPV consistently, divide it by the annuity factor for that project’s own life, and rank comparable repeatable alternatives by EAA. The highest total NPV need not have the highest annual value.

Check Your Understanding

For each problem, assume end-of-year cash flows and use the stated effective annual discount rate. Round dollar answers to the nearest cent.

A two-year project has an NPV of $12,000 at 10%. Calculate its EAA. (Answer: $6,914.29 per year)
A repeatable four-year project has an NPV of $24,000 at 8%. Using $a_{\overline{4}|}=3.312127$, calculate its EAA. (Answer: approximately $7,246.10 per year)
An optional two-year project has an NPV of $-\$4,000$ at 10%. What is its EAA, and should it be accepted if doing nothing is the only alternative? (Answer: $-\$2,304.76$ per year; reject it)
At 10%, repeatable Project M has a three-year life and NPV of $28,000. Project N has a four-year life and NPV of $34,000. Using $a_{\overline{3}|}=2.486852$ and $a_{\overline{4}|}=3.169865$, which has the higher NPV, and which has the higher EAA? (Answer: N has the higher NPV; M has the higher EAA, approximately $11,259.21 per year versus $10,726.01 per year)
Two alternatives provide the same service. At 10%, their present values of costs are $90,000 for a three-year alternative and $120,000 for a five-year alternative. Using $a_{\overline{3}|}=2.486852$ and $a_{\overline{5}|}=3.790787$, calculate the equivalent annual costs and identify the lower-cost alternative. (Answer: approximately $36,190.33 per year and $31,655.70 per year, respectively; choose the five-year alternative)

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