Comparing Mutually Exclusive Projects With Unequal Lives

Why Unequal Lives Create a Comparison Problem

Net present value (NPV) measures the present value of a project's cash inflows and outflows over the period represented by its cash-flow forecast. For a project with cash flow $CF_t$ at the end of year $t$, an effective annual discount rate $i$, and a life of $n$ years, its stand-alone NPV is:

NPV=\sum_{t=0}^{n}\frac{CF_t}{(1+i)^t}

The formula gives a value at time zero, but it does not make every project with a positive NPV directly comparable. If two mutually exclusive projects have different lives, their stand-alone NPVs summarize cash flows over different periods. A project with a longer life may include more years of service and more years of cash flows. A project with a shorter life may require a follow-on investment to keep providing service. A ranking based only on the two stand-alone NPVs can therefore compare unlike alternatives.

Definition: The decision horizon is the period over which the decision-maker needs the project or service being considered. A fair comparison requires evaluating the alternatives over a common horizon, using consistent assumptions about the service each provides and including all relevant cash flows over that horizon.

The horizon may be the useful service period, a fixed planning period, or a period over which projects can be repeated. Which horizon is appropriate depends on the decision. It is not determined automatically by the life of either project.

What Makes a Comparison Fair?

Before comparing unequal-life projects, specify what the decision is meant to accomplish. If the organization needs a service for four years, for example, a two-year project cannot be treated as a complete alternative unless the analysis explains what happens after year two. The service might stop, be provided by a different asset, or be continued by repeating the original project. These possibilities have different cash flows and can produce different rankings.

A sound comparison should state the following assumptions:

Common service requirement: Specify what service each alternative must provide and for how long.
Decision horizon: Set the common period over which the alternatives are evaluated.
Continuation after a project ends: State whether service stops, is obtained elsewhere, or is continued through a new investment. Include its cash flows if specified.
Repeatability: If a project is assumed to be repeated, state whether its cost, operating cash flows, life, and performance are assumed to remain the same.
Consistent valuation: Use the same discount rate when the alternatives have comparable risk, and account for cash flows at the dates they occur.
Horizon-end value: Include any specified salvage value, disposal cost, or other relevant cash flow at the end of the horizon.

If these details are missing, the correct conclusion may be that the data do not determine a preferred project. A positive NPV indicates value relative to the project's own forecast and discount rate; it does not by itself establish that the project is the better choice when another alternative has a different life.

Worked Example: Stand-Alone NPVs Do Not Settle the Choice

Two mutually exclusive equipment proposals require an initial investment of $50,000. Project A generates $30,000 at the end of each of years 1 and 2, then ends. Project B generates $15,000 at the end of each of years 1 through 4, then ends. The effective annual discount rate is $i=8\%$. Calculate the stand-alone NPVs and assess what they establish.

Method. Discount each project's cash flows over its own stated life. This gives the stand-alone NPV for each forecast, but the comparison must still account for the different service periods.

For Project A, the two inflows have present values $30{,}000/1.08=\$27{,}777.78$ and $30{,}000/1.08^2=\$25{,}720.16$. Therefore:

NPV_A=-50{,}000+27{,}777.78+25{,}720.16 =\$3{,}497.94

For Project B, the four discounted inflows are $15{,}000/1.08=\$13{,}888.89$, $15{,}000/1.08^2=\$12{,}860.08$, $15{,}000/1.08^3=\$11{,}907.48$, and $15{,}000/1.08^4=\$11{,}025.45$. Thus:

NPV_B=-50{,}000+13{,}888.89+12{,}860.08+11{,}907.48+11{,}025.45 =-\$318.10

The calculations show that A has the larger stand-alone NPV over its two-year life. They do not establish that A is the better choice if the organization needs four years of service. The problem does not say whether A's service can be continued after year 2, or what that continuation would cost or contribute. Without that information, a four-year comparison is incomplete.

Check. A's two discounted inflows total $53,497.94, which exceeds its $50,000 initial investment by $3,497.94. B's four discounted inflows total $49,681.90, which is $318.10 less than its initial investment.

Comparing Complete Cash-Flow Streams

The underlying reason for using a common horizon is cash-flow timing. The present value of a cash flow depends on both its amount and its date. For a specified decision horizon $H$, the value of a complete alternative is obtained by discounting every relevant cash flow over that horizon:

NPV_H=\sum_{t=0}^{H}\frac{CF_t}{(1+i)^t}

This expression follows from present-value discounting and additivity: each dated cash flow is converted to its value at time zero, and the values are added. If an alternative ends before $H$, its stream is not complete unless the analysis specifies what happens to the service and includes the associated cash flows. Conversely, cash flows after the decision horizon do not belong in the calculation unless they affect the value at the horizon, for example through a stated terminal value.

Thus, unequal-life projects can be compared when a common horizon and complete, comparable cash-flow streams are specified. A shorter project might be paired with known continuation arrangements; a repeatable project might be assumed to be undertaken again. The assumptions matter: they define the actual alternatives being valued, rather than merely adjusting a formula.

Worked Example: Use the Specified Four-Year Horizon

An organization needs equipment service for four years. Project A costs $80,000 now, produces net cash inflows of $50,000 at the end of years 1 and 2, and then ends. The stated plan is to obtain equivalent service under a contract for years 3 and 4, producing net cash inflows of $30,000 at the end of each of those years. Project B costs $80,000 now and produces net cash inflows of $30,000 at the end of each of years 1 through 4. The effective annual discount rate is $i=10\%$, and both alternatives have zero horizon-end value. Compare their four-year NPVs.

Method. Include the cash flows of each complete four-year alternative. For A, the contracted continuation is part of the specified plan; it is not an unstated assumption that the original equipment continues operating.

For A, the discounted inflows are $50{,}000/1.10=\$45{,}454.55$, $50{,}000/1.10^2=\$41{,}322.31$, $30{,}000/1.10^3=\$22{,}539.44$, and $30{,}000/1.10^4=\$20{,}490.40$. Therefore:

NPV_{A,4}=-80{,}000+45{,}454.55+41{,}322.31+22{,}539.44+20{,}490.40 =\$49{,}806.71

For B, the discounted inflows are $\$27{,}272.73$, $\$24{,}793.39$, $\$22{,}539.44$, and $\$20{,}490.40$. Hence:

NPV_{B,4}=-80{,}000+27{,}272.73+24{,}793.39+22{,}539.44+20{,}490.40 =\$15{,}095.96

Under the stated four-year service requirement and cash-flow assumptions, A has the larger horizon NPV. This conclusion depends on the specified continuation arrangement for years 3 and 4. It is not a conclusion from comparing A's two-year stand-alone NPV with B's four-year stand-alone NPV.

Check. A's discounted inflows total $129,806.71; after subtracting $80,000, its NPV is $49,806.71. B's discounted inflows total $95,095.96; after subtracting $80,000, its NPV is $15,095.96.

Repeatability Is an Assumption, Not a Fact

One way to provide service beyond a project's life is to assume the project can be repeated. That assumption may be reasonable for a standard asset that can be purchased again, but it should not be made automatically. A repeated project may face a different purchase price, operating performance, useful life, technology, or demand. If those features are not expected to remain unchanged, repeating the original cash-flow pattern may not represent the available alternative.

A comparison based on repetition also needs a horizon and a clear treatment of its end. If a project is repeated to cover the horizon, state when the new investment occurs, which operating cash flows it generates, and whether any terminal value or disposal cost applies. Do not include a replacement cost after the horizon unless the decision explicitly includes that future investment.

Worked Example: A Repeated Two-Year Project Over Four Years

Two mutually exclusive alternatives provide the same service over a four-year decision horizon. Project A costs $40,000 now and generates $25,000 at each year-end for its two-year life. It can be repeated at the end of year 2 for another $40,000, with the repeated unit generating $25,000 at the end of years 3 and 4. Assume the same cost, life, and performance on repetition, a four-year horizon, zero value at year 4, and an effective annual discount rate of $i=10\%$. Project B costs $70,000 now and generates $23,000 at each year-end for four years, with zero value at year 4. Compare the alternatives.

Method. Build each alternative's cash flows over the same four years. For A, year 2 includes both the original unit's $25,000 inflow and the $40,000 repeat investment, so the net year-2 cash flow is $-\$15{,}000$.

Project A's cash flows are $-\$40{,}000$ at time 0, $+\$25{,}000$ at year 1, $-\$15{,}000$ at year 2, $+\$25{,}000$ at year 3, and $+\$25{,}000$ at year 4. Discounting gives:

\begin{aligned} NPV_{A,4} &=-40{,}000+\frac{25{,}000}{1.10} -\frac{15{,}000}{1.10^2} +\frac{25{,}000}{1.10^3} +\frac{25{,}000}{1.10^4}\\ &=-40{,}000+22{,}727.27-12{,}396.69+18{,}782.87+17{,}075.34\\ &=\$6{,}188.79 \end{aligned}

Project B's four discounted inflows are $\$20{,}909.09$, $\$19{,}008.26$, $\$17{,}280.24$, and $\$15{,}709.31$. Therefore:

NPV_{B,4}=-70{,}000+20{,}909.09+19{,}008.26+17{,}280.24+15{,}709.31 =\$2{,}906.90

Under the stated repeatability and service assumptions, A has the larger four-year NPV. If A could not be repeated on those terms, its four-year cash-flow stream would be different, and this ranking would not be justified.

Check. A's year-2 net cash flow is $\$25{,}000-\$40{,}000=-\$15{,}000$; including both amounts at year 2 produces the same result as discounting them separately.

Common Exam Trap: Ranking Different Horizons

A tempting but incomplete calculation is to compute two stand-alone NPVs and select the larger one, even though the projects provide service for different lengths of time. For example, in the first worked example, the calculation $NPV_A=\$3{,}497.94>NPV_B=-\$318.10$, followed by “choose A,” is an unsupported decision if four years of service are required. The arithmetic is correct; the inference is not. A's two-year forecast does not state what happens in years 3 and 4.

The correction is to identify the required horizon, specify how each alternative provides the required service over it, and compare the resulting complete cash-flow streams. If continuation costs, repeatability, or terminal values are unknown, say so. Do not silently assume that the shorter project can be repeated at its original cost and performance, or that its service simply becomes unnecessary when it ends.

Key takeaway: Unequal-life stand-alone NPVs are values over different forecast periods. A fair choice requires a common decision horizon and explicit assumptions about service, continuation or repetition, discounting, and horizon-end cash flows. When those assumptions are not supplied, the project ranking may be indeterminate.

Check Your Understanding

For each problem, use the stated assumptions and distinguish a stand-alone NPV calculation from a decision over a common horizon.

At $i=10\%$, Project C costs $20,000 and pays $12,000 at each year-end for two years. Project D costs $20,000 and pays $8,000 at each year-end for three years. Calculate both stand-alone NPVs and state why the ranking alone may not settle a choice. (Answer: $NPV_C=\$826.45$, $NPV_D=-\$105.18$; C has the larger stand-alone NPV, but the different service lives leave continuation and horizon requirements unspecified.)
At $i=8\%$, Project E costs $30,000 and pays $34,000 at the end of year 1. Project F costs $30,000 and pays $12,000 at each year-end for three years. Calculate both stand-alone NPVs. If each asset is unique and no service requirement or continuation plan is given, can the NPVs alone establish the better alternative? (Answer: $NPV_E=\$1{,}481.48$, $NPV_F=\$925.16$; no, because the service periods differ and the decision horizon and continuation plan are unspecified.)
A project costs $10,000 and pays $12,000 at each year-end for one year; it can be repeated twice at the same $10,000 cost, with the same performance. Over a three-year horizon, the repeat investments occur at the ends of years 1 and 2. The last unit's year-1 operating inflow occurs in each year it serves, and there is no year-3 replacement. At $i=10\%$, calculate the three-year NPV. (Answer: cash flows are $-\$10{,}000$ at time 0, $+\$2{,}000$ net at year 1, $+\$2{,}000$ net at year 2, and $+\$12{,}000$ at year 3; $NPV_3=-10{,}000+2{,}000/1.10+2{,}000/1.10^2+12{,}000/1.10^3=\$2{,}486.85$.)
Over a four-year horizon and at $i=10\%$, Project G costs $40,000 now and pays $24,000 at the end of each year for two years. It can be repeated at the end of year 2 for $40,000, with the repeated unit paying $24,000 at the end of years 3 and 4. Project H costs $50,000 now and pays $15,000 at each year-end for four years. Both have zero horizon-end value. Calculate the horizon NPVs and identify the preferred alternative under the stated repeatability assumption. (Answer: $NPV_{G,4}=-40{,}000+24{,}000/1.10-16{,}000/1.10^2+24{,}000/1.10^3+24{,}000/1.10^4=\$3{,}018.92$, since year 2 includes the $40,000 repeat cost; $NPV_{H,4}=-50{,}000+15{,}000(3.169865446)= -\$2{,}452.02$. Choose G under the stated assumptions.)
At $i=9\%$, Project J costs $60,000 and pays $35,000 at each year-end for two years. Project K costs $60,000 and pays $18,000 at each year-end for four years. Compute their stand-alone NPVs, then assess whether the larger NPV is enough to choose between them if neither project can be repeated and the service horizon is not specified. (Answer: $NPV_J=-60{,}000+35{,}000/1.09+35{,}000/1.09^2=\$1{,}568.89$; $NPV_K=-60{,}000+18{,}000(3.23971988)=-\$1{,}685.04$. J has the larger stand-alone NPV, but that alone does not settle the choice without a common service horizon.)

What You'll Learn