Comparing Mutually Exclusive Projects

Why the Comparison Basis Matters

When projects are mutually exclusive, choosing one rules out choosing the other. The decision is therefore not whether each project has positive cash flows, or which has the largest benefit in dollars before discounting. The relevant comparison is the value each project creates on a common valuation basis: the same valuation date, the same specified discount rate, and a consistent treatment of project cash flows.

Net present value brings the dated net cash flows of a project to a chosen valuation date. A project may have a large future benefit but also require a large investment; NPV accounts for both. For mutually exclusive choices evaluated under the same assumptions, select the project with the larger NPV. If doing nothing is allowed, its NPV is zero, so a negative-NPV project should not be chosen over that option.

Definition: Projects are mutually exclusive when at most one of them can be undertaken. To compare them, calculate each project's NPV at the same valuation date using the same specified discount rate and consistent cash-flow assumptions. The project with the larger NPV is preferred among the available project alternatives.

NPV at a Common Valuation Date

Assume a project has certain net cash flows $C_k$ at dates $t_k$, and the valuation date is $t_0$. A positive cash flow is an inflow and a negative cash flow is an outflow. If the effective annual discount rate is $i$, is constant over the periods being valued, and the cash flows occur at annual intervals on or after the valuation date, then

\operatorname{NPV}_{t_0} = \sum_k C_k(1+i)^{-(t_k-t_0)}.

A cash flow at the valuation date has exponent zero and is included without discounting. A cash flow one year later is discounted for one year; a cash flow three years later is discounted for three years. The formula assumes the stated rate is effective annually and that the project cash flows and rate are expressed on a consistent basis. It does not decide whether the rate is appropriate for a particular project; it uses the rate specified for the comparison.

A reliable comparison has four parts: identify the valuation date, list each project's net cash flows with dates and signs, discount each project's cash flows on the same basis, and compare the resulting NPVs. Do not compare one project's value at its start date with another project's value at a different date.

Fix the valuation date and rate.
Use the common date and specified effective annual rate for every alternative.

Write each complete net cash-flow schedule.
Include the project's outflows as well as its inflows, with each amount assigned to its payment date.

Calculate each NPV separately.
Discount each dated amount to the common valuation date and sum with its sign.

Apply the decision rule.
Among mutually exclusive projects, prefer the higher NPV. If doing nothing is allowed, include its NPV of zero as an alternative.

Why the Higher-NPV Rule Holds

At a given valuation date and rate, each dated cash flow has a value at that date. The NPV is the sum of those values, with outflows negative. This follows from the additivity of present values: valuing a collection of separate cash flows and adding their values gives the value of the combined cash-flow schedule.

Suppose project A has NPV $V_A$ and project B has NPV $V_B$, each calculated at the same date and rate. If the choice is between undertaking exactly one of them, their values on that common basis are directly comparable. If $V_A>V_B$, project A provides the larger present value under the stated assumptions. The rule does not mean that A has the largest undiscounted receipts or the smallest initial cost; it means its complete net cash-flow schedule has the larger value at the valuation date.

The common basis is essential to this reasoning. A change in valuation date changes the numerical NPV, and a change in discount rate may also change the ranking. Therefore, comparing figures calculated with different dates or rates does not apply the rule.

Worked Example: Compare Two Projects at Today’s Date

Two mutually exclusive projects have certain net cash flows at the end of each year. Project A requires \$1,000 today and returns \$600 at the end of each of the next two years. Project B requires \$1,200 today and returns \$500 at the end of each of the next three years. At an effective annual discount rate of 10%, which project has the larger NPV?

Method. Use today as the common valuation date, assign a negative sign to each current outlay, and discount every future inflow at 10% per year.

For project A, the present value of the year 1 inflow is $600/1.10=\$545.454545$, and the present value of the year 2 inflow is $600/1.10^2=\$495.867769$. Thus,

\operatorname{NPV}_A = -1{,}000+\frac{600}{1.10}+\frac{600}{1.10^2} = -1{,}000+545.454545+495.867769 = \$41.322314.

For project B, the three discounted inflows are $500/1.10=\$454.545455$, $500/1.10^2=\$413.223140$, and $500/1.10^3=\$375.657400$. Therefore,

\operatorname{NPV}_B = -1{,}200+454.545455+413.223140+375.657400 = \$43.425995.

Rounded to the nearest cent, $\operatorname{NPV}_A=\$41.32$ and $\operatorname{NPV}_B=\$43.43$. Project B has the larger NPV, so it is preferred if one of the two must be selected. The calculation avoids the mistake of choosing A just because its initial outlay is smaller: the decision depends on the present value of the complete net cash-flow schedule.

Comparing Projects at a Nonzero Date

The valuation date need not be the present date or the date on which a project first appears in a description. It is the date at which the comparison is requested. Cash flows after that date are discounted back to it; a cash flow exactly on that date is not discounted. Historical cash flows before the valuation date are not future project cash flows to be discounted back as if they were still avoidable costs. Include only the cash flows relevant to the valuation being performed.

For example, if the common valuation date is two years from today, then a cash flow one year after that date is discounted for one year—not for three years. The time in the discount exponent is the interval from the valuation date to the cash-flow date.

Worked Example: Compare at the Project Decision Date

A decision is to be made at time 2. Project A requires \$800 at time 2 and pays \$450 at each of times 3 and 4. Project B requires \$900 at time 2 and pays \$400 at each of times 3, 4, and 5. Assume all amounts are certain, the effective annual discount rate is 5%, and the stated times are years. Compare the NPVs at time 2.

Method. Use time 2 as the common valuation date. The inflow at time 3 is one year after the valuation date, the inflow at time 4 is two years after it, and the inflow at time 5 is three years after it.

For project A, the two discounted inflows are $450/1.05=\$428.571429$ and $450/1.05^2=\$408.163265$. The outlay occurs at the valuation date, so its value remains $-\$800$:

\operatorname{NPV}_{2,A} = -800+\frac{450}{1.05}+\frac{450}{1.05^2} = -800+428.571429+408.163265 = \$36.734694.

For project B, the discounted inflows are $400/1.05=\$380.952381$, $400/1.05^2=\$362.811791$, and $400/1.05^3=\$345.535039$. Hence,

\operatorname{NPV}_{2,B} = -900+\frac{400}{1.05}+\frac{400}{1.05^2}+\frac{400}{1.05^3} = \$189.299211.

Rounded to the nearest cent, project A has NPV $\$36.73$, while project B has NPV $\$189.30$. Project B is preferred. A common timing error would be to discount the time 3 payment for three years because its date is labelled 3. The correct exponent is the interval from time 2 to time 3, which is one year.

NPV Includes Both Benefits and Costs

The present value of a project's inflows is not its NPV when the project also requires an investment. NPV is the sum of all relevant net cash flows, including costs, on the same date basis. This distinction matters when one project has larger gross benefits but also requires more capital.

Worked Example: Larger Benefits Do Not Guarantee Larger NPV

At an effective annual discount rate of 5%, project A costs \$400 today and pays \$500 at the end of year 1 and \$100 at the end of year 2. Project B costs \$700 today and pays \$900 at the end of year 2. The projects are mutually exclusive, and all cash flows are certain. Determine which has the larger NPV.

Method. Calculate each project's NPV, not merely the present value of its inflows. The common valuation date is today.

For project A, the present value of its inflows is $500/1.05=\$476.190476$ and $100/1.05^2=\$90.702948$. Their sum is $\$566.893424$. Subtract the $\$400$ outlay:

\operatorname{NPV}_A = -400+\frac{500}{1.05}+\frac{100}{1.05^2} = \$166.893424.

For project B, the present value of its year 2 inflow is $900/1.05^2=\$816.326531$. Subtract the $\$700$ outlay:

\operatorname{NPV}_B = -700+\frac{900}{1.05^2} = \$116.326531.

Rounded to the nearest cent, project A has NPV $\$166.89$ and project B has NPV $\$116.33$, so project A is preferred. Project B has the larger present value of inflows, $\$816.33$ versus $\$566.89$, but that is not the decision measure. Choosing B on the basis of gross inflows alone ignores its larger investment cost.

Common Mistakes and Decision Checks

Before finalizing a comparison, verify that every amount has been treated consistently. In particular, confirm that both alternatives use the same valuation date and rate, and that each project’s outflows and inflows are included. A small arithmetic check can catch an omitted cost; a timeline check can catch a discounting error.

Comparing NPVs from different dates. Recalculate both at the requested common valuation date. Do not compare one value at time 0 with another at time 2.
Using different discount rates without a stated reason. The basic comparison in this tutorial uses one specified rate for both projects. The rule cannot justify a comparison where the stated assumptions differ.
Ignoring the sign of a cash flow. Investment costs are negative cash flows; subtracting them is part of calculating NPV.
Choosing the larger benefit total. Compare NPVs, not undiscounted receipts or gross present values of inflows.
Assuming the higher NPV means a project must be undertaken. If no project is also an available choice, compare each project's NPV with zero. A project with negative NPV is inferior to doing nothing under the stated valuation assumptions.

Key takeaway: For mutually exclusive projects, calculate each complete NPV at one common valuation date and the specified discount rate. Prefer the larger NPV; if doing nothing is available, include its NPV of zero in the choice.

Check Your Understanding

For each question, assume the cash flows are certain and the stated rate is an effective annual rate. Compare NPVs on the requested common valuation date.

At time 0, project M costs \$300 and pays \$180 at each of times 1 and 2. Project N costs \$320 and pays \$190 at each of times 1 and 2. At 8%, which has the larger NPV, and what are the NPVs to the nearest cent? (Answer: M, with NPV \$20.99 versus N's \$18.82.)
The valuation date is time 1 and the rate is 6%. Project A costs \$1,000 at time 1 and pays \$550 at times 2 and 3. Project B costs \$1,020 at time 1 and pays \$530 at times 2, 3, and 4. Which project has the larger NPV at time 1? (Answer: B; $\operatorname{NPV}_{1,A}=\$8.37$ and $\operatorname{NPV}_{1,B}=\$396.70$, to the nearest cent.)
At 10%, project A has cash flows of $-\$750$ at time 0, $+\$400$ at time 1, and $+\$450$ at time 2. Project B has cash flows of $-\$800$ at time 0, $+\$500$ at time 1, $+\$350$ at time 2, and $+\$200$ at time 3. If exactly one must be chosen, which has the larger NPV? (Answer: B; $\operatorname{NPV}_A=-\$14.46$ and $\operatorname{NPV}_B=\$94.06$, to the nearest cent.)
The valuation date is time 2 and the rate is 4%. Project C costs \$500 at time 2 and pays \$300 at times 3 and 4. Project D costs \$510 at time 2 and pays \$260 at times 3 and 4, plus \$100 at time 5. Which project has the larger NPV at time 2? (Answer: D; $\operatorname{NPV}_{2,C}=\$65.83$ and $\operatorname{NPV}_{2,D}=\$69.28$, to the nearest cent.)
At 12%, project R costs \$600 at time 0 and pays \$350 at times 1 and 2. Project S costs \$700 at time 0 and pays \$400 at times 1, 2, and 3. Which project has the larger NPV, and which should be selected if doing nothing is also allowed? (Answer: S; $\operatorname{NPV}_R=-\$8.48$, $\operatorname{NPV}_S=\$260.73$, so select S rather than doing nothing.)

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