Why Profitability Index Rankings Can Fail

What a Profitability-Index Ranking Can—and Cannot—Tell You

The profitability index (PI) expresses the present value of a project's future cash inflows relative to its initial investment. It is useful for comparing returns per dollar of investment, but a ranking by PI is not a universal rule for selecting projects. A high ratio does not necessarily identify the set of projects with the highest total net present value (NPV).

The distinction matters when a company faces a capital budget, must choose between mutually exclusive alternatives, or has more than one scarce resource. In each case, the decision depends on the feasible choices and the objective being maximized. The examples below use hypothetical projects and show how an attractive PI ranking can lead to a lower-value decision.

Definition: For a project with initial investment $I_0>0$, and future cash flows discounted at the appropriate effective annual rate $i$, define $$ \text{PI}=\frac{\text{present value of future cash inflows}}{I_0} =\frac{\sum_{t=1}^{n}\frac{C_t}{(1+i)^t}}{I_0}. $$ The project's NPV is the present value of future cash inflows minus the initial investment: $$ \text{NPV}=\sum_{t=1}^{n}\frac{C_t}{(1+i)^t}-I_0 =I_0(\text{PI}-1). $$ Thus, for a conventional project with positive initial investment, PI greater than 1 is equivalent to positive NPV. A higher PI, however, is not always the same as a higher NPV or a better portfolio choice.

The formulas assume that the cash-flow estimates and discount rate are appropriate for the decision being made. The discount rate reflects the time value of money and project risk; it is not a device for making projects comparable when their constraints differ. In the examples, all projects have a single cash flow at the end of year 1, the effective annual discount rate is $i=10\%$, and there are no taxes or other cash flows unless stated. The year-1 present value is therefore the cash flow divided by $1.10$.

Why PI Works in a Narrow Case

PI rankings have a clear rationale in the special case of divisible projects competing for one budget. Suppose an amount $y_j$ is invested in project $j$, and its cash flows scale proportionally with investment. If its PI is $\text{PI}_j$, then its NPV contribution is $y_j(\text{PI}_j-1)$. With just one budget constraint, selecting the greatest NPV per dollar is equivalent to selecting the greatest $\text{PI}_j-1$ per dollar. Since subtracting 1 does not change the ranking, PI ranks projects in the same order.

That logic relies on divisibility and a single constrained resource. If projects must be taken in whole units, selecting the top ratio may leave an unusable amount of budget. If a choice is mutually exclusive, the objective is to select the alternative with the greatest NPV, not the greatest ratio. If multiple resources are constrained, each project's attractiveness depends on how much of each resource it uses.

\text{NPV of investment }y_j = y_j(\text{PI}_j-1), \qquad \text{single budget: }\sum_j y_j\leq B.

This derivation also explains why the special case does not justify a general PI rule. When $y_j$ must be either zero or the full project cost, the company cannot freely allocate the budget in proportion to the ratios. When a second resource is scarce, dollars invested alone do not describe the opportunity cost of using that resource.

Indivisible Projects: A High PI Can Leave Value Unused

Worked Example: A Budget That Favors a Lower-Ranked Pair

A firm has a capital budget of $100 and considers three independent, indivisible projects. The initial investment occurs now, and each project pays one cash flow at the end of year 1. At an effective annual discount rate of $i=10\%$, project A costs $60 and pays $99; projects B and C each cost $50 and pay $79.20. No project can be partially undertaken. Which feasible choice maximizes total NPV?

Method. Discount each cash flow, calculate each project's PI and NPV, then compare feasible selections. This tests whether a ratio ranking remains useful when whole projects compete for a fixed budget.

For A, the present value of the year-1 cash flow is $99/1.10=\$90$. Therefore, $\text{PI}_A=90/60=1.50$, and $\text{NPV}_A=90-60=\$30$.

For B, the present value is $79.20/1.10=\$72$. Thus, $\text{PI}_B=72/50=1.44$, and $\text{NPV}_B=72-50=\$22$. Project C has the same figures: $\text{PI}_C=1.44$ and $\text{NPV}_C=\$22$.

A PI-first rule selects A because $1.50>1.44$. A costs $60, leaving $40, which is insufficient to fund either B or C. The resulting portfolio has NPV of $30.

But B and C together cost $50+50=\$100$, exactly the budget. Their combined NPV is $22+22=\$44$, which exceeds $30. The firm should select B and C. The comparison is feasible: A plus either B or C costs $110 and exceeds the budget, while B plus C is affordable.

Check. The selected pair's present value is $72+72=\$144$, less its $100 investment, giving total NPV $144-100=\$44$. This matches the sum of the individual NPVs.

Common mistake. Selecting A solely because it has the largest PI gives a lower total NPV. The correction is to compare feasible whole-project combinations: ratios do not account for the value of a combination or the budget left unusable after a selection.

In an indivisible-project problem, a PI ranking can still be a quick heuristic, but it is not a proof that the selected portfolio is optimal. A lower-ranked project may fit alongside another project and produce greater combined NPV. The relevant comparison is among feasible selections, not just among individual ratios.

Mutually Exclusive Projects: Choose the Best Alternative

Worked Example: The Larger Alternative Has the Greater NPV

A company must choose exactly one of two hypothetical production systems; installing both is not possible. System A costs $40 now and pays $66 at the end of year 1. System B costs $100 now and pays $154 at the end of year 1. Use an effective annual discount rate of $i=10\%$, and assume the stated cash flows are the only project cash flows. Which system should the company select?

Method. For mutually exclusive alternatives, calculate each NPV at the common discount rate and choose the alternative with the larger NPV, also checking it against doing nothing.

System A has present value $66/1.10=\$60$. Its PI is $60/40=1.50$, and its NPV is $60-40=\$20$.

System B has present value $154/1.10=\$140$. Its PI is $140/100=1.40$, and its NPV is $140-100=\$40$.

PI ranks A above B, but B has the greater NPV. Because the systems are mutually exclusive, the company cannot accept A and then use remaining funds for B; it must choose the alternative with the greatest value. B's $40 NPV exceeds A's $20 NPV and the zero NPV of doing nothing, so select B.

Check. The present value of B's cash flow is $140, which exceeds its $100 cost by $40. The calculation is consistent with the NPV comparison.

Common mistake. Treating the larger PI as the decision rule confuses return per dollar with total value created. The correction is to compare NPV for the alternatives on the same basis. PI may describe relative scale efficiency, but it does not replace the mutually exclusive choice criterion.

This issue is especially important when alternatives have different scales. A smaller project can return more present value per dollar while adding less total value. The company generally seeks the alternative that maximizes shareholder value, represented here by NPV, rather than the alternative with the largest percentage-like ratio.

Multiple Binding Constraints: One Ratio Cannot Capture Every Scarce Resource

Worked Example: A High PI Project Blocks a Better Resource Mix

A firm has a $10 capital budget and at most 10 units of specialized technician time. Three hypothetical projects are indivisible, independent, and each pays at the end of year 1. Project A costs $2, uses 8 technician units, and pays $13.20. Projects B and C each cost $5, use 5 technician units, and pay $14.30. Use $i=10\%$; there are no other cash flows. Find the feasible choice with the greatest NPV.

Method. Calculate discounted value, PI, and NPV for each project, then test choices against both the capital and technician limits. This avoids assuming that the capital-based ratio captures all resource scarcity.

For A, the present value is $13.20/1.10=\$12$. Hence, $\text{PI}_A=12/2=6.00$, and $\text{NPV}_A=12-2=\$10$.

For B, the present value is $14.30/1.10=\$13$. Hence, $\text{PI}_B=13/5=2.60$, and $\text{NPV}_B=13-5=\$8$. Project C also has PI 2.60 and NPV $8.

A PI-first choice selects A. But A uses 8 technician units, so neither B nor C can be added: each would require $8+5=13$ units, greater than the limit of 10. A alone is feasible and has NPV $10.

B and C together use $5+5=10$ technician units and cost $5+5=\$10$. This portfolio satisfies both constraints and has NPV $8+8=\$16$, greater than A's $10. Selecting B and C is therefore better.

Check. Their total present value is $13+13=\$26$; subtracting the $10 total investment gives NPV of $16. Their resource use is exactly 10 budget dollars and 10 technician units.

Common mistake. Ranking only by PI treats each dollar as the only scarce input. The correction is to check all resource limits: A's high capital-based PI does not reflect the technician time it uses and the valuable combination that time prevents.

With multiple constraints, project attractiveness depends on the mix of resources consumed, not just initial investment. A capital-based PI may omit the opportunity cost of labor, floor space, production capacity, or another constrained input. Even if every project has a positive NPV, selecting projects in descending PI order need not produce the feasible portfolio with the greatest total NPV.

A Practical Decision Framework

Before using a PI ranking, identify the decision structure. The ratio is most defensible when projects are divisible, returns scale proportionally with investment, and a single budget is the only binding constraint. Outside that setting, treat PI as descriptive rather than decisive.

Calculate value consistently.
Discount each project's cash flows using the appropriate rate, then compute NPV and PI. Do not compare ratios calculated with different discount-rate assumptions.

State the constraints.
Record whether projects are divisible, mutually exclusive, or subject to capital and noncapital resource limits. These conditions determine which combinations are feasible.

Use the objective that matches the choice.
For mutually exclusive alternatives, compare NPVs. For indivisible projects or multiple constraints, compare feasible selections by their total NPV rather than treating a single PI ranking as conclusive.

The central exam trap is to see “capital rationing” and apply a PI ranking automatically. First ask whether the projects are whole-or-nothing and whether capital is the only scarce resource. A ratio that is appropriate for allocating a divisible amount of one resource can mislead when the actual choice is a set of indivisible projects or when several resources bind.

Key takeaway: PI summarizes present value per dollar invested; it does not, by itself, solve every project-selection problem. Use PI rankings as a selection rule only under the assumptions that make the ratio relevant. When projects are mutually exclusive, indivisible, or constrained by multiple resources, evaluate feasible choices using total NPV.

Check Your Understanding

For each hypothetical case, identify the decision that maximizes value under the stated constraints.

Two mutually exclusive one-year projects have $i=10\%$. Project A costs $30 and pays $42 at year 1; project B costs $70 and pays $92.40 at year 1. Calculate which has the higher PI and which should be selected by NPV. (Answer: A has PI $38.1818/30\approx1.2727$ and NPV about $8.18; B has PI $84/70=1.20$ and NPV $14.00, so select B.)
A firm has a $12 budget and considers indivisible one-year projects at $i=10\%$. A costs $7 and pays $10 at year 1. B and C each cost $6 and pay $8.58 at year 1. Which feasible choice has the greatest NPV? (Answer: Select B and C; each has NPV $1.80, so their combined NPV is $3.60, greater than A's NPV of about $2.09.)
A firm has a $10 budget and 10 technician units. At $i=10\%$, project A costs $3, uses 8 technician units, and pays $16.50 at year 1. Projects B and C each cost $5, use 5 technician units, and pay $14.30 at year 1. All are indivisible. Which feasible choice has the greatest NPV? (Answer: Select B and C; their combined NPV is $16, compared with A's $12, and their combined resource use meets both limits exactly.)
Describe the assumptions that make ranking by PI a valid selection rule for capital rationing, and give one reason that it fails outside those assumptions. (Answer: It is valid for divisible projects whose returns scale proportionally when a single budget is the only binding constraint; indivisibility, mutual exclusivity, or another binding resource can make the highest-PI choice fail to maximize total NPV.)

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