Using the Profitability Index Under a Budget Constraint

Why a Budget Constraint Changes the Decision

When every project with positive net present value can be funded, the decision is straightforward: accept each one. A budget constraint changes the question. If the available capital cannot fund all attractive projects, the decision maker must choose how to use each limited dollar. For divisible, independent projects subject to one investment budget, the profitability index measures the present value created per dollar invested and provides a ranking rule.

The rule is conditional, not universal. It relies on projects being scalable in proportion, on there being a single constraint on initial investment, and on project values adding without interactions. Under those conditions, ranking by the index identifies the allocation that maximizes total net present value. The earlier material on capital rationing with indivisible projects addresses the different case in which a project must be accepted or rejected as a whole.

Definition and Decision Model

Assume each project $j$ requires a positive investment $C_j$ at time 0 and produces future net cash flows whose present value, discounted at the appropriate rate for the project, is $B_j$. The initial investment is excluded from $B_j$. Thus, the project's net present value is $B_j-C_j$. The standard profitability index for this setting is the present value of future net cash flows per dollar of initial investment.

Definition: The profitability index of project $j$ is $PI_j=B_j/C_j$, where $B_j$ is the present value of its future net cash flows and $C_j$ is its initial investment. Since $NPV_j=B_j-C_j$, the equivalent expression is $PI_j=1+NPV_j/C_j$. An index above 1 corresponds to positive NPV; an index equal to 1 corresponds to zero NPV; an index below 1 corresponds to negative NPV.

To apply the index under a budget $K$, let $x_j$ be the fraction of project $j$ undertaken. Divisibility means any fraction from 0 to 1 is feasible, and both its investment and its future cash flows scale in the same proportion. Independence means choosing one project does not change another project's cash flows, cost, or feasibility. The single budget limits total time-0 investment.

0 \leq x_j \leq 1,\qquad \sum_j C_jx_j \leq K,\qquad \text{maximize }\sum_j (B_j-C_j)x_j.

The objective is total NPV, not the sum of the selected projects' indices. If all available projects have positive NPV and the budget is binding, the budget is allocated in descending order of $PI_j$, taking each project fully until the remaining budget is too small, then taking the affordable fraction of the next project. A project with $PI_j<1$ should not be funded merely to use the entire budget.

Why the Ranking Works

The profitability index can be read as a benefit per invested dollar: investing $C_jx_j$ in project $j$ produces present value $B_jx_j$. To see why the ranking is optimal, compare two projects when some capital is assigned to the lower-index project while the higher-index project is not yet fully funded.

Compare value per dollar.
Let project $h$ have $PI_h>PI_l$, where $l$ is the lower-index project. For any positive investment amount $\Delta$ that can be shifted between them, the higher-index project produces $\Delta PI_h$ of present value, while the lower-index project produces $\Delta PI_l$.

Shift the investment.
Reduce the investment in project $l$ by $\Delta$ and increase the investment in project $h$ by the same amount. The total budget use is unchanged, and divisibility makes the transfer feasible while both projects remain within their limits.

Check the change in NPV.
The shift increases total NPV by $\Delta(PI_h-PI_l)$, which is positive because $\Delta>0$ and $PI_h>PI_l$. Therefore an allocation that funds the lower-index project while leaving the higher-index project partly unfunded cannot be optimal.

Repeating this exchange gives the descending-index allocation. Equivalently, because $PI_j=1+NPV_j/C_j$, ranking by $PI_j$ is the same as ranking by NPV per dollar of investment. The proof depends on the proportional scaling and single-budget assumptions: those assumptions ensure that a dollar shifted between projects has the same budget cost and that the project's value changes proportionally.

Applying the Index to a Constrained Budget

Worked Example: Allocating a Budget Across Three Projects

A firm has an $80,000 time-0 budget. Three hypothetical independent projects are divisible, and the present values of their future net cash flows have already been calculated using the appropriate project discount rates. Project A requires $60,000 and has a present value of $90,000; Project B requires $50,000 and has a present value of $70,000; Project C requires $40,000 and has a present value of $52,000. Find the index ranking, optimal allocation, and total NPV.

Method. Calculate each present-value-to-investment ratio, then fund projects in descending order until the budget is exhausted. This method avoids ranking by total NPV alone.

For A, $PI_A=90{,}000/60{,}000=1.50$, and $NPV_A=90{,}000-60{,}000=\$30{,}000$. For B, $PI_B=70{,}000/50{,}000=1.40$, and $NPV_B=70{,}000-50{,}000=\$20{,}000$. For C, $PI_C=52{,}000/40{,}000=1.30$, and $NPV_C=52{,}000-40{,}000=\$12{,}000$. The ranking is A, B, then C.

Fund A fully, using $60,000. The remaining budget is $80{,}000-60{,}000=\$20{,}000$. This funds $20{,}000/50{,}000=0.40$ of B. No budget remains for C. The total NPV is A's $30,000 plus 0.40 of B's $20,000: $30{,}000+0.40(20{,}000)=\$38{,}000$.

Check. The allocated investment is $60{,}000+0.40(50{,}000)=\$80{,}000$, exactly the budget. The present value of allocated future net cash flows is $90{,}000+0.40(70{,}000)=\$118{,}000$; subtracting the $80,000 investment gives $38,000 NPV, as calculated.

Calculating Present Values Before Ranking

An index is only as reliable as the present values used to calculate it. For cash flows at different dates, discount each flow to time 0 using the rate appropriate to the project, add the discounted values, and then divide by the time-0 investment. The following example makes the discounting step explicit.

Worked Example: Ranking Projects From Their Cash Flows

At an annual effective discount rate of 8%, a company has a $90,000 budget for two hypothetical independent, divisible projects. Project W costs $75,000 and pays $43,200 at the end of year 1 and $46,656 at the end of year 2. Project Z costs $50,000 and pays $32,400 at the end of year 1 and $34,992 at the end of year 2. Find the index ranking, allocation, and total NPV.

Method. Discount each dated cash flow at 8%, calculate each index, then allocate the budget in index order. Do not compare the undiscounted cash-flow totals with the initial costs.

For W, the year-1 present value is $43{,}200/1.08=\$40{,}000$. Since $1.08^2=1.1664$, the year-2 present value is $46{,}656/1.1664=\$40{,}000$. Therefore, $B_W=40{,}000+40{,}000=\$80{,}000$, $PI_W=80{,}000/75{,}000=1.0667$ to four decimal places, and $NPV_W=80{,}000-75{,}000=\$5{,}000$.

For Z, the year-1 present value is $32{,}400/1.08=\$30{,}000$, and the year-2 present value is $34{,}992/1.1664=\$30{,}000$. Thus $B_Z=\$60{,}000$, $PI_Z=60{,}000/50{,}000=1.20$, and $NPV_Z=\$10{,}000$. Z ranks ahead of W.

Fund Z fully for $50,000, leaving $90{,}000-50{,}000=\$40{,}000$. The funded fraction of W is $40{,}000/75{,}000=8/15$. Total NPV is $10{,}000+(8/15)(5{,}000)=\$12{,}666.67$, rounded to the nearest cent.

Check. The allocation costs $50{,}000+(8/15)(75{,}000)=\$90{,}000$. Its present value of future net cash flows is $60{,}000+(8/15)(80{,}000)=\$102{,}666.67$. Subtracting the budget gives $12,666.67, consistent with the NPV calculation.

Break-Even and Negative-Value Projects

An index of exactly 1 means the present value of future net cash flows equals the initial investment. Funding any fraction of such a project adds zero NPV under the model. An index below 1 means each invested dollar reduces NPV, so it should not be included in an NPV-maximizing allocation when the firm can leave capital unused. The positive-index ranking is therefore applied only after identifying projects with positive NPV.

Worked Example: Deciding Whether to Leave Part of the Budget Unused

A firm has a $70,000 budget and three hypothetical, independent, divisible opportunities. Project D costs $40,000 and has future cash flows with a present value of $52,000. Project E costs $30,000 and has a present value of $30,000. Project F costs $20,000 and has a present value of $18,000. Determine the NPV-maximizing allocation.

Method. Calculate each index and exclude projects with negative NPV rather than spending the budget just because it is available.

For D, $PI_D=52{,}000/40{,}000=1.30$, so $NPV_D=\$12{,}000$. For E, $PI_E=30{,}000/30{,}000=1.00$, so $NPV_E=\$0$. For F, $PI_F=18{,}000/20{,}000=0.90$, so $NPV_F=-\$2{,}000$. Fund D fully, using $40,000. Funding E uses another $30,000 and leaves total NPV at $12,000. Funding any fraction of E does not change NPV; funding F would reduce it. One optimal allocation is therefore D and E in full, with F rejected.

Check. The allocation uses $40{,}000+30{,}000=\$70{,}000$. Its total NPV is $12{,}000+0=\$12{,}000$. If the remaining $20,000 were instead invested in F, total NPV would fall to $12{,}000-2{,}000=\$10{,}000$.

Assumptions, Common Traps, and Practical Use

The index rule is an exact optimization method only for the model described. In particular, “divisible” must mean that taking a fraction of a project proportionally scales both the initial investment and every relevant future cash flow. A project with a fixed setup cost or a minimum viable scale may not have this property. Independence means the value of one project does not depend on which other projects are selected. The budget constraint must be the sole binding resource constraint; a separate limit on staffing, capacity, or another input can change the allocation problem.

The initial costs are assumed to occur at the same decision date, and the benefit numerator is the present value of future net cash flows, discounted consistently to that date. If projects have different risk, their cash flows should be valued using discount rates appropriate to that risk; a comparison built from inconsistent present values is not meaningful. The standard ratio also assumes a clearly identified initial investment denominator. When a project's cash outlays occur at several dates or the project has unusual financing or scale features, the analyst must define the investment measure and decision model carefully rather than apply the simple ratio mechanically.

A frequent exam trap is to rank by total NPV instead of NPV per dollar. In the first worked example, A has NPV $30,000 and B has NPV $20,000, so raw NPV ranks A first; but the relevant comparison under a divisible budget is $30{,}000/60{,}000=0.50$ of NPV per invested dollar for A versus $20{,}000/50{,}000=0.40$ for B. Here the order agrees, but that need not happen. For example, a project costing $100,000 with $40,000 NPV has $PI=1.40$, while one costing $20,000 with $12,000 NPV has $PI=1.60$. Under the stated divisible-project model, the smaller project ranks first because its NPV per dollar is higher. Choosing by total NPV can therefore use scarce capital less effectively.

A related error is to interpret a high index as proof that a project creates the greatest total value. The index is a ratio, while NPV is a dollar amount. The ratio determines the order in which constrained, divisible investments are funded; total NPV measures the dollar contribution at the chosen scale. Always report both the allocation and the resulting portfolio NPV.

Key takeaway: With independent projects that scale proportionally and one time-0 budget, rank positive-NPV projects by $PI=PV(\text{future net cash flows})/(\text{initial investment})$. Fund them in descending order, using a fraction of the marginal project if needed. The conclusion depends on those assumptions.

Check Your Understanding

For each problem, assume the projects are independent and divisible unless stated otherwise. Use the stated present values or discount rate and identify the budget allocation where requested.

A project costs $30,000 and the present value of its future net cash flows is $42,000. If the budget permits only $20,000 of investment in it, what fraction is funded and what NPV does that fraction contribute? (Answer: $PI=1.40$; fraction $=2/3$; NPV contribution $=\$8{,}000$.)
A $70,000 budget is available. Project M costs $40,000 and has a future-cash-flow present value of $52,000; Project N costs $60,000 and has a present value of $72,000. Rank them and determine the optimal allocation and total NPV. (Answer: $PI_M=1.30$, $PI_N=1.20$; fund M fully and half of N; total NPV $=\$18{,}000$.)
At an annual effective rate of 5%, Project R costs $50,000 and pays $31,500 at year 1 and $33,075 at year 2. Project S costs $40,000 and pays $21,000 at year 1 and $22,050 at year 2. With a $60,000 budget, calculate each index and the optimal allocation. (Answer: $B_R=\$60{,}000$, $PI_R=1.20$; $B_S=\$40{,}000$, $PI_S=1.00$; fund R fully and one-quarter of S; total NPV $=\$10{,}000$.)
A $70,000 budget is available for three projects: A costs $40,000 and has present value $52,000; B costs $30,000 and has present value $30,000; C costs $20,000 and has present value $18,000. Which projects should be funded, and what is total NPV? (Answer: fund A and B; reject C; total NPV $=\$12{,}000$.)
Two projects have $C_1=\$100{,}000$, $B_1=\$140{,}000$, $C_2=\$20{,}000$, and $B_2=\$32{,}000$. With a $20,000 budget, which project should be funded under the divisible-project model, and what is the resulting NPV? (Answer: $PI_1=1.40$, $PI_2=1.60$; fund Project 2; NPV $=\$12{,}000$.)

What You'll Learn