Capital Rationing Across Dependent Projects

Why Project Rankings Are Not Enough

When a firm cannot fund every value-creating investment, it faces capital rationing: a limited amount of capital must be allocated among competing projects. The decision is not necessarily to select the project with the highest standalone NPV, or to follow a ranking by profitability index. If projects are indivisible, choosing one project changes how much capital remains for the others. If projects depend on, exclude, or complement one another, their values and feasibility may also depend on the combination selected.

The appropriate decision is to identify the feasible portfolio with the highest total incremental NPV. This tutorial considers a fixed time-zero budget and hypothetical projects. The analysis is aligned with the SOA VEE Accounting and Finance syllabus.

Definition: A capital-rationing problem selects a portfolio of projects subject to a capital constraint. A project is indivisible when it must be accepted in full or rejected; it cannot be scaled to use only part of its required investment. Projects are dependent when accepting one requires accepting another, mutually exclusive when at most one can be accepted, and complementary when their combined value includes an incremental benefit beyond the sum of their separate values.

Let $x_j=1$ if project $j$ is selected and $x_j=0$ otherwise. Let $I_j$ be its required time-zero investment, $V_j$ its NPV, and $B$ the available budget. When project NPVs are additive, the basic model is:

\max \sum_j V_jx_j \quad\text{subject to}\quad \sum_j I_jx_j\le B,\qquad x_j\in\{0,1\}.

The binary restriction represents indivisibility. A prerequisite relationship in which project B requires project A is expressed as $x_B\le x_A$. If A and B are mutually exclusive, the constraint is $x_A+x_B\le1$. Constraints should reflect the actual project relationships; they are not substitutes for valuing project cash flows correctly.

Value the Combinations, Not Just the Standalone Projects

An NPV is the present value of a project's incremental future cash flows, less its required investment, discounted at a rate appropriate to the risk of those cash flows. Use a consistent valuation date, currency, and treatment of taxes and other cash-flow effects. The earlier material on risk-matched discount rates and project cash flows is relevant: a limited budget does not change the method for valuing an individual project's cash flows.

If project cash flows are genuinely separate and are measured relative to the same baseline, the NPV of a portfolio is the sum of the selected NPVs. This additivity is the reason the objective function can be written as a sum. However, if projects share cash flows, one changes another's costs or revenues, or a joint benefit exists, simply adding standalone NPVs may misstate portfolio value. Shared or incremental effects must be incorporated once, in the appropriate combination's value.

Justification: Present value is linear: for cash flows $C_{1,t}$ and $C_{2,t}$ at time $t$, discounted at the same rate $r$, $$ \sum_t\frac{C_{1,t}+C_{2,t}}{(1+r)^t} = \sum_t\frac{C_{1,t}}{(1+r)^t} + \sum_t\frac{C_{2,t}}{(1+r)^t}. $$ If the investments and cash flows are independent in the valuation sense, subtracting both time-zero investments preserves this equality, so the combined NPV equals the sum of the individual NPVs. If a joint effect adds cash flows $S_t$, the portfolio NPV instead includes their present value, $\sum_t S_t/(1+r)^t$, as an additional term. The equality does not justify counting a shared benefit twice.

A common screening measure is the profitability index, defined here for a project with positive initial investment $I_j$ and present value of future inflows $PV_j$ as $PI_j=PV_j/I_j$. Since $V_j=PV_j-I_j$, this is also $1+V_j/I_j$. A high index indicates more present value per dollar invested, but it does not by itself solve an indivisible, constrained selection problem. Ranking by index is most useful under restrictive conditions, such as divisible projects and a single capital constraint. Interdependencies and indivisibility require testing feasible combinations directly.

Worked Examples

Worked Example: A Prerequisite Changes the Best Portfolio

A hypothetical firm has a time-zero capital budget of $100. Four indivisible projects have the following investments and standalone NPVs, all in dollars. Project B can be undertaken only if A is also selected. There are no other interactions. Which feasible portfolio maximizes total NPV, and why is selecting projects in profitability-index order unsafe?

Method. Calculate each index, impose the prerequisite, and compare the feasible portfolios. For each project, $PI=1+V/I$.

Indices. For A, $PI_A=1+20/40=1.50$. For B, $PI_B=1+40/50=1.80$. For C, $PI_C=1+45/60=1.75$. For D, $PI_D=1+24/40=1.60$. The ranking is B, C, D, A.

Apply the dependency. B cannot be selected by itself: $x_B\le x_A$. Selecting B therefore requires A, with total investment $40+50=90$ and NPV $20+40=60$. This leaves only $10, so no other project can be added.

Compare other feasible combinations. C and D together use $60+40=100$ and have NPV $45+24=69$. A and C use 100 and have NPV $20+45=65$. A and D use 80 and have NPV $20+24=44$. A and B are feasible at cost 90 and NPV 60. Combinations including B must also include A, so B cannot be paired with C or D within the budget.

The best portfolio is C and D, with total NPV of $69. The index ranking might lead a decision-maker to start with B and reserve enough capital for its prerequisite A; that feasible choice yields only $60. This example avoids the trap of treating a project ranking as though it automatically accounted for dependency.

Check. C and D use exactly the $100 budget, and their combined NPV, $69, exceeds each other feasible portfolio considered.

Worked Example: Complementary Projects and Incremental Value

A hypothetical firm has a $120 budget. A platform project A costs $70 and has a standalone NPV of $30. A data module B costs $50 and has a standalone NPV of $8. If both are undertaken, integration creates an additional NPV of $25. Independent projects C and D cost $60 each and have NPVs of $32 and $28, respectively. No other interactions apply. Choose the best portfolio.

Method. Compare feasible portfolios, adding the integration benefit exactly once when both A and B are selected.

Value A and B together. Their investment is $70+50=120$, within budget. Their combined NPV is $30+8+25=63$, where $25 is the stated incremental integration value.

Compare the alternatives. C and D together also cost $60+60=120$, and their combined NPV is $32+28=60$. B with C costs $50+60=110$ and has NPV $8+32=40$. B with D costs $50+60=110$ and has NPV $8+28=36$. A with either C or D costs at least $70+60=130$, above budget. Each single project has a lower NPV than the best two-project portfolio.

The optimal choice is A and B, with total NPV $63, which exceeds C and D by $3. A ranking of standalone indices would not reveal the full comparison: $PI_A=1+30/70\approx1.429$, $PI_B=1+8/50=1.160$, $PI_C=1+32/60\approx1.533$, and $PI_D=1+28/60\approx1.467$. C and D rank ahead individually, yet their combined value is lower than the value of the complementary pair.

Check. The $25 integration benefit has been added once, not assigned to both projects and counted twice. The selected portfolio uses exactly $120 and has the highest feasible NPV.

Worked Example: Enumerating a Portfolio With Several Constraints

A hypothetical firm has $150,000 available at time zero. All amounts below are in thousands of dollars. Each project has one cash inflow at the end of year 1; the annual effective discount rate is 10%, and taxes and other cash flows are ignored. Project B requires A. Projects C and D are mutually exclusive. Project data are: A costs 50 and receives 64.9; B costs 70 and receives 103.4; C costs 80 and receives 116.6; D costs 60 and receives 85.8; E costs 40 and receives 57.2. Find the portfolio with the highest NPV.

Method. First calculate the individual NPVs using $NPV=-I+CF_1/(1.10)$. Then enforce the budget, prerequisite, and exclusivity constraints while comparing combinations.

Calculate the NPVs. For A, $64.9/1.10=59$, so $NPV_A=59-50=9$. For B, $103.4/1.10=94$, so $NPV_B=94-70=24$. For C, $116.6/1.10=106$, so $NPV_C=106-80=26$. For D, $85.8/1.10=78$, so $NPV_D=78-60=18$. For E, $57.2/1.10=52$, so $NPV_E=52-40=12$. These values are in thousands.

Test the leading feasible combinations. A and B cost $50+70=120$ and have NPV $9+24=33$. B is valid in this combination because A is included. A, D, and E cost $50+60+40=150$ and have NPV $9+18+12=39$; this portfolio does not violate the C-versus-D restriction. C and E cost $80+40=120$ and have NPV $26+12=38$. A and C cost $50+80=130$ and have NPV $9+26=35$. D and E cost 100 and have NPV $18+12=30$.

Other constraints rule out tempting combinations: C and D cannot be selected together, and B cannot be selected without A. Adding E to A and B would cost $120+40=160$, above the budget. Adding E to A and C would also cost 170. No feasible combination using B and an additional project improves on A and B: A, B, and E exceed the budget, while adding C or D costs still more.

The maximizing portfolio is A, D, and E, with NPV $39,000. It uses the full $150,000 budget. This is $1,000 higher than C and E, the next-best combination identified, and $6,000 higher than A and B.

Check. A is included with D and E, so the prerequisite constraint is satisfied; C is not selected alongside D; and the investments total $50+60+40=150$ thousand.

A Reliable Selection Procedure

For a modest number of projects, a transparent enumeration is often safer than relying on a shortcut. For a larger problem, the same logic can be implemented as a binary optimization model. In either case, the objective is value maximization, not necessarily full use of the budget: unused capital is acceptable if no available feasible project combination adds positive NPV without displacing a more valuable use.

Value each project.
Estimate its incremental cash flows and NPV at the appropriate discount rate, on a consistent valuation basis.

Record the constraints.
State the budget, indivisibility, prerequisites, mutual exclusions, and any other required project relationships explicitly.

Adjust for interactions.
Include combination-specific costs, shared cash flows, or complementary value once, so each candidate portfolio has a correct NPV.

Compare feasible portfolios.
Reject combinations that violate any constraint and select the feasible combination with the greatest total NPV.

Common Traps and Why the Choice Matters

The central trap is to confuse ranking with optimization. A high NPV project may consume capital needed for a better combination; a high profitability index may belong to a project that requires a costly prerequisite; and a modest standalone project may be valuable as part of a complementary bundle. The correct comparison is always between feasible portfolios valued on a consistent basis.

For example, the following calculation is wrong if projects are indivisible and B requires A:

\text{Wrong: select B alone because }PI_B\text{ is largest, so investment}=50,\quad NPV=40.

B alone is not feasible: the prerequisite constraint $x_B\le x_A$ is violated. The correction is to include A and compare the valid bundle A+B, with investment $40+50=90$ and NPV $20+40=60$, against all other feasible portfolios. In the example, C+D costs 100 and yields NPV 69, so it is superior.

Another mistake is to add standalone NPVs when the projects share a benefit or cost, or to count a synergy once in each project's standalone value and again as a bundle adjustment. Define the cash flows relative to a common baseline and account for each incremental effect exactly once. Finally, check units: a budget stated in thousands must be compared with investments stated in thousands, not with dollar amounts left unconverted.

Capital rationing matters because accepting one project can prevent acceptance of another, even when both have positive NPV. A disciplined portfolio comparison preserves the value-maximizing principle while respecting real capital and project constraints. A profitability index can help describe projects, but it is not a substitute for checking feasibility and comparing total portfolio value.

Check Your Understanding

For each problem, identify the best feasible portfolio and its total NPV.

A firm has a $90 budget. Independent, indivisible projects A, B, and C have investments and NPVs of $50 and $28, $40 and $19, and $45 and $24, respectively. Which feasible portfolio maximizes NPV? (Answer: A and B; total NPV $47.)
A firm has a $100 budget. Project S costs $50 and has NPV $26, but requires project R, which costs $40 and has NPV $14. Independent projects T and U cost $60 and $40 and have NPVs $35 and $20. Which portfolio is best? (Answer: T and U; total NPV $55.)
A firm has a $100 budget. Projects A and B cost $55 and $45 and have standalone NPVs $20 and $7; undertaking both creates an additional NPV of $18. Independent projects C and D cost $50 each and have NPVs $23 and $22. What is the maximum portfolio NPV, and is there a tie? (Answer: A and B or C and D; maximum NPV $45, a tie.)
A firm has a $140 budget. Project Q costs $60 and has NPV $25, but requires P, which costs $50 and has NPV $11. Projects R and S each cost $70, have NPVs $29 and $26, and are mutually exclusive. Independent project T costs $40 and has NPV $13. What is the best feasible portfolio? (Answer: R and T; total NPV $42.)
A project requires $80,000 now and pays $95,000 one year later. Using an effective annual discount rate of 8%, calculate its NPV to the nearest cent. If a separate indivisible project requires $60,000 and has NPV $8,000, can both be selected under a $100,000 budget, and what is their combined NPV if they can? (Answer: First project NPV is $7,962.96; both cannot be selected because their total investment is $140,000.)

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