Using Unused Budget When Projects Are Independent

Introduction

A capital budget is a limit on how much a firm can invest, not a requirement to spend every available dollar. If a selected group of projects leaves capital unused, a further project may be worth adding when it is independent, has positive NPV, and its initial investment fits within the remainder. But an unused balance does not turn a value-destroying project into a good investment.

The distinction is important: a project can be affordable without being worthwhile, and it can be worthwhile without fitting the budget left by a particular set of earlier choices. The decision should reflect both the project's contribution to value and the total investment required by the portfolio.

Definition: Let $B$ be the total capital budget, and let $I_j$ be project $j$'s initial investment. For a selected portfolio $S$, the unused budget is $B-\sum_{j\in S} I_j$. A project fits the remainder if its initial investment is no greater than that amount. A positive-NPV project increases portfolio value when added, provided its cash flows are independent of the other selected projects and there are no additional constraints.

Use NPV and the Budget Together

For a project with cash flows $CF_0,CF_1,\ldots,CF_n$ and required return $r$, its NPV is the present value of the cash flows, including the initial outlay. The outlay is ordinarily a negative $CF_0$. If the project instead has an initial investment $I$ stated as a positive amount, the same calculation is written as follows:

\operatorname{NPV}=-I+\sum_{t=1}^{n}\frac{CF_t}{(1+r)^t}

The budget constraint concerns initial investment: a portfolio is feasible when its total initial investment is no greater than $B$. NPV measures the value added by those projects at the required return. These quantities answer different questions, so neither can replace the other.

When projects are independent, accepting one does not change another project's cash flows or NPV. Therefore, portfolio NPV is the sum of the individual NPVs. This follows from the linearity of discounting: the present value of the combined cash flows is the sum of the present values of the separate cash flows. Subtracting the sum of initial investments gives the same result.

\operatorname{NPV}(S) =-\sum_{j\in S} I_j+\sum_{t=1}^{n}\frac{\sum_{j\in S}CF_{j,t}}{(1+r)^t} =\sum_{j\in S}\left(-I_j+\sum_{t=1}^{n}\frac{CF_{j,t}}{(1+r)^t}\right) =\sum_{j\in S}\operatorname{NPV}_j

Thus, with a binding capital budget, the objective is to select a feasible portfolio with the highest total NPV. If a positive-NPV project fits the remaining budget, adding it increases total NPV. If it does not fit, the decision is not automatically settled: replacing one or more selected projects may create a better feasible portfolio. This is why checking the remainder is useful but is not always a substitute for comparing feasible portfolios.

Calculate the selected portfolio's investment and NPV.
Add initial investments to check feasibility, and add NPVs to measure the portfolio's value contribution.

Find the remaining capital.
Subtract the portfolio's total initial investment from the budget.

Test a further project.
Check that it is independent, fits the remaining capital, and has positive NPV. If it does, adding it raises total NPV.

Check alternatives and stop at value destruction.
Compare other feasible combinations if a better allocation may exist. Do not accept a negative-NPV project merely to spend the remainder.

Worked Examples

Worked Example: A Negative-NPV Project Fits the Remainder

A firm has a budget of $120 thousand and considers four independent, one-year projects. Each project requires its investment now and pays its stated cash flow at the end of year 1. The required return is 10% effective annually. The investments and year-end cash flows, in thousands of dollars, are: A, 70 and 105.6; B, 28 and 42.9; C, 17 and 25.3; D, 4 and 3.3. Which projects maximize total NPV, and what should the firm do about any unused budget?

Method. Discount each year-end cash flow at 10%, subtract the initial investment, and compare the total investment and NPV of the projects. Because each cash flow occurs at the end of one year, its present value is the cash flow divided by 1.10.

For A, the present value is $105.6/1.10=96$, so $\operatorname{NPV}_A=96-70=26$ thousand.

For B, the present value is $42.9/1.10=39$, so $\operatorname{NPV}_B=39-28=11$ thousand.

For C, the present value is $25.3/1.10=23$, so $\operatorname{NPV}_C=23-17=6$ thousand.

For D, the present value is $3.3/1.10=3$, so $\operatorname{NPV}_D=3-4=-1$ thousand.

The three positive-NPV projects together require $70+28+17=115$ thousand and have total NPV $26+11+6=43$ thousand. They are within the 120-thousand budget, leaving $120-115=5$ thousand. Project D costs 4 thousand, so it fits that remainder. But adding it changes total NPV from 43 thousand to $43-1=42$ thousand. The highest-value choice is A, B, and C, with 43 thousand of NPV and 5 thousand unused.

Check. The selected investment is 115 thousand, which is no greater than 120 thousand; adding D would bring investment to 119 thousand and total NPV down to 42 thousand. The common mistake is to treat affordability as a reason to accept D. Its negative NPV means it reduces value, even though it fits.

Worked Example: A Small Remainder Does Not Settle Every Portfolio Choice

A firm has a budget of $100 thousand. Four independent, indivisible one-year projects have the following initial investments and NPVs, both in thousands of dollars: A, 65 and 25; B, 22 and 10; C, 12 and 5; D, 2 and 1. The required return used to calculate each stated NPV is 10% effective annually. Find the highest-NPV feasible portfolio and explain what happens to the last unused thousand.

Method. Compare feasible combinations using total investment and the sum of NPVs. The projects are indivisible, so a project either receives its full initial investment or is not selected.

A, B, and C together require $65+22+12=99$ thousand and produce NPV $25+10+5=40$ thousand. They are feasible and leave $100-99=1$ thousand.

Adding D to that portfolio would require $99+2=101$ thousand, so it does not fit. The fact that D has positive NPV does not allow the firm to exceed its budget.

To check whether another feasible portfolio does better, consider combinations that include D. A, B, and D cost $65+22+2=89$ thousand and have NPV $25+10+1=36$ thousand. A, C, and D cost $65+12+2=79$ thousand and have NPV $25+5+1=31$ thousand. B, C, and D cost $22+12+2=36$ thousand and have NPV $10+5+1=16$ thousand. A, C, and B already give the best value among these named alternatives at 40 thousand; the remaining subsets without all three positive projects have no larger total NPV. Therefore, A, B, and C are the highest-NPV feasible portfolio.

Check. Its $99-thousand investment is within the $100-thousand budget, and D cannot be added because it would bring investment to $101 thousand. The 1-thousand remainder is left unused. The common mistake is to infer that every positive-NPV project must be accepted; the budget makes D infeasible with this portfolio, and other feasible combinations have lower total NPV.

Worked Example: Add a Project That Fits the Remainder

A firm with a $100-thousand budget is considering independent, one-year projects P, Q, and R. Each requires an initial investment now and pays one cash flow at the end of year 1. The required return is 10% effective annually. P costs 80 thousand and pays 121 thousand; Q costs 30 thousand and pays 52.8 thousand; R costs 20 thousand and pays 33 thousand. Determine the best feasible portfolio.

Method. Calculate each NPV by discounting its year-end cash flow at 10%. Then compare the feasible combinations. This checks whether a project can be added to the remainder without assuming that the first project considered must be selected.

For P, the present value is $121/1.10=110$ thousand, so $\operatorname{NPV}_P=110-80=30$ thousand. For Q, the present value is $52.8/1.10=48$ thousand, so $\operatorname{NPV}_Q=48-30=18$ thousand. For R, the present value is $33/1.10=30$ thousand, so $\operatorname{NPV}_R=30-20=10$ thousand.

P alone is feasible and leaves $100-80=20$ thousand. R costs 20 thousand, so it fits exactly; P and R together require 100 thousand and have NPV $30+10=40$ thousand. Q does not fit the 20-thousand remainder after P.

For comparison, Q and R together cost $30+20=50$ thousand and have NPV $18+10=28$ thousand. P and Q together cost 110 thousand, and all three together cost 130 thousand, so neither combination is feasible. The best feasible portfolio is P and R, with NPV 40 thousand.

Check. The chosen portfolio uses exactly 100 thousand and its NPV equals the sum of its independent project NPVs. The common mistake is to stop at P because Q does not fit the remainder. R does fit, and adding it raises total NPV.

Why Unused Capital Is Not a Target

A capital budget places an upper bound on investment. It does not create value merely because funds are left over. Under the NPV decision rule, a negative-NPV project has a present value of benefits below its required investment at the stated required return. Accepting it reduces the value contributed by the portfolio. Spending the remaining budget on that project does not reverse the comparison.

Likewise, an unused balance is not automatically evidence that the portfolio is wrong. It may remain because no additional positive-NPV project fits, because every affordable candidate has negative NPV, or because the available indivisible projects do not combine into a higher-value feasible portfolio. A positive-NPV project that fails the remaining-budget test could still be worth considering if replacing some selected projects makes a better feasible portfolio; compare the total NPVs rather than making the decision from the remainder alone.

These conclusions rely on the stated model: the budget constrains initial investment, projects are independent, and the relevant project values are their NPVs at the required return. If projects depend on one another, share resources, or change each other's cash flows, their NPVs may not simply add. Those circumstances require evaluating the relevant combined cash flows and constraints.

Key takeaway: Add an independent positive-NPV project when it fits the available capital and improves the feasible portfolio. Leave funds unused rather than accepting a negative-NPV project simply to exhaust the budget.

Check Your Understanding

For each problem, use the stated budget and NPVs to decide whether the proposed allocation is feasible and whether unused capital should be committed.

A selected independent-project portfolio costs $31 thousand under a $50-thousand budget and has NPV of $12 thousand. A further project costs $18 thousand and has NPV of $4.2 thousand. What is the remainder, and should the further project be added? (Answer: The remainder is $19 thousand; add the project, giving total investment of $49 thousand and total NPV of $16.2 thousand.)
A $80-thousand budget is allocated to project L, costing $49 thousand with NPV $14 thousand, and project M, costing $22 thousand with NPV $6 thousand. Independent project N costs $10 thousand and has NPV $3 thousand. Does N fit after L and M, and what is the highest-NPV feasible portfolio among these projects? (Answer: L and M leave $9 thousand, so N does not fit; L and M have NPV $20 thousand, the highest feasible total.)
A $60-thousand budget is currently allocated to independent projects costing $44 thousand in total and producing NPV $12 thousand. An additional project costs $15 thousand and has NPV $5 thousand. What should the decision-maker do, assuming no other constraints? (Answer: Add it; total investment is $59 thousand, leaving $1 thousand, and total NPV is $17 thousand.)
A firm has $100 thousand to allocate among independent projects: A costs $65 thousand and has NPV $24 thousand; B costs $30 thousand and has NPV $12 thousand; C costs $20 thousand and has NPV $9 thousand; D costs $10 thousand and has NPV $4 thousand. Which feasible portfolio has the highest NPV? (Answer: A, C, and D cost $95 thousand and have NPV $37 thousand, the highest feasible total.)
A firm has $7 thousand of unused budget. An independent project costs $5 thousand and has NPV $-$0.6$ thousand. Should it be accepted solely to use some of the remainder? (Answer: No; it fits, but accepting it reduces portfolio NPV by $0.6 thousand.)

What You'll Learn