Why NPV Ratios Can Misrank Budgeted Projects

Why a High NPV Ratio Does Not Guarantee the Best Portfolio

When a capital budget is limited, a project with a high net present value (NPV) relative to its cost can look especially attractive. But a ratio is not the same as the total value a project adds. If projects must be accepted or rejected as whole units, choosing the highest ratio first can leave too little budget for a better combination. The decision is about the total NPV of a feasible portfolio, not just the ranking of individual projects.

This comparison assumes that each project is independent, can be accepted only in full or rejected, has a known initial cost, and contributes an NPV that does not change depending on which other projects are selected. All NPVs are calculated using the same required return and consistent cash-flow assumptions. If projects depend on one another or their cash flows interact, simply adding their stand-alone NPVs may not give the portfolio NPV.

Definition: For project $j$, let $C_j>0$ be its initial cost and let $NPV_j$ be its NPV. Its NPV-per-dollar ratio is $NPV_j/C_j$, the NPV associated with each dollar of initial cost. For a budget $B$, a feasible portfolio is a set of projects whose total initial cost is no greater than $B$. If projects are independent, the portfolio NPV is the sum of their NPVs.

The ratio can help describe how much NPV a project offers relative to the capital it uses. It does not, by itself, say which combination of indivisible projects produces the greatest total NPV. The difference matters because a project costing $60 cannot be replaced by exactly $60 of another project if that other project is available only as a whole investment.

Set Up the Portfolio Decision

Let $x_j=1$ if project $j$ is selected and $x_j=0$ otherwise. For independent projects, the objective and budget condition are:

\text{Maximize } \sum_j x_j NPV_j \qquad \text{subject to } \sum_j x_j C_j \le B, \qquad x_j\in\{0,1\}.

The binary condition $x_j\in\{0,1\}$ expresses indivisibility. The objective is total portfolio NPV; the ratio does not appear in it. Ranking by $NPV_j/C_j$ is therefore a heuristic for organizing choices, not a general proof that the highest-ranked projects make the best portfolio.

Why a ratio ranking can seem reasonable: If investments were divisible, so that a decision-maker could purchase any fraction of a project, each dollar assigned to project $j$ would add $NPV_j/C_j$ dollars of NPV under a proportional-value assumption. In that different setting, assigning available funds to the highest ratio first would be sensible. With indivisible projects, however, a chosen project uses its entire cost at once. The unused part of the budget after that choice can be too small to fund another attractive project, while a different combination may fit the budget more effectively.

Compare Ratios and Feasible Portfolios

Worked Example: A Higher Ratio Can Produce a Lower Total NPV

A firm has a $100 capital budget and three independent, indivisible projects. Their initial costs and NPVs, already calculated at the same required return, are: project A costs $60 and has an NPV of $54; project B costs $50 and has an NPV of $42; project C costs $50 and has an NPV of $42. Compare a highest-ratio-first selection with the best feasible portfolio.

Method. Calculate each NPV-per-dollar ratio, apply the ranking while respecting the full project costs, and compare the resulting portfolio with all feasible combinations.

Step 1: Calculate the ratios. For A, $54/60=0.90$. For B, $42/50=0.84$. For C, $42/50=0.84$. A ranks first.

Step 2: Apply the ratio-first choice. Select A for $60, leaving $100-60=$40$. Neither B nor C, each costing $50, fits in the remaining budget. The ratio-first portfolio is A alone, with total NPV $54.

Step 3: Compare other feasible portfolios. B and C together cost $50+50=$100$, so they fit exactly. Their combined NPV is $42+42=$84$. A with B costs $60+50=$110$, and A with C also costs $110, so neither pair is feasible. The feasible single-project NPVs are $54, $42, and $42; the best pair is B and C at $84.

Check. The chosen portfolio costs $100 or less, and its total NPV is $84. It beats A alone by $84-54=$30$, even though A has the highest individual ratio.

Common mistake avoided. Selecting A just because its ratio is largest confuses relative NPV with the maximum total NPV. The correction is to check what else fits after each choice and compare feasible portfolio totals.

This example shows the role of project size and budget fit. A strong ratio describes NPV per dollar, but project A consumes $60 of the $100 budget. The two lower-ratio projects together use the whole budget and generate more NPV. There is no contradiction: the ratio and total NPV answer different questions.

Worked Example: The Highest-Ratio Project Need Not Be in the Best Pair

A company has a $120 budget and considers three independent, indivisible investments. Project U costs $70 and has NPV $65; project V costs $50 and has NPV $49; project W costs $60 and has NPV $60. Determine the ratio-first portfolio and the highest-NPV feasible portfolio.

Step 1: Rank by ratio. U has ratio $65/70=0.9286$ to four decimal places. V has ratio $49/50=0.9800$. W has ratio $60/60=1.0000$. Thus the descending order is W, V, U.

Step 2: Select in that order while respecting the budget. W costs $60, leaving $120-60=$60$. V costs $50 and fits, leaving $60-50=$10$. U costs $70 and cannot fit in the remaining $10. The ratio-first portfolio is W and V, costing $60+50=$110$, with NPV $60+49=$109$.

Step 3: Check all pairs. U and V cost $70+50=$120$ and have NPV $65+49=$114$. U and W cost $70+60=$130$, so that pair is over budget. V and W cost $50+60=$110$ and have NPV $49+60=$109$. The single-project NPVs are $65, $49, and $60. Therefore U and V are the best feasible pair, with total NPV $114.

Check. The highest-ratio project, W, is not in the best portfolio. The best feasible pair, U and V, uses the full $120 budget and exceeds the ratio-first total by $114-109=$5$.

Common mistake avoided. It is not enough to identify the first-ranked project and then assume the resulting portfolio is best. After applying the ratio order, compare other feasible combinations, including a pair that excludes the top-ranked project.

Calculate NPVs Before Comparing Ratios

A project ratio is only as reliable as the NPV and cost used to calculate it. When cash flows are given, first discount them at the required return, then compute the NPV and divide by the initial cost. For an initial outlay at time 0 and future cash flows $CF_t$, the standard NPV calculation is:

NPV_j=-C_j+\sum_{t=1}^{n}\frac{CF_{jt}}{(1+i)^t}.

Here $i$ is the effective annual required return, and $CF_{jt}$ is the project’s cash flow at the end of year $t$. This expression assumes annual periods and cash flows at the stated period ends. Use the actual timing of cash flows if they occur on other dates. The formula produces a value today; dividing it by the time-0 cost gives the ratio used in the comparison.

Worked Example: Discount Cash Flows, Then Test the Ratio Ranking

A firm can spend no more than $100 on independent, indivisible projects. The effective annual required return is 10%, and all receipts occur exactly one year after investment. Project A costs $40 and will pay $74.80 at year 1. Project B also costs $40 and will pay $74.80 at year 1. Project L costs $100 and will pay $181.50 at year 1. Compare the ratio-first selection with the best feasible portfolio.

Step 1: Discount each receipt. The one-year discount factor is $1/(1+0.10)=1/1.10$. For A, present value of the receipt is $74.80/1.10=$68.00$. For B it is also $74.80/1.10=$68.00$. For L, present value is $181.50/1.10=$165.00$.

Step 2: Calculate each NPV. For A, $NPV_A=-40+68=$28$. For B, $NPV_B=-40+68=$28$. For L, $NPV_L=-100+165=$65$.

Step 3: Calculate and rank the ratios. A has $28/40=0.70$. B also has $28/40=0.70$. L has $65/100=0.65$. A and B tie for the highest ratio, ahead of L.

Step 4: Compare feasible portfolios. Selecting A and B costs $40+40=$80$ and gives total NPV $28+28=$56$. Selecting L alone costs $100 and gives NPV $65. Either small project alone gives $28, and L cannot be combined with A or B because $100+40>$100$. Thus L alone is the highest-NPV feasible portfolio.

Check. The ratio-first choice of A and B has a total cost below the budget, but its NPV of $56 is less than L’s NPV of $65. The difference is $65-56=$9$.

Common mistake avoided. Do not treat a higher NPV-per-dollar ratio as proof that a combination of smaller projects has greater total value. Calculate the NPVs using the required return, then compare the totals for the feasible choices.

A Reliable Decision Process

For a small number of projects, direct comparison of feasible portfolios is usually clear and safe. A disciplined process also guards against ratio, cost, and budget errors.

Confirm the decision assumptions.
Establish that each project is indivisible, identify its initial cost, use a common required return, and check whether project NPVs can be added without interaction effects.

Calculate each stand-alone NPV.
Discount dated project cash flows using the required return and calculate NPV relative to the initial outlay. Do not compare ratios formed from inconsistent values.

Calculate ratios as a descriptive ranking.
For every project with positive initial cost, divide NPV by initial cost. Keep the ratio’s role clear: it is a comparison measure, not the portfolio objective.

Check feasible combinations.
For each candidate portfolio, add all selected costs and retain it only if the total is no greater than the budget. Then add its NPVs.

Choose the greatest total NPV.
Compare the feasible portfolio totals, including combinations that omit the highest-ratio project. Verify the selected cost and NPV once more.

The reason this final comparison is necessary follows directly from the objective. Under the stated independence assumption, a portfolio’s value is $\sum_j x_jNPV_j$, while its budget use is $\sum_j x_jC_j$. A ratio compares the two quantities for one project at a time. It does not capture how the project’s whole cost fits alongside the costs and NPVs of other indivisible projects.

Common Exam Traps

Stopping at the largest ratio. A high ratio identifies a project with high NPV relative to its cost; it does not establish the best total portfolio. Check feasible alternatives.
Misstating the remaining budget. After selecting a project, subtract its full cost from the budget. A project fits only if its entire cost is no greater than the amount left.
Forgetting a combination that omits the top-ranked project. A best portfolio can exclude the project with the highest ratio, as in the comparison of U, V, and W.
Comparing a feasible pair with an infeasible one. Add the costs before comparing portfolio NPVs. A larger NPV does not matter if the portfolio exceeds the budget.
Mixing up ratio and NPV. A ratio is NPV divided by initial cost; the portfolio objective is the sum of NPVs. They are not interchangeable.

Key takeaway: For independent, indivisible projects under a fixed budget, select the feasible portfolio with the greatest total NPV. NPV-per-dollar ratios can help organize the comparison, but they can misrank projects because the budget must be allocated in whole-project amounts.

Check Your Understanding

For each problem, compare ratio-based ranking with the total NPV of feasible portfolios. Treat projects as independent and indivisible unless the problem states otherwise.

A $90 budget is available. Project A costs $50 and has NPV $45; B costs $40 and has NPV $30; C costs $40 and has NPV $32. Which portfolio does descending-ratio selection choose, and which feasible portfolio has the greatest NPV? (Answer: Ratios are 0.90, 0.75, and 0.80, respectively; ratio-first selects A and C for NPV $77, which is also the greatest feasible NPV.)
A $100 budget is available. Project A costs $70 and has NPV $63; B costs $40 and has NPV $30; C costs $60 and has NPV $50. What does ratio-first select, and what is the highest-NPV feasible portfolio? (Answer: Ratios are 0.90, 0.75, and 0.8333; ratio-first selects A alone for NPV $63, while B and C cost $100 and yield the highest NPV, $80.)
At an effective annual required return of 8%, two projects each cost $45 and each pay $66.96 at the end of year 1. A third project costs $90 and pays $118.80 at the end of year 1. With a $90 budget, compute the NPVs and determine the highest-NPV feasible portfolio. (Answer: Each smaller project has NPV $-45+66.96/1.08=$17$, and the larger project has NPV $-90+118.80/1.08=$20$; the two smaller projects together yield $34 and are best.)
With a $110 budget, project D costs $60 and has NPV $54, project E costs $50 and has NPV $40, and project F costs $55 and has NPV $49. Compare the descending-ratio selection with the best feasible portfolio. (Answer: Ratios are D = 0.90, E = 0.80, and F = 0.8909; ratio-first selects D and E for total NPV $94, which is also the best feasible portfolio.)
A $120 budget is available. Project G costs $80 and has NPV $72; project H costs $60 and has NPV $52; project J costs $60 and has NPV $51. Which portfolio has the greatest NPV, and does it contain the highest-ratio project? (Answer: Ratios are G = 0.90, H = 0.8667, and J = 0.85; H and J cost $120 and give the greatest NPV, $103, so the highest-ratio project G is not in the best portfolio.)

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