How Project Scale Changes NPV

Why Project Scale Matters

Two projects can have similar cash-flow patterns and very different investment sizes. A smaller project may produce a higher surplus relative to its cost, while a larger project may create more total dollars of value. To compare their contribution to value, calculate each project's total net present value using the appropriate required return; do not infer value from the initial investment alone.

The focus here is the dollar amount represented by NPV. The comparison is most direct when projects have comparable risk and use the same required return. If their risks differ, each cash-flow forecast must be discounted at a rate appropriate to that project's risk; the resulting NPVs then reflect both the forecasts and those required returns.

Definition: For a project with cash flow $C_t$ at the end of year $t$, where $C_0$ is the initial cash flow (usually negative), and a constant effective annual required return $r$, its NPV is the present value of all project cash flows: $$ \operatorname{NPV}=\sum_{t=0}^{n}\frac{C_t}{(1+r)^t}. $$ The initial investment is included as $C_0$; do not subtract it again if it is already written as a negative cash flow.

This formula assumes the cash flows are measured in the same currency, occur at the stated dates, and are discounted at a rate that matches their timing and risk. Here, examples use annual periods and a constant effective annual rate. A positive NPV means the discounted cash inflows exceed the initial investment and any other included outflows. Under the NPV accept-or-reject rule, a project with positive NPV adds value relative to the required return.

Scale Changes the Dollar Amount of NPV

Consider a project whose cash flows are all multiplied by a positive factor $k$. This means its initial outlay and every future inflow or outflow are scaled by the same factor. If the dates and required return remain unchanged, NPV scales by that same factor.

\begin{aligned} \operatorname{NPV}_{\text{scaled}} &=\sum_{t=0}^{n}\frac{kC_t}{(1+r)^t}\\ &=k\sum_{t=0}^{n}\frac{C_t}{(1+r)^t}\\ &=k\operatorname{NPV}_{\text{original}}. \end{aligned}

The justification is the linearity of present-value calculations: multiplying each dated cash flow by $k$ multiplies each discounted cash flow by $k$, so their sum is also multiplied by $k$. For instance, tripling every cash flow in a project with NPV of $8,000 produces NPV of $24,000, provided the discount rate and timing are unchanged. This does not mean that every larger project has a larger NPV. A project is not a scaled copy of another merely because its initial investment is larger.

NPV is expressed in dollars, so it measures an amount of value. Dividing NPV by the initial investment gives a ratio, not the total dollar value created. That ratio can help describe value relative to project size, but it answers a different question. When the objective is to compare total value added by otherwise comparable alternatives, use total NPV.

Worked Example: Tripling a Project's Scale

A hypothetical equipment project requires $80,000 now and is forecast to generate $50,000 at the end of year 1 and $75,000 at the end of year 2. A second version has each of those cash flows tripled. The effective annual required return is 25%. Calculate both NPVs and explain the scale effect. Assume the annual cash flows occur at year-end and the same required return applies to both versions.

Method. Discount each dated cash flow at 25%, including the initial outlay as a negative time-0 cash flow. This avoids treating the investment size as a substitute for calculating NPV.

For the original project, the cash flows are $-$80,000$, $50,000, and $75,000:

\begin{aligned} \operatorname{NPV}_{1} &=-80{,}000+\frac{50{,}000}{1.25}+\frac{75{,}000}{(1.25)^2}\\ &=-80{,}000+40{,}000+48{,}000\\ &=\$8{,}000. \end{aligned}

The tripled version has cash flows $-$240,000$, $150,000, and $225,000. Its discounted inflows are $150{,}000/1.25=$120,000$ and $225{,}000/1.5625=$144,000$:

\begin{aligned} \operatorname{NPV}_{2} &=-240{,}000+120{,}000+144{,}000\\ &=\$24{,}000. \end{aligned}

Check. The second project's every cash flow is three times the first project's, and $3\times \$8,000=\$24,000$. The direct calculation agrees with the scale result. Both NPVs are positive, and the tripled project creates three times as much total dollar value under the stated assumptions.

Dollar Value Versus Value Relative to Cost

A project's scale can change the comparison between total NPV and NPV relative to the initial investment. The next example deliberately gives the smaller project the higher NPV-to-investment ratio, but the larger project the higher total NPV. Both projects have one-year cash flows and use the same effective annual required return, so their results can be compared on the same basis.

Worked Example: Higher Total NPV Despite a Lower Ratio

A hypothetical firm can undertake one of two projects with the same risk. Project Alder requires $100,000 now and pays $130,000 at the end of year 1. Project Birch requires $500,000 now and pays $600,000 at the end of year 1. The effective annual required return is 10%. Find each NPV and each NPV-to-investment ratio. If the firm seeks the greater dollar value and has no binding funding limit, which project has the higher NPV?

Method. Discount each one-year inflow by $1.10$, then subtract its initial outlay. For the ratio, divide the resulting NPV by the positive amount invested.

For Alder, the present value of the inflow is $130{,}000/1.10=$118,181.818\ldots$. Therefore:

\begin{aligned} \operatorname{NPV}_{A} &=-100{,}000+\frac{130{,}000}{1.10}\\ &=-100{,}000+118{,}181.818\ldots\\ &=\$18{,}181.82\quad\text{(rounded to cents)},\\ \frac{\operatorname{NPV}_{A}}{100{,}000} &=0.181818\ldots=18.18\%\quad\text{(rounded to two decimals)}. \end{aligned}

For Birch, the present value of the inflow is $600{,}000/1.10=$545,454.545\ldots$. Thus:

\begin{aligned} \operatorname{NPV}_{B} &=-500{,}000+\frac{600{,}000}{1.10}\\ &=-500{,}000+545{,}454.545\ldots\\ &=\$45{,}454.55\quad\text{(rounded to cents)},\\ \frac{\operatorname{NPV}_{B}}{500{,}000} &=0.090909\ldots=9.09\%\quad\text{(rounded to two decimals)}. \end{aligned}

Conclusion and check. Alder has the higher NPV-to-investment ratio, but Birch has the higher total NPV: $45,454.55 versus $18,181.82. If the stated objective is greater dollar value and there is no binding funding limit, Birch is the choice on NPV. The higher percentage for Alder does not overturn that dollar comparison.

Scale Alone Does Not Determine NPV

A larger investment can have a smaller, zero, or negative NPV if its discounted cash flows do not justify its cost. NPV requires the full set of dated cash flows, not just the initial outlay or a label such as “large project.” The same discounting method applies whether the result is positive or negative.

Worked Example: A Larger Project With Negative NPV

Two hypothetical projects have comparable risk and each pays a single cash inflow at the end of year 1. The effective annual required return is 10%. Project Kite costs $50,000 and returns $60,000. Project Harbor costs $500,000 and returns $540,000. Calculate each NPV and determine which one meets the positive-NPV accept-or-reject rule.

Method. Discount each year-1 inflow by $1.10$, then include the negative time-0 outlay. This avoids assuming that the project with the larger investment must create more value.

For Kite:

\begin{aligned} \operatorname{NPV}_{K} &=-50{,}000+\frac{60{,}000}{1.10}\\ &=-50{,}000+54{,}545.4545\ldots\\ &=\$4{,}545.45\quad\text{(rounded to cents)}. \end{aligned}

For Harbor:

\begin{aligned} \operatorname{NPV}_{H} &=-500{,}000+\frac{540{,}000}{1.10}\\ &=-500{,}000+490{,}909.0909\ldots\\ &=-\$9{,}090.91\quad\text{(rounded to cents)}. \end{aligned}

Conclusion and check. Kite has positive NPV and Harbor has negative NPV. Harbor's much larger investment does not make it more valuable: its discounted inflow is less than its cost. The sign check also confirms that Kite meets the positive-NPV rule while Harbor does not.

A Common Scale-Comparison Trap

A common error is to calculate NPV divided by the initial investment, then treat the larger ratio as the larger dollar contribution. In the Alder and Birch example, the incorrect comparison would be: “Alder's 18.18% exceeds Birch's 9.09%, so Alder creates more value.” That conclusion answers a relative-to-cost question, not the question of which project has the greater total NPV.

The correction is to keep the measures distinct. Alder's NPV is $18,181.82; Birch's is $45,454.55. If the comparison is about total dollar value created, Birch has the larger NPV, subject to the assumptions in the example. If a funding limit or other constraint affects which projects can be undertaken, NPV by itself does not settle the full selection problem.

Key takeaway: Calculate NPV from every dated cash flow at the appropriate required return. Total NPV measures dollar value created. Scaling all cash flows by the same positive factor scales NPV by that factor, but a larger initial investment alone says nothing about whether a project's NPV is greater.

Check Your Understanding

Use the stated annual effective required return in each question. Report monetary answers to the nearest cent.

A hypothetical project costs $120,000 now and pays $70,000 at the end of each of years 1 and 2. The required return is 8%. What is its NPV? (Answer: $4,828.53)
Project Elm costs $80,000 and pays $92,000 at the end of year 1. Project Fir costs $240,000 and pays $270,000 at the end of year 1. At a 12% required return, which has the higher total NPV? (Answer: Elm, with NPV $2,142.86; Fir has NPV $1,071.43.)
A project's NPV is $7,500. Every cash flow, including its initial outlay, is multiplied by 1.5, with dates and the required return unchanged. What is the new NPV? (Answer: $11,250.)
At a 10% required return, Project Maple costs $150,000 and pays $80,000 at the end of year 1 and $100,000 at the end of year 2. What is its NPV? (Answer: $5,371.90.)
Two same-risk, one-year projects use an 8% required return. Project North costs $200,000 and pays $224,000 at year-end; Project South costs $600,000 and pays $660,000 at year-end. Which has the higher NPV, and which has the higher NPV-to-investment ratio? (Answer: South has the higher NPV, $11,111.11 versus $7,407.41; North has the higher NPV-to-investment ratio, 3.70% versus 1.85%.)

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