NPVR

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NPVR
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NPVR (net present value ratio) - is the ratio of a project's net present value to the present value of the investment outlay needed to generate it. UNIDO's manual on industrial feasibility studies introduced it as a refinement of plain NPV: "The ratio of the NPV and the present value of the investment (PVI) required will yield a discounted rate of return which should be used for comparing alternative projects"[1]. In short, NPVR tells how much net present value each unit of invested capital produces.

The name varies from book to book. Some authors call it the NPV index or the net present value index, and the UNIDO texts describe the result as a discounted rate of return[1]. That's why the strategic analysis methods lists sometimes place NPVR next to NPV and IRR among the dynamic methods of investment appraisal. It isn't an interest rate, though. It is a pure number, usually shown as a decimal (0.25) or a percentage (25%).

Background

Discounted cash flow appraisal spread through corporate practice in the 1950s and 1960s. NPV itself was never the problem. The problem appeared when a firm had more good projects than money. In 1955 James Lorie and Leonard Savage described three typical rationing situations, among them choosing a set of independent proposals under a fixed budget, and showed that simple rules based on the internal rate of return could lead to the wrong selection[2]. A few years later H. Martin Weingartner recast the same problem as a mathematical programming model, maximising total NPV subject to budget constraints in each period[3].

Ratio rankings sit between those two worlds. They are much simpler than a programming model and much better than ranking by NPV when cash is short. UNIDO's Manual for the Preparation of Industrial Feasibility Studies (1977, prepared by Werner Behrens) wrote the ratio down as NPVR and recommended it for choosing among alternatives[1]. Through UNIDO training workshops and national guidelines, for instance the 1990 guidelines prepared for Papua New Guinea, the term entered the vocabulary of project analysts in many developing and transition economies[4].

NPVR formula

The basic formula:

\[ NPVR = \frac{NPV}{PVI} \]

where:

Written out in full:

\[ NPVR = \frac{\sum_{t=0}^{n} \frac{CF_t}{(1+r)^t}}{\sum_{t=0}^{n} \frac{I_t}{(1+r)^t}} \]

with CFt the net cash flow in year t (after investment), It the investment outlay in year t and r the discount rate. If the whole investment happens at the start, PVI is simply the initial outlay I0. The UNIDO manual makes the same remark: when construction does not exceed one year, the investment does not have to be discounted[1].

Decision rules are straightforward:

  • NPVR > 0 - the project earns more than the discount rate; accept (if money is available),
  • NPVR = 0 - the project earns exactly the discount rate,
  • NPVR < 0 - reject,
  • among competing projects, prefer the higher NPVR, provided all of them are discounted over comparable periods and at the same rate[1].

Note that the accept/reject signal is identical to NPV. NPVR is positive exactly when NPV is positive, because PVI is always positive. The ratio adds information only when projects have to be ranked.

Relation to the profitability index

Here's where the literature gets messy. Two different ratios travel under the name "profitability index":

  • the more common version, used for example by Ross, Westerfield and Jaffe, divides the present value of future cash flows by the initial investment[5],
  • Brealey, Myers and Allen define the profitability index as net present value divided by investment[6]. That's NPVR under another name.

When the entire outlay falls at time 0 and the same base is used, the two versions are linked by a simple identity:

\[ PI = \frac{PV_{inflows}}{I_0} = \frac{NPV + I_0}{I_0} = 1 + NPVR \]

So a PI of 1.25 equals an NPVR of 0.25. The hurdle shifts from 1 to 0, nothing else. Both rank projects identically.

The catch? The identity holds only when PI and NPVR are calculated on the same denominator. If outlays are spread over several years, some analysts put only I0 in the denominator and treat later outlays as negative cash flows in the numerator; others discount all outlays into PVI. The two choices give different numbers and can change the ranking (the example below shows one such case). Whenever a report quotes NPVR, NPV index or PI, it is worth checking which denominator was used. The CEOpedia article on the profitability index uses the PV-of-inflows version.

Where to use NPVR

  • Capital rationing - the main application. If the budget is fixed and projects are independent, ranking by NPVR and funding from the top down picks the projects that deliver the most NPV per unit of scarce capital.
  • Comparing alternative variants of one project - for example different technologies or plant sizes, where the investment differs. UNIDO used it to compare two ways of financing the same project: with and without outside funds[1].
  • Screening proposals in investment proposals and feasibility studies, where a single relative number is easier to communicate than a set of NPVs of very different sizes.
  • Capital allocation between business units, when headquarters wants to see which unit squeezes the most value out of each euro.

When capital is not rationed, NPVR should not override NPV. If a firm can finance every positive-NPV project, it should take all of them, and between mutually exclusive projects it should choose the one with the higher NPV, not the higher ratio.

Steps of NPVR calculation

  1. Estimate the incremental cash flows of each project, year by year (see cash flow analysis).
  2. Choose the discount rate. Use the same rate and a comparable horizon for every project being ranked.
  3. Calculate NPV.
  4. Calculate the present value of all investment outlays (PVI).
  5. Divide NPV by PVI.
  6. Under capital rationing: rank independent projects by NPVR, fund from the top until the budget runs out, and then check whether a different combination would use the leftover money better.

Step 6 is where people cut corners. Don't.

Example of NPVR

Hypothetical example. A manufacturing firm has four independent proposals. The board caps the capital that can be committed this year at 1,000 thousand EUR, measured as the present value of outlays. The discount rate is 10%. Projects A, B and C require a single outlay at the start; project D needs 150 now and 110 a year later, so its PVI = 150 + 110/1.1 = 250. All amounts are in thousands of EUR.

Project Investment Annual cash inflows PV of inflows PVI NPV NPVR IRR
A 700 at t=0 230 in years 1-5 871.88 700 171.88 0.246 19.2%
B 400 at t=0 140 in years 1-5 530.71 400 130.71 0.327 22.1%
C 350 at t=0 60, 110, 130, 150, 150 in years 1-5 438.72 350 88.72 0.253 18.2%
D 150 at t=0, 110 at t=1 85 in years 2-6 292.92 250 42.92 0.172 15.3%

Every project has a positive NPV. Without the budget limit the firm would take all four (total investment 1,700). It can't.

Ranking by NPV. The order is A, B, C, D. A takes 700. B (400) and C (350) no longer fit, D (250) does. Selected: A + D, investment 950, total NPV = 171.88 + 42.92 = 214.81.

Ranking by NPVR. The order is B (0.327), C (0.253), A (0.246), D (0.172). B and C together use 750; A doesn't fit; D fits exactly. Selected: B + C + D, investment 1,000, total NPV = 130.71 + 88.72 + 42.92 = 262.35.

Checking all 15 possible combinations confirms that B + C + D is the best feasible portfolio at a budget of 1,000. The NPVR ranking beats the NPV ranking by about 47.5 thousand EUR of value. The big project A looks most attractive in absolute terms, yet it ties up 70% of the budget while delivering only about 0.25 of NPV per euro. Ranking by IRR (B, A, C, D) happens to give the same portfolio here, but only by coincidence; IRR and NPVR need not agree.

The indivisibility trap. Now raise the budget to 1,100. The NPVR rule still picks B, C, D (1,000 used, 100 left idle, NPV 262.35). But A + B costs exactly 1,100 and gives NPV 302.59. Ranking by the ratio works well when the leftover money is small relative to project sizes; when projects are "lumpy", the top-down rule can miss the optimum, and only checking combinations (or integer programming) finds it.

The denominator problem. For project D, NPVR on full PVI is 42.92 / 250 = 0.172. If someone divides NPV by the initial outlay alone, they get 42.92 / 150 = 0.286, and a "PI on initial outlay" of (292.92 − 100) / 150 = 1.286. On that basis D would jump from last place to second. Same project, same cash flows, different ranking. That's the reason behind the warning above: always state the definition.

Advantages of NPVR

  • Measures efficiency of capital - NPV per unit of investment, exactly what matters when capital is the binding constraint.
  • Consistent with NPV - same discount rate, same cash flows, same accept/reject signal. It ranks rather than replaces.
  • Allows comparison of projects of different size - a 400 project and a 700 project become comparable.
  • Accounts for the time value of money and the whole life of the project, unlike the payback period or the accounting rate of return.
  • No multiple-solution problem - unlike IRR, which can have several values when cash flows change sign more than once, NPVR has one value for a given discount rate.
  • Simple - a spreadsheet column, not an optimisation model. Easy to explain to a board.
  • Recognised in project appraisal manuals - UNIDO's methodology made it standard in many feasibility studies[1][4].

Limitations of NPVR

  • Valid only for a single constraint - the ranking works when capital is limited in one period only. If budgets bind in several years, or if other resources (engineers, plant capacity) are scarce, a single ratio can't capture the trade-offs. Weingartner's programming approach handles multi-period constraints[3], and later reviews argued that the right formulation depends heavily on why capital is rationed in the first place[7].
  • Indivisible projects - as the example shows, ranking may leave money idle and miss a better combination.
  • Wrong tool for mutually exclusive projects - when only one variant can be built and money is not limited, the higher-NPV variant creates more value even if its ratio is lower.
  • Ambiguous denominator - initial outlay or PV of all outlays? Gross or net of later outlays? Different conventions give different results, and textbooks don't agree on the name either.
  • Ignores interdependence - projects that are complements, substitutes or contingent on each other can't be evaluated one ratio at a time.
  • Inherits all weaknesses of NPV - forecasting errors in cash flows and a poorly chosen discount rate distort NPVR in the same way. Sensitivity analysis and scenario analysis remain necessary, as does a risk analysis of the project.
  • Sensitive to the discount rate - rankings can flip when the rate changes, especially between long-lived and short-lived projects.
  • Says nothing about financing side-effects - for that, adjusted present value is more informative.

Other approaches related to NPVR

  • Net present value - the absolute measure of value created; the numerator of NPVR and the final criterion when capital is not rationed.
  • Profitability index - PV of inflows divided by investment; equals 1 + NPVR on the same base.
  • Internal rate of return (IRR) - the discount rate at which NPV = 0. Gives a percentage that businesspeople like, but can mislead under capital rationing and with unconventional cash flows[2].
  • Modified internal rate of return - a variant of IRR that assumes reinvestment at the cost of capital.
  • Adjusted present value - NPV split into base-case value and financing effects.
  • Mathematical programming - linear or integer programming models that maximise total NPV under one or more budget constraints; the exact answer to the rationing problem that NPVR approximates[3].
  • Static methods - payback period, simple rate of return, accounting rate of return; quick, but they ignore discounting.
Criterion What it measures Best use Main weakness
NPV Absolute value created Accept/reject; mutually exclusive projects without rationing Favours large projects when capital is scarce
NPVR NPV per unit of PV of investment Ranking independent projects under a single budget limit Indivisibility; single constraint only
PI PV of inflows per unit of investment Same as NPVR (PI = 1 + NPVR on the same base) Several competing definitions
IRR Break-even discount rate Communicating return as a percentage Multiple IRRs; scale and timing conflicts with NPV

References

Footnotes

  1. ↑ UNIDO, Manual for the Preparation of Industrial Feasibility Studies, UNIDO/ICIS.33, Vienna 1977, pp. 286-287
  2. ↑ Lorie J.H., Savage L.J., Three Problems in Rationing Capital, The Journal of Business 28(4), 1955, pp. 229-239
  3. ↑ Weingartner H.M., Mathematical Programming and the Analysis of Capital Budgeting Problems, Prentice-Hall, 1963
  4. ↑ UNIDO, Guidelines for the Performance of Industrial Feasibility Studies in Papua New Guinea, DP/PNG/86/002, Vienna 1990, pp. 40-41
  5. ↑ Ross S.A., Westerfield R.W., Jaffe J., Corporate Finance, McGraw-Hill/Irwin, 2004
  6. ↑ Brealey R.A., Myers S.C., Allen F., Principles of Corporate Finance, 10th ed., McGraw-Hill/Irwin, 2011
  7. ↑ Weingartner H.M., Capital Rationing: n Authors in Search of a Plot, The Journal of Finance 32(5), 1977, pp. 1403-1431
NPVR — recommended articles
Net present value — Profitability index — Capital rationing — Adjusted present value — Investment appraisal techniques — Payback period — Linear programming — Portfolio of projects
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