Comparing Project Rankings Across Scenarios

Why Project Rankings Can Change

Two mutually exclusive projects are alternatives: under the stated decision, only one can be selected. A project that produces the greatest NPV under one set of assumptions may not do so under another. For example, one project may perform especially well when demand is strong, while another may be less sensitive to weak demand. Comparing the projects across scenarios makes these ranking changes visible.

A scenario is a coherent set of assumptions about conditions that affect project cash flows or their discount rate. Compare projects within the same scenario: using one project’s high-demand cash flows alongside another project’s low-demand cash flows would not be a meaningful comparison. If the scenario changes a market interest rate, apply the rate specified for that scenario to both alternatives unless the problem states otherwise.

Definition: The scenario-specific NPV of project $j$ in scenario $s$ is the present value of its cash flows under that scenario, less its initial investment. For an investment $I_j$ at time 0 and end-of-period cash flows $CF_{j,t,s}$, discounted at an effective annual rate $i_s$,

NPV_{j,s}=-I_j+\sum_{t=1}^{n}\frac{CF_{j,t,s}}{(1+i_s)^t}.

The preferred project in scenario $s$ is the feasible alternative with the largest NPV in that scenario. If rejecting all projects is permitted, that alternative is also available with NPV zero.

This comparison assumes that each project’s cash flows are measured on a consistent basis, occur at the times stated, and are discounted at an appropriate rate for the scenario. Unless specified otherwise, the examples below use effective annual rates and end-of-year cash flows. Mortality assumptions and loss distributions are not relevant to these capital-budgeting examples.

A Direct Comparison Across Operating Scenarios

Calculate NPV separately for every project-scenario pair, then compare the resulting NPVs within each scenario. A table is useful because it displays both the size of the NPV and the ranking; the ranking alone would hide whether the preferred project creates or destroys value.

Worked Example: Ranking Projects Under Three Demand Scenarios

Two mutually exclusive projects each require an initial investment of $1,000. Each has a two-year life, and all receipts occur at each year-end. Use an effective annual discount rate of 10% in all three demand scenarios. Project A’s year-one and year-two receipts are, respectively, $700 and $600 in high demand, $600 and $500 in base demand, and $400 and $350 in low demand. Project B’s corresponding receipts are $350 and $900, $450 and $700, and $300 and $500. Calculate both NPVs in each scenario, rank the projects, and state whether the higher-NPV project is worth undertaking if the decision-maker may reject both.

Method. Apply the two-year NPV formula to each pair of receipts using the same initial investment and discount rate. The discount factors are $1/1.10=0.909091$ for year one and $1/(1.10)^2=1/1.21=0.826446$ for year two.

High demand. For A, the present values of the receipts are $700/1.10=636.3636$ and $600/1.21=495.8678$. Thus its NPV is $636.3636+495.8678-1{,}000=132.2314$, or $132.23. For B, the present values are $350/1.10=318.1818$ and $900/1.21=743.8017$. Its NPV is $318.1818+743.8017-1{,}000=61.9835$, or $61.98. A ranks first.

Base demand. A’s present values are $600/1.10=545.4545$ and $500/1.21=413.2231$, giving $545.4545+413.2231-1{,}000=-41.3224$, or −$41.32. B’s present values are $450/1.10=409.0909$ and $700/1.21=578.5124$, giving $409.0909+578.5124-1{,}000=-12.3967$, or −$12.40. B ranks first, although both projects have negative NPV.

Low demand. A’s present values are $400/1.10=363.6364$ and $350/1.21=289.2562$, giving $363.6364+289.2562-1{,}000=-347.1074$, or −$347.11. B’s are $300/1.10=272.7273$ and $500/1.21=413.2231$, giving $272.7273+413.2231-1{,}000=-314.0496$, or −$314.05. B ranks first.

The ranking changes: A is preferred in high demand, while B is preferred in base and low demand. However, if rejecting both is permitted, the decision-maker should reject both projects in base and low demand because zero exceeds either negative NPV. Check: In high demand, both NPVs are positive and A’s $132.23 exceeds B’s $61.98; in the other scenarios, B’s NPV is less negative than A’s but remains below zero.

Common mistake avoided: Choosing the project with the least negative NPV as though it must be undertaken. “Preferred among the projects” and “worth undertaking” are different conclusions when rejection is possible.

Using Scenario Probabilities Carefully

If a problem assigns probabilities to scenarios, a decision-maker may calculate each project’s probability-weighted NPV. This is the sum of each scenario-specific NPV multiplied by that scenario’s probability. The scenario probabilities must be nonnegative and sum to one. This calculation summarizes the NPVs under the stated probability model; it does not erase the fact that different projects can rank first in different scenarios.

E[NPV_j]=\sum_s P(s)\,NPV_{j,s}.

The formula follows from the definition of an expectation for a discrete outcome: the NPV is a random variable that takes value $NPV_{j,s}$ when scenario $s$ occurs, and each value is weighted by its probability. Because the initial investment is the same in every scenario for a given project, it is included in each scenario NPV before taking the weighted sum.

Worked Example: Scenario Rankings and Probability-Weighted NPV

Projects C and D each require $500 at time 0 and pay one receipt at the end of year one. The effective annual discount rate is 10%. A hypothetical planning model assigns probabilities 0.25, 0.50, and 0.25 to high, base, and low demand, respectively. C’s receipts are $770, $550, and $330 across those scenarios; D’s receipts are $660, $605, and $440. Find the preferred project in each scenario and compare the probability-weighted NPVs. Assume the probabilities describe this decision problem and that no other outcomes are included.

Method. Discount each receipt by $1.10$, subtract the $500 initial investment, and only then weight the scenario NPVs by their probabilities.

High demand. C has NPV $770/1.10-500=700-500=$ $200. D has NPV $660/1.10-500=600-500=$ $100. C ranks first.

Base demand. C has NPV $550/1.10-500=500-500=$ $0. D has NPV $605/1.10-500=550-500=$ $50. D ranks first.

Low demand. C has NPV $330/1.10-500=300-500=$ −$200. D has NPV $440/1.10-500=400-500=$ −$100. D ranks first.

The probability-weighted NPV of C is $0.25(200)+0.50(0)+0.25(-200)=50+0-50=$ $0. For D it is $0.25(100)+0.50(50)+0.25(-100)=25+25-25=$ $25. Therefore D has the higher probability-weighted NPV, even though C ranks first in the high-demand scenario. If rejection is available and the stated decision criterion is to choose the alternative with the highest probability-weighted NPV, D is preferred to rejecting both, whose NPV is zero under every scenario. Check: The probabilities sum to $0.25+0.50+0.25=1$, and each scenario ranking follows directly from its two NPVs.

Common mistake avoided: Weighting the receipts first, then treating the result as a scenario-specific ranking. A weighted NPV is an aggregate decision summary; it is not the preferred project in every possible scenario.

Finding Where the Ranking Switches

A ranking reversal can be analyzed by comparing the projects’ NPVs directly. If two projects have the same initial investment, that common amount cancels when their NPVs are subtracted. The difference then depends on the present values of the cash-flow differences. A tie occurs when this difference is zero.

Result: For two projects A and B evaluated at the same rate, A ranks above B exactly when $NPV_A-NPV_B>0$; they tie when $NPV_A-NPV_B=0$. If the projects have identical initial investments, the initial-investment terms cancel in this difference, but the cash flows must still be discounted at the rate being tested.

Worked Example: Finding a Discount-Rate Ranking Reversal

Projects E and F are mutually exclusive. Each costs $1,000 at time 0 and lasts two years. E pays $900 at the end of year one and $100 at the end of year two. F pays $100 at the end of year one and $1,100 at the end of year two. Calculate their NPVs at effective annual rates of 10%, 25%, and 40%, identify the ranking at each rate, and derive the rate at which they tie. Assume the same rate applies to both projects and all receipts are certain within this comparison.

Method. Discount each year’s receipt at its own time point. Then set $NPV_E-NPV_F=0$ to find the switching rate.

At 10%. E’s NPV is $900/1.10+100/1.21-1{,}000=818.1818+82.6446-1{,}000=-99.1736$, or −$99.17. F’s NPV is $100/1.10+1{,}100/1.21-1{,}000=90.9091+909.0909-1{,}000=0$, or $0.00. F ranks first.

At 25%. The year-one and year-two discount factors are $1/1.25=0.8$ and $1/(1.25)^2=0.64$. E’s NPV is $900(0.8)+100(0.64)-1{,}000=720+64-1{,}000=-216$, or −$216.00. F’s NPV is $100(0.8)+1{,}100(0.64)-1{,}000=80+704-1{,}000=-216$, also −$216.00. They tie.

At 40%. E’s NPV is $900/1.40+100/1.96-1{,}000=642.8571+51.0204-1{,}000=-306.1225$, or −$306.12. F’s NPV is $100/1.40+1{,}100/1.96-1{,}000=71.4286+561.2245-1{,}000=-367.3469$, or −$367.35. E ranks first.

Derivation of the tie rate. Subtract F’s NPV from E’s. Since both projects cost $1,000, their initial investments cancel:

NPV_E-NPV_F =\frac{900-100}{1+i}+\frac{100-1{,}100}{(1+i)^2} =\frac{800}{1+i}-\frac{1{,}000}{(1+i)^2}.

For a tie, set this expression equal to zero. Since $1+i$ is positive for the stated annual rates, multiply both sides by $(1+i)^2$: $800(1+i)-1{,}000=0$. Thus $800+800i=1{,}000$, so $800i=200$ and $i=0.25$, or 25%. This agrees with the direct NPVs at 25%. The earlier cash flow of E is more valuable at higher discount rates; below the tie rate, F’s larger later receipt outweighs that timing advantage. Check: F ranks first at 10%, both NPVs are equal at 25%, and E ranks first at 40%.

Common mistake avoided: Comparing undiscounted receipt totals. E receives $1,000 in total and F receives $1,200, which does not establish which has the higher NPV. At 25%, despite that difference in totals, both projects have the same NPV of −$216.00.

How to Report the Decision

A complete scenario comparison separates three questions: which project has the largest NPV in each scenario, whether that NPV is positive, and whether the decision-maker may reject the projects. When scenario probabilities are supplied, report both the scenario rankings and any probability-weighted result requested. State the discount rate and cash-flow timing used, particularly when market conditions change the rate as well as operating assumptions.

The central discipline is to calculate each scenario-specific NPV consistently before comparing projects. A ranking reversal is not an arithmetic inconsistency: it means that the cash-flow patterns or scenario assumptions favor different alternatives under different conditions. If the ranking does not change over the scenarios provided, that alone does not establish that it could never change under other assumptions.

Key takeaway: Compare like with like: calculate each project’s NPV using the same scenario assumptions, timing convention, and applicable discount rate. Select the highest NPV among feasible choices, but do not confuse that ranking with a recommendation to accept a negative-NPV project when rejection is available.

Check Your Understanding

For each problem, use the stated cash-flow timing and effective annual rate. Round dollar NPVs to the nearest cent.

Two one-year projects each cost $400. At an 8% effective annual rate, project A pays $540 in high demand and $432 in low demand; project B pays $500 in high demand and $468 in low demand. Which project ranks first in each scenario? (Answer: High demand: A, with NPV $100.00 versus B’s $62.96. Low demand: B, with NPV $33.33 versus A’s $0.00.)
At a 10% effective annual rate, project P costs $500 and pays $300 at the end of year one and $330 at the end of year two. Project Q costs $500 and pays $200 and $420 at those year-ends. Which has the higher NPV? (Answer: P, with NPV $45.45; Q’s NPV is $28.93.)
Projects R and S each cost $800 and pay one receipt at the end of year one. At 10%, R pays $990 in high demand and $660 in low demand; S pays $880 and $770, respectively. The scenario probabilities are 0.40 for high demand and 0.60 for low demand. Which project ranks first in each scenario, and which has the higher probability-weighted NPV? (Answer: R ranks first in high demand with NPV $100.00 versus S’s $0.00; S ranks first in low demand with NPV −$100.00 versus R’s −$200.00. The weighted NPVs are −$80.00 for R and −$60.00 for S, so S is higher; if rejection is permitted, rejecting both is better under this weighted-NPV criterion.)
Project E costs $1,000 and pays $900 and $100 at the ends of years one and two. Project F also costs $1,000 and pays $100 and $1,100 at those year-ends. At what effective annual discount rate do their NPVs tie, and which project ranks first at 10% and at 40%? (Answer: They tie at 25%; F ranks first at 10%, and E ranks first at 40%.)

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