Strategic Financial Management · Evaluation of Risky Proposals for Investment Decisions

Portfolio Risk and Project Selection under Risk

Updated 11 October 2026 · Fact-checked

Portfolio risk of projects is the standard deviation of the combined NPV of projects taken together. It depends on each project's risk and on their correlation. Compute expected NPV, standard deviation and coefficient of variation (σ ÷ ENPV), then compare. Lower correlation cuts portfolio risk. Choose the option with the lowest CV within the budget.

Understand Portfolio Risk and Project Selection under Risk

A firm rarely takes one project in isolation. It takes several, and their NPVs move together to some degree. So the risk of the whole set is not just the sum of the risks of the parts.

Correlation (ρ) measures how the NPVs of two projects move together. It lies between −1 and +1. At +1 they move in perfect step and there is no risk reduction. At 0 they are unrelated. At −1 they move in opposite directions and risk can be cancelled out. A firm that takes a project whose returns run against its existing business reduces overall risk.

The expected NPV of a portfolio is simply the sum (or weighted sum) of the individual expected NPVs. Correlation does not affect it. Correlation affects only the standard deviation of the portfolio. That is why diversification helps: you keep the return and lose some of the risk.

To compare projects of different size or return, standard deviation alone misleads. A project with a larger SD may also have a much larger expected NPV. The coefficient of variation (CV) gives risk per rupee of expected NPV. A lower CV means a better risk-return trade-off. Under capital rationing, you rank by CV or by return, then fill the budget with the best feasible combination and check the portfolio risk of that combination.

Key rules to remember

Expected NPV: ENPV = Σ Pi × NPVi; Pi is the probability of each outcome. Probabilities must add up to 1.
Standard deviation of a project: σ = √[Σ Pi × (NPVi − ENPV)²]; Measures total risk of one project, in rupees.
Coefficient of variation: CV = σ ÷ ENPV; Risk per rupee of expected NPV. Lower is better. Use it only when ENPV is positive.
Portfolio expected NPV: ENPVp = w1 × ENPV1 + w2 × ENPV2; If both projects are taken in full, weights are 1 and ENPVp is the plain sum.
Portfolio SD of two projects: σp = √(w1²σ1² + w2²σ2² + 2 × w1 × w2 × ρ12 × σ1 × σ2); With both projects in full (w = 1): σp = √(σ1² + σ2² + 2ρ12σ1σ2).
Covariance: Cov12 = ρ12 × σ1 × σ2; If covariance is given, use it in place of ρσ1σ2.
Limiting cases of correlation: ρ = +1: σp = σ1 + σ2 | ρ = 0: σp = √(σ1² + σ2²) | ρ = −1: σp = |σ1 − σ2|; The last holds for two projects taken in full. ρ always lies between −1 and +1.

How to solve Portfolio Risk and Project Selection under Risk questions

Use this order for any question on portfolio risk or risk-based project selection.

1List each project's ENPV and σ. If a probability table is given, compute ENPV and σ first.
2Note the correlation or covariance between each pair, and whether projects are independent, divisible or mutually exclusive.
3Compute the CV (σ ÷ ENPV) of each project and rank them. Lowest CV ranks first.
4List every feasible combination that fits the budget or other constraint. Projects are usually indivisible unless stated.
5For each combination, add the ENPVs and compute the portfolio σ using the correlation given. Do not add the SDs.
6Compute the portfolio CV for each combination, or compare ENPV against σ as the question asks.
7Recommend the combination with the best trade-off, state the reason in one or two lines, and mention any assumption such as risk-neutral or risk-averse behaviour.

Quickest way: Rank by CV, then test the feasible combinations

When to use it: Use when the question gives ENPV, σ and a budget, and asks which projects to pick. It saves time when there are only three or four projects.

Compute the CV for each project in one line each.
Write the ENPV and σ of only those combinations that fit the budget. Drop the rest at once.
Square the SDs, add 2ρσ1σ2 where ρ is not zero, and take one square root per combination.
Divide σp by the combined ENPV and pick the lowest CV. If the highest ENPV combination also has the lowest CV, you can stop.

Common mistakes in Portfolio Risk and Project Selection under Risk

Adding the standard deviations of two projects to get portfolio σ.
Expected values add, so students assume risk adds too.
Fix: Add SDs only when ρ = +1. Otherwise use the full formula with the correlation term.
Dropping the 2 in the cross term, or writing ρσ1σ2 once instead of 2ρσ1σ2.
The formula is copied from memory under time pressure.
Fix: Write σp² = σ1² + σ2² + 2ρσ1σ2 first, then substitute the numbers. Take the root at the end.
Using variance in CV, or taking the square root before adding the terms.
Variance and SD get mixed up.
Fix: Add the variance terms first, take one square root, and then divide that SD by ENPV.
Ranking projects by SD alone and calling the lowest SD the best.
SD is the first risk number students see.
Fix: Compare risk per rupee of return using CV. Use SD alone only when the ENPVs are equal.
Choosing projects by CV rank without checking the budget.
Students stop once the ranking is done.
Fix: Fill the budget in rank order, then test other feasible combinations. A skipped project may leave unused funds that another combination uses better.
Treating the correlation between two projects as if it said something about their ENPV.
Correlation sounds like a return measure.
Fix: Correlation changes only the portfolio σ. The portfolio ENPV is always the weighted sum.

Worked examples

Example 1

Meridian Components Ltd is considering two projects, A and B, and will take both in full. Project A has ENPV ₹40 lakh and σ ₹12 lakh. Project B has ENPV ₹30 lakh and σ ₹16 lakh. Find the portfolio ENPV, σ and CV if the correlation is (i) +0.5, (ii) 0, (iii) −0.5. Also compare the CV of each project.

Show the solution

Portfolio ENPV = 40 + 30 = ₹70 lakh. It is the same in all three cases.
σ1² = 144 and σ2² = 256, so the sum is 400. The cross term is 2 × ρ × 12 × 16 = 384ρ.
(i) ρ = +0.5: cross term = 192. σp² = 592, σp = ₹24.33 lakh. CV = 24.33 ÷ 70 = 0.348.
(ii) ρ = 0: σp² = 400, σp = ₹20 lakh. CV = 20 ÷ 70 = 0.286.
(iii) ρ = −0.5: cross term = −192. σp² = 208, σp = ₹14.42 lakh. CV = 14.42 ÷ 70 = 0.206.
Individual CVs: A = 12 ÷ 40 = 0.30. B = 16 ÷ 30 = 0.533. B is the riskier project per rupee of return.

Answer: Portfolio ENPV is ₹70 lakh. Portfolio σ is about ₹24.33 lakh at ρ = +0.5, ₹20 lakh at ρ = 0 and ₹14.42 lakh at ρ = −0.5. The CV falls from 0.348 to 0.206 as correlation falls. Lower correlation reduces risk without changing the return.

Example 2

Kaveri Industries has a budget of ₹1,00,00,000. Projects are independent and indivisible. P: outlay ₹50 lakh, ENPV ₹15 lakh, σ ₹6 lakh. Q: outlay ₹40 lakh, ENPV ₹10 lakh, σ ₹6 lakh. R: outlay ₹60 lakh, ENPV ₹18 lakh, σ ₹5.4 lakh. Which projects should it select?

Show the solution

Individual CVs: P = 6 ÷ 15 = 0.40. Q = 6 ÷ 10 = 0.60. R = 5.4 ÷ 18 = 0.30. Rank: R, P, Q.
Feasible combinations within ₹100 lakh: P + Q (₹90 lakh), Q + R (₹100 lakh), or a single project. P + R costs ₹110 lakh and is not feasible.
P + Q: ENPV = ₹25 lakh. Independent, so σ² = 36 + 36 = 72 and σ = ₹8.485 lakh. CV = 8.485 ÷ 25 = 0.339.
Q + R: ENPV = ₹28 lakh. σ² = 36 + 29.16 = 65.16 and σ = ₹8.072 lakh. CV = 8.072 ÷ 28 = 0.288.
R alone: ENPV ₹18 lakh, CV 0.30, and ₹40 lakh stays unused.
Q + R has the highest ENPV, the lowest CV, and uses the whole budget.

Answer: Select projects Q and R. They give ENPV of ₹28 lakh with σ of about ₹8.07 lakh (CV 0.288), better than P + Q (CV 0.339) or R alone (CV 0.30). Project P is rejected because it cannot fit alongside R.

Exam tips

Write the portfolio σ formula first, then substitute. Method marks are usually given even if the arithmetic slips.
Check that the correlation sign is carried into the cross term. A negative ρ must reduce the variance.
End every selection question with a clear recommendation and one line on why. The question asks for a decision, not just numbers.
In MCQs, test the limiting cases quickly. At ρ = +1, σp is the sum of the SDs. At ρ = 0, it is the square root of the sum of squares.
If the question says projects are divisible or gives weights, use the weighted formula. Otherwise assume projects are taken in full.

Practice questions from Evaluation of Risky Proposals for Investment Decisions

Portfolio Risk and Project Selection under Risk in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

CFA Level IPortfolio Selection and Optimal Portfolio

Portfolio Risk and Project Selection under Risk: frequently asked questions

Why does correlation matter in capital budgeting?

Correlation decides how much the NPVs of projects offset each other. The lower the correlation, the lower the portfolio standard deviation for the same expected NPV. A negatively correlated project can cut the firm's overall risk.

How do I compare projects using the coefficient of variation?

Divide each project's standard deviation by its expected NPV. The project with the lower CV carries less risk per rupee of expected return and is preferred. CV is meaningful only when expected NPV is positive.

Does correlation change the expected NPV of a portfolio?

No. Portfolio expected NPV is the sum of the individual expected NPVs (or the weighted sum). Correlation changes only the standard deviation of the portfolio.

How is capital rationing handled when projects are risky?

Rank the projects by a risk-return measure such as CV or by ENPV per rupee of outlay, then list the combinations that fit the budget. Compute the ENPV and portfolio risk of each combination and choose the best trade-off. Indivisible projects must be taken whole.