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Strategic Financial Management · Portfolio Theory and Practice

Portfolio Risk and Return Basics for CMA Final SFM

Updated 11 October 2026 · Fact-checked

Portfolio return is the weighted average of the expected returns of the assets. Portfolio risk is not a weighted average. It is the square root of the portfolio variance, which depends on weights, individual standard deviations and the correlation between assets. Compute weights, expected returns, variance, then take the square root.

Understand Portfolio Risk and Return Basics

Return is what you earn on an investment. Since the future is uncertain, we use expected return: each possible return multiplied by its probability, then added up.

Risk is how far actual returns may stray from the expected return. We measure it with variance (the probability-weighted average of squared deviations) and standard deviation (SD), which is the square root of variance and is in the same unit as return.

When you hold two or more assets, the portfolio's expected return is simply the weighted average of the individual expected returns. Risk behaves differently. It depends on how the assets move together. Covariance shows whether two assets move in the same direction (positive), opposite directions (negative), or unrelated (near zero). Correlation rescales covariance to a range from -1 to +1, so it is easier to read.

This is why diversification works. If correlation is below +1, portfolio SD is less than the weighted average of the individual SDs. At correlation +1 there is no risk reduction. At -1 risk can be eliminated with the right weights.

Key rules to remember

Expected return of a security
E(R) = Σ pᵢ × Rᵢ
pᵢ is the probability of outcome i. Probabilities must add to 1.
Variance and SD of a security
σ² = Σ pᵢ × (Rᵢ − E(R))²; σ = √σ²
With historical data and equal weight, divide by n (or n − 1 if the question says sample).
Portfolio expected return
E(Rp) = w₁E(R₁) + w₂E(R₂) + … + wₙE(Rₙ)
Weights are market-value proportions and add to 1.
Covariance
Cov(A,B) = Σ pᵢ × (RAᵢ − E(RA)) × (RBᵢ − E(RB))
Also equals ρAB × σA × σB.
Correlation
ρAB = Cov(A,B) ÷ (σA × σB)
Always between -1 and +1.
Two-asset portfolio variance
σp² = wA²σA² + wB²σB² + 2 wA wB Cov(A,B)
Replace Cov with ρAB σA σB if correlation is given.
Two-asset portfolio SD
σp = √σp²
Take the root only at the end.
Minimum variance weight (two assets)
wA = (σB² − Cov(A,B)) ÷ (σA² + σB² − 2 Cov(A,B))
wB = 1 − wA.

How to solve Portfolio Risk and Return Basics questions

Use this order for any question on portfolio risk and return. It keeps the working clean and easy to mark.

  1. 1Write down the weights of each asset. If amounts are given, divide each by the total investment.
  2. 2Find each asset's expected return from the data (probabilities, or the average of past returns).
  3. 3Compute portfolio expected return as the weighted average.
  4. 4Find each asset's variance and SD, using deviations from its own expected return.
  5. 5Find covariance or correlation. Use whichever is not given, via Cov = ρ × σA × σB.
  6. 6Put the values into the portfolio variance formula. Write all three terms separately.
  7. 7Take the square root to get portfolio SD. State the final answer in percentage.
  8. 8If asked, comment: compare with the weighted average SD to show the diversification benefit.

Quickest way: Three-term shortcut for two assets

When to use it: Use when the question gives σ, weights and either covariance or correlation, and asks only for portfolio SD.

  1. Convert percentages to decimals, or keep them in % consistently and square them as %².
  2. Compute term 1 = (wA σA)² and term 2 = (wB σB)².
  3. Compute term 3 = 2 × (wA σA) × (wB σB) × ρ.
  4. Add the three terms and take the square root.
  5. Sanity check: the answer must lie between |wAσA − wBσB| and wAσA + wBσB. If not, recheck.

Common mistakes in Portfolio Risk and Return Basics

  • Taking the weighted average of SDs as portfolio risk.

    Return is a weighted average, so students assume risk is too.

    Fix: Always use the variance formula with the covariance term. The weighted average of SDs is only correct when ρ = +1.

  • Forgetting the factor 2 in the covariance term.

    The term wA wB Cov looks complete on its own.

    Fix: Write the formula as three terms before substituting. The cross term appears twice, so 2 is required.

  • Stopping at variance and reporting it as risk.

    Students rush and miss the final square root.

    Fix: Underline 'SD' in the question. If SD is asked, take the root and state the unit as %.

  • Computing deviations from the wrong mean.

    Students use the other asset's mean or the portfolio mean when finding covariance.

    Fix: Each asset's deviation is taken from its own expected return.

  • Using weights that do not add to 1.

    Amounts are given in rupees and students use them directly, or forget the remaining weight.

    Fix: Divide each amount by the total investment and check the weights sum to 1.

  • Mixing percentages and decimals.

    Covariance given as 0.0045 is used with SD in %.

    Fix: Convert everything to decimals first, then convert the final SD back to %.

Worked examples

Example 1

Security X has expected return 12% and SD 10%. Security Y has expected return 18% and SD 20%. A portfolio has ₹6,00,000 in X and ₹4,00,000 in Y. The correlation between X and Y is 0.4. Find the portfolio expected return and SD.

Show the solution
  1. Total investment = ₹10,00,000. wX = 0.6, wY = 0.4.
  2. Expected return = 0.6 × 12% + 0.4 × 18% = 7.2% + 7.2% = 14.4%.
  3. Cov(X,Y) = 0.4 × 0.10 × 0.20 = 0.008.
  4. Term 1 = 0.6² × 0.10² = 0.36 × 0.01 = 0.0036.
  5. Term 2 = 0.4² × 0.20² = 0.16 × 0.04 = 0.0064.
  6. Term 3 = 2 × 0.6 × 0.4 × 0.008 = 0.00384.
  7. Variance = 0.0036 + 0.0064 + 0.00384 = 0.01384.
  8. SD = √0.01384 = 0.11765, about 11.76%.
  9. Weighted average SD would be 0.6 × 10 + 0.4 × 20 = 14%. Actual SD is lower, showing the diversification benefit.

Answer: Expected return = 14.4%; portfolio SD ≈ 11.76%.

Example 2

Returns on two stocks A and B in three equally likely economic states are: Boom: A 20%, B 10%; Normal: A 10%, B 8%; Recession: A 0%, B 6%. Find E(RA), E(RB), σA, σB, covariance and correlation.

Show the solution
  1. Each probability is 1/3.
  2. E(RA) = (20 + 10 + 0) ÷ 3 = 10%. E(RB) = (10 + 8 + 6) ÷ 3 = 8%.
  3. Deviations of A: +10, 0, −10. Deviations of B: +2, 0, −2.
  4. Variance of A = (100 + 0 + 100) ÷ 3 = 66.67 (%²). σA = √66.67 = 8.165%.
  5. Variance of B = (4 + 0 + 4) ÷ 3 = 2.667 (%²). σB = √2.667 = 1.633%.
  6. Covariance = (10 × 2 + 0 × 0 + (−10) × (−2)) ÷ 3 = 40 ÷ 3 = 13.33 (%²).
  7. Correlation = 13.33 ÷ (8.165 × 1.633) = 13.33 ÷ 13.33 = 1.0.

Answer: E(RA) = 10%, E(RB) = 8%, σA ≈ 8.17%, σB ≈ 1.63%, Cov ≈ 13.33 (%²), correlation = +1. Both stocks move perfectly together, so combining them gives no risk reduction.

Exam tips

  • Show the three terms of the variance formula separately. Marks are given for method even if arithmetic slips.
  • Read whether the question gives covariance or correlation. Convert only if needed.
  • Check the answer: portfolio SD at ρ < 1 must be below the weighted average SD.
  • Keep four decimal places in variance working and round only the final SD.
  • In MCQs, test the extremes: ρ = +1 gives the weighted average SD, and ρ = −1 may reduce risk to zero at the right weights.

Practice questions from Portfolio Theory and Practice

Portfolio Risk and Return Basics in other exams

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

Portfolio Risk and Return Basics: frequently asked questions

How do I calculate portfolio standard deviation for two assets?

Find the weights, then compute wA²σA² + wB²σB² + 2wAwBρσAσB. This is the portfolio variance. Take its square root to get the SD.

What is the difference between covariance and correlation?

Covariance shows the direction in which two assets move, but its size depends on units. Correlation divides covariance by the product of SDs, so it always lies between -1 and +1 and is easy to compare.

Is portfolio expected return also affected by correlation?

No. Portfolio expected return is the weighted average of individual expected returns and does not depend on correlation. Only portfolio risk does.

When is portfolio risk zero?

With two assets, risk can be eliminated when correlation is -1 and the weights are chosen suitably, wA = σB ÷ (σA + σB). Otherwise some risk remains.