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Advanced Financial Management · Portfolio Management

Portfolio Return and Risk Basics for CA Final AFM

Updated 5 October 2026 · Fact-checked

Portfolio return is the weighted average of the returns of the assets you hold. Portfolio risk is the standard deviation of that return, and it also depends on the covariance or correlation between assets. For two assets, use σp² = wA²σA² + wB²σB² + 2wAwB·ρ·σA·σB, then take the square root.

Understand Portfolio Return and Risk Basics

Return is what you earn on an investment, as a percentage of what you put in. Future returns are uncertain, so we describe them with expected return: the probability-weighted average of the possible returns. If past data is given instead of probabilities, we use the simple average.

Risk is how far actual returns may stray from the expected return. Variance is the probability-weighted average of squared deviations from the mean. Standard deviation (SD) is the square root of variance. It is in the same unit as return, so it is easier to read. A higher SD means a riskier security.

When you hold two securities, the portfolio return is simply the weighted average of their returns. Portfolio risk is not a weighted average of the SDs. It depends on how the two securities move together.

Covariance measures the direction of that co-movement. Positive means they tend to move together. Negative means they tend to move opposite. Its size is hard to interpret because it depends on the units. Correlation fixes this by dividing covariance by the product of the two SDs. It always lies between −1 and +1.

This is why diversification works. Unless correlation is exactly +1, portfolio SD is lower than the weighted average of the individual SDs. The lower the correlation, the larger the benefit.

Key rules to remember

Expected return (with probabilities)
E(R) = Σ pi × Ri
Probabilities must add up to 1. Use percentages or decimals consistently.
Average return (historical data)
Average R = ΣR ÷ n
Use this when the question gives past returns and no probabilities.
Variance and standard deviation
σ² = Σ pi × (Ri − E(R))²; σ = √σ²
For historical data with no probabilities, divide the sum of squared deviations by n or by n − 1 as the question directs. If unclear, state your assumption.
Covariance
Cov(A,B) = Σ pi × (RA − E(RA)) × (RB − E(RB))
For historical data, sum the products of deviations and divide by n (or n − 1 if the question treats the data as a sample).
Correlation coefficient
ρAB = Cov(A,B) ÷ (σA × σB)
Lies from −1 to +1. Hence Cov(A,B) = ρAB × σA × σB.
Portfolio return
Rp = wA × RA + wB × RB
Weights are proportions of total investment and add up to 1.
Two-asset portfolio variance
σp² = wA²σA² + wB²σB² + 2 × wA × wB × Cov(A,B)
Same as using 2 × wA × wB × ρ × σA × σB for the last term. Portfolio SD = √σp².
Minimum-variance weight (two assets)
wA = (σB² − Cov) ÷ (σA² + σB² − 2 × Cov); wB = 1 − wA
Gives the mix with the lowest portfolio variance.
n-asset portfolio variance
σp² = Σi Σj wi × wj × Cov(i,j)
Cov(i,i) = σi². Each pair appears twice in the double sum.
Coefficient of variation
CV = σ ÷ E(R)
Risk per unit of return. Useful for comparing securities with different expected returns.

How to solve Portfolio Return and Risk Basics questions

Use the same order every time. It keeps your working clear and lets you earn method marks even if an arithmetic slip happens.

  1. 1Read what is given: probabilities or past returns, weights, and whether SD, covariance or correlation is already supplied.
  2. 2Compute the expected return of each security using Σ p × R, or the simple average for past data.
  3. 3Find the deviation of each outcome from its mean. Keep a column for each security.
  4. 4Compute variance and SD for each security. Then compute covariance as the sum of p × deviationA × deviationB.
  5. 5Find correlation if asked: Cov ÷ (σA × σB). Check that it lies between −1 and +1.
  6. 6Compute portfolio return from the weights, then portfolio variance using the two-asset formula. Take the square root for SD.
  7. 7Interpret in one line: compare portfolio SD with the weighted average of SDs, or compare options using return and risk (or CV), and state your conclusion.

Quickest way: Table-and-formula shortcut

When to use it: Use when the question gives SDs and correlation (or covariance) directly, or when you must compare several weight mixes.

  1. Write wA, wB, σA, σB and ρ (or Cov) in one line at the top.
  2. Compute the three terms of the variance separately: wA²σA², wB²σB² and 2wAwBρσAσB. Add them.
  3. Take the square root once, at the end only.
  4. Sanity-check the answer. If ρ = +1, SD equals the weighted average of the SDs. For any lower ρ, your SD must be smaller. For −1 ≤ ρ < 1, it can never exceed that weighted average.
  5. If covariance is given and not correlation, use it directly and skip ρ. If the question gives ρ, do not compute covariance separately unless asked.

Common mistakes in Portfolio Return and Risk Basics

  • Not squaring the weights in the variance formula.

    Students remember wA × σA² from the return formula and carry the habit over.

    Fix: Write the formula before substituting. The weight multiplies the SD in the SD form, and its square multiplies the variance.

  • Taking portfolio SD as the weighted average of the individual SDs.

    Portfolio return is a weighted average, so students assume risk is too.

    Fix: That holds only when ρ = +1. Otherwise use the full variance formula with the covariance term.

  • Forgetting the factor 2 in the covariance term.

    The cross term appears twice (A with B, and B with A), and students overlook this.

    Fix: Always write 2 × wA × wB × Cov. Tick it off before you start calculating.

  • Mixing percentages and decimals.

    A deviation of 9% is squared as 81 in one place and as 0.0081 in another.

    Fix: Stay in percentages throughout, so variance is in %². Then SD comes out directly in %. Do not convert midway.

  • Dividing by the wrong number for historical data.

    Students do not check whether the question treats data as a population (n) or a sample (n − 1).

    Fix: Follow the question's wording. If it is silent, state your assumption clearly and apply it consistently to variance and covariance.

  • Reporting a correlation above +1 or below −1, or mixing up covariance with correlation.

    Covariance is unbounded and students treat it like a ratio. Or the SD used in the denominator is actually a variance.

    Fix: Divide by σA × σB, not σA² × σB². If the result is outside −1 to +1, recheck your SDs.

Worked examples

Example 1

An investor is studying two shares, A and B, under three economic scenarios. Boom has probability 0.3: A returns 20% and B returns 2%. Normal has probability 0.5: A returns 12% and B returns 8%. Slump has probability 0.2: A returns −5% and B returns 18%. Find the expected return, SD, covariance and correlation. Then find the return and SD of a portfolio with 60% in A and 40% in B.

Show the solution
  1. Expected return of A = 0.3×20 + 0.5×12 + 0.2×(−5) = 6 + 6 − 1 = 11%.
  2. Expected return of B = 0.3×2 + 0.5×8 + 0.2×18 = 0.6 + 4 + 3.6 = 8.2%.
  3. Deviations of A: 9, 1, −16. Variance of A = 0.3×81 + 0.5×1 + 0.2×256 = 24.3 + 0.5 + 51.2 = 76. SD of A = √76 = 8.72%.
  4. Deviations of B: −6.2, −0.2, 9.8. Variance of B = 0.3×38.44 + 0.5×0.04 + 0.2×96.04 = 11.532 + 0.02 + 19.208 = 30.76. SD of B = 5.55%.
  5. Covariance = 0.3×(9)(−6.2) + 0.5×(1)(−0.2) + 0.2×(−16)(9.8) = −16.74 − 0.1 − 31.36 = −48.2.
  6. Correlation = −48.2 ÷ (8.72 × 5.55) = −48.2 ÷ 48.35 = −0.997, which is almost perfectly negative.
  7. Portfolio return = 0.6×11 + 0.4×8.2 = 6.6 + 3.28 = 9.88%.
  8. Portfolio variance = 0.36×76 + 0.16×30.76 + 2×0.6×0.4×(−48.2) = 27.36 + 4.92 − 23.14 = 9.15. Portfolio SD = √9.15 = 3.02%.
  9. Interpretation: the weighted average of the SDs is 0.6×8.72 + 0.4×5.55 = 7.45%. The portfolio SD of 3.02% is much lower because the two shares move in opposite directions.

Answer: E(A) = 11%, E(B) = 8.2%; σA = 8.72%, σB = 5.55%; Cov = −48.2; ρ ≈ −0.997. The 60:40 portfolio has expected return 9.88% and SD about 3.02%.

Example 2

A fund manager holds 50% in equity fund E (expected return 14%, SD 16%) and 50% in debt fund D (expected return 9%, SD 10%). The correlation between them is 0.4. Find the portfolio return and SD. Also state the SD if the correlation were +1, and the diversification benefit.

Show the solution
  1. Portfolio return = 0.5×14 + 0.5×9 = 7 + 4.5 = 11.5%.
  2. Covariance = ρ × σE × σD = 0.4 × 16 × 10 = 64.
  3. Portfolio variance = (0.5)²×(16)² + (0.5)²×(10)² + 2×0.5×0.5×64 = 0.25×256 + 0.25×100 + 32 = 64 + 25 + 32 = 121.
  4. Portfolio SD = √121 = 11%.
  5. If ρ = +1, covariance = 160 and variance = 64 + 25 + 80 = 169, so SD = 13%. This equals the weighted average 0.5×16 + 0.5×10 = 13%, which confirms the check.
  6. Diversification benefit = 13% − 11% = 2 percentage points of SD, with no change in expected return.

Answer: Portfolio return = 11.5% and SD = 11%. With ρ = +1 the SD would be 13%, so a correlation of 0.4 reduces risk by 2 percentage points.

Exam tips

  • Show the formula, the substituted values and the result as separate lines. Marks are given for method even if the final figure is slightly off.
  • In case-scenario MCQs, first check whether covariance or correlation is given. Using the wrong one is the commonest trap, so keep ρ × σA × σB handy.
  • Do the ρ = +1 check on every two-asset answer. If your SD is above the weighted average of the SDs, you have made an error.
  • Always end a written answer with a one-line interpretation: which portfolio or security is better on return, risk or CV, and why.
  • Keep SD to two decimal places in working, and avoid rounding SDs before squaring them again. Rounding early causes small mismatches.

Practice questions from Portfolio Management

Portfolio Return and Risk 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 Return and Risk Basics: frequently asked questions

What is the difference between covariance and correlation?

Covariance shows whether two securities move in the same or opposite direction, but its size depends on the units and is hard to compare. Correlation is covariance divided by the product of the two SDs, so it is standardised between −1 and +1. Use correlation to judge the strength of the relationship.

How do I calculate the standard deviation of a two-asset portfolio?

Compute portfolio variance as wA²σA² + wB²σB² + 2wAwB × Cov(A,B), where Cov = ρ × σA × σB. Then take the square root. Do not take a weighted average of the two SDs unless the correlation is +1.

Can portfolio risk be lower than the risk of both securities?

Yes. When correlation is low enough or negative, the portfolio SD can fall below the SD of even the less risky security. Example 1 above shows this: the portfolio SD of about 3.02% is lower than both 8.72% and 5.55%.

Should I divide by n or n − 1 for variance with past returns?

Follow the question. If it treats the data as the whole population, divide by n. If it says sample, divide by n − 1. If it is silent, state your assumption and use the same divisor for variance and covariance.