Introduction to Modern Portfolio Theory

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More units in NISM Series X-A

• Introduction to Personal Financial Planning4
• Time Value of Money14
• Evaluating the Financial Position of Clients5
• Debt Management and Loans2
• Introduction to the Indian Financial Markets4
• Securities Market Segments21
• Introduction to Investment5
• Investing in Stocks20
• Investing in Fixed Income Securities25
• Understanding Derivatives25
• Mutual Funds25
• Portfolio Manager10
• Overview of Alternative Investment Funds (AIFs)15
• Portfolio Construction Process11
• Portfolio Performance Measurement and Evaluation25
• Operational Aspects of Investment Management15
• Key Regulations15
• Ethical Issues15
• Grievance Redress Mechanism15

Practise all of NISM Series X-A

Exam format

All 14 questions on Introduction to Modern Portfolio Theory, with answers

1. Q1In a simplified mean–variance model, Portfolio P and Portfolio Q each have expected return of 10%. Their return standard deviations are 12% and 18% respectively. All other relevant features are identical. Which portfolio would a risk-averse investor prefer within this model?

   • A

     Option A. Neither, because risk-averse investors cannot hold risky investments

     Why it is wrong: Risk aversion is a preference over risk and compensation, not a prohibition on every risky investment.

   • B

     Option B. Q, because higher variability always produces a higher expected return

     Why it is wrong: The question fixes both expected returns at 10%, so the assumed return advantage does not exist.

   • C

     Option C. P, because it offers the same expected return with lower variability — Correct answer

   • D

     Option D. Either, because equal expected returns make risk irrelevant to a risk-averse investor

     Why it is wrong: Ignoring the different variability is inconsistent with the specified risk-averse preference.

   Answer: C. At a fixed expected return, the lower standard deviation is preferred under the stated risk-averse mean–variance comparison. Both portfolios have the same expected return of 10%, so there is no expected-return sacrifice in selecting P. P has standard deviation of 12%, below Q’s 18%. P therefore dominates Q on the two stated mean–variance inputs. This conclusion is confined to the model and identical other features. It does not imply that a risk-averse investor always selects the lowest-risk asset regardless of expected return, liquidity or other constraints. Remember: Compare risk at equal expected return, or expected return at equal risk, before making a mean–variance preference.

   Hint, shown before you answer in practice mode: Common mistake: Interpreting risk aversion as refusing all investments with uncertain returns.

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2. Q2A risk-neutral decision maker must choose between receiving ₹1,000 for certain and a 50% chance of ₹0 plus a 50% chance of ₹2,000. Assume the same payment date and no other differences. Under a criterion based only on expected monetary value, which statement is correct?

   • A

     Option A. The decision maker is indifferent because both expected amounts equal ₹1,000 — Correct answer

   • B

     Option B. The certain amount must be preferred because every decision maker is risk-averse

     Why it is wrong: The problem explicitly specifies risk neutrality rather than imposing a universal risk-averse preference.

   • C

     Option C. The gamble must be preferred because its largest possible payment is higher

     Why it is wrong: The specified criterion uses the weighted expectation, not only the best possible outcome.

   • D

     Option D. The gamble has expected value of ₹2,000 because there are two possible outcomes

     Why it is wrong: Adding outcome amounts without probability weights does not calculate an expected value.

   Answer: A. The certain amount and the probability-weighted uncertain amount have identical expected monetary values. The certain payment has expected value ₹1,000 because that is received in every outcome. The gamble’s expected value is 0.50 × ₹0 + 0.50 × ₹2,000 = ₹1,000. Its variability differs, but its expected monetary amount does not. Because payment timing and other features are identical, the stated risk-neutral criterion implies indifference. An investor with different preferences could make a different choice without changing this calculation. Remember: Expected value uses probabilities. Risk preferences determine how uncertainty affects the choice.

   Hint, shown before you answer in practice mode: Common mistake: Choosing solely from the largest possible payoff rather than the stated decision criterion.

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3. Q3A security has three mutually exclusive, exhaustive one-year outcomes: probability 25% of a −10% return, probability 50% of a 10% return, and probability 25% of a 30% return. What is the expected one-year return?

   • A

     Option A. 15%

     Why it is wrong: This results from changing the −10% loss into a +10% gain: 0.25×10 + 0.50×10 + 0.25×30 = 15%. The negative sign must be retained.

   • B

     Option B. 10% — Correct answer

   • C

     Option C. 30%

     Why it is wrong: This is the most favourable possible outcome, not the average after allowing for all probabilities.

   • D

     Option D. −10%

     Why it is wrong: This is the least favourable possible outcome, not the probability-weighted expected result.

   Answer: B. The weighted sum is −2.5% + 5% + 7.5% = 10%, using probabilities that sum to one. First check that the states cover all outcomes: 25% + 50% + 25% = 100%. The given probabilities can therefore be used directly. Multiply each return by its probability: 0.25 × (−10%) = −2.5%. 0.50 × 10% = 5%. 0.25 × 30% = 7.5%. Add the contributions to obtain 10%. The expected value can match a possible state, as it does here, but its role is still a weighted average rather than an assured payout. Remember: Weight every outcome, including losses, before adding the contributions.

   Hint, shown before you answer in practice mode: Common mistake: Reading an expected return as a guaranteed annual result.

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4. Q4A security’s one-year return is −10%, 10% or 30% with probabilities 25%, 50% and 25%. Its expected return is 10%. Using probability-weighted population variance, what is return standard deviation, rounded to two decimal places?

   • A

     Option A. 16.33%

     Why it is wrong: This corresponds to treating the three states as equally probable, contrary to their given probabilities.

   • B

     Option B. 20.00%

     Why it is wrong: This is the distance of either extreme outcome from the mean, not the distribution’s weighted standard deviation.

   • C

     Option C. 2.00%

     Why it is wrong: This mistakes variance of 0.02 for a standard deviation instead of taking its square root.

   • D

     Option D. 14.14% — Correct answer

   Answer: D. The probability-weighted squared deviations sum to 0.02 in decimal-return units. The square root is approximately 0.1414. Use decimal returns. Deviations from 0.10 are −0.20, 0 and +0.20 for the three states. Variance = 0.25 × (−0.20)² + 0.50 × 0² + 0.25 × 0.20² = 0.02. This is a complete probability distribution, not an unweighted historical sample. Standard deviation = √0.02 = 0.141421, or 14.14%. Do not add a sample-size correction or confuse squared-return units with percentage return units. Remember: Keep probability weights, variance units and the final square-root step explicit.

   Hint, shown before you answer in practice mode: Common mistake: Using equally weighted observations or a sample-variance formula for a supplied probability distribution.

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5. Q5A fully invested two-asset portfolio starts a period with 60% in Asset A and 40% in Asset B. Their expected returns for that period are 12% and 7%. Assume no trades or external cash flows during the period. What is the expected portfolio return?

   • A

     Option A. 9.5%

     Why it is wrong: This gives the two assets equal weight instead of using the 60/40 beginning allocation.

   • B

     Option B. 10% — Correct answer

   • C

     Option C. 19%

     Why it is wrong: This adds asset returns without weighting them by the amounts invested.

   • D

     Option D. 7.2%

     Why it is wrong: This includes only Asset A’s weighted contribution and omits the return contribution from Asset B.

   Answer: B. The 60% holding contributes 7.2 percentage points and the 40% holding contributes 2.8 percentage points. Use weights as fractions: 0.60 and 0.40. Their sum is one, consistent with the fully invested portfolio. Expected contribution from A = 0.60 × 12% = 7.2 percentage points. Contribution from B = 0.40 × 7% = 2.8 percentage points. Adding gives 10%. Correlation is not needed for this expected-return calculation, although it matters when calculating portfolio variability from the same holdings. Remember: Expected portfolio return is linear in beginning weights. Portfolio risk generally is not.

   Hint, shown before you answer in practice mode: Common mistake: Using a simple average when investment weights are unequal.

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6. Q6A portfolio has equal weights in two risky assets with standard deviations of 20% and 10% for the same period. Their return correlation is zero. Using the two-asset variance formula, what is portfolio standard deviation, rounded to two decimals?

   • A

     Option A. 11.18% — Correct answer

   • B

     Option B. 15.00%

     Why it is wrong: This directly averages the two standard deviations and would apply for these nonnegative weights at correlation +1.

   • C

     Option C. 7.50%

     Why it is wrong: This divides the weighted-average volatility again instead of applying squared weights and a square root.

   • D

     Option D. 12.50%

     Why it is wrong: This does not correctly convert the variance of 0.0125 into a standard deviation.

   Answer: A. Variance is 0.5² × 0.20² + 0.5² × 0.10² = 0.0125, whose square root is about 11.18%. For two assets, variance = wA²σA² + wB²σB² + 2wAwBρσAσB. Use returns in decimal form. Here ρ = 0, so the last term is zero. The remaining terms are 0.25 × 0.04 = 0.01 and 0.25 × 0.01 = 0.0025. Total variance is 0.0125. Taking its square root gives 0.111803, or 11.18%, below the 15% weighted-average volatility but still positive. Remember: Zero correlation creates diversification potential without making the assets individually riskless.

   Hint, shown before you answer in practice mode: Common mistake: Averaging standard deviations directly instead of calculating variance first.

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7. Q7Asset A has standard deviation 20% and Asset B 10%. A long-only portfolio assigns them weights of 60% and 40%. Their returns are perfectly positively correlated. What is portfolio standard deviation?

   • A

     Option A. 12.65%

     Why it is wrong: This omits the positive covariance term and instead uses a zero-correlation calculation.

   • B

     Option B. 8%

     Why it is wrong: This subtracts the weighted volatilities, which corresponds to perfect negative rather than positive correlation.

   • C

     Option C. 16% — Correct answer

   • D

     Option D. 30%

     Why it is wrong: This adds unweighted asset standard deviations without accounting for the portfolio’s weights.

   Answer: C. At correlation +1 and nonnegative weights, portfolio standard deviation equals 0.60 × 20% + 0.40 × 10%. Weighted volatility of A is 0.60 × 20% = 12%. Weighted volatility of B is 0.40 × 10% = 4%. With correlation +1, the full variance is the square of their sum. Standard deviation is therefore 12% + 4% = 16%. A return correlation of +1 does not require identical expected returns or identical volatility levels. It specifies perfect positive linear co-movement, which is enough for this risk calculation. Remember: Recognise the +1 correlation boundary as the no-diversification-benefit case for nonnegative weights.

   Hint, shown before you answer in practice mode: Common mistake: Omitting covariance even though the assets are perfectly positively correlated.

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8. Q8In an idealised two-asset model, standard deviations are 20% for A and 10% for B, and correlation is exactly −1. The portfolio puts one-third in A and two-thirds in B. Under this model, what is portfolio standard deviation?

   • A

     Option A. 0% — Correct answer

   • B

     Option B. 10%

     Why it is wrong: This selects one asset’s standalone volatility rather than calculating the combined portfolio risk.

   • C

     Option C. 13.33%

     Why it is wrong: This adds weighted volatilities as though correlation were +1 rather than −1.

   • D

     Option D. 9.43%

     Why it is wrong: This uses the sum of variance terms while omitting the negative covariance term.

   Answer: A. The weighted volatility magnitudes are equal and offset exactly at correlation −1. The weighted magnitudes are (1/3) × 20% and (2/3) × 10%, both equal to 6.6667%. At correlation −1, portfolio standard deviation is the absolute difference of those magnitudes. Their difference is exactly zero using the unrounded fractions. This is a mathematical result for the stated model, not a claim that real assets can be combined into a permanently risk-free investment. Estimation changes and unmodelled risks can invalidate the idealised offset. Remember: A zero-variance combination requires the particular offsetting relationship, not merely owning two assets.

   Hint, shown before you answer in practice mode: Common mistake: Generalising an exact model result to uncertain real-world correlations.

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9. Q9Assets A and B have standard deviations of 0.20 and 0.12, and their correlation is 0.25. What is their covariance?

   • A

     Option A. 0.024

     Why it is wrong: This multiplies only the standard deviations and omits the correlation factor.

   • B

     Option B. 0.080

     Why it is wrong: This multiplies correlation by the sum of the standard deviations, not their product.

   • C

     Option C. 0.006 — Correct answer

   • D

     Option D. 0.250

     Why it is wrong: This reports the unitless correlation rather than the requested covariance.

   Answer: C. Multiplying 0.25 × 0.20 × 0.12 gives 0.006 in the required squared-return units. All inputs already use decimal-return units. Do not turn 0.20 into 20 while leaving the other values unchanged. Multiply the volatilities: 0.20 × 0.12 = 0.024. Apply correlation: 0.25 × 0.024 = 0.006. The positive sign is consistent with positive correlation. Covariance is not a percentage return or a probability. Its units are the product of the two return units. Remember: Keep covariance’s scale and units distinct from the unitless correlation coefficient.

   Hint, shown before you answer in practice mode: Common mistake: Inserting percentage-point volatilities into a formula otherwise expressed in decimal returns.

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10. Q10An equally weighted three-asset portfolio has same-period standard deviations of 30%, 20% and 10%. Every pairwise return correlation is zero. What is portfolio standard deviation, rounded to two decimals?

    • A

      Option A. 20.00%

      Why it is wrong: This averages standard deviations directly and ignores the zero pairwise correlations.

    • B

      Option B. 21.60%

      Why it is wrong: This averages standalone variances using weights rather than squared weights before taking the square root.

    • C

      Option C. 37.42%

      Why it is wrong: This adds the standalone variances without applying any portfolio weights before taking the square root.

    • D

      Option D. 12.47% — Correct answer

    Answer: D. Variance is (0.30² + 0.20² + 0.10²)/9. Its square root is approximately 12.47%. Every weight is exactly 1/3. Each squared weight is therefore 1/9. No intermediate rounded weight is needed. Variance = (1/9) × (0.30² + 0.20² + 0.10²) = 0.14/9 = 0.0155556. Standard deviation = √0.0155556 = 0.124722, or 12.47%. It remains positive because each risky asset contributes variance despite the absence of pairwise correlation. Remember: For multiple assets, square the weights in standalone variance terms and account for every covariance pair.

    Hint, shown before you answer in practice mode: Common mistake: Using weights rather than squared weights in portfolio variance.

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11. Q11Consider four feasible portfolios, with all non-risk/return features identical: A has expected return 9% and standard deviation 12%. B has 9% and 10%. C has 10% and 14%. D has 7% and 8%. Which portfolio is directly dominated by another listed portfolio in a mean–variance comparison?

    • A

      Option A. A — Correct answer

    • B

      Option B. B

      Why it is wrong: None of the other listed portfolios offers at least B’s return with no more than B’s risk and a strict improvement.

    • C

      Option C. C

      Why it is wrong: C has the highest expected return. The lower-risk choices also have lower expected returns, creating trade-offs.

    • D

      Option D. D

      Why it is wrong: D has the lowest standard deviation. Alternatives with higher expected returns also carry higher risk.

    Answer: A. B offers A’s same expected return with strictly lower standard deviation, so A is directly dominated. Compare A and B: each expects 9%, but B’s standard deviation is 10% rather than 12%. B improves risk without sacrificing expected return. C trades higher expected return for higher risk, while D trades lower risk for lower expected return. The listed figures alone do not make either of those trade-offs dominated. These comparisons identify A as inefficient relative to the listed choices. They do not prove that every other choice lies on the full efficient frontier, because other feasible portfolios may exist. Remember: Use dominance to eliminate a strictly inferior risk/return combination without inventing a universal investor preference.

    Hint, shown before you answer in practice mode: Common mistake: Assuming that being undominated within a short list proves global portfolio efficiency.

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12. Q12A mean–variance optimiser changes its recommended weight in one asset from 10% to 70% after that asset’s estimated expected return rises from 8.0% to 8.2%. Which response best addresses this result?

    • A

      Option A. Ignore portfolio constraints because the numerical optimum takes priority

      Why it is wrong: An implementable allocation must respect applicable client and mandate constraints as part of the model and review.

    • B

      Option B. Treat 70% as an objectively correct weight because an optimiser produced it

      Why it is wrong: Optimisation solves the supplied model. It does not make estimated returns and covariances known future facts.

    • C

      Option C. Review input uncertainty and run sensitivity and constraint checks before implementing — Correct answer

    • D

      Option D. Conclude that the asset’s actual future return must rise by exactly 0.2 percentage points

      Why it is wrong: A change in an estimate is not a guaranteed change in the realised investment outcome.

    Answer: C. Large weight changes from a small estimate change call for assessing model sensitivity rather than treating the new allocation as certain. The expected-return input changed by only 0.2 percentage points, while the proposed weight changed by 60 percentage points. That sensitivity makes it important to inspect assumptions, feasible constraints and alternative input scenarios before relying on the allocation. The example does not prove the optimiser is incorrectly programmed. It illustrates that solving a mathematical objective precisely is different from knowing which portfolio will perform best in the future. Remember: A precise optimum is only as reliable as the model, estimated inputs and constraints behind it.

    Hint, shown before you answer in practice mode: Common mistake: Interpreting many decimal places in an allocation as evidence of forecast certainty.

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13. Q13CASE: A portfolio holds 60% equity expected to return 12% and 40% bonds expected to return 6%. What is its expected return?

    • A

      Option A. 9.60% — Correct answer

    • B

      Option B. 15.00%

      Why it is wrong: 15% is not a weighted average of 12% and 6%.

    • C

      Option C. 18.00%

      Why it is wrong: 18% adds the two returns without weights.

    • D

      Option D. 9.00%

      Why it is wrong: 9% is a simple average, not a 60/40 weighted one.

    Answer: A. 0.6 × 12% + 0.4 × 6% = 7.2% + 2.4% = 9.6%. Remember: Keep expected inputs separate from a scenario’s later realised outcomes.

    Hint, shown before you answer in practice mode: Common mistake: Using a realised 25% equity increase when the question asks for the forecast portfolio mean.

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14. Q14CASE: A portfolio holds 60% equity and 40% bonds. Their standard deviations are 20% and 10%, and their correlation is 0.25. What is the portfolio standard deviation?

    • A

      Option A. 1.84%

      Why it is wrong: 1.84% is the variance, 0.0184, not its square root.

    • B

      Option B. 13.56% — Correct answer

    • C

      Option C. 16.00%

      Why it is wrong: 16% adds the weighted standard deviations, as if correlation were 1.

    • D

      Option D. 12.65%

      Why it is wrong: 12.65% leaves out the covariance, as if correlation were 0.

    Answer: B. Variance = 0.36 × 0.04 + 0.16 × 0.01 + 2 × 0.6 × 0.4 × 0.25 × 0.2 × 0.1 = 0.0184. Standard deviation = √0.0184 = 13.56%. Remember: Include both standalone variance terms and the full covariance contribution before taking the square root.

    Hint, shown before you answer in practice mode: Common mistake: Leaving out the factor of two, the correlation term or the final square root.

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Portfolio Construction Process

11 questions on Portfolio Construction Process. Its quiz opens on that unit.