Unit study guide · NISM Series X-A
Introduction to Modern Portfolio Theory
14 questions in this unit
What this unit covers
- In 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 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 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 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 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 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?
- Asset 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?
- In 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?
- Assets A and B have standard deviations of 0.20 and 0.12, and their correlation is 0.25. What is their covariance?
- An 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?
- Consider 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 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?
- CASE: A portfolio holds 60% equity expected to return 12% and 40% bonds expected to return 6%. What is its expected return?
- CASE: 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?
Key ideas
From the practice bank · 1
In 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?
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, or other constraints. Remember: Compare risk at equal expected return, or expected return at equal risk, before making a mean–variance preference.
Hint Common mistake: Interpreting risk aversion as refusing all investments with uncertain returns.
X-A-176From the practice bank · 2
A 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?
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 Common mistake: Choosing solely from the largest possible payoff rather than the stated decision criterion.
X-A-177From the practice bank · 3
A 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?
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 Common mistake: Reading an expected return as a guaranteed annual result.
X-A-178From the practice bank · 4
A 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?
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 Common mistake: Using equally weighted observations or a sample-variance formula for a supplied probability distribution.
X-A-179From the practice bank · 5
A 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?
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 Common mistake: Using a simple average when investment weights are unequal.
X-A-180From the practice bank · 6
A 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?
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 Common mistake: Averaging standard deviations directly instead of calculating variance first.
X-A-181From the practice bank · 7
Asset 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?
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 Common mistake: Omitting covariance even though the assets are perfectly positively correlated.
X-A-182From the practice bank · 8
In 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?
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 Common mistake: Generalising an exact model result to uncertain real-world correlations.
X-A-183From the practice bank · 9
Assets A and B have standard deviations of 0.20 and 0.12, and their correlation is 0.25. What is their covariance?
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 Common mistake: Inserting percentage-point volatilities into a formula otherwise expressed in decimal returns.
X-A-184From the practice bank · 10
An 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?
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 Common mistake: Using weights rather than squared weights in portfolio variance.
X-A-185From the practice bank · 11
Consider 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?
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 Common mistake: Assuming that being undominated within a short list proves global portfolio efficiency.
X-A-186From the practice bank · 12
A 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?
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 Common mistake: Interpreting many decimal places in an allocation as evidence of forecast certainty.
X-A-187From the practice bank · 13
CASE: A portfolio holds 60% equity expected to return 12% and 40% bonds expected to return 6%. What is its expected return?
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 Common mistake: Using a realised 25% equity increase when the question asks for the forecast portfolio mean.
X-A-196From the practice bank · 14
CASE: 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?
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 Common mistake: Leaving out the factor of two, the correlation term or the final square root.
X-A-197Liquidity describes how easily a given quantity can be traded.
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