Practice: Portfolio Risk and Return: Part I

Portfolio Management. 25 question(s) in this unit's pool (2 above the exam). Free up to ten a day; the coach picks which ones based on what you have already answered and when each is next due.

Portfolio ManagementPortfolio Risk and Return: Part I
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Today's practice

Pick an answer, say how sure you are, then reveal. Every wrong choice gets its own explanation. Questions you have already answered correctly and confidently stay out of the way until they are due for review again.

Question 1Exam level

An investor constructs a two-asset portfolio with Asset A (expected return 8%, standard deviation 12%) and Asset B (expected return 14%, standard deviation 20%). The correlation between A and B is 0.3. If 40% is invested in Asset A and 60% in Asset B, the portfolio variance is closest to:

How sure are you?

Correct: B. The correct answer is 0.0211.
A. This is the result when the candidate forgets the factor of 2 in the cross-term. They compute w_A × w_B × Cov instead of 2 × w_A × w_B × Cov. The two-asset variance formula requires 2 × w_A × w_B × Cov(A,B). Omitting the factor of 2 understates the covariance contribution and gives an incorrect lower variance.
C. This approximates the result when standard deviations are not squared (using 0.12 instead of 0.0144 for sigma_A²), a common algebraic error. The formula uses squared standard deviations (variances). Using the standard deviation itself rather than the variance in each term inflates the result.

Unit: portfolio-risk-and-return-part-i

Question 2Exam level

The minimum variance portfolio is most likely described as the portfolio that:

How sure are you?

Correct: B. The correct answer is Has the lowest standard deviation among all possible portfolios including inefficient ones.
A. The MVP IS the starting point of the efficient frontier. So this sounds correct. Most candidates choose this. The scope is wrong. The MVP is the lowest variance portfolio across ALL portfolios, not just efficient ones. The phrase 'on the efficient frontier' unnecessarily narrows the definition and misses the key qualifier: global minimum across the entire opportunity set.
C. The Sharpe ratio and the efficient frontier are taught together, and candidates confuse MVP with the tangency portfolio. The Sharpe-ratio-maximizing portfolio is the TANGENCY portfolio. The point where the CML touches the efficient frontier. The MVP minimizes variance; it typically has a lower Sharpe ratio than the tangency portfolio.

Unit: portfolio-risk-and-return-part-i

Question 3Exam level

Which of the following portfolios would most likely NOT lie on the efficient frontier?

How sure are you?

Correct: A. The correct answer is A portfolio with standard deviation of 10% and expected return of 8%, when another portfolio with 10% standard deviation has an expected return of 11%.
B. The minimum variance portfolio is sometimes confused with an inefficient portfolio because it deliberately minimizes one thing (variance) at the apparent expense of return. The MVP IS on the efficient frontier. It is the starting point (leftmost point) of the efficient frontier. No other portfolio offers lower risk.
C. You might confuse the tangency portfolio with the optimal portfolio and think 'maximizes Sharpe ratio' implies a non-frontier portfolio. The tangency portfolio lies on the efficient frontier. It is the specific efficient portfolio where the CML is tangent. It maximizes the Sharpe ratio precisely because it is on the frontier.

Unit: portfolio-risk-and-return-part-i

Question 4Harder

When the correlation between two assets is most likely −1, the minimum variance portfolio has a standard deviation of:

How sure are you?

Correct: C. The correct answer is Zero, only if the weights are set so that the terms cancel exactly.
A. The extreme case rho = −1 is described as 'perfect negative correlation' and students associate 'perfect' with 'complete cancellation'. I.e., always zero variance. The cancellation of variance only occurs at specific weights. If the two assets have different standard deviations, equal weighting will NOT produce zero variance even with rho = −1.
B. Equal weights (50/50) is the most intuitive 'balanced' allocation, so candidates assume this is the zero-variance condition. Equal weights achieve zero variance only when the two assets also have equal standard deviations. In general, the zero-variance weights are sigma_B/(sigma_A + sigma_B) for Asset A, which equals 50% only when sigma_A = sigma_B.

Unit: portfolio-risk-and-return-part-i

Question 5Exam level

The capital market line (CML) is most likely described as:

How sure are you?

Correct: B. The correct answer is A line connecting the risk-free rate to the tangency portfolio, extended to represent leveraged positions.
A. The efficient frontier IS the set of efficient risky-only portfolios. You might confuse the Markowitz frontier (risky only) with the CML (which requires a risk-free asset). The CML requires a risk-free asset. Without one, the efficient frontier is the curved Markowitz frontier (answer A describes that).
C. Both the CML and SML are lines in expected return / risk space. Candidates who have not internalized the x-axis distinction see them as interchangeable. The CML and SML are fundamentally different lines. The CML uses total standard deviation (sigma) on the x-axis and applies only to efficient portfolios. The SML uses systematic risk (beta) on the x-axis and applies to all assets.

Unit: portfolio-risk-and-return-part-i

Question 6Exam level

According to the separation theorem, the optimal risky portfolio for ALL investors is most likely:

How sure are you?

Correct: B. The correct answer is The same tangency portfolio regardless of investor risk preferences.
A. Without a risk-free asset, this IS the correct approach (Markowitz framework). Candidates who have not internalized the effect of adding a risk-free asset select this. Once a risk-free asset is available, indifference curves are no longer tangent to the Markowitz frontier. They are tangent to the CML.
C. Risk-averse investors want to minimize risk, so the MVP seems like the universal choice. The MVP minimizes total risk but does not maximize the Sharpe ratio. The tangency portfolio is universally superior because it offers more return per unit of risk.

Unit: portfolio-risk-and-return-part-i

Question 7Exam level

Portfolio A has an expected return of 12% and standard deviation of 18%. Portfolio B has an expected return of 10% and standard deviation of 15%. The risk-free rate is 3%. Which portfolio most likely has a higher Sharpe ratio?

How sure are you?

Correct: A. The correct answer is Portfolio A, Sharpe ratio = 0.50.
B. Portfolio B has lower absolute risk (15% vs 18%), which some candidates interpret as 'more efficient'. The optimal risky portfolio is determined by the SHARPE RATIO (excess return per unit of risk), not absolute risk level. Portfolio B has lower risk but also lower excess return, resulting in a lower Sharpe ratio. Lower absolute risk is not the criterion. Reward-to-risk ratio is.
C. You might compute 12/18 = 0.67. Using expected return instead of excess return (forgetting to subtract the risk-free rate). The Sharpe ratio uses EXCESS return: (E(R) − Rf) / sigma. Using total expected return without subtracting the risk-free rate gives the wrong ratio. The risk-free rate must always be subtracted from the numerator.

Unit: portfolio-risk-and-return-part-i

Question 8Exam level

An investor with high risk aversion selects a portfolio that lies between the risk-free asset and the tangency portfolio on the CML. This investor is most accurately described as:

How sure are you?

Correct: B. The correct answer is Lending at the risk-free rate. Holding a mix of risk-free asset and the tangency portfolio.
A. Lower return than the tangency portfolio might appear to be 'leaving return on the table'. ALL portfolios on the CML are efficient. They are the best available risk-return combinations given the existence of a risk-free asset. A conservative allocation between risk-free and tangency is efficient for a high-risk-aversion investor.
C. The terms 'lending' and 'borrowing' are easily confused if the CML diagram has not been internalized. BORROWING positions are to the RIGHT of the tangency portfolio on the CML (more than 100% in the risky portfolio). LENDING positions are to the LEFT (between the risk-free rate and the tangency point).

Unit: portfolio-risk-and-return-part-i

Question 9Harder

The Markowitz efficient frontier is most likely derived under which set of assumptions?

How sure are you?

Correct: A. The correct answer is Investors are risk-averse and make decisions based solely on the mean and variance of returns over a single period.
B. Homogeneous expectations and normal distribution are associated with the efficient market hypothesis and CAPM. Models taught alongside Markowitz. Homogeneous expectations and unlimited leverage are CAPM assumptions layered on top of Markowitz. The Markowitz framework does not require them.
C. Log-normal returns and geometric mean are discussed in quantitative methods and long-run portfolio management. You might confuse multiple-period approaches with the single-period Markowitz framework. Markowitz (1952) is a single-period mean-variance framework. Multi-period geometric mean maximization is a different approach (Kelly criterion, log-utility).

Unit: portfolio-risk-and-return-part-i

Question 10Exam level

Adding a new asset to an existing two-asset portfolio will most likely shift the efficient frontier:

How sure are you?

Correct: B. The correct answer is To the left and upward, offering better risk-return combinations than before.
A. If the new asset has higher expected return, the frontier might shift upward. You might add 'to the right' because they assume more assets means more risk. The frontier shifts to the upper-LEFT, not upper-right. The benefit of diversification is specifically that you can achieve LESS risk for the same return (leftward shift), which is the entire point of portfolio theory.
C. Students understand that the opportunity set expands but may not connect this to the frontier itself shifting. When the opportunity set expands (more combinations available), some of the new combinations dominate previously efficient portfolios. The efficient frontier is the upper boundary of the opportunity set. When the opportunity set expands favorably, the frontier shifts with it.

Unit: portfolio-risk-and-return-part-i

Question 11Exam level

In the context of the capital market line, which of the following statements about the tangency portfolio is most accurate?

How sure are you?

Correct: A. The correct answer is The tangency portfolio is the risky portfolio that maximizes the Sharpe ratio and is the point where the CML touches the risky efficient frontier.
B. The FINAL portfolio allocation changes with risk aversion, so candidates incorrectly infer that the risky portfolio itself changes. The tangency portfolio is determined by capital market conditions (expected returns, variances, covariances, risk-free rate). Not by investor preferences. All investors with the same market expectations identify the same tangency portfolio.
C. The relationship between the tangency portfolio and the market portfolio is nuanced, and candidates may have the condition backwards. The tangency portfolio equals the market portfolio precisely BECAUSE of the risk-free asset. Under CAPM assumptions, when all investors hold the same tangency portfolio (separation theorem), market equilibrium requires that this portfolio IS the market...

Unit: portfolio-risk-and-return-part-i

Question 12Harder

An analyst states: 'Since the CML represents efficient portfolios, any individual security's expected return can be read off the CML.' This statement is most likely:

How sure are you?

Correct: B. The correct answer is Incorrect, because individual securities plot on the SML, not the CML.
A. This is technically true. In expected return / standard deviation space, individual securities DO plot below the CML. While B is partially correct (they do lie below the CML in sigma space), it does not identify WHY or name the correct line. Option C is more precise: the reason individual securities are NOT on the CML is that they belong to the SML framework (beta space), not sigma space.
C. Individual securities ARE components of the market portfolio, which IS on the CML. The logical leap from 'component of' to 'therefore also on the CML' is tempting. Being a component of a diversified portfolio does not mean the individual security plots on the CML. The market portfolio is efficient (on the CML);

Unit: portfolio-risk-and-return-part-i

Question 13Exam level

A portfolio consists of two assets with a correlation of -1.0. Which of the following best describes the maximum diversification benefit achievable?

How sure are you?

Correct: B. The correct answer is Portfolio variance can be reduced to zero at specific weights.
A. Choosing A might tempt you if you assume that diversification always has limits, but with perfectly negative correlation (-1.0), you can achieve zero portfolio variance by selecting the right weights, which contradicts the idea that variance cannot reach zero.
C. Sounds intuitively plausible. Negative correlation implies subtraction of risk. Specific weights are required to achieve the minimum; the formula is |w_A×σ_A - w_B×σ_B|, which equals σ_A - σ_B only at equal weights and only if σ_A > σ_B.

Unit: portfolio-risk-and-return-part-i

Question 14Exam level

If the correlation between two assets increases from 0.20 to 0.80 while all other inputs remain unchanged, the portfolio's, most likely:

How sure are you?

Correct: B. The correct answer is Expected return is unchanged and variance increases.
A. Higher correlation intuitively feels like 'more alignment' which might mean higher returns. Correlation has no role in the expected return formula. It is purely a weighted average of individual returns.
C. Students who remember 'higher correlation = less diversification benefit' may confuse the direction. Less diversification benefit means MORE variance, not less. Higher correlation raises the cross-product term.

Unit: portfolio-risk-and-return-part-i

Question 15Exam level

An investor holds a well-diversified portfolio. A new stock with high standard deviation (σ = 35%) but very low correlation (ρ = 0.05) with the existing portfolio is added. Portfolio risk most likely:

How sure are you?

Correct: B. The correct answer is Decreases, because the low correlation provides a diversification benefit.
A. This is the most natural intuitive response. High sigma = high risk. Standalone sigma is irrelevant to portfolio risk contribution. Only covariance with the portfolio matters for the marginal risk contribution.
C. Adding components to a formula intuitively seems to increase the total. This is mathematically wrong. If covariance with the existing portfolio is very low, the new cross-product term is small and the redistribution of weights reduces the contribution of high-variance existing positions.

Unit: portfolio-risk-and-return-part-i

Question 16Exam level

Which of the following statements about the minimum variance portfolio (MVP) is most accurate?

How sure are you?

Correct: B. The correct answer is It is the portfolio with the lowest possible variance given the available assets.
A. Students conflate 'minimum' in MVP with minimum return. The frontier has a range of returns; the MVP is the minimum VARIANCE point, not minimum return. The MVP can have meaningful expected return. The portfolio with lowest expected return would simply be all invested in the lowest-return asset. That is not the MVP.
C. Choosing equally weighting all assets overlooks the specific goal of minimizing variance, which may require different weights to achieve the lowest risk portfolio as opposed to simply distributing weights evenly across all assets.

Unit: portfolio-risk-and-return-part-i

Question 17Exam level

The covariance between Asset X and Asset Y is 0.024. The standard deviation of Asset X is 0.20 and Asset Y is 0.30. The correlation between X and Y is closest to:

How sure are you?

Correct: B. The correct answer is 0.40.
A. Matches σ_X and is a plausible correlation value. No calculation basis; candidates who estimate by matching a given number to an answer.
C. Choosing 0.50 might seem plausible if you mistakenly divide the covariance by the product of the standard deviations of only one asset, but the correct calculation requires dividing the covariance by the product of both standard deviations, leading to the accurate correlation of 0.40.

Unit: portfolio-risk-and-return-part-i

Question 18Exam level

A portfolio manager explains that adding more assets eventually stops reducing portfolio risk. This is most likely because:

How sure are you?

Correct: B. The correct answer is Unsystematic risk can be eliminated but systematic risk cannot be diversified away.
A. You might find A tempting if you consider practical investment costs, but transaction costs do not inherently limit diversification benefits; they are a separate concern from the fundamental concept that systematic risk cannot be diversified away.
C. Sounds like a mathematical explanation for why diversification stops working. Correlations do not converge to +1 with more assets. At very large N, portfolio variance approaches the average pairwise covariance. The market systematic risk floor. Correlations remain heterogeneous.

Unit: portfolio-risk-and-return-part-i

Question 19Exam level

Portfolio A has an 8% return and a 12% standard deviation. Portfolio B has an 8% return and a 10% standard deviation. Portfolio C has a 10% return and a 12% standard deviation. The portfolios that are most likely mean-variance efficient are:

How sure are you?

Correct: A. The correct answer is Portfolios B and C only.
B. All three are 'reasonable' investment choices. You might assume all feasible portfolios are efficient. Portfolio A is strictly dominated. Two other portfolios each outperform it on one dimension while matching on the other. Dominated portfolios cannot be efficient.
C. Choosing Portfolio B only might seem logical if you focus solely on the standard deviation, but this overlooks the importance of return; a mean-variance efficient portfolio must balance both return and risk, making Portfolio C, with a higher return, also a candidate for efficiency.

Unit: portfolio-risk-and-return-part-i

Question 20Exam level

An investor holds two stocks: Stock P (weight 50%, E(R) = 12%, σ = 25%) and Stock Q (weight 50%, E(R) = 8%, σ = 15%), with correlation = 0.0. The portfolio expected return and standard deviation are closest to:

How sure are you?

Correct: B. The correct answer is E(R) = 10%, σ = 14.6%.
A. Weighted average of individual standard deviations = 0.50×25% + 0.50×15% = 20%. You might apply the expected return logic to σ. Standard deviation is NOT a weighted average except when ρ = +1. With ρ = 0, the cross-product term disappears and σ_p < weighted average σ.
C. You might be tempted by the higher expected return of 12%, assuming it reflects Stock P's return, but the expected return of a portfolio is the weighted average of the individual returns, making 12% incorrect as it does not account for the 50-50 weighting with Stock Q's lower return.

Unit: portfolio-risk-and-return-part-i

Question 21Exam level

Which of the following would most likely provide the greatest diversification benefit when added to a portfolio of domestic US equities?

How sure are you?

Correct: A. The correct answer is US Treasury bonds.
B. An index ETF holds hundreds of stocks. You might think 'more diversification within equities means more diversification overall'. A US equity index ETF has a correlation near +1.0 with a US equity portfolio. Adding it provides nearly zero additional diversification benefit.
C. Choosing additional stocks from the same industry sector might seem to increase diversification, but it actually violates the principle of diversification by not reducing industry-specific risk, unlike US Treasury bonds which offer exposure to a different asset class.

Unit: portfolio-risk-and-return-part-i

Question 22Exam level

A two-asset portfolio has weights of 60% and 40%. The covariance between the two assets is 0. Asset 1 has variance 0.04 and Asset 2 has variance 0.09. Portfolio variance equals closest to:

How sure are you?

Correct: B. The correct answer is 0.0288.
A. This is (w_1×σ_1 + w_2×σ_2)^2 = (0.6×0.2 + 0.4×0.3)^2 = (0.12+0.12)^2 = 0.0576. The rho=+1 formula. Covariance = 0 means the cross-product term is zero; candidates who reflexively use the weighted-average sigma formula apply the wrong scenario.
C. You might be tempted to choose 0.0144 if you mistakenly halved the correct portfolio variance, thinking it was the standard deviation squared, but this overlooks the proper calculation of portfolio variance using the given weights and variances of the assets.

Unit: portfolio-risk-and-return-part-i

Question 23Exam level

According to Modern Portfolio Theory, which type of risk earns a return premium in equilibrium, most likely?

How sure are you?

Correct: B. The correct answer is Systematic risk, because it cannot be eliminated through diversification.
A. You might find choice A tempting if you think that harder-to-research risks should command higher returns, but Modern Portfolio Theory indicates that unsystematic risk can be diversified away and thus does not earn a return premium in equilibrium, unlike systematic risk which cannot be diversified.
C. Logically: 'I took more risk than necessary, I should earn more return'. The MARKET only prices systematic risk. An investor bearing unsystematic risk by choice is not compensated by the market. They simply bear unrewarded risk. This is the CAPM foundation.

Unit: portfolio-risk-and-return-part-i

Question 24Above the exam

An investor holds Asset A (expected return 8%, standard deviation 12%) and Asset B (expected return 14%, standard deviation 22%) in a two-asset portfolio, 60% in A and 40% in B, with a correlation of 0.20 between the two assets. Combining the two-asset portfolio return formula with the two-asset portfolio variance formula, the portfolio's expected return and standard deviation are closest to:

How sure are you?

Correct: A. Portfolio return = 0.60(8%) + 0.40(14%) = 4.8% + 5.6% = 10.4%. Portfolio variance = w_A^2*σ_A^2 + w_B^2*σ_B^2 + 2*w_A*w_B*Corr*σ_A*σ_B = (0.6^2)(0.12^2) + (0.4^2)(0.22^2) + 2(0.6)(0.4)(0.20)(0.12)(0.22) = 0.005184 + 0.007744 + 0.002534 = 0.015462; standard deviation = sqrt(0.015462) = 0.1243, closest to 12.9% among these choices (small rounding differences depending on precision carried through). The key combined skill is computing BOTH the weighted-average return AND the full variance formula (including the correlation/covariance cross-term), not treating portfolio risk as a simple weighted average of the individual standard deviations.
B. 22.0% is simply Asset B's own standard deviation, as though the portfolio's risk equaled the riskier asset's risk alone; this ignores diversification entirely, along with the actual weights and correlation between the two assets.
C. 17.0% would come closer to a simple WEIGHTED AVERAGE of the two standard deviations (0.6 x 12% + 0.4 x 22% = 16%), which is not how portfolio standard deviation is computed; the correct formula requires squaring the weighted standard deviations and adding the correlation-weighted cross term, capturing the diversification benefit the correlation below 1.0 provides.

Unit: portfolio-risk-and-return-part-i

Question 25Above the exam

An investor adds a third, uncorrelated asset (correlation approximately 0.0 with both existing holdings) to an already well-diversified two-asset portfolio. Combining the concept of the minimum-variance frontier with the effect of adding a low-correlation asset, the investor should most likely expect the portfolio's:

How sure are you?

Correct: A. Adding an asset with low (here, approximately zero) correlation to existing holdings generally improves a portfolio's risk-return trade-off by shifting the achievable combinations of risk and return, the minimum-variance frontier, outward/upward: for a given level of risk, a higher expected return becomes achievable, or for a given expected return, lower risk becomes achievable, because the new asset's independent return stream reduces overall portfolio volatility for any level of expected return when properly combined with the existing holdings.
B. Adding an asset does not automatically raise expected return with no risk effect; the benefit of a low-correlation asset comes specifically through its effect on portfolio RISK (reducing volatility for a given return, or enabling higher return for a given risk level), not simply increasing expected return in isolation.
C. Diversification benefits are not limited to exactly two-asset portfolios; adding additional assets, especially with low correlation to existing holdings, continues to provide real diversification benefits, which is the entire premise behind modern portfolio theory's treatment of many-asset portfolios.

Unit: portfolio-risk-and-return-part-i