Corporate Finance Quiz: Diversification And Correlation
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Diversification And CorrelationQuestion 1 of 20

A portfolio consists of two assets, X and Y, with equal weighting. Asset X has an expected return of 10% and a standard deviation of 20%. Asset Y has an expected return of 15% and a standard deviation of 30%. If the two assets are perfectly negatively correlated (ρ=−1\rho = -1), what is the standard deviation of the portfolio?

0.0%
5.0%
12.5%
25.0%
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Corporate Finance Quiz

Corporate Finance Quiz: Diversification And Correlation

Practice Diversification And Correlation in Corporate Finance with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Diversification And Correlation, giving you a quick way to practice the rules, question types, and explanations that matter most for Corporate Finance.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

A portfolio consists of two assets, X and Y, with equal weighting. Asset X has an expected return of 10% and a standard deviation of 20%. Asset Y has an expected return of 15% and a standard deviation of 30%. If the two assets are perfectly negatively correlated (ρ=−1\rho = -1), what is the standard deviation of the portfolio?

  1. 0.0%
  2. 5.0% (correct answer)
  3. 12.5%
  4. 25.0%
Explanation: The formula for the standard deviation of a two-asset portfolio is σp=wX2σX2+wY2σY2+2wXwYρXYσXσY\sigma_p = \sqrt{w_X^2\sigma_X^2 + w_Y^2\sigma_Y^2 + 2w_X w_Y \rho_{XY} \sigma_X \sigma_Y}. With perfect negative correlation (ρ=−1\rho = -1), the formula simplifies to σp=(wXσX−wYσY)2=∣wXσX−wYσY∣\sigma_p = \sqrt{(w_X\sigma_X - w_Y\sigma_Y)^2} = |w_X\sigma_X - w_Y\sigma_Y|. Plugging in the values: σp=∣0.5(0.20)−0.5(0.30)∣=∣0.10−0.15∣=∣−0.05∣=0.05\sigma_p = |0.5(0.20) - 0.5(0.30)| = |0.10 - 0.15| = |-0.05| = 0.05, or 5.0%. A zero-risk portfolio is only possible with ρ=−1\rho = -1 if the weights are chosen specifically to make wXσX=wYσYw_X\sigma_X = w_Y\sigma_Y, which is not the case here. The weighted average of the standard deviations (25.0%) is incorrect as it assumes perfect positive correlation.

Question 2

An investment opportunity set for two assets, a stock fund and a bond fund, is being analyzed. If the correlation of returns between the two funds decreases from 0.5 to -0.5, holding individual asset returns and variances constant, what is the effect on the shape of the investment opportunity set?

  1. The curve will become flatter and shift to the right, offering lower returns for a given level of risk.
  2. The curve will shift vertically upwards, indicating higher expected returns for all possible portfolio combinations.
  3. The curve will become a straight line connecting the two assets, eliminating any benefit from diversification.
  4. The curve will become more pronounced, bending further to the left, expanding the set of available risk-return combinations. (correct answer)
Explanation: When analyzing portfolio diversification, the correlation between assets fundamentally determines the shape and position of the investment opportunity set curve. This curve plots all possible risk-return combinations you can achieve by varying portfolio weights between two assets. As correlation decreases from 0.5 to -0.5, the diversification benefits dramatically increase. With lower correlation, combining the two assets allows you to achieve the same expected return with significantly less risk, or alternatively, access higher returns for any given risk level. This improvement manifests as the curve bending further toward the left (lower risk axis), creating a more pronounced bow shape that expands your available risk-return combinations. Option A is incorrect because the curve shifts left (toward lower risk), not right, and this represents better opportunities, not worse ones. Option B misses the key point—while risk-adjusted returns improve, the individual assets' expected returns remain unchanged, so there's no vertical shift affecting all combinations equally. Option C describes what happens when correlation equals +1.0, where diversification provides no benefit and you get a straight line between the assets. Option D correctly identifies that the curve becomes more pronounced and bends further left. The "expanding set of available risk-return combinations" refers to accessing lower risk levels for given returns, which is the essence of diversification benefit. Remember: lower correlation always improves the investment opportunity set by bending the curve further toward the risk axis (leftward). Perfect negative correlation (-1.0) creates the maximum possible diversification benefit.

Question 3

A financial advisor suggests adding a real estate investment trust (REIT) to a client's existing all-equity portfolio. The REIT has a higher standalone standard deviation than the equity portfolio. Under which condition would the addition of the REIT most likely lead to a reduction in the client's overall portfolio risk?

  1. Only if the correlation between the REIT and the equity portfolio is negative.
  2. It is not possible, as adding a riskier asset will always increase portfolio risk.
  3. If the REIT's expected return is also higher than the equity portfolio's expected return.
  4. If the correlation between the REIT and the equity portfolio is less than +1.0. (correct answer)
Explanation: This question tests your understanding of portfolio diversification and how correlation affects overall portfolio risk. When evaluating whether to add an asset to an existing portfolio, the key isn't just the standalone risk of the new asset—it's how that asset moves relative to your current holdings. The correct answer is D because portfolio risk depends on both the individual asset risks and the correlation between assets. Even when adding a riskier asset, you can reduce overall portfolio risk as long as the correlation is less than perfect positive correlation (+1.0). The portfolio standard deviation formula shows that when correlation is below +1.0, the combined risk is less than the weighted average of individual risks, creating diversification benefits. Let's examine why the other options are wrong. Option A is too restrictive—while negative correlation would indeed reduce risk, it's not the only condition that works. Any correlation below +1.0 provides some diversification benefit. Option B reflects a common misconception that higher standalone risk always increases portfolio risk, ignoring the powerful effects of diversification. Option C confuses risk with return—expected return doesn't determine whether portfolio risk decreases; that depends entirely on correlation. Study tip: Remember that diversification works whenever correlation is less than perfect positive (+1.0). This is a fundamental principle that appears frequently on corporate finance exams. Focus on understanding that portfolio risk isn't just the sum of individual risks—correlation is the crucial factor that determines whether adding an asset helps or hurts your risk profile.

Question 4

Portfolio P1 consists of two assets with a correlation of +1.0. Portfolio P2 consists of two different assets with a correlation of +0.5. Portfolio P3 consists of two different assets with a correlation of -0.5. Assuming all individual assets have positive standard deviation, which statement is most accurate?

  1. The risk of P1 is the simple weighted average of the component assets' standard deviations. (correct answer)
  2. It is possible to construct a zero-risk portfolio using the assets from P2.
  3. The risk of P3 must be lower than the risk of the least risky asset in P3.
  4. The diversification benefits for P2 and P3 are identical because both correlations are 0.5 away from zero.
Explanation: When two assets have a correlation of +1.0, there are no diversification benefits. The portfolio's risk-return combinations form a straight line between the two assets, and the portfolio's standard deviation is the simple weighted average of the individual assets' standard deviations: σp=w1σ1+w2σ2\sigma_p = w_1\sigma_1 + w_2\sigma_2. Choice B is incorrect; a zero-risk portfolio is only possible if correlation is -1.0 (and weights are chosen appropriately). Choice C is not necessarily true; depending on the weights, the portfolio risk could be higher than the least risky asset, although it is possible to achieve a lower risk. Choice D is incorrect; the diversification benefit increases as correlation moves from +1.0 towards -1.0, so P3 offers substantially more diversification than P2.

Question 5

A portfolio is formed by investing 75% of funds in the S&P 500 index fund and 25% in a risk-free Treasury bill with a return of 3%. The S&P 500 has an expected return of 11% and a standard deviation of 20%. What is the standard deviation of this portfolio?

  1. 5.0%
  2. 15.0% (correct answer)
  3. 15.5%
  4. 20.0%
Explanation: A risk-free asset has a standard deviation of zero. Furthermore, its covariance (and correlation) with any risky asset is also zero. The portfolio variance formula σp2=w12σ12+w22σ22+2w1w2ρ12σ1σ2\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1 w_2 \rho_{12} \sigma_1 \sigma_2 simplifies greatly. Let Asset 1 be the S&P 500 and Asset 2 be the T-bill. Since σ2=0\sigma_2=0, the second and third terms are zero. The formula becomes σp2=w12σ12\sigma_p^2 = w_1^2\sigma_1^2. Taking the square root gives σp=w1σ1\sigma_p = w_1\sigma_1. Plugging in the values: σp=0.75×20%=15.0%\sigma_p = 0.75 \times 20\% = 15.0\%. Distractor A is calculated using the weight of the T-bill instead (0.25 * 20% = 5.0%). Distractor C is the weighted average of the returns, not the standard deviation.

Question 6

Two assets, a stock and a bond, have a correlation coefficient of 0. The stock has a variance of 0.09 and the bond has a variance of 0.04. For an equally weighted portfolio of these two assets, what is the portfolio variance?

  1. 0.0225
  2. 0.0325 (correct answer)
  3. 0.0650
  4. 0.1300
Explanation: The formula for portfolio variance is σp2=ws2σs2+wb2σb2+2wswbρsbσsσb\sigma_p^2 = w_s^2\sigma_s^2 + w_b^2\sigma_b^2 + 2w_s w_b \rho_{sb} \sigma_s \sigma_b. When the correlation ρsb\rho_{sb} is 0, the last term drops out. So, σp2=ws2σs2+wb2σb2\sigma_p^2 = w_s^2\sigma_s^2 + w_b^2\sigma_b^2. Given equal weights (w=0.5) and the variances, we have: σp2=(0.52)(0.09)+(0.52)(0.04)=(0.25)(0.09)+(0.25)(0.04)=0.0225+0.01=0.0325\sigma_p^2 = (0.5^2)(0.09) + (0.5^2)(0.04) = (0.25)(0.09) + (0.25)(0.04) = 0.0225 + 0.01 = 0.0325. Distractor A results from forgetting the second term. Distractor C is the simple average of the variances, which incorrectly applies the weights.

Question 7

A portfolio has a 40% weight in Stock A and a 60% weight in Stock B. The standard deviations are 20% for A and 30% for B. If the portfolio's standard deviation is 26%, what must be the covariance between the returns of Stock A and Stock B?

  1. 0.0600 (correct answer)
  2. 0.0500
  3. 0.0400
  4. 0.1200
Explanation: Using the portfolio variance formula: σₚ² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov(A,B). First, calculate: σₚ² = 0.26² = 0.0676; w₁²σ₁² = (0.4²)(0.2²) = 0.0064; w₂²σ₂² = (0.6²)(0.3²) = 0.0324. Substituting: 0.0676 = 0.0064 + 0.0324 + 2(0.4)(0.6)Cov(A,B). Solving: 0.0676 = 0.0388 + 0.48×Cov(A,B), so Cov(A,B) = (0.0676 - 0.0388)/0.48 = 0.06. Alternatively, notice that the portfolio SD (26%) equals the weighted average of individual SDs (0.4×20% + 0.6×30% = 26%), which only occurs when correlation = +1.0. With ρ = +1.0, Cov(A,B) = ρσ₁σ₂ = 1.0 × 0.20 × 0.30 = 0.06.

Question 8

Portfolio A is a single stock with a standard deviation of 40% and a beta of 1.2. Portfolio B is a well-diversified portfolio of 100 stocks with a standard deviation of 22% and a beta of 1.2. Which of the following statements is the most accurate comparison of their risk profiles?

  1. Both portfolios have the same level of total risk because their betas are identical.
  2. Portfolio A has higher systematic risk and higher unsystematic risk than Portfolio B.
  3. Both portfolios have similar levels of systematic risk, but Portfolio A has a much higher level of unsystematic risk. (correct answer)
  4. Portfolio B is riskier because it contains 100 stocks, exposing it to more sources of potential loss.
Explanation: This question tests the distinction between total risk (measured by standard deviation) and systematic risk (measured by beta). Since both portfolios have the same beta (1.2), they have similar levels of non-diversifiable, systematic risk. However, Portfolio A is a single stock, so its total risk (σ\sigma = 40%) is composed of both systematic and a large amount of unsystematic (firm-specific) risk. Portfolio B is well-diversified, meaning most of its unsystematic risk has been eliminated. Its total risk (σ\sigma = 22%) is therefore much closer to its systematic risk component. Thus, Portfolio A has significantly more unsystematic risk than Portfolio B.

Question 9

As the number of assets in a portfolio increases, the importance of each asset's individual variance diminishes, while the importance of the covariance between assets increases. In a very large, equally-weighted portfolio, the total portfolio variance will converge to which of the following measures?

  1. The average variance of all assets in the portfolio.
  2. The variance of the asset with the lowest correlation to the rest of the portfolio.
  3. Zero, as all unsystematic risk is diversified away.
  4. The average covariance between all pairs of assets in the portfolio. (correct answer)
Explanation: When you encounter portfolio theory questions about what happens as the number of assets grows very large, you're dealing with the mathematical properties of diversification and how individual risk components behave in the limit. As portfolio size increases toward infinity in an equally-weighted portfolio, the portfolio variance formula reveals an important pattern. The variance of a portfolio depends on two components: the individual asset variances (weighted by their portfolio weights squared) and the covariances between all pairs of assets. In an equally-weighted portfolio of n assets, each asset gets a weight of 1/n, so individual variances are multiplied by (1/n)². As n approaches infinity, these terms approach zero. However, the covariance terms are only multiplied by (1/n) for each pair, and there are n(n-1) such pairs. The net effect is that covariances dominate. The correct answer is D because mathematically, the portfolio variance converges to the average covariance between all asset pairs. This represents the systematic risk that cannot be diversified away. Answer A is wrong because individual variances become negligible as portfolio size increases—they're divided by n², making them disappear in the limit. Answer B incorrectly focuses on a single asset's correlation rather than the overall covariance structure. Answer C contains a critical error: while unsystematic risk does get diversified away, systematic risk remains, so variance doesn't approach zero. Remember this key insight: diversification eliminates unsystematic risk but cannot eliminate systematic risk, which is captured by the average covariance between assets.

Question 10

Historically, U.S. investors have benefited from international diversification because the correlation between U.S. markets and many foreign markets was significantly less than 1. Which of the following contemporary trends poses the greatest threat to these traditional diversification benefits?

  1. Increasing volatility in foreign exchange rates.
  2. The rise of emerging markets with high standalone risk.
  3. Increased globalization and integration of world financial markets. (correct answer)
  4. Lower transaction costs for international investing.
Explanation: The fundamental benefit of international diversification comes from low correlation between markets. Increased globalization, where economies and financial markets become more interconnected, tends to increase the correlation of returns between markets. As correlations rise, assets in different countries behave more similarly, especially during times of crisis, which reduces the effectiveness of international diversification for risk-reduction purposes. While currency volatility and the risk of emerging markets are relevant risks, the integration of markets directly undermines the low-correlation assumption that is the foundation of the diversification benefit.

Question 11

A key observation in finance is that during market downturns and crises, the correlation between asset classes (e.g., domestic stocks, international stocks, and high-yield bonds) tends to increase significantly. What is the most direct consequence of this phenomenon for investors?

  1. The expected returns of all asset classes tend to converge, reducing the incentive for asset allocation.
  2. Diversification provides the greatest benefit when it is needed the most, as correlations are highest during these times.
  3. The risk-reduction benefits of holding a diversified portfolio are partially eroded during periods of market stress. (correct answer)
  4. Systematic risk is converted into unsystematic risk, allowing it to be diversified away more easily.
Explanation: The primary benefit of diversification is derived from holding assets with low correlations. When correlations increase during a market crisis, the assets tend to move in the same direction (downward). This means that the diversification that was intended to protect the portfolio during a downturn becomes less effective precisely when it is most needed. The risk-reduction benefits are therefore eroded. Choice B states the opposite of what happens. Choice A confuses returns with correlation. Choice D incorrectly describes the relationship between systematic and unsystematic risk.

Question 12

A portfolio manager currently holds a well-diversified portfolio P with an expected return of 12% and a standard deviation of 18%. The manager is considering adding one of two new assets, Asset A or Asset B, to this portfolio.

  • Asset A: Expected Return = 15%, Standard Deviation = 25%, Correlation with P = 0.8
  • Asset B: Expected Return = 13%, Standard Deviation = 28%, Correlation with P = 0.2

Based on the information provided in the passage, which asset would provide a greater reduction in the portfolio's overall risk level, and why?

  1. Asset A, because its expected return is higher, which always leads to better risk-adjusted performance.
  2. Asset A, because its standalone standard deviation is lower than Asset B's, resulting in less added risk.
  3. Asset B, because its lower correlation with the existing portfolio provides superior diversification benefits. (correct answer)
  4. Asset B, because its higher standard deviation indicates greater potential for risk reduction when combined with other assets.
Explanation: The key to risk reduction through diversification is the correlation of a new asset with the existing portfolio. Asset B has a much lower correlation (0.2) with portfolio P than Asset A (0.8). Despite Asset B's higher standalone risk (standard deviation of 28% vs. 25%), its low correlation means it is more likely to move independently of the existing portfolio, providing more significant diversification benefits and thus a greater reduction in the combined portfolio's risk. The standalone risk and expected return are important, but the correlation effect is the dominant factor for diversification.

Question 13

An analyst constructs a large portfolio containing 500 different stocks from a wide range of industries. She continues to add more stocks that are not perfectly correlated with her existing portfolio. Which of the following statements best describes the effect of this continued diversification on portfolio risk?

  1. Both systematic and unsystematic risk will be reduced, eventually approaching zero as more stocks are added.
  2. Unsystematic risk will be substantially reduced, but systematic risk will remain, representing the portfolio's exposure to market-wide factors. (correct answer)
  3. Systematic risk will be reduced, but unsystematic risk associated with the new stocks will keep total risk constant.
  4. The total risk of the portfolio will decrease to the weighted average of the individual stocks' unsystematic risks.
Explanation: Diversification is the process of reducing risk by combining assets. The primary benefit of adding more assets to a portfolio is the reduction of unsystematic (or firm-specific) risk. As the number of assets increases, these unique risks tend to cancel each other out. However, systematic (or market) risk, which is caused by factors affecting the entire market (like interest rate changes or economic recessions), cannot be eliminated through diversification. Therefore, as more stocks are added, the portfolio's total risk will approach the level of its systematic risk.

Question 14

A risk manager observes that during market stress periods, the correlation between two historically uncorrelated asset classes increases from 0.05 to 0.65. If these assets each comprise 30% of a diversified portfolio with individual standard deviations of 15% and 18% respectively, what is the approximate change in the contribution of these two assets to total portfolio variance?

  1. The covariance term increases by approximately 2.9 percentage points, significantly reducing diversification benefits during stress periods. (correct answer)
  2. The covariance term increases by approximately 1.6 percentage points, moderately reducing diversification benefits during stress periods.
  3. The total variance contribution increases by approximately 4.2 percentage points, representing a substantial increase in portfolio concentration risk.
  4. The correlation change has minimal impact since these assets represent only 60% of the portfolio and other correlations remain stable.
Explanation: The correct answer is A. The change in covariance contribution = 2 × w₁ × w₂ × σ₁ × σ₂ × Δρ = 2 × 0.3 × 0.3 × 0.15 × 0.18 × (0.65 - 0.05) = 2 × 0.3 × 0.3 × 0.15 × 0.18 × 0.6 ≈ 0.0292 or 2.9 percentage points. This represents a significant deterioration in diversification benefits. B uses an incorrect calculation, likely omitting the factor of 2 in the covariance formula. C incorrectly adds variance and covariance effects or uses wrong weights. D incorrectly assumes the correlation change has minimal impact, when in fact this represents a substantial shift from near-independence to moderate positive correlation.

Question 15

A portfolio manager constructs a portfolio using three asset classes with the following characteristics: Asset 1 (weight 50%, σ = 12%), Asset 2 (weight 30%, σ = 20%), Asset 3 (weight 20%, σ = 25%). The correlation between Assets 1 and 2 is 0.4, between Assets 1 and 3 is -0.2, and between Assets 2 and 3 is 0.6. If Asset 3's correlation with Asset 1 changes from -0.2 to +0.3 while other parameters remain constant, what is the approximate change in portfolio variance?

  1. Portfolio variance increases by approximately 0.60 percentage points due to the loss of negative correlation benefits between the largest and smallest weighted assets. (correct answer)
  2. Portfolio variance increases by approximately 1.20 percentage points due to the correlation change affecting the highest individual risk asset in the portfolio.
  3. Portfolio variance increases by approximately 0.30 percentage points due to the relatively small weight of Asset 3 limiting the impact of its correlation changes.
  4. Portfolio variance increases by approximately 0.90 percentage points due to the elimination of the only negative correlation in the portfolio structure.
Explanation: The correct answer is A. The change in portfolio variance from the correlation change = 2 × w₁ × w₃ × σ₁ × σ₃ × Δρ = 2 × 0.5 × 0.2 × 0.12 × 0.25 × (0.3 - (-0.2)) = 2 × 0.5 × 0.2 × 0.12 × 0.25 × 0.5 = 0.006 or 0.60 percentage points. B overstates the impact by double-counting or using incorrect weights. C understates the impact by not properly accounting for the 50% weight in Asset 1. D overstates the impact and incorrectly emphasizes the 'only negative correlation' aspect when the calculation depends on specific weights and standard deviations.

Question 16

A risk analyst observes that two hedge fund strategies have individual Sharpe ratios of 1.2 and 0.8 respectively. When combined in a portfolio, the combined Sharpe ratio is 1.5, which exceeds both individual Sharpe ratios. The analyst concludes that the correlation between the strategies must be negative. A colleague argues that this conclusion is incorrect. Who is correct and why?

  1. The analyst is correct because portfolio Sharpe ratios can only exceed individual Sharpe ratios when correlations are negative, creating diversification benefits.
  2. The colleague is correct because portfolio Sharpe ratios can exceed individual ratios even with positive correlations, provided the correlation is sufficiently low. (correct answer)
  3. The analyst is correct because the mathematical properties of Sharpe ratio optimization require negative correlations to achieve the observed improvement level.
  4. The colleague is correct because Sharpe ratio improvements depend on return enhancement rather than risk reduction, making correlation irrelevant to the conclusion.
Explanation: The correct answer is B. Portfolio Sharpe ratios can exceed individual Sharpe ratios when correlations are less than perfect (ρ < 1), not necessarily negative. With optimal weights, even moderately positive correlations (e.g., 0.3-0.6) can produce portfolio Sharpe ratios exceeding individual ratios through risk reduction that outpaces return blending effects. A incorrectly states negative correlation is required. C incorrectly claims mathematical necessity for negative correlations. D incorrectly dismisses correlation's importance - correlation affects risk reduction, which directly impacts Sharpe ratios through the denominator.

Question 17

An institutional investor notices that during the 2008 financial crisis, correlations between previously uncorrelated alternative investments increased significantly. The investor is now designing a portfolio allocation strategy for the next potential crisis period. Based on correlation behavior during stress periods, which diversification approach is most likely to maintain effectiveness?

  1. Increase allocation to alternative investments since their correlations with traditional assets remain lower than traditional asset correlations with each other during stress periods.
  2. Focus on assets with strong negative correlations during normal periods, as these relationships tend to strengthen further during crisis periods, providing enhanced protection.
  3. Emphasize geographic diversification across different regulatory jurisdictions, as these structural differences provide more stable diversification than asset class differences during stress periods.
  4. Concentrate on assets with fundamental economic drivers that are structurally independent, as correlation increases during stress periods primarily affect assets with similar underlying risk factors. (correct answer)
Explanation: The correct answer is D. During stress periods, correlations increase most dramatically between assets that share underlying risk factors (credit risk, liquidity risk, market sentiment). Assets with truly independent fundamental drivers maintain better diversification. A is incorrect because alternative investments often see correlation increases during stress. B is incorrect because negative correlations can break down during crises when normal relationships fail. C is incorrect because geographic diversification often fails during global crises as financial markets become more integrated during stress periods.

Question 18

A portfolio consists of four assets with equal weights. Three assets have pairwise correlations of 0.3 with each other, while the fourth asset has zero correlation with all other assets. All assets have standard deviations of 16%. An alternative portfolio structure maintains the same four assets but changes the weights to 40%, 30%, 20%, and 10% respectively, where the 10% weight is assigned to the uncorrelated asset. Which statement best describes the risk implications of this weight change?

  1. Portfolio risk increases because the uncorrelated asset receives less weight, reducing the diversification benefit it provides to the overall portfolio structure.
  2. Portfolio risk decreases because the higher concentration in the first asset is offset by reduced correlation contribution from the lower-weighted uncorrelated asset.
  3. Portfolio risk increases because higher concentration in correlated assets dominates the weight reduction effect on the uncorrelated asset's variance contribution. (correct answer)
  4. Portfolio risk remains approximately unchanged because the reduction in uncorrelated asset weight is offset by more efficient correlation structure among remaining assets.
Explanation: The correct answer is C. Moving from equal weights (25% each) to unequal weights (40%, 30%, 20%, 10%) increases concentration in the correlated assets while reducing the weight of the diversifying uncorrelated asset. The portfolio variance formula shows that the 40% and 30% weights on correlated assets create larger variance and covariance terms that dominate the small reduction in variance from the uncorrelated asset. A is partially correct about diversification but incomplete about concentration effects. B incorrectly suggests risk decreases and misunderstands the correlation contribution mechanics. D incorrectly assumes offsetting effects when concentration in correlated assets clearly dominates.

Question 19

An investor holds a portfolio of three assets with equal weights. The correlation between assets 1 and 2 is 0.8, between assets 1 and 3 is -0.2, and between assets 2 and 3 is 0.4. All assets have identical standard deviations of 20%. The investor is considering replacing Asset 2 with Asset 4, which has a standard deviation of 25%, correlation of 0.1 with Asset 1, and correlation of -0.4 with Asset 3. What is the primary driver of the change in portfolio risk?

  1. The higher individual standard deviation of Asset 4 will dominate and increase portfolio risk despite improved correlation structure.
  2. The reduction in correlation between the first and second portfolio positions from 0.8 to 0.1 will dominate other effects. (correct answer)
  3. The change in correlation between the second and third positions from 0.4 to -0.4 provides the most significant risk reduction.
  4. The combined effect of all correlation changes will approximately offset the higher standard deviation, leaving portfolio risk unchanged.
Explanation: The correct answer is B. In portfolio variance, the correlation between assets 1 and 2 (positions with the highest individual correlation) changing from 0.8 to 0.1 represents the largest reduction in covariance terms. Since portfolio variance includes weighted covariance terms, the dramatic reduction from high positive correlation to near-zero correlation has the most significant impact. A is incorrect because correlation effects often dominate individual asset risk in diversified portfolios. C is incorrect because while the correlation change from 0.4 to -0.4 is beneficial, the magnitude of covariance reduction is smaller than the 1-2 correlation change. D is incorrect because the correlation improvements significantly outweigh the modest increase in individual asset risk.

Question 20

A portfolio consists of five assets with equal weights and equal standard deviations of 18%. Currently, all pairwise correlations equal 0.5. A portfolio optimizer suggests changing the allocation to create two groups: three assets with 25% weights each (maintaining 0.5 correlation within this group) and two assets with 12.5% weights each (with 0.2 correlation between them and 0.3 correlation with the first group). What is the primary benefit of this reallocation?

  1. The lower correlations in the second group and with the first group reduce overall portfolio variance more than the concentration increase in the first group.
  2. The equal total weights between groups (75% vs 25%) maintain portfolio balance while optimizing the correlation structure for risk reduction benefits.
  3. The reduced variance contribution from lower-weighted assets dominates the correlation improvements, leading to overall portfolio risk reduction through weight optimization.
  4. The correlation reduction from 0.5 to 0.3 and 0.2 creates diversification benefits that outweigh the modest increase in concentration among the higher-weighted assets. (correct answer)
Explanation: The correct answer is D. The key insight is that the reallocation reduces multiple correlation terms from 0.5 to lower values (0.3 and 0.2), creating substantial diversification benefits. While concentration increases slightly in the first group, the correlation reductions across multiple asset pairs provide greater risk reduction. A incorrectly weighs the relative effects. B mischaracterizes the optimization as 75% vs 25% groups when it's actually about correlation structure. C incorrectly emphasizes weight effects over correlation effects, when correlation changes dominate in this scenario.