A portfolio currently has an expected return of 10% and a standard deviation of 15%. A risk analyst suggests adding a new asset to the portfolio. This new asset has a standard deviation of 25% and a very low correlation with the existing portfolio. A junior analyst objects, stating, 'We cannot add an asset with 25% risk to our 15%-risk portfolio; it will automatically increase the portfolio's total risk.' The risk analyst's suggestion is:
Answer: Plausible, because the diversification effects from the low correlation could outweigh the new asset's higher standalone risk.
The junior analyst's reasoning is flawed. The change in a portfolio's risk depends not only on the standalone risk of the added asset but also, crucially, on its correlation with the existing portfolio. If the correlation is sufficiently low (or negative), the risk-reducing diversification effect can be powerful enough to offset the high standalone risk of the new asset, potentially leading to a decrease in the overall portfolio's standard deviation. Therefore, the risk analyst's suggestion is plausible.
A portfolio manager oversees a well-diversified fund containing 100 different stocks. The manager is considering adding one more stock to the fund. Four candidates are being evaluated. Which stock would provide the greatest marginal diversification benefit?
Answer: The stock with the lowest correlation with the existing fund.
For a well-diversified portfolio, the total risk is dominated by systematic (market) risk. The diversification benefit of adding a new asset is determined by how that asset's returns move in relation to the existing portfolio. An asset with a low (or negative) correlation will provide the greatest reduction in portfolio risk for a given level of expected return. Standalone risk (standard deviation) and expected return are less important than the correlation effect in this context. A high beta stock would add more systematic risk, which is counterproductive to diversification.
A portfolio consists of two stocks, A and B, which have a correlation of 0.7. A portfolio manager considers adding a third stock, C. The correlation between C and A is -0.5, and the correlation between C and B is -0.6. Which of the following is the most likely impact of adding Stock C to the portfolio?
Answer: The portfolio's standard deviation will decrease due to the negative correlations of the new asset.
The initial portfolio of A and B is not well-diversified because of the high positive correlation (0.7). Stock C is negatively correlated with both existing components. Adding an asset that is negatively correlated with the existing portfolio provides significant diversification benefits. The negative covariance terms involving stock C will help offset the positive covariance between A and B and the individual variances, leading to a reduction in the overall portfolio standard deviation.
An investor holds two assets, A and B, whose returns are perfectly negatively correlated (ρ=−1.0). Asset A has a standard deviation of 10% and Asset B has a standard deviation of 15%. What weight must be placed in Asset A to construct a zero-risk portfolio?
Answer: 60%
For a two-asset portfolio with ρ=−1.0, the standard deviation is σp=∣wAσA−wBσB∣. To create a zero-risk portfolio (σp=0), the weights must satisfy wAσA=wBσB. Given wA+wB=1, we can write wB=1−wA. Substituting this gives wAσA=(1−wA)σB. Plugging in the standard deviations: wA(0.10)=(1−wA)(0.15). Solving for wA: 0.10wA=0.15−0.15wA⇒0.25wA=0.15⇒wA=0.15/0.25=0.60. Therefore, the weight in Asset A must be 60%.
An analyst is comparing two potential two-asset portfolios, Portfolio X (Stock A and Stock B) and Portfolio Y (Stock A and Stock C). All three stocks have identical expected returns and standard deviations. The correlation between returns for Stocks A and B is 0.8, while the correlation between returns for Stocks A and C is -0.2. Assuming equal weights in both portfolios, which of the following statements is most accurate?
Answer: Portfolio Y will have a lower standard deviation than Portfolio X due to its lower correlation.
The benefit of diversification is inversely related to the correlation between the assets in the portfolio. A lower correlation coefficient leads to greater risk reduction. Since the correlation between A and C (-0.2) is significantly lower than between A and B (0.8), combining A and C will result in a portfolio with a lower standard deviation (less risk), all else being equal. Therefore, Portfolio Y will have a lower standard deviation than Portfolio X.
An investor replaces a stock in her two-stock portfolio with a new stock that has the exact same expected return and standard deviation as the original, but a much lower correlation with the other stock in the portfolio. Which of the following describes the effect on the new portfolio's expected return and standard deviation?
Answer: The expected return will be unchanged, and the standard deviation will decrease.
The expected return of a portfolio is the weighted average of the expected returns of its component assets. Since the new stock has the same expected return as the one it replaced, the portfolio's expected return will remain unchanged. However, the portfolio's standard deviation is a function of the individual standard deviations, the weights, and the correlation coefficient. A lower correlation coefficient leads to a lower portfolio standard deviation (greater diversification benefit). Therefore, the standard deviation will decrease.
A portfolio manager is considering adding either a gold mining stock or a broad market S&P 500 index fund to a portfolio that currently consists primarily of technology and industrial stocks. Historically, gold has performed well during economic downturns when the broader market has fallen. What is the most likely correlation relationship and diversification outcome?
Answer: The gold stock is likely to be negatively correlated with the portfolio, offering strong diversification benefits.
Gold is often considered a 'safe-haven' asset. During periods of economic uncertainty or stock market declines, investors often move capital into gold, causing its price to rise. This creates a negative correlation between gold and the general stock market. Since the existing portfolio is composed of market-sensitive stocks, adding the gold stock, with its likely negative correlation, would provide the strongest diversification benefits by hedging against market downturns. The S&P 500 fund would be highly positively correlated with the existing stocks.
As an investor adds more randomly selected stocks to a portfolio, which of the following is the most accurate description of the effect on portfolio risk?
Answer: Unsystematic risk is reduced, but systematic risk remains largely unaffected.
The total risk of a portfolio consists of two components: systematic (market) risk and unsystematic (firm-specific) risk. Diversification works by reducing unsystematic risk, as the unique risks of individual companies tend to cancel each other out in a large portfolio. Systematic risk, which is driven by broad market factors (e.g., interest rates, economic cycles), is non-diversifiable and affects all stocks to some degree. Therefore, as more stocks are added, the portfolio's total risk will decline as unsystematic risk is eliminated, but it will approach the level of systematic risk as a lower bound.
Consider the formula for the variance of a two-asset portfolio: σp2=w12σ12+w22σ22+2w1w2ρ12σ1σ2. Which component of this formula captures the primary source of risk reduction from diversification?
Answer: The interaction term, 2w1w2ρ12σ1σ2, because its value depends on the correlation.
The first two terms represent the weighted contribution of each asset's individual variance. The third term, which is equivalent to 2w1w2Cov(1,2), captures how the assets move together. The diversification benefit comes from this interaction. When the correlation (ρ12) is less than 1, this term is smaller than it would otherwise be, reducing the total variance. If correlation is negative, this term becomes negative, actively reducing the portfolio variance below the weighted sum of individual variances. Therefore, this term is the source of the diversification benefit.
Two portfolios, X and Y, are created using assets with identical expected returns and standard deviations. Portfolio X combines assets with a correlation of 0.8. Portfolio Y combines assets with a correlation of -0.4. Both portfolios are equally weighted. Which statement accurately compares the risk and return of these portfolios?
Answer: Portfolio Y will have the same expected return as Portfolio X but a lower standard deviation.
The expected return of a portfolio is the weighted average of the individual assets' expected returns. Since the underlying assets and weights are identical for both portfolios, their expected returns will be the same. Portfolio risk (standard deviation), however, is highly dependent on correlation. A lower correlation provides greater risk reduction. Since Portfolio Y's assets have a much lower correlation (-0.4) than Portfolio X's assets (0.8), Portfolio Y will exhibit a lower overall standard deviation.
When two assets in a portfolio have a correlation coefficient of -1.0, which of the following is true?
Answer: It is possible to create a portfolio of the two assets that has zero standard deviation.
A correlation coefficient of -1.0 represents perfect negative correlation, which offers the maximum possible diversification benefit. With a specific set of weights, the positive returns on one asset will perfectly offset the negative returns on the other, leading to a portfolio with zero risk (standard deviation of zero). The weights are determined by the individual assets' standard deviations. This is a special case; it is not true that any combination will have zero risk, only a specific one. The weighted average of standard deviations applies only when correlation is +1.0.
A portfolio is constructed with a 60% allocation to Asset A and a 40% allocation to Asset B. Asset A has a standard deviation of 20% and Asset B has a standard deviation of 30%. The returns of the two assets are uncorrelated (ρ=0). What is the standard deviation of this portfolio?
Answer: 16.9%
The formula for the variance of a two-asset portfolio is σp2=wA2σA2+wB2σB2+2wAwBρABσAσB. Since correlation (ρ) is 0, the last term drops out. σp2=(0.60)2(0.20)2+(0.40)2(0.30)2=(0.36)(0.04)+(0.16)(0.09)=0.0144+0.0144=0.0288. The portfolio standard deviation is the square root of the variance: σp=0.0288≈0.1697, or 16.97%. Distractor A is the weighted average of the standard deviations, which is only correct if ρ=1. Distractor B is the weighted average of the variances. Distractor C is the simple average of the standard deviations.
In the context of portfolio theory, what is the relationship between statistical independence and correlation?
Answer: If two assets are statistically independent, their correlation must be zero.
Statistical independence is a stronger condition than zero correlation. If two random variables (like asset returns) are independent, it means that the outcome of one has no influence on the outcome of the other. A consequence of independence is that the covariance, and therefore the correlation, between them is zero. However, the reverse is not necessarily true. Zero correlation only means there is no linear relationship between the variables. A strong non-linear relationship could exist between two variables that are uncorrelated. Therefore, independence implies zero correlation, but not vice-versa.
An equally weighted portfolio of two assets has a standard deviation of 18%. Asset A has a standard deviation of 20% and Asset B has a standard deviation of 30%. What is the implied correlation coefficient (ρAB) between the two assets?
Answer: -0.01
We use the portfolio variance formula: σp2=wA2σA2+wB2σB2+2wAwBρABσAσB. We are given σp=0.18, wA=wB=0.5, σA=0.20, and σB=0.30. Plugging in the values: (0.18)2=(0.5)2(0.20)2+(0.5)2(0.30)2+2(0.5)(0.5)ρAB(0.20)(0.30). 0.0324=(0.25)(0.04)+(0.25)(0.09)+0.5ρAB(0.06). 0.0324=0.01+0.0225+0.03ρAB. 0.0324=0.0325+0.03ρAB. −0.0001=0.03ρAB. ρAB=−0.0001/0.03≈−0.0033, which is very close to -0.01.
Which of the following statements best describes the marginal benefit of diversification as the number of assets in a portfolio increases?
Answer: The greatest risk reduction occurs when moving from a portfolio of one asset to two assets.
The benefits of diversification are most dramatic for the first few assets added to a portfolio. The reduction in portfolio standard deviation is substantial when going from one stock to two, and from two to three, and so on. However, this benefit diminishes as the portfolio grows larger. The risk reduction from adding the 31st stock to a 30-stock portfolio is much smaller than the reduction from adding the 2nd stock to a 1-stock portfolio. Therefore, the greatest marginal benefit occurs with the first additions.
An investor has a well-diversified portfolio of global equities. A financial crisis begins, causing a market downturn. Historically, the investor's portfolio has benefited from low correlations among its holdings. Which of the following is the most likely effect of the crisis on her portfolio's risk profile?
Answer: The risk-reducing benefits of her portfolio's diversification will likely diminish as asset correlations increase.
Empirical evidence shows that during periods of market stress and financial crises, the correlation between asset classes and individual securities tends to increase and move towards +1.0. This phenomenon is sometimes called 'correlation breakdown.' As correlations rise, the benefits of diversification are reduced because assets begin to move in the same direction. Therefore, the portfolio's standard deviation will not be reduced as much as it was in normal market conditions.
An analyst created the following scatter plot of monthly returns for Asset X and Asset Y over the past five years.
Answer: The assets appear to be negatively correlated, offering significant diversification benefits when combined.
The scatter plot shows a clear downward-sloping trend. When Asset X has high positive returns (right side of the x-axis), Asset Y tends to have negative returns (bottom of the y-axis), and vice versa. This pattern indicates a negative correlation. Combining assets with negative correlation provides substantial diversification benefits by reducing portfolio volatility. The points do not fall on a perfect straight line, so the correlation is negative but not perfectly -1.0.
When evaluating a single stock's contribution to the risk of a large, well-diversified portfolio, which of the stock's characteristics is most important?
Answer: Its covariance with the other assets in the portfolio.
In a well-diversified portfolio, the firm-specific (unsystematic) risks of individual stocks have been largely eliminated. The dominant risk is systematic risk. A single stock's contribution to the portfolio's total risk is not its own volatility (total risk), but rather how much it moves with the rest of the portfolio. This co-movement is measured by its covariance with the other assets. This concept is fundamental to the Capital Asset Pricing Model (CAPM), where beta is a standardized measure of this covariance.
The covariance of returns between Stock X and Stock Y is 0.0108. The standard deviation of Stock X is 20% and the standard deviation of Stock Y is 30%. Based on this information, an investor combining these two stocks should expect:
Answer: Positive diversification effects, as the assets' returns have a low positive correlation.
First, calculate the correlation coefficient using the formula ρXY=Cov(X,Y)/(σXσY). ρXY=0.0108/(0.20×0.30)=0.0108/0.06=0.18. A correlation of +0.18 is a low positive correlation. Since the correlation is less than +1.0, combining the assets will provide diversification benefits, reducing the portfolio's unsystematic risk. The effect is positive (i.e., beneficial), although not as strong as it would be with a negative correlation.
An analyst is constructing an efficient frontier using two assets with positive expected returns. If the correlation coefficient between the two assets decreases from +0.5 to -0.5, how will the investment opportunity set change?
Answer: The curve will become more bowed to the left, expanding the set of low-risk portfolio opportunities.
The curvature of the investment opportunity set (and the efficient frontier) is determined by the correlation between the assets. A correlation of +1.0 results in a straight line. As the correlation decreases, the curve bends or bows more to the left (towards the y-axis of returns). This indicates that for any given level of expected return, a portfolio with lower risk (standard deviation) can be formed. A decrease in correlation from +0.5 to -0.5 will therefore make the curve more bowed, expanding the set of attainable low-risk portfolios.
