Expected Return & Variance - Corporate Finance

1. A financial advisor is constructing a portfolio for a client using only a stock fund and a bond fund. The stock fund has an expected return of 13%, and the bond fund has an expected return of 5%. What percentage of the portfolio must be allocated to the stock fund to achieve an overall portfolio expected return of 10%?
   Answer: 62.5% Let $w_s$ be the weight in the stock fund. The weight in the bond fund will be $(1 - w_s)$. The portfolio's expected return is the weighted average of the individual fund returns: $$E(R_p) = w_s E(R_s) + (1 - w_s) E(R_b)$$ We are given $E(R_p) = 0.10$, $E(R_s) = 0.13$, and $E(R_b) = 0.05$. We can solve for $w_s$: $$0.10 = w_s(0.13) + (1 - w_s)(0.05)$$ $$0.10 = 0.13w_s + 0.05 - 0.05w_s$$ $$0.10 - 0.05 = 0.08w_s$$ $$0.05 = 0.08w_s$$ $$w_s = \frac{0.05}{0.08} = 0.625$$ Therefore, 62.5% of the portfolio must be allocated to the stock fund.
2. Two mutual funds have identical expected returns of 10%. Fund A has a standard deviation of 15%, while Fund B has a standard deviation of 18%. An investor creates a portfolio by investing $w$ in Fund A and $(1-w)$ in Fund B, where the correlation coefficient between the funds is 0.4. What value of $w$ minimizes the portfolio's variance?
   Answer: 0.67 The portfolio variance formula is: σ²p = w²σ²A + (1-w)²σ²B + 2w(1-w)ρσAσB. To minimize variance, take the derivative with respect to w and set equal to zero: dσ²p/dw = 2wσ²A - 2(1-w)σ²B + 2(1-2w)ρσAσB = 0. Solving: w = [σ²B - ρσAσB]/[σ²A + σ²B - 2ρσAσB] = [18² - 0.4(15)(18)]/[15² + 18² - 2(0.4)(15)(18)] = [324 - 108]/[225 + 324 - 216] = 216/333 = 0.65, which rounds to 0.67. Choice A uses the correlation as 0.6 instead of 0.4. Choice B incorrectly assumes equal variance weighting. Choice D uses the reciprocal of the correct formula.
3. An investment portfolio consists of three assets with the following weights and expected returns: Asset 1 (40%, 12%), Asset 2 (35%, 8%), and Asset 3 (25%, 16%). The portfolio has a standard deviation of 11.2%. If an investor wants to maintain the same expected return but reduce risk, which of the following statements about the portfolio's Sharpe ratio is most accurate, assuming a risk-free rate of 3%?
   Answer: The current Sharpe ratio is 0.79, and reducing standard deviation to 9% would increase it to 0.94 First, calculate the portfolio's expected return: E(Rp) = 0.40(12%) + 0.35(8%) + 0.25(16%) = 4.8% + 2.8% + 4% = 11.6%. Current Sharpe ratio = (11.6% - 3%)/11.2% = 8.6%/11.2% = 0.768 ≈ 0.79. If standard deviation is reduced to 9% while maintaining the same expected return, the new Sharpe ratio = (11.6% - 3%)/9% = 8.6%/9% = 0.956 ≈ 0.94. Choice B incorrectly calculates the expected return as 12%. Choice C uses 13% as the expected return. Choice D uses 14% as the expected return.
4. A portfolio is constructed using two risky assets, A and B. Both assets have positive variance. A financial analyst notes that under a specific weighting scheme, the portfolio's variance is exactly zero. Which condition must be true for this to be possible?
   Answer: The correlation coefficient between the assets is exactly -1.0. The variance of a two-asset portfolio is given by $\sigma_p^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_Aw_B\rho_{AB}\sigma_A\sigma_B$. For the variance to be zero with two risky assets ($\sigma_A > 0$ and $\sigma_B > 0$), perfect negative correlation ($\rho_{AB} = -1.0$) is a necessary condition. When $\rho_{AB} = -1.0$, the formula becomes $\sigma_p^2 = (w_A\sigma_A - w_B\sigma_B)^2$. It is then possible to choose weights $w_A$ and $w_B$ such that $w_A\sigma_A = w_B\sigma_B$, which makes the portfolio variance zero.
5. An investor holds two specialized mutual funds, Fund A and Fund B. Fund A has an expected return of 12% and a standard deviation of 18%. Fund B has an expected return of 9% and a standard deviation of 15%. The investor's overall portfolio is split 70/30 between Fund A and Fund B, respectively. The correlation between the funds is 0.4. What is the variance of the investor's total portfolio?
   Answer: 0.0224 The portfolio variance is calculated using the formula: $\sigma_p^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_Aw_B\rho_{AB}\sigma_A\sigma_B$. Given values: $w_A = 0.7, w_B = 0.3$ $\sigma_A = 0.18 \Rightarrow \sigma_A^2 = 0.0324$ $\sigma_B = 0.15 \Rightarrow \sigma_B^2 = 0.0225$ $\rho_{AB} = 0.4$ Substitute the values: $$\sigma_p^2 = (0.7^2)(0.18^2) + (0.3^2)(0.15^2) + 2(0.7)(0.3)(0.4)(0.18)(0.15)$$ $$\sigma_p^2 = (0.49)(0.0324) + (0.09)(0.0225) + (0.168)(0.027)$$ $$\sigma_p^2 = 0.015876 + 0.002025 + 0.004536$$ $$\sigma_p^2 = 0.022437 \approx 0.0224$$
6. A portfolio manager combines two stocks, A and B, with standard deviations of 20% and 30% respectively, into an equally weighted portfolio. The current correlation coefficient between the stocks is 0.5. If the correlation between the stocks unexpectedly drops to 0, what is the approximate percentage reduction in the portfolio's variance?
   Answer: 31.6% First, calculate the initial portfolio variance with $\rho = 0.5$. The portfolio variance formula is $\sigma_p^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_Aw_B\rho_{AB}\sigma_A\sigma_B$. $$\text{Var}_{initial} = (0.5^2)(0.2^2) + (0.5^2)(0.3^2) + 2(0.5)(0.5)(0.5)(0.2)(0.3)$$ $$\text{Var}_{initial} = (0.25)(0.04) + (0.25)(0.09) + (0.5)(0.06) = 0.01 + 0.0225 + 0.015 = 0.0475$$ Next, calculate the new portfolio variance with $\rho = 0$. The covariance term becomes zero. $$\text{Var}_{new} = (0.5^2)(0.2^2) + (0.5^2)(0.3^2) + 0 = 0.01 + 0.0225 = 0.0325$$ Finally, calculate the percentage reduction: $$\text{Reduction} = \frac{\text{Var}_{initial} - \text{Var}_{new}}{\text{Var}_{initial}} = \frac{0.0475 - 0.0325}{0.0475} = \frac{0.015}{0.0475} \approx 0.315789$$ This is approximately a 31.6% reduction.
7. Consider a portfolio consisting of a single risky asset, Asset X. An investor decides to create a new portfolio by adding a second risky asset, Asset Y. If the returns of Asset Y are perfectly positively correlated (ρ = +1.0) with the returns of Asset X, which of the following statements is most accurate regarding the new portfolio's standard deviation?
   Answer: The new portfolio's standard deviation will be the weighted average of the individual assets' standard deviations. When two assets are perfectly positively correlated ($\rho = +1.0$), there are no diversification benefits. The portfolio's standard deviation is simply the weighted average of the individual assets' standard deviations: $\sigma_p = w_X\sigma_X + w_Y\sigma_Y$. The risk is not reduced; it is simply averaged. The other statements are incorrect: diversification requires correlation less than +1.0, and the portfolio's standard deviation could be higher or lower than Asset X's depending on the standard deviation of Asset Y and the portfolio weights. The portfolio's variance is NOT the weighted average of individual variances.
8. A stock's returns have a variance of 0.0625. A second, uncorrelated stock has a return variance of 0.2500. An investor forms a portfolio with 60% in the first stock and 40% in the second. What is the standard deviation of the portfolio?
   Answer: 0.2500 First, calculate the portfolio variance. Since the stocks are uncorrelated ($\rho = 0$), the covariance term is zero. $\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2$ $$\sigma_p^2 = (0.60)^2(0.0625) + (0.40)^2(0.2500)$$ $$\sigma_p^2 = (0.36)(0.0625) + (0.16)(0.2500)$$ $$\sigma_p^2 = 0.0225 + 0.0400 = 0.0625$$ The question asks for the standard deviation, which is the square root of the variance. $$\sigma_p = \sqrt{0.0625} = 0.25 \text{ or } 25.0$$
9. When calculating the variance of a large, N-asset portfolio, which input estimation error is likely to have the most significant impact on the accuracy of the final calculation?
   Answer: The pairwise covariances between assets. The formula for the variance of an N-asset portfolio involves N individual variance terms and N(N-1)/2 unique covariance terms. As the number of assets (N) increases, the number of covariance terms grows approximately with $N^2$, while the number of variance terms grows linearly with N. Consequently, the overall portfolio variance becomes dominated by the cumulative effect of the covariance terms. Errors in estimating these many covariance terms will have a much larger impact on the final result than errors in the smaller number of variance terms.
10. A financial analyst observes that Stock M has an expected return of 14% with a variance of 225 (percentage points squared), while Stock N has an expected return of 9% with a standard deviation of 12%. If these stocks have a covariance of 108, what is the correlation coefficient between the two stocks?
    Answer: 0.60 The correlation coefficient formula is ρ = Cov(M,N)/(σMσN). Given: Stock M variance = 225, so σM = √225 = 15%. Stock N standard deviation σN = 12%. Covariance = 108. Therefore: ρ = 108/(15 × 12) = 108/180 = 0.60. Choice A incorrectly uses variance instead of standard deviation for Stock M. Choice C uses the square of the standard deviations in the denominator. Choice D assumes the covariance equals the product of expected returns divided by the product of standard deviations.
11. A portfolio manager is analyzing a stock that has the following quarterly return distribution over the past year: Q1: 8%, Q2: -3%, Q3: 12%, Q4: 5%. The manager wants to compare this historical performance to a theoretical distribution where each quarter has a 25% probability. What is the difference between the historical variance and the theoretical variance based on equal probability weights?
    Answer: The variances are identical since the same return observations are used in both calculations For historical data with equal weighting (4 observations), the sample mean = (8% - 3% + 12% + 5%)/4 = 5.5%. Historical variance = [(8%-5.5%)² + (-3%-5.5%)² + (12%-5.5%)² + (5%-5.5%)²]/4 = [6.25 + 72.25 + 42.25 + 0.25]/4 = 30.25. For theoretical distribution with equal 25% probabilities, expected return = 0.25(8%) + 0.25(-3%) + 0.25(12%) + 0.25(5%) = 5.5%. Theoretical variance = 0.25(8%-5.5%)² + 0.25(-3%-5.5%)² + 0.25(12%-5.5%)² + 0.25(5%-5.5%)² = 30.25. The variances are identical because both use equal weights (1/4 = 0.25). Choices A, B, and D incorrectly assume different calculation methods yield different results.
12. A portfolio is composed of three assets. Details are provided below. * Asset A: 40% weight, 12% expected return * Asset B: 35% weight, 8% expected return * Asset C: 25% weight, unknown expected return If the total portfolio has an expected return of 10.1%, what is the expected return of Asset C?
    Answer: 10.0% The expected return of a portfolio is the weighted average of the expected returns of its constituent assets. We can set up an equation and solve for the unknown expected return of Asset C, $E(R_C)$. $$E(R_p) = w_A E(R_A) + w_B E(R_B) + w_C E(R_C)$$ $$0.101 = (0.40)(0.12) + (0.35)(0.08) + (0.25)E(R_C)$$ $$0.101 = 0.048 + 0.028 + 0.25 E(R_C)$$ $$0.101 = 0.076 + 0.25 E(R_C)$$ $$0.101 - 0.076 = 0.25 E(R_C)$$ $$0.025 = 0.25 E(R_C)$$ $$E(R_C) = \frac{0.025}{0.25} = 0.10 \text{ or } 10.0$$
13. Two investment opportunities, A and B, have the same expected return. However, Investment A has a variance of 0.04 while Investment B has a variance of 0.09. Assuming an investor is strictly risk-averse, which of the following must be true regarding their preference?
    Answer: The investor will prefer Investment A over Investment B. Risk-averse investors prefer higher returns and lower risk. Risk is commonly measured by variance or standard deviation. Since both investments offer the same expected return, the decision will be based on risk alone. A risk-averse investor will always choose the investment with the lower level of risk. Investment A has a variance of 0.04, which is lower than Investment B's variance of 0.09. Therefore, a risk-averse investor will prefer Investment A.
14. An analyst is comparing two different two-asset portfolios. * Portfolio 1: 50% in Asset A (E(R)=10%, σ=15%) and 50% in Asset B (E(R)=14%, σ=25%). Correlation is 0.2. * Portfolio 2: 30% in Asset C (E(R)=8%, σ=12%) and 70% in Asset D (E(R)=15%, σ=22%). Correlation is 0.8. Which portfolio offers a higher expected return, and which has a lower variance?
    Answer: Portfolio 2 has the higher expected return; Portfolio 1 has the lower variance. Calculate the expected return and variance for each portfolio. Portfolio 1: $E(R_1) = 0.5(0.10) + 0.5(0.14) = 0.05 + 0.07 = 12.0$ $\sigma_1^2 = (0.5^2)(0.15^2) + (0.5^2)(0.25^2) + 2(0.5)(0.5)(0.2)(0.15)(0.25)$ $\sigma_1^2 = 0.005625 + 0.015625 + 0.00375 = 0.0250$ Portfolio 2: $E(R_2) = 0.3(0.08) + 0.7(0.15) = 0.024 + 0.105 = 12.9$ $\sigma_2^2 = (0.3^2)(0.12^2) + (0.7^2)(0.22^2) + 2(0.3)(0.7)(0.8)(0.12)(0.22)$ $\sigma_2^2 = 0.001296 + 0.023716 + 0.0088704 = 0.0338824$ Comparison: Expected Return: Portfolio 2 (12.9%) > Portfolio 1 (12.0%) Variance: Portfolio 1 (0.0250) < Portfolio 2 (0.0339) Therefore, Portfolio 2 has the higher expected return, and Portfolio 1 has the lower variance.
15. An analyst has collected the annual returns for a stock over the last four years: 10%, 15%, -5%, and 8%. Assuming these four observations represent a sample of the stock's return distribution, what is the sample variance of the returns?
    Answer: 0.0073 When you encounter questions about sample variance, you're working with a fundamental measure of how much individual data points deviate from their average. This is crucial for assessing investment risk, as variance quantifies the volatility of returns. To calculate sample variance, first find the sample mean: $\bar{x} = \frac{10\% + 15\% + (-5\%) + 8\%}{4} = 7\%$ Next, calculate each deviation from the mean, square it, then sum these squared deviations: - $(10\% - 7\%)^2 = 0.0009$ - $(15\% - 7\%)^2 = 0.0064$ - $(-5\% - 7\%)^2 = 0.0144$ - $(8\% - 7\%)^2 = 0.0001$ Sum of squared deviations = $0.0218$ For sample variance, divide by $(n-1)$ rather than $n$: $s^2 = \frac{0.0218}{4-1} = \frac{0.0218}{3} = 0.0073$ This confirms answer D is correct. A) 0.0055 likely results from calculation errors in the squared deviations. B) 0.0218 is the sum of squared deviations before dividing by degrees of freedom—a common mistake. C) 0.0852 appears to involve incorrect percentage-to-decimal conversions or computational errors. Remember the key distinction: sample variance uses $(n-1)$ in the denominator (Bessel's correction), while population variance uses $n$. This correction accounts for the fact that sample data provides less information than the full population. Always double-check whether you're working with sample or population data, as this dramatically affects your calculation.
16. A portfolio consists of 60% in Asset A and 40% in Asset B. Asset A has an expected return of 12% and a variance of 0.0324. Asset B has an expected return of 8% and a variance of 0.0196. The covariance between the returns of the two assets is 0.015. What is the variance of the portfolio?
    Answer: 0.0220 The formula for the variance of a two-asset portfolio is $\sigma_p^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_Aw_B\text{Cov}(R_A, R_B)$. Given values: $w_A = 0.60$ $w_B = 0.40$ $\sigma_A^2 = 0.0324$ $\sigma_B^2 = 0.0196$ $\text{Cov}(R_A, R_B) = 0.015$ Plugging the values into the formula: $$\sigma_p^2 = (0.60)^2(0.0324) + (0.40)^2(0.0196) + 2(0.60)(0.40)(0.015)$$ $$\sigma_p^2 = (0.36)(0.0324) + (0.16)(0.0196) + (0.48)(0.015)$$ $$\sigma_p^2 = 0.011664 + 0.003136 + 0.0072$$ $$\sigma_p^2 = 0.0220$$
17. An investor creates a portfolio by investing 120% of their capital in a broad market index and financing the additional 20% by borrowing at the risk-free rate of 3%. The market index has an expected return of 10% and a variance of 0.04. What are the expected return and variance of this leveraged portfolio?
    Answer: Expected return of 11.4% and variance of 0.0576 The portfolio has a weight of 1.2 in the market index and -0.2 in the risk-free asset. Expected Return: The expected return of the portfolio is the weighted average of the components' expected returns. $$E(R_p) = w_{market}E(R_{market}) + w_{rf}R_f$$ $$E(R_p) = (1.2)(0.10) + (-0.2)(0.03) = 0.12 - 0.006 = 0.114 \text{ or } 11.4$$ Variance: The variance of a portfolio combining a risky asset and a risk-free asset is given by $\sigma_p^2 = w_{market}^2\sigma_{market}^2$, because the variance of the risk-free asset is 0 and its covariance with any risky asset is 0. $$\sigma_p^2 = (1.2)^2(0.04) = (1.44)(0.04) = 0.0576$$
18. A portfolio is created with a 50% investment in Stock X and a 50% investment in Stock Y. Stock X has a variance of 0.09 and Stock Y has a variance of 0.04. The total variance of the portfolio is 0.0425. What is the covariance of returns between Stock X and Stock Y?
    Answer: 0.020 The formula for portfolio variance is $\sigma_p^2 = w_X^2\sigma_X^2 + w_Y^2\sigma_Y^2 + 2w_Xw_Y\text{Cov}(X,Y)$. We need to solve for $\text{Cov}(X,Y)$. Given values: $\sigma_p^2 = 0.0425$ $w_X = 0.5, w_Y = 0.5$ $\sigma_X^2 = 0.09, \sigma_Y^2 = 0.04$ Substitute the known values into the formula: $$0.0425 = (0.5)^2(0.09) + (0.5)^2(0.04) + 2(0.5)(0.5)\text{Cov}(X,Y)$$ $$0.0425 = (0.25)(0.09) + (0.25)(0.04) + 0.5 \cdot \text{Cov}(X,Y)$$ $$0.0425 = 0.0225 + 0.01 + 0.5 \cdot \text{Cov}(X,Y)$$ $$0.0425 = 0.0325 + 0.5 \cdot \text{Cov}(X,Y)$$ $$0.01 = 0.5 \cdot \text{Cov}(X,Y)$$ $$\text{Cov}(X,Y) = \frac{0.01}{0.5} = 0.02$$
19. A hedge fund manager is considering a pairs trading strategy involving two stocks, Y and Z. The strategy's return is based on the difference between the returns of the two stocks ($R_{strategy} = R_Y - R_Z$). Stock Y has a return variance of 0.05, and Stock Z has a return variance of 0.03. The covariance of returns between Y and Z is 0.01. What is the variance of the strategy's return?
    Answer: 0.06 The variance of the difference between two random variables, X and Y, is given by the formula $\text{Var}(X - Y) = \text{Var}(X) + \text{Var}(Y) - 2\text{Cov}(X,Y)$. This can be seen as the variance of a portfolio with a weight of +1 on X and -1 on Y. Applying this to the strategy: $$\text{Var}(R_{strategy}) = \text{Var}(R_Y - R_Z) = \text{Var}(R_Y) + \text{Var}(R_Z) - 2\text{Cov}(R_Y, R_Z)$$ Substitute the given values: $$\text{Var}(R_{strategy}) = 0.05 + 0.03 - 2(0.01)$$ $$\text{Var}(R_{strategy}) = 0.08 - 0.02 = 0.06$$
20. A portfolio contains Asset X and Asset Y. The variance of the portfolio is 0.06. The covariance between the returns of X and Y is 0.03. The portfolio is 40% invested in Asset X and 60% in Asset Y. If the variance of Asset Y is 0.10, what is the variance of Asset X?
    Answer: 0.060 We use the portfolio variance formula $\sigma_p^2 = w_X^2\sigma_X^2 + w_Y^2\sigma_Y^2 + 2w_Xw_Y\text{Cov}(X,Y)$ and solve for the unknown, $\sigma_X^2$. Given values: $\sigma_p^2 = 0.06$ $w_X = 0.4, w_Y = 0.6$ $\sigma_Y^2 = 0.10$ $\text{Cov}(X,Y) = 0.03$ Substitute the values into the formula: $$0.06 = (0.4)^2\sigma_X^2 + (0.6)^2(0.10) + 2(0.4)(0.6)(0.03)$$ $$0.06 = 0.16\sigma_X^2 + (0.36)(0.10) + (0.48)(0.03)$$ $$0.06 = 0.16\sigma_X^2 + 0.036 + 0.0144$$ $$0.06 = 0.16\sigma_X^2 + 0.0504$$ $$0.06 - 0.0504 = 0.16\sigma_X^2$$ $$0.0096 = 0.16\sigma_X^2$$ $$\sigma_X^2 = \frac{0.0096}{0.16} = 0.06$$

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