Portfolio Expected Return & Variance - Finance

1. A portfolio is composed of 60% Asset A and 40% Asset B. Asset A has an expected return of 10% and a variance of 0.0400. Asset B has an expected return of 15% and a variance of 0.0900. The covariance of returns between Asset A and Asset B is 0.0240. What is the approximate standard deviation of the portfolio?
   Answer: 20.1% The formula for portfolio variance is $\sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B Cov(R_A, R_B)$. 1. Identify the given values: $w_A = 0.60$, $w_B = 0.40$, $\sigma_A^2 = 0.0400$, $\sigma_B^2 = 0.0900$, and $Cov(R_A, R_B) = 0.0240$. 2. Substitute these values into the formula: $$\sigma_p^2 = (0.60)^2(0.0400) + (0.40)^2(0.0900) + 2(0.60)(0.40)(0.0240)$$ $$\sigma_p^2 = (0.36)(0.0400) + (0.16)(0.0900) + (0.48)(0.0240)$$ $$\sigma_p^2 = 0.0144 + 0.0144 + 0.01152$$ $$\sigma_p^2 = 0.04032$$ 3. The portfolio standard deviation is the square root of the variance: $$\sigma_p = \sqrt{0.04032} \approx 0.2008$$ Thus, the portfolio's standard deviation is approximately 20.1%.
2. A portfolio consists of a 70% allocation to a stock fund and a 30% allocation to a bond fund. The stock fund has a standard deviation of 25%, and the bond fund has a standard deviation of 10%. If the total portfolio variance is 0.033625, what is the correlation coefficient between the stock and bond funds?
   Answer: 0.20 The problem requires solving for the correlation coefficient ($\rho$) within the portfolio variance formula. 1. Formula: $\sigma_p^2 = w_S^2 \sigma_S^2 + w_B^2 \sigma_B^2 + 2 w_S w_B \rho_{SB} \sigma_S \sigma_B$ 2. Given values: $\sigma_p^2 = 0.033625$, $w_S = 0.7$, $w_B = 0.3$, $\sigma_S = 0.25$, $\sigma_B = 0.10$. 3. Calculate the individual variance components: * $w_S^2 \sigma_S^2 = (0.7)^2 (0.25)^2 = 0.49(0.0625) = 0.030625$ * $w_B^2 \sigma_B^2 = (0.3)^2 (0.10)^2 = 0.09(0.01) = 0.0009$ 4. Substitute known values into the main formula: $$0.033625 = 0.030625 + 0.0009 + 2(0.7)(0.3)\rho_{SB}(0.25)(0.10)$$ 5. Simplify and solve for $\rho_{SB}$: $$0.033625 = 0.031525 + (0.42) \rho_{SB} (0.025)$$ $$0.033625 = 0.031525 + 0.0105 \rho_{SB}$$ $$0.0021 = 0.0105 \rho_{SB}$$ $$\rho_{SB} = \frac{0.0021}{0.0105} = 0.20$$
3. A portfolio is constructed with allocations of 50% to Asset A, 30% to Asset B, and the remainder to Asset C. The expected returns are 12% for A and 8% for B. The covariance of returns between A and B is 0.03, and the correlation between B and C is -0.4. If the portfolio's overall expected return is 10.0%, what is the expected return of Asset C?
   Answer: 8.0% The information about covariance and correlation is irrelevant for calculating the portfolio's expected return and is included to distract the test-taker. 1. Determine the weight of Asset C: The weights must sum to 1.0. $$w_C = 1.0 - w_A - w_B = 1.0 - 0.5 - 0.3 = 0.20$$ 2. Use the formula for portfolio expected return: $$E[R_p] = w_A E[R_A] + w_B E[R_B] + w_C E[R_C]$$ 3. **Substitute the known values and solve for $E[R_C]$:** $$0.100 = (0.50)(0.12) + (0.30)(0.08) + (0.20)E[R_C]$$ $$0.100 = 0.060 + 0.024 + 0.20 E[R_C]$$ $$0.100 = 0.084 + 0.20 E[R_C]$$ $$0.016 = 0.20 E[R_C]$$ $$E[R_C] = \frac{0.016}{0.20} = 0.08$$ Thus, the expected return of Asset C is 8.0%.
4. A portfolio manager combines two assets, X and Y, in equal proportions. Asset X has a standard deviation of 18% and Asset Y has a standard deviation of 26%. The correlation between their returns is 0.5. By approximately how much does diversification reduce the portfolio's standard deviation compared to the weighted average of the individual asset standard deviations?
   Answer: 2.84% This problem requires comparing the calculated portfolio standard deviation to a simple weighted average of the component standard deviations. 1. Calculate the weighted average standard deviation: $$\text{Weighted Avg } \sigma = w_X \sigma_X + w_Y \sigma_Y = 0.5(0.18) + 0.5(0.26) = 0.09 + 0.13 = 0.22 \text{ or } 22.00\%$$ 2. **Calculate the actual portfolio standard deviation ($\sigma_p$):** First, calculate the portfolio variance, $\sigma_p^2$. $$\sigma_p^2 = (0.5)^2(0.18)^2 + (0.5)^2(0.26)^2 + 2(0.5)(0.5)(0.5)(0.18)(0.26)$$ $$\sigma_p^2 = 0.25(0.0324) + 0.25(0.0676) + 0.5(0.5)(0.0468)$$ $$\sigma_p^2 = 0.0081 + 0.0169 + 0.0117 = 0.0367$$ Now, find the standard deviation: $$\sigma_p = \sqrt{0.0367} \approx 0.19157 \text{ or } 19.16\%$$ 3. Calculate the reduction: $$\text{Reduction} = \text{Weighted Avg } \sigma - \sigma_p = 22.00\% - 19.16\% = 2.84\%$$
5. A portfolio is 50% invested in Stock A and 50% in Stock B. Stock A has a standard deviation of 16%. The correlation between the stocks is 0.25. If the portfolio's standard deviation is exactly 15%, what must be the approximate standard deviation of Stock B?
   Answer: 21.7% This problem requires solving a quadratic equation for the unknown standard deviation, $\sigma_B$. 1. Set up the variance formula: $$\sigma_p^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_Aw_B\rho\sigma_A\sigma_B$$ Given: $\sigma_p=0.15 \implies \sigma_p^2 = 0.0225$, $w_A=w_B=0.5$, $\sigma_A=0.16$, $\rho=0.25$. 2. Substitute known values: $$0.0225 = (0.5)^2(0.16)^2 + (0.5)^2\sigma_B^2 + 2(0.5)(0.5)(0.25)(0.16)\sigma_B$$ $$0.0225 = 0.25(0.0256) + 0.25\sigma_B^2 + 0.5(0.25)(0.16)\sigma_B$$ $$0.0225 = 0.0064 + 0.25\sigma_B^2 + 0.02\sigma_B$$ 3. **Rearrange into a standard quadratic form ($ax^2+bx+c=0$) where $x=\sigma_B$:** $$0.25\sigma_B^2 + 0.02\sigma_B - 0.0161 = 0$$ 4. **Solve using the quadratic formula $x = [-b \pm \sqrt{b^2-4ac}] / 2a$:** $$\sigma_B = \frac{-0.02 \pm \sqrt{(0.02)^2 - 4(0.25)(-0.0161)}}{2(0.25)}$$ $$\sigma_B = \frac{-0.02 \pm \sqrt{0.0004 + 0.0161}}{0.5} = \frac{-0.02 \pm \sqrt{0.0165}}{0.5}$$ $$\sigma_B = \frac{-0.02 \pm 0.12845}{0.5}$$ Since standard deviation cannot be negative, we take the positive root: $$\sigma_B = \frac{0.10845}{0.5} \approx 0.2169$$ The standard deviation of Stock B is approximately 21.7%.
6. A portfolio is constructed with equal weights in two stocks, General Industries (GI) and Counter-Cyclical Ventures (CCV). GI has a standard deviation of 40% and CCV has a standard deviation of 60%. The correlation between their returns is -1.0. What is the standard deviation of this portfolio?
   Answer: 10% With perfect negative correlation ($\rho = -1.0$), the portfolio standard deviation formula simplifies to $\sigma_p = |w_A\sigma_A - w_B\sigma_B|$. For an equally weighted portfolio, $w_{GI} = w_{CCV} = 0.5$. Using the general formula: $$\sigma_p^2 = w_{GI}^2 \sigma_{GI}^2 + w_{CCV}^2 \sigma_{CCV}^2 + 2 w_{GI} w_{CCV} \rho \sigma_{GI} \sigma_{CCV}$$ $$\sigma_p^2 = (0.5)^2(0.4)^2 + (0.5)^2(0.6)^2 + 2(0.5)(0.5)(-1.0)(0.4)(0.6)$$ $$\sigma_p^2 = (0.25)(0.16) + (0.25)(0.36) - (0.5)(0.24)$$ $$\sigma_p^2 = 0.04 + 0.09 - 0.12$$ $$\sigma_p^2 = 0.01$$ $$\sigma_p = \sqrt{0.01} = 0.10$$ The portfolio standard deviation is 10%. Note that zero risk is only possible with a specific weighting (60% in GI, 40% in CCV), not with equal weights.
7. A portfolio is equally weighted between two stocks, A and B. Both stocks have a standard deviation of 30%. If the portfolio's standard deviation is also exactly 30%, what must be the correlation coefficient ($\rho_{AB}$) between the returns of Stock A and Stock B?
   Answer: 1.0 When the correlation coefficient between two assets is +1.0, there are no diversification benefits, and the portfolio's standard deviation is simply the weighted average of the individual assets' standard deviations. Let's prove this with the formula: $$\sigma_p = \sqrt{w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \rho_{AB} \sigma_A \sigma_B}$$ Given: $\sigma_p = 0.30$, $\sigma_A = 0.30$, $\sigma_B = 0.30$, $w_A = 0.5$, $w_B = 0.5$. $$0.30 = \sqrt{(0.5)^2(0.3)^2 + (0.5)^2(0.3)^2 + 2(0.5)(0.5)\rho_{AB}(0.3)(0.3)}$$ Square both sides: $$0.09 = 0.25(0.09) + 0.25(0.09) + 0.5 \rho_{AB} (0.09)$$ $$0.09 = 0.0225 + 0.0225 + 0.045 \rho_{AB}$$ $$0.09 = 0.045 + 0.045 \rho_{AB}$$ $$0.045 = 0.045 \rho_{AB}$$ $$\rho_{AB} = 1.0$$
8. A portfolio is formed by investing 130% of its value in Stock A and shorting an amount equal to 30% of its value in Stock B. Stock A has a variance of 0.05, Stock B has a variance of 0.02, and their covariance is 0.01. What is the variance of this portfolio?
   Answer: 0.0785 This problem involves a portfolio with a short position, so one of the weights is negative. 1. Identify weights and parameters: * Long position in A: $w_A = 1.30$ * Short position in B: $w_B = -0.30$ * $\sigma_A^2 = 0.05$, $\sigma_B^2 = 0.02$, $Cov(A,B) = 0.01$ 2. Apply the portfolio variance formula: $$\sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B Cov(A, B)$$ $$\sigma_p^2 = (1.30)^2(0.05) + (-0.30)^2(0.02) + 2(1.30)(-0.30)(0.01)$$ $$\sigma_p^2 = (1.69)(0.05) + (0.09)(0.02) + (-0.78)(0.01)$$ $$\sigma_p^2 = 0.0845 + 0.0018 - 0.0078 = 0.0785$$ The variance is 0.0785. Note that the negative covariance term reduces portfolio risk despite the short position.
9. Portfolio P consists of a 50% allocation to Asset X and a 50% allocation to Asset Y. Both assets have a standard deviation of 20%. If the covariance between the assets is -0.02, what is the portfolio's standard deviation?
   Answer: 14.1% The problem provides covariance directly, so there is no need to calculate it from correlation. 1. Identify parameters: $w_X=0.5$, $w_Y=0.5$, $\sigma_X=0.2$, $\sigma_Y=0.2$, $Cov(X,Y)=-0.02$. 2. Calculate variances of assets: $\sigma_X^2 = (0.2)^2 = 0.04$; $\sigma_Y^2 = (0.2)^2 = 0.04$. 3. Calculate portfolio variance: $$\sigma_p^2 = w_X^2 \sigma_X^2 + w_Y^2 \sigma_Y^2 + 2 w_X w_Y Cov(X, Y)$$ $$\sigma_p^2 = (0.5)^2(0.04) + (0.5)^2(0.04) + 2(0.5)(0.5)(-0.02)$$ $$\sigma_p^2 = (0.25)(0.04) + (0.25)(0.04) + (0.5)(-0.02)$$ $$\sigma_p^2 = 0.01 + 0.01 - 0.01 = 0.02$$ 4. Calculate portfolio standard deviation: $$\sigma_p = \sqrt{0.02} \approx 0.1414$$ Thus, the portfolio standard deviation is approximately 14.1%.
10. An analyst compares two portfolios. Portfolio 1 is an equally-weighted portfolio of uncorrelated ($\rho=0$) assets A and B, which have standard deviations of 20% and 40%, respectively. Portfolio 2 consists of a 70% allocation to Asset C ($\sigma_C=15\%$) and 30% to a risk-free asset. Which portfolio has a higher standard deviation, and by how much?
    Answer: Portfolio 1 is higher by 11.8% This problem requires calculating the standard deviation for two different portfolios and then comparing them. 1. **Calculate $\sigma$ for Portfolio 1:** * Assets A and B are uncorrelated ($\rho=0$), so the covariance term is zero. * $\sigma_{p1}^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 = (0.5)^2(0.2)^2 + (0.5)^2(0.4)^2$ * $\sigma_{p1}^2 = 0.25(0.04) + 0.25(0.16) = 0.01 + 0.04 = 0.05$ * \sigma_{p1} = \sqrt{0.05} \approx 0.2236 \text{ or } 22.36\%\] 2. **Calculate \(\sigma for Portfolio 2:** * A risk-free asset has zero variance and zero covariance with any risky asset. * $\sigma_{p2}^2 = w_C^2\sigma_C^2 + w_{RF}^2\sigma_{RF}^2 + 0 = (0.7)^2(0.15)^2 + 0$ * $\sigma_{p2}^2 = 0.49(0.0225) = 0.011025$ * \sigma_{p2} = \sqrt{0.011025} = 0.105 \text{ or } 10.5\%\] 3. **Compare the two portfolios:** * Portfolio 1 has a higher standard deviation. * Difference = \(\sigma_{p1} - \sigma_{p2} = 22.36\% - 10.5\% = 11.86\%.
11. A portfolio holds two assets, a stock index fund and a bond index fund. The stock fund has a variance of 0.0484 and the bond fund has a variance of 0.0064. The correlation between them is 0.15. The portfolio is weighted to achieve a standard deviation of 15%. Which of the following is a possible weight for the stock fund?
    Answer: 65% This problem requires testing the given weights to see which one results in the target portfolio standard deviation. Let $w_S$ be the weight in stocks. 1. Identify parameters: $\sigma_S^2=0.0484 \implies \sigma_S=0.22$; $\sigma_B^2=0.0064 \implies \sigma_B=0.08$; $\rho=0.15$; $\sigma_p=0.15 \implies \sigma_p^2=0.0225$. 2. Test each weight: Let's test the correct answer, $w_S = 0.65$. Then $w_B = 0.35$. $$\sigma_p^2 = w_S^2\sigma_S^2 + w_B^2\sigma_B^2 + 2w_Sw_B\rho\sigma_S\sigma_B$$ $$\sigma_p^2 = (0.65)^2(0.0484) + (0.35)^2(0.0064) + 2(0.65)(0.35)(0.15)(0.22)(0.08)$$ $$\sigma_p^2 = (0.4225)(0.0484) + (0.1225)(0.0064) + (0.455)(0.15)(0.0176)$$ $$\sigma_p^2 = 0.020449 + 0.000784 + 0.0012012$$ $$\sigma_p^2 = 0.0224342$$ $$\sigma_p = \sqrt{0.0224342} \approx 0.14978$$ This is approximately 15%, so a 65% weight in the stock fund is a valid solution.

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