Finance Quiz: Portfolio Expected Return And Variance
20 questions · exam conditions
0:00
Portfolio Expected Return And VarianceQuestion 1 of 20

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?

0%
10%
20%
50%
← Back to quizzes

Finance Quiz

Finance Quiz: Portfolio Expected Return And Variance

Practice Portfolio Expected Return And Variance in 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 Portfolio Expected Return And Variance, giving you a quick way to practice the rules, question types, and explanations that matter most for Finance.

How to use this quiz

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.

All questions

Question 1

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?

  1. 0%
  2. 10% (correct answer)
  3. 20%
  4. 50%
Explanation: With perfect negative correlation (ρ=−1.0\rho = -1.0), the portfolio standard deviation formula simplifies to σp=∣wAσA−wBσB∣\sigma_p = |w_A\sigma_A - w_B\sigma_B|. For an equally weighted portfolio, wGI=wCCV=0.5w_{GI} = w_{CCV} = 0.5. Using the general formula: σp2=wGI2σGI2+wCCV2σCCV2+2wGIwCCVρσGIσCCV\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} σp2=(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 = (0.5)^2(0.4)^2 + (0.5)^2(0.6)^2 + 2(0.5)(0.5)(-1.0)(0.4)(0.6) σp2=(0.25)(0.16)+(0.25)(0.36)−(0.5)(0.24)\sigma_p^2 = (0.25)(0.16) + (0.25)(0.36) - (0.5)(0.24) σp2=0.04+0.09−0.12\sigma_p^2 = 0.04 + 0.09 - 0.12 σp2=0.01\sigma_p^2 = 0.01 σp=0.01=0.10\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.

Question 2

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?

  1. 16.0%
  2. 18.5%
  3. 21.7% (correct answer)
  4. 25.4%
Explanation: This problem requires solving a quadratic equation for the unknown standard deviation, σB\sigma_B.
  1. Set up the variance formula: σp2=wA2σA2+wB2σB2+2wAwBρσAσB\sigma_p^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_Aw_B\rho\sigma_A\sigma_B Given: σp=0.15  ⟹  σp2=0.0225\sigma_p=0.15 \implies \sigma_p^2 = 0.0225, wA=wB=0.5w_A=w_B=0.5, σA=0.16\sigma_A=0.16, ρ=0.25\rho=0.25.
  2. Substitute known values: 0.0225=(0.5)2(0.16)2+(0.5)2σB2+2(0.5)(0.5)(0.25)(0.16)σB0.0225 = (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)+0.25σB2+0.5(0.25)(0.16)σB0.0225 = 0.25(0.0256) + 0.25\sigma_B^2 + 0.5(0.25)(0.16)\sigma_B 0.0225=0.0064+0.25σB2+0.02σB0.0225 = 0.0064 + 0.25\sigma_B^2 + 0.02\sigma_B
  3. Rearrange into a standard quadratic form (ax2+bx+c=0ax^2+bx+c=0) where x=σBx=\sigma_B: 0.25σB2+0.02σB−0.0161=00.25\sigma_B^2 + 0.02\sigma_B - 0.0161 = 0
  4. Solve using the quadratic formula x=[−b±b2−4ac]/2ax = [-b \pm \sqrt{b^2-4ac}] / 2a: σB=−0.02±(0.02)2−4(0.25)(−0.0161)2(0.25)\sigma_B = \frac{-0.02 \pm \sqrt{(0.02)^2 - 4(0.25)(-0.0161)}}{2(0.25)} σB=−0.02±0.0004+0.01610.5=−0.02±0.01650.5\sigma_B = \frac{-0.02 \pm \sqrt{0.0004 + 0.0161}}{0.5} = \frac{-0.02 \pm \sqrt{0.0165}}{0.5} σB=−0.02±0.128450.5\sigma_B = \frac{-0.02 \pm 0.12845}{0.5} Since standard deviation cannot be negative, we take the positive root: σB=0.108450.5≈0.2169\sigma_B = \frac{0.10845}{0.5} \approx 0.2169 The standard deviation of Stock B is approximately 21.7%.

Question 3

An analyst compares two portfolios. Portfolio 1 is an equally-weighted portfolio of uncorrelated (ρ=0\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 (σC=15%\sigma_C=15\%) and 30% to a risk-free asset. Which portfolio has a higher standard deviation, and by how much?

  1. Portfolio 1 is higher by 11.8% (correct answer)
  2. Portfolio 1 is higher by 22.4%
  3. Portfolio 2 is higher by 10.5%
  4. Portfolio 2 is higher by 30.0%
Explanation: This problem requires calculating the standard deviation for two different portfolios and then comparing them.
  1. Calculate σ\sigma for Portfolio 1:
    • Assets A and B are uncorrelated (ρ=0\rho=0), so the covariance term is zero.
    • σp12=wA2σA2+wB2σB2=(0.5)2(0.2)2+(0.5)2(0.4)2\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
    • σp12=0.25(0.04)+0.25(0.16)=0.01+0.04=0.05\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.
    • σp22=wC2σC2+wRF2σRF2+0=(0.7)2(0.15)2+0\sigma_{p2}^2 = w_C^2\sigma_C^2 + w_{RF}^2\sigma_{RF}^2 + 0 = (0.7)^2(0.15)^2 + 0
    • σp22=0.49(0.0225)=0.011025\sigma_{p2}^2 = 0.49(0.0225) = 0.011025
    • $\sigma_{p2} = \sqrt{0.011025} = 0.105 \text{ or } 10.5%]
  3. Compare the two portfolios:
    • Portfolio 1 has a higher standard deviation.
    • Difference = (\sigma_{p1} - \sigma_{p2} = 22.36% - 10.5% = 11.86%$.

Question 4

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 (ρAB\rho_{AB}) between the returns of Stock A and Stock B?

  1. -1.0
  2. 0.0
  3. 0.5
  4. 1.0 (correct answer)
Explanation: 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: σp=wA2σA2+wB2σB2+2wAwBρABσAσB\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: σp=0.30\sigma_p = 0.30, σA=0.30\sigma_A = 0.30, σB=0.30\sigma_B = 0.30, wA=0.5w_A = 0.5, wB=0.5w_B = 0.5. 0.30=(0.5)2(0.3)2+(0.5)2(0.3)2+2(0.5)(0.5)ρAB(0.3)(0.3)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)+0.5ρAB(0.09)0.09 = 0.25(0.09) + 0.25(0.09) + 0.5 \rho_{AB} (0.09) 0.09=0.0225+0.0225+0.045ρAB0.09 = 0.0225 + 0.0225 + 0.045 \rho_{AB} 0.09=0.045+0.045ρAB0.09 = 0.045 + 0.045 \rho_{AB} 0.045=0.045ρAB0.045 = 0.045 \rho_{AB} ρAB=1.0\rho_{AB} = 1.0

Question 5

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?

  1. 0.00%
  2. 2.84% (correct answer)
  3. 6.19%
  4. 19.16%
Explanation: 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: Weighted Avg σ=wXσX+wYσY=0.5(0.18)+0.5(0.26)=0.09+0.13=0.22 or 22.00%\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 (σp\sigma_p): First, calculate the portfolio variance, σp2\sigma_p^2. σp2=(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 = (0.5)^2(0.18)^2 + (0.5)^2(0.26)^2 + 2(0.5)(0.5)(0.5)(0.18)(0.26) σp2=0.25(0.0324)+0.25(0.0676)+0.5(0.5)(0.0468)\sigma_p^2 = 0.25(0.0324) + 0.25(0.0676) + 0.5(0.5)(0.0468) σp2=0.0081+0.0169+0.0117=0.0367\sigma_p^2 = 0.0081 + 0.0169 + 0.0117 = 0.0367 Now, find the standard deviation: σp=0.0367≈0.19157 or 19.16%\sigma_p = \sqrt{0.0367} \approx 0.19157 \text{ or } 19.16\%
  3. Calculate the reduction: Reduction=Weighted Avg σ−σp=22.00%−19.16%=2.84%\text{Reduction} = \text{Weighted Avg } \sigma - \sigma_p = 22.00\% - 19.16\% = 2.84\%

Question 6

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?

  1. 0.0650
  2. 0.0785 (correct answer)
  3. 0.0847
  4. 0.0941
Explanation: 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: wA=1.30w_A = 1.30
    • Short position in B: wB=−0.30w_B = -0.30
    • σA2=0.05\sigma_A^2 = 0.05, σB2=0.02\sigma_B^2 = 0.02, Cov(A,B)=0.01Cov(A,B) = 0.01
  2. Apply the portfolio variance formula: σp2=wA2σA2+wB2σB2+2wAwBCov(A,B)\sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B Cov(A, B) σp2=(1.30)2(0.05)+(−0.30)2(0.02)+2(1.30)(−0.30)(0.01)\sigma_p^2 = (1.30)^2(0.05) + (-0.30)^2(0.02) + 2(1.30)(-0.30)(0.01) σp2=(1.69)(0.05)+(0.09)(0.02)+(−0.78)(0.01)\sigma_p^2 = (1.69)(0.05) + (0.09)(0.02) + (-0.78)(0.01) σp2=0.0845+0.0018−0.0078=0.0785\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.

Question 7

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?

  1. 17.0%
  2. 18.3%
  3. 19.6%
  4. 20.1% (correct answer)
Explanation: The formula for portfolio variance is σp2=wA2σA2+wB2σB2+2wAwBCov(RA,RB)\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: wA=0.60w_A = 0.60, wB=0.40w_B = 0.40, σA2=0.0400\sigma_A^2 = 0.0400, σB2=0.0900\sigma_B^2 = 0.0900, and Cov(RA,RB)=0.0240Cov(R_A, R_B) = 0.0240.
  2. Substitute these values into the formula: σp2=(0.60)2(0.0400)+(0.40)2(0.0900)+2(0.60)(0.40)(0.0240)\sigma_p^2 = (0.60)^2(0.0400) + (0.40)^2(0.0900) + 2(0.60)(0.40)(0.0240) σp2=(0.36)(0.0400)+(0.16)(0.0900)+(0.48)(0.0240)\sigma_p^2 = (0.36)(0.0400) + (0.16)(0.0900) + (0.48)(0.0240) σp2=0.0144+0.0144+0.01152\sigma_p^2 = 0.0144 + 0.0144 + 0.01152 σp2=0.04032\sigma_p^2 = 0.04032
  3. The portfolio standard deviation is the square root of the variance: σp=0.04032≈0.2008\sigma_p = \sqrt{0.04032} \approx 0.2008 Thus, the portfolio's standard deviation is approximately 20.1%.

Question 8

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?

  1. 25%
  2. 45%
  3. 65% (correct answer)
  4. 85%
Explanation: This problem requires testing the given weights to see which one results in the target portfolio standard deviation. Let wSw_S be the weight in stocks.
  1. Identify parameters: σS2=0.0484  ⟹  σS=0.22\sigma_S^2=0.0484 \implies \sigma_S=0.22; σB2=0.0064  ⟹  σB=0.08\sigma_B^2=0.0064 \implies \sigma_B=0.08; ρ=0.15\rho=0.15; σp=0.15  ⟹  σp2=0.0225\sigma_p=0.15 \implies \sigma_p^2=0.0225.
  2. Test each weight: Let's test the correct answer, wS=0.65w_S = 0.65. Then wB=0.35w_B = 0.35. σp2=wS2σS2+wB2σB2+2wSwBρσSσB\sigma_p^2 = w_S^2\sigma_S^2 + w_B^2\sigma_B^2 + 2w_Sw_B\rho\sigma_S\sigma_B σp2=(0.65)2(0.0484)+(0.35)2(0.0064)+2(0.65)(0.35)(0.15)(0.22)(0.08)\sigma_p^2 = (0.65)^2(0.0484) + (0.35)^2(0.0064) + 2(0.65)(0.35)(0.15)(0.22)(0.08) σp2=(0.4225)(0.0484)+(0.1225)(0.0064)+(0.455)(0.15)(0.0176)\sigma_p^2 = (0.4225)(0.0484) + (0.1225)(0.0064) + (0.455)(0.15)(0.0176) σp2=0.020449+0.000784+0.0012012\sigma_p^2 = 0.020449 + 0.000784 + 0.0012012 σp2=0.0224342\sigma_p^2 = 0.0224342 σp=0.0224342≈0.14978\sigma_p = \sqrt{0.0224342} \approx 0.14978 This is approximately 15%, so a 65% weight in the stock fund is a valid solution.

Question 9

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?

  1. 8.0% (correct answer)
  2. 9.5%
  3. 10.0%
  4. 10.5%
Explanation: 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. wC=1.0−wA−wB=1.0−0.5−0.3=0.20w_C = 1.0 - w_A - w_B = 1.0 - 0.5 - 0.3 = 0.20
  2. Use the formula for portfolio expected return: E[Rp]=wAE[RA]+wBE[RB]+wCE[RC]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[RC]E[R_C]: 0.100=(0.50)(0.12)+(0.30)(0.08)+(0.20)E[RC]0.100 = (0.50)(0.12) + (0.30)(0.08) + (0.20)E[R_C] 0.100=0.060+0.024+0.20E[RC]0.100 = 0.060 + 0.024 + 0.20 E[R_C] 0.100=0.084+0.20E[RC]0.100 = 0.084 + 0.20 E[R_C] 0.016=0.20E[RC]0.016 = 0.20 E[R_C] E[RC]=0.0160.20=0.08E[R_C] = \frac{0.016}{0.20} = 0.08 Thus, the expected return of Asset C is 8.0%.

Question 10

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?

  1. 10.0%
  2. 14.1% (correct answer)
  3. 17.3%
  4. 20.0%
Explanation: The problem provides covariance directly, so there is no need to calculate it from correlation.
  1. Identify parameters: wX=0.5w_X=0.5, wY=0.5w_Y=0.5, σX=0.2\sigma_X=0.2, σY=0.2\sigma_Y=0.2, Cov(X,Y)=−0.02Cov(X,Y)=-0.02.
  2. Calculate variances of assets: σX2=(0.2)2=0.04\sigma_X^2 = (0.2)^2 = 0.04; σY2=(0.2)2=0.04\sigma_Y^2 = (0.2)^2 = 0.04.
  3. Calculate portfolio variance: σp2=wX2σX2+wY2σY2+2wXwYCov(X,Y)\sigma_p^2 = w_X^2 \sigma_X^2 + w_Y^2 \sigma_Y^2 + 2 w_X w_Y Cov(X, Y) σp2=(0.5)2(0.04)+(0.5)2(0.04)+2(0.5)(0.5)(−0.02)\sigma_p^2 = (0.5)^2(0.04) + (0.5)^2(0.04) + 2(0.5)(0.5)(-0.02) σp2=(0.25)(0.04)+(0.25)(0.04)+(0.5)(−0.02)\sigma_p^2 = (0.25)(0.04) + (0.25)(0.04) + (0.5)(-0.02) σp2=0.01+0.01−0.01=0.02\sigma_p^2 = 0.01 + 0.01 - 0.01 = 0.02
  4. Calculate portfolio standard deviation: σp=0.02≈0.1414\sigma_p = \sqrt{0.02} \approx 0.1414 Thus, the portfolio standard deviation is approximately 14.1%.

Question 11

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?

  1. 0.20 (correct answer)
  2. 0.40
  3. 0.50
  4. 0.65
Explanation: The problem requires solving for the correlation coefficient (ρ\rho) within the portfolio variance formula.
  1. Formula: σp2=wS2σS2+wB2σB2+2wSwBρSBσSσB\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: σp2=0.033625\sigma_p^2 = 0.033625, wS=0.7w_S = 0.7, wB=0.3w_B = 0.3, σS=0.25\sigma_S = 0.25, σB=0.10\sigma_B = 0.10.
  3. Calculate the individual variance components:
    • wS2σS2=(0.7)2(0.25)2=0.49(0.0625)=0.030625w_S^2 \sigma_S^2 = (0.7)^2 (0.25)^2 = 0.49(0.0625) = 0.030625
    • wB2σB2=(0.3)2(0.10)2=0.09(0.01)=0.0009w_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)ρSB(0.25)(0.10)0.033625 = 0.030625 + 0.0009 + 2(0.7)(0.3)\rho_{SB}(0.25)(0.10)
  5. Simplify and solve for ρSB\rho_{SB}: 0.033625=0.031525+(0.42)ρSB(0.025)0.033625 = 0.031525 + (0.42) \rho_{SB} (0.025) 0.033625=0.031525+0.0105ρSB0.033625 = 0.031525 + 0.0105 \rho_{SB} 0.0021=0.0105ρSB0.0021 = 0.0105 \rho_{SB} ρSB=0.00210.0105=0.20\rho_{SB} = \frac{0.0021}{0.0105} = 0.20

Question 12

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?

  1. 17.0%
  2. 18.3%
  3. 19.6%
  4. 20.1% (correct answer)
Explanation: The formula for portfolio variance is σp2=wA2σA2+wB2σB2+2wAwBCov(RA,RB)\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: wA=0.60w_A = 0.60, wB=0.40w_B = 0.40, σA2=0.0400\sigma_A^2 = 0.0400, σB2=0.0900\sigma_B^2 = 0.0900, and Cov(RA,RB)=0.0240Cov(R_A, R_B) = 0.0240.
  2. Substitute these values into the formula: σp2=(0.60)2(0.0400)+(0.40)2(0.0900)+2(0.60)(0.40)(0.0240)\sigma_p^2 = (0.60)^2(0.0400) + (0.40)^2(0.0900) + 2(0.60)(0.40)(0.0240) σp2=(0.36)(0.0400)+(0.16)(0.0900)+(0.48)(0.0240)\sigma_p^2 = (0.36)(0.0400) + (0.16)(0.0900) + (0.48)(0.0240) σp2=0.0144+0.0144+0.01152\sigma_p^2 = 0.0144 + 0.0144 + 0.01152 σp2=0.04032\sigma_p^2 = 0.04032
  3. The portfolio standard deviation is the square root of the variance: σp=0.04032≈0.2008\sigma_p = \sqrt{0.04032} \approx 0.2008 Thus, the portfolio's standard deviation is approximately 20.1%.

Question 13

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 (ρAB\rho_{AB}) between the returns of Stock A and Stock B?

  1. -1.0
  2. 0.0
  3. 0.5
  4. 1.0 (correct answer)
Explanation: 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: σp=wA2σA2+wB2σB2+2wAwBρABσAσB\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: σp=0.30\sigma_p = 0.30, σA=0.30\sigma_A = 0.30, σB=0.30\sigma_B = 0.30, wA=0.5w_A = 0.5, wB=0.5w_B = 0.5. 0.30=(0.5)2(0.3)2+(0.5)2(0.3)2+2(0.5)(0.5)ρAB(0.3)(0.3)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)+0.5ρAB(0.09)0.09 = 0.25(0.09) + 0.25(0.09) + 0.5 \rho_{AB} (0.09) 0.09=0.0225+0.0225+0.045ρAB0.09 = 0.0225 + 0.0225 + 0.045 \rho_{AB} 0.09=0.045+0.045ρAB0.09 = 0.045 + 0.045 \rho_{AB} 0.045=0.045ρAB0.045 = 0.045 \rho_{AB} ρAB=1.0\rho_{AB} = 1.0

Question 14

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?

  1. 0.20 (correct answer)
  2. 0.40
  3. 0.50
  4. 0.65
Explanation: The problem requires solving for the correlation coefficient (ρ\rho) within the portfolio variance formula.
  1. Formula: σp2=wS2σS2+wB2σB2+2wSwBρSBσSσB\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: σp2=0.033625\sigma_p^2 = 0.033625, wS=0.7w_S = 0.7, wB=0.3w_B = 0.3, σS=0.25\sigma_S = 0.25, σB=0.10\sigma_B = 0.10.
  3. Calculate the individual variance components:
    • wS2σS2=(0.7)2(0.25)2=0.49(0.0625)=0.030625w_S^2 \sigma_S^2 = (0.7)^2 (0.25)^2 = 0.49(0.0625) = 0.030625
    • wB2σB2=(0.3)2(0.10)2=0.09(0.01)=0.0009w_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)ρSB(0.25)(0.10)0.033625 = 0.030625 + 0.0009 + 2(0.7)(0.3)\rho_{SB}(0.25)(0.10)
  5. Simplify and solve for ρSB\rho_{SB}: 0.033625=0.031525+(0.42)ρSB(0.025)0.033625 = 0.031525 + (0.42) \rho_{SB} (0.025) 0.033625=0.031525+0.0105ρSB0.033625 = 0.031525 + 0.0105 \rho_{SB} 0.0021=0.0105ρSB0.0021 = 0.0105 \rho_{SB} ρSB=0.00210.0105=0.20\rho_{SB} = \frac{0.0021}{0.0105} = 0.20

Question 15

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?

  1. 8.0% (correct answer)
  2. 9.5%
  3. 10.0%
  4. 10.5%
Explanation: 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. wC=1.0−wA−wB=1.0−0.5−0.3=0.20w_C = 1.0 - w_A - w_B = 1.0 - 0.5 - 0.3 = 0.20
  2. Use the formula for portfolio expected return: E[Rp]=wAE[RA]+wBE[RB]+wCE[RC]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[RC]E[R_C]: 0.100=(0.50)(0.12)+(0.30)(0.08)+(0.20)E[RC]0.100 = (0.50)(0.12) + (0.30)(0.08) + (0.20)E[R_C] 0.100=0.060+0.024+0.20E[RC]0.100 = 0.060 + 0.024 + 0.20 E[R_C] 0.100=0.084+0.20E[RC]0.100 = 0.084 + 0.20 E[R_C] 0.016=0.20E[RC]0.016 = 0.20 E[R_C] E[RC]=0.0160.20=0.08E[R_C] = \frac{0.016}{0.20} = 0.08 Thus, the expected return of Asset C is 8.0%.

Question 16

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?

  1. 0.0650
  2. 0.0785 (correct answer)
  3. 0.0847
  4. 0.0941
Explanation: 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: wA=1.30w_A = 1.30
    • Short position in B: wB=−0.30w_B = -0.30
    • σA2=0.05\sigma_A^2 = 0.05, σB2=0.02\sigma_B^2 = 0.02, Cov(A,B)=0.01Cov(A,B) = 0.01
  2. Apply the portfolio variance formula: σp2=wA2σA2+wB2σB2+2wAwBCov(A,B)\sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B Cov(A, B) σp2=(1.30)2(0.05)+(−0.30)2(0.02)+2(1.30)(−0.30)(0.01)\sigma_p^2 = (1.30)^2(0.05) + (-0.30)^2(0.02) + 2(1.30)(-0.30)(0.01) σp2=(1.69)(0.05)+(0.09)(0.02)+(−0.78)(0.01)\sigma_p^2 = (1.69)(0.05) + (0.09)(0.02) + (-0.78)(0.01) σp2=0.0845+0.0018−0.0078=0.0785\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.

Question 17

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?

  1. 10.0%
  2. 14.1% (correct answer)
  3. 17.3%
  4. 20.0%
Explanation: The problem provides covariance directly, so there is no need to calculate it from correlation.
  1. Identify parameters: wX=0.5w_X=0.5, wY=0.5w_Y=0.5, σX=0.2\sigma_X=0.2, σY=0.2\sigma_Y=0.2, Cov(X,Y)=−0.02Cov(X,Y)=-0.02.
  2. Calculate variances of assets: σX2=(0.2)2=0.04\sigma_X^2 = (0.2)^2 = 0.04; σY2=(0.2)2=0.04\sigma_Y^2 = (0.2)^2 = 0.04.
  3. Calculate portfolio variance: σp2=wX2σX2+wY2σY2+2wXwYCov(X,Y)\sigma_p^2 = w_X^2 \sigma_X^2 + w_Y^2 \sigma_Y^2 + 2 w_X w_Y Cov(X, Y) σp2=(0.5)2(0.04)+(0.5)2(0.04)+2(0.5)(0.5)(−0.02)\sigma_p^2 = (0.5)^2(0.04) + (0.5)^2(0.04) + 2(0.5)(0.5)(-0.02) σp2=(0.25)(0.04)+(0.25)(0.04)+(0.5)(−0.02)\sigma_p^2 = (0.25)(0.04) + (0.25)(0.04) + (0.5)(-0.02) σp2=0.01+0.01−0.01=0.02\sigma_p^2 = 0.01 + 0.01 - 0.01 = 0.02
  4. Calculate portfolio standard deviation: σp=0.02≈0.1414\sigma_p = \sqrt{0.02} \approx 0.1414 Thus, the portfolio standard deviation is approximately 14.1%.

Question 18

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?

  1. 25%
  2. 45%
  3. 65% (correct answer)
  4. 85%
Explanation: This problem requires testing the given weights to see which one results in the target portfolio standard deviation. Let wSw_S be the weight in stocks.
  1. Identify parameters: σS2=0.0484  ⟹  σS=0.22\sigma_S^2=0.0484 \implies \sigma_S=0.22; σB2=0.0064  ⟹  σB=0.08\sigma_B^2=0.0064 \implies \sigma_B=0.08; ρ=0.15\rho=0.15; σp=0.15  ⟹  σp2=0.0225\sigma_p=0.15 \implies \sigma_p^2=0.0225.
  2. Test each weight: Let's test the correct answer, wS=0.65w_S = 0.65. Then wB=0.35w_B = 0.35. σp2=wS2σS2+wB2σB2+2wSwBρσSσB\sigma_p^2 = w_S^2\sigma_S^2 + w_B^2\sigma_B^2 + 2w_Sw_B\rho\sigma_S\sigma_B σp2=(0.65)2(0.0484)+(0.35)2(0.0064)+2(0.65)(0.35)(0.15)(0.22)(0.08)\sigma_p^2 = (0.65)^2(0.0484) + (0.35)^2(0.0064) + 2(0.65)(0.35)(0.15)(0.22)(0.08) σp2=(0.4225)(0.0484)+(0.1225)(0.0064)+(0.455)(0.15)(0.0176)\sigma_p^2 = (0.4225)(0.0484) + (0.1225)(0.0064) + (0.455)(0.15)(0.0176) σp2=0.020449+0.000784+0.0012012\sigma_p^2 = 0.020449 + 0.000784 + 0.0012012 σp2=0.0224342\sigma_p^2 = 0.0224342 σp=0.0224342≈0.14978\sigma_p = \sqrt{0.0224342} \approx 0.14978 This is approximately 15%, so a 65% weight in the stock fund is a valid solution.

Question 19

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?

  1. 0.00%
  2. 2.84% (correct answer)
  3. 6.19%
  4. 19.16%
Explanation: 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: Weighted Avg σ=wXσX+wYσY=0.5(0.18)+0.5(0.26)=0.09+0.13=0.22 or 22.00%\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 (σp\sigma_p): First, calculate the portfolio variance, σp2\sigma_p^2. σp2=(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 = (0.5)^2(0.18)^2 + (0.5)^2(0.26)^2 + 2(0.5)(0.5)(0.5)(0.18)(0.26) σp2=0.25(0.0324)+0.25(0.0676)+0.5(0.5)(0.0468)\sigma_p^2 = 0.25(0.0324) + 0.25(0.0676) + 0.5(0.5)(0.0468) σp2=0.0081+0.0169+0.0117=0.0367\sigma_p^2 = 0.0081 + 0.0169 + 0.0117 = 0.0367 Now, find the standard deviation: σp=0.0367≈0.19157 or 19.16%\sigma_p = \sqrt{0.0367} \approx 0.19157 \text{ or } 19.16\%
  3. Calculate the reduction: Reduction=Weighted Avg σ−σp=22.00%−19.16%=2.84%\text{Reduction} = \text{Weighted Avg } \sigma - \sigma_p = 22.00\% - 19.16\% = 2.84\%

Question 20

An analyst compares two portfolios. Portfolio 1 is an equally-weighted portfolio of uncorrelated (ρ=0\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 (σC=15%\sigma_C=15\%) and 30% to a risk-free asset. Which portfolio has a higher standard deviation, and by how much?

  1. Portfolio 1 is higher by 11.8% (correct answer)
  2. Portfolio 1 is higher by 22.4%
  3. Portfolio 2 is higher by 10.5%
  4. Portfolio 2 is higher by 30.0%
Explanation: This problem requires calculating the standard deviation for two different portfolios and then comparing them.
  1. Calculate σ\sigma for Portfolio 1:
    • Assets A and B are uncorrelated (ρ=0\rho=0), so the covariance term is zero.
    • σp12=wA2σA2+wB2σB2=(0.5)2(0.2)2+(0.5)2(0.4)2\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
    • σp12=0.25(0.04)+0.25(0.16)=0.01+0.04=0.05\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.
    • σp22=wC2σC2+wRF2σRF2+0=(0.7)2(0.15)2+0\sigma_{p2}^2 = w_C^2\sigma_C^2 + w_{RF}^2\sigma_{RF}^2 + 0 = (0.7)^2(0.15)^2 + 0
    • σp22=0.49(0.0225)=0.011025\sigma_{p2}^2 = 0.49(0.0225) = 0.011025
    • $\sigma_{p2} = \sqrt{0.011025} = 0.105 \text{ or } 10.5%]
  3. Compare the two portfolios:
    • Portfolio 1 has a higher standard deviation.
    • Difference = (\sigma_{p1} - \sigma_{p2} = 22.36% - 10.5% = 11.86%$.