Given the following information, what is the standard deviation of the returns o

Question

Given the following information, what is the standard deviation of the returns on a portfolio that is
invested 35 percent in both stocks A and C, and 30 percent in stock B?
                                                                  

                                                             rate of return if state occurs
state of               Probability of state       stock a     stock b     stock c        
boom                           .20                          16.4%    31.8%    11.4%
Normal                         .80                          11.2%   19.6%     7.3%

Explanation / Answer

According to the given information: Weight of stock-A (Wa) = 35% Weight of stock-B (Wb) = 30% Weight of stock-C (Wc) = 35% The portfolio return when the economy booms is calculated as: E(Rp) = (Wa X Ra) + (Wb X Rb) + (Wc X Rc) where Ra, Rb and Rc are the return of the stocks when the economy is in the state of boom. E(Rp) = (0.35 X 0.164) + (0.3 X 0.318) + (0.35 X 0.114)            = 0.0574 + 0.0954 + 0.04            = 0.1928 or 19.28% Similarly calculating the Portfolio return when the economy is in the normal state: E(Rp) = (Wa X Ra) + (Wb X Rb) + (Wc X Rc) where Ra, Rb and Rc are the return of the stocks when the economy is in the normal state. E(Rp) = (0.35X 0.112) + (0.3 X 0.196) + (0.35 X 0.073)            = 0.0392 + 0.0588 + 0.02555            = 0.1235 or 12.35% To calculate the portfolio expected return, we have to calculate the expected return for each individual stock. Expected return for Stock-A is calculated as: E(Ra) = (Pb X Rb) + (Pn X Rn) where Pb and Pn are the probabilities in the boom state and the normal state.            Rb and Rn are the returns of stock-A in the boom state and normal state. E(Ra) = (0.2 X 0.164) + (0.8 X 0.112)            = 0.0328 + 0.0896            = 0.1224 or 12.24% Expected return for Stock-B is calculated as: E(Rb) = (Pb X Rb) + (Pn X Rn) where Pb and Pn are the probabilities in the boom state and the normal state.            Rb and Rn are the returns of stock-B in the boom state and normal state. E(Rb) = (0.2 X 0.318) + (0.8 X 0.196)            = 0.0636 + 0.1568            = 0.2204 or 22.04% Expected return for Stock-C is calculated as: E(Rc) = (Pb X Rb) + (Pn X Rn) where Pb and Pn are the probabilities in the boom state and the normal state.            Rb and Rn are the returns of stock-C in the boom state and normal state. E(Rc) = (0.2 X 0.114) + (0.8 X 0.073)            = 0.0228 + 0.0584            = 0.0812 or 8.12% Calculating the expected return on portfolio: E(Rp) = [Wa X E(Ra)] + [Wb X E(Rb)] + [Wc X E(Rc)]            = [0.35 X0.1224] + [0.3 X 0.2204] + [0.35 X 0.0812]            = 0.04284 + 0.06612 + 0.02842            = 0.1374 or 13.74% Therefore, the expected return on portfolio is 13.74% Calculating the variance of the portfolio: The portfolio variance is calculated by multiplying the probability with the return that comes after deducting the expected return in the boom state from the total portfolio expected return and squaring the result. Portfolio variance = [0.2 X {(0.1928 - 0.1374)^2}] + [0.8 X {(0.1235 - 0.1374)^2}]                                 = [0.2 X 0.00306] + [0.8 X 0.000154]                                 = 0.000614 + 0.00016                                 = 0.000772 We know that the square root of variance is the standard deviation: Portfolio standard deviaiton = 0.000772                                                 = 0.0277 or 2.77% Therefore, the portfolio standard deviaiton value is 2.77% where Ra, Rb and Rc are the return of the stocks when the economy is in the normal state. E(Rp) = (0.35X 0.112) + (0.3 X 0.196) + (0.35 X 0.073)            = 0.0392 + 0.0588 + 0.02555            = 0.1235 or 12.35% To calculate the portfolio expected return, we have to calculate the expected return for each individual stock. Expected return for Stock-A is calculated as: E(Ra) = (Pb X Rb) + (Pn X Rn) where Pb and Pn are the probabilities in the boom state and the normal state.            Rb and Rn are the returns of stock-A in the boom state and normal state. E(Ra) = (0.2 X 0.164) + (0.8 X 0.112)            = 0.0328 + 0.0896            = 0.1224 or 12.24% Expected return for Stock-B is calculated as: E(Rb) = (Pb X Rb) + (Pn X Rn) where Pb and Pn are the probabilities in the boom state and the normal state.            Rb and Rn are the returns of stock-B in the boom state and normal state. E(Rb) = (0.2 X 0.318) + (0.8 X 0.196)            = 0.0636 + 0.1568            = 0.2204 or 22.04% Expected return for Stock-B is calculated as: E(Rb) = (Pb X Rb) + (Pn X Rn) where Pb and Pn are the probabilities in the boom state and the normal state.            Rb and Rn are the returns of stock-B in the boom state and normal state. E(Rb) = (0.2 X 0.318) + (0.8 X 0.196)            = 0.0636 + 0.1568            = 0.2204 or 22.04% Expected return for Stock-C is calculated as: E(Rc) = (Pb X Rb) + (Pn X Rn) where Pb and Pn are the probabilities in the boom state and the normal state.            Rb and Rn are the returns of stock-C in the boom state and normal state. E(Rc) = (0.2 X 0.114) + (0.8 X 0.073)            = 0.0228 + 0.0584            = 0.0812 or 8.12% Calculating the expected return on portfolio: E(Rp) = [Wa X E(Ra)] + [Wb X E(Rb)] + [Wc X E(Rc)]            = [0.35 X0.1224] + [0.3 X 0.2204] + [0.35 X 0.0812]            = 0.04284 + 0.06612 + 0.02842            = 0.1374 or 13.74% Therefore, the expected return on portfolio is 13.74% Calculating the variance of the portfolio: The portfolio variance is calculated by multiplying the probability with the return that comes after deducting the expected return in the boom state from the total portfolio expected return and squaring the result. Portfolio variance = [0.2 X {(0.1928 - 0.1374)^2}] + [0.8 X {(0.1235 - 0.1374)^2}]                                 = [0.2 X 0.00306] + [0.8 X 0.000154]                                 = 0.000614 + 0.00016                                 = 0.000772 We know that the square root of variance is the standard deviation: Portfolio standard deviaiton = 0.000772                                                 = 0.0277 or 2.77% Therefore, the portfolio standard deviaiton value is 2.77% Therefore, the expected return on portfolio is 13.74% Calculating the variance of the portfolio: The portfolio variance is calculated by multiplying the probability with the return that comes after deducting the expected return in the boom state from the total portfolio expected return and squaring the result. Portfolio variance = [0.2 X {(0.1928 - 0.1374)^2}] + [0.8 X {(0.1235 - 0.1374)^2}]                                 = [0.2 X 0.00306] + [0.8 X 0.000154]                                 = 0.000614 + 0.00016                                 = 0.000772 We know that the square root of variance is the standard deviation: Portfolio standard deviaiton = 0.000772                                                 = 0.0277 or 2.77% Therefore, the portfolio standard deviaiton value is 2.77%

Related Questions

Navigate

• Browse (All)
• Browse G
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.