FINE-4110 Chp6.3 - 6.4 Efficient Diversification

Asset A has an expected return of 20% and a standard deviation of 25%. The risk-free rate is 12%. What is the reward-to-variability ratio?

(0.20 − 0.12)/0.25 = 0.32

The portfolio with the lowest standard deviation for any risk premium is called the __________.

global minimum variance portfolio

An investor can design a risky portfolio based on two stocks, A and B. The standard deviation of return on stock A is 20%, while the standard deviation on stock B is 15%. The correlation coefficient between the returns on A and B is 0%. The rate of return for stocks A and B is 20% and 10% respectively. The expected return on the minimum-variance portfolio is approximately __________.

wA = σ^2B/(σ^2B + σ^2A) = 0.15^2/0.15^2 + 0.20^2 = 0.36// wB = 1 − wA = 1 − 0.36 = 0.64// E(rP) = 0.20 × 0.36 + 0.10 × 0.64 = 0.136 = 13.6%

A portfolio is composed of two stocks, A and B. Stock A has a standard deviation of return of 21%, while stock B has a standard deviation of return of 27%. Stock A comprises 70% of the portfolio, while stock B comprises 30% of the portfolio. If the variance of return on the portfolio is 0.045, the correlation coefficient between the returns on A and B is __________.

0.045 = (0.7^2)(0.21^2) + (0.3^2)(0.27^2) + 2(0.7)(0.3)(0.21)(0.27) ρ; ρ = 0.707

Your assistant gives you the following diagram as the efficient frontier of the group of stocks you asked him to analyze. The diagram looks a bit odd, but your assistant insists he double-checked his analysis. a. Would you trust him? b. Is it possible to get such a diagram?

No, it is not possible to get such a diagram. Even if the correlation between A and B were 1.0, the frontier would be a straight line connecting A and B.

An investor can design a risky portfolio based on two stocks, A and B. Stock A has an expected return of 18% and a standard deviation of return of 20%. Stock B has an expected return of 14% and a standard deviation of return of 5%. The correlation coefficient between the returns of A and B is 0.50. The risk-free rate of return is 10%. The standard deviation of return on the optimal risky portfolio is __________.

(0.18 − 0.10) × 0.052 − (0.14 − 0.10) × 0.05 × 0.20 × 0.50(0.18 − 0.10) × 0.052 − (0.14 − 0.10) × 0.202 − (0.18 − 0.10 + 0.14 − 0.10) × 0.05 × 0.20 × 0.5 = 0 → = (0.18 − 0.10) × 0.052 − (0.14 − 0.10) × 0.05 × 0.20 × 0.50(0.18 − 0.10) × 0.052 − (0.14 − 0.10) × 0.202 − (0.18 − 0.10 + 0.14 − 0.10) × 0.05 × 0.20 × 0.5 = 0 → wB = 1 − wA = 1wB = 1 - wA = 1 σP = (0 + 12 × 0.052 + 0 )0.5 = 0.05 = 5.00%

A portfolio's expected return is 12%, its standard deviation is 20%, and the risk-free rate is 4%. Which of the following would make for the greatest increase in the portfolio's Sharpe ratio?

A 1% point increase in expected return and 1% point decrease in the risk-free rate will have the same impact of increasing Sharpe ratio from 0.40 to 0.45 SP = (rP − rf)/σP = (0.12 − 0.04)/0.20 = 0.40 SP = (0.13 − 0.04)/0.20 = 0.45 SP = (0.12 − 0.03)/0.20 = 0.45 SP = (0.12 − 0.04)/0.19 = 0.42

A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a sure rate of 5.5%. The probability distributions of the risky funds are: The correlation between the fund returns is 0.25. What is the Sharpe ratio of the best feasible CAL?

Milk

An investor can design a risky portfolio based on two stocks, A and B. Stock A has an expected return of 18% and a standard deviation of return of 20%. Stock B has an expected return of 14% and a standard deviation of return of 5%. The correlation coefficient between the returns of A and B is 0.50. The risk-free rate of return is 10%. The proportion of the optimal risky portfolio that should be invested in stock A is __________.

wA = [E(rA) − rf]σ2B − [E(rB) − rf]σBσAρBA[E(rA) − rf]σ2B − [E(rB) − rf]σ2A − [E(rA) − rf + E(rB) − rf]σBσAρBAwA = [E(rA) − rf]σB2 − [E(rB) − rf]σBσAρBA[E(rA) − rf]σB2 − [E(rB) − rf]σA2 − [E(rA) − rf + E(rB) − rf]σBσAρBA = (0.18 − 0.10) × 0.052 − (0.14 − 0.10) × 0.05 × 0.20 × 0.50(0.18 − 0.10) × 0.052 − (0.14 − 0.10) × 0.202 − (0.18 − 0.10 + 0.14 − 0.10) × 0.05 × 0.20 × 0.5 = 0

The efficient frontier represents a set of portfolios that:

maximize expected return for a given level of risk.

An investor can design a risky portfolio based on two stocks, A and B. The standard deviation of return on stock A is 20%, while the standard deviation on stock B is 15%. The correlation coefficient between the returns on A and B is 0%. The rate of return for stocks A and B is 20% and 10% respectively. The standard deviation of return on the minimum-variance portfolio is ___________.

12% wA = σ^2B/(σ^2B + σ^2A) = 0.15^2/0.15^2 + 0.20^2 = 0.36// wB = 1 − wA = 1 − 0.36 = 0.64// E(rP) = 0.20 × 0.36 + 0.10 × 0.64 = 0.136 = 13.6% σP =(0.20^2 ×0.36^2 +0.15^2 ×0.64^2 + 0)0.5 = 0.12 = 12.00%//

The standard deviation of return on investment A is 26%, while the standard deviation of return on investment B is 21%. If the covariance of returns on A and B is 0.003, the correlation coefficient between the returns on A and B is __________.

Correlation = 0.003/[0.26(0.21)] = 0.055

An investor can design a risky portfolio based on two stocks, A and B. The standard deviation of return on stock A is 24%, while the standard deviation on stock B is 14%. The correlation coefficient between the returns on A and B is 0.35. The expected return on stock A is 25%, while on stock B it is 11%. The proportion of the minimum-variance portfolio that would be invested in stock B is approximately __________.

Cov(rA,rB) = ρABσAσB = 0.35 × 0.24 × 0.14 = 0.01176 wB = σ^2A − Cov(rA,rB)/ (σ^2B + σ^2A − 2Cov(rA,rB)) = (0.24^2 − 0.01176)/(0.14^2 + 0.24^2 − 2 × 0.01176) = 0.8539 = 85.39%

An investor can design a risky portfolio based on two stocks, A and B. Stock A has an expected return of 21% and a standard deviation of return of 39%. Stock B has an expected return of 14% and a standard deviation of return of 20%. The correlation coefficient between the returns of A and B is 0.4. The risk-free rate of return is 5%. The proportion of the optimal risky portfolio that should be invested in stock B is approximately __________.

wB = [E(rB) − rf]σ^2A − [E(rA) − rf] x σB xσA x ρBA / [E(rB) − rf]σ^2A − [E(rA) − rf]σ^2B − [E(rB) − rf + E(rA) − rf]σBxσAxρBA = {(0.14 − 0.05) (0.39)^2 − (0.21 − 0.05) (0.20) (0.39) (0.40) divide by (0.14 − 0.05) × (0.39)^2 − (0.21 − 0.05) × 0.20^2 − (0.14 − 0.05 + 0.21 − 0.05) × 0.20 × 0.39 × 0.40 = (.18-.10)(.05)^2 - (.14-.10)(.20)(.50)Divided By (.18-.10)(.05)^2 + (.14-.10)(.20)^2 -(.18-.10+.14-.10)(.05)(020)(.5) 0.008697 / -0.000511 =0.71 = 71%

A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a sure rate of 5.5%. The probability distributions of the risky funds are: Solve numerically for the proportions of each asset and for the expected return and standard deviation of the optimal risky portfolio.

Cheese

The expected return of a portfolio is 9.4%, and the risk-free rate is 4%. If the portfolio standard deviation is 17%, what is the reward-to-variability ratio of the portfolio?

Reward-to-variability ratio = (0.094 − 0.040)/0.17 = 0.32

The expected return of a portfolio is 8.9%, and the risk-free rate is 3.5%. If the portfolio standard deviation is 12%, what is the reward-to-variability ratio of the portfolio?

S= (0.089 − 0.035)/0.12 = 0.45

Suppose that many stocks are traded in the market and that it is possible to borrow at the risk-free rate, rƒ. The characteristics of two of the stocks are as follows: a. Calculate the expected rate of return on this risk-free portfolio? (Hint: Can a particular stock portfolio be formed to create a "synthetic" risk-free asset?) b. Could the equilibrium rƒ be greater than rate of return?

Since Stock A and Stock B are perfectly negatively correlated, a risk-free portfolio can be created and the rate of return for this portfolio in equilibrium will always be the risk-free rate. To find the proportions of this portfolio [with wA invested in Stock A and wB = (1 − wA) invested in Stock B], set the standard deviation equal to zero. With perfect negative correlation, the portfolio standard deviation reduces to: σP = ABS[wAσA − wBσB]0 = 40 × wA − 60 × (1 − wA) ⇒⇒ wA = 0.60 The expected rate of return on this risk-free portfolio is: E(r) = (0.60 × 0.08) + (0.40 × 0.12) = 9.60% E(r) = 9.60% Therefore, the risk-free rate must also be 9.60%.

An investor can design a risky portfolio based on two stocks, A and B. Stock A has an expected return of 16% and a standard deviation of return of 21%. Stock B has an expected return of 10% and a standard deviation of return of 16%. The correlation coefficient between the returns of A and B is 0.5. The risk-free rate of return is 5%. The proportion of the optimal risky portfolio that should be invested in stock B is approximately __________.

WB = ((0.10 − 0.05)(0.212) − (0.16 − 0.05)(0.16)(0.21)(0.5))/((0.10 − 0.05)(0.212) + (0.16 − 0.05)(0.162) − (0.10 − 0.05 + 0.16 − 0.05)(0.16)(0.21)(0.5)) WB= 15%

Suppose that many stocks are traded in the market and that it is possible to borrow at the risk-free rate, rƒ. The characteristics of two of the stocks are as follows: a. Calculate the expected rate of return on this risk-free portfolio? (Hint: Can a particular stock portfolio be formed to create a "synthetic" risk-free asset?) b. Could the equilibrium rƒ be greater than rate of return?

a. Since Stock A and Stock B are perfectly negatively correlated, a risk-free portfolio can be created and the rate of return for this portfolio in equilibrium will always be the risk-free rate. To find the proportions of this portfolio [with wA invested in Stock A and wB = (1 − wA) invested in Stock B], set the standard deviation equal to zero. With perfect negative correlation, the portfolio standard deviation reduces to: σP = ABS[wAσA − wBσB] 0 = 40 × wA − 60 × (1 − wA) ⇒⇒ wA = 0.60 The expected rate of return on this risk-free portfolio is: E(r) = (0.60 × 0.08) + (0.40 × 0.13) = 10.0% b. E(r) = 10.0% Therefore, the risk-free rate must also be 10.0%