Final 322 Investments
A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term bond fund, and the third is a money market fund that provides a safe return of 8%. The characteristics of the risky funds are as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12 15 The correlation between the fund returns is 0.10. 1.What are the investment proportions in the minimum-variance portfolio of the two risky funds? 2. What is the expected value and standard deviation of the minimum-variance portfolio rate of return?
1. Portfolio invested in the stock 0.1739 Portfolio invested in the bond 0.8261 2. Expected return= 0.1339 Standard deviation= 0.1392 1) wMin (S)=σB2 − Cov(rS, rB) = 225 − 45 =0.1739 σS2 + σB2 − 2Cov(rS, rB) =900 + 225 − (2 × 45) Min (B)=1 − 0.1739 = 0.8261 2) E(rMin) = (0.1739 × 0.20) + (0.8261 × 0.12) = 0.1339 σMin=[wS2σS2 + wB2σB2 + 2wSwB Cov(rS, rB)]1/2 =[(0.17392 × 900) + (0.82612 × 225) + (2 × 0.1739 × 0.8261 × 45)]1/2 =0.1392
You manage a risky portfolio with an expected rate of return of 17% and a standard deviation of 36%. The T-bill rate is 6%. Your client chooses to invest 75% of a portfolio in your fund and 25% in an essentially risk-free money market fund. What are the expected return and standard deviation of the rate of return on his portfolio? Expected Return Standard Deviation
Expected Return = 14.25% Standard Deviation = 27% Expected return = (0.75 × 17%) + (0.25 × 6%) = 14.25% Standard deviation = 0.75 × 36% = 27.00%
Consider a portfolio that offers an expected rate of return of 10% and a standard deviation of 23%. T-bills offer a risk-free 5% rate of return. What is the maximum level of risk aversion for which the risky portfolio is still preferred to T-bills? Maximum level of risk aversion must be
Less Than, 1.89 U = 0.10 − 0.5 × A × (0.23)2 = 0.10 − 0.0265 × A 0.10 − 0.0265A > 0.05 ⇒⇒ A < 0.05/0.0265 = 1.89
Compute the mean and standard deviation of the HPR on stocks.
Mean: 10.71% Standard Deviation: 26.69% E(r)=[0.28 × 49.0%] + [0.22 × 17.0%] + [0.50 × (-13.5%)] = 0.1071 or 10.71% σ2=[0.28 × (49.0 − 10.71)2] + [0.22 × (17.0 − 10.71)2] + [0.50 × (−13.5 − 10.71)2] = 712.281σ=0.2669 or 26.69%
A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term bond fund, and the third is a money market fund that provides a safe return of 8% (let this represent the risk-free rate of return in this market). The characteristics of the risky funds are as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12 15 The correlation between the fund returns is 0.10.What is the Sharpe ratio of the best feasible CAL?
Sharpe Ratio= 0.4603 E(rP)=(0.4516 × 0.20) + (0.5484 × 0.12) =0.1561 σp=[(0.45162 × 900) + (0.54842 × 225) + (2 × 0.4516 × 0.5484 × 45)]1/2 =0.1654 [E(rp) − rf] / [σp] = [0.1561 - 0.08] /0.1654 = 0.4603
You manage a risky portfolio with an expected rate of return of 19% and a standard deviation of 30%. The T-bill rate is 4%. Your client chooses to invest 75% of a portfolio in your fund and 25% in a T-bill money market fund. Suppose that your risky portfolio includes the following investments in the given proportions: Stock A 45% Stock B 35% Stock C 20% What are the investment proportions of your client's overall portfolio, including the position in T-bills
T-Bills= 25% Stock A= 33.8% Stock B= 26.3% Stock C= 15% Investment proportions: 25.0% in T-bills 0.75 × 45%=33.8% in Stock A 0.75 × 35%=26.2% in Stock B 0.75 × 20%=15.0% in Stock C
You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate is 8%. Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market fund. What is the reward-to-volatility (Sharpe) ratio (S) of your risky portfolio? Your client's?
Your reward-to-volatility ratio = 0.3571 Client's reward-to-volatility ratio= 0.3571 Your reward-to-volatility ratio: S =(0.18 − 0.08) / 28 = 0.35710 Client's expected return = (0.7 × 18%) + (0.3 × 8%) = 15% Client's standard deviation = 0.7 × 28% = 19.6% Client's reward-to-volatility ratio: S =(0.15 − 0.08) / 0.196 = 0.35710
Given an optimal risky portfolio with expected return of 6%, standard deviation of 23%, and a risk free rate of 3%, what is the slope of the best feasible CAL? a) 0.13 b) 0.39 c) 0.64 d) 0.36 e) 0.08
a) 0.13 Slope = (6 − 3)/23 = 0.1304
You manage a risky portfolio with an expected rate of return of 20% and a standard deviation of 36%. The T-bill rate is 5%. Your client's degree of risk aversion is A = 1.6, assuming a utility function U = E(r) − ½Aσ². 1. What proportion, y, of the total investment should be invested in your fund? 2. What are the expected value and standard deviation of the rate of return on your client's optimized portfolio?
a. 0.7234 b. E(rC)=0.05 + 0.15 × y* = 0.05 + (0.7234 × 0.15) = 0.1585 = 15.85% σC=0.7234 × 36% = 26.04%
You manage a risky portfolio with an expected rate of return of 19% and a standard deviation of 33%. The T-bill rate is 7%. Suppose that your client prefers to invest in your fund a proportion y that maximizes the expected return on the complete portfolio subject to the constraint that the complete portfolio's standard deviation will not exceed 19%. 1. What is the investment proportion, y? 2. What is the expected rate of return on the complete portfolio?
a. σC = y × 33%If your client prefers a standard deviation of at most 19%, then:y = 19/33 = 0.5758 = 57.58% invested in the risky portfolio. b. E(rc)=(1 − y) × T-bill rate + (y) × Risky rate 13.91%=(1 − 0.5758) × 0.07 + 0.5758 × 0.19
Given the capital allocation line, an investor's optimal portfolio is the portfolio that a) minimizes both her risk and return. b) maximizes her risk. c) None of the options are correct. d) maximizes her expected utility. e) maximizes her expected profit.
d) maximizes her expected utility.
Suppose that there are many stocks in the security market and that the characteristics of stocks A and B are given as follows: Stock Expected Return Standard Deviation A 10% 5% B 15 10 Correlation = -1 Suppose that it is possible to borrow at the risk-free rate, rf. What must be the value of the risk-free rate? (Hint: Think about constructing a risk-free portfolio from stocks A and B.)
σP = Absolute value [wAσA - wBσB] 0 = 5 × wA − [10 × (1 - wA)] => wA = 0.6667 The expected rate of return for this risk-free portfolio is:E(r) = (0.6667 × 10) + (0.3333 × 15) = 11.667% Therefore, the risk-free rate is: 11.667%
The variance of a portfolio of risky securities a) is the weighted sum of the securities' variances and covariances. b) is the sum of the securities' variances. c) is a weighted sum of the securities' variances. d) None of the options are correct. e) is the sum of the securities' covariances.
a) is the weighted sum of the securities' variances and covariances.
According to the Capital Asset Pricing Model (CAPM), overpriced securities have a) negative alphas. b) zero alphas. c) positive alphas. d) positive betas.
a) negative alphas.
Market risk is also referred to as a) systematic risk or nondiversifiable risk. b) unique risk or diversifiable risk. c) unique risk or nondiversifiable risk. d) systematic risk or diversifiable risk.
a) systematic risk or nondiversifiable risk.
The market risk, beta, of a security is equal to a) the covariance between the security's return and the market return divided by the variance of the market's returns. b) the variance of the security's returns divided by the variance of the market's returns. c) the covariance between the security and market returns divided by the standard deviation of the market's returns. d) the variance of the security's returns divided by the covariance between the security and market returns.
a) the covariance between the security's return and the market return divided by the variance of the market's returns.
A fair game a) will not be undertaken by a risk-averse investor and is a risky investment with a zero risk premium. b) is a risky investment with a zero risk premium. c) will not be undertaken by a risk-averse investor. d) will not be undertaken by a risk-averse investor and is a riskless investment. e) is a riskless investment.
a) will not be undertaken by a risk-averse investor and is a risky investment with a zero risk premium.
The continuously compounded annual return on a stock is normally distributed with a mean of 13% and standard deviation of 20%. With 95.45% confidence, we should expect its actual return in any particular year to be between which pair of values? a) −27.0% and 53.0% b) −17.0% and 53.0% c) −13.6% and 39.6% d) −6.9% and 29.4%
a) −27.0% and 53.0%
Which statement is not true regarding the market portfolio? a) All securities in the market portfolio are held in proportion to their market values. b) All of the options are true. c) It includes all publicly-traded financial assets. d) It is the tangency point between the capital market line and the indifference curve. e) It lies on the efficient frontier.
d) It is the tangency point between the capital market line and the indifference curve.
The risk-free rate is 7%. The expected market rate of return is 15%. If you expect a stock with a beta of 1.3 to offer a rate of return of 12%, you should a) None of the options, as the stock is fairly priced. b) sell the stock short because it is underpriced. c) buy the stock because it is underpriced. d) buy the stock because it is overpriced. e) sell short the stock because it is overpriced.
e) sell short the stock because it is overpriced.
A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term bond fund, and the third is a money market fund that provides a safe return of 8%. The characteristics of the risky funds are as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12 15 The correlation between the fund returns is 0.10. Solve numerically for the proportions of each asset and for the expected return and standard deviation of the optimal risky portfolio. Portfolio Invested in Stock= Portfolio Invested in Bonds= Expected Return= Standard Deviation=
[(0.20 − 0.08) × 225] − [(0.12 − 0.08) × 45] / [(0.20 − 0.08) × 225] + [(0.12 − 0.08) × 900] − [(0.20 − 0.08 + 0.12 − 0.08) × 45] = 0.4516 wB=1 − 0.4516 = 0.5484 E(rP)=(0.4516 × 0.20) + (0.5484 × 0.12) =0.1561 σp=[(0.45162 × 900) + (0.54842 × 225) + (2 × 0.4516 × 0.5484 × 45)]1/2 =0.1654
Which of the following statement(s) is(are) true regarding the selection of a portfolio from those that lie on the capital allocation line? I) Less risk-averse investors will invest more in the risk-free security and less in the optimal risky portfolio than more risk-averse investors. II) More risk-averse investors will invest less in the optimal risky portfolio and more in the risk-free security than less risk-averse investors. III) Investors choose the portfolio that maximizes their expected utility. a) II and III b) III only c) II only d) I and III e) I only
a) II and III
Which of the following statements regarding risk-averse investors is true? a) They only accept risky investments that offer risk premiums over the risk-free rate. b) They only care about the rate of return, and they accept investments that are fair games. c) They accept investments that are fair games. d) They only care about the rate of return. e) They are willing to accept lower returns and high risk.
a) They only accept risky investments that offer risk premiums over the risk-free rate.
The capital allocation line can be described as the a) line on which lie all portfolios that offer the same utility to a particular investor. b) investment opportunity set formed with two risky assets. c) investment opportunity set formed with a risky asset and a risk-free asset. d) line on which lie all portfolios with the same expected rate of return and different standard deviations.
c) investment opportunity set formed with a risky asset and a risk-free asset.
Which statement is not true regarding the capital market line (CML)? a) The CML always has a positive slope. b) The CML is also called the security market line. c) The CML is the line from the risk-free rate through the market portfolio. d) The CML is the best attainable capital allocation line. e) The risk measure for the CML is standard deviation.
b) The CML is also called the security market line.
Suppose that you have $1 million and the following two opportunities from which to construct a portfolio: Risk-free asset earning 12% per year. Risky asset with expected return of 30% per year and standard deviation of 40%. If you construct a portfolio with a standard deviation of 30%, what is its expected rate of return?
σP= 30 =y×σ= 40 ×y=>y= 0.75 E(rP)= 12 + 0.75(30 − 12) = 25.5%
Other things equal, diversification is most effective when a) securities' returns are high. b) securities' returns are positively correlated. c) securities' returns are positively correlated and high. d) securities' returns are uncorrelated. e) securities' returns are negatively correlated.
e) securities' returns are negatively correlated.
The line representing all combinations of portfolio expected returns and standard deviations that can be constructed from two available assets is called the a) capital allocation line. b) efficient frontier. c) portfolio opportunity set. d) risk/reward tradeoff line. e) Security Market Line.
c) portfolio opportunity set.
The risk premium on the market portfolio will be proportional to a) the average degree of risk aversion of the investor population. b) the average degree of risk aversion of the investor population and the risk of the market portfolio as measured by its beta. c) the average degree of risk aversion of the investor population and the risk of the market portfolio as measured by its variance. d) the risk of the market portfolio as measured by its variance. e) the risk of the market portfolio as measured by its beta.
c) the average degree of risk aversion of the investor population and the risk of the market portfolio as measured by its variance.
Portfolio theory as described by Markowitz is most concerned with a) active portfolio management to enhance returns. b) the elimination of systematic risk. c) the effect of diversification on portfolio risk. d the identification of unsystematic risk.
c) the effect of diversification on portfolio risk.
In a return-standard deviation space, which of the following statements is(are) true for risk-averse investors? (The vertical and horizontal lines are referred to as the expected return-axis and the standard deviation-axis, respectively.) I) An investor's own indifference curves might intersect. II) Indifference curves have negative slopes. III) In a set of indifference curves, the highest offers the greatest utility. IV) Indifference curves of two investors might intersect. a) None of the options are correct. b) I and II only c) II and III only d) III and IV only e) I and IV only
d) III and IV only
The separation property refers to the conclusion that a) investors are separate beings and will, therefore, have different preferences regarding the risk-return tradeoff. b) the choice of inputs to be used to determine the efficient frontier is objective, and the choice of the best CAL is subjective. c) the determination of the best CAL is objective, and the choice of the inputs to be used to determine the efficient frontier is subjective. d) the determination of the best risky portfolio is objective, and the choice of the best complete portfolio is subjective. e) the choice of the best complete portfolio is objective, and the determination of the best risky portfolio is objective.
d) the determination of the best risky portfolio is objective, and the choice of the best complete portfolio is subjective.
According to the mean-variance criterion, which one of the following investments dominates all others? a) E(r) = 0.10; Variance = 0.25 b) E(r) = 0.10; Variance = 0.20 c) None of these options dominates the other alternatives. d) E(r) = 0.15; Variance = 0.25 e) E(r) = 0.15; Variance = 0.20
e) E(r) = 0.15; Variance = 0.20
Which statement is true regarding the market portfolio? I) It includes all publicly traded financial assets. II) It lies on the efficient frontier. III) All securities in the market portfolio are held in proportion to their market values. IV) It is the tangency point between the capital market line and the indifference curve. a) IV only b) II only c) I only d) III only e) I, II, and III
e) I, II, and III
You manage a risky portfolio with an expected rate of return of 20% and a standard deviation of 29%. The T-bill rate is 5%. Your risky portfolio includes the following investments in the given proportions: Stock A 27% Stock B 32% Stock C 41% Suppose that your client decides to invest in your portfolio a proportion y of the total investment budget so that the overall portfolio will have an expected rate of return of 17%. 1. What is the proportion y? 2. What are your client's investment proportions in your three stocks and the T-bill fund? 3. What is the standard deviation of the rate of return on your client's portfolio?
a.E(rC) = rf + y × [E(rP) − rf] = 0.05 + y × (0.20 − 0.05)If the expected return for the portfolio is 17%, then: 17% = 5% + 15% × y ⇒ y =0.17 − 0.05= 0.80 or 80.00% Therefore, in order to have a portfolio with expected rate of return equal to 17%, the client must invest 80% of total funds in the risky portfolio and 20.00% in T-bills. b.Client's investment proportions: 20.00% in T-bills 0.80 × 27%=21.60% in Stock A 0.80 × 32%=25.60% in Stock B 0.80 × 41%=32.80% in Stock C c. σC = 0.80 × σP = 0.80 × 29% = 23.20%
Elias is a risk-averse investor. David is a less risk-averse investor than Elias. Therefore, a) for the same return, Elias tolerates higher risk than David. b) for the same return, David tolerates higher risk than Elias. c) for the same risk, Elias requires a lower rate of return than David. d) Cannot be determined. e) for the same risk, David requires a higher rate of return than Elias.
b) for the same return, David tolerates higher risk than Elias.
The presence of risk means that a) terminal wealth will be less than initial wealth. b) more than one outcome is possible. c) the standard deviation of the payoff is larger than its expected value. d) investors will lose money. e) final wealth will be greater than initial wealth.
b) more than one outcome is possible.
The security market line (SML) is a) All of the options. b) the line that represents the expected return-beta relationship. c) the line that is tangent to the efficient frontier of all risky assets. also called the capital allocation line. d) the line that describes the expected return-beta relationship for well-diversified portfolios only.
b) the line that represents the expected return-beta relationship.
The efficient frontier of risky assets is a) the portion of the minimum-variance portfolio that represents the highest standard deviations. b) the portion of the minimum-variance portfolio that lies above the global minimum variance portfolio. c) the set of portfolios that have zero standard deviation. d) the portion of the minimum-variance portfolio that includes the portfolios with the lowest standard deviation.
b) the portion of the minimum-variance portfolio that lies above the global minimum variance portfolio.
The global minimum variance portfolio formed from two risky securities will be riskless when the correlation coefficient between the two securities is a) any negative number. b) −1.0. c) 0.5. d) 0.0. e) 1.0.
b) −1.0.
You invest $1,000 in a risky asset with an expected rate of return of 0.17 and a standard deviation of 0.40 and a T-bill with a rate of return of 0.04. The slope of the capital allocation line formed with the risky asset and the risk-free asset is equal to a) 0.407. b) 0.912. c) 0.325. d) 0.675. e) Cannot be determined.
c) 0.325. (0.17 − 0.04)/0.40 = 0.325.
The risk-free rate and the expected market rate of return are 0.04 and 0.12, respectively. According to the capital asset pricing model (CAPM), the expected rate of return on security X with a beta of 1.4 is equal to a) 6% b) 12% c) 15.2% d) 18% e) 14.4%
c) 15.2% E(R) = 4% + 1.4(12% − 4%) = 15.2%.
Use the below information to answer the following question. Investment Expected Return Standard Deviation 1 0.12 0.13 2 0.15 0.15 3 0.21 0.16 4 0.24 0.21 U = E(r) − (A/2)s2, where A = 4.0. Based on the utility function above, which investment would you select? a) 1 b) Cannot be determined from the information given. c) 3 d) 2 e) 4
c) 3 U(c) = 0.21 − 4/2(0.16)2 = 15.88 (highest utility of choices)
In the context of the Capital Asset Pricing Model (CAPM), the relevant measure of risk is a) variance of returns. b) unique risk. c) beta. d) standard deviation of returns.
c) beta.
