Solution: The optimal position for an investor with a coefficient of risk aversion A = 5 in the risky asset is y*:


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1 Problem 1. Consider a risky asset. Suppose the expected rate of return on the risky asset is 15%, the standard deviation of the asset return is 22%, and the riskfree rate is 6%. What is your optimal position in the risky asset if the degree of risk aversion is 5? 1. What are the expected rate of return and the standard deviation of your complete portfolio? 2. What is the risk premium of the complete portfolio in this case? Solution: The optimal position for an investor with a coefficient of risk aversion A = 5 in the risky asset is y*: So, this particular investor will invest 37% of the investment budget in the risky asset, and 63% in the riskfree asset. This is the value of y for which utility is maximized. With 37% invested in the risky asset, the rate of return of the complete portfolio will have an expected return and standard deviation as follows: So, the risk premium of the complete portfolio is RP = 9.35%  6% = 3.35%, which is obtained by taking on a portfolio with a standard deviation of 8.18%. Problem 2. Suppose the economy can be.in one of the following two states: (i) Boom or good state and (ii) Recession or bad state. Each state can occur with an equal opportunity (50%). The annual return on the market and a certain security X in the two states of the economy are as follows: Market: at the end of the year, the market is expected to yield a return of 30% in the good state and a return of (10%) in the bad state;
2 Security X: at the end of the year, the security is expected to yield a return of 40% in the good state and a return of (35%) in the bad state; Furthermore, assume that annual riskfree rate of return is 5%. (a) Calculate the beta of security X relative to the market. (b) Calculate the alpha of security X. (c) Draw the security market line (SML). Please, label the axes and all points (including the market portfolio, the riskfree security, and security X) in the graph clearly. Identify alpha in the graph. Solution: (a) Beta of security X is: Market portfolio: (1/2) probability of getting 30% and (1/2) probability of getting  10%. So, expected/mean return and the standard deviation of the market are: Security X: (1/2) probability of getting 40% and (1/2) probability of getting 35%. So, expected/mean return and the standard deviation of security X are: The covariance between the market return and the security X return is: Therefore, the beta of security X will be: (b) Security s alpha is:
3 Problem 3. Consider the following table, which gives a security analyst s expected return on two stocks for two particular market returns: Market return Aggressive stock (A) Defensive stock (D) 4% 1% 6% 20% 33% 10% (a) What is the expected rate of return on each stock if the market return is equally likely to be 4% or 20%? (b) If the riskfree rate is 6%, and the market return is equally likely to be 4% or draw the SML for this security. (c) Plot the two securities on the SML graph. What are the alphas of each security? (Note: beta of stock A is 2.0 and beta of stock D is 0.25) Solution: (a) With the two scenarios equal likely, the expected rate of return is an average of the two possible outcomes: E(rA) = 0.5 (1% + 33%) = 17.0% E(rD) = 0.5 (6% + 10%) = 8.0% The SML is determined by the following: riskfree rate = 6% with a beta equal to zero, beta for the market is 1.0, and the expected rate of return for the market is: 0.5 (20% + 4%) = 12.0% The equation for the security market line is: E(r) = 6% + β(12% 6%) The aggressive stock has a fair expected rate of return of: E(rA) = 6% + 2.0(12% 6%) = 18.0% The security analyst s estimate of the expected rate of return is 17%. Thus the alpha for the aggressive stock is: αa = actual expected return required return predicted by CAPM αa = 17% 18% = 1.0%
4 Similarly, the required return for the defensive stock is: E(rD) = 6% (12% 6%) = 7.5% The security analyst s estimate of the expected return for D is 8%, and hence: αd = 8% 7.5% = 0.50% Multiple Choice Questions: 1. When two risky securities that are positively correlated but not perfectly correlated are held in a portfolio, A. the portfolio standard deviation will be greater than the weighted average of the individual security standard deviations. B. the portfolio standard deviation will be less than the weighted average of the individual security standard deviations. C. the portfolio standard deviation will be equal to the weighted average of the individual security standard deviations. D. the portfolio standard deviation will always be equal to the securities' covariance. E. both the portfolio standard deviation will be greater than the weighted average of the individual security standard deviations and it will always be equal to the securities' covariance. Whenever two securities are less than perfectly positively correlated, the standard deviation of the portfolio of the two assets will be less than the weighted average of the two securities' standard deviations. There is some benefit to diversification in this case.
5 2. Given an optimal risky portfolio with expected return of 14% and standard deviation of 22% and a risk free rate of 6%, what is the slope of the best feasible CAL? A B C D E Slope = (146)/22 = The standard deviation of a twoasset portfolio is a linear function of the assets' weights when A. the assets have a correlation coefficient less than zero. B. the assets have a correlation coefficient equal to zero. C. the assets have a correlation coefficient greater than zero. D. the assets have a correlation coefficient equal to one. E. the assets have a correlation coefficient less than one. When there is a perfect positive correlation (or a perfect negative correlation), the equation for the portfolio variance simplifies to a perfect square. The result is that the portfolio's standard deviation is linear relative to the assets' weights in the portfolio. 4. A twoasset portfolio with a standard deviation of zero can be formed when A. the assets have a correlation coefficient less than zero. B. the assets have a correlation coefficient equal to zero. C. the assets have a correlation coefficient greater than zero. D. the assets have a correlation coefficient equal to one. E. the assets have a correlation coefficient equal to negative one. When there is a perfect negative correlation, the equation for the portfolio variance simplifies to a perfect square. The result is that the portfolio's standard deviation equals waσa  wbσb, which can be set equal to zero. The solution wa = σb/(σa + σb) and wb = 1  wa will yield a zerostandard deviation portfolio. 5. The separation property refers to the conclusion that A. the determination of the best risky portfolio is objective and the choice of the best complete portfolio is subjective. B. the choice of the best complete portfolio is objective and the determination of the best risky portfolio is objective. C. the choice of inputs to be used to determine the efficient frontier is objective and the choice of the best CAL is subjective. D. the determination of the best CAL is objective and the choice of the inputs to be used to determine the efficient frontier is subjective. E. investors are separate beings and will therefore have different preferences regarding the riskreturn tradeoff.
6 The determination of the optimal risky portfolio is purely technical and can be done by a manager. The complete portfolio, which consists of the optimal risky portfolio and the riskfree asset, must be chosen by each investor based on preferences. 6. Which statement is not true regarding the market portfolio? A. It includes all publicly traded financial assets. B. It lies on the efficient frontier. C. All securities in the market portfolio are held in proportion to their market values. D. It is the tangency point between the capital market line and the indifference curve. E. it lies on a line that represents the expected riskreturn relationship. The tangency point between the capital market line and the indifference curve is the optimal portfolio for a particular investor. 7. Which statement is true regarding the market portfolio? A. It includes all publicly traded financial assets. B. It lies on the efficient frontier. C. All securities in the market portfolio are held in proportion to their market values. D. It is the tangency point between the capital market line and the indifference curve. E. It includes all publicly traded financial assets, lies on the efficient frontier, and all securities in the market portfolio are held in proportion to their market values. The tangency point between the capital market line and the indifference curve is the optimal portfolio for a particular investor. 8. According to the Capital Asset Pricing Model (CAPM), underpriced securities A. have positive betas. B. have zero alphas. C. have negative betas. D. have positive alphas. E. have negative alphas. According to the Capital Asset Pricing Model (CAPM), underpriced securities have positive alphas. 9. According to the Capital Asset Pricing Model (CAPM), which one of the following statements is false? A. The expected rate of return on a security increases in direct proportion to a decrease in the riskfree rate. B. The expected rate of return on a security increases as its beta increases. C. A fairly priced security has an alpha of zero. D. In equilibrium, all securities lie on the security market line. E. All of these are correct. The statement that the expected rate of return on a security increases in direct proportion to a decrease in the riskfree rate is false.
7 10. In a well diversified portfolio A. market risk is negligible. B. systematic risk is negligible. C. unsystematic risk is negligible. D. nondiversifiable risk is negligible. E. risk does not exist. Market, systematic, or nondiversifiable, risk is present in a diversified portfolio; the unsystematic risk has been eliminated. 11. Your personal opinion is that a security has an expected rate of return of It has a beta of 1.5. The riskfree rate is 0.05 and the market expected rate of return is According to the Capital Asset Pricing Model, this security is A. underpriced. B. overpriced. C. fairly priced. D. cannot be determined from data provided. E. can either be overpriced or underpriced but not fairly priced. 11% = 5% + 1.5(9%  5%) = 11.0%; therefore, the security is fairly priced. 12. You invest $600 in a security with a beta of 1.2 and $400 in another security with a beta of The beta of the resulting portfolio is A B C D E (1.2) + 0.4(0.90) = A security has an expected rate of return of 0.10 and a beta of 1.1. The market expected rate of return is 0.08 and the riskfree rate is The alpha of the stock is A. 1.7%. B. 1.7%. C. 8.3%. D. 5.5%. E. 5.5%. 10%  [5% + 1.1(8%  5%)] = 1.7%.
8 14. Given the following two stocks A and B If the expected market rate of return is 0.09 and the riskfree rate is 0.05, which security would be considered the better buy and why? A. A because it offers an expected excess return of 1.2%. B. B because it offers an expected excess return of 1.8%. C. A because it offers an expected excess return of 2.2%. D. B because it offers an expected return of 14%. E. B because it has a higher beta. A's excess return is expected to be 12%  [5% + 1.2(9%  5%)] = 2.2%. B's excess return is expected to be 14%  [5% + 1.8(9%  5%)] = 1.8%. 15. Standard deviation and beta both measure risk, but they are different in that A. beta measures both systematic and unsystematic risk. B. beta measures only systematic risk while standard deviation is a measure of total risk. C. beta measures only unsystematic risk while standard deviation is a measure of total risk. D. beta measures both systematic and unsystematic risk while standard deviation measures only systematic risk. E. beta measures total risk while standard deviation measures only nonsystematic risk. Standard deviation and beta both measure risk, but they are different in that beta measures only systematic risk while standard deviation is a measure of total risk. 16. The risk premium on the market portfolio will be proportional to A. the average degree of risk aversion of the investor population. B. the risk of the market portfolio as measured by its variance. C. the risk of the market portfolio as measured by its beta. D. both the average degree of risk aversion of the investor population and the risk of the market portfolio as measured by its variance. E. both the average degree of risk aversion of the investor population and the risk of the market portfolio as measured by its beta. The risk premium on the market portfolio is proportional to the average degree of risk aversion of the investor population and the risk of the market portfolio measured by its variance.
9 17. Assume that a security is fairly priced and has an expected rate of return of The market expected rate of return is 0.11 and the riskfree rate is The beta of the stock is. A B C. 1. D E % = [4% +β(11%  4%)]; 13% = β(7%); β = Discuss the assumptions of the capital asset pricing model, and how these assumptions relate to the "real world" investment decision process. The assumptions are: (a) The market is composed of many small investors, who are pricetakers; i. e., perfect competition. In reality this assumption was fairly realistic until recent years when institutional investors increasingly began to influence the market with their large transactions, especially those transactions via program trading. (b) All investors have the same holding period. Obviously, different investors have different goals, and thus have different holding periods. (c) Investments are limited to those that are publicly traded. In addition, it is assumed that investors may borrow or lend any amount at a fixed, riskfree rate. Obviously, investors may purchase assets that are not publicly traded; however, the dollar volume of publicly traded assets is considerable. The assumption that investors can borrow or lend any amount at a fixed, riskfree rate obviously is false. However, the model can be modified to incorporate different borrowing and lending rates. (d) Investors pay no taxes on returns and incur no transaction costs. Obviously, investors do pay taxes and do incur transaction costs. The tax differentials across different types of investment income and across different income levels have been lessened as a result of the income tax simplification of Obviously, investors should consider aftertax, not beforetax, returns; however, the notax assumption of the model is not a serious departure from reality. In addition, any investment vehicle should stand on its own merits, not its tax status (again, less of a problem with the tax simplification of 1986). Compared to other investment alternatives, such as real estate, transaction costs for securities are relatively low, unless the investor is an active trader. The active trader should be sure that he or she is not trading himself/herself out of a profit situation and into a loss situation and making profits for the broker. In general, these assumptions are not serious violations of "real world" scenarios. (e) All investors are meanvariance efficient. This assumption implies that all investors make decisions based on maximizing returns available at an acceptable risk level; most investors probably make decisions in this manner. However, some investors are pure wealth maximizers (regardless of the risk level); and other investors are so risk averse that avoiding risk is their only goal. (f) All investors have homogeneous expectations, meaning that given the same data all investors would process the data in the same manner, resulting in the same risk/return assessments for all investment alternatives. Obviously, we do not have homogenous expectations; one only has to read the differing recommendations of various analysts to realize that we have heterogeneous expectations. However, modeling heterogeneous expectations would require multiple, specific models; the homogenous expectations assumption allows the development of a generalized model, the CAPM.
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