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Unformatted text preview: FINS1613 BUSINESS FINANCE TUTORIAL WEEK 9 (Based on Lecture 8, RTBWJ Chapter 11) [1] Read and ponder on possible solutions to questions under CRITICAL THINKING AND CONCEPTS REVIEW 11.1, 11.3, 11.5, 11.7, 11.8 1. Some of the risk in holding any asset is unique to the asset in question. By investing in a variety of assets, this unsystematic portion of the total risk can be eliminated at little cost. On the other hand, there are systematic risks that affect all investments. This portion of the total risk of an asset cannot be costlessly eliminated. In other words, systematic risk can be controlled, but only by a costly reduction in expected returns. 3. a. systematic b. unsystematic c. both; probably mostly systematic d. unsystematic e. unsystematic f. systematic 5. No to both questions. The portfolio expected return is a weighted average of the asset returns, so it must be less than the largest asset return and greater than the smallest asset return. 7. Yes, the standard deviation can be less than that of every asset in the portfolio. However, portfolio beta cannot be less than the smallest beta because it is a weighted average of the individual asset betas. 8. Yes. It is possible, in theory, to construct a zero beta portfolio of risky assets whose return would be equal to the risk‐free rate. It is also possible to have a negative beta; the return would be less than the risk‐free rate. A negative beta asset would carry a negative risk premium because of its value as a diversification instrument. 1 [2] Solve these problems from Chapter 11 under QUESTIONS AND PROBLEM Q10, Q12, Q19, Q22, Q33 10. a. This portfolio does not have an equal weight in each asset. We first need to find the return of the portfolio in each state of the economy. To do this, we will multiply the return of each asset by its portfolio weight and then sum the products to get the portfolio return in each state of the economy. Doing so, we get: Boom: E(Rp) = .30(.30) + .40(.45) + .30(.33) E(Rp) = .3690 or 36.90% Good: E(Rp) = .30(.12) + .40(.10) + .30(.15) E(Rp) = .1210 or 12.10% Poor: E(Rp) = .30(.01) + .40(–.15) + .30(–.05) E(Rp) = –.0720 or –7.20% Bust: E(Rp) = .30(–.20) + .40(–.30) + .30(–.09) E(Rp) = –.2070 or –20.70% And the expected return of the portfolio is: E(Rp) = .15(.3690) + .45(.1210) + .35(–.0720) + .05(–.2070) E(Rp) = .0743 or 7.43% b. To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, and then sum. The result is the variance. So, the variance and standard deviation of the portfolio is: p2 = .15(.3690 – .0743)2 + .45(.1210 – .0743)2 + .35(–.0720 – .0743)2 + .05(–.2070 – .0743)2 p2 = .02546 p = (.02546)1/2 P = .1596 or 15.96% 12. The beta of a portfolio is the sum of the weight of each asset times the beta of each asset. If the portfolio is as risky as the market, it must have the same beta as the market. Since the beta of the market is one, we know the beta of our portfolio is one. We also need to remember that the beta of the risk‐free asset is zero. It has to be zero since the asset has no risk. Setting up the equation for the beta of our portfolio, we get: p = 1.0 = 1/3(0) + 1/3(1.65) + 1/3(X) Solving for the beta of Share X, we get: X = 1.35 19. There are two ways to correctly answer this question. We will work through both. First, we can use the CAPM. Substituting in the value we are given for each share, we find: E(RDingo) = .055 + .080(1.50) 2 E(RDingo) = .1750 or 17.50% It is given in the problem that the expected return of Dingo is 16 percent, but according to the CAPM, the return of the share based on its level of risk, the expected return should be 17.50 percent. This means the share return is too low, given its level of risk. Dingo’s share plots below the SML and is overvalued. In other words, its price must decrease to increase the expected return to 17.50 percent. For Shark, we find: E(RShark) = .055 + .080(0.70) E(RShark) = .1110 or 11.10% The return given for Shark is 11.5 percent, but according to the CAPM the expected return of the share should be 11.10 percent based on its level of risk. Shark’s share plots above the SML and is undervalued. In other words, its price must increase to decrease the expected return to 11.10 percent. We can also answer this question using the reward‐to‐risk ratio. All assets must have the same reward‐to‐risk ratio. The reward‐to‐risk ratio is the risk premium of the asset divided by its . We are given the market risk premium, and we know the of the market is one, so the reward‐to‐risk ratio for the market is 0.08, or 8 percent. Calculating the reward‐to‐risk ratio for Dingo, we find: Reward‐to‐risk ratio for Dingo = (.16 – .055) / 1.50 Reward‐to‐risk ratio for Dingo = .0700 The reward‐to‐risk ratio for Share Dingo is too low, which means the share plots below the SML, and the share is overvalued. Its price must decrease until its reward‐to‐risk ratio is equal to the market reward‐to‐risk ratio. For a Share in Shark, we find: Reward‐to‐risk ratio for Shark = (.115 – .055) / .70 Reward‐to‐risk ratio for Shark = .0857 The reward‐to‐risk ratio for Share Shark is too high, which means the share plots above the SML, and the share is undervalued. Its price must increase until its reward‐to‐risk ratio is equal to the market reward‐to‐risk ratio. 22. Here, we are given the expected return of the portfolio and the expected return of the assets in the portfolio and are asked to calculate the dollar amount of each asset in the portfolio. So, we need to find the weight of each asset in the portfolio. Since we know the total weight of the assets in the portfolio must equal 1 (or 100%), we can find the weight of each asset as: E[Rp] = .12 = .16wHomestead + .095(1 – wHomestead) wHomestead = 0.3846 wLimestone = 1 – wHomestead wLimestone = 1 – .3846 wLimestone = .6154 So, the dollar investment in each asset is the weight of the asset times the value of the portfolio, so the dollar investment in each asset must be: Investment in Homestead = 0.3846($250,000) Investment in Homestead = $96,153.85 Investment in Limestone = 0.6154($250,000) 3 Investment in Limestone = $153,846.15 33. a. We need to find the return of the portfolio in each state of the economy. To do this, we will
multiply the return of each asset by its portfolio weight and then sum the products to get the
portfolio return in each state of the economy. Doing so, we get:
Boom: E(Rp) = .40(.05) + .40(.25) + .20(.60)
E(Rp) = .2400 or 24.00%
Normal: E(Rp) = .40(.09) + .40(.12) + .20(.20)
E(Rp) = .1240 or 12.40%
Bust:
E(Rp) = .40(.12) + .40(–.13) + .20(–.40)
E(Rp) = –.0840 or –8.40%
And the expected return of the portfolio is:
E(Rp) = .25(.2400) + .55(.1240) + .20(–.0840)
E(Rp) = .1114 or 11.14%
To calculate the standard deviation, we first need to calculate the variance. To find the variance,
we find the squared deviations from the expected return. We then multiply each possible
squared deviation by its probability, and then sum. The result is the variance. So, the variance
and standard deviation of the portfolio are:
2p = .25(.2400 – .1114)2 + .55(.1240 – .1114)2 + .20(–.0840 – .1114)2
2p = .01186
p = (.01186)1/2
p = .1089 or 10.89%
b. The risk premium is the return of a risky asset, minus the riskfree rate, so:
RPi = E(Rp) – Rf
RPi = .1114 – .0425
RPi = .0689 or 6.89% 4 [3] Answer the following Multiple‐choice questions 1. Stock A has a beta of 1.5 and Stock B has a beta of 0.5. Which of the following statements must be true about these securities? (Assume the market is in equilibrium.) a. When held in isolation, Stock A has greater risk than Stock B. b. Stock B would be a more desirable addition to a portfolio than Stock A. c. Stock A would be a more desirable addition to a portfolio than Stock B. d. The expected return on Stock A will be greater than that on Stock B. e. The expected return on Stock B will be greater than that on Stock A. Answer is (d) 2. Stock X has a beta of 0.5 and Stock Y has a beta of 1.5. Which of the following statements is most correct? a. Stock Y's return this year will be higher than Stock X's return. b. Stock Y's return has a higher standard deviation than Stock X. c. If expected inflation increases (but the market risk premium is unchanged), the required returns on the two stocks will increase by the same amount. d. If the market risk premium declines (leaving the risk‐free rate unchanged), Stock X will have a larger decline in its required return than will Stock Y. e. If you invest $50,000 in Stock X and $50,000 in Stock Y, your portfolio will have a beta less than 1.0, provided the stock returns on the two stocks are not perfectly correlated. Answer is (cc). Statement a is false; Y has a higher required return because it is more risky, but it may still end up actually earning a lower return than X. Statement b is false; beta tells us about the covariance of the stock with the market. It tells us nothing about the stocks' individual standard deviations. Statement c is correct from the CAPM: ks = kRF + (kM ‐ kRF)b. Statement d is false from the CAPM. Statement e is false; the portfolio beta, bp, is calculated as (0.5 x 0.5) + (0.5 x 1.5) = 1.0. 3. Which of the following statements is most correct? a. If you add enough randomly selected stocks to a portfolio, you can completely eliminate all the market risk from the portfolio. b. If you formed a portfolio that included a large number of low‐beta stocks (stocks with betas less than 1.0 but greater than ‐1.0), the portfolio would itself have a beta coefficient that is equal to the weighted average beta of the stocks in the portfolio, so the portfolio would have a relatively low degree of risk. c. If you were restricted to investing in publicly traded common stocks, yet you wanted to minimize the riskiness of your portfolio as measured by its beta, then according to the CAPM theory you should invest some of your money in each stock in the market. That is, if there were 10,000 traded stocks in the world, the least risky portfolio would include some shares in each of them. d. Diversifiable risk can be eliminated by forming a large portfolio, but normally even highly‐
diversified portfolios are subject to market risk e. Statements b and d are correct. Answer is (e) 4. Stock A and Stock B both have an expected return of 10 percent and a standard deviation of 25 percent. Stock A has a beta of 0.8 and Stock B has a beta of 1.2. The correlation coefficient, r, between 5 the two stocks is 0.6. Portfolio P is a portfolio with 50 percent invested in Stock A and 50 percent invested in Stock B. Which of the following statements is most correct? a. Portfolio P has a coefficient of variation equal to 2.5. b. Portfolio P has more market risk than Stock A but less market risk than Stock B. c. Portfolio P has a standard deviation of 25 percent and a beta of 1.0. d. All of the statements above are correct. e. None of the statements above is correct. Answer is (b). This is a good question to get you thinking about the construction of a portfolio with respect to the stocks within that portfolio. It is worth taking some time over this and making sure you are clear on how the stocks will interact. The standard deviation of the portfolio will be less than the weighted average of the two stocks' standard deviations because the correlation coefficient is less than one. Therefore, although the expected return on the portfolio will be the weighted average of the two returns (10 percent), the CV will not be equal to 25%/10%. Therefore, statement (a) is false. Remember, market risk is measured by beta. The beta of the portfolio will be the weighted average of the two betas; therefore, it will be less than the beta of the high‐beta stock (B), but more than the beta of the low‐beta stock (A). Therefore, the market risk of the portfolio will be higher than A's, but lower than B's. Therefore, statement (c)b is correct. Because the correlation between the two stocks is less than one, the portfolio's standard deviation will be less than 25 percent. Therefore, statement (c) is false. 5. Stock A has a beta of 0.8, Stock B has a beta of 1.0, and Stock C has a beta of 1.2. Portfolio P has equal amounts invested in each of the three stocks. Each of the stocks has a standard deviation of 25 percent. The returns of the three stocks are independent of one another (i.e., the correlation coefficients all equal zero). Assume that there is an increase in the market risk premium, but that the risk‐free rate remains unchanged. Which of the following statements is most correct? a. The required return of all three stocks will increase by the amount of the increase in the market risk premium. b. The required return on Stock A will increase by less than the increase in the market risk premium, while the required return on Stock C will increase by more than the increase in the market risk premium. c. The required return of all stocks will remain unchanged since there was no change in their betas. d. The required return of the average stock will remain unchanged, but the returns of riskier stocks (such as Stock C) will decrease while the returns of safer stocks (such as Stock A) will increase. e. The required return of the average stock will remain unchanged, but the returns of riskier stocks (such as Stock C) will increase while the returns of safer stocks (such as Stock A) will decrease. Answer is (b). Another good question for helping you understand how stocks and portfolios interest and respond to changing circumstances. The correct answer is statement b. Remember, the market risk premium is the slope of the Security Market Line. This means high‐beta stocks experience greater increases in their required returns, while low‐beta stocks experience smaller increases in their required returns. Statement (a) is incorrect. Statement b is correct; stocks with a beta less than 1 increase by less than the increase in the market risk premium, and vice versa. Statement c is incorrect; since the market risk 6 premium is changing, required returns must change too. Statements d and e are incorrect for the same reason that statement (c) is incorrect. 7 ...
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This note was uploaded on 10/20/2011 for the course COMMERCE 3502 taught by Professor All during the One '11 term at University of New South Wales.
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