《⾦融学（第⼆版）》讲义⼤纲及课后习题答案详解⼗⼆章CHAPTER 12CHOOSING AN INVESTMENT PORTFOLIOObjectivesTo understand the process of personal investing in theory and in practice.To build a quantitative model of the tradeoff between risk and reward.Outline12.1 The Process of Personal Portfolio Selection12.2 The Trade-off between Expected Return and Risk12.3 Efficient Diversification with Many Risky AssetsSummaryThere is no single portfolio selection strategy that is best for all people.Stage in the life cycle is an imp ortant determinant of the optimal composition of a person’s optimal portfolio of assets and liabilities.Time horizons are important in portfolio selection. We distinguish among three time horizons: the planning horizon, the decision horizon, and the trading horizon.In making portfolio selection decisions, people can in general achieve a higher expected rate of return only by exposing themselves to greater risk.One can sometimes reduce risk without lowering expected return by diversifying more completely either withina given asset class or across asset classes.The power of diversification to reduce the riskiness of an investor’s portfolio depends on the correlations among the assets that make up the portfolio. In practice, the vast majority of assets are positively correlated with each other because they are all affected by common economic factors. Consequently, one’s ability to reduce risk through diversification among risky assets without lowering expected return is limited.Although in principle people have thousands of assets to choose from, in practice they make their choices from a menu of a few final products offered by financial intermediaries such as bank accounts, stock and bond mutual funds, and real estate. In designing and producing the menu of assets to offer to their customers theseintermediaries make use of the latest advances in financial technology.Solutions to Problems at End of Chapter1. Suppose that your 58-year-old father works for the Ruffy Stuffed Toy Company and has contributed regularly to his company-matched savings plan for the past 15 years. Ruffy contributes $0.50 for every $1.00 your father puts into the savings plan, up to the first 6% of his salary. Participants in the savings plan can allocate their contributions among four different investment choices: a fixed-income bond fund, a “blend” option that invests in large companies, small companies, and the fixed-income bond fund, a growth-income mutual fund whose investments do not include other toy companies, and a fund whose sole investment is stock in the Ruffy Stuffed Toy Company. Over Thanksgiving vacation, Dad realizes that you have been majoring in finance and decides to reap some early returns on that tuition money he’s been investing in your education. He shows you the most recent quarterly statement for his savings plan, and you see that 98% of its current value is in the fourth investment option, that of the Ruffy Company stock..a.Assume that your Dad is a typical risk-averse person who is considering retirement in five years. Whenyou ask him why he has made the allocation in this way, he responds that the company stock has continually performed quite well, except for a few declines that were caused by problems in a division that the company has long since sold off. Inaddition, he says, many of his friends at work have done the same. What advice would you give your dad about adjustments to his plan allocations? Why?b.If you consider the fact that your dad works for Ruffy in addition to his 98% allocation to the Ruffy stockfund, does this make his situation more risky, less risky, or does it make no difference? Why? SOLUTION:a.Dad has exposed himself to risk by concentrating almost all of his plan money in the Ruffy Stock fund. This is analogous to taking 100% of the money a family has put aside for investment and investing it in a single stock.First, Dad needs to be shown that just because the company stock has continually performed quite well is no guarantee that it will do so indefinitely. The company may have sold off the divisions which produced price declines in the past, but future problems are unpredictable, and so is the movement of the stock price. “Past performance is no guarantee of future results” is the lesson.Second, Dad needs to hear about diversification. He needs to be counseled that he can reduce his risk by allocating his money among several of the options available to him. Indeed, he can reduce his risk considerably merely by moving all of his money into the “blend” fund because it is diversifi ed by design: it has a fixed-income component, a large companies component, and a small companies component. Diversification isachieved not only via the three differing objectives of these components, but also via the numerous stocks that comprise each of the three components.Finally, Dad’s age and his retirement plans need to be considered. People nearing retirement age typically begin to shift the value of their portfolios into safer investments. “Safer” normally connotes less variability, so that the risk of a large decline in the value of a portfolio is reduced. This decline could come at any time, and it would be very unfortunate if it were to happen the day before Dad retires. In this example, the safest option would be the fixed-income bond fund because of its diversified composition and interest-bearing design, but there is still risk exposure to inflation and the level of interest rates. Note that the tax-deferred nature of the savings plan encourages allocation to something that produces interest or dividends. As it stands now, Dad is very exposed to a large decline in the value of his savings plan because it is dependent on the value of one stock.Individual equities over time have proven to produce the most variable of returns, so Dad should definitely move some, probably at least half, of his money out of the Ruffy stock fund. In fact, a good recommendation given his retirement horizon of five years would be to re-align the portfolio so that it has 50% in the fixed- income fund and the remaining 50% split between the Ruffy stock fund (since Dad insists) and the “blend” fund.Or, maybe 40% fixed-income, 25% Ruffy, 15% growth-income fund, and 20% “blend” fund. This latterallocation has the advantage of introducing another income-producing component that can be shielded by the tax-deferred status of the plan.b.The fact that Dad is employed by the Ruffy Company makes his situation more risky. Let’s say that the companyhits a period of slowed business activities. If the stock price declines, so will th e value of Dad’s savings plan. If the company encounters enough trouble, it may consider layoffs. Dad’s job may be in jeopardy. At the same time that his savings plan may be declining in value, Dad may also need to look for a job or go onunemployment. Thus, Dad is exposed on two fronts to the same risk. He has invested both his human capital and his wealth almost exclusively in one company.2. Refer to Table 12.1.a.Perform the calculations to verify that the expected returns of each of the portfolios (F, G, H, J, S) in thetable (column 4) are correct.b.Do the same for the standard deviations in column 5 of the table.c.Assume that you have $1million to invest. Allocate the money as indicated in the table for each of the fiveportfolios and calculate the expected dollar return of each of the portfolios.d.Which of the portfolios would someone who is extremely risk tolerant be most likely to select? SOLUTION:d.An extremely risk tolerant person would select portfolio S, which has the largest standard deviation but also thelargest expected return.3. A mutual fund company offers a safe money market fund whose current rate is4.50% (.045). The same company also offers an equity fund with an aggressive growth objective which historically has exhibited an expected return of 20% (.20) and a standard deviation of .25.a.Derive the equation for the risk-reward trade-off line.b.How much extra expected return would be available to an investor for each unit of extra risk that shebears?c.What allocation should be placed in the money market fund if an investor desires an expected return of15% (.15)?SOLUTION:a.E[r] = .045 + .62b.0.62c.32.3% [.15 = w*(.045) + (1-w)*(.020) ]4. If the risk-reward trade-off line for a riskless asset and a risky asset results in a negative slope, what does that imply about the risky asset vis-a-vis the riskless asset?SOLUTION:A trade-off line wit h a negative slope indicates that the investor is “rewarded” with less expected return for taking on additional risk via allocation to the risky asset.5. Suppose that you have the opportunity to buy stock in AT&T and Microsoft.a.stocks is 0? .5? 1? -1? What do you notice about the change in the allocations between AT&T andMicrosoft as their correlation moves from -1 to 0? to .5? to +1? Why might this be?b.What is the variance of each of the minimum-variance portfolios in part a?c.What is the optimal combination of these two securities in a portfolio for each value of the correlation,assuming the existence of a money market fund that currently pays 4.5% (.045)? Do you notice any relation between these weights and the weights for the minimum variance portfolios?d.What is the variance of each of the optimal portfolios?e.What is the expected return of each of the optimal portfolios?f.Derive the risk-reward trade-off line for the optimal portfolio when the correlation is .5. How much extraexpected return can you anticipate if you take on an extra unit of risk?SOLUTION:a.Minimum risk portfolios if correlation is:-1: 62.5% AT&T, 37.5% Microsoft0: 73.5% AT&T, 26.5% Microsoft.5: 92.1% AT&T, 7.9% Microsoft1: 250% AT&T, short sell 150% MicrosoftAs the correlation moves from -1 to +1, the allocation to AT&T increases. When two stocks have negativec orrelation, standard deviation can be reduced dramatically by mixing them in a portfolio. It is to the investors’benefit to weight more heavily the stock with the higher expected return since this will produce a high portfolio expected return while the standard deviation of the portfolio is decreased. This is why the highest allocation to Microsoft is observed for a correlation of -1, and the allocation to Microsoft decreases as the correlationbecomes positive and moves to +1. With correlation of +1, the returns of the two stocks will move closely together, so you want to weight most heavily the stock with the lower individual standard deviation.b. Variances of each of the minimum variance portfolios:62.5% AT&T, 37.5% Microsoft Var = 073.5% AT&T, 26.5% Microsoft Var = .016592.1% AT&T, 7.9% Microsoft Var = .0222250% AT&T, short 150% Microsoft Var = 0c. Optimal portfolios if correlation is:-1: 62.5% AT&T, 37.5% Microsoft0: 48.1% AT&T, 51.9% Microsoft.5: 11.4% AT&T, 88.6% Microsoft1: 250% AT&T, short 150% Microsoftd. Variances of the optimal portfolios:62.5% AT&T, 37.5% Microsoft Var = 048.1% AT&T, 51.9% Microsoft Var = .022011.4% AT&T, 88.6% Microsoft Var = .0531250% AT&T, short 150% Microsoft Var = 0e. Expected returns of the optimal portfolios:62.5% AT&T, 37.5% Microsoft E[r] = 14.13%48.1% AT&T, 51.9% Microsoft E[r] = 15.71%11.4% AT&T, 88.6% Microsoft E[r] = 19.75%250% AT&T, short 150% Microsoft E[r] = -6.5%f.Risk-reward trade-off line for optimal portfolio with correlation = .5:E[r] = .045 + .66/doc/31dbf23b580216fc700afd59.html ing the optimal portfolio of AT&T and Microsoft stock when the correlation of their price movements is 0.5, along with the results in part f of question 12-5, determine:a.the expected return and standard deviation of a portfolio which invests 100% in a money market fundreturning a current rate of 4.5%. Where is this point on the risk-reward trade-off line?b.the expected return and standard deviation of a portfolio which invests 90% in the money market fundand 10% in the portfolio of AT&T and Microsoft stock.c.the expected return and standard deviation of a portfolio which invests 25% in the money market fundand 75% in the portfolio of AT&T and Microsoft stock.d.the expected return and standard deviation of a portfolio which invests 0% in the money market fundand 100% in the portfolio of AT&T and Microsoft stock. What point is this?SOLUTION:a.E[r] = 4.5%, standard deviation = 0. This point is the intercept of the y (expected return) axis by the risk-rewardtrade-off line.b.E[r] = 6.03%, standard deviation = .0231c.E[r] = 15.9%, standard deviation = .173d.E[r] = 19.75%, standard deviation = .2306. This point is the tangency between the risk-reward line from 12-5part f and the risky asset risk-reward curve (frontier) for AT&T and Microsoft.7. Again using the optimal portfolio of AT&T and Microsoft stock when the correlation of their price movements is 0.5, take $ 10,000 and determine the allocations among the riskless asset, AT&T stock, and Microsoft stock for:a. a portfolio which invests 75% in a money market fund and 25% in the portfolio of AT&T and Microsoftstock. What is this portfolio’s expected return?b. a portfolio which invests 25% in a money market fund and 75% in the portfolio of AT&T and Microsoftstock. What is this portfolio’s expect ed return?c. a portfolio which invests nothing in a money market fund and 100% in the portfolio of AT&T andMicrosoft stock. What is this portfolio’s expected return?SOLUTION:a.$7,500 in the money-market fund, $285 in AT&T (11.4% of $2500), $2215 in Microsoft. E[r] = 8.31%, $831.b.$2,500 in the money-market fund, $855 in AT&T (11.4% of $7500), $6645 in Microsoft. E[r] = 15.94%, $1,594.c.$1140 in AT&T, $8860 in Microsoft. E[r] = 19.75%, $1,975.8. What strategy is implied by moving further out to the right on a risk-reward trade-off line beyond the tangency point between the line and the risky asset risk-reward curve? What type of an investor would be most likely to embark on this strategy? Why?SOLUTION:This strategy calls for borrowing additional funds and investing them in the optimal portfolio of AT&T and Microsoft stock. A risk-tolerant, aggressive investor would embark on this strategy. This person would be assuming the risk of the stock portfolio with no risk-free component; the money at risk is not onl y from this person’s own wealth but also represents a sum that isowed to some creditor (such as a margin account extended by the investor’s broker).9. Determine the correlation between price movements of stock A and B using the forecasts of their rate of return and the assessments of the possible states of the world in the following table. The standard deviations for stock A and stock B are0.065 and 0.1392, respectively. Before doing the calculation, form an expectation of whether that correlation will be closer to1 or -1 by merely inspecting the numbers.SOLUTION:Expectation: correlation will be closer to +1.E[r A] = .05*(-.02) + .15*(-.01) + .60*(.15) + .20*(.15) = .1175, or, 11.75%E[r B] = .05*(-.20) + .15*(-.10) + .60*(.15) + .20*(.30) = .1250, or, 12.50%Covariance = .05*(-.02-.1175)*(-.20-.125) + .15*(-.01-.1175)*(-.10-.125) +.60*(.15-.1175)*(.15-.125) + .20*(.15-.1175)*(.30-.125) =.008163Correlation = .008163/(.065)*(.1392) = .90210.Analyze the “expert’s” answers to the following questions:a.Question:I have approx. 1/3 of my investments in stocks, and the rest in a money market. What do you suggestas a somewhat “safer” place to invest another 1/3? I like to keep 1/3 accessible for emergencies.Expert’s answer:Well, you could try 1 or 2 year Treasury bonds. You’d get a little bit more yie ld with no risk.b.Question:Where would you invest if you were to start today?Expert’s answer:That depends on your age and short-term goals. If you are very young – say under 40 –and don’tneed the money you’re investing for a home or college tuition or such, you would put it in a stockfund. Even if the market tanks, you have time to recoup. And, so far, nothing has beaten stocks overa period of 10 years or more. But if you are going to need money fairly soon, for a home or for yourretirement, you need to play it safer.SOLUTION:a.You are not getting a little bit more yield with no risk. The real value of the bond payoff is subject to inflationrisk. In addition, if you ever need to sell the Treasury bonds before expiration, you are subject to the fluctuation of selling price caused by interest risk.b.The expert is right in pointing out that your investment decision depends on your age and short-term goals. In addition, the investment decision also depends on other characteristics of the investor, such as the special character of the labor income (whether it is highly correlated with the stock market or not), and risk tolerance.Also, the fact that over any period of 10 years or more the stock beats everything else cannot be used to predict the future.。
