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1、CHAPTER 12CHOOSING AN INVESTMENT PORTFOLIOObjectives To understand the process of personal investing in theory and in practice. To build a quantitative model of the tradeoff between risk and reward.Outline12.1The Process of Personal Portfolio Selection12.2The Trade-off between Expected Return and Ri
2、sk12.3Efficient Diversification with Many Risky AssetsSummary There is no single portfolio selection strategy that is best for all people. Stage in the life cycle is an important determinant of the optimal composition of a persons optimal portfolio of assets and liabilities. Time horizons are import
3、ant 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 some
4、times reduce risk without lowering expected return by diversifying more completely either within a given asset class or across asset classes. The power of diversification to reduce the riskiness of an investors portfolio depends on the correlations among the assets that make up the portfolio. In pra
5、ctice, the vast majority of assets are positively correlated with each other because they are all affected by common economic factors. Consequently, ones ability to reduce risk through diversification among risky assets without lowering expected return is limited. Although in principle people have t
6、housands 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 these intermed
7、iaries 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 e
8、very $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-i
9、ncome 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 tha
10、t tuition money hes 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 co
11、nsidering retirement in five years. When you 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. In addition, he say
12、s, 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 stock fund, does this make his situation more risky, less ris
13、ky, 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. 父亲将自己完全置身于冒险中，通过将几乎所有的计划资金
14、投资于Ruffy存货基金。这就相当于将一个家庭所有的金钱100%的用于投资且是单一的股票投资。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 prob
15、lems are unpredictable, and so is the movement of the stock price. “Past performance is no guarantee of future results” is the lesson.首先，父亲需要被展示这个公司的股票持续良好的态势，并且（is no guarantee that it will do so indefinitely）这个公司在过去也许已经出售完价格下跌的部分，但是未来的问题是不确定的，股票接个的变动也如此。“过去的表现并不能保证未来的结果。”就是讲述这样的道理。Second, Dad need
16、s 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 diversified by design: it has
17、 a fixed-income component, a large companies component, and a small companies component. Diversification is achieved not only via the three differing objectives of these components, but also via the numerous stocks that comprise each of the three components.其次，父亲必须知道有关多样化的信息。他需要被告知，他可以通过将金钱在对他而言可得到的
18、多种选择中进行分配来分散风险。他可以减少风险，通过移动他的金钱到“混合”的基金中，因为这样就保证了计划的多样性：包括固定收入部分，一家大的公司部分和几家小的公司部分。多样化不仅通过不同的三个组成部分得到实现，还可以通过组成这三个部分的众多股份得到实现。Finally, Dads 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.
19、 “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
20、 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.最后，父亲的年龄以及他的退休计划需要被考虑进去。接近退休年龄的人们都开始将资金倾向于更安全的投资。“更安全”通常意味着更少的可变性，以至于投资价值较大减少这样的风险可以被减小。这样的下降可以在任何时间到来，并且如果这一天在父亲退休前到来的话会非常的不幸。在这个实例中，最安全的选择是固定
21、收入债券基金因为多样的组成部分和（interest-bearing ）方面的设计，但是仍然存在通货膨胀和利率水平的风险。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
22、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.注意到储蓄计划的延期征税特性鼓励了倾向于利息和股息的分配。立足于此，父亲很容易遭受储蓄计划价值的大幅度下降因为这仅仅依赖于一种股票。随着时间流逝，个人的资产净值将产生不定的回报，
23、因而父亲必然需要将一些，很可能是至少一半的金钱移到Ruffy存货基金。 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 4
24、0% fixed-income, 25% Ruffy, 15% growth-income fund, and 20% “blend” fund. This latter allocation has the advantage of introducing another income-producing component that can be shielded by the tax-deferred status of the plan.事实上，一个好的建议是在他五年的退休范围内将个人投资进行重组，以达到他有505的固定收入基金，剩下的50%在Guffy存货基金（既然父亲坚持）和“混合
25、”基金之间进行分配。或者说，大约405的固定收入，25%的Ruffy基金，15%的收入增长基金，还有20%的“混合”基金。后一种分配拥有引入另一种增加收入的成分的优势，来抵御计划中延期税收的情形。b. The fact that Dad is employed by the Ruffy Company makes his situation more risky. Lets say that the company hits a period of slowed business activities. If the stock price declines, so will the valu
26、e of Dads savings plan. If the company encounters enough trouble, it may consider layoffs. Dads 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 on unemployment. Thus, Dad is exposed on two fronts to the same risk. He
27、 has invested both his human capital and his wealth almost exclusively in one company.事实是父亲被Ruffy公司雇佣使得他的情形更加的冒险。即我们所说的这家公司对迟钝的商业活动进行采样。如果股票的价格下降，父亲的储蓄计划的价值也会下降。如果这家公司遭遇了困境，它将会考虑到解雇。父亲的工作将会面临危险。同时他的储蓄计划也会价值下降，父亲可能会再找一份工作或者是继续失业。因而，父亲面临着两个冒险。他将他的人力资本和财富都专一的投资进了同一家公司。2.Refer to Table 12.1.a. Perform t
28、he calculations to verify that the expected returns of each of the portfolios (F, G, H, J, S) in the table (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
29、the five portfolios 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:a.c.PortfolioExpected Return Standard DeviationExpected Dollar Returnof $1million investmentF.00*(.14) + 1.
30、0*(.06) = .060*(.20) = 0$60,000G.25*(.14) + .75*(.06) = .08.25*(.20) = .05$80,000H.50*(.14) + .50*(.06) = .10.50*(.20) = .10$100,000J.75*(.14) + .25*(.06) = .12.75*(.20) = .15$120,000S1.0*(.14) + .00*(.06) = .141.0*(.20) = .20$140,000d.e.f.g.h.i.j.k.l.m.d An extremely risk tolerant person would sele
31、ct portfolio S, which has the largest standard deviation but also the largest expected return. 一个完全的风险爱好者会选择投资S，这项投资有最大的标准差，同时也有最大的期望收益。3. A mutual fund company offers a safe money market fund whose current rate is 4.50% (.045). The same company also offers an equity fund with an aggressive growth o
32、bjective 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 she bears?c. What allocation shoul
33、d be placed in the money market fund if an investor desires an expected return of 15% (.15)?SOLUTION:a. Er = .045 + .62 sb. 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
34、 risky asset vis-a-vis the riskless asset?SOLUTION:A trade-off line with a negative slope indicates that the investor is “rewarded” with less expected return for taking on additional risk via allocation to the risky asset.一条斜率倾斜的交易线表明了这个投资者将面临较少的期望收益因为倾向于冒险的资产分配。5.Suppose that you have the opportuni
35、ty to buy stock in AT&T and Microsoft. AT&TMicrosoftMean.10.21Standard Deviation.15.25a. What is the minimum risk (variance) portfolio of AT&T and Microsoft if the correlation between the two stocks is 0? .5? 1? -1? What do you notice about the change in the allocations between AT&T and Microsoft as
36、 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 fu
37、nd 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
38、 for the optimal portfolio when the correlation is .5. How much extra expected 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% Microsoft 0:73.5% AT&T, 26.5% Microsoft.5:92.1% AT&T, 7.9% Microsoft 1:250% AT&T, s
39、hort sell 150% MicrosoftAs the correlation moves from -1 to +1, the allocation to AT&T increases. When two stocks have negative correlation, 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 e
40、xpected 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 correlation becomes positive and m
41、oves 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.随着相关性从-1移动到+1，投资于AT-T的分额增加。当两支股票呈现负相关性时，将他们进行混合投资组合会使得标准差大幅度的降低。这对于期待较高的预期收益的投资者是有益的如果他加重投资的分量因为预期收益会随着标准差的降低而增加。这就是
42、为什么将最高的分额投资到微软时相关性为-1，然后投资进微软的分额随着相关性有负数移向+1而减少。在相关性为+1的时候，两种股票所带来的收益是接近相同的，所以你希望将绝大部分投资于自身标准差较低的股票。b.Variances of each of the minimum variance portfolios:62.5% AT&T, 37.5% MicrosoftVar = 073.5% AT&T, 26.5% MicrosoftVar = .016592.1% AT&T, 7.9% MicrosoftVar = .0222250% AT&T, short 150% MicrosoftVar =
43、 0c.Optimal portfolios if correlation is:-1:62.5% AT&T, 37.5% Microsoft 0:48.1% AT&T, 51.9% Microsoft.5:11.4% AT&T, 88.6% Microsoft 1:250% AT&T, short 150% Microsoftd.Variances of the optimal portfolios:62.5% AT&T, 37.5% MicrosoftVar = 048.1% AT&T, 51.9% MicrosoftVar = .022011.4% AT&T, 88.6% Microso
44、ftVar = .0531250% AT&T, short 150% MicrosoftVar = 0e.Expected returns of the optimal portfolios:62.5% AT&T, 37.5% MicrosoftEr = 14.13% 48.1% AT&T, 51.9% MicrosoftEr = 15.71%11.4% AT&T, 88.6% MicrosoftEr = 19.75%250% AT&T, short 150% MicrosoftEr = -6.5%f. Risk-reward trade-off line for optimal portfo
45、lio with correlation = .5:Er = .045 + .66 s6. Using 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
46、a money market fund returning 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 fund and 10% in the portfolio of AT&T and Microsoft stock.c. the expected return and standard
47、 deviation of a portfolio which invests 25% in the money market fund and 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 fund and 100% in the portfolio of AT&T and Microsoft stock. What point is this?S
48、OLUTION:a. Er = 4.5%, standard deviation = 0. This point is the intercept of the y (expected return) axis by the risk-reward trade-off line.b. Er = 6.03%, standard deviation = .0231c. Er = 15.9%, standard deviation = .173d. Er = 19.75%, standard deviation = .2306. This point is the tangency between the risk-reward line from 12-5 part 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 correl
