CHAPTER 12
CHOOSING 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.Outline
12.1 The Process of Personal Portfolio Selection12.2 The Trade-off between Expected Return and Risk12.3 Efficient 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 person’s optimal portfolio of assets andliabilities.
Time horizons are important in portfolio selection. We distinguish among three time horizons: the planning horizon, thedecision horizon, and the tradinghorizon.
In making portfolio selection decisions, people can in general achieve a higher expected rate of return only by exposingthemselves to greater risk.
One can sometimes reduce risk without lowering expected return by diversifying more completely either within a givenasset class or across asset classes.
The power of diversification to reduce the riskiness of an investor’s portfolio depends on the correlations among the assetsthat make up the portfolio. In practice, the vast majority of assets are positively correlated with each other because they areall 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 afew final products offered by financial intermediaries such as bank accounts, stock and bond mutual funds, and real estate. Indesigning and producing the menu of assets to offer to their customers these intermediaries make use of the latest advancesin financial technology.
Solutions to Problems at End of Chapter
1. 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 savingsplan, up to the first 6% of his salary. Participants in the savings plan can allocate their contributions among four differentinvestment choices: a fixed-income bond fund, a “blend” option that invests in large companies, small companies, and thefixed-income bond fund, a growth-income mutual fund whose investments do not include other toy companies, and a fundwhose sole investment is stock in the Ruffy Stuffed Toy Company. Over Thanksgiving vacation, Dad realizes that you havebeen 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 thefourth 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. When you ask him why he has made the allocation in this way, he responds that the company stockhas continually performed quite well, except for a few declines that were caused by problems in a division that the companyhas long since sold off. In addition, 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 stock fund, 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 moneyin the Ruffy Stock fund. This is analogous to taking 100% of the money afamily 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 thatit 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 guaranteeof 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 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 hismoney into the “blend” fund because it is diversified by design: it has a fixed-income component, a large companies
component, and a small companies component. Diversification is achieved not only via the three differing objectives of thesecomponents, 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, Dad’s age and his retirement plans need to be considered. People nearing retirement age typically begin to shift thevalue of their portfolios into safer investments. “Safer” normally connotes less variability, so that the risk of a large decline inthe value of a portfolio is reduced. This decline could come at any time, and it would be very unfortunate if it were to happenthe day before Dad retires. In this example, the safest option would be the fixed-income bond fund because of its diversifiedcomposition 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 thevalue 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, soDad 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，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 40% fixed-income, 25% Ruffy, 15% growth-income fund, and 20% “blend” fund. This latter allocation has theadvantage of intro ducing
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”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. Let’s say that the company hits a period of slowed business activities. If the stock price declines, so will the value ofDad’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 wealthalmost 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 the calculations to verify that the expected returns of each of theportfolios (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 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 bemost likely to select?SOLUTION:
d.e.f.g.h.i.j.
k.l.m.
d An extremely risk tolerant person would select 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 objective which historically hasexhibited 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 foreach unit of extra risk that she bears?
c.What allocation should be placed in the money market fund if an investordesires an expected return of 15% (.15)?SOLUTION:a.E[r] = .045 + .62b.0.62
c.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 aboutthe 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.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 opportunity to buy stock in AT&T and Microsoft.
a.
the correlation between the two stocks is 0? .5? 1? -1? What do you notice about the change in the allocations betweenAT&T and Microsoft 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 noticeany 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 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% Microsoft0: 73.5% AT&T, 26.5% Microsoft.5: 92.1% AT&T, 7.9% Microsoft
1: 250% AT&T, short sell 150% Microsoft
As 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 isto the investors’ benefit to
weight more heavily the stock with the higher expected return since this will produce a high portfolio expected return whilethe 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 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.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% 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
http://www.doczj.com/doc/615882588102d276a20029bd783e09127dbe.html ing the optimal portfolio of AT&T andMicrosoft 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 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 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 andMicrosoft 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-reward trade-off line.b.E[r] = 6.03%, standard deviation = .0231c.E[r] = 15.9%, standard deviation = .173
d.E[r] = 19.75%, standard deviation = .2306. This point is the tangency betweenthe 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 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 whichinvests 75% in a money market fund and 25% in the
portfolio of AT&T and Microsoft stock. What is this portfolio’s expected return?b. a portfolio which invests 25% in a money market fund and 75% in theportfolio of AT&T and Microsoft stock. What is this portfolio’s expected return?c. a portfolio which invests nothing in a money market fund and 100% in theportfolio of AT&T and Microsoft 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 inMicrosoft. E[r] = 8.31%, $831.
b.$2,500 in the money-market fund, $855 in AT&T (11.4% of $7500), $65 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 pointbetween the line and the risky asset risk-reward curve? What type of an investor would be most likely to embark on thisstrategy? Why?SOLUTION:
This strategy calls for borrowing additional funds and investing them in the optimal portfolio of AT&T and Microsoft stock. Arisk-tolerant, aggressive investor would embark on this strategy. This person would be assuming the risk of the stock portfoliowith no risk-free component; the money at risk is not only 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).
这个策略需要借进额外的基⾦并将它们投资于最理想的AT-T和微软股票的组合。⼀个承受风险并且有闯劲的投资者会从事这项投资。这个⼈将会接受这个股票组合所带来的风险并且不理会没有风险的投资组合；处于风险之中的⾦钱不仅来⾃于他的个⼈财富，还代表了⼀部分⾪属于债券⼈的数⽬（正如被投资者的经纪⼈扩⼤或延长的证券交易中的顾客保证⾦户）。9. Determine the correlation between price movements of stock A and B using the forecasts of their rate of return and theassessments 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.相关性将接近于+1E[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) =.008163
Correlation = .008163/(.065)*(.1392) = .902
10.A nalyze 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 moneymarket. What do you suggest as a somewhat “safer” place to investanother 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 yield 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’t need the money you’reinvesting for a home or college tuition or such, you would put it in a stock fund. Even if the
market tanks, you have time to recoup. And, so far, nothing has beaten stocks over a period of 10 years or more. But if youare going to need
money fairly soon, for a home or for your retirement, 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 inflation risk. In addition, if you ever need to sell the Treasury bonds before expiration, you aresubject to the fluctuation of selling price caused by interest risk.
你将⽆法得到更多的收益如果没有冒险的话。债券收益的真实价值会遭受通货膨胀的风险。另外，如果你在到期之前必须卖出你的长期国债，你将可能遭受由于利息风险引起的卖价的波动。
b.The expert is right in pointing out that your investment decision depends onyour 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 highlycorrelated with the stock market or not), and risk tolerance.
Also, the fact that over any period of 10 years or more the stock beatseverything else cannot be used to predict the future.
这位专家是正确的，他指出了你的投资决策应该依赖于你的年龄和短期⽬标。另外，投资决策也取决于投资者的其他因素，正如劳动收⼊的特性（⽆论它是否与股票市场有相当的联系），以及风险承受⼒。当然，事实上在10年或者更长的时间内股票胜出并不能预⽰未来。
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