金融学第二版讲义大纲及课后习题答案详解十二章

CHAPTER 12

CHOOSING AN INVESTMENT PORTFOLIO

Objectives

« To un dersta nd the process of pers onal in vest ing in theory and in practice.

« To build a qua ntitative model of the tradeoff betwee n risk and reward.

Outline

The Process of Pers onal Portfolio Selecti on

The Trade-off betwee n Expected Retur n and Risk Efficie nt Diversificati on with Many Risky Assets

Summary

* There is no sin gle portfolio selecti on strategy that is best for all people.

« Stage in the life cycle is an imp orta nt determ inant of the optimal compositi on of a pers on

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 mak ing portfolio selecti on decisi ons, people can in gen eral achieve a higher expected rate of retur n only by expos ing themselves to greater risk. *

One can sometimes reduce risk without loweri ng expected return by diversify ing more completely either within a give n asset class or across asset classes. * The power of diversificati on to reduce the risk in ess of an in vestor ' scpw r efodti m depe nds on the

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 com mon econo mic factors. Con seque ntly, one

' s ability to red

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 fun ds, and real estate. In desig ning and produc ing the menu of assets to offer to their customers these

in termediaries make use of the latest adva nces in finan cial tech no logy.12.1 12.2 12.3 s optimal portfolio of

In structor s Ma nual Chapter 12 Page 144

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 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. 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 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 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. 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 becaeudsbeyitdiessdigivne:rist ihfias 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.

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 vari 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 ”

Or, maybe 40% 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.

b. The fact that Dad is employed by the Ruffy Company makes his situation more risky. Let 's say that the

hits 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 on unemployment. 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.

In structor s Ma nual Chapter 12 Page 145

a. Perform the 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 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:

d. An extremely risk tolera nt pers on would select portfolio S, which has the largest sta ndard deviati on but also the largest expected retur

n.

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 has exhibited an expected return of 20% (.20) and a standard deviation of .2

5.

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 should be placed in the money market fund if an investor desires an expected return of 15% (.15)?

SOLUTION:

a. E[r] = .045 + .62 :二

b. 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 about the risky asset vis-a-vis the riskless asset?

SOLUTION:

rewarded with less expected return for tak ing A trade-off line wit h a negative slope indicates that the investor is additional risk via

allocation to the risky asset.

In structor s Ma nual Chapter 12 Page 146

5. Suppose that you have the opportunity to buy stock in AT&T and Microsoft.

a. 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 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?

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