CHAPTER 12
CHOOSING AN INVESTMENT PORTFOLIO


Objectives
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 Selection
12.2 The Trade-off between Expected Return and Risk
12.3 Efficient Diversification with Many Risky Assets
Summary
 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 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 within
a 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 these
intermediaries make use of the latest advances in financial technology.
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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. “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 diversified by design: it has a fixedincome 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 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 fixedincome 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 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 company
hits a period of slowed business activities. If the stock price declines, so will the 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.
Instructor’s Manual
a.
b.
c.
d.
d.
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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.
Do the same for the standard deviations in column 5 of the table.
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.
Which of the portfolios would someone who is extremely risk tolerant be most likely to select?
SOLUTION:
a.–c.
Portfolio
F
G
H
J
S
Chapter 12
Expected Return
.00*(.14) + 1.0*(.06) = .06
.25*(.14) + .75*(.06) = .08
.50*(.14) + .50*(.06) = .10
.75*(.14) + .25*(.06) = .12
1.0*(.14) + .00*(.06) = .14
Standard Deviation
0*(.20) = 0
.25*(.20) = .05
.50*(.20) = .10
.75*(.20) = .15
1.0*(.20) = .20
Expected Dollar
Return
of $1million
investment
$60,000
$80,000
$100,000
$120,000
$140,000
An extremely risk tolerant person would select portfolio S, which has the largest standard deviation but also the
largest expected return.
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 .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 should be placed in the money market fund if an investor desires an expected return of
15% (.15)?
SOLUTION:
a. E[r] = .045 + .62 
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:
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.
Instructor’s Manual
5.
a.
b.
c.
d.
e.
f.
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Suppose that you have the opportunity to buy stock in AT&T and Microsoft.
AT&T
Microsoft
Mean
.10
.21
Standard Deviation
.15
.25
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?
What is the variance of each of the minimum-variance portfolios in part a?
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?
What is the variance of each of the optimal portfolios?
What is the expected return of each of the optimal portfolios?
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% Microsoft
0:
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 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 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.
b. Variances of each of the minimum variance portfolios:
62.5% AT&T, 37.5% Microsoft
Var = 0
73.5% AT&T, 26.5% Microsoft
Var = .0165
92.1% AT&T, 7.9% Microsoft
Var = .0222
250% AT&T, short 150% Microsoft
Var = 0
c. 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% Microsoft
d. Variances of the optimal portfolios:
62.5% AT&T, 37.5% Microsoft
Var = 0
48.1% AT&T, 51.9% Microsoft
Var = .0220
11.4% AT&T, 88.6% Microsoft
Var = .0531
250% AT&T, short 150% Microsoft
Var = 0
e. 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 
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6. 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 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 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-reward
trade-off line.
b. E[r] = 6.03%, standard deviation = .0231
c. E[r] = 15.9%, standard deviation = .173
d. E[r] = 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 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 Microsoft
stock. 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 Microsoft
stock. What is this portfolio’s expected return?
c. a portfolio which invests nothing in a money market fund and 100% in the portfolio 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 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 only from this person’s own
wealth but also represents a sum that is owed to some creditor (such as a margin account extended by the investor’s
broker).
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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 are 0.065 and 0.1392, respectively. Before doing the calculation, form an expectation
of whether that correlation will be closer to 1 or -1 by merely inspecting the numbers.
State of the
Economy
Moderate recession
Slight recession
2% growth
3% growth
Probability
.05
.15
.60
.20
Stock A:
Rate of Return
-.02
-.01
.15
.15
Stock B:
Rate of Return
-.20
-.10
.15
.30
SOLUTION:
Expectation: correlation will be closer to +1.
E[rA] = .05*(-.02) + .15*(-.01) + .60*(.15) + .20*(.15) = .1175, or, 11.75%
E[rB] = .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. 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 suggest
as 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 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’re investing 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 you are 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 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.