OBJECTEVES
To understand the process of personal portfolio selection in theory and in practice.
· To build a quantitative model of the trade-off between risk and reward .
·
CONTENTS
12.1
12.2
The Process of Personal Portfolio Selection
The Trade-Off between Expected Return and Risk
12.3
Efficient Diversification with Many Risky Assets
This chapter is about how people should invest their wealth, a process called portfolio selection. A person’s wealth
portfolio includes all of his or her assets (stocks, bonds, shares in unincorporated businesses, houses or apartments,
pension benefits, insurance policies, etc.) and all of his or her liabilities (student loans, auto loans, home mortgages,
etc.).
There is no single portfolio selection strategy that is best for all people. There are, however, some general
principles, such as the principle of diversification that apply to all risk-averse people. In chapter 11 we discussed
diversification as a method of managing risk. This chapter extends that discussion and analyzes the quantitative
trade-off between risk and expected return.
Section 12.1 examines the role of portfolio selection in the context of a person’s life-cycle financial planning
process and shows why there is no single strategy that is best for all people. It also examines how the investor’s time
horizon and risk tolerance affect portfolio selection. Section 12.2 analyzes the choice between a single risky asset and a
riskless asset, and Section 12.3 examines optimal portfolio selection with many risky assets.
12.1 THE PROCESS OF PERSONAL PORTFOLIO SELECTION
Portfolio selection is the study of how people should invest their wealth. It is a process of trading off risk and expected
return to find the best portfolio of assets and liabilities. A narrow definition of portfolio selection includes only
decisions about how much to invest in stocks, bonds, and other securities. A broader definition includes decisions about
whether to buy or rent one's house, what types and amounts of insurance to purchase, and how to manage one’s
liabilities. An even broader definition includes decisions about how much to invest m one’s human capital (e.g., by
furthering one’s professional education). The common element in all of these decisions is the trade-off between risk
and expected return.
This chapter is devoted to exploring the concepts and techniques you need to know to evaluate risk-reward
trade-offs and to manage your wealth portfolio efficiency. A major theme is that although there arc some general rules
for portfolio selection that apply to virtually everyone, there is no single portfolio or portfolio strategy that is best for
everyone. We begin by explaining why.
12.1.1 The Life Cycle
In portfolio selection the best strategy depends on an individual's personal circum- stances (age, family status,
occupation, income, wealth, etc.). For some people holding a particular asset may add to their total risk exposure, but
for others the same asset may be risk reducing. An asset that is risk reducing at an early stage in the life cycle may not
be at a later stage.
For a young couple starting a family it may be optimal to buy a house and take out a mortgage loan. For an older
couple about to retire it may be optimal to sell their house and invest the proceeds in some asset that will provide a
steady stream of income for as long as they live.
Consider the purchase of life insurance. The optimal insurance policy for Miriam, a parent with dependent
children, will differ from the policy appropriate for Sanjiv, a single person with no dependents, even if the two people
are the same in all other respects (age, income, occupation, wealth, etc.). Miriam would be concerned about protecting
her family in the event of her death and would, therefore, want a policy that provides cash benefits payable to her
children upon her death. Sanjiv, on the other hand, would not be concerned about benefits payable if he dies; therefore,
the purchase of life insurance would not be risk reducing for him. At a later stage in her life, Miriam too may find that
her children can provide for them- selves and no longer need the protection afforded by life insurance.
Now consider the situation of Miriam and Sanjiv after they reach retirement age. Miriam has children and is
happy to have them inherit any assets that are left after she dies-If she should live an extraordinarily long time and
exhaust her own wealth, she is confident her children will provide financial support for her.
Sanjiv is a loner with no one to whom he cares to leave a bequest. He would like to consume all of his wealth
during his own lifetime but is concerned that if he increases his spending he will exhaust his wealth if he
happens to live an extraordinarily long time. For Sanjiv buying an insurance policy that guarantees him an income
for as long as he lives would be risk reducing; for Miriam it would not be. Such an insurance policy is
called a life annuityAs these examples make clear, even people of the same age, with the same income and wealth, may
have different perspectives on buying a house or buying insurance. The same is true of investing in stocks, bonds, and
other securities. There is no single portfolio that is best for all people.
To see this, consider two different individuals of the same age and family status. Chang is 30 years old and
works as a security analyst on Wall Street. His current and future earnings are very sensitive to the performance of the
stock market. Obi is also 30 years old and teaches English in the public school system. Her current and future earnings
are not very sensitive to the stock market. For Chang investing a significant proportion of his investment portfolio in
stocks would be z=t risky than it would be for Obi.
12.1.2 Time Horizons
In formulating a plan for portfolio selection you begin by determining you goals and time horizons. The planning
horizon is the total length of time for which one plans.
The longest time horizon would typically correspond to the retirement goal and would be the balance of one’s
lifetime. Thus, for a 25-year-old who expects live to age 85, the planning horizon would be 60 years. As one ages, the
planning horizon typically gets shorter and shorter (see Box 12.1).
There are also shorter planning horizons that correspond to specific financial goals, such as paying for a child's
education. For example, if you have a child who is three years old and plan to pay for her college education when she
reaches age 18, the planning horizon for this goal is 15years.
The decision horizon is the length of time between decisions to revise the portfolio. The length of the decision
horizon is controlled by the individual within certain limits.
Some people review their portfolios at regular intervals, for example, once a month (when they pay their bills),
or once a year (when they file income tax forms ). People of modest means with most of their wealth invested in bank
accounts might review their portfolios very infrequently and at irregular intervals determined by some triggering event
such as getting married or divorced, having a child, or receiving a bequest. A sudden rise or fall in the price of an asset
a person owns might also trigger a review of the portfolio.
People with substantial investments in stocks and bonds might review their portfolios every day or even more
frequently. The shortest possible decision horizon is the trading horizon, defined as the minimum time interval over
which investors can revise their portfolios.
The length of the trading horizon is not under the control of the individual. Whether the trading horizon is a
week, a day, an hour, or a minute is determined by the structure of the markets in the economy (e.g., when the
securities exchanges are open or whether organized off---exchange markets exist).
In today's global financial environment trading in many securities can be carried on somewhere on the globe
around the clock. For these securities at least the trading horizon is very short.
Portfolio decisions you make today are influenced by what you think might happen tomorrow. A plan that takes
account of future decisions in making current decisions is called a strategy.
How frequently investors can revise their portfolios by buying or selling securities is an important consideration
in formulating investment strategies. If you know that you can adjust the composition of your portfolio frequently, you
may invest differently than if you cannot adjust it.
For example, a person may adopt a strategy of investing "extra" wealth in stocks, meaning wealth in excess of
the amount needed to insure a certain the old standard of living. If the stock market goes up over time, a person will
increase the proportion of his or her portfolio invested in stocks. However, if the stock market goes down, a person will
reduce the proportion invested in stocks. If the stock market fails to the point at which the person's threshold standard
of living is threatened, he or she will get out of stocks altogether. An investor pursuing this particular strategy is more
likely to have a higher threshold if stocks can only be infrequently.
12.1.3 Risk Tolerance
A person's tolerance for bearing risk is a major determinant of portfolio choices. We expect risk tolerance to be
influenced by such characteristics as age, family status, job status, wealth, and other attributes that affect people's
ability to maintain their standard of living in the face of adverse movements in the market value of their investment
portfolio. One's attitude towards risk also plays a role in determining a person's tolerance for bearing risk. Even among
people with the same apparent personal, family, and job characteristics, some may have a greater willingness to take
risk than others.
When we refer to a person's risk tolerance in our analysis of optimal portfolio selection, we do not distinguish
between capacity to bear risk and attitude toward risk. Thus, whether a person has a relatively high tolerance for risk
because he is young or rich, because he handles stress we11, or because he was brought up to believe that taking
chances is the morally right path, all that matters in the analysis to follow is that he is more willing than the average
person to take on additional risk to achieve a higher expected return.
12.1.4 The Role of Professional Asset Managers
Most people have neither the knowledge nor the time to carry out portfolio optimization. Therefore they hire an
investment advisor to do it for them or they buy a "finished product” from a financial intermediary. Such finished
products include various investment accounts and mutual funds offered by banks, securities a vestment companies, and
insurance companies.
When financial intermediaries decide what asset choices to offer to households, they are in a position analogous
to a restaurant deciding on its menu. There are many ingredients available (the basic stocks, bonds, and other securities
issued by firms and governments) and an infinite number of possible ways to combine them, but only a limited number
of items will be offered to customers. The portfolio theory developed in the rest of this chapter offers some guidance in
finding the least number of items to offer that still cover the full array of customer demands.
12.2 THE TRADE-OFF BETWEEN EXPECIED REIURN AND RISK
The next two sections present the analytical framework used by professional port- folio managers for
examining the quantitative trade-off between risk and expected return. The objective is to find the portfolio
that offers investors the highest expected rate of return for any degree of risk they arc willing to tolerate.
Throughout the analysis we will refer to risky assets without specifically identifying them as bonds, stocks,
options, insurance policies, and so on. This is because, as explained in the preceding sections of this chapter,
the riskiness of a particular asset depends critically on the specific circumstances of the investor.
Portfolio optimization is often done as a two-step process: (1) Find the optimal combination of risky
assets, and (2) mix this optimal risky-asset portfolio with the riskless asset. For simplicity, we start with the
second step: mixing a single risky-asset portfolio and a riskless asset. (We discuss the identity of the
riskless asset in the next section.) The single risky-asset portfolio is composed of many risky assets chosen
in an optimal way. In section 12.3.4we investigate how the optimal composition of this risky-asset portfolio
is found.
12.2.I What Is the Riskless Asset?
In chapter 4 we discussed interest rates and showed that there is a different riskless asset that corresponds to each
possible unit of account (dollars, yen, etc.) and to each possible maturity. Thus, a 10-year, dollar-denominated,
zero-coupon bond that offers a default-free yield-to-maturity of6%per year is riskless only in terms of dollars and only
if held to maturity. The dollar rate of return on that same bond is uncertain if it is sold before maturity because the price
to be received is uncertain. And even if held to maturity, the bond's rate of return denominated in yen or in terms of
consumer purchasing power is uncertain because future exchange rates and consumer prices are uncertain.
In the theory of portfolio selection the riskless asset is defined as a security that offers a perfectly predictable
rate of return in terms of the unit of account selected for the analysis and the 1ength of the investor's decision horizon.
When no specific investor is identified, the riskless asset refers to an asset that offers a predictable rate of return over
the trading horizon (i.e., the shortest possible decision horizon).
Thus if the U.S. dollar is taken as the unit of account and the trading horizon is a day, the riskless rate is the
interest rate on U.S-Treasury bills maturing the next day.
12.2.2
Combining the RiskIess Asset and a Single Risky Asset
Suppose that you have $100,000 to invest. You are choosing between a riskless set with an interest rate of .06per year
and a risky asset with an expected rate of return of .14 per year and standard deviation of .20.3 .How much of your
$100,000 should you invest in the risky asset?
We examine all of the risk-return combinations open to you with the aid of Table 12.1and Figure 12.1. Start with
the case in which you invest a11of your money in the riskless asset. This corresponds to the point labeled F in Figure
12.1 and the first row inTable12.1.Column 2 in Tabie12.1 gives the proportion of the portfolio invested in the risky
asset (0) and column 3 the proportion invested in the riskless asset (100%). The proportions always add to 100%.
Columns 4 and 5 of Table 12.1 give the expected return and standard deviation that correspond to portfolio F:E (r)
of .06 per year and σ of 0.00.
The case in which you invest all of your money in the risky asset corresponds to the point labeled S in Figure
12.1and the last row in Table 12.1. Its expect return is .14 and its standard deviation 20.
In Figure 12.1 the portfolio expected rate of return, E (r), is measured L: the vertical axis and the standard
deviation, σ ,along the horizontal axis. The portfolio proportions are not explicitly shown in Figure 12.1; however,
we know what they are from Table 121.
Figure 12.1graphically illustrates the trade-off between risk and reward. The line connecting points F, G, H, I,
and S in Figure 12.1represents the set of alternatives open to you by choosing different combinations (portfolios) of the
risky asset and riskless asset. Each point on the line corresponds to the mix of these two assets given in columns 2 and
3 ofTable12.1.
At point R which is on the vertical axis in Figure 12.1, with E(r) of .06 per and σ of zero, all of your money is
invested in the riskless asset. You face EC and your expected return is .06per year. As you shift money out of the
riskless asset and into the risky asset, you move to the right along the trade-off line and face both a higher expected rate
of return and a greater risk. If you invest all of your money in the risky asset, you would be at point S with expected
return, E(r),of .14 and standard deviation, σ ,of .20.
Portfolio H (corresponding to the third row of Table 12.1) is half invested in the riskless asset and half in the
risky asset. With $50,000 invested in the risky asset and $50,000 invested in the riskless asset, you would have an
expected rate of return that is halfway between the expected return on the ail-stock portfolio (.14) and the riskless rate
of interest (.06). The expected rate of return of.10 is shown in column 4 and the standard deviation of .10 in column 5.
Now let us show how we can find the portfolio composition for any point lying on the trade-off line in Figure
12.1, not only the points listed in Table 121.For example, suppose we want to identify the portfolio that has an expected
rate of return of .09. We can tell from Figure 12.1 that the point corresponding to such a portfolio lies on the trade-off
line between points G and H. But what is the portfolio's composition and what is its standard deviation? In answering
this question we shall also derive the formula for the trade-off line connecting ail of the points in Figure 12.1.
Step 1: Relate the portfolio's expected return to the proportion invested in the risky
Asset.
Let w denote the proportion of the $100,000 investment to be allocated to the risky asset. The remaining
proportion, 1-w is to be invested in the riskless asset. The expected rate of return on any portfolio, E(r) , is given by:
E(r)=wE(rs)+(1-w)rf
=rf+ w[E(rs)-rf]
Where E(rs) denotes the expected rate of return on the risky asset and rf is the z- less rate. Substituting .06for rf
and .14for E (rs) we get:
E(r)=.06+w(.14 - .06)
=.06+.08w
Equation 12.1is interpreted as follows. The base rate of return form for any portfolio is the riskless rate (.06in our
example). In addition, the portfolio is expected to earn a risk premium which depends on (1) the risk premium on the
risk premium on the risky asset, E(r) –r f (.08in our example) and (2) the proportion of the portfolio invested in the
risky asset, denoted by w.
To find the portfolio composition corresponding to an expected rate of return of .09, we substitute in
equation12.1 and solve for w:
.09=. 06+. 08w

.09  .06
 .375
.08
Thus, the portfolio mix is 37.5% risky asset and 62.5% riskless asset.
Step2: Relate the portfolio standard deviation to the proportion invested in the risky asset.
When we combine a risky and a riskless asset in a portfolio, the standard deviation of that portfolio is the
standard deviation of the risky asset times the weight of that asset in the portfolio. Denoting the standard deviation of
the risky asset σ s we have an expression for the portfolio's standard deviation:
σ= σsw=.2w
To find the standard deviation corresponding to an expected rate of return of .09, we substitute .375 for w in
equation 12.2 and solve for σ :
σ=σsw=.2 × .375=.075
Thus, the portfolio standard deviation is .075.
Finally, we can eliminate w to derive the formula directly relating the expected rate of return to standard
deviation along the trade-off line.
Step3: Relate the portfolio expected rate of return to its standard deviation.
To derive the exact equation for the trade-off line in Figure 12.1, we rearrange equation 12.2 to find that
w= σ / σ s. By substituting for w in equation 12.1, we have that:
E (r )  rf 
E (rs )  rf
s
 .06  .40
In words, the portfolio's expected rate of return expressed as a function of its standard deviation is a straight line,
with an intercept rf=. 06 and a slope:
E (rs )  rf

.08
 .40
.2
s
12.2.3 Achieving a Target Expected Return: 1
Find the portfolio corresponding to an expected rate of return of .11per year. What is its standard deviation?
Solution:
To find the portfolio composition corresponding to an expected rate of return of 0.11, we substitute in equation
12.1 and solve for w. 11=.06 十 .08w
W
.11  .06
 .625
.08
Thus, the portfolio mix is 62.5% risky asset and 37.5% riskless asset.
To find the standard deviation corresponding to an expected rate of return of .11, we substitute .625 for w in
equation 12.2 and solve for σ
.
σ =.2w=.2 × .625=.125
Thus, the portfolio standard deviation is .125.
12.2.4 Portfolio Efficiency
An efficient portfolio is defined as the portfolio that offers the investor the highest possible expected rate of return at a
specified level of risk.
The significance of the concept of portfolio efficiency and how to achieve it are illustrated by adding a second
risky asset to our previous example. Risky Asset 2 has an expected rate of return of .08 per year and a standard
deviation of .15 and ‘ represented by point R in Figure 12.2.
An investor requiring an expected rate of return of .08 per year could achieve this by investing all of his or her
money in Risky Asset 2 and, thus, would be at point R. But point R is inefficient because the investor can get the
same expected rate of return of .08 per year and a lower standard deviation at point G.
From Table 12.1we know that at point G the standard deviation is only C.7 and that this is achieved by holding
25% of the portfolio in Risky Asset 1and 75 in the riskless asset. Indeed, we can see that a risk-averse investor would
be better off at any point along the trade-off line connecting points G and S than at point R. All of these points are
feasible and are achieved by mixing Risky Asset 1 with LK riskless asset. For example, portfolio has a standard
deviation equal to that of Risky Asset 2 ( σ =.1.5) , but its expected return is.12 per year rather than .08. From Table
12.1we know that its composition is 75% Risky Asset 1and 25% riskless asset.
We can use equations12.1and 122to find the composition of other efficient portfolios that lie between points G
and J and, therefore, have both a higher expected rate of return and a lower standard deviation than Risky Asset 2. For
example, consider a portfolio that consists of 625% Risky Asset 1 and 37.5% riskless asset. Its expected rate of return
is .11 per year and its standard deviation is .125.
12.3 EFFICIENT DEVERSEFICATEON WITH MANY RESKY ASSETS
Although holding Risky Asset 2by itself is inefficient, what about holding portfolios that mix the two risky assets? Or
portfolios that mix the two risky assets nth the riskless asset?
We will explore the ways to efficiently combine the three assets in two steps. The first step is to consider the risk
and return combinations attainable by mixing only Risky Assets 1and 2, and then in the second step we add the riskless
asset.
12.3.1PortfoEiosof Two Risky Assets
Combining two risky assets in a portfolio is similar to combining a risky asset with a riskless asset discussed in section
12.2. (Take a moment to review Table 12.1, Figure 12.1, and equations 12.1 and 122.) When one of the two assets is
riskless, the standard deviation of its rate of return and its correlation with the other asset are zero. When both assets arc
risky, the analysis of the risk and return trade-off is somewhat more involved.
The formula for the mean rate of return of any portfolio consisting of a proportion w in Risky Asset 1 and a
proportion l-w in Risky Asset 2 is:
E(r)=wE (rl)+(1-w) E(r2)
(12.4)
And the formula for variance is:
 2  w2 12  (1  w) 2  22  2w(1  w)  1 2
These two equations should be compared to equations 12.1 and 122.
Equation 12.4 is essentially the same as
equation 12.1 with the expected return on Risky Asset 2, E ( r 2), substituted for the Interest rate on the riskless asset, rf.
Equation 12.5 is a more general form of equation 12.2. When asset 2 is riskless, then σ 2=0, and equation 12.5
simplifies to equation 12.2.
Table 122summarizes our assessments of the probability distribution of the rates of return on Risky Assets 1and
2. Note that we assume the correlation coefficient is zero ( ρ =O).
Table 12.3 and FIgure12.3 show the combinations of mean and standard deviation of returns attainable by
combining Risky Asset 1 and Risky Asset 2. Point Sin Figure 12.3 corresponds to a portfolio consisting entirely of
Risky Asset 1and point R to a portfolio entirely of Risky Asset 2.
Let us demonstrate how the expected rates of return and standard deviations in Table 12.3 were computed using
the formulas in equations 12.4 and 12.5. Consider portfolio C, which consists of 25% Risky Asset 1 and 75% Risky
Asset 2.
Table
12.2
Distribution of Rates of Return on Risky Assets.
Risk Asset 1
Mean
.14
Standard deviation
.20
Correlation
0
Risk Asset 2
.08
.15
0
Substituting into equation 12.4, we find the expected rate of return at point C to be 0.095 per year:
E (r)=25E(rl)+. 75E(r2)
=.25 × .14+.75 × .08
=.095
And substituting for w into equation 12.5, we find the standard deviation to be:
 2  w2 12  (1  w) 22  2w(1  w)  1 2
 .252  .2 2  .752  .152  0
 .01515625
  .01515625  .1231
Let us follow the curve connecting points R and S in Figure 12.3 with the aid of Table 12.3. Start at point R and
move some of our money from Risky Asset 2 to Risky Asset 1. Not only does the mean rate of return go up, but the
standard deviation goes down. It keeps going down until we reach a portfolio that has 36% invested in Risky Asset 1
and 64% invested in Risky Asset 2.
This point is the minimum-variance portfolio of Risky Asset 1 and Risky Asset 2.4 Increasing the proportion
invested in Risky Asset 1 beyond 36% causes the standard deviation of the portfolio to increase.
12.3.2 The Optimal Combination of Risky Assets
Now let us consider the risk-reward combinations we can obtain by combining the riskless asset with Risky Asset 1 and
Risky Asset 2. Figure 12.4 presents a graphical description of all possible risk-reward combinations and also illustrate
how one locates the optimal combination of risky assets to mix with the riskless asset.
First consider the straight line connecting point F with point S. This should be familiar to you as the risk-reward
trade-off line we looked at in Figure 12.1. It represents the risk-reward combinations that can be obtained by mixing the
riskless asset with Risky Asset l.
A straight line connecting point F to any point along the curve connecting points R and S represents a
risk-reward trade-off line involving a particular mix 34 Risky Assets 1 and 2 and the riskless asset. The highest
trade-off line we can get to is the one connecting points F and Z Point T is the point of tangency between a straight line
from point F drawn to the curve connecting points R and S. We call this particular risky portfolio, which corresponds to
the tangency point T in Figure 12.4, the optimal combination of risky assets. It is the portfolio of risky assets that is
then mixed with the riskless asset to achieve the most efficient portfolios. The formula for finding the portfolio
proportions at point T is:
w1 
[ E (r1 )  rf ] 22  [ E (r2 )  rf ] 1 2
[ E (r1 )  rf ] 22  [ E (r2 )  rf ] 12  [ E (r1 )  rf  E (r2 )  rf ] 1 2
w2  1  w
Substituting into equation 12.6, we find that the optimal combination of risky assets (the tangency portfolio) is
composed of 69.23% Risky Asset 1 and 30.77% Risky Asset 2. Its mean rate of return, E( r T), and standard deviation,
σT, are:
E (rT )  .122
 T  .146
Thus the new efficient trade-off line is given by the formula:
E (r )  rf  w[ E (rf )  rf ]
 rf 
[ E (rT  rf )]
T
.122  .06

.146
 .06  .42
 rf 
Where the slope, the reward-to-risk ratio, is .42.
Compare this to the formula for the old trade-offline connecting points F and S :
E(r)=.06+.40σ
where the slope is .40.Clearly the investor is better off now because he or she can achieve a higher expected rate of
return for any level of risk he or she is willing to tolerate.
12.3.3
Selecting the Preferred Portfolio
To complete the analysis, let us now consider the investor's choice of his or her preferred portfolio along the efficient
trade-off line. Recall from our discussion in section 12.1 that a person's preferred portfolio wi1l depend on his or her
stag in the life cycle, planning horizon, and risk tolerance. Thus, an investor might choose to be at a point that is
halfway between points F and 7: Figure 12.5 shows this as point E. The portfolio that corresponds to point E consists of
50% invested in the tangency portfolio and 50% invested in the riskless asset. By transforming equations12.1 and 12.2
to reflect the fact that the tangency portfolio is now the single risky asset to combine with the riskless asset, we find
that the expected return and standard deviation of portfolio E are:
E (r f )  rf  .5  [ E (rT )  rf ]
 .06  .5  (.122  .06)  .091
 E  .5   T
 .5  .146  .073
Noting that the tangency portfolio is itself composed of 692%Risky Asset 1 and 30.8%Risky Asset 2,
the composition of portfolio E is found as follows:
Weight in riskless asset Weight in Risky Asset 1 Weight in Risky Asset 2 Total
5 × 69.2%= 5 × 30.8%=
50.0% 34.6% 15.4%
100.0%
Thus, if you were investing $100,OOO in portfolio E, you would invest $50,000 in the riskless asset,
$34,600 in Risky Asset 1,and $15,400 in Risky Asset 2.
Let us now summarize what we have learned about creating efficient portfolios when the raw materials are
two risky assets and a riskless asset. There is a single portfolio of the two risky assets that H is best to
combine with the riskless asset. We call this particular risky portfolio, which corresponds to the tangency point T in
Figure 12.4, the optimal combination or risky assets. The preferred portfolio is always some combination
of this tangency portfolio and the riskless asset.
12.3.4 Achieving a Target Expected Return: 2
Suppose you have $100,000 to invest and want an expected rate of return of .L: year. Compare the standard
deviation you would have to tolerate under the old risk-reward trade-off line (connecting points F and S) with the
standard deviation under the new trade-off line (connecting points F and T). What is the composition of each of the two
portfolios you are comparing?
Solution:
First, let us write down the formula relating the expected return on the portfolio to the proportion invested in
risky assets and solve it to find the proportion to invest in risky assets. For the new trade-off line using the optimal
combination of two risky assets it is:
E (r )  E (rT ) w  rf (1  w)
E (r )  .122w  .06(1  w)
EE
Setting the expected rate of return on the portfolio equal to 0.10 and solving for we find:
E (r )  .06  .062w  .10
.10  .06
w
 .65
.062
Thus, 65% of the $100,000 must be invested in the optimal combination of risky assets and 35% in the riskless asset.
The standard deviation of this portfolio is given by:
  w T
 .65  .146  .095
Because the optimal combination of risky assets is itself composed of 69.2% Risky Asset 1and 30.8% Risky
Asset 2, the composition of the final desired portfolio with expected return of .10 per year is found as follows:
For the old trade-off line with a single risky asset the formula relating the return and w was:
E (r )  E (rs ) w  rf (1  w)
E (r )  .14w  .06(1  w)
Setting the expected rate of return on the portfolio equal to 0.10 and solving for w we find:
E (r )  .06  .08w  .10
.10  .06
w
 .50
.08
Thus, 50%of the $100,000 must be invested in Risky Asset 1and 50% in the riskless asset.
The standard deviation of this portfolio is given by:
  w s
 .5  .2  .10
It is important to note that in finding the optimal combination of risky assets we do not need to know anything
about investor wealth or preferences. The composition of this portfolio depends only on the expected rate of returns and
standard deviations of Risky Asset 1 and Risky Asset 2 and on the correlation between them. This implies that all
investors who agree on the probability distributions for rates of return will want to hold this same tangency portfolio in
combination with the riskless asset.
This is a general result that carries over to the case in which there are many risky assets in addition to Risky
Asset 1and Risky Asset 2:
12.3.5 Portfolios of Many Risky Assets
When there are many risky assets we use a two-step method of portfolio construction similar to the one used in the
previous section. In the first step, we consider portfolios constructed from the risky assets only, and in the second step
we find the tangency portfolio of risky assets to combine with the riskless asset. Because the computation involves a lot
of number crunching, it is best done using computersFigure 12.6 illustrates the inputs and outputs of an electronic spreadsheet used to carry out portfolio optimization.
The individual basic assets are Risky Asset 1, Risky Asset 2, and so on. They are represented as shaded points in the
diagram on the left. The curved colored line lying to the northwest of these points is called the efficient portfolio
frontier of risky asset. It is defined as the set of portfolios of risky assets offering the highest possible expected rate of
return for any given standard deviation:
The reason the individual basic assets lie inside the efficient frontier is that there is usually some combination
of two or more basic securities that has a higher expected rate of return than the basic security for the same standard
deviation.
The optimal combination of risky assets is then found as the point of tangency between a straight line from the
point representing the riskless asset (on the vertical axis) and the efficient frontier of risky assets. The straight line
connecting the riskless asset and the tangency point representing the optimal combination of risky assets is the best
risk-reward trade-off line achievable.
We now return to the issue raised in Section 12.1. How can a financial intermediary, such as a firm
offering mutual funds to the investing public, decide on the menu of asset choices to offer to its customers? We
just showed that the composition of the optimal combination of risky assets depends only on the expect returns
and standard deviations of the basic risky assets and on the correlations among them. It does not depend on
investor preferences. Therefore, one doesn’t need to know anything about investor preferences in order to create
this portfolio.
If customers delegate the task of forecasting expected asset returns, standard deviations, and correlations
to a financial intermediary that specializes in doing it and they delegate the task of combining the basic assets
in the optimal proportion, then the only choice the customers need to make is the proportion to invest in the
optimal risky portfolio.
The static mean-variance model thus leads to an elementary theory of mutual fund financial intermediation.
Since the late 1960s, academic research on optimal portfolio selection has gone beyond that model to dynamic
versions that integrate intertemporal optimization of the life-cycle consumption-saving decisions with the
allocation of those savings among alternative investments. In these models, the demands for individual assets
depend on more than just optimal diversification as presented here; they also come from the desire to hedge
various risks not included in the original model. Some of the risks identified as creating these hedging demands
in port- folio decisions are mortality risk and stochastic changes in interest rates and in the trade-off between
expected return and risk, in returns to human capital, and in relative consumption goods prices. These models
provide a richer theory for the role of securities and financial intermediation than the static mean-variance
model-6.
The basic mean-variance approach to quantitative investment management is still the dominant one used in the
practice of asset management. However, this is changing. The more complete models of portfolio selection provide
design guidance for investment firms to offer a wider "family" of mutual funds beyond just the optimal combination of
risky assets and the riskless asset. Those additional funds represent optimal hedging portfolios more tailored to the
needs of different clienteles. The investment firm can create integrated products from its funds by putting various
combinations of its member funds together in proportions that reflect the right mix for customers in various stages of
their life cycle.
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.