5
CHAPTER
The Theory
of Portfolio Allocation
If you turn to Section C of The Wall Street Journal, you will see ads for financial services companies, offerings for bonds or stocks, and listings of financial assets ranging
from the common stock of Fortune 500 firms to junk bonds used to finance leveraged
buyouts to a staggering array of mutual funds. Each of these assets offers savers the
potential for future returns and a reward in the form of interest or capital gains for postponing consumption. How can savers choose among these financial assets in deciding
where to invest their funds?
Individuals may be motivated to save for several reasons: to smooth spending over
time, to purchase durable goods, to accumulate precautionary (or emergency) funds for
retirement, and to leave bequests. In meeting these needs, savers are concerned about
the expected return on their savings. They also care about how easily their savings can
be converted into a secure and steady source of income to finance future spending.
In this chapter, we explore portfolio allocation to understand how savers decide to
allocate their wealth among alternative assets. The theory of portfolio allocation
describes why savers behave as they do when selecting one asset rather than another.
But this is not the only choice savers must make. Suppose you find the perfect financial
asset—bonds issued by Golden Horizons, Inc. Should you put all your savings into
Golden Horizons? The answer is no. The second theme of this chapter is to demonstrate why investing in a group of assets, or a portfolio, allows investors to reduce their
risk. The decisions they make about asset allocation affect the performance of the entire
portfolio. A third theme of this chapter is to describe why investing in a portfolio of
assets allows investors to reduce their risk. Our subsequent analysis in later chapters of
interest rate determination, the behavior of financial institutions, and innovation in
financial markets and institutions builds on the concepts presented in this chapter.
Determinants of Portfolio Choice
Web Site Suggestions:
http://www.federal
reserve.gov/releases/Z1
Z1, Flow of Funds
describes households’
portfolio allocation.
The financial system offers savers an array of assets from which to choose. Such assets
are stores of value; that is, they can be sold when the saver needs the funds to spend on
goods and services. The types of financial assets that savers have held, on average, are
shown in Fig. 5.1. Americans in 2003 held 1.7% of their financial assets in checkable
deposits, another 16.3% in bank savings and time deposits, 11.2% in equity mutual
funds, 17.9% in stocks directly held, 4.2% through life insurance reserves, and 34.1%
through pension fund reserves. A generation ago, things were different. In 1970, households kept a larger share of their savings in bank checking and saving accounts and life
insurance reserves and a smaller share in mutual funds and pension reserves. Two generations ago, in 1950, households held most of their financial assets in bank accounts,
government securities, and stocks; mutual funds and pension reserves were not major
stores of household wealth.
85
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Interest Rates
Figure 5.1 shows two patterns. The first pattern is that savers divide their assets among
different financial assets, and the second pattern is that these allocations change over time.
Our focus in this section is on the decisions that an individual saver makes in deciding which
assets to include in a portfolio, or collection of assets, and how much a saver will devote to
each asset in a portfolio. Later in the chapter, we describe the reason for the changes in asset
allocation over time.
To begin, how would you invest $1000? You might choose to invest in stocks, a
bond, a money market fund, or physical assets (such as commodities, real estate, gold,
machines, or paintings) or hold it as cash. What influences your choice?
The theory of portfolio allocation seeks to answer questions about portfolio choice
and predicts how a saver distributes his or her savings across alternative investments.
According to this theory, savers evaluate five criteria when deciding what investments
to make and how much to invest in each alternative. These determinants of portfolio
choice are
• the saver’s wealth or total stock of savings to be allocated,
• the expected return from the investment as compared with the expected return
from other investments,
• the degree of risk of the asset as compared to the risk of other assets,
• the liquidity of the asset as compared to other assets, and
• the cost of acquiring information about the asset as compared to gathering
information about other assets.
FIGURE 5.1
Portfolios of U.S. Households: 2003, 1970, 1950
Percentage of household
financial assets
Source: Board of Governors of the Federal Reserve System, Flow of Funds Accounts, various issues.
40
36.1
2003
35
34.1
1970
29.8
1950
30
25
22.5
20
17.9
16.3
15.6
15.2
15
12.8
12.6
12.7
11.2
10
6.9
6.3
5
3.1
1.7
0 0
0
Checkable Savings
deposits and time
deposits
5.6
5.4
4.6
2.7
2.4 2.1
4.0
2.7
3.7
1.91.6
0.5
Money
U.S.
TaxCorporate Mortages
market government exempt
bonds
mutual
securities securities
funds
4.2
2.4
0.8
Equity
mutual
funds
Equities
Life
insurance
reserves
Pension
reserves
CHAPTER 5
The Theory of Portfolio Allocation
87
Wealth
As people become wealthier, the size of their portfolio of assets increases because they
have more savings to allocate to the acquisition of assets. As people grow richer, they
do not increase the quantities of all they have in their portfolios; rather, they choose to
increase their purchases of some assets rather than others. Which assets do they
choose? To begin answering this question, consider the asset mix you would purchase
if your total wealth were $1000; you might hold 10%, or $100, in cash. If your total
wealth were $1 million, however, you probably would not hold $100,000 in cash. Cash
holdings would make up a smaller percentage of your wealth, and you would increase
your relative holdings in other assets, such as stocks. Although holding many shares of
different high-quality stocks with wealth of only $1000 is difficult, you might well own
a variety of stocks with $1 million.
The wealth elasticity of demand describes how responsive the percentage change in
the quantity of an asset chosen is to a percentage change in wealth. The wealth elasticity of demand does not depend on the actual dollar value of your wealth. Rather, it
equals the percentage increase in the quantity of an asset you demand divided by the percentage increase in your total wealth; that is,
% change in quantity demanded of the asset
Wealth elasticity of demand
.
% change in wealth
for an asset
Let’s use this definition to find the wealth elasticity of your demand
for cash when your wealth increases from $1000 to $1 million. With $1000, you might
hold 10%, or $100, of your wealth as cash if your total wealth were $1000. When your
wealth reaches $1 million (be optimistic!), you might hold 0.1%, or $1000, in cash. Thus
your wealth elasticity of demand for cash is 0.01, which is less than 1. Hence an increase
in wealth generates a decrease in percentage terms of cash held. However, your wealth
elasticity of demand for stocks and other assets is greater than 1. Hence an increase in
wealth generates an increase in percentage terms of stocks and other assets held.
A necessity asset is one for which the wealth elasticity of demand is less than 1. Savers
demand necessity assets, such as cash or checking accounts, in order to conduct regular
transactions. A luxury asset, however, is one for which the wealth elasticity of demand
exceeds 1. What makes some assets luxuries? They are assets, such as stocks, that are held
for investment rather than for facilitating transactions. Savers also must consider the high
fixed cost of owning a luxury asset, such as real estate taxes and insurance costs for a $1
million house, or the high transactions costs of acquiring the asset, such as stockbroker or
dealer fees for stocks. For savers with less wealth, these costs make up a larger percentage
of their investment than for savers with more wealth. Thus acquiring some assets, such as
buying a famous painting, is feasible only for wealthy individuals.
To summarize, as wealth increases, savers hold more of their wealth in luxury
assets and less in necessity assets.
Expected Returns on Assets
What factors determine how savers, whether wealthy or poor, choose to allocate their
wealth among assets? Given the choice between two otherwise similar assets, a saver
will pick the one with the higher expected return. The correct measure of expected
return is the expected real rate of return.
Savers must assess the impact of inflation on returns because changes in the value of
money will affect the real value of returns. We encountered the problem of converting
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nominal returns to real returns in Chapter 4 when we studied rates of returns on bonds.
The expected real return savers consider equals the nominal return less expected inflation.
Savers use real returns as the measure of their gain in an investment because those returns
adjust for changes in purchasing power.
Because savers care about expected returns for financing current and future spending, they also focus on the amount that they can keep after taxes; that is, they compare
expected real after-tax returns. Taxation of returns on savings varies significantly in the
United States. Interest on private corporate bonds, bank deposits, and dividends from
holdings of corporate stock are taxed at the federal, state, and local levels (though dividends are taxed at a lower rate than interest). Interest received from U.S. Treasury securities is subject to federal income taxation but not to state and local income taxation.
The obligations of state and local governments (called municipal bonds) generally are
exempt from all taxation and often are called tax-exempt bonds. All such differences
affect savers’ portfolio decisions.
When assets are similar—that is, all other factors being held constant—an increase
in the expected return on one asset relative to other assets makes the asset more desirable to savers. The remaining three determinants of portfolio choice—risk, liquidity,
and information costs—are attributes that we use to compare two assets.
C H E C K P O I N T
Interest received from municipal bonds is tax-exempt in the United States. Should
you switch your savings from taxable assets to municipal bonds to take advantage
of their favorable tax treatment? Not necessarily. Investors compare expected
after-tax returns when making their investment decisions. Suppose that a taxable bond
pays an interest rate of 10%. If investors face a tax rate of 30%, this after-tax return
is equivalent to the return on the tax-exempt bond of 10% 0.3(10%) 7%. If the
tax-exempt rate were higher (say, 8%), the expected return on the tax-exempt bond
would exceed that on the taxable bond. Investors would increase their demand for taxexempt bonds, bidding up their price and reducing their yield. (Recall that bond prices
and yields are inversely related.) At a tax rate of 15%, investors would prefer to invest
in the taxable bond because they could receive an expected after-tax return of 10% 0.15(10%) 8.5%. This is greater than the 7% expected rate of return on the taxexempt bond. Investors who are subject to high tax rates are more likely to invest in
tax-exempt bonds than are investors who are subject to low tax rates. ♦
Risk Associated with Asset Returns
In making investment decisions, savers evaluate the variability of (fluctuations up and
down in) the expected return as well as the size of the return. Because households use
their assets largely to smooth their spending over time, they want to avoid having assets
fall in value just when they need funds.
To demonstrate the impact of variability on expected returns, suppose that you
have $1000 to invest in stocks and are comparing the shares of Solid Enterprises and
Rollercoaster Industries. Solid Enterprises’ shares yield a return of 10% all the time
(with certainty), whereas Rollercoaster Industries’ shares yield a return of 20% half the
time and 0% half the time. We calculate the expected return on Rollercoaster’s shares
using a weighted average of its possible returns:
1
1
a b 10.202 a b 102 0.10,
2
2
or
10%.
CHAPTER 5
The Theory of Portfolio Allocation
89
Solid’s expected return, 10%, is the same. How do you choose between the two investments?
The answer lies in the degree of risk or variability associated with the two investments. Although Solid and Rollercoaster have equal expected returns, the likelihood that
an investor in Rollercoaster will earn 10% is less certain than the return from an equivalent investment in Solid. The greater potential variation in Rollercoaster’s return means
that it is a riskier investment. The investor’s view of risk determines which asset the
investor will buy. Most people are risk-averse savers. They seek to minimize variability
in the return on their savings and prefer security in their investments. A risk-averse saver
would even accept a lower return from Solid Enterprises because of this desire for stability. Risk-neutral savers judge assets only on their expected returns; variability of
returns is not a concern. A few individuals are risk-loving savers, who actually prefer to
gamble by holding a risky asset with the possibility of maximizing returns.
Empirical evidence on expected returns from financial markets confirms the riskaverse behavior of most investors. For example, annual real rates of return (adjusted
for inflation) on U.S. common stocks averaged 8.2% from 1926 to 1999, while annual real rates of return on long-term government bonds averaged only 2.0%.✝ Why do
investors accept such low returns on government bonds when they could earn more by
investing in stocks? The principal reason is that government bonds have less risk.
Stocks offer higher potential returns to compensate savers for taking the higher risk
associated with equity investment. Nevertheless, many economists have argued that the
equity premium implied by the gap between the return on stocks and bonds is too large
to be explained by risk considerations alone.
Because savers are generally risk-averse, an increase in the risk of one asset relative
to other assets leads to a decline in the quantity of that asset chosen.
Liquidity of Assets
Assets with greater liquidity help savers to smooth spending over time or to draw down
funds for emergencies. For example, if you maintain some savings in financial assets to
meet unanticipated medical expenses, you want to be able to sell those assets quickly
if you need the money for an operation.
Obviously, cash is the most liquid asset. Many marketable securities, such as U.S.
government bonds or shares of IBM, are very liquid assets because finding a buyer for
them with minimal transactions costs is easy. Real estate, coins, and fine paintings are
relatively illiquid assets because their sale incurs substantial transactions costs. For
example, a saver who wants to sell a house might need to wait months or even years
before finding a buyer who is willing to pay the full asking price.
C H E C K P O I N T
Certificates of deposit (CDs) offered by banks have a penalty for early withdrawal. For example, if you invest $1000 in a one-year CD paying 5% interest, you
receive less interest if you withdraw your savings before the end of a year. Why
are investors willing to accept a lower interest rate (say, 3.25%) on savings
accounts without a penalty for early withdrawal? Savers are generally willing to
sacrifice some portion of expected return to be able to convert an asset to cash quickly to finance unplanned or emergency spending. ♦
✝ The calculations are based on data from Roger C. Ibbotson and Rex A. Sinquefield, Stocks, Bonds, and
Inflation: 2000 Yearbook, Chicago: Ibbotson Associates, 2000.
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CASE
STUDY
How Much Risk Should You
Tolerate in Your Portfolio?
Although all investments are risky, you can take steps to understand and manage risk when building
your own portfolio. Financial planners encourage their clients to evaluate their financial situation and
their willingness to bear risk in determining whether an investment is appropriate.
In assessing the volatility of returns, financial planners recommend that you determine how far ahead
your savings goal extends.
The longer your time horizon, the more you can focus on the growth potential of investments in
stocks. Over the period from 1926 to 1999, one-year returns on stocks ranged between 43% and
54%, while 20-year returns ranged between 3% and 18%. For most people, an important savings goal
is retirement, and retirement savings make up a significant component of their wealth. If your retirement
is many years away, you can take advantage of the long-term gains from riskier investments such as common stock without much concern for short-term variability in returns. Then as you approach retirement,
you should adopt a more conservative strategy to reduce the risk of losing a substantial portion of your
savings. Here are two typical financial plans that differ in the time horizon and savings goal of younger
and older savers:
Younger Saver
Older Saver
Description:
Below age 50 and wishes to build his or her
net worth over a relatively long time.
Description:
Close to retirement age with a portfolio at or
near the amount needed to retire.
Goal:
Accumulate funds by earning high long-term return.
Goal:
Conserve existing funds to earn a return slightly
above the inflation rate.
Portfolio plan:
Select portfolio based on maximizing expected real
return with only limited concern for variability.
Portfolio plan:
Reduce risk by selecting safe assets to earn an
expected real return of about zero.
Finally, in assessing the volatility of your returns, you must consider the effects of inflation and taxes.
Your investment returns are (generally) subject to taxation, and your real returns lop off inflation. With
these considerations in mind, over the period mentioned above, “safe” nominal government bonds would
have brought you a much lower and less-than-volatile average annual yield relative to common stocks.✝
Understanding the types of risk that influence your investment will help you to reduce emotional reactions to market volatility and to make informed investment decisions.
✝The calculations are based on data from Roger C. Ibbotson and Rex A. Sinquefield, Stocks, Bonds, and Inflation: 2000
Yearbook, Chicago: Ibbotson Associates, 2000.
The average investor favors assets that are liquid over those that are not. The
investor must weigh the benefits of liquidity against the lower returns that are generally available on liquid assets when selecting assets for a portfolio.
Costs of Acquiring Information
Savers seek to lower the risk associated with an asset but want to do so without devoting time or resources to assessing the issuer’s creditworthiness or monitoring the borrower’s actions. For some assets, such as cash or government securities, information is
CHAPTER 5
TA B L E 5 . 1
The Theory of Portfolio Allocation
91
Determinants of Asset Allocation
An increase in …
Causes the quantity
of the asset in the
portfolio chosen to …
wealth
rise
savers have greater stock of savings
to allocate
expected return on asset
relative to expected returns
on other assets
rise
savers gain more from holding asset
risk (variability of returns)
fall
savers are generally risk-averse
liquidity (ease with which
an asset can be converted to cash)
rise
the asset can be cheaply converted to
cash to finance consumption
information costs
fall
savers must spend more resources
acquiring and analyzing data on the
asset and its returns
Because …
readily available to the public at low cost. For example, if you want to buy a government bond, you can easily find prices and returns in The Wall Street Journal. Similarly,
savers can gather information about the stocks and bonds of large corporations inexpensively because financial analysts publicize information about these assets.
If a new company issues financial claims, however, investors must spend time and
resources to collect and analyze information about the company before deciding to
invest. Therefore savers prefer to hold assets with low information costs. An increase
in the information cost for an asset raises the required rate of return on the asset; a
decrease in information costs reduces the required rate of return. Specialists in the
financial system acquire and analyze information about issuers of financial assets and
make this information available for a fee. Other factors being held constant, a higher
cost of information for an asset relative to other assets leads to a decrease in the quantity of that asset demanded.
Table 5.1 summarizes the principal determinants of portfolio choice. The underlying assumption for the effects listed in the second column of the table is that all other
factors remain constant. Given the discussion of each factor, you can think about how
these determinants shape your own financial decisions. As we discuss in Part 4, the
same factors affect the portfolio allocation decisions of businesses and financial intermediaries.
Advantages of Diversification
Savers purchase assets with the expectation that these assets will increase in value over
time, but not all assets appreciate in this way. From our description of a saver’s behavior in selecting an asset, you might expect that a saver could sort through the maze of
available assets and find one that does increase in value and is a “best” choice for the
saver’s investment dollars. But implicit in our discussion of the theory of portfolio allocation is the importance given to each asset the saver decides to include in a portfolio.
Why did we assume that the saver would hold more than one asset if he or she could
locate the perfect investment? The answer is that the real world is full of uncertainty, and
despite all the analysis and careful decision making, a saver cannot be certain that an
asset will perform as expected. To compensate for the inability to find a perfect asset,
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Interest Rates
individuals typically hold various types of assets, including financial instruments, property, and durable goods. Even within categories of financial assets (stocks, for example),
investors usually hold many individual issues. Allocating savings among many different
assets is known as diversification.
In the real world, returns on assets do not move together perfectly because their risks are imperfectly correlated. That is, assets do not all fare well or poorly
at the same time. Thus the return on a diversified portfolio is more stable than the returns
on the individual assets making up the portfolio. Diversification effectively allows the
investor to divide risk into smaller and thus less potentially harmful pieces. Research on
the benefits of diversification led to three Nobel Prizes in economics: to James Tobin of
Yale University, Harry Markowitz of Baruch College, and William Sharpe of Stanford
University.
How does diversification reduce portfolio risk? Let’s use an example to answer this
question. Suppose that you want to invest $1000 in stocks and are choosing between two
investments—shares in Boomco, Inc., and shares in Bustco, Inc.—whose returns vary
with the economy’s performance in different ways. Suppose that Boomco does well half
the time and not so well the other half. When the economy does well, Boomco prospers.
Its shares have a rate of return of 20%, and you earn $200. But in a weak economy, sales
of Boomco’s products are poor. Then the stock’s rate of return is 0%, and you have nothing. The expected amount you will earn on your $1000 investment in Boomco is
1
a 1 b 1$2002 a b 1$02 $100.
2
2
If you invest only in Boomco, you can expect a rate of return of 10% (100/1000).
Suppose that Bustco’s returns follow an opposite pattern. The rate of return on
Bustco shares is high (20%) when the economy is weak, and you earn $200. When the
economy does well, you earn nothing (0%). Like Boomco shares, Bustco shares have
an expected return of
a 1 b 1$02 a 1 b 1$2002 $100.
2
2
If you invest only in Bustco, you can expect a rate of return of 10%. Of course, if you
invest only in Boomco shares or only in Bustco shares, you incur risk because the
returns vary with the economy’s performance.
Now consider what happens if you invest equal amounts in Boomco and Bustco
shares. In good times, Boomco’s rate of return is 20% and Bustco’s rate of return is 0%.
Therefore your total rate of return in good times is
1
a 1 b 1Boomco return2 a b 1Bustco return2 10%,
2
2
or
1$500210.202 1$5002102 $100.
Similarly, in bad times, you earn
a 1 b 1Boomco return2 a 1 b 1Bustco return2 10%,
2
2
CHAPTER 5
The Theory of Portfolio Allocation
or
93
1$5002102 1$500210.202 $100.
By this strategy you earn the same expected return (10%) as you would earn from buying the shares of only one of the companies. However, you lessen the risk affecting your
portfolio’s returns by limiting the influence of one source of variability: the economy.
The strategy of dividing risk by holding multiple assets ensures steadier income.
Savers cannot eliminate risk entirely because assets share some common risk called
market (or systematic) risk. For example, general fluctuations in economic conditions
can increase or decrease returns on stocks collectively. Assets also carry their own
unique risk called idiosyncratic (or unsystematic) risk. For example, the price of an
individual stock may be influenced by factors such as discoveries, strikes, or lawsuits
that influence the profitability of the firm and its share value. Diversification can eliminate idiosyncratic risk but not systematic risk.✝
Diversification reduces the riskiness of the return on a portfolio unless assets’ returns
move together perfectly. The less the returns on assets move together, the greater the benefit savers reap from diversification in reducing portfolio risk. Because savers generally
are risk-averse, they amass portfolios containing an array of different assets.
Figure 5.2 illustrates the results of a study to determine how much risk can be eliminated through diversification in a portfolio of stocks traded on the New York Stock
Exchange. It illustrates the relationship between the average annual variability on equally weighted portfolios and the different numbers of stocks (selected randomly) in the portfolios. Although a single security had an average annual variability (measured by the standard deviation) of about 49%, holding two stocks reduced the variability by about onequarter, to just over 37%.✝✝ Holding eight stocks cuts the average annual variability in
half, to just less than 25%. Increasing the number of assets to 20 cuts the average annu-
Average annual variability
(standard deviation, %)
FIGURE 5.2
Reducing Risk through Stock
Portfolio Diversification
Increasing the number of New
York Stock Exchange–listed
stocks held in a portfolio decreases the variability of the portfolio’s
return. While diversification can
reduce individual risk, there is a
certain amount of risk that cannot
be reduced.
49.2
37.0
25.0
Idiosyncratic risk
21.7
19.2
Source: Based on calculations presented
in Meir Statman, “How Many Stocks Make
a Diversified Portfolio?” Journal of
Financial and Quantitative Analysis,
22:353–364, 1980.
Market risk
0
1
2
8
20
1000
Number of stocks in portfolio
✝ Indeed,
even if asset returns are independent (completely uncorrelated), increasing the number of assets held
in a portfolio reduces overall risk.
✝✝ The variance of a portfolio return is the squared deviation from the expected return. The standard deviation is the square root of the variance.
94
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Interest Rates
al variability further to about 21.7%. Holding the entire portfolio of stocks listed on the
New York Stock Exchange reduces the average annual variability to 19.2%. This remaining variability is traceable to market risk and cannot be eliminated by holding additional stocks; that is, the risk is nondiversifiable. In the (unlikely) event that returns are perfectly and positively correlated, adding additional assets does not reduce the variability of
the portfolio.
To measure systematic risk, financial economists calculate a variable called beta,
the responsiveness of a stock’s expected return to changes in the value of the complete
market portfolio of that stock—that is, the collection of all stocks. For example, if a
1% increase in the value of the market portfolio leads to a 0.5% increase in the value
of the asset, the asset’s beta is calculated to be 0.5. If the value of the asset rises by 1.5%
when the market portfolio rises by 1%, then the asset’s beta equals 1.5.
The market portfolio faces no idiosyncratic risk, only systematic risk. Hence, when
an asset has a high value of beta, its return has a lot of systematic risk. This systematic risk is scaled by beta; a beta of 1.5 implies three times the systematic risk as a beta
of 0.5. Because systematic risk cannot be diversified away, investors are less willing to
hold an asset with a high beta, all other factors being equal. Hence assets with higher
values of beta must have a higher expected return to compensate investors for their
higher risk.
Savers’ ability to diversify is limited by the cost of acquiring information about
alternative assets and the transactions costs of buying and selling individual assets.
Another potential limit to diversification comes from legal restrictions on the assets
that can be held by individual savers or by certain financial intermediaries on their
behalf. For example, individuals face limits on their investments in derivative securities
depending on their net worth and financial sophistication. Commercial banks are not
allowed to invest depositors’ funds in corporate equities.
Putting It All Together:
Explaining Portfolio Allocation
At the beginning of the chapter we looked at current and past patterns of asset allocation by households. We can use the determinants of portfolio choice and the principle
of diversification to explain how portfolio composition changes over time.
One pattern in Fig. 5.1 is that, in looking at the allocation of savings to different
assets over time, the popularity of some assets has increased among savers, whereas that
of others has declined. For example, there has been a sharp decrease in the proportion of
savers’ portfolios devoted to checkable deposits. One reason for this is the increase in
wealth that has occurred over the post–World War II period. Checking accounts are an
example of a necessity asset. As U.S. households have become much wealthier over the
postwar period, their balances of checking accounts have fallen relative to other asset
holdings.
Another reason for the changes is the tax treatment of different assets. Savers compare expected returns on alternative assets when making decisions about portfolio allocation. One component of differences in expected returns is different tax treatment.
When a household buys stocks or bonds directly, it pays income taxes on returns.
Savings through life insurance reserves receive favorable tax treatment because no tax
is paid as earnings on reserves accrue. Pension fund reserves receive similarly favorable
tax treatment and an additional benefit: households’ contributions to pension plans are
The Theory of Portfolio Allocation
CHAPTER 5
CASE
95
STUDY
Modeling Risk Premiums in Financial Markets
An increase in the risk of an asset’s return makes investors less willing to hold the asset, other things
being equal. This useful qualitative prescription has stimulated major efforts to model risk premiums. In
what follows, we consider two of these developments: the capital asset pricing model (CAPM) and the
arbitrage pricing theory (APT).
The CAPM was developed by William Sharpe of Stanford University and the late John Lintner of
Harvard University in the 1960s. It begins with the idea that the risk contributed by an individual asset
(say, Exxon stock) to a well-diversified portfolio of stocks reflects the magnitude of its systematic risk.
This magnitude is measured by beta. The larger is beta, the greater is the systematic risk and therefore
the higher is the expected return required by investors for being willing to hold the asset. In the CAPM
the expected return on asset j, Rej , depends on the default-risk-free interest rate, Rf , and the expected
e . Specifically,
return on the market portfolio, Rm
e
=
Rj
{
Expected
return on
asset j
Rf
{
βj
+
Risk-free
rate
{
(14243)
e
×
Beta of
asset j
Rm − Rf
Risk premium on
the market portfolio
144
42444
3
Risk premium on asset j
In other words, the expected return on asset has two components: the default-risk-free rate and the
risk premium for that asset. The risk premium compensates the investor for the risk that the security will
not generate the expected return. In the CAPM the risk premium equals the risk premium on the market portfolio scaled by beta. Suppose, for example, that an asset has a beta of 1.0. Then its risk premie – R ), or the risk premium on the market portfolio. If the asset has a beta of 1.5,
um equals 1.0 (Rm
f
then its risk premium is one and one-half times that of the market portfolio. In this case, for example, if
the expected return on the market portfolio is 8% and the risk-free rate is 2%, then the risk premium
on the market portfolio is 8%, and the risk premium on the individual asset is 1.5 6%, or 9%.
A key element in the simplicity of the CAPM is its assumption of a single source of market risk—that
is, the systematic risk of the market portfolio. There may, however, be multiple sources of systematic
risk in the economy. Examples include aggregate inflation or aggregate output. In the arbitrage pricing
theory, developed by Stephen Ross of Yale University, each of these factors has an associated beta. The
beta values are found by estimating how sensitive the expected return is to a change in the factor. To
e the expected return from
see how the APT refines the CAPM, we can substitute for the market return Rm
e
each factor RFAC. Hence in the arbitrage pricing theory,
e
Rj
{
Expected return
on asset j
=
Rf
{
Risk-free
rate
+
(42443)
14
e
␤j 1 R FAC 1 − R f
+
(42443)
14
e
␤j 2 R FAC 2 − R f
+ . . .
Contribution to risk
Contribution to risk
premium from factor 1 premium from factor 2
(42443)
14
e
␤ jN R FACN − R f
Contribution to risk
premium from factor N
14444444444
42444444444444
3
Risk premium on asset j
In this theory, an increase in the sensitivity of the assets to each factor increases the risk premium.
While they have been extended in many ways, both the CAPM and the APT are used by practitioners
in financial markets. Although each theory offers technical interpretations for calculating risk premiums,
the central insight is common: An increase in systematic risk raises an asset’s risk premium and the
return investors require for holding the asset.
96
PART 2
Interest Rates
from pre-tax dollars; no individual-level tax is paid on the earnings contributed. Since
1950, as pension plan eligibility has expanded, households have held more assets
through pension plans to take advantage of higher after-tax returns. Direct holdings of
U.S. government bonds, corporate bonds, and equities have declined in relative importance over the same period.
In 1950, U.S. households held less than 1% of their financial assets in stock mutual funds; that figure more than doubled by 1970 to 2.4%, then more than quadrupled
again by 2003 to 11.2%. One reason for their increase is that mutual funds reduce risk
and information costs for small investors and increase the liquidity of owning stock.
One way in which mutual funds help savers to reduce overall exposure to risk is by
facilitating diversification. One barrier to diversification is the cost of buying and selling
financial assets. On the one hand, transactions costs of direct purchases and sales can be
high for small savers. For example, brokerage commissions to buy a few shares of GM
stock are very high. On the other hand, investors in a mutual fund buy shares in diversified portfolios of assets from financial markets. Because a mutual fund has a great quantity of funds to invest (the collective funds of the individual savers), it offers lower transactions costs. Thus mutual funds can offer diversified portfolios of stocks. Mutual funds
offer diversification possibilities beyond stocks—to government or corporate bonds and
money market instruments such as Treasury bills or commercial paper.
In addition to offering a way for savers to pool risk, mutual funds allow maintenance
of liquidity through low transactions costs and, in some cases, check-writing features. They
OTHER
TIMES,
OTHER
PLACES
...
long been known, most savers
nonetheless hold the vast majority of
their wealth in domestic assets. The
In domestic equity markets, investors study by French and Poterba of the
can reduce their exposure to risk by world’s five largest stock markets—
diversifying—that is, by holding many United States, Japan, United
individual stocks whose returns do Kingdom, Germany, and France—indinot rise or fall at the same time. cates that domestic investors account
Investors can apply this principle even for about 90% of ownership, with the
further by diversifying in international exception of Germany, which has
markets: Because equities in different about 80% domestic ownership.
national markets do not always move
precisely together, investors can Why are investors missing the potential
improve their portfolio diversification for significant gains from more global
by holding stocks from many coun- diversification of their portfolios? One
tries. Kenneth French of Dartmouth possible explanation is institutional barCollege and James Poterba of M.I.T. riers, such as government regulations
have shown that equity returns on that limit investment abroad. Although
stocks from the United States, Japan, this explanation may have been plausiUnited Kingdom, France, Germany, ble in the 1970s when such controls
and Canada do not move together, were widespread, capital controls are
indicating that investors could signifi- not widely used by industrial countries
cantly reduce risk by holding a portfo- today. Another theory suggests that
lio made up of stocks from more than there are different transactions costs
one of these countries.✝ Although the between domestic and foreign markets
benefits of global diversification have and that investors seek the lowest
Are Investors (Globally)
Well Diversified?
transactions costs. However, since such
costs are lower in very liquid markets
such as New York, this explanation
does not fit the fact that most investors
keep their investments in their home
market. The most likely explanation
relates to information costs: Investors
may assign extra “risk” to foreign
stocks simply because they know less
about foreign firms and markets.
Therefore they choose the less risky
option of domestic investment.
Country-specific mutual funds, which
give investors access to broad groups
of foreign stocks and have been growing in popularity, might help to overcome this barrier to diversification.
✝See
Kenneth R. French and James M. Poterba,
“Investor Diversification and International Equity
Markets,” American Economic Review,
81:222–226, 1991.
CHAPTER 5
The Theory of Portfolio Allocation
97
also reduce savers’ information costs by economizing on costs of research and information
collection about the assets in the portfolio.
Why do mutual funds provide these services that have accounted for their rapid
growth? The funds earn a profit for the fund managers, as investors are willing to sacrifice some of the expected return on investments to obtain these benefits. While diversification is a worthwhile investment goal, be careful not to invest in too many similar funds.
When you buy a mutual fund, you are already diversifying by investing in a group of
securities. If you invest in too many similar funds, you may be racking up extra costs in
fees and recordkeeping.
To summarize, we can explain trends in the way in which savers allocate their
funds among different assets by applying the theory of portfolio choice and diversification. As the wealth of the population grows, investors are more likely to substitute
luxury assets for necessity assets. Investments that reduce risk and information costs
and increase liquidity become popular vehicles for savers who seek high expected
returns. Shifts in taxation or differences in taxation among assets cause investors to
favor some securities over others.
KEY TERMS AND CONCEPTS
Beta
Determinants of portfolio choice
Diversification
Idiosyncratic (unsystematic) risk
Market (systematic) risk
Risk-averse savers
Risk-loving savers
Risk-neutral savers
Theory of portfolio allocation
Wealth elasticity of demand
SUMMARY
1. The theory of portfolio allocation helps to predict
how savers select assets to hold as investment. A
saver’s allocation of savings in a particular asset is
determined by (1) wealth (with greater responsiveness
for luxury assets than for necessity assets); (2) expected return on the asset relative to expected re-turns on
other assets; (3) risk of the return on the asset relative
to the returns on other assets; (4) liq-uidity of the
asset relative to other assets; and (5) cost of gathering
information about the asset relative to information
costs associated with other assets.
2. Diversification (holding more than one asset) reduces
the risk of the return on a portfolio unless the returns
on the individual assets move together perfectly. The
less the returns on assets move together, the greater is
the reduction in risk provided by diversification. This
reduction in risk is valued by risk-averse savers, who
are concerned not only about the expected return on
their savings (portfolio of assets), but also about the
variability of that return.
REVIEW QUESTIONS
QUIZ
1. What are the five key determinants of demand for a
particular asset?
2. What is the difference between a necessity asset and a
luxury asset? Give some examples of each.
3. What are the differences in being risk-averse, risk-neutral, and risk-loving? Which type of saver is likely to
own only stocks and stock options? Which type of
saver is likely to hold more bonds and cash than stocks?
4. U.S. citizens invest mostly in the U.S. stock market;
Japanese citizens invest mostly in the Japanese stock
market. Why do they do so if there are gains to diversification?
5. The saying “You shouldn’t put all your eggs in one
basket” is an example of what principle in investing?
What does it mean?
98
PART 2
Interest Rates
MOVING
FROM
THEORY
TO
PRACTICE
THE WALL STREET JOURNAL
...
SEPTEMBER 18, 2002
Europe’s Tender Equity Culture
While the bear market has bear market of the early
prompted greater regulato- 1970s, according to Credit
ry scrutiny in the U.S., it Suisse First Boston.
could have a deeper effect a
Fear of market forces
on Europe.
has returned. “We didn’t
Evidence is growing have a real risk culture
that big and small business, here,” says Bodo Heiss, the
government and individuals founder and chief executive
are retreating from a push of Tonxx. . . .
toward stock-market capiThe last time Europe
talism that began about five suffered such stock-market
years ago. . . .
woes, some 70 years ago, a
Certainly, a bear mar- seemingly well-developed
ket is a normal part of a stock-market culture unravmature equity culture, and eled. In 1913, initial public
Europe may quickly resume offerings, equity issues and
its nascent move toward bank deposits in several
stock-market
capitalism European countries all exonce markets turn around. ceeded those in the U.S., rel“What alternative is there?” ative to GDP. But then the
asks Jean Lemierre, the 1929 stock-market crash
president of the European and ensuing Depression, the
Bank for Reconstruction rise of the trade-union moveand Development.
ment, and World War II
But
for
many turned sentiment and govEuropeans, the past few ernments in Europe against
years have been the worst market forces. Over the next
bear market of their life- two decades, the role of govtimes. Since its peak in ernment in European econoMarch 2000, the stock- b mies grew, as leaders sought
market fall has wiped
to protect citizens from the
away more than $5 trillion whims of stock prices and
from the value of European foreign capital. . . .
companies. As a percentage
The market slump is
of gross domestic product, undermining another huge
that’s a bigger decrease than undertaking:
revamping
either the 1987 crash or the Europe’s pension systems.
In recent years, governments have acknowledged
that the widely used pay-asyou-go system—in which
workers and employers pay
a percentage of wages each
month to support current
retirees—isn’t sustainable.
When the equity culture
was spreading a few short
years ago, governments
began introducing marketfinanced elements, similar c
to the U.S. 401(k) plan. As
the stock market boomed in
the late 1990s, even skeptical unions ceded ground.
Europe’s new start-up
culture may need a while to
get going again as well.
Venture-capital firms invested nearly $50 billion in
European start-ups in 2000,
seven times the total of just
four years earlier. This year,
money has dried up.
All of this is crimping
Europe’s banks, which went
on a hiring binge as IPO
volume rose from 1995 to
2000. Then everything
stopped. Banks are in the
midst of laying off tens of
thousands. . . .
CHAPTER 5
ANALYZING
THE
The Theory of Portfolio Allocation
NEWS
Investors compare alternative assets
by considering expected returns, risk,
liquidity, and information costs. Many
financial planners in the United States
recommend that individuals place a
significant portion of their retirement
savings (the bulk of most households’
financial assets) in stocks. A majority
of U.S. households own stocks either
directly or through mutual funds.
Likewise, stockholding is popular in
the United Kingdom as well.
Continental European portfolio allocation remains much less weighted
toward equities. A budding equity culture in the 1990s was dealt a harsh
blow by the decline in stock indices
around the world over the 2000–2002
period. European households may be
passing up higher expected returns
and diversification over the long run
(and a weak equity market makes it
difficult for companies to raise equity
capital). The determinants of portfolio
choice offer some explanations.
99
...
a One possibility is that European for households, intermediaries such as
investors (say, in France or
Germany) have a different attitude
toward risk than U.S. investors—in particular, Germans are more risk-averse.
If European investors dislike risk,
stocks—with sometimes quite variable
returns—will be less preferred than
bonds.
b
mutual funds face lower transactions
costs, which, when passed on to individual investors, increase the demand
for stocks. Investors in the United
States went through this transformation during the 1960s–1990s.
For further thought…
Why might households’ unwillingness
When a stock market is illiquid,
to hold stocks depress venture capital
investors incur high costs of buying
funding of emerging companies?
and selling shares, lowering their
returns. Because the market for many Source: Excerpted from Christopher Rhoads, "Europe’s Tender Equity
European government bonds has much Culture," The Wall Street Journal, September 18, 2002. Copyright ©
2002 by Dow Jones and Co. Reprinted by permission.
lower costs of buying and selling for
individual investors, European households may be more likely to choose
bonds.
c
Tax breaks for retirement saving—like IRAs and 401(k) plans in the
United States—can give a stimulus to
the mutual fund business. While transactions costs for equities may eliminate the expected return differential
100
PART 2
Interest Rates
6. If the equity premium implied by the gap between the
return on stocks and the return on bonds is too large
to be explained by risk considerations alone, how
might it be explained?
7. What advantages do small investors see in owning
stock mutual funds as opposed to holding individual
stocks?
8. Despite the stock market boom of the 1990s, the fraction of household financial assets directly invested in
equities declined by 10 percentage points between
1970 and 1998. Can you think of reasons why this
happened?
9. What is the difference between market risk and idiosyncratic risk? Which type of risk can be reduced by
diversification?
10. Why don’t all risk-averse investors hold a fully diversified portfolio?
11. Would you expect the variability of returns on individual stocks traded on the New York Stock
Exchange to be greater or less than the variability of
the return on a portfolio consisting of all stocks traded on the exchange? Why or why not?
ANALYTICAL PROBLEMS
QUIZ
12. Suppose that your wealth elasticity of demand for
IBM stock is 2, you own 1000 shares of IBM stock,
and your total wealth is $1 million. You earn a
$100,000 bonus at work. How much more IBM
stock will you buy?
13. Suppose that you are an investor with a choice of
three assets that are identical in every way except in
their rate of return and taxability. Which asset yields
the highest after-tax return?
Asset 1: interest rate 10%, interest taxed at a 40% rate
Asset 2: interest rate 8%, interest taxed at a 25% rate
Asset 3: interest rate 6.5%, no tax on interest
14. Suppose that Asset 1 in Problem 13 had a return of 11%.
Would your answer change? If so, in what way?
15. U.S. government bonds with 30-year maturities used
to be sold with a call provision: After 25 years, the
government could call the bonds and make a final
interest payment plus principal repayment. When the
government eliminated the call provision in 1985, it
found that it could offer a different interest rate on
the bonds than it could before. Was the interest rate
higher or lower? Why? When was the government
likely to call the outstanding callable bonds?
16. Suppose that you have invested $1000 in Acme
Widget. Half of the time you earn a 20% return on
your Acme shares and half of the time you earn a 0%
return. Suppose that you have studied the returns on
the shares of Amalgamated Gidget and it turns out
that in years when the return on Acme’s shares was
20%, half the time the return on Amalgamated’s
shares was also 20% and half the time the return was
0%. Similarly, in years when the return on Acme’s
shares was 0%, half the time the return on
Amalgamated’s shares was 20% and half the time the
return was 0%. Would you be better off selling $500
worth of your Acme shares and investing the funds in
Amalgamated or would you be better off keeping all
of your funds invested in Acme?
17. In the mid-1980s, a new technique was developed
that divides payments on government coupon bonds
into two parts: One part consists of the coupon interest payments on the bonds, and the other part consists of the principal repayment on the bonds. Sold
separately, the two parts are worth more to investors
than the entire bond. Why?
18. Suppose that you are investing money in a portfolio
of stocks and are choosing from among Badrisk
Company, which returns 30% in good years and loses
50% in bad years; Worserisk Company, which
returns 30% in good years and loses 75% in bad
years; Norisk Company, which returns 10% all the
time; and Lowrisk Company, which returns 20% in
good years and loses 5% in bad years.
a. If you were completely risk-averse and your only
goal was to minimize your risk, which stock(s)
would you buy?
b. If you were risk-neutral, and good years and bad
years each occurred half the time, which stock(s)
would you buy?
c. If you were somewhat risk-averse, would you ever
have both Badrisk and Worserisk in your portfolio? Why or why not?
d. If you decided on a portfolio consisting of one- third
Badrisk, one-third Norisk, and one-third Lowrisk,
what would be your rate of return in good years? In
bad years? What would be your average rate of return
over all years if good years and bad years each
CHAPTER 5
The Theory of Portfolio Allocation
occurred half the time? If good years occurred 80% of
the time and bad years 20% of the time?
19. Suppose that you want to hold a stock portfolio for just
one year. You have $1000 to invest in stocks, and you
can choose to invest in Topgunner, Inc., which has
returns of 20% in good years and –10% in bad years,
or in Lowrunner, Inc., which has returns of 35% in
good years and –15% in bad years.
a. What is your return in a good year if you buy just
Topgunner? In a bad year? What is your return in
a good year if you buy just Lowrunner? In a bad
year? What is your return in a good year if you put
half your money in Topgunner and half in
Lowrunner? In a bad year?
b. Now suppose that, for every stock you buy, you
must pay transactions costs equal to $50. Repeat (a)
with your return reduced by these transactions costs.
What happens to your portfolio choice?
20. You are a member of an investment club that owns
shares in a firm that manufactures men’s clothing.
Explain the arguments for and against buying
a. shares in a company that manufactures women’s
clothing.
101
b. shares in a chemical manufacturing concern.
Which investment is more likely to decrease the overall risk of your club’s portfolio? Why?
21. Using the theory of portfolio allocation, state why
you would be more willing or less willing to buy a
share of IBM stock if you
a. win $1 million in the state lottery.
b. expect that stock prices will become more volatile.
c. expect the price of IBM shares to increase over the
next year.
d. read about new developments increasing the
liquidity of the bond market.
22. Using the theory of portfolio allocation, state why
you would be more willing or less willing to buy corporate bonds if you
a. expect interest rates on bonds to rise.
b. expect a large capital loss next month on the sale
of your house.
c. learn that the transactions costs of selling bonds
will increase.
d. expect inflation to increase significantly in the
future.
DATA QUESTIONS
23. You can find information about mutual funds through
advertisements in The Wall Street Journal. Such ads
invite you to write to the fund manager to obtain a
copy of the prospectus, which contains information
about the fund’s portfolio, management strategy, and
fees. Find a prospectus on the World Wide Web, or
locate one in your library for a fund specializing in
equities, and examine the list of stocks held. Is the fund
well diversified? How can you tell?
folio allocation decisions. Table 100 of each issue
breaks down this information on an annualized basis
over the most previous eight-year period. Look at this
table in the most recent release of this report at
http://www.federalreserve.gov/releases/z1/current/
data.htm. How has households’ allocation of various
investment vehicles changed during this time period?
How do these changes relate to changes in the business cycle during this period of time?
24. The Federal Reserve periodically publishes a summary of assets and liabilities of U.S. households and businesses. Locate a recent copy of Balance Sheets for the
U.S. Economy in your library. Calculate for the most
recent year available the ratio of foreign corporate
equities held to total corporate equities held. Using
the theory of portfolio allocation, explain why such a
small fraction of equity holdings of U.S. residents are
in non–U.S. stocks.
26. Information on mutual funds is ubiquitous. A popular internet site is Morningstar’s. Look at
Morningstar’s “Family Fund Data Pages” via
h
t
t
p
:
/
/
w
w
w
.
morningstar.com/Cover/Funds.html?topnav=funds
and click on a cross section of funds listed. Look at
each fund’s sector breakdown, and then evaluate the
degree of portfolio allocation in terms of risk and
return. Which funds would be preferred by investors
most concerned about wealth maximization?
Expected return? Degree of risk? Explain.
25. The Federal Reserve’s Flow of Funds Accounts report
provides quarterly information on household’s port-