220 Part TWO Portfolio Theory

With $130 million to invest, Index will place .5556 $130 million $72.22 million in

BU shares. Table 7.4 shows a few other points on Index™s demand curve for BU shares. The

second column of the table shows the proportion of BU in total stock market value at each as-

sumed price. In our two-stock example, this is BU™s value as a fraction of the combined value

of BU and TD. The third column is Index™s desired dollar investment in BU and the last col-

umn shows shares demanded. The bold row corresponds to the case we analyzed in Table 7.1,

for which BU is selling at $39.

Index™s demand curve for BU shares is plotted in Figure 7.2 next to Sigma™s demand, and

in Figure 7.3 for TD shares. Index™s demand is smaller than Sigma™s because its budget is

smaller. Moreover, the demand curve of the index fund is very steep, or “inelastic,” that is, de-

mand hardly responds to price changes. This is because an index fund™s demand for shares

does not respond to expected returns. Index funds seek only to replicate market proportions.

As the stock price goes up, so does its proportion in the market. This leads the index fund to

invest more in the stock. Nevertheless, because each share costs more, the fund will desire

fewer shares.

Equilibrium Prices and the Capital Asset Pricing Model

Market prices are determined by supply and demand. At any one time, the supply of shares of

a stock is fixed, so supply is vertical at 5,000,000 shares of BU in Figure 7.2 and 4,000,000

shares of TD in Figure 7.3. Market demand is obtained by “horizontal aggregation,” that is,

for each price we add up the quantity demanded by all investors. You can examine the hori-

zontal aggregation of the demand curves of Sigma and Index in Figures 7.2 and 7.3. The equi-

librium prices are at the intersection of supply and demand.

However, the prices shown in Figures 7.2 and 7.3 will likely not persist for more than an

instant. The reason is that the equilibrium price of BU ($40.85) was generated by demand

curves derived by assuming that the price of TD was $39. Similarly, the equilibrium price of

TD ($38.41) is an equilibrium price only when BU is at $39, which also is not the case. A full

equilibrium would require that the demand curves derived for each stock be consistent with

the actual prices of all other stocks. Thus, our model is only a beginning. But it does illustrate

the important link between security analysis and the process by which portfolio demands,

market prices, and expected returns are jointly determined.

In the next section we will introduce the capital asset pricing model, which treats the prob-

lem of finding a set of mutually consistent equilibrium prices and expected rates of return

across all stocks. When we argue there that market expected returns adjust to demand pres-

sures, you will understand the process that underlies this adjustment.

Current BU Market-Value Dollar Investment* Shares

TA B L E 7.4 Price Proportion ($ million) Desired

Calculation of

$45.00 .5906 76.772 1,706,037

index demand

for BU shares 42.50 .5767 74.966 1,763,908

40.00 .5618 73.034 1,825,843

39.00 .5556 72.222 1,851,852

37.50 .5459 70.961 1,892,285

35.00 .5287 68.731 1,963,746

*Dollar investment BU proportion $130 million.

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7 Capital Asset Pricing and Arbitrage Pricing Theory

7.2 THE CAPITAL ASSET PRICING MODEL

The capital asset pricing model, or CAPM, was developed by Treynor, Sharpe, Lintner, and capital asset

Mossin in the early 1960s, and further refined later. The model predicts the relationship be- pricing model

tween the risk and equilibrium expected returns on risky assets. We will approach the CAPM (CAPM)

in a simplified setting. Thinking about an admittedly unrealistic world allows a relatively easy A model that relates

leap to the solution. With this accomplished, we can add complexity to the environment, one the required rate

step at a time, and see how the theory must be amended. This process allows us to develop a of return for a

security to its risk as

reasonably realistic and comprehensible model.

measured by beta.

A number of simplifying assumptions lead to the basic version of the CAPM. The funda-

mental idea is that individuals are as alike as possible, with the notable exceptions of initial

wealth and risk aversion. The list of assumptions that describes the necessary conformity of

investors follows:

1. Investors cannot affect prices by their individual trades. This means that there are many

investors, each with an endowment of wealth that is small compared with the total

endowment of all investors. This assumption is analogous to the perfect competition

assumption of microeconomics.

2. All investors plan for one identical holding period.

3. Investors form portfolios from a universe of publicly traded financial assets, such as

stocks and bonds, and have access to unlimited risk-free borrowing or lending

opportunities.

4. Investors pay neither taxes on returns nor transaction costs (commissions and service

charges) on trades in securities. In such a simple world, investors will not care about the

difference between returns from capital gains and those from dividends.

5. All investors attempt to construct efficient frontier portfolios; that is, they are rational

mean-variance optimizers.

6. All investors analyze securities in the same way and share the same economic view of the

world. Hence, they all end with identical estimates of the probability distribution of

future cash flows from investing in the available securities. This means that, given a set

of security prices and the risk-free interest rate, all investors use the same expected

returns, standard deviations, and correlations to generate the efficient frontier and the

unique optimal risky portfolio. This assumption is called homogeneous expectations.

Obviously, these assumptions ignore many real-world complexities. However, they lead to

some powerful insights into the nature of equilibrium in security markets.

Given these assumptions, we summarize the equilibrium that will prevail in this hypo-

thetical world of securities and investors. We elaborate on these implications in the following

sections.

1. All investors will choose to hold the market portfolio (M), which includes all assets of market portfolio

the security universe. For simplicity, we shall refer to all assets as stocks. The proportion The portfolio for

of each stock in the market portfolio equals the market value of the stock (price per share which each security is

times the number of shares outstanding) divided by the total market value of all stocks. held in proportion to

its market value.

2. The market portfolio will be on the efficient frontier. Moreover, it will be the optimal

risky portfolio, the tangency point of the capital allocation line (CAL) to the efficient

frontier. As a result, the capital market line (CML), the line from the risk-free rate

through the market portfolio, M, is also the best attainable capital allocation line. All

investors hold M as their optimal risky portfolio, differing only in the amount invested in

it as compared to investment in the risk-free asset.

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3. The risk premium on the market portfolio will be proportional to the variance of the

market portfolio and investors™ typical degree of risk aversion. Mathematically

2

E(rM) rf A* (7.1)

M

where M is the standard deviation of the return on the market portfolio and A* is a scale

factor representing the degree of risk aversion of the average investor.

4. The risk premium on individual assets will be proportional to the risk premium on the

market portfolio (M) and to the beta coefficient of the security on the market portfolio.

This implies that the rate of return on the market portfolio is the single factor of the

security market. The beta measures the extent to which returns on the stock respond to

the returns of the market portfolio. Formally, beta is the regression (slope) coefficient of

the security return on the market portfolio return, representing the sensitivity of the stock

return to fluctuations in the overall security market.

Why All Investors Would Hold the Market Portfolio

Given all our assumptions, it is easy to see why all investors hold identical risky portfolios. If

all investors use identical mean-variance analysis (assumption 5), apply it to the same universe

of securities (assumption 3), with an identical time horizon (assumption 2), use the same se-

curity analysis (assumption 6), and experience identical tax consequences (assumption 4),

they all must arrive at the same determination of the optimal risky portfolio. That is, they all

derive identical efficient frontiers and find the same tangency portfolio for the capital alloca-

tion line (CAL) from T-bills (the risk-free rate, with zero standard deviation) to that frontier,

as in Figure 7.4.

With everyone choosing to hold the same risky portfolio, stocks will be represented in the

aggregate risky portfolio in the same proportion as they are in each investor™s (common) risky

portfolio. If GM represents 1% in each common risky portfolio, GM will be 1% of the aggre-

gate risky portfolio. This in fact is the market portfolio since the market is no more than the

aggregate of all individual portfolios. Because each investor uses the market portfolio for the

optimal risky portfolio, the CAL in this case is called the capital market line, or CML, as in

Figure 7.4.

F I G U R E 7.4 E(r)

The efficient frontier

and the capital

market line

CML

M

E(rM)

rf

σ

σM

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7 Capital Asset Pricing and Arbitrage Pricing Theory

Suppose the optimal portfolio of our investors does not include the stock of some company,

say, Delta Air Lines. When no investor is willing to hold Delta stock, the demand is zero, and

the stock price will take a free fall. As Delta stock gets progressively cheaper, it begins to look

more attractive, while all other stocks look (relatively) less attractive. Ultimately, Delta will

reach a price at which it is desirable to include it in the optimal stock portfolio, and investors

will buy.

This price adjustment process guarantees that all stocks will be included in the optimal

portfolio. The only issue is the price. At a given price level, investors will be willing to buy a

stock; at another price, they will not. The bottom line is this: If all investors hold an identical

risky portfolio, this portfolio must be the market portfolio.

The Passive Strategy Is Efficient

A passive strategy, using the CML as the optimal CAL, is a powerful alternative to an active

strategy. The market portfolio proportions are a result of profit-oriented “buy” and “sell” or-

ders that cease only when there is no more profit to be made. And in the simple world of the

CAPM, all investors use precious resources in security analysis. A passive investor who takes

a free ride by simply investing in the market portfolio benefits from the efficiency of that port-

folio. In fact, an active investor who chooses any other portfolio will end on a CAL that is less

efficient than the CML used by passive investors.

We sometimes call this result a mutual fund theorem because it implies that only one mu- mutual fund

tual fund of risky assets”the market portfolio”is sufficient to satisfy the investment de- theorem

mands of all investors. The mutual fund theorem is another incarnation of the separation States that all

property discussed in Chapter 6. Assuming all investors choose to hold a market index mutual investors desire the

fund, we can separate portfolio selection into two components: (1) a technical side, in which same portfolio of

risky assets and can

an efficient mutual fund is created by professional management; and (2) a personal side, in

be satisfied by

which an investor™s risk aversion determines the allocation of the complete portfolio between

a single mutual

the mutual fund and the risk-free asset. Here, all investors agree that the mutual fund they fund composed

would like to hold is the market portfolio. of that portfolio.

While different investment managers do create risky portfolios that differ from the market

index, we attribute this in part to the use of different estimates of risk and expected return.

Still, a passive investor may view the market index as a reasonable first approximation to an

efficient risky portfolio.

The logical inconsistency of the CAPM is this: If a passive strategy is costless and effi-

cient, why would anyone follow an active strategy? But if no one does any security analysis,

what brings about the efficiency of the market portfolio?

We have acknowledged from the outset that the CAPM simplifies the real world in its

search for a tractable solution. Its applicability to the real world depends on whether its pre-

dictions are accurate enough. The model™s use is some indication that its predictions are rea-

sonable. We discuss this issue in Section 7.4 and in greater depth in Chapter 8.

<

1. If only some investors perform security analysis while all others hold the market Concept

portfolio (M), would the CML still be the efficient CAL for investors who do not en-

CHECK

gage in security analysis? Explain.

The Risk Premium of the Market Portfolio

In Chapters 5 and 6 we showed how individual investors decide how much to invest in the

risky portfolio when they can include a risk-free asset in the investment budget. Returning

now to the decision of how much to invest in the market portfolio M and how much in the

risk-free asset, what can we deduce about the equilibrium risk premium of portfolio M?

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We asserted earlier that the equilibrium risk premium of the market portfolio, E(rM ) rf ,

will be proportional to the degree of risk aversion of the average investor and to the risk of the

2

market portfolio, M. Now we can explain this result.

When investors purchase stocks, their demand drives up prices, thereby lowering expected

rates of return and risk premiums. But if risk premiums fall, then relatively more risk-averse

investors will pull their funds out of the risky market portfolio, placing them instead in the

risk-free asset. In equilibrium, of course, the risk premium on the market portfolio must be just

high enough to induce investors to hold the available supply of stocks. If the risk premium is

too high compared to the average degree of risk aversion, there will be excess demand for se-

curities, and prices will rise; if it is too low, investors will not hold enough stock to absorb the