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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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Essentials of Investments, and Arbitrage Pricing Companies, 2003
Fifth Edition Theory

7 Capital Asset Pricing and Arbitrage Pricing Theory

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
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
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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222 Part TWO Portfolio Theory

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
E(rM) rf A* (7.1)

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




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Essentials of Investments, and Arbitrage Pricing Companies, 2003
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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-
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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224 Part TWO Portfolio Theory

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
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


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