1. The market portfolio is efficient.
While these two statements often are thought of as complementary, they are actually sub-
stitutes because one can be derived from the other (one is true if and only if the other is
as well). We have focused on one direction, proceeding from the efficiency of the market
portfolio to the mean-beta equation. We now proceed from the mean return–beta relation-
ship to the efficiency of the market portfolio using the index-model market structure we
Deriving the CAPM is even more intuitive when starting from a single-index market.
Rather than beginning with investors who all apply the Markowitz algorithm to identical
input lists, suppose instead that they all face a market where excess stock returns, R
302
P A R T I I I
Equilibrium in Capital Markets
normally distributed and driven by one systematic factor. The effect of the macro factor is
assumed captured by the return on a broad, value-weighted stock-index portfolio, M.
The excess return on any stock is described by Equation 8.11 and restated here.
R
i
5 a
i
1 b
i
R
M
1 e
i
(9.9)
Each firm-specific, zero-mean residual,
e
i
, is uncorrelated across stocks and uncorrelated
with the market factor, R
M
. Residuals represent diversifiable, nonsystematic, or unique
risk. The total risk of a stock is then just the sum of the variance of the systematic com-
ponent, b
i
R
M
, and the variance of e
i
. In sum, the risk premium (mean excess return) and
variance are:
E( R
i
)
5 a
i
1 b
i
E(
R
M
)
s
i
2
5 b
i
2
s
M
2
1 s
2
(e
i
)
(9.10)
The return on a portfolio,
Q, constructed from
N stocks (ordered by
k 5 1, . . . ,
N ) with
a set of weights, w
k
, must satisfy Equation 9.11, which states that the portfolio alpha, beta,
and residual will be the weighted average of the respective parameters of the component
securities.
R
Q
5 a
N
k
51
w
k
a
k
1 a
N
k
51
w
k
b
k
R
M
1 a
N
k
51
w
k
e
k
5 a
Q
1 b
Q
R
M
1 e
Q
(9.11)
Investors have two considerations when forming their portfolios: First, they can diversify
nonsystematic risk. Since the residuals are uncorrelated, residual risk, s
2
(e
Q
) 5
g
N
k
51
w
k
2
s
2
(e
k
),
becomes ever smaller as diversification reduces portfolio weights. Second, by choosing stocks
with positive alpha, or taking short positions in negative-alpha stocks, the risk premium on Q
can be increased.
9
As a result of these considerations, investors will relentlessly pursue positive alpha stocks,
and shun (or short) negative-alpha stocks. Consequently, prices of positive alpha stocks will
rise and prices of negative alpha stocks will fall. This will continue until all alpha values
are driven to zero. At this point, investors will be content to minimize risk by completely
eliminating unique risk, that is, by holding the broadest possible, market portfolio. When all
stocks have zero alphas, the market portfolio is the optimal risky portfolio.
10
9
The systematic part of the portfolio is of no relevance in this endeavor, since, if desired, the beta of Q can be
increased by leverage (borrow and invest in
M ), or decreased by including in
Q a short position in
M. The pro-
ceeds from the short position in M can be invested in the risk-free asset, thus leaving the alpha and nonsystematic
risk unchanged.
10
Recall from Chapter 8 that the weight of a stock in an active portfolio will be zero if its alpha is zero (see
Equation 8.20); hence if all alphas are zero, the passive market portfolio will be the optimal risky portfolio.
9.2
Assumptions and Extensions of the CAPM
Now that we understand the basic insights of the CAPM, we can more explicitly iden-
tify the set of simplifying assumptions on which it relies. A model consists of (i) a set of
assumptions, (ii) logical/mathematical development of the model through manipulation
of those assumptions, and (iii) a set of predictions. Assuming the logical/mathematical
manipulations are free of errors, we can test a model in two ways, normative and positive.
Normative tests examine the assumptions of the model, while positive tests examine the
predictions.
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C H A P T E R
9
The Capital Asset Pricing Model
303
If a model’s assumptions are valid, and the development is error-free, then the predic-
tions of the model must be true. In this case, testing the assumptions is synonymous with
testing the model. But few, if any, models can pass the normative test. In most cases, as with
the CAPM, the assumptions are admittedly invalid—we recognize that we have simplified
reality, and therefore to this extent are relying on “untrue” assumptions. The motivation for
invoking unrealistic assumptions is clear; we simply cannot solve a model that is perfectly
consistent with the full complexity of real-life markets. As we’ve noted, the need to use sim-
plifying assumptions is not peculiar to economics—it characterizes all of science.
Assumptions are chosen first and foremost to render the model solvable. But we prefer
assumptions to which the model is “robust.” A model is robust with respect to an assump-
tion if its predictions are not highly sensitive to violation of the assumption. If we use
only assumptions to which the model is robust, the model’s predictions will be reason-
ably accurate despite its shortcomings. The upshot of all this is that tests of models are
almost always positive—we judge a model on the success of its empirical predictions.
This standard brings statistics into any science and requires us to take a stand on what are
acceptable levels of significance and power.
11
Because the nonrealism of the assumptions
precludes a normative test, the positive test is really a test of the robustness of the model
to its assumptions.
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