Morningstar - Summer 2019 - 62

Strategies

EXHIBIT 3

Beta vs. Expected Return
Expected Return
15%

O
KET PORTFOLI
T TO THE MAR
WITH RESPEC

Security 1

Security 2
WIT

S
H RE

HE
TO T
PECT

T

EN
ANG

OR
CY P

TFOL

10

Security 3

IO

5

0
0
Beta

0.5

1.0

1.5

2.0

Source: Morningstar.

two portfolios to determine the unconstrained
efficient frontier line.
Returning to E X H I B I T 1 , note the location of the
portfolios of two investors, Investor 1 and
Investor 2. Investor 1, being somewhat risk-averse,
holds a portfolio on Efficient Frontier Part I.
Investor 2, being less risk-averse and being subject
to the constraints against short selling, holds a
portfolio along Efficient Frontier Part II Constrained.
Now suppose that these two investors have equal
wealth and comprise the entire market. Then,
as E X H I B I T 1 shows, the market portfolio is not on
the efficient frontier. We can also see this in
E X H I B I T 2 where the market portfolio lies below
both efficient frontiers.
What This Means for Beta
A major conclusion of the CAPM (which Markowitz
calls C2) is that there is a linear relationship
between beta and expected return, beta being by
the sensitivity of a security's return to the
return on the market portfolio. In fact, if we use
any portfolio on the unconstrained efficient as the
benchmark, there will be a linear relationship
between beta with respect to that benchmark and
expected return. In our example, because
the tangency portfolio is on Efficient Frontier Part I,
which is part of the unconstrained frontier,
as E X H I B I T 3 shows, there is a linear relationship
between beta with respect to the tangency

62

Morningstar Summer 2019

portfolio. In the CAPM, because the market
portfolio is on the efficient frontier, the linear
relation holds.

assets. But due to restrictions on short selling,
even the true market portfolio could be inefficient.
2 Because there is no linear relationship between

However, in our example, the market portfolio
is not efficient. Therefore, the linear
relationship between beta with respect to the
market portfolio and expected return does
not hold. Markowitz illustrated this by drawing
the dotted line segments that connect the
points for the three securities in EXHIB IT 3 .
To drive the point home, in EXHIB IT 3 , I plot
beta with respect to the market portfolio versus
expected return on the 66 long-only portfolios
with allocations that are multiples of 10%,
such as 10% in Security 1, 30% in Security 2, and
60% in Security 3. The scattering of points
clearly shows that if the market portfolio is not
on the unconstrained efficient frontier, there
is no relationship between beta and expected
return as predicted by the CAPM.
Markowitz's Conclusions
Based on his analysis that I have restated here,
Markowitz draws the following conclusions:
1 The frequent explanation for the empirical failure

of the CAPM's prediction that expected returns
are a linear function of beta is that the empirical
tests use benchmarks that are not complete
representations of the true market portfolio of all

beta and expected return, the practice of
using beta to form a risk-adjusted expected return
as a cost of capital or discount rate in corporate
finance or valuation practice is not valid.
3 Despite its drawbacks, the CAPM should still be

taught as a base case. But finance students should
also be shown the effects of prohibiting short
selling on the conclusions of the CAPM.
In a future issue of Quant U, I will explore the
CAPM's assumption that there is a risk-free
rate that investors can borrow and lend at without
limit. I will also explore the consequences of
investors holding different options on the expected
returns of securities. K
Paul D. Kaplan, Ph.D., CFA, is director of research with
Morningstar Canada. He is a member of the editorial board
of Morningstar magazine.
References
Friedman, M. 1953. Essays in Positive Economics. Chicago:
University of Chicago Press.
Markowitz, H.M. 2005. "Market Efficiency: A Theoretical
Distinction and So What?" Financial Analysts Journal,
Vol. 61, Issue 5 (September/October), PP. 17-30.
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