Morningstar - Fall 2019 - 42

Strategies

Why the CAPM Falls Flat
Fischer Black exposed a bias in a
measure of risk-adjusted return.

QUANT U

Paul D. Kaplan

The Capital Asset Pricing Model is one of the
most influential models in finance. It makes
strong assumptions about investor behavior from
which it derives strong conclusions about
portfolio construction and expected returns. Its
assumptions are:
A1 Transaction costs and other illiquidities can

be ignored.
A2 All investors hold mean-variance efficient portfolios.

Security B) are risky securities with expected
returns and standard deviation of returns shown in
EXHIB IT 1 . The third security is the risk-free
asset. As EXHIB IT 1 shows, the risk-free asset has
a return of 3% and standard deviation of returns
of zero.

In this issue, I discuss an alternative version of the
CAPM in which A4 does not hold, leading
to different conclusions. This version of the CAPM
was developed by the late Fischer Black (1972),
so it is sometimes called the Black CAPM. For
reasons that will become evident in my discussion
here, it is also called the Zero-Beta CAPM.
A Three-Security Example
In the previous Quant U, I illustrated the impact of
dropping A4' with a three-security example.
Here, I demonstrate the impact of dropping A4
with another three-security example.1 In the
example here, two of the securities (Security A and

Because there are only two risky securities, the
efficient frontier of risky securities consists
entirely of combinations of them. The curve in
EXHIB IT 1 represents all combinations of Security A
and Security B, starting with negative 20%
in Security A and 120% in Security B at the upper
part of the curve and ending with 130% in Security
A and negative 30% in Security B at the lower
part of the curve. The minimum variance portfolio,
labeled "Min. Var. Port.," is at about 86% in
Security A and 14% in Security B. Points along the
curve above the minimum variance portfolio
constitute the efficient frontier. This is why I made
the part of the curve above the minimum variance
portfolio solid and the part below it dotted.
In the standard CAPM, there is a risk-free asset
that investors can combine with risky assets

A3 All investors hold the same (correct) beliefs about

means, variances, and correlations of returns
of securities.
A4 Every investor can lend all she or he has or can
borrow all she or he wants at the risk-free rate.
A4' Investors can sell short without limit and use the
proceeds of the sale to buy long positions.

EXHIBIT 1

Risk vs. Expected Return With Two Risky Securities and a Risk-Free Asset
Expected Return

Its two main conclusions are:
C1 Each investor holds the market portfolio of risk

assets in combination with a risk-free asset
(long or short).
C2 The expected return in excess of the risk-free
return on each security is the security's
systematic risk with respect to the market portfolio
(beta) times the expected excess return on the
market portfolio.
In the Summer 2019 issue of Quant U, I discussed
how Harry Markowitz, the father of portfolio
theory, showed that the conclusions of the CAPM
do not hold if the assumption A4' is violated.

1 Both illustrations are based on Markowitz (2005).

42

Morningstar Fall 2019

CAP

Max. Sharpe Ratio
Security A

E F F IC IE

I TA L

NT
NT FRO

M

IE R

ET
ARK

14%

LINE

Security B

7

Market Port.

Min. Var. Port.
Zero-Beta Portfolio

Risk-Free Asset

0
0%
Standard Deviation
Source: Morningstar.

10

20

30

40
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