Morningstar Magazine - June/July 2018 - 76

Strategies

From these assumptions, the following
conclusions emerge:

lend, depending on their risk preference. Because
in such a world we would only need two funds
(an index fund that tracks the market portfolio
and a money market fund), this result is sometimes
called the two-fund separation theorem.2

1 From among all possible portfolios of risky assets,

the market portfolio (that is, the capitalizationweighted combination of all risky assets
in the market) maximizes the Sharpe ratio; hence,
it is on the efficient frontier.

E X H I BI T 1 illustrates conclusions one and two
graphically. It shows a CAPM equilibrium in which
there are three investors (1, 2, and 3) and five
securities (A, B, C, D, and E). It shows that,
under the CAPM, the market portfolio is on the
MVO efficient frontier. Its location is the
point of tangency between the capital market line
and efficient frontier. The capital market line
is the line of tangency that emanates from
the risk-free rate on the vertical axis. As the chart
shows, not only is the market portfolio on the
capital market line, but so are the portfolios
of all investors. Investors who take more risk than
the market portfolio have portfolios above
the line, indicating that they hold levered positions
in it. In this example, investor 3 is the sole
investor who does this. Investors who take less

2 Each investor combines the market portfolio with

long or short positions in the risk-free asset.
Hence, investors do not actually need to perform
MVO to construct optimal portfolios.
3 The expected excess return of each security

is proportional to its systematic risk with respect
to the market portfolio (beta).
Conclusions one and two form the theoretical
basis for index investing with a single,
broad, cap-weighted market index. According
to the CAPM, all investors should hold
the market portfolio and either borrow or

risk than the market portfolio have portfolios below
the line, indicating that they hold delevered
positions in it. In this example, investors 1 and 2 do
this. Furthermore, because they have the
same risk tolerance, they have the same portfolio
of risky assets and cash.
The third conclusion of the CAPM is the bestknown. To state it formally, let:
E [ r˜j ] = the expected return on security j
= the risk-free rate of return
rf
= the systematic risk of security j
j
E [ r˜M ] = the expected return on the
market portfolio
According the CAPM, systematic risk is the only
factor that has an impact on expected return:
E [ rj ] = rf +

E [ rM ] - rf

As we shall see, in the PAPM, pthere can be
E [ rj ] = rf + j E [ rM ] - rf + ∑ jk πk
additional
factors that affect kexpected
returns.
=1
The PAPM
: Assumptions and Conclusions
n

EXHIBIT 2

Equilibrium Under the PAPM
Securities

Expected Return
25%

CMk PAPM
= ∑ xMj
The
is Ca jkgeneralization of the CAPM in which
j=1
securities have characteristics other than risk
and expected return that investors are concerned
about.
are:
c assumptions
- Cjk
= Its
jk
j Mk
1 Each
security has a bundle of characteristics.
m

Maximum Sharpe Ratio Portfolio

Investor 1

Investor 2

IO T
R AT
E
P
HAR
MS
IMU
TIER
X
A
M
FRON
T
N
D
CIE
EFFI
C
B

EN
ANG

NE
T LI

Investor 3

∑ wi = 1

i=1

2 Investors have preferences regarding these

characteristics
1 in addition to their preferences
=
M
m
wi expected return.
regarding
risk
∑ i = 1 and
i

3 All investors use a generalized form of MVO
E (least popular)

m
that incorporates
their preferences regarding
πk = M ∑ wi ik
security characteristics.
i=1 i

4 All investors have the same forecasts;

A (most popular)

0
0%
5
Standard Deviation

Source: Morningstar.

2 The two-fund separation theorem was first discovered by James Tobin (1958). It later became associated with the CAPM.

Morningstar June/July 2018

E [ rj ] = rf + j E [ rM ] - rf
that is, the same capital-market assumptions
= 2.00% + j 6.66%
(expected returns,
standard deviations,
and correlations).

E [ on
rM ] what
- rf +thej πcharacteristics
rj ] = rf + agree
E [ investors
5 All
j
of
securities
are. +
= the
2.00%
+ j 5.65%

4.50%

Table of Contents for the Digital Edition of Morningstar Magazine - June/July 2018