Morningstar - Winter 2019 - 41

EXHIBIT 6

6

7

8

Investor 6's short position in Security F is due to
her strong negative preference for its exposure
to Characteristic Y.

The PAPM With Homogeneous Expectations
Expected Return
20%

The market portfolio under the PAPM with
heterogeneous expectations is the aggregation of
the portfolios of all market participants. In
general, this is not on the efficient frontier drawn
using correct expectations. (More on this later.)

CAP

Investor 1
Investor 5
Max. Sharpe Ratio

The portfolio under the PAPM Het that maximizes
the Sharpe ratio is on the efficient frontier
drawn using correct expectations.4 (Again, more
on this later.)

Investors 3 & 4
Investor 2

Market
Investor 6

Security B

Plotting Risk and Expected Return
E X H I B I T S 4 through 7 show the expected returns
and standard deviation of the securities
and the investor portfolios under each of the four
models (CAPM Hom, CAPM Het, PAPM Hom,
and PAPM Het).

All of the points shown in these four
exhibits are plotted using correct expectations.
In addition to the points representing
the securities and investor portfolios, each
exhibit includes:
gThe efficient frontier of risky securities.
gThe market portfolio.
gThe portfolio that maximizes the Sharpe ratio.
gThe capital market line; i.e., the line that
runs through the risk-free asset and that is tangent
to the efficient frontier at the portfolio that
maximizes the Sharpe ratio.
There are important and subtle differences
across the four charts.
We start with the CAPM Hom model in E X H I BI T 4 .
Here, the market portfolio and Sharperatio-maximizing portfolio are one and the same.
Because Investors 3, 4, 5, and 6 each have
the same moderate level of risk aversion as
the market average level of risk aversion,
they all hold the market/Sharpe-ratio-maximizing
portfolio. Investor 1 has lower-than-average

Risk-Free Asset

ICI
EFF

L
I TA

FR
ENT

MA

ONT

RKE

IER

T

E
LIN

Security A

Security F
Security D
Security C

10

Security E

Investor 5 likes Security E and Investor 6
dislikes Security F; thus, Security E
has a relatively low expected return,
Security F has a relatively high expected
return, and both investor portfolios
are off the efficient frontier.

0
0%
Standard Deviation

10

20

30

Source: Idzorek, Kaplan, and Ibbotson (2019).

risk aversion and chooses a levered position
in the market portfolio. Investor 2 has higher-thanaverage risk aversion and chooses an unlevered
position in the market portfolio.
Looking across EXHIB IT S 5 , 6 , and 7 , the risk and
expected return of at least two securities
are different relative to what EXHIB IT 4 shows;
as such, each of the corresponding efficient
frontiers is different. The three efficient frontiers
result in Sharpe-ratio-maximizing portfolios
that no longer correspond with the original
market portfolio nor the corresponding market
portfolio in a given model. This leads to
three distinct capital market lines, all of which
dominate the corresponding market portfolio
and show that the market portfolio is not
efficient when one incorporates either unique
forecasts or popularity preferences. (In the case

of PAPM Hom, which is shown in EXHIBIT 6 , the
plot point of the market portfolio turns out to
be very close to the plot point of the Sharpe-ratiomaximizing portfolio, making it difficult to
tell visually that it is beneath the tangency line.)
EXHIB IT 5 shows the CAPM Het model. We see
that all six of the investors are making at
least one forecasting error, creating portfolios that
aren't optimal. Investors 1, 2, 5, and 6 have
the same correct estimates of expected returns
for Securities A, B, E, and F, but they all
have estimated the wrong expected returns for
both Securities C and D. Investor 3 is the only
investor who recognizes Security C is underpriced,
and Investor 4 is the only investor who
recognizes Security D is overpriced-they are
more skilled and their portfolios are closer to the
efficient frontier.

4 The Sharpe ratio is the ratio of expected excess return (expected return minus the riskless rate) to standard deviation. Under CAPM Hom, the market portfolio is the portfolio that maximizes the
Sharpe ratio. In other models, the market portfolio and the portfolio that maximizes the Sharpe ratio differ.

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Morningstar - Winter 2019

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