Morningstar - Winter 2019 - 42

Strategies

EXHIBIT 7

The PAPM With Heterogeneous Expectations
Expected Return

Max. Sharpe Ratio
EF

IE
FIC

N
RO

TIE

Security C

Investor 4
Market

A
PIT

R
MA

INE

20%

Security A

Investor 3
Security F

Investor 1

Investor 2

Security E

Investor 5
Security B

Investor 6

For both the CAPM and the PAPM, allowing
heterogeneous forecasts means that not
all investors have correct forecasts, making the
market less than perfectly efficient.

Security D

This graph shows the full impact
of heterogeneous expectations
and preferences.

Risk-Free Asset

0
0%
Standard Deviation

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

shows the PAPM Hom, Investors 5
and 6 have distinct preferences for the
Characteristics X and Y exhibited by Securities
E and F, respectively, which cause them to
no longer hold the market portfolio. Instead, they
form unique portfolios reflecting their
preferences, resulting in an equilibrium that
differs from that of the standard CAPM.
Notice that the plot points of Securities E and F
differ from what they are in E X H I BI T 4 .
Investor 5 and 6 arrive at unique portfolios, and
Investors 1, 2, 3, and 4 find the Sharpe-ratiomaximizing portfolio that Investor 1 levers and
Investor 2 unlevers.
EXHIBIT 6

Finally, in E X H IB IT 7 , which shows the PAPM
Het, we can see the combined effects
of heterogeneous expectations and popularity
preferences simultaneously-that is, the
unique expected returns of Investors 3 and 4
for Securities C and D, as well as the preferences
of Investors 5 and 6 for Securities E and F.
Securities C, D, E, and F all have plots points

Morningstar Winter 2019

leads to investors arriving at unique portfolios that
reflect their own unique forecasts. This is vastly
different from the standard CAPM world, in
which all investors hold a portion of their wealth
in the market portfolio. Relaxing the same
assumption for the PAPM has no major impact on
the conclusions of the model-only that the
rationale for holding a unique portfolio is
expanded beyond unique preferences to include
unique forecasts. To put it bluntly, most of
the elegance and ease of the CAPM falls apart
when we allow heterogeneous forecasts, while
the PAPM is nearly unaffected.

that differ from what they are in the
standard CAPM ( EXHIB IT 4 ), and no investor
holds portfolios on the capital market line.
More Realistic Assumptions
The original CAPM and PAPM models with
homogeneous expectations have very
similar assumptions, yet due to the PAPM's ability
to consider investor preferences beyond risk
and return, the two models lead to very different
conclusions. Under the CAPM, the market
portfolio is the Sharpe-ratio-maximizing portfolio.
Under the PAPM, it is not. Likewise, under the
CAPM, mean-variance optimization is not
necessary to arrive at an investor's ideal portfolio,
but under the PAPM, it is. Finally, under
the CAPM, the expected excess return of each
security is proportional to its systematic
risk with respect to the market portfolio. Under
the PAPM, it is not.

Relaxing the CAPM assumption that all investors
share the same heterogeneous forecasts

Finally, the PAPM with heterogeneous
expectations can be viewed as an extension of
the standard CAPM, but it may be more useful
to view the CAPM as a special case of the
PAPM Het. By incorporating investor preferences
and allowing for heterogeneous forecasts,
the PAPM Het takes two major steps toward
realistic asset-pricing assumptions. 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
Black, F. 1972. "Capital Market Equilibrium with Restricted
Borrowing." Journal of Business, Vol. 45, No. 3 (July),
PP. 444 - 455.
Ibbotson, R.G., Idzorek, T.M., Kaplan, P.D., & Xiong, J.X.
2018. Popularity: A Bridge Between Classical and Behavioral
Research. CFA Institute Research Foundation.
Idzorek, T.M., Kaplan, P.D., & Ibbotson, R.G. 2019. "The
PAPM With Heterogeneous Preferences and Expectations."
Working paper, August.
Lintner, J. 1969. "The Aggregation of Investor's Diverse
Judgments and Preferences in Purely Competitive
Security Markets." Journal of Financial and Quantitative
Analysis, 4 (4), PP. 347- 400.
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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