Morningstar Magazine - June/July 2018 - 78

Strategies

∑ wi = 1

i=1

M

=

πk =

1
wi

m

∑i = 1

m

M

∑

wi

i=1

i

E [ rj ] = rf +

EXHIBIT 4

= 2.00% +
w
Expected Return Equations
πk = MUnder
∑ i ik the CAPM and the PAPM
m

i=1

E [ rM ] - rf

j

E [ rj ] = rf +

E [ rM ] - rf

6.66%
j

Beta

= 2.00% +

j

Expected
Return (%)

Beta

Popularity
Loading

Expected
Return (%)

E [ rM ]6.23
- rf + j π

0.63

-0.38

3.85

1.25

0.00

9.03

B

= 2.00%
1.25 +

C

1.14

9.62

1.15

0.23

9.54

D

1.24

10.25

1.26

0.50

11.38

E

0.94

8.28

0.97

0.69

10.61

Market

1.00

8.66

1.00

0.00

7.65

5.65% 10.33
+ j 4.50%

Source: Morningstar.

where ik is investor i's preference parameter for
characteristic k. This equation shows how
investors' preferences for popularity characteristics
have an impact on the popularity premiums
in proportion to their contributions to the market
level of risk aversion and the distribution
of wealth.
E X H I B I T 4 compares the equations for expected
returns under the CAPM and under the
PAPM in the example. Note how the most popular
security, A, has a negative popularity loading. In
addition, note how the three least-popular
securities, C, D and E, have positive popularity
loadings so these can be categorized as popular
securities. Also, note how, under the PAPM,
differences in popularity loadings can lead to
differences in expected returns larger than would
be expected given the differences in betas. For
example, security E has a higher expected
return than security B (10.61% versus 9.03%) even
though security E has a much lower beta than
security B (0.97 versus 1.25). This is consistent
with many empirical findings that differences in
betas do not explain differences in returns
and demonstrates how the theory of popularity
can explain this phenomenon.

c
q

M

4.50%
j

E [ r0.64
] = rf +
j
j

78

Morningstar June/July 2018

y
q

1+

c
q

=

(

P y; c,

n
q

)

-j

In the PAPM, an equilibrium emerges in which
the expected excess return of each security
is a linear function of its systematic risk
with respect to the market portfolio (beta) and
its popularity loadings, which measure the
popularity of the security based on its characteristics relative to those of the beta-adjusted
market portfolio.
As I illustrate, differences in popularity loadings
can cause differences in expected returns
greater than would be expected given differences
in betas. The coefficients on the popularity
loadings-the popularity premiums-are the
aggregated attitudes of investors toward
the nonrisk security characteristics. Furthermore,
the market portfolio does not maximize the Sharpe
ratio, as it is merely the aggregation across
investors of each investor's customized portfolio.
These conclusions have important practical
implications. One implication is that it
may be possible, by focusing only on risk and
expected return, to create portfolios that
are more efficient than market-weighted indexes.
Thus, one can profit by trading against
investors who consider security characteristics
other than risk and expected return. K

n
q

J.X. 2018. Popularity: A Bridge Between Classical
and Behavioral Finance. CFA Institute Research Foundation.
y
Pc ( y ) = y1 1 + q
Forthcoming.

(

(

Idzorek, T.M. & Ibbotson, R.G. 2017. "Popularity
= y1 1 +
Dc ( y ) and
Asset Pricing." The Journal of Investing, Vol. 26, Spring,
pp. 46-56.

f

)-

f
q

P ( y; c, T )

Markowitz, H.M. 1952. "Portfolio Selection." Journal of D ( y; c, T )
Dm (y; c, T ) =
y
Finance, Vol 7, pp. 77-91.
1+ q
Markowitz, H.M. 1959. Portfolio Selection: Efficient
∆P
Diversification of Investments. New York: John
≈ - Dm ∆y
P
Wiley & Sons.
Markowitz, H.M. 1987. Mean-Variance Analysis in PortfolioD
DV 01 = P
= P * Dm
Choice and Capital Markets, Oxford: Basil Blackwell Ltd.1 + y
q

Tobin, J. 1958. "Liquidity Preference as Behavior
Towards Risk." Review of Economic Studies, Vol. 25, No. 2
(February), pp. 65-86.
P ( y; c, T )

C (y; c, T ) =

P ( y; c, T )

C ( y; c, T ) =
c
q

n

∑ j =1

( )(
f-j
q

f- j-1
q

+

(

1
T T+ q

) (1 + )
y
q

y
q

- qT - 2

P ( y; c, T )

2
∆P
≈ - Dm ∆y + 12 C ( ∆y )
P

Wt = Ht + Ft
Wt = (1 + r) (Wt -1 - ct-1 )
T

-t

∑ (1 + r) ct-1 = W0

t=0

c0 = ∆ 0 W0

f- j

) (1 + )

P ( y; c, T )

-1

Paul D. Kaplan, Ph.D., CFA, is director of research with
Morningstar Canada. He is a member of the editorial board
of Morningstar magazine.

y
q

)

c, T ) =
Dm (y;
Kaplan, P.D. 2018. "Build Bridges, Not Walls."
Morningstar,
P ( y; c, T )
April/May, pp. 8-9.

Practical Implications

I formed the PAPM by extending the CAPM to
include investor preferences for security
characteristics other than risk and expected return.

) 1+

( )
=
μ
P ( y; c, )
Ibbotson, R.G., Idzorek, T.M., Kaplan, P.D., & Xiong,

A

j

y
q

References
P ( y; c, T )
Ibbotson, R.G. & Idzorek, T.M. 2014. "Dimensions
of Popularity." Journal of Portfolio Management, Vol. 40,
y
Special 40th Anniversary Issue, pp. 68-74.
1+ q

E [ rM ] - rf + j π

5.65% +
j

n
q

f- j

(1 + )

c
q

6.66%
j

PAPM

E [ rj ] = rf +

Security

j

i

CAPM

= 2.00% +

(

P ( y; c, T ) = P y; c,

ik

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