Morningstar Magazine - June/July 2018 - 77

EXHIBIT 3

Investor and Market Portfolios Under the CAPM and the PAPM

popularity loading ( jk) and the corresponding
marketwide popularity premium (πk).

Investor 1 (%)

Investor 2 (%)

Investor 3 (%)

Market (%)

CAPM

PAPM

CAPM

PAPM

CAPM

PAPM

CAPM

PAPM

A

30

35

30

55

81

-17

35

36

B

23

23

23

24

62

59

27

27

C

18

17

18

16

47

54

21

20

D

12

11

12

9

31

42

14

13

E

3

2

3

0

8

20

3

3

Cjk = security j's value of characteristic k
xMj = weight of security j in the
E [ rj ] = market
rf + j portfolio
E [ rM ] - rf

14

11

14

-3

-139

-58

0

0

n

Security/Statistic

Cash
Source: Morningstar.

6 All investors can borrow and lend at the same

risk-free rate without limit.
The conclusions of the PAPM are:
1 The market portfolio does not maximize the Sharpe

ratio among all portfolios of risk assets.
2 Each investor forms a customized portfolio of the

risk assets that reflects his or her attitudes
toward each security characteristic. This portfolio
is combined with long or short positions in
the risk-free asset. Portfolio optimization is required
to find the overall investor-specific portfolio.

Note that the conclusions of the PAPM are
nearly the exact opposite of those of
the CAPM. Additionally, conclusion two is much
more consistent with observed investor portfolios.
E X H I B I T 2 illustrates conclusions one and two.
In this example, each of the five securities

= the number of securities in the market
p

π
E [ rjvalue
] = rfof+ characteristic
E [ rM ] - rf k+for
∑ the
The
j
jk kmarket
k=1
portfolio is:
has a single popularity characteristic, with security
A being the most popular, E being the most
unpopular, and B, C, and D falling in between.
The market portfolio is not on the Sharpe
ratio-maximizing tangent line. Neither are the
portfolios of investors 1 and 2, as is the case
in the CAPM. However, investor 3's portfolio is on
the tangent line because this investor is not
only risk-averse but has no other preferences for
security characteristics. Therefore, investor
3 holds an efficient portfolio.
E X H I BIT 3 shows the portfolios of the three
investors and the market portfolio under both the
CAPM and the PAPM in this example.

3 The expected excess return of each security

is a linear function of its beta and its popularity
loadings, which measure the popularity of the
security based on its characteristics relative
to the those of the beta-adjusted market portfolio.
The popularity loadings are multiplied by the
popularity premiums, which are aggregations of
the preferences of the investors regarding the
characteristics. (I explain popularity loadings and
premiums in detail below.) In this way, the market
aggregates investor preferences in determining
the influence of security characteristics on
the expected returns and prices of the securities.

The popularity loading of security j on
characteristic k is the difference between the
beta-adjusted market characteristic and the
security-specific characteristic. To see this, let:

In creating this example, I assumed that investors
1 and 2 both care about the popularity
characteristic, with investor 2 caring more about
it than investor 1. This is why investor 2 holds more
of the most popular stock A than investor 1.
In contrast, investor 3, who is unconcerned about
the popularity characteristic, holds security
A short, holds the other securities long in large
allocations, and levers the portfolio.
Expected Returns Under the PAPM
E [ rj ] = rf + j E [ rM ] - rf
Mathematically, conclusion 3 of the PAPM is:

E [ rj ] = rf +

p

j

E [ rM ] - rf + ∑

k=1

π

jk k

The firstn part of this equation is the same as the
CMk = . ∑Where
xMj Cjkthe PAPM differs is in the last
CAPM
j=1
term, which consists of p popularity characteristic
terms, each the product of a security-specific
= j cMk - Cjk
jk
m

∑ wi = 1

i=1

p

n

[ rj =] =∑ rxf + C j E [ rM ] - rf + ∑
CEMk
Mj jk

π

k=1

j=1

jk k

n

The
of security j on
CjkMk=popularity
= ∑j cMk
xMj-CCjkloading
jk
j=1
characteristic
k is:
m

∑ wi = 1
i =jk1 = j cMk - Cjk
1
m
Thus,
∑M w=i in= general,
1 m wi for a given characteristic,
∑
i=1
ithat
= 1 are
securities
more popular than the market
i
portfolio have negative popularity loadings,
1
=
and
less popular than the market
M thosemmthat
wi ware
i
π
=
∑
∑
ikE [ rE []r-
] r=f rf + j loadings.
E [ rM ] - rf
E k[ rj ] =M ihave
r=fi 1=+1 i jpositive
portfolio
i
M j popularity
E [ rj ] = rf + j E [ rM ] - rf
To see
how
m the popularity premiums are
p
Eπ[ r=j ] = ∑
r +wi j E [ rEM[]r-] r=f r + p E [ r ] - r + ∑ π
j r
f+ ∑ j ofπ M
f
jk k
ikthe
aggregations
preferences
r
-
E k[ rj ] =M i r=ff 1+ i of
E
[
]
k=1
j
M
f
jk k
k=
= 2.00%
+ jregarding
6.66% the characteristics,
p1
the
investors
we
E [ r ] = r + j E [ rM ] - rf + ∑ jk πk
first jneed fto define
the market
level of risk
n
k=1
E
[ rj ] =n rLet:
+ j E [ rCMMk] -= rf∑ xMj Cjk
f
aversion.
j=1
C = ∑ xMj Cjk
=EMk
j =n 1 r++ j 6.66%
E
[
r
]
-
r
+ jπ
rj ] =
[2.00%
f
j
M
f
Ci Mk == ∑ the
xMj risk-aversion
Cjk
parameter of investor i
= 2.00%
j = 1 + j 5.65% += j 4.50%
c - Cjk
jk
j Mk
m = the number
of investors
c
-
C
=
Mk
jk
wEjki[ r ]==j the
rf + fraction
E [ rMmof
] -wealth
rf + jheld
π by
j
j
investor
i;
∑
w
=
1
c
-
C
=
i
jk
i = 1+
=mjk2.00%j Mk
+ j 5.65%
4.50%
j
∑ wi = 1
i = 1 market level of risk aversion is:
The
1
m
=
M
m
w
∑ wi = 1
∑i = 1 i
i=1
i
1
=
M
m
wi
∑ i = 11
m
i
=
M
πk = M ∑ wi ik
m
w
∑i = 1 i
i=1 i
i
The popularity
premium
for characteristic k is:
m
w
i
πk = M ∑
ik
E [ rj ] = rf + j E [ rM ] - rf
i =m1 i
πk = M ∑ wi ik = 2.00% + 6.66%
j
i=1

i

E [ rj ] = rf +

j

= 2.00% +

6.66%

E [ rM ] - rf

E [ rj ] = rf + j j E [ rEM[]r-] r=f r + E [ r ] - r + π
global.morningstar.com/Morningstarmagazine
77
f
j
M
f
j
j
= 2.00% +

j

6.66%= 2.00% +

j

5.65% +

j

4.50%
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