Morningstar Magazine - June/July 2017 - 70

Strategies

A Primer on Factor Models and Portfolio
Optimization, Part III
In this wrap-up, we look at crosssectional regression and statistical
factor models.
QUANT U

Paul D. Kaplan

The February/March issue of Quant U was
the first of a three-part series on factor models
and their applications, especially portfolio
optimization. In the April/May issue, I discussed
some of the various multifactor models that
have been developed over the years, with
the goal of identifying factors that are pervasive
throughout the equity markets.
Broadly speaking, equity factor models fall into
three categories: (1) time-series regression
models, (2) cross-sectional regression models, and
(3) statistical factor models. In the April/May
issue, I discussed some of the time-series models.
In this issue, I discuss cross-sectional
regression and statistical factor models. I then
switch gears and discuss a popular factor
model for managed investment products and
portfolios, returns-based style analysis. I conclude
this issue (and this series) by discussing how
the results of a returns-based style analysis can
be used as inputs to an optimizer that forms
a portfolio of managed investment products based
on a target asset allocation.
Review of Factor Model Basics
As I discuss in the previous installments of this
series, all factor models have the following form:

m

r˜ it =

it

m

f˜ + u˜ it

+

The Morningstar Global Risk Model, discussed by
Lee Davidson (2016) in the October/November
issue of this publication, is a good example
of a cross-sectional regression model. Each month,
Morningstar estimates the realized values
of the factors using cross-sectional regressions of
stock returns on 36 factor loadings organized
in four groups, which are derived from the
variables listed in EXHIB IT 1 . How these variables
are translated into factor loadings varies by
group as follows:

=

ikt kt
k=1

Bikt

Akt

k=1

n
r˜ it = it + i r˜Mt + u˜ it
k = ∑ i=1 xi Bik
where:
= the excess return (total return
r˜it
less return on cash) of security i for
period t
=
the intercept for security i
it
in period t
m
= the number of factors
= the factor loading of factor k for
ikt
security i in period t
= the realization of factor k in period t
f˜kt
= the idiosyncratic component
u˜ it
of the return on security i in period t

Building a factor model consists mainly of
estimating the factors and the factor loading.
Cross-Sectional Regression Models

In cross-sectional regression models, in each
period t, the factor loadings (the ikt's) are
derived from fundamental and market
characteristics of the securities such as priceearnings ratio and market capitalization.
The realized values of the factors in period t
(the f˜ kt's) are estimated by treating the
above equation as a cross-sectional regression
model. In effect, these cross-sectional
regressions estimate factor realizations from
the returns on the stocks (the r˜it's).
2

=

i

2
M

ij

=

i

j

2
P

=

i

2

+

i

2

+ u˜ it

Style Factor Loadings
Morningstar normalizes each style factor loading
variable into a factor loading by subtracting
m
its cross-sectional
mean and dividing
by the
2
RBMit = Bik RAkt
i = Var[˜ui]
cross-sectional
standard
deviation.
The
resulting
k =1
factor loading of each stock, known as a
2
2
2
z-score, is the number of standard
A = deviations
M + S
from the average.
m

m

( ksign
- k )of( the
M = variables,
For some
q - underlying
q ) k q kq
k=1 ∑ q=1the
variable is flipped so that a positive factor
n
2
2 exposure to a factor that
loading
positive
i
S = means
i=1 i
has a positive premium. For example, according
to the size effect, small-cap stocks have a
positive premium over large-cap stocks. Therefore,
we want small-cap stocks to have a positive
factor loading on the size variable. For this
reason, the variable for size is minus the logarithm
of market capitalization.
Sector Factor Loadings
The 11 sector factors measure the economic
exposure of a company to the 11 Morningstar
sectors. For a given stock, the 11 factor
loadings are 11 sector weights, each between
0% and 100% and all summing to 100%.
Morningstar estimates these weights using a
technique that takes a 100% allocation to
the sector assigned by Morningstar analysts as
a starting point for a regression of the returns
of the stock on the returns on the Morningstar
Sector indexes.¹
Region Factors
The seven region factors represent the economic

IRSt

PRSt
TRSt
PZt-∆

TRZt (

IRZt (

IRZt (

PRZt (

PBt -∆
+ 1

2
M

1 The technique is called Bayesian regression. For a general discussion of Bayesian ideas, see Kaplan (2016b).
n

70

Morningstar June/July 2017

n

n

wi wj
i =1 j =1

i

2
j M

+

wi 2
i =1

i

2

TRBt (



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