Morningstar - Q3 2021 - 64

Strategies
A Ruler for Risk
The Morningstar Portfolio Risk Score can
be applied in many investment settings.
QUANT U
Paul D. Kaplan
Diversification is said to be the only free lunch
in investing. A diversified portfolio of investment
products representing different asset classes
has lower risk than does the portfolio-weighted
average of the risk of its components.
To measure the risk of a portfolio and its
constituents, Morningstar developed
the Morningstar Portfolio Risk Score, which is
based on the following principles:
1 There are two sources of risk: systematic
and unsystematic.
2 Asset allocation determines systematic risk.
3 Unsystematic risk is unrelated to asset allocation.
It can occur as the result of an undiversified
exposure within an asset class or to active
management.
The risk score is an investor-friendly ruler for risk.
Because it accounts for both the systematic
and unsystematic sources of risk, it can be applied
to single-asset-class products, such as
mutual funds and exchange-traded funds, and
to multi-asset-class portfolios.
The risk score is calculated in three steps:
1 Using returns-based style analysis and regression
analysis on the historical returns of a portfolio
or investment product, we estimate the portfolio's
asset allocation and unsystematic risk.
We conduct the returns-based style analysis to
estimate asset allocation using the constituents
of the Morningstar Target Allocation Indexes.
2 We compare the standard deviation of
the estimated asset allocation to the standard
deviations of the Morningstar Target Allocation
Indexes and find a blend that has the same
standard deviation as the fund or portfolio
in question. The equity allocation of the blend of
the indexes is the base risk score.
3 We calculate how much leverage would need
to be applied to the blend of Morningstar
Target Allocation Indexes to have the same total
risk (systematic and unsystematic) as the fund
or portfolio. We then multiply the leverage
ratio by the base risk score to get the Morningstar
Portfolio Risk Score.
Below, I describe these steps in detail and then
illustrate with an example.
Estimating Systematic and Unsystematic Risk
To estimate the systematic risk of a fund
or portfolio, we first estimate its asset allocation.
We generally do this by using returns-based
style analysis, followed by a regression analysis.
Returns-based style analysis is a method
of estimating a fund's or portfolio's asset class
exposures using historical returns. It is similar
to a regression model:
RPt = ∑ m
∑ m
i =1 xPj Rjt
A
+ et
1 In some applications, if there are less than 48 months of returns on a fund being analyzed or on the constituents of a portfolio being analyzed, the missing returns are filled in with the returns
on proxies.
i =1 xPi = 1, xPi ≥ 0
RPt = ∑ NP
64
Morningstar Q3 2021
ˆ
RPt = ∑ m
k =1 wk Rkt
i =1 xPj Rjt
A
max
A
,
B
s.t. =
U EW
(
A'
(T +A B ) VW (T +A B ))
A ≥ 0,
,
,
B' = 1 - ,
B ≥ 0
P
where:
Rpt
m
xPi
Rit
et
A
= the return on the fund or portfolio
in month t
= the number of asset classes
= the estimated weight on asset class i
for the fund or portfolio
= the return on the index for asset class i
in month t
= the error term in month t
In each market in which we calculate the
Morningstar Portfolio Risk Score, we
use the indexes that are the constituents of the
Morningstar Target Allocation Indexes in
that market. EXHIBIT 1 shows the five constituent
indexes of the target allocation series for
the United States.
Returns-based style analysis finds the asset
class weights that minimize the variance
of et
RPt = ∑ m
∑ m
A
are non-negative and sum to 100%:
+ et
i =1 xPi = 1, xPi ≥ 0
RPt = ∑ NP
RPt = ∑ m
We perform the returns-based analysis with up
to 48 months of returns.1
k =1 wk Rkt
RPt = ∑ m xPj Rjt
ˆ
For a portfolio, we form a time series of returns
from its constituents using the most recent set of
portfolio weights:
i =1 xPj Rjt
i =1
∑ mR = + + ut
Pt
S =
i =1 xPi = 1, xPi ≥ 0
P V P
RˆPt
RPt = ∑ NP
P =
ˆ
RPt = ∑ m
RS =P
where:
NP
Pt
wk
Rkt
S =
If the asset allocation of the portfolio is specified
without reference to any actual investments,
neither holdings-based analysis nor returns-based
style analysis is necessary.
A RSP
P =
S + u
RS =P
AS =P
RW
P RSP
B
2
2
˜ = F RF
W
P ˜ + H RH
˜
W
-
W
L RL
˜
f
n
P
k =1 wk Rkt
2
S + u
2
i =1 xPj Rjt
P RSP
R = + + ut
AS =P
B
2
2
= the number of portfolio constituents
A RSP
RˆPt
= the weight on constituent k
P V P
P
= the return on constituent k
2
2
A
A
A + et
, subject to the constraints that the weights
i =1 xPj Rjt

Morningstar - Q3 2021

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Contents
Morningstar - Q3 2021 - Cover1
Morningstar - Q3 2021 - Cover2
Morningstar - Q3 2021 - 1
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Morningstar - Q3 2021 - Cover3
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