Morningstar - Q3 2021 - 65

EXHIBIT 1
Morningstar Target Allocation Indexes and Their Constituent Weights (%)
The equity allocation of each target allocation
index is at the midpoint of the equity range
of its corresponding target allocation category.
Left
Asset Class
U.S. Equity
Morningstar Index
US Market
Developed Markets ex U.S. Equity DM x US
Emerging-Markets Equity EM
U.S. Bonds
Non-U.S. Bonds
Cash
Equity (Risk Score)
Source: Morningstar. Data as of 06/30/2020.
US Core Bond
Global x US Core Bond GR Hedged
US Cash
Extension
100
Morningstar Target Allocation Category
US Fund
Allocation-
15% to 30%
Equity
US Conservative
Target
US Fund
Allocation-
30% to 50%
Equity
Morningstar Target Allocation Index
US Moderately
Allocation
17.0
5.5
0.0
57.0
12.0
8.5
22.5
Conservative Target
Allocation
28.5
9.5
2.0
45.5
8.5
6.0
40.0
US Fund
Allocation-
50% to 70%
Equity
US Moderate
Target
Allocation
47.5
10.0
2.5
30.0
4.5
5.5
60.0
US Fund
Allocation-
70% to 85%
Equity
US Moderately
Aggressive Target
Allocation
56.0
17.0
4.5
15.0
3.0
4.5
77.5
US Fund
Allocation-
85%+
Equity
US Aggressive
Target
Allocation
69.0
19.0
4.5
3.5
0.0
4.0
92.5
Right
Extension
82.1
22.6
5.4
0.0
0.0
-10.0
110.0
Next, we run a simple regression of returns
on the returns-based style analysis benchmark
on the returns of the fund or portfolio
being analyzed to estimate additional parameters
needed to calculate the Morningstar Portfolio
Risk Score. To do this, we first calculate the time
series of returns on the benchmark:
i =1 xPi = xPi ≥ 0
RPt = ∑ m
∑ m
RPt = ∑ m
i =1 xPj Rjt
i =1 xPi = 1, xPi
i =1 xPj Rjt
∑ mRPt = ∑ NP
ˆ
ˆ
RPt = ∑ m
RPt = ∑ NP
Then, we run the following simple regression:2
RPt = ∑ m
S =
R = + + ut
A
Pt
RˆPt
i =1 xPj Rjt
R = + + ut
2
Pt
We estimate three parameters using this regression:
P RSP
P =
S =
1 . This is how much the benchmark needs to be
leveraged or unleveraged for the best fit between
its returns and those of the fund or portfolio. If
returns-based style analysis worked well, is very
close to 1.
RS =P
P =
AS =P
RS =P
AS =P
S + u
2
2
B
2
P
B
A RSP
P RSP
2
A RSP
P
2
P V P
2
S + u
RˆPt
2
P V P
A
+ et
A≥ 0
+ et
k =1 1,wk Rkt
A
i =1 xPj Rjt
k =1 wk Rkt
2
. This is the standard error of the regression
and the estimate of idiosyncratic risk.
u
3 R2
. This is the goodness-of-fit measure of
the regression and of the returns-based style
analysis model.
RPt = ∑ m
∑ m
We estimate systematic risk from the results of
the returns-based style analysis, from the
regression, and the covariance matrix for the asset
class indexes, estimated from long-term historical
monthly returns on the indexes, which we
denote V. Letting P
A
i =1 xPj Rjt
i =1 xPi = 1, xPi ≥ 0
o
RPt = ∑ NP
ˆ
RPt = ∑ m
Pt
k =1 wk Rkt
i =1 xPj Rjt
R = + + ut
RˆPt
S =
Calculating the Base Risk Score
The base risk score is measured against an
extended set of Morningstar Target Allocation
2
P =
RS =P
AS =P
˜ = F RF
W
W
˜ = F RF
RW
RW
˜ + H RH
˜
W
˜ + H RH
˜
W
-
-
W
W
L RL
˜
L RL
˜
S + u
2
P RSP
B
A RSP
P
2
P V P
2
class weights from the returns-based style analysis.
The systematic standard deviation of the fund
or portfolio is:
denote the vector of asset
o
A
+ et
Indexes. Morningstar publishes a set of indexes
for the U.S. and other markets that serve as
category benchmarks for each market's
target allocation categories. As EXHIBIT 1 shows,
there are five target allocation categories
for the U.S., each defined by an equity allocation
range. The equity allocation of each index is
the midpoint of its category's range. For example,
the midpoint of the 15% to 30% range is 22.5%.
Hence, the equity allocation of the Conservative
index is 22.5%.
f = n- qT
n = qT
f = n- qT
n = qT
n
P =
c
q
P =P(y; c, T) = ∑(1+( )
j=1
c
q
∑
∑
n
c
q
j=1
While the category's midpoint determines each
index's equity allocation, Morningstar bases
the allocations among the equity and fixed-income
asset classes on the average allocations
of the funds within each category. This is done
annually each July.
P(y; c, T) = ∑(( ) (1+
f-j
D(y; c, T) =
D(y; c, T) =
For the purpose of calculating the base risk score,
we extend the Morningstar Target Allocation
Index family by adding an index that is 100% cash
(the left extension) and one that is 110% equity
P(y; c, T) = P( )( (1+ ) )
1+ f
c
q
(
P(y; c, T)
1+
y
q
)
c
q
=
P(y; c, T)
1+
y
q
2 If the asset mix of the fund or portfolio came about through either (1) holding-based analysis or (2) by specifying the asset mix apart from any actual investments, we set
y; c,
c
q
=c (y)
P =
c
q
P =
( ( ) )=
P( )
(
1+ q
y; c,
1+ q
morningstar.com/lp/magazine )- q
c (y)
D =
c (y)
P( )
(
1
y
1
y
1
y
(
(
n
q
1+
1+ y
y
)
q
y
q
1+ y
q
)
)
f
P (y; c T)
f-j
y
n
q
-j
f
= 1,
f
65
u = 0, and R2
= 1.
-j
c
q
f-j
=y; c,
c
q
n
q
y; c,
(
y; c,
y
q
n
q
y
q
1+
n
q
y
q
)
P( )(1+
f-j
P( )(1+
y
q
y
q
)
)
f
f
c
q
∑
n
j=1
j=1
∑ n
j=1
j -f
q
y; c,
n
q
1+
j -f
q
y
q
c
q
)
P(y; c, T) = P( )( P(y; c, T)
( ) (1+
1+
c
q
y
q
)
f
f-j
P(y; c, T)
y
q
)
+ 1+( ) )
f-j
y
q
y
q
) ( (1+
f-j
y
q
y
q
y
q
+ T 1+
- qT
o+ T 1+
(
(
-j
j=1
n
+ 1+ + 1+ ) )
-n
f-j
y
q
- qTf
-j
+ 1+
(
y
q
)
-n
(
1+
y
q
)
f
y
q
y
q
)
)
- qT
- qT
http://www.morningstar.com/lp/magazine
Morningstar - Q3 2021

Table of Contents for the Digital Edition of Morningstar - Q3 2021
