Morningstar - Q4 2021 - 60

Strategies
EXHIBIT 5
Addendum
In the previous issue of Quant U, I described the
methodology for calculating the Morningstar Portfolio Risk
Score. Here, I give a brief summary.
The Morningstar Portfolio Risk Score is tied to the
multi-asset class families of Morningstar Target Allocation
Indexes. We expand the indexes' family to include
an index that is 100% cash and one that is 110% equity.
We calculate the portfolio risk score in three steps. In the
first step, we use Returns-Based Style Analysis, or
RBSA, to estimate the asset allocation of the investment
being analyzed.2
Morningstar Portfolio Risk Score Versus Expected Geometric Mean Return
With Risk-Comfort Range for Average Investor
Morningstar Portfolio Risk Score = 25
We run RBSA using the constituent asset
class indexes of the Target Allocation Indexes over 48
months. From the results of RBSA, we form an RBSA-based
benchmark. We then estimate unsystematic risk
estimated by regressing the RBSA-based benchmark
returns on the investment being analyzed.
) MPRST
-
For the second step, we use long-term return data on the
asset class indexes that comprise the Target Allocation
Indexes to form an asset-class covariance matrix. Then, from
the RBSA weights, beta from the post-RBSA regression,
and this covariance matrix, we estimate the systematic risk
of the investment being analyzed. We also estimate the
standard deviation of each Target Allocation Index from the
asset-class weights and the covariance matrix. We then
find the blend of two consecutive Target Allocation Indexes
that have the standard deviation as the beta-adjusted
RBSA benchmark. We call the equity allocation of the
blended benchmark the Base Risk Score. However, if the
systematic risk of the investment being analyzed is
greater than that of the Target Allocation Index that is 110%
equity, the Base Risk Score is 110.
∑ m
i =1 xPj Rjt
A
+ et
)
2
In the third and final step, we adjust the Base Risk Score
to account for the unsystematic risk that we estimated
in the the first step. We do this by applying leverage
to the blend of Target Allocation Indexes that form the Base
Risk Score. The leverage ratio that we use compares
the total risk (systematic risk and unsystematic risk taken
together) of the investment being analyzed to the risk
of the blend of Target Allocation Indexes. The Morningstar
Portfolio Risk Score is the Base Risk Score times the
leverage ratio. Mathematically, we have:
(
s.t.
EW (T + ,
A' = ,
A
B ) VW
(T +A
A
B ))
B' = 1 - , ≥ 0,
B ≥ 0
Morningstar Portfolio Risk Score = 65
MPRS =
Base
×
Risk Score
Sys. Risk2 + Unsys. Risk2
Blended Benchmark Risk
Leverage
Adjustment
US Large-Cap Growth Fund
US Large-Cap Value Fund
Foreign Large-Cap Growth Fund
Emerging-Markets Fund
US Short-Term Bond Fund
US Intermediate-Term Bond Fund
US High-Yield Bond Fund
Source: Morningstar.
c
q
(
P(y; c, T)
1+
y
q
)
=
c
q
(
c (y)
P =
D =
c (y)
y
1+ q
P(y; c,
(
1
y
1
y
(
f-j
=
)
n
q
-j
)
1+ y
q
y
q
)
+ B ))
60
Morningstar Q4 2021
Dm (y; c, T) =
f
0%
1+ )- q
f
50
100
2 RBSA was proposed by Nobel laureate William Sharpe as a means of estimating asset allocation. See Sharpe, W.F. 1988. " Determining a Fund's Effective Asset Mix. " Investment Management
Review, December, and Sharpe, W.F. 1992. " Asset Allocation: Management Style and Performance Measurement. " The Journal of Portfolio Management, Winter.
P(y; c, T)
P (y; c, T)
c
q
P(y; c, T) = P(y; c,
US Large-Cap Growth Fund
US Large-Cap Value Fund
Foreign Large-Cap Growth Fund
Emerging-Markets Fund
US Short-Term Bond Fund
US Intermediate-Term Bond Fund
US High-Yield Bond Fund
0%
Morningstar Portfolio Risk Score = 50
US Large-Cap Growth Fund
US Large-Cap Value Fund
Foreign Large-Cap Growth Fund
Emerging-Markets Fund
US Short-Term Bond Fund
US Intermediate-Term Bond Fund
US High-Yield Bond Fund
f = n- qT
n = qT
Morningstar Portfolio Risk Score = 55
P =
n
o
US Large-Cap Growth Fund
US Large-Cap Value Fund
Foreign Large-Cap Growth Fund
Emerging-Markets Fund
US Short-Term Bond Fund
US Intermediate-Term Bond Fund
US High-Yield Bond Fund
c
q
∑
j=1
n
P(y; c, T) = ∑(1+
j=1
c
q
D(y; c, T) =
∑ n
j=1
j -f
q
y
q
)
0%
n
q
)(1+
(
P(y; c,
AS Not Taken Into Account
f-j
y
q
)
1+
n
q
y
q
)
)(1+
y
q
)
f
f
AS Taken Into Account
f-j
+ 1+
(
( ) (1+
c
q
y
q
)
P(y; c, T)
50
100
y
q
f-j
)
- qT
-j
0%
+ 1+
(
AS Not Taken Into Account
1+
y
q
)
-n
(
y
q
)
f
50
AS Taken Into Account
100
AS Not Taken Into Account
50
AS Taken Into Account
100
AS Not Taken Into Account
AS Taken Into Account
+ 1+T
(
y
q
)
- qT
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2021q3/index.php#/p/64
Morningstar - Q4 2021

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