Morningstar - Q3 2021 - 67

RPt = ∑ k =1
ˆ
RPt = ∑ m
Pt
wk Rkt
RPt = ∑ i =1
i =1 xPj Rjt
R = + + ut
RˆPt
S =
P =
S + u
2
P V P
Let RSP denote the base risk score of the fund or
portfolio. The Morningstar Portfolio Risk Score is:
2
R =SP
AS =P
P RSP
B
2
EXHIBIT 2 illustrates how the Morningstar Portfolio
Risk Score is calculated. The line emanating
from cash and running through the base asset mix
shows possible combinations of the base
asset mix and cash, either long or short. Points
along the line to the right of the base asset
mix represent leveraged positions. The point
labeled " Morningstar Portfolio Risk Score "
represents the leveraged position with the same
risk as the total risk of the fund or portfolio
in question.
A RSP
P
2
˜ = F RF
W
RW
˜ + H RH
˜
W
-
W
L RL
˜
max
A
,
B
s.t. =
U EW
(
A'
(T +A B ) VW (T +A B ))
A ≥ 0,
,
,
B' = 1 - ,
RPt = ∑ NP
ˆ
RPt = ∑ m
Pt
k =1 wk Rkt
in question relative to the base asset mix
is due to (1) differences in asset allocation (misfit
risk) and (2) unsystematic risk. These two
taken together constitute active risk, which
I denote
i =1 xPj Rjt
R = + + ut
2
RˆPt
S =
a fraction of total risk in units of the Morningstar
Portfolio Risk Score:
P RSP
P =
RS =P
B
2
AS =P
A RSP
P
2
An Example
To demonstrate the Morningstar Portfolio Risk
Score, I collected historical returns on seven
U.S. funds, each representing a different
Morningstar Category, and calculated their risk
scores, which is shown in EXHIBIT 3 . As expected,
the four equity funds have relatively high
scores and the three bond funds have relatively
low scores.
B ≥ 0
u =(c)
1 +
1
ct
{ln(c), O = 1
ct+1
~
ct
- 1
O
c -1
1-
1- 1
O
1
O
=
Qt+1
~t
, O ≠ 1
I then combined these funds to form a family
of target-risk portfolios ranging from a conservative
portfolio to an aggressive portfolio. EXHIBIT 3
shows the weights that I assigned to the funds.
I then calculated the weighted average risk
score of each portfolio, the actual score of each
portfolio, and the difference between the
two. This difference measures the reduction in
risk from diversification. In all cases, the
diversification benefit is positive but lower for
the riskier portfolios.
ct+1
~
=
~
R =
Ot+1
Et
ct+1
~
Et
=
(1 + )
1
O
( )
Qt+1
~t
( )
( )
Qt+1
~t
Qt+1
~t
Qt+1
~t
- O
1 - O
- 1
ct+1
~
=
The Alignment Score
ct
In addition to the Morningstar Portfolio Risk
Score, we calculate a complementary measure,
the Alignment Score, which measures how
closely aligned the fund or portfolio in question
is with the base asset mix used to calculate
the risk score. The base asset mix selected along
the continuum of Morningstar Target Allocation
Indexes when calculating the base risk score
serves as a custom benchmark for the portfolio
being analyzed. The risk of the fund or portfolio
( )
(1 + )O
1 - O
(
1 + R )
~
Ot+1
~
R =
Ot+1
Et
ct+1
~
ct
Short-Term
Bond Fund
U.S. Et
=
ct
- O
EXHIBIT 4 plots the Morningstar Portfolio Risk
Score versus the Alignment Score for the seven
funds and five portfolios presented in EXHIBIT 3 .
As you can see, while the Alignment Score
for single-asset-class funds can be quite large,
the scores of diversified portfolios of funds
can be very low. This is because diversification
both reduces unsystematic risk, and results in
asset mixes that are similar to those represented
by the Target Allocation Index continuum.
,
˜ = F RF
W
RW
˜ + H RH
˜
W
22.5% Equity
Conclusion
The Morningstar Portfolio Risk Score measures risk
with respect to benchmarks derived from a set
max
A
,
B
s.t. =
U EW
(
A'
(T +A B ) VW (T +A B ))
A ≥ 0,
,
B' = 1 - ,
B ≥ 0
DV01 P=
0.0% Equity (Cash)
EXHIBIT 4
u =(c)
0%
1 +
1
Morningstar Portfolio Risk and Alignment Scores for Funds and Portfolios
Standard Deviation
=
{ln(c), O = 1
3
ct+1
~
ct
(1 + )
1
- 1
O
c -1
1-
1- 1
O
1
O
Qt+1
~t
O
( )
Qt+1
~t
( )
( )
Qt+1
~t
Qt+1
~t
Qt+1
~t
- O
1 - O
- 1
U.S. High-Yield Bond Fund
U.S. Intermediate-Term
Bond Fund
( )
(1 + )O
1 - O
(
Conservative
Portfolio
Moderately
Conservative
Portfolio
Ot+1
40
Morningstar Portfolio Risk Score
Source: Morningstar.
1 + R )
~
Moderate
Portfolio
Foreign Large-Cap Growth Fund
Moderately
Aggressive
Portfolio
80
t = 0
T
∆P
P
Aggressive
Portfolio
- O
C (y; c, T) =
, O ≠ 1
6
9
12
C(y; c, T) =
c
q
∑ n
Emerging-Markets Fund )(1+
j=1
( )(
f -j
q
T T+ )(
(
+
P(y; c, T)
1+ y
1
q
q
≈ -Dm∆y + C(∆y)
1
2
Wt = + Ft
Ht
120
Wt = (1+r) (Wt-1
∑ (1+r) ct-1
- t
c0 = ∆0 W0
- ct-1
= W0
morningstar.com/lp/magazine
-1
67
)
f -j- 1
q
P(y; c, T)
)
- qT-2
40
2
U.S. Large-Cap Value Fund
U.S. Large-Cap Growth Fund
160
20
y
q
)
P(y; c, T)
P (y; c, T)
15
f-j-2
18
Alignment Score
60
Base Risk Score 49.28
Morningstar Portfolio Risk Score
-
W
L RL
˜
40% Equity
A
2
.
P V P
The Alignment Score (ASP
S + u
2
) measures active risk as
A
A
∑ m
i =1 xPi = 1, xPi ≥ 0
o
c
q
xPj Rjt
+ et
q
∑
j=1
n
P(y; c, T) = ∑(1+
j=1
D(y; c, T) =
∑ n
j=1
j -f
q
y
q
)
f-j
+ 1+
(
( ) (1+
c
q
P(y; c, T) = P( )(1+
y; c,
c
q
=
c
q
n
q
(
P(y; c, T)
1+
y
q
)
=
There are several additional applications of the
risk score, one of which is to use it in conjunction
with a risk tolerance tool to help advisors
recommend suitable portfolios to investors. I will
discuss this application in the next issue of
Quant U. K
c
q
c (y)
P =
D =
c (y)
77.5% Equity
60% Equity
Fund
Asset Allocation
∆P
P
≈ -Dm∆y
1+ y = Dm
P
D
*
q
2
1
P( )
(
y; c,
1
y
1
y
n
q
1+ y
q
(
y
q
)
Paul D. Kaplan, Ph.D., CFA, is director of research with
Morningstar Canada. He is a member of the editorial board
of Morningstar magazine.
92.5% Equity
Dm (y; c, T) =
Dm (y; c, T) =
P(y; c, T)
P (y; c, T)
The author thanks Tom Idzorek and Shawn Brayman for their
helpful edits and comments.
Morningstar
Target Allocation
Index Continuum
D(y; c, T)
1+ y
q
3
f
1+ )110% Equity
f
- q
(
f-j
(
y
q
)
y
q
)
f
of asset mixes that reflect how target allocation
funds generally invest. It is based on the
well-established principles of asset allocation
and the separation of systematic and unsystematic
risk. It can be applied to funds and portfolios
of funds to measure risk and to quantify the
benefits of diversification. Hence, it provides a
ruler for risk that can be applied in many
investment settings.
P(y; c, T)
o
1+
y; c,
n
q
y
1+ q
)
-j
y
q
)
f-j
P( )(1+
y
q
)
f
Expected Return
7%
6
5
4
y
q
f-j
)
- qT
(
q
)
(
q
)
+ T 1+
(
y
q
)
- qT
LEVERAGE LINE
http://www.morningstar.com/lp/magazine
Morningstar - Q3 2021

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