Morningstar - Q4 2021 - 58

Strategies
EXHIBIT 2
in constructing a portfolio. Investors often have
preferences regarding asset allocation that
they may be willing to forgo some expected return
to satisfy. For example, a Canadian investor
may have a strong preference for Canadian and
U.S. stocks over stocks of developed countries
outside of North America and of emerging
markets, even if the expected returns on North
American stocks is less than that of stocks or other
regions. Below, I discuss how the Alignment
Score can be used in portfolio construction to take
asset-allocation preferences into account.
Expected Geometric Mean
The expected geometric mean (GM) is calculated
from the expected arithmetic mean of the
portfolio (µ) and the portfolio's standard deviation
( ), representing the total risk of the portfolio,
which includes both systematic and unsystematic
risk. For a given portfolio of funds, the expected
arithmetic mean is calculated from (1) the
asset allocation of the portfolio, (2) the expected
returns on the asset classes, (3) the portfolio
weights on the funds, and (4) fund alphas. Here,
I set all alphas to zero, so that µ reflects asset
class expected returns.
To calculate GM from µ and , I assume that
portfolio returns are lognormally distributed. Under
this assumption, we have:
GM =
(1 +µ 2
2
)
(1 +µ +)
max Q ( ) =
2
- 1
Alignment Score
In the portfolio construction framework I present
here, I use the Alignment Score (AS) to take
asset-allocation preferences into account.
AS measures how closely aligned the portfolio in
question is to the blend of Morningstar Target
Allocation Indexes that has the same systematic
risk as the portfolio. Misalignment can be
due to differences in asset allocation as well
as unsystematic risk. Perfect alignment is
when AS= 0.
-
n
s.t. xi ≥ 0,
∑ m
∑
j=1
RPt = ∑ NP
ˆ
RPt = ∑ m
x = 1i
i =1 xPi = 1, xPi ≥ 0
(
k =1 wk Rkt
i =1 xPj Rjt
A
RPt = + RˆPt
S =
+ ut
A
GM GM ( )
MPRS
-
AS AS
(MPRS ( ) - MPRST
)
2
Each target-allocation index has a fixed equity/
fixed-income split, but the allocations within
MPRS =
equity and fixed income are based on averages
of target allocation funds within a given
Morningstar Category. Thus, the indexes' asset
allocations represent consensus views of
professional managers and serve as good proxies.
So, the lower the AS, the better.
The Optimization Problem
Let be a vector of portfolio weights on a set
of funds, where a portfolio weight is simply
the percentage of the total portfolio allocated to a
given fund. The vector or list of portfolio
weights sums to 100%. By applying to the funds'
historical monthly returns, we have a return
series based on a given set of portfolio weights
that we can use to calculate both the Morningstar
Portfolio Risk Score (MPRS) and AS, as well
as total risk, enabling us to calculate GM. Hence,
we can treat GM, MPRS, and AS as functions
of .
GM =
We determine by solving the following
optimization problem:
(1 +µ 2
2
)
(1 +µ +)
max Q ( ) =
2
- 1
-
n
s.t. xi ≥ 0,
∑ m
where MPRST is the target value of MPRS from the
∑ mxPj Rjt
A
i =1
risk score range set by the Risk-Comfort Range,
and
i =1 xPi = 1, xPi ≥ 0
, and
+ et
GM, MPRS
RPt = ∑ NP
ˆ
RPt = ∑ m
k =1 wk Rkt
EW (T + ,
To solve the optimization problem, the optimizer
is programmed to change the various
portfolio weights, seeking the optimal weights that
maximize GM minus two different penalties:
(1) a penalty for an MPRS of the portfolio
that differs from the target MPRS, and (2) a penalty
based on AS.
i =1 xPj Rjt
RPt = + RˆPt
S =
P =
B ) VW
s.t. A' = , R =SP
A =SP
Risk Score=× F RF
˜
Base
RW
58 = Morningstar Q4 2021
P
RS
S + u
2
P RS
2
max U EW (T + B ) VW
A
(
,
B
s.t.
A
Leverage
Adjustment
,
A' = ,
S + u
2
(T +A
B' = 1 - , ≥ 0,
2
An Example
To illustrate how the Morningstar Portfolio Risk
Score can be used to construct portfolios,
P RSP
B
2
A
A RSP
P
Sys. Risk2 + Unsys. Risk2
˜ + H RH
˜
-
W
W Blended Benchmark Risk
W
L RL
˜
(T +A
A
B ))
B' = 1 - , ≥ 0,
B ≥ 0
=
c
q
B ≥ 0
+ ut
P V P
2
B ))
o
MPRS =
AS are non-negative
A
(
A
coefficients for controlling the relative importance
of the three terms of the objective function.
s.t. A' = ,
∑
j=1
x = 1i
GM GM ( )
MPRS
-
AS AS
(MPRS ( ) - MPRST
)
2
EXHIBIT 3 shows MPRS (from 0 to 110) versus
GM for (1) the continuum of blended pairs
of target-allocation indexes (which I call the
target-allocation frontier), (2) the efficient frontier
of optimized portfolios in which AS is not
taken into account ( GM
o
= MPRS = 1 and AS
EW (T + ,
B ) VW
B' = 1 - , ≥ 0, = 1).
=
f = n- qT
n = qT
(T +A
n
Base
Risk Score
EXHIBIT 3 also shows the MPRS comfort ranges
for an investor who scored 50 on the Morningstar
Risk Profiler. From these ranges, we can see
what set of optimal portfolios might be suitable
for what in this case would be an average
investor. In general, we would use the ranges
corresponding to the investor's risk profiler score.
P(y; c, T)
P =
×
c
q
∑
j=1
Sys. Risk2 + Unsys. Risk2
n
P(y; c, T) = ∑(1+
Blended Benchmark Risk + 1+
j=1
c
q
Leverage
Adjustment
D(y; c, T) =
∑ n
j=1
j -f
q
y
q
)
(
c
q
y
q
)
P(y; c, T) = P(y; c,
1 A common misconception is that taking on more risk means having a higher expected return. But this may not be the case, as we shall see in the examples I present here.y; c,
P V P
c
q
P(y; c, T)
1+
(
(
y
q
)
y
1+ q
P(y; c,
=
)
n
q
-j
)
c
q
P(
f-j
n
q
)(1+
(
y
q
)
f
( ) (1+
f-j
y
q
f-j
)
- qT
c
q
+ 1+T
(
y
q
EXHIBIT 4 shows MPRS (from 0 to 110) versus
AS for optimized portfolios that take AS into
account and for those that do not. Like EXHIBIT 3 ,
it also shows the MPRS comfort ranges for
1+
n
q
y
q
)
f-j
)(1+
y
q
)
f
Dm
)
- qT
=
Pc
Dc
-j
+ 1+
(
y
q
)
-n
(
1+
y
q
)
f
MPRS = AS
A
B ))
= 0),
and (3) the efficient frontier of optimized
portfolios in which AS is taken into account
( GM
B ≥ 0
P
D
P
P
Risk-Comfort Ranges for
Morningstar Risk Profiler Score of 50
Risk-Comfort Range
Too Little Risk
Marginal (Low End)
Comfort Range
Marginal (High End)
Too Much Risk
Source: Morningstar.
Morningstar Portfolio
Risk Score Range
≤25
26-35
36-55
56-65
>65
∑ m
I build off the example I presented in the previous
issue of Quant U in which I calculated MPRS
and AS for a set of portfolios of seven mutual
funds, each from a different Morningstar Category.
Based on their categories, I called the funds:
U.S. Large-Cap Growth, U.S. Large-Cap Value,
Foreign Large-Cap Growth, Emerging-Markets
Fund, U.S. Short-Term Bond, U.S. Intermediate
Bond, and U.S. High-Yield Bond Fund.
i =1 xPj Rjt
A
+ et
f
n
Dm

Morningstar - Q4 2021

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