Morningstar Magazine - June/July 2018 - 64

MISFIT RISK

Spotlight: Packaged Investing
6.97%

0
Risk Score

1

RISK SCORE

2

0

3

4

5

6

EXHIBIT 2

Calculation of Risk Score and
Misfit Risk The minimal values of each
gives the portfolio's overall risk
score, which in this example is 2.6.
Misﬁt Risk
300bps

Portfolio

MISFIT RISK
RISK SCORE

0
1
Risk Score

2

3

4

5

200

100

Let ( ) be the vector of asset allocations of the
blended target-risk portfolio as we described here.

6

As you can see in E X H I B I T 1 , the five target-risk
indexes have a risk score of 1 through 5, respectively. To the left of the Conservative portfolio, is
the Ultra Conservative portfolio with a risk score
of 0. We form this portfolio by rescaling
the fixed-income allocations of the Conservative
portfolio to sum to 100%. To the right of the
Aggressive portfolio is the Ultra Aggressive portfolio with a risk score of 6. We form this
portfolio by rescaling the equity allocations
of the Aggressive portfolio to sum to 110% and set
the cash allocation to negative 10%.
The risk score is a continuous number on a scale
from 0 to 6. When it is an integer, it refers
to one of the seven asset-allocation models of
the extended target-risk family. If it has a decimal,
it refers to a blend of two consecutive targetrisk indexes. For example, a portfolio with
a risk score of 2.6 refers to a blend of 40% in the
Moderately Conservative portfolio (with a
risk score of 2) and 60% in the Moderate portfolio
(with a score of 3). We can formalize this as:

(

)w +
ik

wi(k+1)

Let be the risk score. It consists of an integer
D( )(k)
= and a fractional
( ) - 'V
)-
part
part ( () so
that = k + . In our example, if = 2.6, then
k= 2 and = 0.6.

(

DBudget ( ) =
64

Let V be the covariance matrix of the asset classes.
We estimate V using monthly returns on indexes
representing asset classes for the U.S. model from
January 2002 to a recent date, here December 2017.
wi ( ) = 1 - wik + wi(k+1)
Misfit
risk, as a function of a given risk score, is:

(

Source: Morningstar.

wi ( ) = 1 -

To explain how the engine calculates the risk score,
we use vector and matrix notation.
Let denote the vector of asset allocations for the
fund or composite portfolio. This is the result of
returns-based style analysis, as we discussed earlier.

0

) (

+

Morningstar June/July 2018

)

These two scenarios capture two critical
dimensions of asset allocation: the domestic-bondversus-international-bond allocation and the
domestic-equity-versus-international-equity
allocation. To account for the overall fixed-incomeversus-equity split, we take the weighted average
of these two misfits where the weights are
simply based on the overall fixed-income-versusequity split of the index. We plotted the risk score
of the five target-risk indexes versus the weighted
average misfits and fitted a regression line.
As we might expect, the line is upward sloping,
indicating that in general, as we move from left
to right from the Conservative index to the
Aggressive index, the calculated weighted-average
misfit risk from the two scenarios increases.

Let wik denote the allocation to asset class i
in target-risk portfolio k and let wi ( ) denote the
allocation to asset class i in the blend target-risk
portfolio corresponding to .

D( ) =

Arguably, it is intuitive that asset-allocation misfits
associated with aggressive asset allocations
(dominated by more volatile asset classes) should
be greater than those of conservative
asset allocation (dominated by less volatile asset
classes), and likewise, investors with high
wi ( ) = 1 - wik + wEi(k+1)
[ rj be
] =willing
( Etake
[ rM ] -onrf )
rf + j to
overall
risk tolerance would

)

(

( )-

) 'V (

( )-

)

(

)

The risk score, *, is the value of that minimizes
). Misfit
D(
DBudget
( ) = risk+is minimal value of D ( *).
E X H I BI T 2 illustrates how this works. It depicts the
portfolio in our example with a risk score of 2.6.

more asset-allocation misfit risk relative to a
target asset allocation.
p
D( ) =
( ) - 'V
( )-
E [ rj ] = rf + j ( E [ rM ] - rf ) + ∑ jk πk
k=1
We can write the regression line as:

Assess Alignment With the Target-Risk Indexes
The engine assesses how aligned the fund or
portfolio's asset allocation is with the Morningstar
Target Risk Indexes by setting thresholds
for misfit risk. To establish these thresholds, we
studied the tracking error associated with a
wide variety of common asset-allocation misfits.
Here, we describe the analysis for calculating
the asset-allocation misfit tracking error limit for
the U.S. version of the risk-scoring system.

DBudget ( ) =

(

)

n

+

CMk = ∑ xMj Cjk
j=1

We form three zones of alignment (High, Medium,
= j cMk - Cjk
jk
and Low) by applying a wedge,
w, to DBudget .
m

If D( *) < (1 - w) DBudget ( ∑*),
w the
= 1alignment is High.
i=1

i

If D( *) > (1 + w) DBudget ( *), alignment is Low.
1
=
M
m
wi
If (1 - w) DBudget ( *) ≤ D( *) ≤∑(1i =+1 w)i DBudget ( *),
alignment is Medium.

Working with a panel of Morningstar experts,
we identified two scenarios to help calibrate
a dynamic tracking error cutoff that 1) changes
through time with evolving market volatility
and 2) adjusts for the overall fixed-income-versusequity split across the five Morningstar Target
Risk Indexes:

m
wi
πk = Medium,
displays the High,
M ∑
ik and
i=1 i
Low zones.

EXHIB IT 3

E [ rj ]Funds
= rf +
Results for U.S. Target-Risk

μ = 2.00% +
j M
n = qT
To illustrate how the risk-scoring engine works,
we obtained
returns
f = n-monthly
qT
E [ rj ] on
= rthe
+ oldest
μ + jπ
f
j M
share classes ofn funds in the five U.S. target-risk
- n 5.65% f4.50%
=- j2.00% +
c
y
y
j
categories
+ 1Direct
+ q over
+qj
P = from
1the
q Morningstar
j =1from January 2014 to December
48-month period

gWithin fixed income, allocate 50% to non-U.S.
E [ rj ] = rf + j ( E [ rM ] - rf )
bonds, and
gWithin equities, allocate 70% to non-U.S. equities.
E [ rj ] = rf +

) (

(

∑

p

( E [ rM ] - rf ) + ∑
j

k=1

) (

n

π
jk k

P ( y; c, T ) =

c
q

f- j

)

- qT

∑ (1 + ) + (1 + )
y
q

y
q

j =1
n

j -f

c

y

f- j

y

- qT

j

6.66%

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