# Morningstar Magazine - June/July 2017 - 72

Strategies

EXHIBIT 2

models as they apply to individual securities. I now
turn to a factor model that was specifically
designed for managed products such as mutual
funds, particularly for those that invest in multiple
asset classes. This model, returns-based style
analysis, or RBSA, was developed by Sharpe
(1988, 1992) as a method of estimating how much
exposure a fund has to various asset classes.

Manager Structure Optimization Efficient Frontier
Alpha
0.6
0.5
0.4

RBSA is often contrasted with holdings-based
style analysis, or HBSA. HBSA requires the
identification of each security being held in the
portfolio. Because holding data can be much
more expensive and difficult to obtain than returns
data, RBSA is often seen as an inexpensive
and simple alternative to HBSA.

Whether by RBSA or HBSA, we need to know
how much of each fund is allocated to each asset
class in order to assemble a portfolio of funds
with an overall asset-allocation target. In the last
section of this issue of Quant U, I discuss how
to do this.

0.3
0.2
0.1
0.0
0.5
Active Risk

1.0

1.5

2.0

it

3.0

3.5

4.0

4.5

0.0

Source: Morningstar.

1 All of the asset-class exposures are non-negative:

Sharpe's RBSA Model
Sharpe's RBSA model is very much like a
time-series regression model:

2.5

Bik ≥ 0 for k = 1, 2, ..., m
2 The asset-class exposures must sum to 100%:
Bik = 1

the mean-variance model of Markowitz (1952,
1991), which can be extended to "Markowitz 2.0."
(See Kaplan 2015, 2017.)

The manager selection stage can also be carried
out with ∑
an optimization model that Waring
m
m
2
et
al.
(2000)
call manager structure optimization,
RBMit = Bik RAkt
= Bikt Akt + u˜ it
i
k=1
k =1
or MSO. In MSO, the target asset allocation is
given (hence, the first stage), and the
2
A
n
managers are chosen for their forecast
where:
B
=
x
∑
k
i=1 i ik
= the total return on fund i in period t
outperformance from the benchmarks (alphas)
R˜ it
but combined in such a way as to keep
m
= the number of asset classes
m
m
∑
M =
k=1 ∑ q=1 ( k - k ) ( q - q ) k q kq
TRSt = IRSt + PRSt
m class k
m
the assetI allocation of the overall benchmark close
Bik m = fund i's exposure to asset
St
2
RBMit = Bik RAkt
˜ it
r˜ it = R˜ it + = ikt f˜the
u˜ it return on index
= for
Bikt asset
IRto
= target
i = Var[˜ui]
kt + total
Akt + u
St the
Akt
n
PSt- qT- ∆t asset allocation. The standard
2
2
k=1
k=1
k =1
yst -∆
i
i
S =
i=1
t (T )
class k in period t
deviation
of the difference between the portfolio
PZt-∆ t T; yst -∆ t (T ) = 1 +
q
of managers
= the part of the return on fund i
u˜ it
2
2
2
PSt and a portfolio of indexes
A = M + S
n
PRrepresenting
-1
r˜ it = it + i r˜Mt +inu˜ period
∑ i=1 xi Bik by the
St =
How useful the results of RBSA are depends on
t that is not
k = explained
it
PSt - ∆t the target asset allocation is the
P
(T
-
∆t)
T
-
∆t;
y
Zt
st
how well the custom benchmark explains
asset-class index returns
tracking error or active risk of the portfolio
TRZt (T ) =
-1
m
m
the
fund.
This
is
measured
by
the
R
-squared
of
Just as the mean-variance model
=
(
-
)
(
-
)
∑
M
k q kq
PZt -∆ t T ; yst -∆ t (TTR) managers.
k
k
q
q
q=1
k=1
St = IRSt + PRSt
of the model. This is the square of the correlation
As with a time-series regression, we estimate the
traces out a trade-off between expected
- qT
n the
return and standard deviation of return,
between
on the fund (R˜ i)
asset-class exposures (the Bik's) from a time
2 return
2
yst -∆ t (T )
i
S =
i=1 i
+
P
=
T;
y
(T
)
1
and the return on the custom benchmark
(R
).
MSO
traces
out
the
between
alpha
series of returns on the fund (R˜ it) and time series of
st -∆ t
PZt TBMit
) t
- ∆t; yst -∆ t (TZt-∆
q
˜
=
IR
-
1
and active risk, as EXHIB IT 2 illustrates.
returns on the asset-class indexes (the RAkt's)
Zt (T )
PZt -∆ t T ; yst -∆ t (T )
Using RBSA Outputs as Inputs to Manage
over a historical period. The estimation procedure
PZt T - ∆t; yst (T - ∆t)
Structure Optimization
provides
estimates for the manager-specific
minimizes the empirical variance of the residuals
TRRBSA
-1
Zt (T ) =
(T )
Portfolios are often built in two stages: asset
parametersPof
active
risk;
the m asset(˜uit). What makes RBSA different from conventional
Zt -∆
t T;y
st -∆ t namely,
allocation and fund or manager selection.
regression is that it imposes constraints on the
class exposures of each fund i (the Bik's)
A standard model for the asset-allocation stage is
and the variance of the unsystematic component
asset-class exposures, namely:
PZt T -∆t; yst -∆ t (T )
=
IR
-1
(T
)
Zt
q∆t
PZt -∆ t T ; yst -∆ t (T )
yst -∆ t (T )
72 Morningstar June/July 2017
IRZt (T ) = 1 +
-1
q
These constraints mean that the asset-class
ISt
exposures
can be interpreted
IRStas=weights on a long= Var[˜
ui]
PSt - ∆t
only portfolio of the asset-class indexes, which
can be interpreted as a custom multiasset-class
PSt
= M2 + S2
benchmark for the fund. The
= t return
PRStperiod
- 1on the
PSt - ∆t
custom benchmark is:

(

)

(

(

)

(

)

(

)(
)

)

(

(

(

)

(

)
)

)

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