166 M. La Rocca and D. Vistocco
In the classical approach, portfolio style is determined by estimating the influence
of style exposure on expected returns. Extracting information at places other than
the expected value should provide useful insights as the style exposure could affect
returns in different ways at different locations of the portfolio returns distribution. By
exploiting QR, a more detailed comparison of financial portfolios can then be achieved
as QR coefficients are interpretable in terms of sensitivity of portfolio conditional
quantile returns to constituent returns [5]. The QR model for a given conditional
quantileθcan be written as:
Qθ(rport|Rconst)=Rconstwconst(θ) s.t.:wconst(θ)≥ 0 , 1 Twconst(θ)= 1 ,∀θ,whereθ( 0 <θ< 1 )denotes the particular quantile of interest.
As for the classical model, thewconsti(θ)coefficient of the QR model can be
interpreted as the rate of change of theθth conditional quantile of the portfolio returns
distribution for a unit change in theith constituent returns holding the values ofR·constj,j=i
constant.
The conditional quantiles are estimated through an optimisation function minimis-
ing a sum of weighted absolute deviation where the choice of the weight determines
the particular conditional quantile to estimate. We refer to Koenker and Ng [20] for
computing inequality constrained quantile regression.
The use of absolute deviations ensures that conditional quantile estimates are
robust. The method is nonparametric in the sense that it does not assume any specific
probability distribution of the observations. In the following we use a semiparametric
approach as we assume a linear model in order to compare QR estimates with the
classical style model. Moreover we restrict our attention to the median regression
by settingθ = 0 .5. As previously stated, it is worthwile to mention that the use of
different values ofθallows a set of conditional quantile estimators to be obtained that
can be easily linearly combined in order to construct an L-estimator, in order to gain
efficiency [15, 19].
4 Simulation results
In this section the finite sample properties of the proposed procedure are investigated
via a Monte Carlo study. Artificial fund returns are simulated using the following
data-generation process:
rtport=rconstt ′wconst+σet, t= 1 , 2 ,...,T.In particular, we considered a portfolio with 5 constituents generated by using
GARCH(1,1) processes to simulate the behaviour of true time series returns. The true
style weights have been set towconsti =^0.^2 ,i=^1 ,^2 ,...,5, thus mimicking a typ-
ical “buy and hold” strategy. This allows a better interpretation of simulation results
whereas the extension to different management strategies does not entail particular
difficulties. The scaling factorσhas been fixed in order to haveR^2 close to 0. 90
whileet∼N( 0 , 1 ). We considered additive outliers at randomly chosen positions