Robust estimation of style analysis coefficients 1652 Sharpe-style regression model
The Sharpe-style analysis model regresses portfolio returns on the returns of a variety
of investment class returns. The method thus identifies the portfolio style in the time
series of its returns and of constituent returns [12]. The use of past returns is a Hobson’s
choice as typically there is no other information available to external investors.
Let us denote byrportthe random vector of portfolio returns along time and by
Rconstthe matrix containing the returns along time of theithportfolio constituent on
theithcolumn (i= 1 ,...,n). Data refer toTsubsequent time periods. The style
analysis model regresses portfolio returns on the returns of thenconstituents:
rport=Rconstwconst+e s.t.:wconst≥^0 ,^1 Twconst=^1.The random vectorecan be interpreted as the tracking error of the portfolio, where
E(Rconste= 0 ).
Style analysis models can vary with respect to the choice of style indexes as well
as with respect to the specific location of the response conditional distribution they
are estimating. The classical style analysis model is based on a constrained linear
regression model estimated by least squares [25, 26]. This model focuses on the
conditional expectation of portfolio returns distributionE(rport|Rconst): estimated
compositions are interpretable in terms of sensitivity of portfolio expected returns to
constituent returns.
The presence of the two constraints imposes the coefficients to be exhaustive
and non-negative, thus allowing their interpretation in terms of compositional data:
the estimated coefficients mean constituent quotas in composing the portfolio. The
Rconstwconstterm of the equation can be interpreted as the return of a weighted
portfolio: the portfolio with optimised weights is then a portfolio with the same
style as the observed portfolio. It differs from the former as estimates of the internal
composition are available [8, 9]. We refer the interested reader to the paper of Kim et
al. [14] for the assumptions on portfolio returns and on constituent returns commonly
adopted in style models.
In the following we restrict our attention to the strong style analysis model, i.e.,
the model where both the above constraints are considered for estimating style coef-
ficients. Even if such constraints cause some problems for inference, the strong style
model is nevertheless widespread for the above-mentioned interpretation issues.
3 A robust approach to style analysis
Quantile regression (QR), as introduced by Koenker and Basset [18], can be viewed
as an extension of classical least-squares estimation of conditional mean models to
the estimation of a set of conditional quantile functions. For a comprehensive review
of general quantile modelling and estimation, see [16].
The use of QR in the style analysis context was originally proposed in [5] and
revisited in [2] and [3]. It offers a useful complement to the standard model as it
allows discrimination of portfolios that would be otherwise judged equivalent [4].