Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

(Romina) #1

Staying ahead on downside risk 151


straightforward to extend the analysis to the case in which we target a preas-
signed portfolio alpha given a set of asset alphas. The crucial point is that the
method described here allows portfolio expectiles to change over time. In the
mean – variance world, this corresponds to a minimum variance allocation cou-
pled with a dynamic estimator of the covariance matrix.
How does one compute the optimal weights? Manganelli (2007) suggests a
solution in the context of conditional autoregressive expectile models. A spe-
cial case is the constant expectile. The problem can be expressed as a linear
programming one, similar to the approach of Rockafellar and Uryasev (2000).
Assume that n assets are traded on the market and we have observed the his-


tory of returns R  [ r (^) 1, ... , r (^) T ] , a T  n matrix that is taken to have full rank.
The risk tolerance parameter^05 <<ω and the signal noise ratio q 0 are
taken as given. Furthermore, I assume that an m 1  n matrix A and an m 2  n
matrix G are given, both of full rank, for positive integers m 1 , m 2. Finally, b
and g are given m 1  1 and m 2  1 vectors. It is also assumed that n m 1 and
T n  m 1  m 2  1.
For any vector of n portfolio weights, λ , historical portfolio returns are
obtained as λ  R.
The matrix A is used to impose equality constraints, e.g., to force the sum
of weights to be equal to one (fully invested portfolio). I obtain this result by
optimizing on a vector of dimension n  m 1 , λ *.
The matrix G is used to impose inequality constraints, e.g., long-only con-
straints or a minimum holding in any given asset.
The aim is to solve:
max ˆ (( ); )




  • μωT^1 y
    subject to
    λλAb


  • and
    Gii•λ gii∀ 1,,n
    where μˆT 1 is the predicted ω -expectile of y T (^)  1  λ r T (^)  1 , namely the portfolio
    return one step ahead.
    It is worth stressing that the objective function depends on the parameter q ,
    although I omitted the parameter from the formula in order to keep the nota-
    tion simple. In fact, μˆT^1 is the prediction obtained from the dynamic model,
    which takes as inputs the (univariate) time series of portfolio returns y ( λ ) and
    the parameters ω and q.
    The objective function μωˆ;T 1 (( ) )yλ is equal to the prediction from a ran-
    dom walk model of the unobserved expectile, which is in turn equal to the



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