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

152 Optimizing Optimization

estimated expectile at the end of the sample period, μˆT. From the previous
discussion, we know that it satisfies:

μˆ,TT ˆ ,ˆ

[()]()Ωλμ λμλDR DR R

1

where the subscript on the inverse matrix indicates the T -th row. Recall that
each diagonal element of DR()λμ, can only take on two values, ω and 1  ω.
As a result, on any set Λ  R n such that DRˆ()λ is constant ∀ λ  Λ , the objec-
tive function is linear in λ. De Rossi (2009) shows that the objective function
μωˆ;T 1 (( ) )yλ , viewed as a function of λ * , is piecewise linear and convex. This
implies that the solution to the maximization problem must be on a vertex. The
simplex algorithm starts from an arbitrary vertex and moves along the edges
of the “ surface ” until it reaches the maximum. At each step, the ω -expectiles
of the portfolios corresponding to the new vertices are estimated and used for
prediction one step ahead. The absence of local minima makes the optimiza-
tion procedure robust and reliable.
It is straightforward to see that the procedure can be used to deal with
objective functions that include alpha terms. Call α a vector of expected asset
returns. We can impose the additional inequality constraint λα r , where r
is a preassigned constant. The asset allocation procedure would then minimize
risk subject to an expected return of at least r.
The parameters q and ω are inputs to the asset allocation procedure. How
should a portfolio manager choose their values?
First , q can be estimated from the data as argued above, e.g., from a time
series of returns to the minimum variance portfolio. The parameter can be
viewed as a signal to noise ratio: The higher q , the more information content
can be found in the observations. In addition, q is related to turnover. The
more responsive the model is to changes in expectiles over time, the larger
the amount of trading. By looking at historical data, the value of q can be
fine-tuned to match the desired turnover level.
As for the parameter ω , Manganelli (2007) highlights the fact that the
maximum expectile objective can be interpreted in a utility maximization
framework. Minimizing the EVaR is equivalent to maximizing:

EU Y((μ ))

where Y is the portfolio return and

Ux

xx

μ xx

μ

μ

ω

ω

μμ

()

if

()if











1

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⎪

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If we interpret U(x) as a utility function, then it is easy to see that as ω → 0.5,
the preferences approach risk neutrality: all the investor cares about is the
expected return (which is the 50% expectile). For ω 0.5, however, lower lev-
els of ω correspond to higher risk aversion.