Staying ahead on downside risk 149
Equivalently , the problem can be cast in a nonparametric framework. The
goal is to find the optimal curve f ( t ), plotted in Figure 6.1 , that fits the obser-
vations. It is worth stressing that f ( t ) is a continuous function, so here the argu-
ment t , with a slight abuse of notation, is allowed to be a positive real number
such that 0 t T. The solution minimizes:
[ ( )]ft dt q (y fss ( ))
sT T
’^2(^01)
∫ ∑ρω
(6.3)
with respect to the whole function f , within the class of functions having square
integrable first derivative. At any point t 1,..., T , we then set μ (^) t ( ω ) f ( t ).
The first term in Equation (6.3) is a roughness penalty. Loosely speaking,
the more the curve f ( t ) wiggles, the higher the penalty. The second term is the
objective function for the expectile, which as noted above gives asymmetric
weights on positive and negative errors.
The constant q represents the relative importance of the expectile criterion.
As q grows large, the objective function tends to become influenced less and
less by the squared first derivative. In the limit, only the errors y t μ (^) t ( ω ) mat-
ter and the solution becomes μ (^) t ( ω ) y t , i.e., the estimated expectile coincides
with the corresponding observation. As q tends to zero, instead, the integral of
the squared derivative is minimized by a straight line. As a result, the solution
in the limit is to set all expectiles equal to the historical expectile. The role of q
is illustrated in Figure 6.2.
It can be shown that the optimal curve is a piecewise linear spline.
Computationally, finding the optimal spline boils down to solving a nonlinear
system in μ , the vector of T expectiles. After some tedious algebra, the first-
order conditions to minimize Equation (6.3) turn out to be:
[()] ()ΩμμDy,,Dy yμ
where D(y, μ ) is diagonal with element ( t , t ) given by
|(ωμIytt<^0 )|
and
Ω
q^1
11 00 0
1210 0
01 21 0
021
011
...
...
...
...
...
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