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