How to scale $\alpha$, trading costs in a standard portfolio optimization problem

Question

In the usual "portfolio optimization problem under linear constraints".

Let me define the terms here. $$ \text{Find } w^*=\underset{w}{\text{argmax}} \ \ r^Tw - \lambda w^{T} \Sigma w - tradingCost(|w-w_0|)\\ \text{uc.} \ \ l_b \leq Aw \leq u_b $$ where $w \in \mathbb{R}^n$ is the final position of a n-assets portfolio, $w_0 \in \mathbb{R}^n$ the initial position, $r \in \mathbb{R}^n$ is vector of expected returns over a time period (say $[0,T]$), $\Sigma$ is the $\mathbb{M}_{n,n}(\mathbb{R})$ matrix of covariance of the asset returns. Second line are constraints.

My problem here is that $r$ is an estimation with a low correlation to ex-post returns (that I am trying to predict). The trading cost on the other hand are much likely to be a more more accurate prediction of real trading cost.

1. Is there a usual way to scale those quantities ?
2. Is there references describing a theory/methodology ?

Answer 1

One standard approach is to shrink your forecasts towards zero (or to some reasonable value as in the Black-Littermann model). Shrinking towards zero is done by:

$$w^*=\underset{w}{\text{argmax}} \ \ \lambda_{\alpha} r^Tw - \lambda_r w^{T} \Sigma w - tradingCost(|w-w_0|)\\$$

$$0\leq\lambda_{\alpha}\leq1$$ Shrinkage coefficient $\lambda_{\alpha}$ is best backtested using simulation. If you are really careful about overfitting, you'd also want to run an on-line algorithm for choosing the best coefficients.

Alternatively you can use a shrinked regression to come up with return predictions as L1 or L2 regularization (Lasso or Ridge regression).

EDIT

I'm afraid you won't find much better than fitting one more parameter, although I'd hope to read a new solution here. By on-line algorithm I mean a causal expert combination, which won't overfit your data. Have a look at Empirical log-optimal portfolio selections: a survey., in particular "Kernel/Histogram/Nearest Neighbort based strategy" sections. Using a grid wisely will avoid over-fitting completely and result in un-biased (or pessimistic) return estimates.

W.r.t. L1 and L2 regularization (try Lasso or Ridge first, it's more simple than elastic net)... once you chose the right shrinkage coefficient in your regression by cross-validation or backtesting, you won't have this problen in your portfolio optimizaiton anymore.