Scaling Return Forecasts Against Risk and Trading Costs
Summary
The document considers how to balance uncertain expected-return forecasts against covariance risk and trading costs in a constrained portfolio optimization problem. Its proposed method is to multiply the return signal by a shrinkage coefficient between zero and one, reducing the influence of forecasts that may correlate weakly with realized returns while leaving the cost estimate unchanged.
The coefficient can be selected through simulation or backtesting, with an online causal expert-combination method suggested to reduce overfitting. Another route is to regularize the return-prediction regression, for example with Lasso or Ridge, and tune that regularization through cross-validation or backtesting. The discussion offers methods rather than empirical comparisons or a worked result. It does not establish a universal scaling rule: performance depends on the forecast and cost models, the evaluation design, and the risk objective, while tuning coefficients on historical data can itself overfit.
Key ideas
- Shrink uncertain return forecasts toward zero by reducing their weight in the portfolio objective.
- Tune the forecast scale using simulation or backtesting, while guarding against overfitting.
- Online causal expert combinations are suggested as one way to choose coefficients over time.
- Regularized return regressions can reduce forecast instability before portfolio optimization.
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# How to scale $\alpha$, trading costs in a standard portfolio optimization problem
# How to scale $\alpha$, trading costs in a standard portfolio optimization problem
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.
- Is there a usual way to scale those quantities ?
- Is there references describing a theory/methodology ?
## Answer by Mark Horvath (score 1)
https://quant.stackexchange.com/a/18943
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.