Given a set of monthly alpha forecasts (i.e. standardized z-scores from a multi-factor return model) and a non-linear market impact model (or more specifically, its piecewise-linear approximation), is there a generally accepted "standard" method of setting their objective weights in an optimization so they are scaled appropriately?
Based on historical returns of a back-test, some of the methods that have been suggested include:
Let,
a = average monthly gross portfolio alpha forecast
R = annual realized gross portfolio return
XO = average annual portfolio turnover
- Set t-cost objective $= (12 \times a) / R$, set alpha objective $= 1$. This effectively converts the arbitrary z-score to an actual return forecast (or set the alpha objective = the reciprocal of this value, and the t-cost $= 1$).
- Set t-cost objective $= (a / R) * XO$, set alpha objective $= 1$. This treats t-cost as an annual number, and adjusts the holding period horizon based on annual expected turnover.
- Set t-cost objective $=$ the full period average of [stock-level alpha / forward return]. Cross-sectionally more accurate and removes the effect of large portfolio weights on long-term averages.
- Re-scale alphas using Grinold approach: $\alpha = IC \times \text{volatility} \times \text{z-score}$, then objective weight for both alpha and t-cost $= 1$.
