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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

1) Set t-cost objective = (12 * 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).

2) 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.

3) 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.

4) Re-scale alphas using Grinold approach: alpha = IC * volatility * z-score, then objective weight for both alpha and t-cost = 1.

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