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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 $= (12 \times a) / R$, 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 $= 1$).
  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: $\alpha = IC \times \text{volatility} \times \text{z-score}$, then objective weight for both alpha and t-cost $= 1$.
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  • $\begingroup$ I'm just wondering if you also have access to bid ans ask prices. You then would be able to model something based on the quoted half spread which I think would be more accurate and straight forward then the options outlined above, especially in the small cap markets. $\endgroup$
    – Tim
    Jan 16 '17 at 17:18

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