Position Sizing from Return Forecasts

Article Quant Q&A · Author: Fiddod

Summary

The document asks how to convert daily return forecasts and their forecast variances from a Kalman-filtered stock-market model into an investment allocation. It defines forecast return, realized return, and forecast error, and assumes realized return is normally distributed around the prediction with the stated prediction variance. It then proposes maximizing a Sharpe-like ratio of allocated returns, with allocation allowed to depend on the forecast and its uncertainty.

The author reports a weak correlation between forecast and realized returns and notes that forecast-error variance is much larger than forecast variance. They conjecture that scaling the forecast by its standard deviation might be useful, but cannot establish that this is optimal. No solution, empirical test, or evidence for a best allocation function appears in the document. The stated likelihood criterion for estimating the model does not by itself settle the separate portfolio objective, and the ratio definition should not be treated as a standard Sharpe ratio without further assumptions.

Key ideas

The model supplies a daily return prediction and an associated forecast-error variance.
The proposed allocation uses only those forecast quantities and determines both position direction and size.
The author defines a Sharpe-like objective using the mean and variance of allocated returns.
The reported forecast-to-realized-return correlation is weak, while prediction uncertainty is comparatively large.
Scaling the forecast by uncertainty is presented as a conjecture, not a demonstrated optimum.

Tags

Equities Statistics Position sizing Risk management Machine learning

Full text

# Looking for an optimal investment function from stock market model


# Looking for an optimal investment function from stock market model












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Note: this question was written in plain text

From a stock market price prediction model, I am looking for an optimal investment function to check performance sustainability.  
  

1. Background  

1.1. Using Kalman filtering on a market state model, the model provides u_pred and uvar_pred each daytime t, where :

> u_pred(t) = [price_pred(t) -price_prev(t)] /price_prev(t),  
> uvar_pred(t) = u_pred_error(t) variance, where :
> u_pred_error(t) = u_mes(t) -u_pred(t),  
> u_mes(t) = [price_mes(t) -price_prev(t)] /price_prev(t),  

> price_pred(t) = market price predicted by state model at t,  
> price_mes(t) = market price measured (ie: real market price) at t,  
> price_prev(t) = previous price measured at t, ie: price_prev(t) = price_mes(t-1),  

1.2. From the previous notations, the daily yield obviously is u_mes(t), so that in summary, we know that the future yield u_predmes is deterministically defined as :  
u_predmes(t) =u_pred(t) + z(t), where z(t) is a random, normally distributed function with 0-mean and variance = uvar_pred(t).  

1.3. The model predictive performance is nonetheless very weak since the correlation coefficient between u_pred and u_mes variables lies in [0.10 ; 0.15] range. Which means that uvar_pred is one order of magnitude greater than variance(u_pred).

2. Objective  

2.1. Now, knowing u_pred(t) and its variance uvar_pred(t), I am looking for alpha(t), an investment allocation function which maximizes the (what I abusively call) investment Sharpe, defined by:
Sharpe = mean[ alpha(t) * u_mes(t) ] / variance[ alpha(t) * u_mes(t) ]**0.5, where 
variance[ alpha(t) * u_mes(t)] = mean[ (alpha(t) * u_mes(t))**2 ] -mean[ alpha(t) * u_mes(t) ]**2  

2.2. alpha(t) obviously defines the investment sign (+: long, -: short) as well as quantity ; for instance, if alpha(t) is a constant equal to 1, this corresponds to a passive, long investment position in the stock market. Ideally, alpha(t) should only depend on u_pred(t) and uvar_pred(t) computed by the model.
  

3. Supposition and question  

The optimized state model, obtained from the log-likelihood maximization, implies that :
mean [ log( uvar_pred(t) ) + u_pred(t)**2 /uvar_pred(t) ] is minimum.  
So, an intuition consists in replacing alpha(t) with u_pred(t) /uvar_pred(t)**0.5 but I cannot demonstrate it is the optimal solution.
If a practical solution is feasible, I suspect a much more complex function that well-versed statisticians/quants could formalize.

Thank you for your help. Any additional information (research articles, etc.) welcome.
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Choosing Position Size from Noisy Return Forecasts

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