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Suppose we have a universe of $n$ stocks, and for each time period $t$ we have $n$ predictions for their future returns. Now we can calculate the information coefficient for our predictions in two ways:

  1. Time series $IC_{ts}$: for each stock, calculate the Pearson correlation of its prediction and realized return over time, then average over all stocks
  2. Cross section $IC_{cs}$: for the time period, calculate the Pearson correlation of all stocks' predictions and realized returns, then average over time

The above two calculations motivate two different strategies:

  1. For each stock we chase its time series performance, which is similar to CTA strategies that profit from momentum and reversals. Stocks are treated independently as different commodities, but there could be an overall framework to control portfolio risk and to manage positions.

  2. We can do a long short strategy, optimize portfolio weights while controlling for certain risk measures or maintaining beta/dollar neutral.

My question is, if the two ICs are at similar scales, (actually in my experience $IC_{cs}$ is generally larger than $IC_{ts}$ with the same set of predictions and same lookback time), what are the pros and cons for each strategy?

My intuition is that the time series strategy has more degrees of freedom since each stock runs independently. As a result the strategy could suffer from more larger drawdowns and could often be overexposed to certain sectors. On the contrary, the cross section strategy is more rigid and has easier and better risk control, so its Sharpe could be higher. However, if we manage to control for everything else, and only look at predictions, are the two strategies actually the same?

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  • $\begingroup$ Can you clarify what you mean by "2.... calculate the Pearson correlation of all stocks' predictions and realized returns... ". What two random variables is the correlation calculated between? $\endgroup$ Jun 7, 2022 at 8:20
  • $\begingroup$ Pontus, I'm referring to the correlation between all the predictions and the real returns over a universe of stocks, so we will have a prediction array for each stock and a return array for each stock, and this is calculated cross-sectional for each time step. This basically measures how well you can rank the stocks. $\endgroup$ Jun 7, 2022 at 9:56
  • $\begingroup$ So if X(i, t) is the return at time t of stock i, and Y(i, t) is the predicted return, you take sample correlation corr(X(i, t), Y(i, t)) over all i? the problem is that X(i, t) is not the same process as X(k, t), and so when calculating sample correlation you use samples from different processes. For the 1. Case it makes more sense since X(i, t) is the same process as X(i, t+s) $\endgroup$ Jun 7, 2022 at 11:06
  • $\begingroup$ The predictions are from the same model trained over all historical data, so I suppose they should be comparable in this sense. Nevertheless, it does not affect my question. The correlation is just a way to evaluate how good I could rank the stocks for my Long Short strategy to work. $\endgroup$ Jun 8, 2022 at 3:28

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1. Basic model assumptions

Let $X_t^{(i)}$ denote the return at time $t=1,2,...,T$ for stock $i=1,2,...,m$. Assume that each stock return series is a stationary series and that $X_t^{(i)}\sim \mathcal{N}(\mu_i, \sigma_i^2)$. Now assume that you have a prediction $Y_t^{(i)}\sim \mathcal{N}(\nu_i, \eta_i^2)$ and define $IC_{ts}^{(i)}:=\text{corr}(X^{(i)}, Y^{(i)})$. The conditional distribution of the return given the prediction is then $X_t^{(i)}|Y_t^{(i)}=y_t^{(i)} \sim \mathcal{N}\left(\mu_i + IC_{ts}^{(i)} \frac{\sigma_i}{\eta_i}(y_t^{(i)}-\nu_i), \left(1-{IC_{ts}^{(i)}}^2\right)\sigma_i^2\right)$.

2. Your two ways of measuring $IC$, what are we measuring?

Measure 1. Time series $IC_{ts}$: for each stock, calculate the Pearson correlation of its prediction and realized return over time, then average over all stocks.

We have that $IC_{ts}^{(i)}:=\text{corr}[X^{(i)},Y^{(i)}]$, which is the correlation between two stationary time-series, hence mean and variance of the two are well-defined and constant through time. Now create an equally weighted portfolio of the stocks which will have return $\widetilde X_t=\tfrac{1}{m}\sum_{i=1}^m X_t^{(i)}$ and prediction $\widetilde Y_t=\tfrac{1}{m}\sum_{i=1}^m Y_t^{(i)}$. Furthermore assume that all return series have the same variance, $\sigma_i=\sigma$ for all $i$, and all predictions have the same variance, $\eta_i=\eta$ for all $i$, and that $cov[X^{(i)},X^{(j)}]=0$ for $i\neq j$. Correlation between the portfolio return and portfolio prediction is then $\widetilde{IC}_{ts}:=\text{corr}(\widetilde X_t, \widetilde Y_t) = \tfrac{1}{m}\sum_{i=1}^m IC_{ts}^{(i)}$, which is the arithmetic average across stocks of individual return-prediction correlations. Hence, this method of calculating correlation is the $IC_{ts}$ for a portfolio with special constraints on weights, variances, and cross-covariances.

Measure 2. Cross section $IC_{cs}$: for the time period, calculate the Pearson correlation of all stocks' predictions and realized returns, then average over time

Now this method is problematic, because the sample series $\{X_t^{(i)}\}_{i=1}^m$ is not from the same population. Remember that a sample statistic (mean, variance, covariance, correlation) is meant to describe a population from which the samples are taken. Obviously, nothing stops you from assuming that they are from the same distribution and calculate sample statistics, but inference becomes problematic. Therefore, I will not discuss this method further.

3. Two strategies, different but the same

Strategy 1. For each stock we chase its time series performance, which is similar to CTA strategies that profit from momentum and reversals. Stocks are treated independently as different commodities, but there could be an overall framework to control portfolio risk and to manage positions.

Strategy 2. We can do a long short strategy, optimize portfolio weights while controlling for certain risk measures or maintaining beta/dollar neutral.

I would say that these two "strategies" can both be performed by using Measure 1. Under Strategy 1 you simply assume that stocks are independent, and you form a weighted portfolio of returns $\widetilde{X_t} = \sum_i w_i X_t^{(i)}$ and then you find your weights $w_i$ by optimizing over some or combined criteria, such as dollar-neutral, utility maximization (max Sharpe). Each stock is still an independent bet, but you combine them in order to decide where you allocate capital. Either the prediction can be used as an estimate for the return, or the conditional return given the estimate, when optimizing the criteria used. Strategy 2 works the same but you do not assume that stocks are independent. E.g. the variance of your portfolio now also includes covariances between stock returns. The rest is the same.

My question is, if the two ICs are at similar scales, (actually in my experience ICcs is generally larger than ICts with the same set of predictions and same lookback time), what are the pros and cons for each strategy?

Given the reasonable model assumption I made, I explained that $IC_{cs}$ made little sense, and that you should stick with $IC_{ts}$ but probably not perform an arithmetic average but rather a weighted one based on your portfolio weights. The reason why you observe different values of them is because that correlation is not a linear function of the samples. Under some assumptions, like equal variances and zero correlations, $IC_{ts}=IC_{cs}$. But don't worry, just use $IC_{ts}$ as it is the only one that makes sense.

With regards of assuming that stock returns are independent. Advantage is that you have fewer parameters to estimate (no covariance matrix estimation). You mention that you have $T=m$, i.e. the same number of return samples as you have different stocks. To estimate a covariance matrix you would like to have $T>m$ and preferably $T>10\cdot m$. Disadvantage is that you do not take into account that the returns are "related", so you might have trouble with constructing an efficient long-short portfolio that is e.g. market neutral. But it also depends on your prediction model, if it includes systematic risk.

4. Extended model assumptions

Let us extend this model by assuming that returns follow an $m$-dimensional Gaussian distribution, $\bf X_t \sim \mathcal{N}_m(\vec{\bf{0}}, \bf\Sigma_{xx})$, and prediction vector $\bf Y_t \sim \mathcal{N}_m(\vec{\bf{0}}, \bf\Sigma_{yy})$. Then the conditional return given the prediction is

\begin{align} X_t|Y_t &=y \sim \mathcal{N}_m(\mu_{x|y}, \Sigma_{x|y}), \\ \mu_{x|y} &= \Sigma_{xy}\Sigma_{yy}^{-1} y, \\ \Sigma_{x|y} &= \Sigma_{xx}-\Sigma_{xy}\Sigma_{yy}\Sigma_{xy}, \end{align} where return covariance $\Sigma_{xx}$ is estimated by historical returns, and prediction covariance $\Sigma_{yy}$ depends on your prediction model, and cross covariance $\Sigma_{xy}$ depends on how well your prediction model covariates (correlates via $IC_{ts}$) with the returns. To simplify, assume that the prediction only $\Sigma_{xy}=\text{diag}_i(\sigma_i\eta_i IC_{ts}^{(i)})$, i.e. prediction for asset $i$ only covariates with return $i$, and assume $\Sigma_{yy}=\text{diag}_i(\eta_i^2)$, i.e. the prediction only depends on the associated idiosyncratic stock mechanics.

The conditional expected returns is then distributed with mean $\mu_{x|y}=\text{diag}_i\left(\frac{\sigma_i}{\eta_i} IC_{ts}^{(i)}\right)y = \text{vector}_i\left(\frac{\sigma_i}{\eta_i} IC_{ts}^{(i)}y_i\right)$ and covariance $\Sigma_{x|y} = \Sigma_{xx}-\text{diag}_i\left(\sigma_i^2 {IC_{ts}^{(i)}}^2\right)$.

You can now use this for your portfolio optimization, e.g. maximizing Sharpe ratio under CAPM with or without constraints on dollar-neutral etc.

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  • $\begingroup$ I think it could make sense to use $t=1,\dots,T$ instead of $k=1,\dots,n$, it would be easier to follow. $\endgroup$ Jun 15, 2022 at 11:20

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