Thread by @macrocephalopod

Correlation between your signal and future returns is an important metric in quant trading. But what is a “good” correlation? Here’s a simple way to think about it.

We’ll use a simple model where future returns y over some time period tau are normally distributed with a mean of beta * x and a daily volatility of sigma (here x is a signal with std deviation 1)

We can easily work out the correlation between signal and returns and use that to express beta as a function of correlation, volatility and forecast horizon.

The key insight is that it should be easy to find signals that are not profitable if you take trading costs into account, since you won’t be able to action them anyway.

If we require that even a three standard deviation signal is unprofitable then we can bound the correlation —

What does that tell us? Say that we are interested in a stock with 3% daily volatility, trading cost of 5 bps and a forecast horizon of one day. Then we expect to easily find signals with a correlation of 0.5% with future returns

If we use factor hedging to remove non-idiosyncratic risk, we might halve the volatility and double the cost to trade (since we need to trade the factor hedge as well) so we expect to be able to easily find signals with 2% correlation with future idiosyncratic return.

Alternatively if we are trying to predict a short term (1 minute) fx return which costs 0.2bps to trade and has a daily vol of 0.3% then we expect to easily find alphas with a much higher correlation of 8.5%

You can think of these as absolute minimum correlations, you need to exceed these to be able to trade profitably. As a rule of thumb, a correlation which is 1.5x the minimum would be ok (you will have a trade to do in about 5% of periods) and 2x the minimum is very good (you will be able to trade 20-30% of the time)

This is one of the many ways that you can extend the law of active management to be more relevant to real world trading, a topic for another time maybe.