cephalopod

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

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.

There’s nothing which forces it to be less than 1 but for any realistic set of parameters it will be.

People are saying I am not posting enough finance content. FINE.

Once you have decent alphas a lot of the skill in running a systematic strategy is reducing turnover. Alpha is unknown but trading costs are certain. Keep trading to the minimum needed to monetise your alpha.

If you have a univariate strategy (all assets independent, no portfolio considerations) a simple heuristic is to use a no trade region. Your strategy outputs a target position x at all times and you compare it to your current position y.

Introduce a new buffer parameter b. If y &lt; x - b then trade until your position is x - b. If y &gt; x + b then trade until x + b. Otherwise you are in the “no trade region” and you don’t trade.

Another way to say it — you have an interval of width 2b around your target position, and if your position is outside that interval you trade until you are at the edge of it. Larger values of b will reduce turnover more.

How to pick b? A few ideas

1. Optimise it in a backtest (eg to maximise net Sharpe ratio)

2. Pick it to target a particular turnover, eg if your forecast horizon is ten days you hope to turn over ~20% of your book per day, so pick b to achieve this

3. Pick a sensible value based on your max position size, eg 5-25% of max position size.

I’m normally against optimising parameters based on a backtest but this is quite a benign optimisation. The main danger is that if your edge is overestimated, your optimisation will pick a value of b that is too low, so you should be conservative (eg double the “best” value of b)

In practise I normally do either 1 or 2, but 3 is a good check that you have selected a sensible value.

There is a mathematical model that produces this trading rule btw! If you have linear alpha, quadratic risk and trading costs proportional to trade size (eg a fixed bid offer spread) this is the trading rule that results. Can run through the math if there’s demand. Fin.

This is a sort of “folk knowledge” in quant finance. It’s rarely discussed in books/papers because it’s so basic that it’s assumed everyone knows it, so I couldn’t find a good reference for it. Derivation is below.

Nice thing about knowing the derivation is that it’s straightforward to extend it eg other t-cost forms (3/2 or quadratic), multiple assets (sigma becomes a covariance matrix) or introduce constraints on position size or turnover. Most extensions need to be solved numerically.