Introduction

X (formally Twitter) user @macrocephalopod ¹ describes an interesting derivation of the minimum required correlation between a trading signal and the target return for profitable trading. It is defined as a function of the signal Z-score, forecast horizon volatility and transaction costs.

This post consists of a summary of this technique and applies it to historical Bitcoin price data. We visualise the minimum correlation for different combinations of transaction costs (exchange fee tiers) and target forecast horizons (via historical volatility).

Correlation, Volatility and Costs

A quick recap on correlation before we continue, the (symmetric) Pearson correlation coefficient ($ \rho$) tells us if $x$ $(y)$ increases by one standard deviation, what is the expected response in increase of standard deviations of $y$ $(x)$:

$$ Corr(x,y) = \frac{\sum_{i=1}^{N} (x_i - \bar{x})(y_i - \bar{y})} {\sigma_x \cdot \sigma_y} $$

Grinold & Kahn ² introduce a “refined forceast” $ \alpha $ in units of target return, where Volatility is the target asset volatility (standard deviation at specific horizon), Corr is the correlation between a trading signal and the target returns and Score is the Z-score of a trading signal: $$ \alpha = Volatility \cdot Corr \cdot Score $$ In practise the $ \alpha $ value is quoted without the score coefficient as it is variable over sequential evaluations of the model. The score is commonly assumed to be drawn from a normal distribution, in which case we can make assumptions about how often we expect to see a signal over a certain magnatude (68-95-99.7 rule ³). It’s important that the signal reaches a magnitude large enough that the forecast as a whole exceeds trading costs and that it does so frequently enough for us to acheive our target return on capital.

The expanded version of the “refined forecast” is below, it looks at a target return $(y)$ and a signal $(x)$. It exploits the correlation relationship described above, by multiplying the Z-score of the signal and the correlation we get the expected response of $y$ in standard deviations, by multiplying this with $\sigma_y$ we get the response in units of $y$:

$$ \alpha = \sigma_y \cdot Corr(x, y) \cdot (\frac{x-E(x)}{\sigma_x}) $$

@macrocephalopod expands on this reasoning and forumlates the minimum correlation required for profitable trading - to overcome the trading costs $c$ (eg. exchange fees, slippage). He describes regressing forecast returns $y$ where the volatility coefficient has been expanded to allow for a time-varying lookahead, paramaterised by $\tau$. He then shows how the correlation formula can be re-written when $x$ has unit variance to realise $ \beta $ as the product of correlation and volatility (see original post for detail), similar to $ \alpha $ above (without the Z-score of signal $x$):

$$ y = \beta \cdot x + \sigma \sqrt{\tau} \cdot \epsilon $$ $$ \beta = \rho \cdot \sigma \cdot \sqrt{\tau} $$

Next, he introduces costs $c$ and calculates the signal correlation at which a $3\sigma$ signal will only cover costs, we need to exceed this correlation to make profits:

$$ 3\beta \le c $$ $$ \Rightarrow 3\rho\sigma\sqrt{\tau} \le c $$ $$ \Rightarrow \rho \le \frac{c}{3\sigma\sqrt{\tau}} $$

Assuming the signal follows a normal distribution, it will only exceed a magnatude of $ 3\sigma $ in 0.3% of periods - not enough to trade consistently. As such, as stated by the author values of 1.5x or 2x the minimum correlation is neccessary to realistically trade profitably - this allows us to reduce the signal threshold and exceed costs in a greater number of periods.

Evaluation on Binance Bitcoin price data

We look at the following (reduced) Binance fee tiers for taker orders and combine them with a constant slippage of 5 basis points (bps) for our cost variable $c$. The Total Fee is this estimate doubled, to account for both entering and exiting the position.

Using year-to-date (2024) minute bar close price data from Binance we calculate the one minute volatility as 7.5bps and utilise the $ \sqrt{\tau} $ multiplier to scale this to longer forecast horizons.

The notebook used to generate these plots is available here [^4].

Fee tier burn in

As an interesting exercise we investigate the “burn in” cost of a strategy which is only profitable at a higher fee tier and has to invest in an initial loss to advance to the required tier. We need to set some exta parameters: forecast horizon is 5 minutes, maximum time to complete the burn in is one week (2016 five minute periods) and the capacity of each trade is $30,000.

Example one - Regular to VIP3

We evaluate a signal with $ \rho = 0.496$, with an alpha of $8.3bps$, this is unable to beat Regular tier transaction costs at the $3\sigma$ level ($\alpha = 24.9$, $c = 25$) however on the VIP3 fee tier we can overcome transaction costs when $\sigma \gt 2.29$ and at this threshold we can trade in just over 2% of periods.

To go from the base tier to VIP3 we need to do 20 million of dollar volume, with our time and capacity constraints we need to trade 672 (33% of total period) times, to acomplish this effectively we can only trade a signal of (approx.) absolute value $ \ge 1 \sigma $.

At a minimum signal strength of $ x = 1 \sigma $, alpha is $8.3bps$ - after costs this is a loss of 16bps per trade. 16bps of 20 million - we lose approx ＄32,000 before advancing to VIP3.

Example two - Regular to VIP9

With the previous examples capacity and time constraints we are unable to achieve the target volume of 40 billion. We update them to continue, time window is 4 weeks and trade capacity is ＄1,500,000. Given the increase in capacity, it’s a given that the slippage costs would also increase and affect our total fee values - but we will leave these as they are for simplicity.

We evaluate a signal with $ \rho = 0.405 $ and $ \alpha = 6.8bps $ - this allows us to exceed costs on VIP9 with a $ \ge 2\sigma $ signal (trading in approx. 5% of periods) and to exceed costs on VIP6 with a $ \ge 2.12\sigma $ signal (approx. 3% of periods). We do not exceed costs in the lower tiers, even at a $ 3 \sigma $ signal.

TODO: How to account for profitable signals instead of hardcoding 1sigma??

Summary

TODO