# Sharpe from signal to daily return correlation

A few years back in an interview I remember being asked to derive the Sharpe ratio from the correlation between a pre-open daily signal and the open-close returns. I think you had to make some assumptions regarding the risk/reward and accuracy but you could start with a correlation of say 0.05 and back out a sharpe ratio.

I'm interested in revisiting this and can't seem to find anything similar online. Anyone know of any resources?

Update: Happy with comment answer, thanks!

• The 'correlation profile' exercise in Short Sharpe Course does this computation, finding that the Sharpe is approximately equal to the correlation under a 'small angle approximation'. – steveo'america Sep 17 '20 at 22:27
• Thank you! So we have the optimal allocation being $c \frac{\mu + \rho \sigma f_t}{(\mu + \rho \sigma f_t)^2 + (1 - \rho^2) \sigma^2}$, from the next page I use $c = \rho$ and that's the sharpe? Testing for $\rho = 0.1$, $\mu = 0$, $\sigma = 0.01$, $f_t = 1$, $c = \rho$ gives a sharpe of 1 which seems reasonable. Is that correct? – Quantoisseur Sep 18 '20 at 12:15
• $c$ is a positive constant which adjusts the allocation to reach a desired level of volatility. The (optimal) signal-noise ratio is $\zeta_*$ and that's approximately equal to $|\rho|$. – steveo'america Sep 18 '20 at 22:41
• Okay, so then I'm confused at how to arrive at the Sharpe ratio which shouldn't be scalable. – Quantoisseur Sep 18 '20 at 23:11
• It's a small angle approximation, so the Sharpe is only approximately $|\rho|$ when that number is small. That's a reasonable assumption when looking at a daily signal, and the kind of alpha available to mere mortals. – steveo'america Sep 19 '20 at 5:13