# Variance of the sample mean and impact on optimal leverage

02 Dec 2022## Assumptions

- Stable distributions
- Distributions are normal
- Assume \(\sigma\) is known
- There is no autocorrelation in logreturns

Let the *observed* logreturn \(\mu\) be estimated as a mean of a series of \(n\) independent logreturns.

Using the variance of the sample mean, \(\mu\) is drawn from a normal distribution, \(\mu \sim \mathcal{N}(m, \sigma_{\mu}^2)\), with \(\sigma_\mu = \sigma / \sqrt{n}\).

The error is then

\[\mu - m \sim \mathcal{N}\left(0, \sigma^2 + \sigma_\mu^2\right) = \mathcal{N}\left(0, \left( \frac{1}{n} + 1 \right) \sigma^2 \right) .\]The Kelly leverage \(l\) is then \(l \leq \frac{\mu}{(1/n + 1) \sigma^2}\).

## Reality

- Distributions and the joint distribution are not stable
- Distributions are not normal
- \(\sigma\) is not known

Of these, the first matters by far the most. In practice the estimation of \(\mu\) is both more important and more uncertain than that of \(\sigma\). In addition, non-normal distributions require a larger number of samples to characterise.