# Variance of the sample mean and impact on optimal leverage

## Assumptions

1. Stable distributions
2. Distributions are normal
3. Assume $$\sigma$$ is known
4. 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

1. Distributions and the joint distribution are not stable
2. Distributions are not normal
3. $$\sigma$$ 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.