Following on from the prior article, we will now study the effect of leverage on the severity of drawdowns. This is done for a series with a positive mean daily return over the last decade — so it should produce profitable investments in expectation. We will investigate the effect of leverage on the maximum drawdown in the series by using the bootstrap to elucidate the mean relationship rather than looking at one specific realization.
Our approach is as before, with the exception that we pick the leverage for each trial at randome between zero and four. Four is the maximum leverage permitted to a Pattern Day Trader, and so that seems a sensible limit for our analysis.
The above chart shows the observed relationship between the leverage used and the maximum drawdown within the nine years of trading represented by each simulation. The curve fitted is to the expression below (and is done by non-linear least squares).
=100\%\left(1-e^{-gL}\right)^\kappa)
(Here M is the maximum drawdown and L is the leverage; I don't assert that this relationship is anything other than a convenient representation of the data.) We see that with a standard margin account (2× leverage permitted), we expect a maximum drawdown of around 75% of capital and the probability of a drawdown exceeding 50% of capital is of order unity. Near the higher levels of leverage, complete drawdown (maxmimum drawdown exceeding 99% of capital) is increasingly certain. We shall present an empirical model for these probabilities in the next post.