  # Give me a moment: Optimal leverage in the presence of volatility, skewness, and kurtosis

This paper discusses the role that volatility, skewness, and kurtosis play in the distribution of wealth for a series of returns with reinvestment. We begin with a simplistic model to explain how volatility shapes the wealth distribution of investments. (Page 1)

Derivatives pricing often uses a binomial tree to simplify the price change of an asset. The asset price starts at today’s level, say \$100 per share, and can move in two states only: up or down by a return increment with a probability (Page 2)

As we increase the volatility of our asset, we create paths with far-right tails but move more of the outcomes to the left-side of \$100. This phenomenon shows the volatility drag concept in continuous return space and how it becomes more pronounced at higher volatility. As volatility increases, the distribution shifts to where more of paths are losers and a small outlier of very large winners emerge. All this occurs despite an expected value of \$100 across all three distributions (Page 4)

The goal of investing is to find positive expected returns and take risk. But as we saw previously, high volatility can create a profile where most outcomes are negative, while only a few lucky paths receive stellar returns. (Page 4)

Suppose we own a stock with a 10% expected return and 20% annualized volatility2. Also, suppose we have the ability for near infinite leverage, we can trade at any time scale, transaction costs are zero, and leverage has no cost3. In the context of Kelly, how do we maximize the expected value of the logarithm of wealth? (Page 6)

Next, we use real world return data to confirm this property. We selected gold (proxied by the ETF: GLD) which between 2017 and 2021 had average annualized returns of 9.4% and volatility of 13.5%. If we knew these values for certain ex-ante, how much should we have been willing to risk? (Page 7)

Solving for optimal leverage we calculate r/σ2 = 5.17; that is, we should borrow 4.17x our wealth and own 517% of our wealth in the asset, rebalancing back to that leverage amount as the price of the asset increases or decreases. (Page 7)

We show that, for investments with identical reward to risk, investors can apply more leverage to positively skewed returns and generate larger ending wealth. Meanwhile, negatively skewed distributions require investors to use less leverage, thus limiting wealth. (Page 8)

The graph above shows that despite the expected return and volatility being equivalent for all three distributions, the leverage L can be increased further before wealth peaks for the positively skewed return distribution. On the other hand, leverage for the negatively skewed return distribution peaks at lower L and lower wealth than either the zero or positive skewed distributions. (Page 10)

Part I of this paper described the role that volatility plays in shaping the distribution of an asset or strategy over time. We saw that increasing volatility shifts the peak of wealth distributions to the left and creates a more skewed right tail. We then linked volatility’s effect to Kelly’s definition of optimal leverage, showing that too much volatility relative to return moves wealth sub-optimally. (Page 11)

Positively skewed return distributions allow more leverage, and thus higher wealth, than non-skewed or negatively skewed distributions. Any estimate of optimal leverage is far from perfect as the non-stationarity of games played in the financial markets reduces their effectiveness. It is difficult enough to predict forward looking returns and volatility of an asset or strategy. Predicting skewness and kurtosis with true precision is an even taller task. Even if our crystal ball were perfectly accurate, trading at leverage multiples that maximize wealth will lead to exceptionally large volatility that likely exceed bounds tolerated by investors and professional allocators. Nonetheless, this paper describes the influence that third and fourth moments of the return distribution have on wealth and proposes a measure to estimate optimal leverage for those bold enough to trade there. (Page 11)