# Optimal leverage for strategy with normal returns

Given a strategy with normal returns with mean 5% and standard deviation 10% what is the optimal leverage (up to a maximum of 2x) to maximize the expected wealth? With the same setting, if trading is discretized in 5 rounds, what is the optimal leverage?

With a Monte Carlo approach, the optimal leverage seems to be the maximum allowed and that also makes seem to make sense intuitively.

Assuming you are compounding returns, the optimal strategy is given by:

μ/σ^2

For your strategy this is 0.05/0.01 = 5x, but because of your constraint of a maximum of 2x, then you'd choose 2x. From a first principles basis, the formula is derived by:

1. Assuming that the change in asset price, dS, is given by: dS = Sμ * dt + Sσ * dW

Since leverage, λ, is instantaneously linear for returns (+1% on 2x leverage = 2%), levered returns are given by:

λ * dS/S = λ * (μ * dt + σ * dW)

1. In the continuous case, the log returns are given by: ln((S_t + dS)/S_t) = (μ - σ^2/2) * dt + σ * dW

The '- σ^2/2' term comes from the second term of the Taylor series expansion of ln(x+1).

In the levered case, the leverage is squared in the variance term, meaning the levered log/continuously compounded returns are given by:

μ - λ^2σ^2/2) * dt + λ*σ * dW

1. The expected levered return is given by: (λμ - λ^2σ^2/2) * dt

Intuitively, one can see that if leverage or variance is too high, the expected return turns negative. One can differentiate the function with respect to λ:

dR_λ/dλ = (μ - λ*σ^2)*dt

Setting to 0 to find the maximum:

μ = λ*σ^2

Solving for λ:

μ/σ^2 = λ

Note that this is only the optimal leverage when returns are definitely defined by a normal distribution, with no sampling error of the mean or volatility. In markets we rarely see returns globally defined by their long-run means or variances (though, perhaps locally we do with more frequent measurements), and with higher kurtosis and non-zero skewness, it is ill-advised to lever as high as the formula would indicate. Half is a popular alternative.

• Does the finite discretization effect the problem in any way? Mar 6, 2023 at 11:03
• Do you have any books or papers to reference about the proof? Mar 6, 2023 at 11:11
• No. I assume your dt is ~ 1 day? The approximation will hold fine. If your volatility * sqrt(dt) ~= 30% then the approximation will begin to break down. For almost every asset simulating at daily volatility you won't ever touch the sides. Mar 6, 2023 at 11:13
• Sure. The wikipedia page is good: en.wikipedia.org/wiki/Kelly_criterion#Stock_investments Mar 6, 2023 at 11:14