# 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.

## 1 Answer

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 at 11:03
• Do you have any books or papers to reference about the proof? Mar 6 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 at 11:13
• Sure. The wikipedia page is good: en.wikipedia.org/wiki/Kelly_criterion#Stock_investments Mar 6 at 11:14