Choosing Optimal Leverage for Normally Distributed Returns
Summary
The document derives the leverage that maximizes expected continuously compounded wealth when returns follow a normal model. It starts with an asset whose returns have mean 5% and standard deviation 10%, then shows that leverage scales the mean linearly while scaling the variance quadratically. Maximizing the resulting expected log-return expression gives leverage equal to the mean divided by variance, or 5x in the example; a 2x cap therefore makes 2x optimal under those assumptions.
This result assumes continuous compounding and known, stable return parameters. The question also mentions trading in five discrete rounds, but the response does not derive a separate discrete-round solution. The answer cautions that real returns may have estimation error, skewness, and heavier tails than a normal distribution, so the formula can imply excessive exposure; using half the calculated leverage is mentioned as a common alternative.
Key ideas
- For normally distributed returns under continuous compounding, expected log return is maximized at mean divided by variance.
- Leverage increases expected return linearly but increases the variance penalty quadratically.
- With the stated 5% mean, 10% standard deviation, and 2x cap, the formula selects the maximum permitted leverage.
- The derivation assumes known parameters and a normal return distribution.
- Estimation error, skewness, and heavy tails can make the theoretical leverage too aggressive.
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Full text
# Optimal leverage for strategy with normal returns # 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. ## Answer by Newquant (score 3, accepted) https://quant.stackexchange.com/a/74846 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: - 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) - 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: (λμ - λ^2σ^2/2) * dt + λ*σ * dW - 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.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.