Optimal leverage for a return stream

Question

A strategy earns an expected excess return of $\mu = 8\%$ per year with volatility $\sigma = 16\%$ . You can lever it up (a fraction $f$ of capital exposed, where $f$ may exceed $1$ ).

What leverage maximizes long-run growth, and what growth rate does it deliver? Comment on whether you would actually run it.

Accepted Answer

For a continuous return stream, applying leverage f gives an expected log-growth rate g(f) = f\mu - 	frac{1}{2}f^2\sigma^2, where the second term is the volatility drag that leverage amplifies quadratically. Maximizing: g'(f) = \mu - f\sigma^2 = 0 \implies f^ = \frac{\mu}{\sigma^2}. Plugging in: f^ = \dfrac{0.08}{0.16^2} = \dfrac{0.08}{0.0256} = 3.125. Full Kelly says lever about 3.1 to 1. The resulting growth rate is g(f^) = \frac{\mu^2}{2\sigma^2} = \frac{0.08^2}{2(0.0256)} = \frac{0.0064}{0.0512} = 0.125, that is 12.5\% per year. Notice this equals 	frac{1}{2}	ext{(Sharpe)}^2: with Shar…

Optimal leverage for a return stream

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