Zou Yilin's Blog

Kelly Criterion for Continuous Probability Distribution: From Gambling to Leveraged ETFs

By Zou Yilin

Principle of the Kelly criterion

If Kamala has a 40% chance of winning in the presidential election and a bet of 1 dollar wins 2 dollars, what is the optimal betting proportion? The result is directly given by the Kelly criterion:

\[ f = \frac{bp - q}{b} = \frac{2 \times 0.4 - 0.6}{2} = 0.1 \]


Thus, the optimal betting proportion is 10%. Here, \(b = 2\) is the odds, \(p = 0.4\) is the probability of winning, and \(q = 1 - p = 0.6\) is the probability of losing.

In fact, the result of multiple bets is the product of the results of individual bets. Therefore, the correct optimization goal for a single bet is to maximize the expectation of \(\log(\text{result})\). Based on this, the Kelly criterion can be derived through simple calculations.

Let the betting proportion be \(f\), then the gambling result is \(1 + bf\) or \(1 - f\), and the expectation of the logarithm of the result is:

\[ g(f) = p \log(1 + bf) + q \log(1 - f) \]


Taking the partial derivative with respect to \(f\):

\[ \frac{\partial}{\partial f} g(f) = \frac{bp}{1 + bf} - \frac{q}{1 - f} = 0 \]


Thus,

\[ f = \frac{bp - q}{b} \]

Extension to Continuous Probability Distribution

The classic Kelly criterion used for gambling only considers two outcomes: win or lose. Using the same calculation method, it can be extended to cases with continuous probability distributions.

Let the daily return rate of the index tracked by a leveraged ETF be \(r\), with a probability density function of \(p(r)\). The leveraged ETF goes long \(f\) times its daily return. Thus, the daily return rate is \(f r\). Since this is also a case of continuous multiple gambling, the optimal betting proportion \(f\) is still derived from maximizing the expectation of \(\log(\text{result})\). In this case, the expectation of the logarithm of the result is:

\[ g(f) = \int_{-1}^\infty p(r) \log(1 + f r)\, \mathrm{d}r \]


Taking the derivative with respect to \(f\):

\[ \frac{\partial}{\partial f} g(f)= \int_{-1}^\infty p(r) \frac{r}{1 + f r}\, \mathrm{d}r = 0 \]


For the daily return rate, we have \( r \ll 1 \). Approximating the fraction gives:

\[ \frac{r}{1 + f r} \approx r - f r^2 \]


Therefore, the equation can be approximated as:

\[ \int_{-1}^\infty p(r) (r - f r^2)\, \mathrm{d}r = 0 \]


Thus,

\[ f = \frac{\int_{-1}^\infty p(r) r\, \mathrm{d}r}{\int_{-1}^\infty p(r) r^2\, \mathrm{d}r} = \frac{\mathbb{E}[r]}{\mathbb{E}[r^2]} = \frac{\mu}{\mu^2 + \sigma^2} \]


where \( \mu \) is the expected daily return rate and \( \sigma \) is the standard deviation of the daily return rate (volatility).

Result Verification

Using different time windows to calculate the optimal daily leverage for the Nasdaq 100 index according to the above formula, the results fall between \(2.5\) and \(3.5\). This aligns with the historical performance of TQQQ, which significantly outperforms QLD and QQQ.