Kelly criterion

Gamblers often state the size of their bets relative to the Kelly criterion. A full Kelly bet is a bet made at the Kelly Criterion. A half Kelly bet is half the size of a full Kelly bet. A quarter Kelly bet is a quarter of the size of a full Kelly. Gamblers would use less than full Kelly in order to reduce the chance of ruin, reduce volatility, and account for model error. Due to the high drawdowns, gamblers in practice find fractional Kellies much better emotionally than full Kelly. This reduced volatility is a tradeoff, as it increases the time to reach an intended wealth or decreases the wealth growth rate. It has been found that betting an amount larger than the Kelly amount increases the risk of ruin. == Kelly criterion for binary return rates ==

In a system where the return on an investment or a bet is binary, so an interested party either wins or loses a fixed percentage of their bet, the expected growth rate coefficient yields a very specific solution for an optimal betting percentage. Gambling Formula Where losing the bet involves losing the entire wager, the Kelly bet is: : f^* = p-\frac{q}{b} = p - \frac{1 - p}{b} where: • f^{*} is the fraction of the current bankroll to wager. • p is the probability of a win. • q=1-p is the probability of a loss. • b is the proportion of the bet gained with a win. E.g., if betting $10 on a 2-to-1 odds bet (upon win you are returned $30, winning you $20), then b = \$20/\$10 = 2.0. As an example, if a gamble has a 60% chance of winning (p = 0.6, q = 0.4), and the gambler receives 1-to-1 odds on a winning bet (b=1), then to maximize the long-run growth rate of the bankroll, the gambler should bet 20% of the bankroll at each opportunity (f^{*} = 0.6-\frac{0.4}{1} = 0.2). If the gambler has zero edge (i.e., if b = q / p), then the criterion recommends the gambler bet nothing (see gambler's ruin). If the edge is negative (b ), the formula gives a negative result, indicating that the gambler should take the other side of the bet. Investment formula A more general form of the Kelly formula allows for partial losses, which is relevant for investments: : f^{*} = \frac{p}{l}-\frac{q}{g} where: • f^{*} is the fraction of the assets to apply to the security. • p is the probability that the investment increases in value. • q is the probability that the investment decreases in value ( q = 1 - p). • g is the fraction that is gained in a positive outcome. The Kelly criterion maximizes the expected value of the logarithm of wealth (the expectation value of a function is given by the sum, over all possible outcomes, of the probability of each particular outcome multiplied by the value of the function in the event of that outcome). We start with 1 unit of wealth and bet a fraction f of that wealth on an outcome that occurs with probability p and offers odds of b. The probability of winning is p, and in that case the resulting wealth is equal to 1+fb. The probability of losing is q=1-p and the odds of a negative outcome is a. In that case the resulting wealth is equal to 1-fa. Therefore, the geometric growth rate r is: : r=(1+fb)^p\cdot(1-fa)^{q} We want to find the maximum r of this curve (as a function of f), which involves finding the derivative of the equation. This is more easily accomplished by taking the logarithm of each side first; because the logarithm is monotonic, it does not change the locations of function extrema. The resulting equation is: : E = \log(r) = p \log(1+fb)+q\log(1-fa) with E denoting logarithmic wealth growth. To find the value of f for which the growth rate is maximized, denoted as f^{*}, we differentiate the above expression and set this equal to zero. This gives: : \left.\frac{dE}{df}\right|_{f=f^*}=\frac{pb}{1+f^{*}b}+\frac{-qa}{1-f^{*}a}=0 Rearranging this equation to solve for the value of f^{*} gives the Kelly criterion: : f^{*} = \frac{p}{a}-\frac{q}{b} To be thorough, we should also consider the behaviour as f approaches the boundaries -\frac{1}{b} and \frac{1}{a} since there can be a maximum there without the derivative being 0. But E tends to −∞ for both. Finally, we need to show that the critical point found is not a minimum, this can be easily shown by computing the second derivative which is strictly negative for all f in the domain. Notice that this expression reduces to the simple gambling formula when a=1=100\%, when a loss results in full loss of the wager. == Kelly criterion for non-binary return rates ==

If the return rates on an investment or a bet are continuous in nature the optimal growth rate coefficient must take all possible events into account. Application to the stock market In mathematical finance, if security weights maximize the expected geometric growth rate (which is equivalent to maximizing log wealth), then a portfolio is growth optimal. The Kelly Criterion shows that for a given volatile security this is satisfied when f^* = \frac {\mu - r} {\sigma^2} where f^* is the fraction of available capital invested that maximizes the expected geometric growth rate, \mu is the expected growth rate coefficient, \sigma^2 is the variance of the growth rate coefficient and r is the risk-free rate of return. Note that a symmetric probability density function was assumed here. Computations of growth optimal portfolios can suffer tremendous garbage in, garbage out problems. For example, the cases below take as given the expected return and covariance structure of assets, but these parameters are at best estimates or models that have significant uncertainty. If portfolio weights are largely a function of estimation errors, then Ex-post performance of a growth-optimal portfolio may differ fantastically from the ex-ante prediction. Parameter uncertainty and estimation errors are a large topic in portfolio theory. An approach to counteract the unknown risk is to invest less than the Kelly criterion. Rough estimates are still useful. If we take excess return 4% and volatility 16%, then yearly Sharpe ratio and Kelly ratio are calculated to be 25% and 150%. Daily Sharpe ratio and Kelly ratio are 1.7% and 150%. Sharpe ratio implies daily win probability of p=(50% + 1.7%/4), where we assumed that probability bandwidth is 4 \sigma = 4\% . Now we can apply discrete Kelly formula for f^{*} above with p=50.425\%, a=b=1\% , and we get another rough estimate for Kelly fraction f^{*} = 85\% . Both of these estimates of Kelly fraction appear quite reasonable, yet a prudent approach suggest a further multiplication of Kelly ratio by 50% (i.e. half-Kelly). A detailed paper by Edward O. Thorp and a co-author estimates Kelly fraction to be 117% for the American stock market SP500 index. Significant downside tail-risk for equity markets is another reason to reduce Kelly fraction from naive estimate (for instance, to reduce to half-Kelly). == Proof ==

A rigorous and general proof can be found in Kelly's original paper We give the following non-rigorous argument for the case with b = 1 (a 50:50 "even money" bet) to show the general idea and provide some insights. on gambling on many mutually exclusive outcomes, such as in horse races. Suppose there are several mutually exclusive outcomes. The probability that the k-th horse wins the race is p_k, the total amount of bets placed on k-th horse is B_k, and : \beta_k=\frac{B_k}{\sum_i B_i}=\frac{D}{1+Q_k} , where Q_k are the pay-off odds. D=1-tt, is the dividend rate where tt is the track take or tax, \frac{D}{\beta_k} is the revenue rate after deduction of the track take when k-th horse wins. The fraction of the bettor's funds to bet on k-th horse is f_k. Kelly's criterion for gambling with multiple mutually exclusive outcomes gives an algorithm for finding the optimal set S^o of outcomes on which it is reasonable to bet and it gives explicit formula for finding the optimal fractions f^o_k of bettor's wealth to be bet on the outcomes included in the optimal set S^o. The algorithm for the optimal set of outcomes consists of four steps: but in some cases the solution obtained may be infeasible. For single assets (stock, index fund, etc.), and a risk-free rate, it is easy to obtain the optimal fraction to invest through geometric Brownian motion. The stochastic differential equation governing the evolution of a lognormally distributed asset S at time t (S_t) is : dS_t/S_t=\mu dt+\sigma dW_t whose solution is : S_t = S_0\exp\left( \left(\mu - \frac{\sigma^2}{2} \right)t + \sigma W_t\right) where W_t is a Wiener process, and \mu (percentage drift) and \sigma (the percentage volatility) are constants. Taking expectations of the logarithm: : \mathbb{E} \log(S_t)=\log(S_0)+\left(\mu- \frac {\sigma^2}{2}\right)t. Then the expected log return R_s is : R_s = \left(\mu -\frac{\sigma^2}{2}\,\right). Consider a portfolio made of an asset S and a bond paying risk-free rate r, with fraction f invested in S and (1-f) in the bond. The aforementioned equation for dS_t must be modified by this fraction, i.e. \frac{dS_t'}{S_t'} =f\frac{dS_t}{S_t}, with associated solution : S'_t = S'_0\exp\left( \left(f\mu - \frac{(f\sigma)^2}{2} \right)t + f\sigma W_t\right) the expected one-period return is given by : \mathbb{E}{\left(\left[\frac{S'_1}{S'_0} - 1\right] + ( 1 - f)r \right)} = \mathbb{E}{\left( \left[\exp\left( \left(f\mu - \frac{(f\sigma)^2}{2} \right) + f\sigma W_1\right) - 1 \right]\right)} + (1 - f) r For small \mu, \sigma, and W_t, the solution can be expanded to first order to yield an approximate increase in wealth : G(f) = f\mu - \frac {(f\sigma)^2}{2} + (1-f)\ r. Solving \max (G(f)) we obtain : f^* = \frac {\mu - r} {\sigma^2}. f^* is the fraction that maximizes the expected logarithmic return, and so, is the Kelly fraction. Thorp == Bernoulli ==

In a 1738 article, Daniel Bernoulli suggested that, when one has a choice of bets or investments, one should choose that with the highest geometric mean of outcomes. This is mathematically equivalent to the Kelly criterion, although the motivation is different (Bernoulli wanted to resolve the St. Petersburg paradox). An English translation of the Bernoulli article was not published until 1954, but the work was well known among mathematicians and economists. == Criticism ==

Although the Kelly strategy's promise of doing better than any other strategy in the long run seems compelling, some economists have argued strenuously against it, mainly because an individual's specific investing constraints may override the desire for optimal growth rate. In colloquial terms, the Kelly criterion requires accurate probability values, which isn't always possible for real-world event outcomes. When a gambler overestimates their true probability of winning, the criterion value calculated will diverge from the optimal, increasing the risk of ruin. In the stock market, the Kelly bet can be thought as time diversification, which is taking equal risk during different sequential time periods (as opposed to taking equal risk in different assets for asset diversification). There is a difference between time diversification and asset diversification, which was raised by Paul A. Samuelson. There is also a difference between ensemble-averaging (utility calculation) and time-averaging (Kelly multi-period betting over a single time path in real life). The debate was renewed by evoking ergodicity breaking. Yet the difference between ergodicity breaking and Knightian uncertainty should be recognized. == See also ==