Wikipedia / Kelly criterion
Wikipedia / Kelly criterion
In finance, the Kelly criterion (or Kelly strategy or Kelly bet) is a formula for risk allocation with the sizing a sequence of bets by maximizing the long-term expected value of the logarithm of wealth, which is equivalent to maximizing the long-term expected geometric growth rate. John Larry Kelly Jr., a researcher at Bell Labs, described the criterion in 1956.
The practical use of the formula has been demonstrated for gambling, and the same idea was used to explain diversification in investment management. In the 2000s, Kelly-style analysis became a part of mainstream investment theory and the claim has been made that well-known, successful investors including Warren Buffett and Bill Gross use Kelly methods (also see intertemporal portfolio choice). It is also the standard replacement of statistical power in anytime-valid statistical tests and confidence intervals, based on e-values and e-processes.
Relative bet sizes
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.
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.
Formula
The Kelly bet is: where:
is the fraction of the assets to apply to the security.
is the probability that the investment increases in value.
).
is the fraction that is gained in a positive outcome..
is the fraction that is lost in a negative outcome.. In gambling, typically the entire wager is lost, which corresponds to .
Note that the Kelly criterion is perfectly valid only for fully known outcome probabilities, which is almost never the case with investments. In addition, risk-averse strategies invest less than the full Kelly fraction.
where:
is the win-loss probability ratio, which is the ratio of probabilities of winning to losing bets.
is the reward-to-risk ratio of outcome sizes, which is the winning skew.
.
). This happens somewhat counterintuitively, because the Kelly fraction formula compensates for a small losing size with a larger bet. However, in most real situations, there is high uncertainty about all parameters entering the Kelly formula. In the case of a Kelly fraction higher than 1, it is theoretically advantageous to use leverage to purchase additional securities on margin.
Example
In a study, each participant was given $25 and asked to place even-money bets on a coin that would land heads 60% of the time. Participants had 30 minutes to play, so could place about 300 bets, and the prizes were capped at $250. But the behavior of the test subjects was far from optimal:
Using the Kelly criterion and based on the odds in the experiment (ignoring the cap of $250 and the finite duration of the test), the right approach would be to bet 20% of one's bankroll on each toss of the coin, which works out to a 2.034% average gain each round. This is a geometric mean) if it were not capped.
In this particular game, because of the cap, a strategy of betting only 12% of the pot on each toss would have even better results (a 95% probability of reaching the cap and an average payout of $242.03).
Proof
Heuristic proofs of the Kelly criterion are straightforward. The Kelly criterion maximizes the expected value. Therefore, we are trying to maximize the utility for which the utility is maximized, we differentiate the above expression with respect to in the domain.
For this value of , the wealth after a win would be and the wealth after a loss would be .
The expected geometric growth rate is , which is .
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
is the variance 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 6% and volatility 15% then the yearly Sharpe ratio is . The Kelly fraction (i.e., the appropriate investment) for these values is . (That is, one could achieve the Kelly criterion by investing in a diverse set of stocks with a beta of 2.67.) However the market invests 100% in itself so the market is apparently making its investment decisions using of the Kelly criterion value.
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 or in some of the other references listed below. Some corrections have been published. (a 50:50 "even money" bet) to show the general idea and provide some insights. but the proportion of winning bets will eventually converge according to the weak law of large numbers to zero, which means following the Kelly strategy. This illustrates that Kelly has both a deterministic and a stochastic component. If one knows K and N and wishes to pick a constant fraction of wealth to bet each time (otherwise one could cheat and, for example, bet zero after the Kth for a stretch, but in the long run, Kelly always wins. The heuristic proof for the general case proceeds as follows. Edward O. Thorp provided a more detailed discussion of this formula for the general case..
Multiple outcomes
Kelly's criterion may be generalized. The algorithm for the optimal set of outcomes consists of four steps:
will be the best bet.
(the empty set will be considered first.
Repeat:
, and stop the repetition.
One may prove where the right hand-side is the reserve rate may be interpreted-th horse wins. The binary growth exponent is
and the doubling time
.
Stock investments
The second-order Taylor polynomial can be used as a good approximation of the main criterion. Primarily, it is useful for stock investment, where the fraction devoted to investment is based on simple characteristics that can be estimated from existing historical data – expected value and variance. This approximation may offer similar results as the original criterion, but in some cases the solution obtained may be infeasible.
For single assets (stock, index fund, etc.), and a risk-free rate, it is straightforward to obtain the optimal fraction to invest through geometric Brownian motion. The stochastic differential equation) is
whose solution is
is a Wiener process, with associated solution
the expected one-period return is given by
, the solution can be expanded to first order to yield an approximate increase in wealth
is the fraction that maximizes the expected logarithmic return, and so, is the Kelly fraction. Thorp. Confusing this is a common mistake made by websites and articles talking about the Kelly Criterion.
and the rest is invested in the bond. Without loss of generality Expanding this with a Taylor series Thus we reduce the optimization problem to quadratic programming are the vector of means and the matrix of second mixed noncentral moments of the excess returns. There is also a numerical algorithm for the fractional Kelly strategies and for the optimal solution under no leverage and no short selling constraints.
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. One such economist is Paul Samuelson, who stated that a flaw of the Kelly Criterion was risk tolerance. He stated that, while maximizing log growth results in large growth rates, it is not worth the losses from a wrong bet. This preference for lower risk is a critique of maximizing log growth and the Kelly Criterion, as the negative utility of losses would be too much to bear for those individuals.
The conventional alternative is expected utility theory which says bets should be sized to maximize the expected utility of the outcome (to an individual with logarithmic utility, the Kelly bet maximizes expected utility, so there is no conflict; moreover, Kelly's original paper clearly states the need for a utility function in the case of gambling games which are played finitely many times). Even Kelly supporters usually argue for fractional Kelly (betting a fixed fraction of the amount recommended by Kelly, which is more rigorously calculated with the utility function when ) for a variety of practical reasons, such as wishing to reduce volatility, or protecting against non-deterministic errors in their advantage (edge) calculations. 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
References
External links