John Kelly answered the question of how a gambler with an edge should act in order to maximize his bankroll growth. The resulting trade sizing scheme (the Kelly criterion) shows the optimal fraction of the bankroll to be allocated to each opportunity.
The Kelly criterion has a number of desirable properties:
- As we only ever invest a fraction of our wealth, we can never go bankrupt.
- The strategy is guaranteed to asymptotically outperform any essentially different strategy.
- The time for the bankroll to reach any fixed amount is asymptotically smallest with this strategy.
However, the scheme also possesses some undesirable properties. The biggest concern is overestimating the sizing fraction which can lead to disaster. There is a critical level of the invested fraction (approximately equal to twice the optimal proportion) where the growth rate of the portfolio becomes negative. In this case, the bankroll asymptotically tends toward zero.
In some situations, the Kelly ratio can be calculated exactly. However, in most cases this is not possible. When applying the Kelly criterion to sports gambling or trading financial instruments, the optimal ratio needs to be estimated by analyzing historical data.
As we will only ever have a finite amount of data, this is a case where we are attempting to estimate a population parameter from a sample. Inevitably, this means our estimate will have some degree of uncertainty.
Traders know this. They often arbitrarily reduce the size recommended by the Kelly criterion. This is known as fractional Kelly betting. This idea reduces risk, but doesn't eliminate it, because we are still just scaling the estimated fraction. There is even the possibility that the true bet size should be negative, and we should not bet at all. Also, arbitrarily reducing the fraction like this does not take into account the actual uncertainty of our estimate of the betting fraction.
Here we will derive the distribution of the investment fraction. This will allow us to calculate confidence intervals around our point estimate, and in particular let us estimate a scaling factor that gives us only a given chance of overbetting. This connects the fractional Kelly heuristic to statistical sampling.
This will also directly address the common argument against applying the Kelly criterion: that because we can never precisely know the true Kelly ratio we should not trust the idea at all (Although here I also feel compelled to mention that the numbers needed to estimate the Kelly fraction are the same as those needed to estimate expected value, so if you don't know enough to calculate the Kelly ratio you probably shouldn't be trading at all).
First we consider the case where the outcome of the bets are independent random variables X n which take the value W for a win (with probability p) and L for a loss (with probability q=1-p). We assume that these parameters are such that the bet has positive expected value to the gambler.
The way to make the most money is to wager the entire bankroll on each bet in the sequence. However if we follow this strategy, the first losing bet will bankrupt us. Ideally, we want a strategy that makes as much money as possible while also avoiding the risk of bankruptcy.
The way to do this is to bet only a fraction of the bankroll on each opportunity. For any fraction less than one it will be impossible to go bankrupt.
Further, we can choose the fraction to maximize the exponential rate of growth of capital. This bet sizing scheme is the Kelly criterion.
The fraction, f, which maximizes the growth rate is given by
The growth rate's dependence on f is shown in Figure 01. This
shows that overbetting, investing more than f max ,
decreases the growth rate and if we bet greater than
approximately 2 times
f max the growth rate becomes negative.
The relationship between Kelly sizing and bankruptcy is very different in theory and practice. In any scheme where we bet only a fraction of our current bankroll, it will be impossible to ever actually go bankrupt.
However, by overbetting (betting a fraction greater than that given by the Kelly ratio) we will reduce our growth rate and possibly make it negative. This will asymptotically reduce our bankroll below any given level, which in practical situations is as good as being bankrupt.
Figure 01:The growth rate of the bankroll for the case p=0.55, W=L=1
While we could explore the impact of uncertainty in the case of binary betting, the more relevant situation in finance is where the outcome of a trade is known to have a certain continuous distribution. Again we bet a fraction, f, of our wealth at the start of each period.
If we maximize over the bankroll fraction, f, we find that the optimal value is the one that satisfies
Where µ is the drift, σ the volatility, and Λ 3 is the third raw moment of the trade result distribution.
Therefore, if f is small, we can truncate the series after the first term to get
And if µ is small we can further approximate by
But it is important to remember that f max is an estimator and has a probability distribution.
First, consider the case of a normal distribution of trade results. Here, the estimation errors of mean, μ and variance, σ 2 can be approximated by
These estimation errors in the mean and variance will lead to estimation errors in f.
Now we apply the delta method for calculating the variance of a function. This states that the variance of a function f(θ), where θ=(µ,σ 2 ), is
So evaluating equation 7 gives the asymptotic variance of our estimate of the Kelly ratio as
In the case of a general distribution of trade results, we need to make use of the result (Zhang, 2007) that
Where λ 3 is the third central moment of the population distribution. Now we can get
We now use an example of real trading results to show the importance of including estimation error in trade sizing. The trade results are from a short volatility option trading strategy. It is somewhat typical of many such strategies in that it has positive expected value but large negative skewness. The summary statistics for these trade results are given in Table 01 and the distribution of results in shown in Figure 02.
|Table 01: Summary statistics for the option trade|
|Standard Deviation||1.137 USD|
We can rearrange equation 12 to give an explicit expression for the estimated standard deviation of the Kelly ratio.
where the denominator of n-1 is due to the fact that we have a sample rather than a complete population.
Because of the Central Limit Theorem we know that the distribution of f is normal, so we can calculate the probability that f is actually below any critical value f *.
where Z is the cumulative distribution function of the standard normal distribution.
Figure 02: The distribution of the option trade results
Equation 4 gives the Kelly ratio as 0.045, but equation 13 tells us that the standard deviation of this point estimate is 0.036, so our point estimate is only 1.25 standard deviations above zero. Another way of stating this result is that there is an 8.9% chance that the true Kelly ratio of the population is less than zero.
Having an expression for the sampling distribution also allows us to estimate the chance that we are overbetting so much that our growth rate is negative. This case corresponds to the true value of f being less than half the estimated value. Equation 14 tells us this is 25%.
This leads us to a complementary way to use the information. We can use equation 14 to solve for a benchmark, given that we want a certain chance of overbetting. For example, we have just seen that, in the case of the option trade, using a benchmark of half the measured Kelly fraction (i.e. betting at 'half-Kelly') still implies a 25% chance that we will be overbetting. Table 02 shows the probabilities of overbetting for various fractional Kelly schemes.
So in order to introduce a margin of safety we would need to scale the measured Kelly ratio by a considerable amount. This is in line with the practice of professional gamblers. Much of this need for scaling is due to the presence of negative skewness. If the returns were normally distributed the scaling could be reduced. This is shown in Table 03.
Table 03: Fractional schemes corresponding to various probabilities of overbetting when setting skewness of the trading results to zero
We have derived the estimation error in the Kelly ratio that stems from errors in estimation of parameters of the underlying trade. This allows us to assign confidence intervals around our point estimate of the optimal trading ratio, and derive a scaling factor that gives a defined probability of overbetting.
A new interpretation of information rate.
Bell System Technical Journal, 35, 917-926.
A Note on the Delta Method.
American Statistician, 46, 27-29.
Sample Mean and Sample Variance: Their Covariance and Their
American Statistician, 61, 159-160.
Optimal gambling systems for favorable games.
Fourth Berkeley Symposium on Probability
and Statistics, Vol. 1, 65-78
Confidence Intervals for the Kelly Criterion.
Journal of Investment Strategies