### How Kelly bet size and Number of bets affect Max drawdown

In this post we vary the fractions of Kelly bet size and the number of plays to see how they affect the maximum drawdown. Betting on Kelly size optimizes the long term returns. People can alternatively use various fractions of Kelly size such as full, half, quarter etc based on their risk preference. In addition, we will study the effect of number of plays on maximum drawdown. Maximum drawdown is a measure of risk,  historically the largest loss one would have sustained if one had the misfortune of investing at the peak and redeeming at the trough. We further divide the number of plays into three categories short run ( 0 to 100 bets) , mid run ( 100 to 1000 bets ) and long run ( > 1000 bets).

By maximizing one’s bet size using the Kelly criterion, one can maximize the return over the long-run without losing the entire bankroll, but the path can be quite bumpy. To evaluate the bumpiness of the ride we consider the number and size of drawdowns which are the  losses during a peak-to-valley decline.

The graph below shows how one’s bankroll might increase if one played a gamble 500 times. Here the gamble is (P = 0.55, a = 1, b = -1), so the chance of winning is 0.55 and one can win or lose 1 dollar in each play. The returns are based on an optimal bet size of 0.1 for each play. Green stands for the drawdowns, red stands for the maximum drawdown and black stands for plays when one is not in a drawdown.  In this hypothetical history one would have experienced many drawdowns, but each is followed by a recovery. This situation repeats itself numerous times with drawdowns of various sizes. Note, however, that one spends the vast majority of the plays within drawdown periods with long drawdown periods separated by brief excursions to new highs.

In this simulation, the drawdowns are painful, but the recovery continues to new highs. To see drawdowns in a longer-term context, we extend the series to 10,000 plays in the plot below. In that plot, even large drawdowns appear small relative to the size of the cumulative return. Note, however, as a whole, longer play lengths will yield larger maximum drawdowns in both dollar terms and percentage terms.

To develop a feel for the expected maximum drawdowns as the number of plays increases, we simulated 100,000 scenarios of various lengths. The boxplot below describes how the expected maximum drawdown increases as the number of plays increases. Here, box represents interquartile range, the central line represents median, the whisker represents 1.5 times the interquantile range and the small dots represent outliers.

By construction, the optimal Kelly bet size insures that one will never reach a 100 percent drawdown, however, the expected size of the maximum drawdown converges to 1.0 if one plays long enough. For shorter series, the expected size of the maximum drawdown can be much less than 1.0, but the variability becomes larger. Besides being bounded above by 1.0, the maximum drawdown is bounded below by zero, which only occurs when winning bets make up the entire sequence. Other than the all-win case, the next smallest maximum drawdown is the amount bet on each play. In this case, each bet was 0.10 of the bankroll, so the smallest non-zero drawdown is 10%. This value is marked by a dashed line.

We also observe that the interquartile range of the maximum drawdown first increases slightly as the number of plays gets larger and then converges to 0. For instance, if one plays 30 trials one has little chance of experiencing a very large drawdown. By contrast, if one plays more, one will have a greater chance of a very large drawdown and as the number of plays gets very large, one will almost be guaranteed to experience at least one near-catastrophic drawdown.

Next we consider the case for decreasing one’s risk by lowering the size of each bet. In particular, we show how the maximum drawdown decreases if we were to halve the size of our bets in what we will call a half Kelly bet size.

The boxplot below shows how maximum drawdowns evolve as the number of plays increases when one uses a half Kelly bet size as well as a Kelly. The pink boxes stand for betting with Kelly size and the red boxes stand for betting with half Kelly size. The red dashed line is 0.05, half of the Kelly bet size.

For any number of plays, the maximum drawdown for the half Kelly size is smaller on average than for the Kelly size. However, the vertical distance between the medians in each red box and each related pink box follows the same pattern as the interquartile range follows: increases first and then decreases. So the advantage of half Kelly size is more visible if one plays a mid-run (in the plot below mid-run falls between 100 and 1000 of plays) rather than a long-run or a short-run.

For example, if one bets 30 times, the median of full Kelly is around 0.38 and median of half Kelly is around 0.2. The difference between the two is 0.18. However, if one bets 1000 times, the median of full Kelly is around 0.9 and median of half Kelly is around 0.6. The difference between the two is 0.3. Lastly, if one bets 100,000 times, the median of full Kelly is around 1 and median of half Kelly is around 0.9. The difference between the two is 0.1. By this example we see that the difference between the medians initially increases and then decreases as one increases the number of bets. We have verified that other gambles share the similar pattern.

We have seen how halving the bet size decreases the expected maximum drawdown across various lengths of play. Alternatively we can fix the number of plays and determine how decreasing the size of the bet also decreases the expected maximum drawdown. In the plot below, we show how the median of  maximum drawdown for a set of 10 plays decreases as we lower the leverage from the full Kelly bet size to 0

As an aside, the break in whiskers between 0.48 and 0.50 arises due to the discrete nature of the distribution of maximum drawdowns. The quantile cutoffs can be jumpy as they move from a discrete value to the next, as shown in the next plot, which is the distribution of maximum drawdown if the size is 0.48 Kelly size.

Now we show how the distribution of maximum drawdowns changes for a long series of plays (100,000 bets) over a range of leverage from 0 to the full Kelly bet size. Instead of the linear relationship, between bet size and maximum drawdown  for the short run ( 10 bets), we see a convex relationship in the yellow boxplot below. Therefore, if the bet size is close to Kelly size, one cannot reduce the risk of maximum drawdown effectively unless one decreases the leverage dramatically. For instance, if one decreases the leverage by 60% (to 40% of Kelly size), one gets a 20% reduction (to 80%) of the expected maximum drawdown. On the other hand, when the bet size is close to 0,the expected maximum drawdown would decrease dramatically as the leverage decreases.

As an aside, regardless of the number of plays, if we extend the leverages from smaller than Kelly bet size to multiples of Kelly bet size, the overall picture shares the same convex relationship between maximum drawdown and Kelly bet size.

Conclusion

We found that maximum drawdown depends on both the Kelly size and the number of bets. For example,  the differences between the medians of the maximum drawdown of full Kelly and half Kelly were 0.18 , 0.3 and 0.1 respectively for 30 , 1000 and 100,000 bets. Therefore the half Kelly has a larger effect on reduction of maximum drawdown in the mid range (100 to 1000 bets) compared to other ranges. If one plans to play 100 bets, using the half Kelly size is a good strategy to reduce the maximum drawdown. However, if one plans to play more than 1000 times, the half Kelly may not have as much of an effect on reducing maximum drawdown.

In addition, for small number of plays (10 bets) the relationship between maximum drawdown and Kelly size is close to linear but if we increase the number of plays(>1000 bets) this relationship becomes convex, which means a greater decrease in leverage is needed to reduce the average maximum drawdown. If one plans to play a large number of bets, then one must reduce one’s bet size by a larger percentage to reduce the maximum drawdown, than if one plays a small number of bets.