Trade Sizing II of IV

Shall we play a game? Maximizing for log wealth

Your rich uncle has 10 times as much money as you. He proposes a game. He’ll flip a coin; if the coin comes up heads, you receive 1/3 of what was wagered. If the coin lands tails, he takes 3/10 of what was bet. He chooses your bet size per play. The coin is fair. The game ends when either player drops below a 1$. Should you play? How much do you stand to win?

The game, per iteration, is clearly positive expected value. You’ve got a 50% chance to win 1/3 of a bet and a 50% chance to lose only 3/10 of that bet. The expected return of each bet is

Each bet is for edge! The game can be expressed in the same Kelly framework from the first part of this series. We can express the long term capital value of the game as

Where p/q are odds of win/loss and w/l are payoff for win/loss. f is once again fraction invested. N is number of plays. W0 is initial wealth.

Our uncle, since he chooses our bet size, will opt to make sure the fraction we invest is maximally suboptimal. If we assume that our uncle will force us to bet our whole fraction each play, such that f = 1

Plugging in our numbers.

We notice that, when we simplify,

Even though our game has theoretically positive expected return per play, our expected wealth decays playing it by over 3% per step!

Monte Carlo simulation of our game verifies that the game does, in fact, not go well:

Note that the 95th percentile is well below the mean performance. We start at $1000 in these simulations.

In a game where our expected value is positive

Our optimal Kelly fraction is

Plugging it all in, we arrive at a Kelly fraction of 1/6. As previously discussed, playing at any fraction above twice our Kelly fraction will drive growth negative. If our uncle was generous, and allowed us to wager at the optimal Kelly instead, observe:

Playing instead at Kelly (1/6th) will instead generate a 4x return (on average).

Whereas even at f=1/2, the game degenerates quickly into negative long term expected value:

Mean loss ~ 50%, median loss 99.5%!

We must be very cautious of our intuition when it comes to the applicable statistics for these sorts of problems. While the game, at first glance, seems like a no-brainer given the positive edge, the total wealth of our portfolio follows a geometric and not an arithmetic process. That is to say, our wealth is multiplicative of its current level.

Any process that either increases by 33% per step or decreases by 30% with equal probability, given time, decays to zero. In trading, if you lose 20% on a trade, you have to make 25% to get back to your original level (high water mark effect).

A bet, trade, asset, opportunity with positive expected value, which in our simplified framework, is simply

Any such opportunity has some fraction f (Kelly) that, if invested, yields positive long-term growth. Investing at larger fractions, even when the opportunity is positive edge, does not guarantee growth.

Log utility

Discussions of Kelly often, quite reasonably, ask:

Is maximizing long term exponential growth rate optimal? Why? Doesn’t this imply an arbitrary fondness for log utility of wealth?

While we avoided the explicit implication of taking the log of our wealth in our initial derivations, it is implicit in our assumption of maximizing growth rate.

Maximizing the log of the final wealth is implicit in maximizing growth rate. We’re still in the log-wealth world.

Direct maximization of terminal wealth is problematic reminiscent of the St Petersburg Paradox. Rare outliers comprise the bulk of net expected value.

Consider the following pseudocode for a similar game but against an opponent of infinite wealth (a reasonable proxy for the market)

When we compare the median and mean of our terminal wealth at varying investment fractions, we see that as the fraction increases so does the spread between mean and median performance.

When we simulate and look at the individual paths, the root cause is more clear. A few outliers drive the mean significantly.

Comparison of wealth: mean and the mean less the greatest value.

Unless you happen to be living in one of the luckier possible worlds, optimizing directly for wealth ends poorly.

Another common way to think about wealth maximization is minimizing the time to reach some goal (which we’ll call G). Kelly, as it maximizes growth rate, intuitively minimizes the time to reach said goal.

Benefits of maximizing for log wealth (instead of wealth):

As noted, maximizing for log wealth equivalently maximizes for long term growth.¹ The expected time to reach a goal G is minimized (without limit on how high G is) by maximizing for log wealth.² The log wealth investor never faces ruin (as he only bets a fraction) The amount wagered is monotone increasing with wealth. The Kelly investor maximizes median wealth by maximizing log wealth.³

Conclusion

Thinking about games of this nature provides valuable insight into risk management. Our intuitions are not necessarily robust and we’ve got to be very cautious describing outcomes via only their mean (“average”) result. Trade sizing is inseparable from modeling the distribution of trade outcomes. While log wealth utility is not a necessary truth, it has many nice properties for valuing possible futures.

References