Kelly Criterion Limitations

I've recently written about a way to base asset allocation decisions - the Kelly Criterion. This suggest a way to maximize your returns while still positioning your portfolio for relative safety.

Another link also shows a way to apply the same criterion in more complex situations, that also offers some utility in business situations, or asymmetrical investing situations. However, it can be a time-consuming effort to apply this to every investing situation. Probably a better way to do it is to calculate the Kelly Criterion for your own portfolio and then use that as a baseline guide for your own investing behaviour.

However, using it in this simpler way requires some thinking about it's limitations. For example, calculating my own Kelly, suggests that based on my historical purchases, and assuming equal weightings, I should purchase about 11% of my portfolio into each position. Approximating my actual historical purchase amounts, suggests my Kelly in that instance is about 14.5%, indicating I was successful in properly overweighting more promising positions.

However, these Kelly calculations presume - to some extent - that our investment opportunities are symmetrical (as shown in the Mauboussin example), whereas most investors begin to recognize that investment opportunities are asymmetrical. In other words, sometimes you'll spot opportunities that you clearly feel are much better than some others.

For instance, in analyzing most of the trades I made to calculate my own Kelly, I had bit of a mixed bag in terms of wins and loses. Excluding my five largest purchases (of an initial 37 or so), I had 18 wins out of 32 purchases (37-5), which is a 56% win/loss ratio.

However, adding those five large purchases I made into the mix significantly changed both the absolute and relative results on the entire portfolio, and so are worth considering on their own.

On four of the five, I made significantly outsized gains, something I believed was possible ex ante. This expectation was clearly achieved. I also believed, ex ante, that my inherent risk in those five investments was much lower than my portfolio average. Even the one investment that wasn't quite as good as I thought, still turned out to be a break-even proposition.

So, while the Kelly Criterion would have maximized my gains if all the opportunities were highly similar, it couldn't quite close the gap here. Using your common sense in this instance would have helped you take advantage of this situation, just as I did, and just as I intend to do so in the future.

A couple of quick other limitations I can think of with the Kelly, when you analyze your portfolio trades the way I did:

1. A rising tide lifts all boats; it's only when the tide goes out that you get to see who has been swimming naked - in other words - be careful to think about whether your Kelly covers a bear portion of the market as well as a bull portion - otherwise it may cause you to under or over state your Kelly.
2. Analyzing the Kelly considers your history over the time considered - if you are getting to be a better investor (as shown by your more recent investments), then you might want consider upping your Kelly to properly reflect that. Of course, the opposite applies too ...
3. Finally, you could consider segregating your portfolio analysis, so that - for instance - if you allow yourself 1/3 of your portfolio to be invested in small caps, 1/3 in mega caps, and 1/3 in ETFs, you could consider your Kelly on each of those segments. This might help you with portfolio risk and returns into the future.
   

The Kelly Criterion - not the be all and end all - but a useful tool ...

JW

The Confused Capitalist

Asset allocation: how much is enough/too much?

Successfully outperforming the index requires identifying sufficient favorable trends to give you a distinct advantage, and then pressing that advantage to the maximum within a reasonable context of overall portfolio safety.

Said another way, for consistent outperformance, it's not good enough just to identify favorable investment opportunities, but you must also identify how much of your assets to allocate to the take full advantage of the opportunity. In other words, how much to weight your portfolio with that particular holding.

Generally, amateur investors like myself who hold focused portfolios, have some instinctual understanding that positions must be overweighted to outperform the index. We may reference general comments and schools of thought, like knowing that superinvestor Warren Buffett once held over 50% of his net worth in a single stock (GEICO), and also once invested more than 25% of his public stock holdings in American Express to help us with our portfolio weightings.

But the question remains of how much, scientifically, to allocate to a particularly favorable opportunity in our own portfolios. The Kelly Criterion (link also provides an example of how to use it, based on your own historical trading success and patterns) provides one such answer. I'm going to borrow a different and simpler example (using the Kelly Criterion) to illustrate the point.

Suppose, on a coin toss, you were to receive $2 for every time the coin turned up heads, but had to pay $1 for every time it turned up tails. How much should you allocate to maximize winnings, while ensuring that a few coin tosses don't send you to ruin? The Kelly Criterion says you can effectively maximize winnings by using this formula:

• Edge/Odds = Allocation percentage (the answer we're seeking)
  

The edge is calculated by comparing how much of an advantage, over an infinite period of time, this particular coin toss strategy has. It's calculated by comparing the odds of winning, against the odds of losing. In this case, you can expect that 50% of the time, you'll win $2. This is part [a]: (50% x $2 = $1). However, in 50% of cases, you'll lose $1. This is part [b]: (50% x $1 = $0.50). You now subtract [b] from [a] to determine your edge, thusly: $1.00 - $0.50 = $0.50. This is your edge ($0.50), the top part of the equation.

The odds are calculated by knowing how often this the event will turn out favorably. Since we know the coin has only two sides, and one sides value is double the other, then we know that the odds are 2:1 (ie. $2/$1). So the odds figure is $2.

We then divide one figure (edge, $0.50), by the other (odds, $2), to arrive at the suggested allocation for the portfolio, or "the bet". In this instance, $0.50/$2.00 = 25%. This suggests we should allocate 25% of our portfolio in each particular round, to this particular "bet" (assuming all odds and edge factors remain the same).

This system has several noteworthy features:

1. It's theoretically impossible to go bankrupt, given that money is theoretically infinitely divisible (down to the 1 cent level anyway);
2. The system produces the maximum return in the shortest period of time, on average;
3. The returns are very noticeable and lumpy - for example, if your first three coin tosses were negative, and you started with a $10 bankroll, you'd be down to $4.22.
   

If you want to use this system, I suggest you study the materials in both links I've provided until you have a good understanding of the benefits and drawbacks of it.

Hat tip to Abnormal Returns for sending us out there ...


JW

The Confused Capitalist