I wrote something in the past on the Kelly criterion (KC) which is an algorithm for optimizing the geometric return (or, rather, optimizing the bet sizes) in a sequence of independent gambling wagers
where the wager-er has an real edge and known odds. Note that I am not an expert on this and that the countervailing theory that economists seem to like is utility based. I just wanted to try it out on a short options strategy, where my risk return is upside down, to see what happens. [1]
KC has some affinities with investing propositions and particularly systematic rules based systems and especially for sizing the risk of trading positions. It doesn't translate perfectly but it's usable. The basic distilled math is: size of bet = edge / odds or more specifically f = (bp-q)/b where f is the fraction to bet, b is net odds received, p is the probability of winning and q is the probability of losing or 1-p.
Since there are a bunch of reasons that investing is not the same as gambling (usually) the full Kelly criterion can be a little dicey while still mathematically interesting. Here is an off-the-top-of-the-head list of what one might worry about:
- in investing the bets are not necessarily independent and sequential
- in trading and investing the bets can be parallel and overlapping and hopefully diversified
- payoff estimation can be unstable
- edge estimation can be unstable
- payoff estimates are based on a distribution
- short run volatility can be crushingly and psychologically difficult
- the time to recover from short term vol can be pretty long, maybe longer than the planned horizon
- total wealth cannot be realistically be the betting bankroll
- there might be liquidity constraints
- etc...
For these reasons and others, lot's of folks appear to retreat to a half-kelly wager where 3/4 of the return can be achieved for a lot less volatility. That means I'll use KC/2.
First, let's swag a KC for a "traditional" trading strategy I had in the past. The odds were supposed to be 3:1 but turned out to be more like 2.8:1 (sometimes less) for a lot of reasons. The probability of winning was about 30% though if brutally honest it was drifting downward. The KC in this case would be (2.8*.3-.7)/2.8 = .05 and KC/2 = .025. This is consistent with my pre-kelly uninformed guideline of not more than 2.5% risk on any single trade and not more than 5% aggregate risk. It is also consistent with un-findable articles I read early in my trading career that use a 2.5% bank-roll risk guideline for individual trades.
Now let's apply it to my short options strategy to see if I am setting my risk to the right level using the 2.5% rule of thumb guideline I have blindly used until now. Initially I thought that it (KC result) would be necessarily different because of the upside down risk-return but now that I think about it the KC has only edge and odds so "necessarily" is a stupid concept. It's all up to the edge and the odds; could be the same, could be different. In this case the "planned" risk return is 1:3 or .3333 which I'll call the odds received. In practice my odds are currently higher than this but that creates results I can't mentally process.[2] The probability of winning has been hovering around 80% since the middle of 2012 so we'll go with that. That makes KC, if I have it right (not 100% confident on that), equal to (.3333*.80 - .20)/.3333 = ~.2 with KC/2 = ~.10.
Taking this at face value, it looks like I am leaving money on the table by undersizing my risk by quite a bit, mostly due to the fairly high probabilities involved. On the other hand, for the reasons above and because the results are fairly sensitive to variability in the odds, I am not going to jump into the deep end and quadruple my bet size...yet. But what I can do is tweak my notional capital. Since I am not explicitly committing capital to a separate account but rather using my margin capacity to sell premium, I get to decide arbitrarily what my capital commitment is. If it were 100k, a 2.5% risk is 2500 and 10% is 10k. So, rather than raise risk from something like 2.5k to 10, I can keep the absolute risk size the same and change the notional capital to a lower about to get to the 10% risk size. This feels to me like a type of leverage (which it would be, all else being equal, but all else is not equal) but I guess it isn't since I am merely right-sizing capital and risk. At least this will free up capital (as long as I don't lose it) for other purposes. Then, if the experienced odds trend against me I'll still have the capital (hopefully) to back myself and then I'll change the notional amount to something else. In a year I'll figure it out all over again. Either way, the take away for me is that I can probably deploy capital more efficiently than I have been. The linchpin though is not the math, it's the psychology, but I'm thinking that is a different post.
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[1] by upside down I mean the typical trading strategy risks $1 to make 3 where in short options $3 is risked to make 1, but with higher probability.
[2] The working odds so far result in a KC of .53 but there is no universe, yet, where I am going to bet 1/2 or even 1/4 of my committed capital even though mathematically it might make sense. In this case I lean on the idea that odds and edge are unstable and that my scale of trade is increasing which feeds back to the instability and therefore conservatism wins the day.
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