Comment by xiphias2
4 years ago
A simple description of the Kelly criterion is that if you want to grow wealth over a long period time, at each decision point take the one that maximizes your average expected log wealth.
I'm trying to use it in real life, though sometimes the decisions are quite scary, as it's hard to estimate the probability of outcomes. Also my wealth is much more volatile than most people can stomach, but I look at it like a game.
It doesn't seem obvious that this is a good strategy for personal wealth management because besides maximizing expected wealth, there's another very important criterion: minimizing probability of going broke. I only get to play one game, after all. Obviously you can't go entirely broke if you always bet a fraction of your portfolio, but are there results of how these strategies compare in, say, the probability of dipping below 10%, or 1%, of the starting value?
I can't tell you about the 1% version, but when it dipped to 15%, it was a strange feeling that I made a bad decision with the thinking that I'm making a great decision (or more trying not to think about it and trust the decision that I made earlier). It's a mental game at that point that you have to wait through. At least with investing it's just about waiting through those periods, being a CEO of a company and making decisions in that state would have been much harder.
I love the quote “the money in investing isn’t in the buying and selling but in the waiting”.
I know for me I had the moment where things had gone down to roughly 15% and I questioned my decision making. Learning to wait through those periods is super important. Years ago I made the repeated mistakes of not waiting through those periods and missed out on log gains in favor of linear gains.
Agree that it’s easier as an investor and not a CEO to manage that experience day to day.
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Betting with 'full' Kelly-calculated stakes is highly volatile. If I'm remembering correctly, if you get your probabilities/edge exactly right, you will still have a 50/50 chance of losing half of your bank at some point in the future (i.e. after some number of future bets) It's very common to bet just some fraction of the Kelly stakes in order to smooth out the roller coaster ride.
Sure, I've gone through losing more than 80% of my wealth multiple times by being 100% in BTC, so I got used to that already. At the same time it stresses my friends out a lot. I'm expecting to lose more than 50% of my wealth, but at this point it doesn't really change my life style.
E log X strategies are known for Being very volatile.
However, there are two things that take the scariness out of estimating probabilities for me:
- You're often maximising something that looks like a quadratic function. This means you're aiming at a plateau more than a peak: if you make small errors in either direction it doesn't affect growth that much.
- You always have the safe option of underestimating. The E log X strategy forms an "efficient frontier" (to borrow terminology from MPT) of linear combinations from the risk-free rate to the full Kelly bet (and even past it into leveraged Kelly strategies.) You can always mix in more of the risk-free rate and get lower growth but at higher safety.
These two properties makes the Kelly criterion very forgiving to estimation. (In contrast to MPT style mean--variance estimations, and other less principled strategies.)
I find both mean variance and Kelly to be very poor in practice due to the dependence on the expected return term. Like, if I knew that, I wouldn't be wasting my time with all this math! (half joking)
Kelly does not depend on the expected return -- it depends on the joint distribution of outcomes. That is a big difference!
Accidentally thinking that "E log X" and "log E X" are the same thing is a common mistake, but Jensen's inequality tells us it can be a costly one.
Of course, your general point still stands: if we only knew the joint distribution of outcomes the battle would be over already.
Do you have examples of how that would be used in real life decisionmaking?
One simple example is buying 2X S&P index ETF instead of 1x. There was a great article about the Kelly optimal S&P allocation, and with all the fees included it's about 2x. Of course there's increased execution risk for the ETF itself, which needs to be estimated.
Another thing where I may look stupid from outside is that I started to take some loan against my BTC and use that to finance my lifestyle, as currently (under $100k BTC price) my estimate of the Kelly optimal BTC allocation is more than 1. This is of course a personal estimate, I don't suggest other people to do the same thing, and again there's a lot of execution risk, so I do this only with a part of my portfolio.
I have an old blog post about the subject: https://cryptm.org/posts/2019/10/04/vol.html
Optimal over my time period was 2.99x, but the expense ratio was not accounted for.
For people who earn a wage and don't just make money by investing, the Kelly Criterion can't be applied in its basic form, since it means your capital gain has both constant and linear components, instead of just being linear as the formula assumes, which complicates matters a lot.
Plus for low probability high reward bets you have the additional complication that you probably can't make them often enough to get a decent chance of hitting the jackpot.
For people who expect to have stable earnings with the current interest rates being below the real inflation the Kelly optimal strategy is to be in debt use it to finance investments (of course this works only if the future earnings are really stable).
As a business example startups are starting to apply for loans against their future subscription earnings to reinvest in their companies. Debt against your salary is the personal version of the same strategy.