Kelly Formula (Part 1)

Most of my smart winners have been small capital bets while the losers have seen larger bets of capital. One of the reasons for this could be that catching a high beta asset at its low point (about to breakout; Lower Bollinger  touch; RSI < 20%, whatever) has given a higher alpha to me since the price move (due to high beta) has been swift and definitive. The low beta stocks, giving the illusion of secure staidness, have killed often by bleeding a part of my portfolio to death. Since these are the blue chip, low beta shares, I have been tempted to invest a larger proportion of my capital as compared to the mid/small cap sprightly upstarts being lured by the illusion of safety. As I look into my trading log, it is the multiple smaller bets that have really been multi baggers for me. So, I spent a good part of the long weekend reading on optimal bet sizes!

It is a common belief that a high amount of beginning traders do better with paper trading than once actually live. The reason being the realization of losing ‘real’ money. Knowing how to deal with profits doesn’t make one a successful trader insomuch as being able to handle losses. Every trade can have five potential outcomes – big profits, small profits, par, small losses and big losses. Taking the big losses out of the picture will logically give any trader less chances of frowning. Big losses can come due to decision paralysis (and waiting in the hope that the price will recover) or large capital outlay. The topic of this post is to ponder on understanding what should be the right size of trades. While its natural to think that the size of trades depends on a) your personal strengths and weaknesses, b) amount of capital, b) trust in an investment thesis, c) degree of diversification etc but like all good things there is a formula for this too! Namely, the Kelly Formula. This was designed by John Kelly who worked for AT&T and devised this for use in long distance communication – signal loss; signal to noise ratio and all those nice things. Since there are two basic components to the formula: win probability and win/loss ratio, this found application in the world of trading as well. The Kelly Formula states that for any given stock, you should invest the probability of winning times the payoff minus the probability of losing divided by the payoff. It is represented by the equation:

 where, P = payoff (i.e. odds offered by the betting syndicate); W = probability of winning and L = probability of losing

So while the debate on chance vs. skill in investing continues, here are some of my investigations and thoughts on whatever I have read and managed to understand so far. At the core of all this line of enquiry is a desire to figure out a personal method to ensure that I statistically end up putting more money on my winning ideas as compared to my duds. If I get to do that I am pretty sure, I would  be all :)s.

So back to the ruminations on the Kelly Formula: If you are worth 1 crore (say) then it’s clearly stupid to be risk averse for small amounts like INR 5,000 (say). You should regard “gaining 5,000” and “losing 5,000” as equal-but-opposite faces of the same coin. But it’s very rational to be risk averse for 50 lakhs. Whatever I have seen and heard from people with modest wealth around me suggests that people in fact behave in a predictably irrational manner when faced with these little choices of chance that their investment process throws at them. Penny wise and pound foolish…

But what if we step out of the frame a bit and focus on the really long term? Personally, if my health (mental as well as physical) stays with me, I very well have a further three decades of investing ahead of me. However, let’s take a hypothetical case – say you inherit a sum of money at a young age and to invest it for a really long time. [if wishes were true…] You always have a choice between a safe investment (treasuries) and a myriad of risky bets whose probabilities of outcomes is also known to you. [this is possible in a casino, but never on Dalal Street, but please lets go ahead with the logic]. Also, lets visualize that you are breaking up your investment time horizon into finite smaller periods (could be years, months or even weeks or days) at the end of which you take stock (i.e. P&L) and start again with whatever you’ve got at that time (i.e. rebalance your portfolio at periodic rests). The Kelly Formula gives you an explicit rule regarding how to divide your investments to maximize long-term growth rate.

There is this book called Fortune’s Formula by William Poundstone which I need to read to understand this better. Part two of this post will come after I have read that book and some more.