I very much recommend listening to the podcast episode of Masters in Business with Ed Thorp, through iTunes or through this Bloomberg link. It might well be one of the greatest podcasts of 2017. Thorp talks about how much he liked working with Claude Shannon, the father of information theory (my favorite class in uni). Shannon helped Thorp to optimize his position sizing in blackjack, given the odds and the estimated edge calculated by Thorp’s own card counting system.
- When Thorp didn’t ace a chemistry competition because he did not bring along the right slide rule to the exam, he learned that there’s many different ways to fail in real life. Takeaway being that in investing one should think creatively about different downside scenarios.
- Thorp was asked whether managing money did not get scary when the stakes went higher. His response was that he gradually got used to new amounts of money and that the fundamental problem of investing does not change along with scale (although the type of successful investing changes because of scaling). I relate a lot to this argument that investors should always start small to get comfortable with managing money. This could be one of the reasons why many of my friends that were not investing as students feel uncomfortable now with the higher amount of money that they accumulated throughout their careers as they did not “grow” into investing. For my personal portfolio, I started out with a small amount of savings 10 years ago and I am managing an amount almost two orders of magnitudes higher today. Compounding has the property to feel slow in the short-term, which is why I feel completely at ease managing this new order of magnitude. I also make abstraction of absolute money amounts as I agree with Buffett (and Thorp!) that I do not really need the money.
- Thorp, being both a great practitioner and scientist, likes to use the rule of 72.
“Don’t confuse the cost of living with the standard of living.” – Buffett & Thorp
For those that are looking for position sizing literature, I recommend Thorp’s work on Kelly betting (Wikipedia). For example, this paper by Thorp (page 27-28) shows that in a world with uncertainty of Kelly parameters (i.e. not gambling but the real world), it is better to use fractional Kelly betting as one gets penalized twice in case one’s estimate of risk/reward is too good (once for extra risk and once for less growth). As I will explain below, the way I think about which fraction to choose is besides the point.
Limits of the Kelly criterion for investors
I think the use of full Kelly betting is very dangerous in the stock market as the parameters are (very) uncertain. Too many quants go bankrupt by applying full Kelly when it turns out they had overconfidence in their ability to estimate the risk and reward parameters. This is what Nassim Taleb (a great fan of Thorp, he wrote the foreword to his biography) calls victims of the ludic fallacy.
How I use the Kelly criterion in practice
If (full) Kelly betting cannot be done in the real world with uncertainty of parameters, and we don’t know how to choose our fraction, you might ask why bother. I let some years pass to think about this (as I was not able to find someone that addresses this), and I think the point of fractional Kelly is that, although we never know which fraction to pick, we should try to do our relative position sizing between individual portfolio positions proportional to the expected value versus risk (or signal-to-noise for the information theory folks) of each pick.
For example, the way I use the criterion goes like this: if I estimate that position A has three times better risk/reward ratio than position B, it should be sized three times the size of B. This is what I call internal position sizing consistency, and I try applying this in my portfolio.
Lastly, even in a portfolio that is for example 20% cash and 80% invested with internal position sizing consistency, it is of course dangerous to trust one’s relative risk-reward estimates if some position sizes go beyond 2-5x of others and reach high absolute sizes in the portfolio, because of idiosyncratic risk. Thus, position sizes should also be capped on an absolute basis (for the real Buffetts among us this is probably in the ballpark of 30-50% in exceptional circumstances, for the rest it might well be in the 15-20% range).
If we estimate that position A has three times better risk/reward ratio than position B, it should be sized three times the size of B.
I follow Greenwood Investors letters with great interest. However, I find the following in their latest letter a bit distasteful:
At quarter-end, our ratio of reward-to-risk stood at 38.3x, which is marginally better than the 38.2x at the beginning of this year. – Q2 2017 letter Greenwood Investors
- first of all, I find it absurd to report a difference in reward-to-risk of 38.3x Q2 vs 38.2x Q1, this looks like pseudo-accuracy to me
- secondly, if reward-to-risk for the portfolio would really be so huge as 38x, it would take huge uncertainty on these parameters to justify not having a huge gross exposure like 500 – 2000% for a rational investor like Ed Thorp (and which I trust Greenwood doesn’t have), which kind of proves my first point.
Remark 2: W. Poundstone popularized the Kelly criterion in his book Fortune’s formula, easy read.
Til next time,