Kelly Formula Meet Portfolio Management
The Kelly formula or criterion is best known as a bet optimisation tool. Popularised by Ed Thorpe, the formula which is named after its creator, John Kelly, is used by gamblers to determine the optimal bet based on given odds. A year ago I mentioned that I use Kelly criterion in my portfolio management and I promised to expand on that. While I have thought in terms of probabilities for a long time it was only a couple years ago that Mohnish Pabrai, in in his book The Dhando Investor, introduced me to the Kelly Criterion. Let’s get stuck into how I use a betting tool for money and portfolio management and then I cover some Kelly background. If you’re a Kelly novice you may want to read the second part first.
Using the Kelly Formula for Portfolio Management
I have two main uses for the Kelly formula, neither of which are the intended purpose. My third and less important use is the intended purpose of Kelly, an aid in determining my individual investment size. I use Kelly as a comparative tool and as and aid to focusing on probabilities, risk and returns. Before we dive into each of those I should explain how I use a method that is designed to be used with binary outcomes with known odds for investing, where there are multiple possible outcomes with unknown odds. I mentioned yesterday how I was struggling to articulate my use of the Kelly Criterion. The reason for this lies in my own confusion. I could not reconcile how my simplistic spreadsheet calculations mapped to the Kelly Formula. I have spent countless minutes trying to reconcile my spreadsheet to this site which computes the Kelly Criterion for multiple outcomes. If only I’d annotated my spreadsheet with my reasoning I’d have saved myself those minutes. I took a different tact this morning and looked at my spreadsheet and thought “what the hell was I thinking and what are those figures telling me.” I then recalled I had dismissed the use of Kelly and settled on a much simpler approach of expected average return. I’ll call it Kelly for dummies. Here are the steps in my process and (hopefully) an embedded spreadsheet to illustrate.
- For each current and possible investment determine three target prices. I unimaginatively call the targets, low, medium and high. You may wish to call them forkit, it’lldo and yeforkingha, I’ll leave that detail up to you. The idea is you conservatively determine the worst, best and most probable outcomes (low, high, medium). In investing the worst outcome is always zero, i.e. a 100% loss. You can decide what to use for worst, but I suggest continuously using 100% would not be very useful in the calculation or you analysis of probable downside. Think of worst as worst probable outcome.
- Then give each outcome a probability. My default is 30%, 40%, 30% for worst, probable and best respectively. I alter those defaults when I have a strong conviction.
- Then based on the current asset price your spreadsheet calculates the average expected return.
If missing try this link.
I use the average expected return for an investment to compare all my actual and possible investments. The relatively higher the expected return the more likely I am to invest and the larger my investment will be. Steps one and two are by far the most important. They are the focusing steps. Determining targets and especially the downside worst case focuses my efforts on a key aspect in investing, managing risk and reward. Ascribing percentages to those targets is a double check. In general I find a lot of investors spend too much time focusing on intrinsic value to the detriment of risk and reward analysis. Investing is an exercise in probability not in mathematics. The expected returns is also part of my selling criteria. It quickly highlights any stocks I should be considering selling. That’s a topic I’ll cover another day. Another benefit of this approach is avoiding delusion. You don’t have to be delusional for you mind to fool you into distorting history. We all like to think we’re better than we are and our mind is happy to oblige with lashings of hindsight bias. Recording your targets and other observations up front and then revisiting them is an excellent way to keep hindsight bias at bay.
Putting it all together
Here is an example from my investment in Biota. I starting investing in Biota in October 2008, with the price around $0.40. I calculated the worst probable outcome as $0.30 with a probability of 40%, the probable outcome of $0.75 and the high of $2.00 both with 30% probabilities. [Note: I did publish those figures, though I actually presented four outcomes.] The average expected return from that was a high 136% or an average target price of $0.95. For me that is a very high average return and combined with other factors led me to make a large investment in what many people would have considered a speculative stock. So what were some of those other factors?
- Return to risk. With the exception of my dividend producing core stocks, I don’t get out of bed for a return to risk of less than three. That is I’m looking for a return of three times my risk, 3:1. For Biota I commented “Like beauty valuations are also subjective, a combination of art and science. My range of values for Biota range from $53M – $215M, with per share $0.70 – $1.20 looking like fair value and $0.30 as downside value.” So return was $0.30 to $0.80 with risk $0.10 for a return to risk ratio between 3:1 and 8:1. Not only do I get out of bed for that I leap out and sprint naked to my computer to place a buy order. I apologise for that mental image 😉
- Buffett’s Rule Number Two: Don’t forget rule number one, never loss money. I admit to playing fast and loose with those rules. As my teachers often commented “could try harder”. Despite not being great at implementing rule number two, I do get excited when I find an opportunity that has very little downside and plenty of upside. The investing world is full of opportunities with considerable upside, but they are predominately accompanied by plenty of risk of loss of capital. Opportunities like Biota are beautifully illustrate what Mohnish Pabrai meant when he coined the phrase “Heads you win a lot, Tails you lose a little“. Low risk opportunities with possible high returns. In general I consider myself a value growth investor. I buy growth when it is on sale.
- Confirmation bias is considered a cognitive bias and as such is tainted. Heck who wants a cognitive bias! While I don’t want a bias I am happy to utilise them. The only cognitive bias I am terrified of is bias blind spot — the tendency not to compensate for one’s own cognitive biases. While I am a smart guy, I recognise the investing world if full of smart people and that being the case I am unlikely to be the first person to “discover” a wonderful opportunity. I like to look at a share register to see if any investors I respect are on the register. When I looked at Biota I discovered not only was management buying shares back at almost twice the current price, but Hunter Hall International (HHL.AX, who I mentioned yesterday) had also been buying shares at considerably higher prices and then held 12% of shares. As nothing had fundamentally changed since their recent purchases I took a big hit on the confirmation bias bong.
- Circle of confidence. Tick, I know and like biotech investing.
- Possible catalysts. Tick, tick, tick. While the swine flu outbreak was not predictable a large one quarter uptick in Relenza sales certainly was probable and Biota had a pipeline which could deliver good news.
- There’s more, but I am now so far off topic that I must stop and get back to Kelly.
Here is an example spreadsheet which you can copy to Excel. I did this in Zoho sheet so you can play around with a few examples and easily copy it to Excel. Unfortunately Zoho sheet has a few limitations so if you’re entering new shares you need to copy down the formulas and change the current formula to your ticker (unlike Google Docs, zoho does not let you reference other cells for stocks updates). Also note, I plugged in the first numbers that sprung to mind, these are not intended to be actual target for any of the stocks.
[UPDATE: I have now updated the above spreadsheet. Check out how I use Kellyesk calculations in my sizing of individual stock positions for an explanation of the new columns.]
Kelly Background and some Notes I’ve copied over the years.
I think Munger’s recent book recommendation, Fortune’s Formula, and many of the papers referenced in the bibliography are pertinent to this discussion. The book is about Kelly‘s criterion, which is a formula for sizing bets to maximize long-term compound return from a series of bets where the winnings are reinvested. Kelly‘s criterion has two components: edge and odds. Edge is the amount of profit that YOU BELIEVE that you will make if you could repeat this bet many times with the same probability. Odds are the market place or tote board odds. Your odds must be different from the market place odds or you don’t have an edge.The optimal return is obtained by betting a fraction of your portfolio equal to the edge/odds. Overbetting and underbetting result in sub-optimal results. However, overbetting is more serious because it leads to a large variation in returns and to eventual blowups (LTC and Eifuyu were examples of overbetting). Underbetting reduces returns and variance. The book presented Ed Thorpe’s (one of the best Kelly criterion investors) hedge fund results which were spectacular for the degree of over-performance and the small variance over a 20 year period. Ed Thorpe always underbet – specifically made bets of half the Kelly criterion because he was worried about being too confident about his assumptions in investing and unconsciously overbetting. Long story short, the book and especially the papers show that if you really good at finding investments with large edges, you could get decent returns without a lot risk holding only four or five positions. If you can only find investments with smaller edges, you will get less return and need more positions to reduce risk. From reading the book and the papers, I found that I was intuitively doing the right thing. However, I am planning to now track the edge and odds on investments to see if I can improve. via TMF : Liquid Lounge
Basically the idea is that if a single play is not acceptable, then no sequence should be acceptable, when the goal is to maximize the expected utility. The theory is basically that people, when thinking about the law of large numbers, tend to forget that even if you play 50 million or more you still don’t have full certainty that you will win, but at the same time your potential losses grow accordingly (you could sequentially play 100 million times and still not win, losing a lot of money if the idea was to be risk averse). Mathematically the expected value of return is the same no matter how many times you would play, it’s a simple multiplication; but psychological humans behave very different. I don’t remember the source off the top of my head, but there have been some experiments asking what people were willing to pay for the probability of winning something (i.e. how much they would pay to increase a single percentage point in their chance of winning). The increase from 0% to 1% (lottery) or from 99% to 100% (insurance, the assurance of winning) will always command much higher prices than, for example, paying from 34% to 35%; even though the expected return is the same in all cases. also via TMF: Liquid Lounge
But wait, there’s more, how about a set of steak knives.. Each of the following articles and sites are excellent resources well worth taking a look at.
- For those disappointed with my spreadsheet here is the Kelly Bet Sizing for Multiple Outcomes program.
- The best site on the web I know of for papers and discussion of the Kelly Formula. The site includes a link to the original 1956 article, A New Interpretation of Information Rate, which appeared in the Bell Systems Technical Journal.
- Ed Thorp The Kelly Criterion in Blackjack, Sports Betting, and the Stock market.
- Michael Mauboussin Size Matters – The Kelly Criterion and the Importance of Money Management
- I already linked Risk and Uncertainty: A Fallacy of Large Numbers by Paul Samuelson, but in case you missed it, there it is again.
- A Business Week review of Fortune’s Formula with good background into Kelly.
- A suggested method for using the Kelly Formula in money management.
[KELLY PART TWO: Check out how I use Kellyesk calculations in my sizing of individual stock positions.]