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The math that dictates optimal portfolio allocations is complicated and an overly simplistic approach introduces a lot of unnecessary risk. The math of “gambling” and the math of investing share a lot of similarities. I believe the math presented below is equally applicable to both worlds. While EV (Expected Value) is a critical concept, it is meaningless without the concept of EG (Expected Growth). Chip Kelly is not the guy the “Kelly Criterion” is named after, but his presence creates interesting football betting opportunities. As a guy who many would consider to be a “professional gambler,” the concept of bet sizing has been something that I have spent a lot of time thinking about. I firmly believe trading stocks and derivatives for great portfolio managers is not all that different from playing poker for elite poker players. Individual investments for a portfolio manager and individual bets of a poker player (or elite sports handicapper) may be extremely risky, but the entire set of investments that make up a portfolio or long series of bets over time by a “professional gambler” are likely to yield a high return with a relatively low risk over the long run. This article is in response to an Instablog written by one of the most interesting contributors on this site, Chris DeMuth . He gives a relatively simple methodology of how to allocate capital. He presents the basic premise that he will invest around 1.25% of his portfolio in an investment he likes and will increase his position in the stock if he continues to love it as the price declines. He will continue to add to this position until a maximum of 10% of his portfolio is allocated to the individual investment. Adding to an investment that is becoming more undervalued relative to its fair market value makes a lot of sense. However, the exact portfolio allocations he suggests seem to be quite arbitrarily chosen instead of meticulously calculated. Based on my user name, it is probably clear that I have spent a lot of time in my life thinking about the fancy math of endeavors most would consider to be reckless gambling. I would like to introduce the idea of the Kelly Criterion, the most fundamental formula for elite sports gamblers. You can read about it here . What this magic formula does is tell you how much of your portfolio (bankroll) you should invest (bet) on a particular investment in order to maximize the growth of your portfolio given your estimate of the probability of winning and the odds received on the wager. The formula is written below: Every investor (bettor) is familiar with the concept of EV (Expected Value). Everyone knows that positive EV bets are wonderful. However, there is a corresponding concept that is much less well understood. It is the idea of Expected Growth, and frankly, it is equally important to understand as Expected Value when thinking about portfolio allocations. Let’s consider an investment where you are allowed to bet on a game of flipping quarters. The odds of picking a winning bet are 50% when flipping a quarter one time. Let’s also assume that in this generous game that for each flip of the quarter, you are getting a +200 payout. For those of you not familiar with common sports gambling notation, this means that you are given 2-1 on your wager. If you wager $1 on this bet and lose, you will lose $1. However, if you win, you will receive back $3 ($1 for your initial investment and a $2 profit). Better yet, let us assume that there are no caps on how much we are allowed to bet. This is a wonderful game that I would love to play forever everyday if it were readily available. Let’s now further assume you have a bankroll of $1,000,000. You are allowed to play this game only two times. In this game, there are four distinct possible outcomes (the sample space) that each have the same probability of occurring. The 4 possible outcomes are as follows: Win both the first and second bets. Win the first bet, lose the second bet. Lose the first bet, win the second bet. Lose both the first and second bets. Let’s assume that you are conservative and wager 1% of your bankroll on each coin flip. These are the possible outcomes of the size of your bankroll after playing the game of 2 quarter flips. Bankroll = $1,000,000 x (1 + (2 * 0.01)) x (1 + (2 * 0.01)) = $1,040,400 Bankroll = $1,000,000 x (1 + (2 * 0.01)) x (1 – (1 * 0.01)) = $1,009,800 Bankroll = $1,000,000 x (1 – (1 * 0.01)) x (1 + (2 * 0.01)) = $1,009,800 Bankroll = $1,000,000 x (1 – (1 * 0.01)) x (1 – (1 * 0.01)) = $980,100 Since each of these results is equally likely, the Expected Value of the outcome of these sequential bets is a profit of $10,025 (or 1.0025%). Expected Growth (EG) is a little bit more tricky. In order to figure it out (without having the formula in front of you), you must see what the expected outcome is. Since the odds of the game are always 50/50 for each coin flip, the expected outcome is simply winning once for each time you lose. Since we are flipping the coin twice, the expected outcome is to win once and lose once. The order that you win or lose doesn’t matter as the bankroll ends up at the same number either way in this game. The bankroll, based on the calculations above, in the expected outcome is $1,009,800, which is a profit of $9,800 (or 0.98%). This 0.98% is the EG. For those that are interested in generalized equations, here they are: Bankroll after Expected Outcome = (Initial Bankroll) * (1 + (Decimal Odds – 1) * (Bet Size / Initial Bankroll))p * (1 – (Bet Size / Initial Bankroll))(1 – p) EG = (1 + (Decimal odds – 1) * (Bet Size / Initial Bankroll))p * (1 – (Bet Size / Initial Bankroll))(1-p) – 1 EV = ((p * Decimal Odds) – 1) * (Bet Size / Initial Bankroll) where p is the probability of winning Let’s look at a fun example of playing that original game but instead of betting 1% of your bankroll on each coin flip, you want to make a bet with a higher EV and bet 90% of your bankroll. (For those of you still reading at this point, that is far greater than what the Kelly criterion says you should bet.) The 4 possible outcomes are as follows again: Win both the first and second bets. Win the first bet, lose the second bet. Lose the first bet, win the second bet. Lose both the first and second bets. Bankroll = $1,000,000 x (1 + (2 * 0.90)) x (1 + (2 * 0.90)) = $7,840,000 Bankroll = $1,000,000 x (1 + (2 * 0.90)) x (1 – (1 * 0.90)) = $280,000 Bankroll = $1,000,000 x (1 – (1 * 0.90)) x (1 + (2 * 0.90)) = $280,000 Bankroll = $1,000,000 x (1 – (1 * 0.90)) x (1 – (1 * 0.90)) = $10,000 Since each of the 4 outcomes is equally likely, that yields an EV of $2,102,500 or a profit of $1,102,500 (or 110.25%). However, the EG here is a loss of 74%! That means that although you would be making bets with higher expected value, you would end up with (significantly) worse expected growth. In fact, you now expect a significant shrink in the size of your portfolio (or bankroll). Extending this logic out further, if you were to go “all in” on every single bet even if the coin were weighted in a manner such that you win 99.999999999% of the time, the EG = -100% if you are allowed to flip the coin infinite many times despite the fact that your EV would be exploding to infinity. The math of investing isn’t as simple as winning and losing as in the case, I present above. You get to input distributions of possible results with probability distributions of those results. In the end, you get to the indisputable truth that BET SIZING MATTERS (portfolio allocation sizing matters). The math gets way more complicated than this and frankly, I don’t truly understand it yet. The key takeaways from this fun math end up being quite intuitive: The better the investment (in terms of expected return and likelihood of success), the greater the percentage of your portfolio that you should allocate to this investment. If you overbet (oh what a horrible screen name to have for this discussion) or underbet, you will not maximize your expected growth. If you underbet in a +EV situation, you still will expect to grow your portfolio. If you overbet in a +EV situation, expected growth of your portfolio can be positive or negative. The penalty of underbetting (in general) is less severe than the penalty of overbetting. While I think the guidelines Chris DeMuth lays out are not necessarily all that bad in practice, I would caution taking an overly simplistic approach to a (very) complex problem. Scalper1 News
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