How large an investment position can you take in a stock that offers exceptional value?

It’s a question that is debated rigorously amongst the analysts at Intelligent Investor, because it can mean the difference between returns that are mediocre, exceptional or disastrous. In Fortune’s Formula, author William Poundstone provides a fascinating account of a mathematical approach to position sizing, known as Kelly’s criterion, which has been employed with some success by a range of hedge funds since the 1970’s. There are some useful lessons for stockmarket investors.

The approach is best demonstrated by an example. Suppose you are given the opportunity to bet on the toss of a coin you know to be biased. The likelihood of a ‘head’ is 55%, and if you correctly guess the result of the coin toss you win $1 for every $1 bet. If you were offered 500 flips of the coin but had only $100 to bet, how much should you wager on each toss of the coin?

Because you know the odds favour betting heads, you might approach this game by betting as heavily as possible to maximise your exposure to the attractive opportunity. Taken to its extreme, it’s not difficult to see the folly of this approach. If you bet 100% of your available funds on each toss of the coin, you are almost certain to lose all of your money well before the 500^th flip. You can’t compound zero.

Through trial and error you discover, as you can see in the attached spreadsheet, that even if you only bet 20% of your funds, you still tend to lose out with depressing regularity. On the other hand, if you only bet a small portion of your funds, say 1%, you tend to end up ahead, but are you missing out on a higher potential return with such a timid approach?

The problem with large bets is that the inevitable losses damage your wealth beyond the point of recovery. Reducing the bet size mitigates the problem, but also limits your exposure to an attractive value transaction.

Using some fairly simple algebra (Matt’s workings are also in the spreadsheet), we can plot the projected results as the size of our wager increases. The chart plots the coin flipping example we have been using and shows projected returns increase up until you begin to wager 10% of your current funds, at which point your returns start to collapse. Regular bets of more than 20% are likely to send you broke in short order.

John Kelly derived the optimum wager size for a more general version of this problem in 1956. Kelly’s criterion states that the optimal position size to maximise your return should equal expected net winnings divided by the net proceeds of a win (if you remember your high school algebra, to find the optimal wager look for the point on the chart where the slope equals zero).

Applying Kelly’s criterion to our example, expected net winnings = 55% x $1 – 45% x $1 = $0.10. The net proceeds of a win = $1, and the position size is identified as $0.1/$1 = 10%.

Here’s another example. Assume you have been offered $6 for each $1 bet on a six-sided dice that you know is biased. You know that the number 1 will come up one third of the time, instead of the theoretical one in six. How much should you bet? Well, your maximum expected payoff will arise when you bet 22% of your funds on each roll of the dice ([33% x $6 - 67% x $1]/$6).

This principle has been applied by punters to a wide variety of gambling pursuits, but it also has great significance to the stockmarket. Whilst in practice we never know with certainty the expected ‘net winnings’ of a stock or security, there are three straight-forward aspects of Kelly’s criterion that are directly applicable to portfolio position sizing:

1.      For the same probability of loss, the more attractively priced the opportunity, the larger the position size you can take (if heads paid $2.05 rather than $2, the optimal position size moves from 10% to 12.1%).

You can see this is in the graph below. The maximum expected payoff is obviously higher, but the curve also shifts to the right. 

2.      For the same level of attractiveness, the less risky the opportunity the larger the position you can take (you can allocate a greater position size to a 3 to 1 horse that is a 50% chance of winning, than to a 4 to 1 horse that is a 40% chance of winning).

3.      Even an attractively priced bet can send you broke if you get your portfolio allocation wrong (the coin toss example is after all a value bet, but a larger than 20% stake still tends to ruin you)

Finally, a word on risk. The position size is optimised from a return perspective, but may be totally inappropriate from a risk perspective. In our original coin tossing example, the 10% portfolio allocation maximises the projected return but also produces a lot of volatility. Even over 500 throws, you can expect to lose money 13% of the time.

At all points greater than a 10% allocation, you get more volatility and a lower return. To the left of 10%, though, you get lower return but also lower risk. All of these points might be optimal for different investors, depending on their risk tolerance.

It's all too mathematical for any explicit use by stockmarket investors, but these are some very useful concepts nonetheless.