### Optimal Allocation or Position Sizing

Imagine you found a stock to invest in. Also imagine that you have earmarked some amount of money for stock investments.

You have some idea about the chances for a positive return from this stock. Also you have some inklings of risks due to which there will be capital loss.

So in this situation, is there a scientific method to find out the optimal allocation into that stock?

Turns out there is. And such a method is used by successful investors including Warren Buffett and Bill Gross.

I first heard about this method from a fantastic book called ‘Dhandho Investor‘ written by Mohnish Pabrai.

The method is called Kelly criterion. This was developed by John Larry Kelly, Jr. (1923–1965), a scientist who worked at Bell Labs. Kelly criterion is a formula to determine what proportion of wealth to risk in a sequence of positive expected value bets so as to maximize the rate of growth of wealth.

Optimal fraction of your bankroll to bet on a favor-able bet is:

Edge/odds = Fraction of your bankroll you should bet each time.

Edge is the overall expected payout, taking into account all possible probabilities and payouts

E.g: assume you’re offered a coin toss where heads means you get Rs 2 and tails costs you Rs 1.

Edge = [0.5 x 2Rs] + [0.5 x -1Rs] = 0.5

Odds are what you win, if you win. So in this case, odds are Rs 2.

So applying Kelly formula, what percentage of your funds must you invest in each toss in the above example?

Edge/odds = 0.5/2 = 0.25 = 25%

So you must put 25% of your funds each time to maximize your returns.

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Let us come back to our Stock. We must look at each investing opportunity as a game of probabilities. Calculate your ‘Edge’ taking into account all probabilities that you can think of.

In our example, let us suppose the probability of this stock giving two times returns in next three years (that requires the stock to appreciate 27% annually for next three years ) is 50%. The probability of this stock giving only bank FD returns over next three years (which means it will grow 1.25 times in 3 years) is 40%. And there is a 10% probability that some unimaginable will happen and the company will shutdown and the entire investment will become worthless over the next three years.

What percentage of our bankroll should we invest in this opportunity for optimum returns?

Applying Kelly formula, we get the optimum allocation of the above as;

[(0.5 x 2)+(0.4 x 1.25)+ (0.1 x -1)] / 2 = 70%

This means, given the probabilities described above, allocation must not exceed 70%.

Many investors use only a fraction of the maximum allocation suggested by Kelly criterion, again due to their own ‘margin of safety’ constraints.