We have seen several "experts" predict stock market prices with their impeccable analysis and forecasting tools. We have seen them value stocks with their seemingly infallible techniques. We have seen them claim their own glory in beating the market 85 % of the time. And those who have observed them for long enough have seen them fail miserably. Which brings us to the question - is it really impossible to predict stock prices?

With whatever I have observed, I tend to believe that it is impossible to do so. Not because people lack the ability to do so; we have enough brains and enough computing power to accurately determine anything that can be determined. It is because the market is not an exact science. And in such cases, it is wiser to use a probabilistic model. When we say that there is a 90 % chance of something happening, we are also saying that there is a 10 % chance of it not happening. Strangely, humans are wired to ignore the latter. We confuse a 90 % probability with certainty and that heralds our doom.

There have been innumerable cases where people have invested huge sums on a particular bet and it just so happened that the 10 % chance of the event not happening prevailed. Consequently, large sums of money were lost. This is the very reason why we must learn one very important lesson in our investing journey - position sizing.

Having said that, it is noteworthy that when things are uncertain, it is prudent to be open to change. In an uncertain environment, nothing is etched in stone. Not even the size of our positions. Whether the stock moves in our favour or otherwise, we must keep recalculating our bets and add/reduce our position accordingly. We will look through a very detailed example and understand how to get a hold of this technique.

We derive our formula from the famous Kelly Criterion. For those having an inclination towards mathematical research papers, I recommend this book that will explain the concept in ample detail. The Kelly Criterion is a precise tool to calculate the size of our investment in a given trade. While it has its applications in other areas as well, we will confine ourselves to its use in investing in the stock markets and other asset classes. 

The formula is given as follows :



where:

f * is the fraction of our entire funds that can be invested
b is the winning multiple, i.e. the factor by which our initial investment will be multiplied
p is the probability of winning
q is the probability of losing, which is 1 − p

As a simple illustration, let us consider a simple investment in a CFD issued by a company which has a 5 % chance of going bankrupt and not paying out its commitments. We assume the CFD to pay a 5-year return of 8.75 % which means our b is 1.0875, p is 0.95 and q turns out to be 0.05. Kelly suggests that we wager 90.402 % of our funds in such a scenario. However, things get far more complicated when we trade stocks or derivatives since we cannot determine the odds of our winning in any situation. In fact, we are not even sure of the winning multiple in such cases because we never really know for certain how far the stock price will move even in the case of a favourable move. In order to discount this uncertainty, we will use a 50 % probability of winning.

Obviously, the winning chances are higher in every trade, which is why we enter the trade in the first place. Had it not been so, trading in stocks would be akin to a gamble. Traders use their judgement and select a particular stock from a sea of available options because they have some justification in that choice. Which explains why the chances of a win are certainly higher than 50 %. But since we are never sure of the exact return that trade might deliver, we bring down the win chance to 50 % in order to account for this uncertainty. On the other hand, we will use a multiplier that reflects our expectation of where prices might reach in the ensuing move. 


In the above example, let us say we buy at 57.50 with an expectation that the price will reach 70.35. The emphasis here is not on how we figure out the target price. That varies widely depending on the analysis technique used. What we are trying to look at is the multiplier that we shall consider for our calculation. In this case, the winning multiplier will be 1+ {(70.35 - 57.50)/57.50} which is 1.2234. Using the Kelly equation, we see that the suggested investment size is 9.13 % of our corpus.

Talking of the long term, let us take a look at what this seemingly simple formula does for us. Here are some equity curves that we must look at.





We simulated a sequence of 1000 trades where the win/loss is decided at random and two scenarios are evaluated. One equity curve is drawn assuming the investment size is determined using the Kelly Investment Criterion while the other is drawn assuming that the investor uses 30 % of his corpus for each trade. We have considered that each winning trade delivers a return of 5 % and each losing trade delivers a loss of 95 %. This again is the worst case scenario since no investor would invest where the risk is equal to the reward. Surprisingly, it is observed that for a non-Kelly investor, it takes less than 100 trades to wipe out his capital. If we use 50 % of our corpus each time, the wipeout will occur even sooner. On the other hand, by the use of the Kelly Factor, the investor successfully retains all of his capital even in the worst sequence of trades. In this simulation, we have also assumed that the probability of our winning the trade is just 50 %, which means we go wrong half the time. This simulation gives us a picture of the margin of safety this formula gives us even in the worst of cases.

Evidently, it makes enough sense to use this tool in all our investment decisions. Whether it be the capital markets, forex trades, or investments in projects, the Kelly Investment criterion is the first tool one must use. The tool seems to work best for asset classes where the loss of capital erosion is very high in the case of a losing trade. For example, trading in naked options. Here is a link to download the spreadsheet and play around with the figures, should you want to experiment with some more scenarios. Feel free to share your findings with us in the comments. Happy investing!