A reflection on the importance of probability, outcome, and expected return.

Why probability alone is useless, the challenges posed by tail outcomes, and our inability to determine distributions.

It’s All About Expected Return

I am currently reading Nassim Taleb’s Fooled By Randomness. Early on, Taleb makes an important distinction between probabilities, and expected returns. This is particularly key when there is an asymmetric distribution of outcomes. 

Here is a brief summary of Taleb’s example. Taleb considers the stock market. For the purposes of the example, say there is a 70% chance the market (index) will go up, and a 30% chance the market will go down. Using these probabilities - without considering the outcomes - we would arrive at the conclusion that one ought to invest in the index. Now let us take the same odds, and attach outcomes to each probability; there is a 70% probability that the market will go up by 1, and a 30% probability that the market will go down by 10. In the following table, expected return is calculated:

Probability	Outcome	Expected returns
70%	1	0.7
30%	-10	-3
	Total expected return:	-2.3

The expected return is -2.3, suggesting that it would be wiser to short the index. This is because if I were to make this decision - in the same theoretical conditions - an infinite number of times, I would only be able to generate a positive return by doing this.

Low Probabilities and Finite Deaths

Whilst I have just used Taleb’s example to show that expected return is a more useful metric than probability, I must strike a cautionary note. This is because in reality, we do not always have the ability to repeatedly make the low probability bet in this example. In effect, it is possible for you to make decisions that are correct on an “expected return” basis and still suffer an “unacceptable outcome”. This is because we can only make a finite number of ‘bets’. If the more likely outcome were to occur in each of your ‘bets' you could lose all capital. This would be a finite death. Thus, perhaps it is best to invest only when the probability and expected return are both in your favour.

We have now discussed how investing solely in low probability events that can be justified on an expected return basis could result in an indigestible outcome. However, it must be said that ignoring tail risks (of very low P) that have a great magnitude could have equally - if not more - devastating consequences. In my post about Confidence Game, I distinguish between improbability and impossibility. This is an important distinction, because events with very low probabilities can produce unacceptable outcomes when they do occur. 

Effectively, it is important to consider low probability events in a careful, balanced manner as they may be associated with unacceptable outcomes.

Imagining Odds and Outcomes

In the example Taleb refers to, there are two definite probabilities, and two specific outcomes. This serves the purpose of illustration; however, assuming that things are as easily predicted in reality, would be a grave mistake. 

In reality, we must begin by coming up with a range of outcomes. This itself is very challenging. Now, after we have determined a range of outcomes, it is important to attach a probability to each outcome. It is very difficult to attach accurate probabilities. People normally overestimate the amount by which things will change in a year, and underestimate the magnitude of change that can be achieved through slow, steady progress in a decade. This is just one systematic error we are prone to making. 

Imagining the outcomes, and then attaching probabilities to these outcomes is an art form. It must be undertaken with diligence, caution, and an awareness of the difficulty of the task. Investing does not reward ‘accuracy’ - indeed some say this is impossible. Instead, it pays you for (i) being right about the ‘story’ of a business, and (ii) seeking a margin of safety.