Unknowingly all of us constantly use probabilities to make decisions in our day-to-day lives. We use such similar type of probabilistic thinking with investing.

We know that stock returns are hard to predict on a yearly basis. Over the course of several decades, however, we know stock returns tend to be positive.

## Unknowingly all of us constantly use probabilities to make decisions in our day-to-day lives.

Likewise, we know that owning bonds will likely reduce our long-term investment returns, but will provide a much smoother ride.

By looking at historical investment returns, we can make some reasonable guesses at how different asset classes will probably perform in the future.

Using probability to analyse how returns impact your investment portfolio
When historically comparing how the best and worst years for the S&P 500 impacted different-sized portfolios, it is evident that how smaller portfolios are barely moved by massive market gyrations.

For example, let’s say you had an stock investment portfolio worth Dh5,000. During the S&P500’s worst ever year when it recorded a loss of 37 per cent, you would have been hit with a Dh1,828 million loss.

Whereas if your portfolio size was Dh1 million, your loss would equate Dh365,500. Similarly, the same probability rule applies when the S&P 500 posted its biggest ever 53 per cent return.

You would then have pocketed a profit of Dh2,628 if you had invested Dh5,000 and a profit of Dh525,600 if you had Dh 1 million invested.

### Probability, a lens for all things investment

In order to make consistently profitable investment decisions, we have to be able to assess and manage our expectations about risk and reward.

We do this by analyzing all relevant information, and then adjusting the probabilities and payouts that we assign to various outcomes.

## Probabilities are the lens through which we must view all things investment related.

The framework through which we do this is known as probability theory, and a basic understanding of probabilities is important not just for investment purposes, but for routine finances in general.

Understanding probability theory, and its counterpart – expected value – will go a long way in helping you to become an excellent decision maker.

Rules that govern probability
Let’s go over two basic principles of probability, and then we’ll look at some key points about probability theory.

There are two properties that govern probability. The first is that a given probability must be between 0 (impossible) and 1 (certain).

Or in percentage terms, a probability must fall between 0 per cent and 100 per cent. The second is that all possible outcomes must have probabilities that add up to 1 (or 100 per cent). Remember to calculate risks associated with mortgage-backed securities Image Credit: Shutterstock

Let’s use a basic coin flip as an example to highlight these two rules. When you flip a coin, the probability of landing heads-up is 50 per cent, and the probability of landing tails-up is 50 per cent.

Excluding the possibility of maybe the coin landing on its edge (as it’s an unlikely event), the only other two possible outcomes are heads and tails.

These outcomes must add to 100 per cent, which is why we assign them equal probabilities (50 per cent).

What if there was a legitimate chance of a coin landing on its edge?
If there was a legitimate chance of a coin landing on its edge, say 1 per cent, we would have to reduce the probabilities of the coin landing heads or tails. In this case, we could say that the probabilities are:

Heads – 49.5%, Tails – 49.5%, Edge – 1%. Here again, all possible outcomes add to 1 (or 100%).

Alright, now let’s get to the heart of probability theory. And while we’re at it, let’s explore a key element of probability theory: Expected Value.

### Determining Expected Value

To help understand this concept better, here is the question worth discussing:

## Would you rather have Dh2,000 right now, or a 70 per cent chance of winning Dh4,000?

Your answer to the above question will involve a number of factors, one of which should be probability theory. Also how you choose to go about it will show how risk averse you may or may not be.

Turning to a probability theory to answer the question!
The first step in answering this question is creating a fair comparison.

We know how much Dh2,000 is worth, but how does that compare against a 60 per cent chance of winning Dh4,000? To find an answer, we turn to probability theory, or more specifically, a concept called expected value.

In probability theory, the value of a probability-based outcome is known as the expected value.

The expected value of a particular outcome is equal to the probability of receiving a value, multiplied by the value received. In short hand: Expected Value = Probability of Receiving Value x Value Received

## The expected value of a particular outcome is equal to the probability of receiving a value, multiplied by the value received.

So, in our case, we can see that the expected value of a 70 per cent chance of winning Dh4,000 is Dh2,800. The formula: Expected Value = 0.70 x Dh4,000 = Dh2,800

Why is it called expected value?
Because this is the value you could expect to receive if you were to run this scenario multiple times, as let’s say the stock market.

In reality, for now the value to you would be either Dh4,000 (if you won) or Dh0 (if you lost).

Now that you have two values to compare for your decision (\$2,000 and \$2,800), how do you decide?

## Your individual decision will be influenced by a number of psychological factors, one such being risk aversion.

Your individual decision will be influenced by a number of psychological factors, one such being risk aversion.

How much risk can you take on?
If you’re risk-averse, you probably chose to keep the Dh2,000, and if you’re risk-seeking, or have more money than you know what to do with, you might have chosen to gamble for the Dh4,000.

Either decision can be justified. It becomes more apparent if you were told that you we were going to run this scenario several number of times.

The key to allowing probability theory to work in your favor is having enough opportunities for probability theory to take effect.

If there is a 70 per cent chance of an event occurring, it may not happen on the first instance, or the second, but given enough opportunities, that outcome will occur close to 70 per cent of the time.

Let’s say this scenario was run some 200 times, and you chose the 70 per cent chance of winning Dh4,000 option each time.

Then there is a very high likelihood that you would end up with close to Dh280,000. This is calculated as follows: Expected Value = 0.70 x Dh2,000 x 200 = Dh280,000

## The key to allowing probability theory to work in your favor is having enough opportunities for probability theory to take effect.

What do you take away from this?
In the long run this option is much more profitable than choosing to receive the guaranteed Dh2,000 each time, which would leave you with Dh200,000 after 200 rounds.

Probability theory never plays out over one event, or even a few. When this scenario is played only once, it is understandable for holding onto the Dh2,000 and not gambling it away as some cash is better than none.

But as an investor if you chose the guaranteed Dh2,000 each time, there is a much lower probability to achieving long-term financial success.

So, being a successful investor boils down to understanding the nature of decisions under uncertainty, and that is governed by basic probability theory.

## Being a successful investor boils down to understanding the nature of decisions under uncertainty!

### Real world probabilities not an exact science

Although we went through in detail the exact nature of how probabilities work with regard to investing, in the real world working with exact probabilities is unheard of. Instead, all we have to go on are subjective probabilities.

What are subjective probabilities?
Subjective probabilities are based not on the actual odds of an event occurring, but on the probabilities an individual uniquely assigns to an event occurring.

Subjective probabilities are unique to the person making them. This carries two implications. First, recognize that the probabilities you assign to various outcomes will be different than that of others.

Second, the accuracy of the probabilities you or anyone else assigns is based entirely on your knowledge and experience. The more you know, the more accurately you will be able to forecast the likelihood that certain outcomes will occur.

### Real world illustration on how probability applies to business decisions

Let's say the executives of a food company must decide whether to launch a new packaged cereal and they need to then submit a paln to management.

They have come to a conclusion that five factors are the deciding variables: advertising and promotion expense, total cereal market, share of market for this product, operating costs, and new capital investment.

## On the basis of the “most likely” estimate for each of these variables, the picture looks very bright-a healthy 30 per cent return.

On the basis of the “most likely” estimate for each of these variables, the picture looks very bright-a healthy 30 per cent return. This future, however, depends on whether each of these estimates actually comes true.

If each of these educated guesses has, for example, a 60 per cent chance of being correct, there is only a minimal chance that all five will be correct. So the “expected” return actually depends on a rather unlikely coincidence.

Decision makers need to know a great deal more
The decision makers need to know a great deal more about the other values used to make each of the five estimates, learning about what they stand to gain or lose from various combinations of these values.

This simple example illustrates that the rate of return actually depends on a specific combination of values of a great many different variables.

Thus predicting a single most likely rate of return gives precise numbers that do not tell the whole story.

The expected rate of return represents only a few points of possible combinations of future happenings.

## Predicting a single most likely rate of return gives precise numbers that do not tell the whole story.

It is a bit like trying to predict the outcome in a dice game by saying that the most likely outcome is a 7. The description is incomplete because it does not tell us about all the other things that could happen.

Applying the concept to a business environment can get increasingly complex!
For instance, in some dice games we normally see the odds on throws of two dice having 6 sides each. Now imagine a scenario supposing that each of six sides of the dice has another 100 sides.

This is a situation more comparable to business investment, where the company’s market share might become any 1 of 100 different sizes and where there are factors such as pricing, promotion, and so on that can affect the outcome. Beware of hidden fees for savings plans that could eat away from your retirement nest egg. Image Credit: Stock photo

## Investors should embrace probabilities to help improve decision making, especially when it comes to investments.

But remember investors can be irrational – such as the tendency to neglect probabilities when considering extreme scenarios, and our tendency to either give too much importance or ignore small probabilities.

Use of probability is more of a measure of confidence, than a precise science!
Applying specific probabilities to a range of scenarios can appear overly scientific – what does it mean to believe the likelihood of an event is 17 per cent?

This view, however, misses the value of applying probabilities. Its use is not as a precise figure but as a measure of confidence and a means to monitor how our views evolve through time.

The notion, however, that we should not talk about probability because we have limitations in this regard is entirely spurious.