Calculating Expected Return: Probability Distribution Explained

Imagine you are deciding whether to invest in a new venture, perhaps a local coffee shop or a promising tech startup. You know there are different possible outcomes. Things could go exceptionally well, just okay, or unfortunately, maybe even poorly. Each of these outcomes isn’t equally likely. Some are more probable than others. To make a smart decision, you need a way to weigh these possibilities and understand the overall potential return you might expect. This is where the concept of expected return comes into play, especially when we’re given a probability distribution.

Think of a probability distribution like a detailed weather forecast for your investment’s future returns. Instead of predicting sunshine or rain, it lays out the possible return percentages you might experience and the likelihood of each of those percentages actually happening. For example, a probability distribution might tell you there is a 20% chance your investment will yield a 10% return, a 50% chance it will yield a 5% return, and a 30% chance it will result in a 2% loss. This distribution is crucial because it doesn’t just give you a single possible outcome; it gives you a range of outcomes and their associated probabilities.

Now, how do we use this probability distribution to calculate the expected return? The expected return is essentially a weighted average of all possible returns, where the weights are the probabilities of each return occurring. It’s like calculating your grade point average. Each grade you receive in a course has a certain credit weight. To find your GPA, you multiply each grade by its credit weight, sum these products, and then divide by the total credit hours. Expected return is conceptually similar, but instead of course grades and credit hours, we’re dealing with possible investment returns and their probabilities.

To calculate the expected return, you perform a straightforward calculation. First, you identify each possible return from the probability distribution. Let’s say in our coffee shop example, the possible returns are 15%, 8%, and -5% representing a great year, an average year, and a poor year, respectively. Next, you need to know the probability associated with each return. Perhaps there’s a 30% chance of a great year with a 15% return, a 50% chance of an average year with an 8% return, and a 20% chance of a poor year with a -5% return.

The calculation process then involves multiplying each possible return by its corresponding probability. So, you would multiply 15% by 30%, then 8% by 50%, and finally -5% by 20%. These multiplications give you the weighted return for each scenario. After you have these weighted returns for each possible outcome, you simply add them all together. This sum is the expected return.

Let’s do the math for our example. Fifteen percent times 30% is 4.5%. Eight percent times 50% is 4%. Negative five percent times 20% is negative 1%. Adding these together, 4.5% plus 4% minus 1%, gives us 7.5%. Therefore, the expected return for this coffee shop investment, based on our probability distribution, is 7.5%.

It’s important to remember that the expected return is not a guarantee. It is not saying that you will definitely earn exactly 7.5% on your investment. Instead, it’s a statistical measure that represents the average return you could expect to receive over the long run if this probability distribution holds true and you were to make this investment many, many times. Think of it as the long-run average outcome, not a prediction for any single instance.

Expected return is a valuable tool for comparing different investment opportunities. If you are considering two different projects, each with its own probability distribution of returns, you can calculate the expected return for each and compare them. Generally, a higher expected return is more desirable, assuming you are comfortable with the associated risks reflected in the probability distribution. However, remember that expected return is just one factor to consider in investment decisions. It’s crucial to also consider the level of risk, your personal financial goals, and your investment timeline. But understanding how to calculate expected return from a probability distribution is a fundamental step in making informed investment choices and understanding the potential outcomes of various ventures.