Understanding Variance of Returns using Probability Distribution

Imagine you are considering an investment, perhaps in a new tech company or a promising stock. You know that investments can go up or down, and you want to understand how much the returns might fluctuate. This is where the concept of variance comes into play, specifically when we’re looking at the probability distribution of returns.

Think of a probability distribution as a map of possible outcomes for your investment, along with how likely each outcome is. Let’s say you are thinking about investing in a new ice cream shop. You might envision several scenarios for its first year: it could be a huge success, a moderate success, break even, or even lose money. Each of these outcomes, these potential returns on your investment, has a certain probability of happening. Maybe there is a 10 percent chance of a big loss, a 20 percent chance of breaking even, a 50 percent chance of moderate success, and a 20 percent chance of a huge success. This spread of possible returns, with their associated probabilities, is your probability distribution.

Now, variance helps us measure the spread or dispersion of these potential returns around the average return we expect. It tells us how much the actual returns are likely to deviate from that average. A high variance suggests that the returns are widely scattered, meaning there is a greater range of possible outcomes, from very high to very low, making the investment riskier. Conversely, a low variance indicates that the returns are clustered closer to the average, implying a more predictable and potentially less risky investment.

To calculate this variance using the probability distribution, we follow a few key steps. First, we need to determine the expected return. The expected return is essentially the average return you would anticipate getting over the long run, considering all the possible outcomes and their probabilities. To calculate this, you multiply each possible return by its probability and then sum up these products. Continuing with our ice cream shop example, let’s say a big loss is -10 percent return, breaking even is 0 percent, moderate success is 10 percent, and huge success is 25 percent. The expected return would be calculated as follows: (-10 percent times 0.10) plus (0 percent times 0.20) plus (10 percent times 0.50) plus (25 percent times 0.20). This gives us an expected return, which is like the central point of our distribution.

Once we have the expected return, we move to the next step which is to calculate how much each possible return deviates from this expected return. For each possible return, we subtract the expected return from it. This difference is called the deviation. For example, if our expected return from the ice cream shop calculation turns out to be 8 percent, and one possible outcome is a 10 percent return, the deviation for that outcome is 10 percent minus 8 percent, which is 2 percent.

After calculating the deviations for each possible return, we square each of these deviations. Squaring the deviations is important because it makes all the deviations positive, and it also gives more weight to larger deviations. A return that is far from the expected return will have a much larger squared deviation than a return that is close to the expected return. In our example, the squared deviation for the 10 percent return outcome would be 2 percent squared, which is 4 percent squared.

Finally, to get the variance, we take each squared deviation and multiply it by the probability of that return occurring. Then, we sum up all these probability-weighted squared deviations. This sum represents the variance of the returns. It is essentially the average of the squared deviations, weighted by the probabilities of each deviation occurring. This final number, the variance, quantifies the overall dispersion of the returns around the expected return, providing a measure of the investment’s riskiness. A higher variance indicates a wider spread of potential outcomes and thus greater risk. Understanding variance helps in making informed decisions about investments by giving a clearer picture of the potential volatility of returns.