Covariance: Key to Portfolio Risk in Large Portfolios

Imagine building a team for a tug-of-war competition. If you only have a few strong individuals, the overall strength of your team, and thus your chance of winning, depends heavily on the strength of each individual. However, what happens when you have a very large team, say a hundred people? Suddenly, the strength of any single person becomes much less important. Instead, what truly matters is how well everyone is pulling together, in the same direction, at the same time. This ‘pulling together’ is analogous to covariance in the world of investments.

In finance, when we talk about portfolio risk, we’re essentially talking about the uncertainty of your investment returns. We want to minimize this uncertainty, or risk, while still achieving our desired returns. One way to reduce risk is through diversification, which is like adding more players to your tug-of-war team. By holding a mix of different assets, like stocks from various companies and industries, or even bonds and real estate, you are spreading your investment across multiple sources. The idea is that if one investment performs poorly, hopefully, others will perform well, offsetting the losses and smoothing out your overall portfolio returns.

Now, let’s consider an equally weighted portfolio. This means you invest the same amount of money in each asset you choose. As you add more and more assets to this equally weighted portfolio, something interesting happens to the sources of risk. The risk of your portfolio is driven by two main factors: the individual riskiness of each asset, and how these assets move in relation to each other. The individual riskiness of an asset is measured by its variance, which essentially tells you how much its price tends to fluctuate on its own. Think of it as how wobbly each individual tug-of-war player is on their feet.

The relationship between assets is measured by covariance. Covariance tells us how two assets move together. If two assets tend to move in the same direction, like umbrellas and rain boots, they have a positive covariance. If they tend to move in opposite directions, like ski equipment and beachwear, they have a negative covariance. If they move independently of each other, their covariance is close to zero. In our tug-of-war analogy, covariance is like how well each player coordinates their pulling with their teammates.

When you have a small portfolio with only a few assets, the overall portfolio risk is influenced significantly by the individual variances of those assets. If you have a few very volatile assets, they can significantly impact your portfolio’s overall risk. However, as you keep adding more and more assets to an equally weighted portfolio, the weight or importance of each individual asset becomes smaller and smaller. Imagine dividing your investment pie into more and more slices; each slice gets thinner. Consequently, the impact of any single asset’s individual volatility on the overall portfolio risk diminishes.

At the same time, as you add more assets, you are also adding more pairs of assets, and therefore, many more covariance relationships. The number of covariance terms in a portfolio grows much faster than the number of variance terms as you increase the number of assets. Even though the weight of each individual covariance also decreases in an equally weighted portfolio, the sheer volume of covariance terms becomes overwhelming. In a very large portfolio, the overall portfolio risk becomes primarily determined by the average covariance between all the assets in the portfolio.

Essentially, in a large, diversified portfolio, it is not so much about how volatile each individual asset is in isolation. Instead, it’s about how these assets move together on average. If, on average, the assets in your portfolio tend to move in the same direction, your portfolio will still be quite risky, even if it’s very large and diversified. This is because when one asset goes down, many others are likely to go down as well, leading to a significant overall portfolio loss. Conversely, if the average covariance is low or even negative, meaning assets tend to move independently or in opposite directions, your portfolio will be less risky, as losses in some areas are likely to be offset by gains in others.

Therefore, as portfolio size grows in an equally weighted setting, the average covariance becomes the dominant factor in determining portfolio risk. It is the ‘teamwork’ or coordination, represented by covariance, that ultimately dictates the overall portfolio performance and risk, much like in our large tug-of-war team where collective coordination outweighs individual strength. Focusing on understanding and managing the average covariance between assets is crucial for effective risk management in large, diversified portfolios.