Calculating Two-Asset Portfolio Variance: Variance and Covariance Explained

Imagine you are creating a balanced meal, not for your body, but for your financial future. Let’s say you decide to combine two key ingredients, perhaps a stock and a bond, to build an investment portfolio. Just like each ingredient in a meal has its own characteristics, each asset in your portfolio has its own level of risk, or what we call variance.

Variance, in simple terms, is a measure of how much an asset’s price tends to fluctuate. Think of it like the wobble in a spinning top. A stock known for big price swings has high variance, like a top wobbling wildly. A bond, generally more stable, has lower variance, like a top spinning smoothly with minimal wobble. When you hold just one asset, its variance is simply its own wobble.

However, when you combine two assets into a portfolio, the overall risk of the portfolio isn’t just the sum of the individual wobbles. It’s more nuanced than that. This is where the concept of portfolio variance comes in, and it cleverly takes into account not only the individual variances of each asset but also how these assets move in relation to each other. This relationship is called covariance.

Covariance is like understanding if our two ingredients, say stock and bond, dance together or move independently. Do they tend to go up and down at the same time, or when one goes up, does the other sometimes go down? If they tend to move in the same direction, that’s positive covariance. Imagine two dancers who mirror each other’s moves. If they move in opposite directions, like when one rises while the other falls, that’s negative covariance. Think of dancers in a counterpoint, complementing each other but moving differently. If they move with no predictable relationship, that’s near zero covariance, like two separate dancers in different rooms.

Now, let’s put it all together to calculate the variance of our two-asset portfolio. The formula might seem a bit intimidating at first, but we can break it down into understandable pieces. To find the portfolio variance, we consider three main components.

First, we look at the variance of the first asset, let’s say the stock. We multiply this variance by the square of the proportion of the portfolio invested in that stock. Think of the proportion as the weight of that ingredient in our meal; if stock makes up 60% of your portfolio, its weight is 0.6. Squaring this weight emphasizes the impact of the stock’s own risk on the portfolio, adjusted for how much of the portfolio it occupies.

Second, we do the same for the second asset, let’s say the bond. We take the variance of the bond and multiply it by the square of the proportion of the portfolio invested in the bond. So if bonds are 40% of your portfolio, its weight is 0.4, and we square that and multiply it by the bond’s variance.

Finally, and this is the crucial part that makes portfolio variance different from just adding up individual variances, we consider the covariance between the stock and the bond. We multiply this covariance by two times the proportion invested in the stock and times the proportion invested in the bond. This covariance term adjusts the overall portfolio variance based on how these two assets move in relation to each other.

Putting it all together, the portfolio variance is calculated as follows: take the weight of asset one, squared, times the variance of asset one, plus the weight of asset two, squared, times the variance of asset two, plus two times the weight of asset one, times the weight of asset two, times the covariance between asset one and asset two.

The magic of diversification comes into play when the covariance is low or even negative. If our stock and bond tend to move in opposite directions or with little correlation, the covariance term becomes smaller, or even negative, reducing the overall portfolio variance. This means that by combining assets that don’t move in perfect lockstep, we can actually create a portfolio that is less risky than simply holding either asset alone. It’s like combining ingredients that balance each other out, creating a more stable and well-rounded meal, or in this case, a more stable and potentially rewarding investment portfolio. Understanding portfolio variance and covariance is key to building a resilient and diversified investment strategy.