What Is Covariance?

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What Is Covariance?

Let me explain covariance directly: it's a statistical measurement that shows the relationship between two variables, and in finance, I usually see it applied to the returns of two assets. As a tool, covariance tells you how much these variables change together. If you get a positive covariance, the asset returns are moving in the same direction; a negative one means they're going opposite ways. You calculate it by looking at standard deviations from expected returns or by multiplying the correlation between the variables by each one's standard deviation.

Key Takeaways

Covariance calculates the relationship between two random variables, so when two stocks move together, they have positive covariance, and when they move inversely, it's negative. Remember, this is different from the correlation coefficient, which measures the strength of that relationship. In modern portfolio theory, covariance helps decide which securities belong in a portfolio. You can reduce risk and volatility by pairing assets with negative covariance.

Understanding Covariance

Covariance looks at how the mean values of two random variables move together. Take stocks, for instance: if stock A's return goes up whenever stock B's does, and the same when they drop, that's positive covariance. In finance, we use covariances to diversify holdings. Beyond finance, you'll find covariance in statistics and machine learning too.

Formula for Covariance

When you have price data from a stock or fund, use this formula to calculate covariance: Covariance = Σ [(Ret_abc - Avg_abc) × (Ret_xyz - Avg_xyz)] / (Sample Size - 1), where Ret_abc is the day's return for ABC stock, Avg_abc is ABC's average return over the period, Ret_xyz is for XYZ stock, Avg_xyz is its average, and Sample Size is the number of days sampled. This setup lets you compute it step by step with your data.

Types of Covariance

The covariance equation determines if two variables move in the same or opposite directions, based on whether the value is positive or negative. For positive covariance, the variables tend to be higher or lower together—if stock one is above average when stock two is, and vice versa, the data points slope upward on a graph. Negative covariance means an inverse relationship: when one is below average, the other is above, and so on.

Applications of Covariance

Covariance plays a big role in finance and modern portfolio theory. In the capital asset pricing model, for example, it helps calculate beta, which measures a security's volatility against the market—drawing from covariance to assess risk. Portfolio theory uses it to lower overall risk through diversification informed by covariance. If assets have similar covariances, that doesn't diversify much, so mix in varying ones for better protection against volatility.

Covariance vs. Variance

Covariance relates to variance, which measures the spread of data points around a mean. Variance looks at one axis, while covariance checks the directional tie between two variables. In investing, covariance shows how assets perform relative to each other—positive means they do well together, negative means opposites, which investors use for diversification.

Covariance vs. Correlation

Covariance differs from correlation: it measures direction, but correlation measures strength via a coefficient from -1 to +1. Covariance doesn't indicate strength; for that, use correlation. A coefficient near +1 or -1 shows a strong link, while near zero means a weak one.

Example of Covariance Calculation

Let's walk through an example with four days of returns for stocks ABC and XYZ: Day 1: 1.2% and 3.1%; Day 2: 1.8% and 4.2%; Day 3: 2.2% and 5.0%; Day 4: 1.5% and 4.2%. First, find averages: ABC 1.675%, XYZ 4.125%. Then compute deviations and products for each day, sum them to 0.943, and divide by 3 (sample size minus 1) to get 0.314. This positive covariance suggests similar return patterns.

What Does a Covariance of 0 Mean?

A zero covariance means no clear directional relationship—the variables could pair high with high or low just as easily.