investment risk in Stock Investing: Variance, Covariance, and Portfolio Volatility

Measuring risk in the stock market requires more than intuition. It demands quantitative tools that translate the unpredictability of daily price movements into numbers that can be compared, analysed, and acted upon. Two such tools, variance and covariance, form the backbone of modern portfolio risk assessment. This chapter introduces variance first, before moving on to covariance and ultimately to portfolio volatility.

Variance

Variance is a statistical measure that captures how widely a stock’s daily returns are spread around their average. In simple terms, it answers the question: how much does this stock’s daily performance deviate from what is typical for it? A stock that returns almost exactly the same amount each day has very low variance. A stock whose daily returns swing sharply in either direction has high variance, and by extension, higher risk.

The formula for variance is as follows.

Sigma squared = Sum of (X minus mu) squared, divided by N

Where sigma squared represents variance, X represents each daily return, mu represents the average of all daily returns, and N represents the total number of observations.

To illustrate this, consider the following five consecutive days of returns for a hypothetical stock.

Day 1: +0.75%. Day 2: +1.25%. Day 3: -0.55%. Day 4: -0.75%. Day 5: +0.80%.

The average daily return across these five days is +0.30%. To calculate variance, the deviation of each day’s return from this average is computed and then squared. Squaring the deviations serves two purposes: it eliminates negative values, and it amplifies larger deviations so they carry proportionately greater weight in the final figure.

The sum of all squared deviations in this example comes to 0.0318000 per cent. Dividing this by five, the total number of observations, produces the variance.

0.0318000 divided by 5 = 0.00636000 per cent

This figure tells an investor how much the stock’s daily returns tend to stray from their average. A large variance signals greater unpredictability and therefore greater risk. A smaller variance suggests the stock’s returns are more stable and consistent. With only five days of data, the variance calculated here would be considered relatively high, as a small sample size tends to amplify observed deviations.

Variance and Standard Deviation

Variance and standard deviation are directly related through a simple mathematical step. The standard deviation of a set of returns is simply the square root of the variance. This conversion is useful because standard deviation is expressed in the same units as the original returns, making it more intuitive to interpret.

Applying this to the example above, the square root of the five-day variance yields a standard deviation of approximately 0.8 per cent. This means that on a typical day, the stock’s return deviated from its average by roughly 0.8 per cent in either direction.

Standard deviation is also commonly referred to as volatility in the context of financial markets. When a financial advisor or market commentator describes a stock as highly volatile, they are, in quantitative terms, describing a stock with a high standard deviation of returns.

Why Variance Matters for Equity Investment

Variance is not merely an academic exercise. It serves as a practical input into some of the most important calculations in portfolio construction, including the portfolio variance equation, which measures the overall risk of a combination of stocks rather than any single holding in isolation.

The significance of variance extends further when one considers that different stocks held together in a portfolio do not behave independently. Their returns interact with one another in ways that can either amplify or reduce overall portfolio risk. Understanding how individual variances combine requires an additional concept, covariance, which measures the degree to which two stocks move in relation to each other.

Both variance and covariance feed into the portfolio variance equation, which ultimately determines the standard deviation, or volatility, of a portfolio as a whole. This figure is one of the most important metrics any investor engaged in equity investment can monitor, and it will be examined in detail in the chapters that follow.

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