How to Calculate the Variance of Returns

The variance of returns is a fundamental statistical measure in finance, quantifying the dispersion or spread of a set of returns around their average value. It provides insight into how much individual returns deviate from the central tendency. Understanding this measure helps in assessing the consistency of an asset’s or portfolio’s performance over time.

This statistical tool is widely used to analyze the historical behavior of investments. By calculating variance, financial professionals can gain a quantitative understanding of past performance fluctuations. It reflects the degree to which an asset’s returns have historically moved away from its average return. Variance is an important component in the broader framework of financial analysis.

Required Data for Calculation

Calculating the variance of returns necessitates specific historical data related to the asset or portfolio. The primary requirement is a series of individual returns recorded over a defined period, such as daily, weekly, monthly, or annual figures. Each individual return reflects the percentage change in the asset’s value over its respective period.

In addition to individual returns, the total count of these observations, denoted as ‘N’, is also a necessary input. This number represents the sample size of the historical data. For instance, if monthly returns for a stock over five years are being considered, there would be 60 individual return observations.

Historical return data can be sourced from various financial databases. Publicly available company reports, such as annual filings, also contain information from which returns can be derived. Furthermore, historical stock prices, available online, can be used to compute the individual returns required for this calculation.

Determining the Average Return

Before calculating the variance, it is necessary to establish the average, or mean, return of the dataset. This average return represents the central point around which individual returns are expected to fluctuate. The variance calculation specifically measures deviations from this central tendency, making the average return an indispensable preliminary step.

The formula for calculating the arithmetic mean of a series of returns involves summing all individual returns in the dataset. This sum is then divided by the total number of observations, ‘N’, in the dataset.

For example, if you have five monthly returns, you would add these five percentages together. The resulting sum is then divided by five to yield the average monthly return. This calculation provides the baseline against which the spread of individual returns will be measured.

The Variance Calculation Steps

Once the average return has been determined, the calculation of variance proceeds through several steps. First, compute the deviation of each individual return from the average return by subtracting the average from each observation in your dataset. These deviations indicate how far each data point lies from the central tendency.

Next, square each of these deviations. Squaring the differences serves two main purposes: it eliminates any negative values, ensuring that all deviations contribute positively to the total, and it amplifies larger deviations, giving them more weight in the final variance figure.

After squaring each deviation, sum all the squared deviations. To arrive at the variance, divide this sum of squared deviations by ‘N-1’, where ‘N’ is the total number of observations. For historical financial returns, which are typically considered a sample of a larger population, using ‘N-1’ provides a more accurate, unbiased estimate of the population variance.

The complete formula for sample variance (s²) is expressed as: s² = Σ(Xi – X̄)² / (N – 1), where Xi represents each individual return, X̄ is the average return, and N is the number of observations. This formula systematically captures the average of the squared differences from the mean.

A Practical Calculation Example

Consider a hypothetical set of monthly returns for an investment over five months: 2%, -1%, 3%, 0%, and 4%. The first step in calculating variance is to determine the average return for this dataset. Summing these returns (2 + (-1) + 3 + 0 + 4 = 8%) and dividing by the number of observations (5) yields an average return of 1.6% (8% / 5).

Next, calculate the deviation of each return from this average and then square each deviation. For the first return of 2%, the deviation is 2% – 1.6% = 0.4%, and squaring this yields 0.0016. For -1%, the deviation is -1% – 1.6% = -2.6%, and squaring results in 0.000676. The return of 3% has a deviation of 3% – 1.6% = 1.4%, which squares to 0.000196.

The 0% return results in a deviation of 0% – 1.6% = -1.6%, squaring to 0.000256. Finally, the 4% return has a deviation of 4% – 1.6% = 2.4%, and its square is 0.000576. The next step involves summing these squared deviations: 0.0016 + 0.000676 + 0.000196 + 0.000256 + 0.000576 = 0.003304.

To complete the variance calculation, divide this sum by ‘N-1’. Since there are 5 observations, ‘N-1’ is 4. Therefore, the variance is 0.003304 / 4, which equals 0.000826. This final numerical value represents the variance of the given set of returns.

Interpreting the Variance Value

The calculated variance value quantifies the spread or volatility of an asset’s returns around its average. It provides a numerical representation of how much individual returns tend to deviate from the mean return. A higher variance indicates that the individual returns are widely dispersed from the average, suggesting greater historical volatility or inconsistency in performance.

Conversely, a lower variance implies that the individual returns are clustered more closely around the average. This suggests a more consistent and less volatile historical performance for the asset. Variance serves as a direct indicator of the degree of fluctuation an investment has experienced.

While variance provides a measure of dispersion, its unit is the square of the original return unit, which can make direct interpretation challenging. For easier understanding, the standard deviation is often preferred, as it is simply the square root of the variance. Standard deviation is expressed in the same units as the original returns, making it more intuitive to interpret the extent of typical deviation.