# How to calculate the Variance of Returns?

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## What is Variance?

Variance is a metric that is needed to estimate the squared deviation of any random variable from the mean value. In the portfolio theory, the variance of return is called the measure of risk inherent in a singular or in an asset of portfolios.

In general, the higher the value of variance, the bigger is the squared deviation of return of the given portfolio from the expected rate. The higher values show a larger risk, and low values indicate a lower inherent risk.

## Formula: How to Calculate Variance

We have two different approaches to calculate the variance of returns −

• Probability Approach
• Historical Return Approach

## Probability Approach

The probability approach for determining variance is used when the complete set of possible outcomes is available. This means the probability distribution of the asset or portfolio is known in advance.

The equation of variance formula in the Probability approach can be written as follows −

$$\mathrm{\sigma^2 =\displaystyle\sum\limits_{i=1}^n {}{(𝑟𝑖 − ERR)^2 × p_{𝑖}}}$$

Where,

• 𝑟𝑖 is the rate of return achieved at ith outcome,

• ERR is the expected rate of return,

• 𝑝𝑖 is the probability of ith outcome, and

• n is the number of possible outcomes.

## Historical Return Approach

The historical return approach is more generally used in investing and finance. Using finite data set of the history of the investment in an asset or a portfolio, the return is calculated with assumptions that each possible outcome has the same probability. Thus, the variance of return on a single asset or portfolio is measured as −

$$\mathrm{\sigma^2 =\frac{\sum_{\substack{i=1}}^n {(𝑟𝑖 − ERR)^2}}{N}}$$

where N is the size of the entire population.

The above formula considers the idea that a dataset represents the entire population, but in numerous practical situations, a sample of the given population is used instead of the entire population which may be very large. Therefore, a sample variance is an estimation of the variance of the entire population −

$$\mathrm{{\sigma_{\substack{s}}^2}=\frac{\sum_{\substack{i=1}}^n {(𝑟𝑖 − ERR_{s})^2}}{N− 1}}$$

where ERRS is the expected rate of return of a sample or sample mean, and N is the size of the sample.

## Favorable vs. Unfavorable Variance

As variance analysis is done for both revenues and expenses, it’s crucial to carefully distinguish between the two sides of impacts – the positive or negative impact.  For this reason, the terms favorable and unfavorable are used instead of saying positive, negative, over or under, etc., as they make the point clearer.

For example, if a cost has higher than expected difference to the forecast, that’s an unfavorable variance as it’s worse to have costs higher.

## Variance in Budgeting and Forecasting

The variance formula is very useful in budgeting and forecasting. It gives a clear picture when analyzing results.  It helps the financial analyst to perform his duties appropriately and with utmost accuracy.

The role of Financial Planning & Analysis (FP&A) department is to provide the management with accurate, timely, and insightful information so that managers can take proactive decisions about the company. Working with variance is therefore crucial for the FP&A department.

Updated on 17-Sep-2021 08:51:05