 # Variance of Return By Yuriy Smirnov Ph.D.

## Definition

Variance is a metric used in statistics to estimate the squared deviation of a random variable from its mean value. In portfolio theory, the variance of return is the measure of risk inherent in investing in a single asset or portfolio. In other words, the higher the variance, the greater the squared deviation of return from the expected rate of return. The higher values indicate a greater amount of risk, and low values mean a lower inherent risk.

The Greek symbol used to designate the variance is σ2 “squared sigma.” It can also be referred to as Var (X), where X is a random variable.

## Formula

### Probability Approach

The probability approach is used when there is a complete set of possible outcomes. In other words, the probability distribution for the return on a single asset or portfolio is known in advance. The equation of variance can be written as follows: 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 commonly used in the practice of investing. It uses the finite data set of a historical investment’s returns and assumes that each possible outcome has the same probability. Thus, the variance of return on a single asset or portfolio can be estimated as follows: where N is the size of the entire population.

Using the formula above assumes that a data set represents the entire population, but in many practical situations a sample of the population is used instead of the entire population. Therefore, sample variance is used to estimate the variance of the entire population: where ERRS is the expected rate of return of a sample or sample mean, and N is the size of the sample.

## Calculation Examples

### Example 1

The asset manager of a mutual fund is estimating the possibility of adding two securities in a portfolio. Security A has five possible scenarios. The return of Security B has three possible outcomes. The first step in computing the variance of return starts by calculating the expected rate of return for each security and then computing the squared deviation from the expected rate of return (ERR) under each scenario.

ERR of Security A = 0.10×(-8%) + 0.15×(-2%) + 0.40×4% + 0.30×10% + 0.05×16% = 4.30%

ERR of Security B = 0.25×(-10%) + 0.45×7% + 0.30×25% = 8.15%

The squared deviations from the expected rate of return for Security A are as follows:

Scenario 1 = (-8% - 4.3%)2 = 151.29

Scenario 2 = (-2% - 4.3%)2 = 39.69

Scenario 3 = (4% - 4.3%)2 = 0.09

Scenario 4 = (10% - 4.3%)2 = 32.49

Scenario 5 = (16% - 4.3%)2 = 136.89

The second step is to weigh each squared deviation by its probability.

σ2 Security A = 151.29×0.10 + 39.69×0.15 + 0.09×0.40 + 32.49×0.30 + 136.89×0.05 = 37.71

Let’s perform the same calculation for Security B.

Scenario 1 = (-10% - 8.15%)2 = 329.42

Scenario 2 = (-7% - 8.15%)2 = 1.32

Scenario 3 = (25% - 8.15%)2 = 283.92

σ2 Security B = 329.42×0.25 + 1.32×0.45 + 283.92×0.30 = 168.13

As we can see, Security A has a lower expected rate of return than Security B but also has lower inherent risk as was measured by variance of return.

### Example 2

The mutual fund manager needs to estimate three securities. Their performance during the last five years is shown in the table below. The historical approach assumes that the expected rate of return is an average value.

ERR of Security A = (2.3% + 8.0% + (-3.0%) + 5.5% + 4.1%) ÷ 5 = 3.38%

ERR of Security B = (6.7% + 2.1% + 4.0% + (-2.0%) + 5.1%) ÷ 5 = 3.18%

ERR of Security C = (-3.2% + 1.5% + 7.5% + (-1.0%) + 2.2%) ÷ 5 = 1.40%

The squared deviations from the expected rate of return of each security are shown in the table below. Finally, the estimated sample variance of return of each security is as follows:

σ2 Security A = 68.23 ÷ (5-1) = 17.06

σ2 Security B = 44.75 ÷ (5-1) = 11.19

σ2 Security C = 64.78 ÷ (5-1) = 16.20