## What is ‘Variance’

Variance is a measurement of the spread between numbers in a data set. The variance measures how far each number in the set is from the mean. Variance is calculated by taking the differences between each number in the set and the mean, squaring the differences (to make them positive) and dividing the sum of the squares by the number of values in the set.

## Explaining ‘Variance’

Variance is used in statistics for probability distribution. Since variance measures the variability (volatility) from an average or mean and volatility is a measure of risk, the variance statistic can help determine the risk an investor might take on when purchasing a specific security. A variance value of zero indicates that all values within a set of numbers are identical; all variances that are non-zero will be positive numbers. A large variance indicates that numbers in the set are far from the mean and each other, while a small variance indicates the opposite.

## Variance in Investing

Variance is one of the key parameters in asset allocation. Along with correlation, variance of asset returns helps investors to develop optimal portfolios by optimizing the return-volatility trade-off in investment portfolios. Risk or volatility is often expressed as a standard deviation rather than variance because the former is more easily interpreted.

## Example of Variance

Returns for a stock are 10% in year 1, 20% in year 2 and -15% in year 3. The average of these three returns is 5%. The differences between each return and the average are 5%, 15%, and -20% for each consecutive year. Squaring these deviations yields 25%, 225% and 400%, respectively; summing these squared deviations gives 650%. Dividing the sum of 650% by the number of returns in the data set (3 in this case) yields the variance of 216.67%. Taking the square root of the variance yields the standard deviation of 14.72% for the returns.

## Further Reading

- Finiteness of variance is irrelevant in the practice of quantitative finance – onlinelibrary.wiley.com [PDF]
- In search of the exchange risk premium: A six-currency test assuming mean-variance optimization – www.sciencedirect.com [PDF]
- Prices and asymptotics for discrete variance swaps – www.tandfonline.com [PDF]
- Markowitz's mean-variance asset–liability management with regime switching: A multi-period model – www.tandfonline.com [PDF]
- The effect of jumps and discrete sampling on volatility and variance swaps – www.worldscientific.com [PDF]
- Testing and locating variance changepoints with application to stock prices – www.tandfonline.com [PDF]
- The origins of the mean-variance approach in finance: revisiting de Finetti 65 years later – link.springer.com [PDF]
- Are Asian stock markets efficient? Evidence from new multiple variance ratio tests – www.sciencedirect.com [PDF]
- Valuing Volatility and Variance Swaps for a Non‐Gaussian Ornstein–Uhlenbeck Stochastic Volatility Model – www.tandfonline.com [PDF]
- Variance-optimal hedging for time-changed Lévy processes – www.tandfonline.com [PDF]