Variance

Variance is a statistical measure of the dispersion of observations around their mean. It measures the average squared deviation from the mean.

In finance, variance is commonly used to measure the variability of investment returns and is an important component of volatility, portfolio risk, and portfolio theory.

Population variance

For a population of $N$ observations, variance is defined as:

${\sigma }^2=\frac{1}{N}\sum _{i=1}^N(x_i-\mu {)}^2$

where:

• ${\sigma }^2$ = population variance;
• $x_i$ = individual observation;
• $\mu$ = population mean;
• $N$ = number of observations.

The population mean is:

$\mu =\frac{1}{N}\sum _{i=1}^N{x}_i$

Sample variance

When observations represent a sample from a larger population, the sample variance is commonly calculated as:

$s^2=\frac{1}{n-1}\sum _{i=1}^n(x_i-\overline{x}{)}^2$

where:

• $s^2$ = sample variance;
• $x_i$ = individual observation;
• $\overline{x}$ = sample mean;
• $n$ = number of observations.

The use of $n-1$ provides the commonly used unbiased estimator of population variance under standard assumptions.

Variance of investment returns

In finance, variance can be calculated from a series of periodic investment returns.

A simplified formula is:

$s_R^2=\frac{1}{n-1}\sum _{t=1}^n(R_t-\overline{R}{)}^2$

where:

• $s_R^2$ = sample variance of returns;
• $R_t$ = return during period $t$;
• $\overline{R}$ = average return;
• $n$ = number of return observations.

A higher return variance indicates greater dispersion of returns around their average.

Variance and standard deviation

Standard deviation is the square root of variance:

$\sigma =\sqrt{{\sigma }^2}$

Equivalently:

${\sigma }^2=(\sigma {)}^2$

Variance is expressed in squared units, whereas standard deviation is expressed in the same units as the original observations.

For example, if returns are measured in percentage points, variance is expressed in squared percentage points while standard deviation is expressed in percentage points.

Alternative representation

Variance can also be expressed as the difference between the expected value of the squared variable and the square of its expected value:

$\mathrm{Var}(X)=E(X^2)-[E(X){]}^2$

This identity is equivalent to the definition based on squared deviations from the mean.

Variance and covariance

Variance is a special case of covariance in which the two variables are the same:

$\mathrm{Var}(X)=\mathrm{Cov}(X,X)$

For two variables:

$\mathrm{Cov}(X,Y)=E[(X-E[X])(Y-E[Y])]$

Thus, variance measures the dispersion of one variable, while covariance measures the joint variation of two variables.

Variance and correlation

Variance is related to correlation through covariance:

${\rho }_{XY}=\frac{\mathrm{Cov}(X,Y)}{{\sigma }_X{\sigma }_Y}$

Since:

${\sigma }_X=\sqrt{\mathrm{Var}(X)}$

and:

${\sigma }_Y=\sqrt{\mathrm{Var}(Y)}$

correlation can also be written as:

${\rho }_{XY}=\frac{\mathrm{Cov}(X,Y)}{\sqrt{\mathrm{Var}(X)\mathrm{Var}(Y)}}$

Portfolio variance

Variance is an important measure of portfolio risk because the variance of a portfolio depends on both the variances of individual assets and their covariances.

For a two-asset portfolio:

${\sigma }_p^2=w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2+2w_1{w}_2\mathrm{Cov}(R_1,R_2)$

where:

• ${\sigma }_p^2$ = portfolio variance;
• $w_1,w_2$ = portfolio weights;
• ${\sigma }_1^2,{\sigma }_2^2$ = individual asset variances;
• $\mathrm{Cov}(R_1,R_2)$ = covariance between the two asset returns.

Using correlation, the same equation can be written as:

${\sigma }_p^2=w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2+2w_1{w}_2{\sigma }_1{\sigma }_2{\rho }_{12}$

The covariance term means that portfolio risk cannot generally be determined by simply averaging the individual asset variances.

Covariance matrix representation

For a portfolio containing multiple assets, portfolio variance can be expressed using a covariance matrix:

${\sigma }_p^2={𝐰}^{𝖳}𝜮𝐰$

where:

• $𝐰$ = vector of portfolio weights;
• $𝜮$ = covariance matrix of asset returns;
• ${\sigma }_p^2$ = portfolio variance.

For a two-asset portfolio, the covariance matrix is:

$𝜮=\left[\begin{array}{cc}{\sigma }_1^2& \mathrm{Cov}(R_1,R_2)\\ \mathrm{Cov}(R_2,R_1)& {\sigma }_2^2\end{array}\right]$

Variance and diversification

Variance is central to diversification because combining assets with different covariance characteristics can reduce portfolio variance.

For two assets, the diversification effect depends partly on their correlation:

${\rho }_{12}<1\Rightarrow \text{Potential Portfolio Risk Reduction}$

The greatest potential reduction from correlation alone occurs when the correlation approaches −1, although the actual portfolio risk also depends on the assets' individual volatilities and portfolio weights.

Annualized variance

When returns are independently and identically distributed, variance can be scaled across time.

If the variance for one period is ${\sigma }_p^2$ and there are $N$ periods in a year:

${\sigma }_{\text{annual}}^2=N{\sigma }_p^2$

Correspondingly, standard deviation scales as:

${\sigma }_{\text{annual}}={\sigma }_p\sqrt{N}$

The square-root-of-time relationship requires assumptions about the return process and should not be applied automatically to all financial time series.

Limitations

Variance has several limitations as a financial risk measure:

• it treats positive and negative deviations equally;
• it is sensitive to extreme observations;
• it is expressed in squared units;
• historical variance may not represent future variance;
• it does not directly describe the probability of a particular loss;
• it does not distinguish between temporary and persistent changes in risk.

Because of these limitations, variance is often considered together with measures such as Standard deviation, Value at risk, Expected shortfall, Maximum drawdown, and Downside risk.

See also

• Standard deviation
• Covariance
• Correlation
• Pearson correlation coefficient
• Volatility (finance)
• Portfolio (finance)
• Diversification (finance)
• Modern portfolio theory
• Risk management
• Value at risk
• Expected shortfall
• Downside risk