Standard deviation

Standard deviation is a statistical measure of the dispersion of observations or returns around their mean. In finance, it is commonly used to measure the variability of an asset's returns and is often used as a measure of total risk or volatility.

A higher standard deviation indicates that observations are more widely dispersed around the mean, while a lower standard deviation indicates that observations are more closely concentrated around the mean.

Population standard deviation

For a population of $N$ observations, the standard deviation is:

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

where:

• $\sigma$ = population standard deviation;
• $x_i$ = individual observation;
• $\mu$ = population mean;
• $N$ = number of observations.

The population mean is:

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

Sample standard deviation

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

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

where:

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

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

Standard deviation of investment returns

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

A simplified formula is:

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

where:

• ${\sigma }_R$ = standard deviation of returns;
• $R_t$ = return during period $t$;
• $\overline{R}$ = average return;
• $n$ = number of return observations.

The measure indicates how widely individual returns vary around their average return.

Example

Suppose an investment has three annual returns:

$10\mathrm{\%},5\mathrm{\%},15\mathrm{\%}$

The average return is:

$\overline{R}=\frac{10\mathrm{\%}+5\mathrm{\%}+15\mathrm{\%}}{3}=10\mathrm{\%}$

The deviations from the mean are:

$0\mathrm{\%},-5\mathrm{\%},5\mathrm{\%}$

The sample variance is:

$s^2=\frac{(0\mathrm{\%}{)}^2+(-5\mathrm{\%}{)}^2+(5\mathrm{\%}{)}^2}{3-1}=25$

Using percentage points, the sample standard deviation is:

$s=\sqrt{25}=5\mathrm{\%}$

Standard deviation and volatility

In financial analysis, standard deviation of returns is commonly used as a measure of historical volatility.

A simplified relationship is:

$\text{Volatility}\approx \text{Standard Deviation of Returns}$

However, standard deviation and volatility are not necessarily interchangeable in every context. Volatility can refer more broadly to different statistical or market-based measures of price variability.

Annualized standard deviation

When standard deviation is calculated from periodic returns, it can be annualized when appropriate.

If ${\sigma }_p$ is the standard deviation of returns for a period and there are $N$ such periods in a year, the annualized standard deviation is:

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

For example, when using daily returns and assuming 252 trading days per year:

${\sigma }_{\text{annual}}={\sigma }_{\text{daily}}\sqrt{252}$

The square-root-of-time relationship assumes conditions under which returns are appropriately aggregated in this manner.

Standard deviation and portfolio risk

Portfolio standard deviation depends on the standard deviations of individual assets, their portfolio weights, and the correlations between their returns.

For a two-asset portfolio:

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

where:

• ${\sigma }_p$ = portfolio standard deviation;
• $w_1,w_2$ = portfolio weights;
• ${\sigma }_1,{\sigma }_2$ = standard deviations of the two assets;
• ${\rho }_{12}$ = correlation between the two assets.

The correlation term means that combining assets with less-than-perfect positive correlation can reduce portfolio standard deviation.

Standard deviation and risk-adjusted return

Standard deviation is used in several financial performance measures.

For example, the Sharpe ratio uses the standard deviation of portfolio returns as a measure of total risk:

$\text{Sharpe Ratio}=\frac{R_p-R_f}{{\sigma }_p}$

where:

• $R_p$ = portfolio return;
• $R_f$ = risk-free rate;
• ${\sigma }_p$ = standard deviation of portfolio returns.

The Information ratio similarly uses the standard deviation of active returns:

$\text{Information Ratio}=\frac{R_p-R_b}{\sigma (R_p-R_b)}$

where $R_b$ is the benchmark return.

Limitations

Standard deviation has several limitations when used as a measure of financial risk.

It:

• treats positive and negative deviations equally;
• does not distinguish between upside and downside volatility;
• may not adequately capture extreme tail events;
• can be sensitive to the measurement period;
• relies on historical observations when calculated from historical returns;
• may not fully represent changing market conditions.

For this reason, standard deviation is often considered alongside other measures such as beta, Value at risk, Expected shortfall, Maximum drawdown, and Downside risk.

See also

• Variance
• Mean
• Volatility (finance)
• Risk-adjusted return
• Sharpe ratio
• Information ratio
• Sortino ratio
• Beta (finance)
• Correlation
• Covariance
• Portfolio (finance)
• Risk management
• Value at risk
• Expected shortfall