Overview

One common measure of portfolio risk, pioneered by Harry Markowitz, is the portfolio standard deviation of returns. (see variance for information about the statistical measure of variance and covariance)

Measuring Portfolio Standard Deviation

There are several different methods that can be employed to measure the standard deviation of asset returns.

• Portfolio as Time Series measures portfolio variance using standard time series methods.
• Using Asset Covariance Matrix : measures the volatilities and correlations between assets in a portfolio and then aggregates the result.
• Single Index Model : The single index model seeks to simplify the measurement of asset covariances by assuming that the correlation between assets is driven by a single risk factor, such as the market risk.
  {% r_i = \alpha_i + \beta_i \times r_m + e_i %}
  {% r_i %} is the rate of return of a given asset and {% r_m %} is the return on the market. The Capital Asset Pricing Model is an example of a single index model.
• Multi Index Model
• Arbitrage Pricing Theory/Risk Factor Approach : The arbitrage pricing theory is an extension of the CAPM approach that hypothesizes that the market is driven by any number of factors, as opposed to a single systemic risk factor.

Model Portfolios

There are several portfolio construction techniques that utilize the standard deviation of returns as an input.

• Minimum Variance Portfolio : The minimum variance portfolio is the portfolio with cash fully invested that has a minimum variance among all such portfolios. It is not a typical portfolio that firms invest in, but is instructive in its construction and can be used as a benchmark.
• Active Portfolio : An active portfolio is a portfolio that has assets weights that differ from a given benchmark. Such portfolios are typically measured with respect to the benchmark portfolio.
• Mean Variance Portfolio :
• Growth Variance Portfolio :

Additional Topics

• Equity Models - equity models are the primary area where risk as standard deviation is applied. There are a multitude of different factor models that are used to measure equity risk.
• As a measure of risk, the standard deviation is often used to risk adjust a set of measured returns. (see risk adjusted performance)
• Stochastic Market Model