The Theory and Practice of Investment Management: Asset Allocation, Valuation, Portfolio Construction, and Strategies, Second Edition

MEASURING PORTFOLIO RISK

Investors have used a variety of definitions to describe risk. Harry Markowitz quantified the concept of risk using the well-known statistical measure: the standard deviation and the variance. The former is the intuitive concept. Most of any probability distribution is between its average plus or minus two standard deviations. Variance is standard deviation squared. Computations are simplest in terms of variance. Therefore, it is convenient to compute the variance of a portfolio and then takes its square root to obtain standard deviation.¹³

Variance and Standard Deviation as a Measure of Risk

The variance of a random variable is a measure of the dispersion or variability of the possible outcomes around the expected value (mean). In the case of an asset’s return, the variance is a measure of the dispersion of the possible rate of return outcomes around the expected return.

The equation for the variance of the expected return for asset i, denoted var(R_i), is

var(R_i) = p₁[r₁–E(R_i)]² + p₂[r₂–E(R_i)]² + . . . + p_N[r_N–E(R_i)]²

or

(3.5)

Using the probability distribution of the return for stock XYZ, we can illustrate the calculation of the variance:

The variance associated with a distribution of returns measures the tightness with which the distribution ...