Subject 7. The Univariate and Multivariate Distributions

To this point, the focus has been on distributions that involve only one variable, such as the binomial, uniform, and normal distributions. A univariate distribution describes a single random variable. For example, suppose that you would like to model the distribution of the return on an asset. Such a distribution is a univariate distribution.

A multivariate distribution specifies the probabilities for a group of related random variables. It is used to describe the probabilities of a group of continuous random variables if all of the individual variables follow a normal distribution.

Each individual normal random variable would have its own mean and its own standard deviation, and hence its own variance. When you are dealing with two or more random variables in tandem, the strength of the relationship between (or among) the variables assumes huge importance. You will recall that the strength of the relationship between two random variables is known as the correlation.

When there is a group of assets, the distribution of returns on each asset can either be modeled individually or on the assets as a group. A multivariate normal distribution for the returns on n stocks is completely defined by three lists of parameters:

  • The list of the mean returns on the individual securities (n means in total).
  • The list of the securities' variances of return (n variances in total).
  • The list of all the distinct pairwise return correlations (n(n-1)/2 distinct correlations in total).

The higher the correlation values, the higher the variance of the overall portfolio. In general, it is better to build a portfolio of stocks whose prices are not strongly correlated with each other, as this lowers the variance of the overall portfolio.

It is the correlation values that distinguish a multivariate normal distribution from a univariate normal distribution. Consider a portfolio consisting of 2 assets (n = 2). The multivariate normal distribution can be defined with 2 means, 2 variances, and 2 x (2-1)/2 = 1 correlation. If an analyst has a portfolio of 100 securities, the multivariate normal distribution can be defined with 100 means, 100 variances, and 100 x (100 - 1)/2 = 4950 correlations. Portfolio return is a weighted average of the returns on the 100 securities. A weighted average is a linear combination. Thus, portfolio return is normally distributed if the individual security returns are (joint) normally distributed. In order to specify the normal distribution for portfolio return, analysts need means, variances, and the distinct pairwise correlations of the component securities.

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