**2.9 PORTFOLIO STATISTICS**

A portfolio of financial securities can be represented by a collection of N numbers, denoted as:

where each number a_{i} is the proportion of wealth invested in security *i*. Sometimes, a_{i} is called the **share** of security *i* in the portfolio. The sum of these shares must equal one. Here, we show you how to calculate the expected returns and variance of a portfolio.

If *x* is a random variable, then so is any multiple of *x* by some number a. We illustrate this computation for the population mean and population variance of a*x*:

Similarly, for some other random variable *y*, and any number b, then b*y* is also a random variable with

.

Finally, for the sum of any two random variables a *x* + b *y*, we can calculate the expectation and variance as follows:

These last two formulas are very useful when calculating the expected returns and variances of portfolios.

We describe risk and return characteristics of a portfolio in Summary of Portfolio Statistics, followed by examples.

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