The Rules of Expectations

As promised, here are some rules for working with expected values, first in words and then in math. All of these can be verified in excel. I’ll also include a list of things that are not rules for working with expectations. These are all especially relevant in valuation if you have multiple uncertain parameters you are combining through probabilities. These also come into play while making total rows at the bottom of complex schedules. Sometimes they won’t foot, and this post will help you understand why.

The expected value of a constant is a constant.

E[c] = c

The expected value of two random values is equal to the sum of the expected values.

E[X + Y] = E[X] + E[Y]

The expected value of a random variable times a constant is equal to the constant times the expected value of the random variable.

E[cX] = cE[X]

Please note that this doesn’t hold if c is not a constant.

E[XY] \neq E[X]E[Y]

Remember these rules, as you will be tested on them.