What is Expected Value?

A random variable is a value which takes different values with certain probabilities. Per math convention, random variables are capital letters while values random variables may equal have lower case letters. Deeper questions like “what is probability” may be addressed in a future post.

Suppose you have a random variable X that can take any of the following values:


Working through the first line, there is a 75\% chance that X will equal \$ 10. The product of these two figures, \$ 7.50 is known as a “partial expectation.” The sum of all partial expectations, \$ 13.50 is known as the expected value of X, or E[X].

For discrete random variables (which can take a finite number or countably infinite number of values), this is denoted.

E[X] = \sum_{i} p_i \cdot x_i

Where p_i is the probability of scenario i occurring, and x_i is the value of scenario i. Examples of discrete random variables include the number of days a stock will increase in a row, the number of deposits in a bank account in a month or the sum of two rolled dice. Note that the first two are theoretically infinite.

For continuous random variables (which can take an infinite number of values), expected value is denoted as:

E[X] = \int\limits_{-\infty}^{\infty} x \cdot f(x) dx

Where f(x) = P(X = x) (the probability X = x. Examples of continuous random variables include the earnings of a company, the dollar value of deposits in a month or the time until someone in a family flips over a Monopoly board. Note that the lattermost is only theoretically infinite.

So that’s a mathematical explanation of expected value. What’s an intuitive explanation?

The expected value of X is the weighted average of x across all possible scenarios where the weights are based on the probability of a scenario occurring. This isn’t strictly accurate for the continuous case (since the probability of any specific individual outcome occurring is zero), but the intuition still applies.

Expected values are important because if you simulate X an infinite number of times, then you will average a return of E[X]. This is important for valuation because if you believe a cash flow will be worth X, then you will break even in the long run if you pay E[X] for it. If you need to make a profit to compensate yourself for opportunity cost, then you will have to pay less than E[X].

In modern finance, diversification and securitization have made expected value an increasingly important concept since it is far easier to own a large number of assets. However, as far as valuation is concerned, it is worth pointing out that expected value is but the first “moment” that can be used to describe a distribution. Higher moments (such as variance, skewness and kurtosis) play an important role, especially in financial contexts when diversification is limited.

When valuing companies, it is standard practice to develop “best, worst, likely” scenarios and subjectively determine probabilities for each outcome. The company is then valued as the expected value of each of the three discounted cash flows. This is valid as long as the discount rate adequately covers the higher moment concerns I alluded to above.

Next, you can expect to see a post on the mathematical rules of working with expectations.

The Math Constant e

In my prior discussion of log returns, I made an offhand reference to the math constant e. Do you remember that kid who memorized 100 digits of \pi because that’s what he thought smart people did? He should have memorized 100 digits of e, because it’s far cooler than \pi. Actually, he should have practiced programming and memorized nothing, but I digress.

I think it’s worth taking the time to explain e in detail. There are other ways to get to e, but for aspiring financiers, this is usually how e approaches you.

Say you had $100 in a bank account that paid 100% interest over a period of time t. How much money would you have at the end of the period? Using simple interest theory: Continue reading →

Arithmetic vs. Logarithmic Rates of Return

Say you hold a stock as it increases from $100 to $105. Usually, this is reported as a return of 5%. The formula for this return (which we’ll call arithmetic) is as follows:

r_\alpha = \frac{FV}{PV} - 1 = \frac{FV - PV}{PV} = \frac{105}{100} - 1 = 5 \%

This simple definition of return serves us well for most uses, but there are some quirks that make arithmetic returns difficult to use in some academic and valuation settings. For example, continuously compounded arithmetic returns are not symmetric. If a position appreciates 15% and then depreciates 15%, the total change is -2.25%.

FV = PV(1 + r_\alpha)

FV = PV(1+.15)(1 - .15) = PV(0.9775)

\frac{FV}{PV} - 1 = 0.9775 - 1 = -2.25 \%

To avoid this quirk, practitioners sometimes use log returns, which are defined as follows: Continue reading →