# Expected Value

4th August 2021

Expected value (also known as EV, expectation, average, or mean value) is a long-run average value of random variables. It also indicates the probability-weighted average of all possible values.

Expected value is a commonly used financial concept. In finance, it indicates the anticipated value of an investment in the future. By determining the probabilities of possible scenarios, one can determine the EV of the scenarios. The concept is frequently used with multivariate models and scenario analysis. It is directly related to the concept of expected return.

Formula for Expected Value

The first variation of the expected value formula is the EV of one event repeated several times (think about tossing a coin). In such a case, the EV can be found using the following formula:

EV = P(x) *n

Where:

EV: The expected value

P(X): The probability of the event

N: The number of the repetitions of the event

The expected value (EV) is an anticipated value for an investment at some point in the future. In statistics and probability analysis, the expected value is calculated by multiplying each of the possible outcomes by the likelihood each outcome will occur and then summing all of those values. By calculating expected values, investors can choose the scenario most likely to give the desired outcome.

EV=∑P(Xi)×Xi

Example:

Examples of using expected value

It turns out that all events have some aspect of risk and value. Insurance companies use this to determine how much to charge you for your premiums. They add up everyone in your reference class, and determine how much it costs them on average in payouts. They then add a little on the top to make a profit, which makes buying insurance net negative (the costs minus the benefits to you) on expectation, just like buying a lottery ticket. However, this doesn’t mean getting insurance is a bad idea! A lot of people don’t like taking on excessive risk (a small chance of becoming bankrupt feels much worse than paying up for insurance you might never need), so buying insurance is rational. Another way to put this is that we have diminishing marginal returns to extra money (or concave utility functions, for the mathematically inclined).

Pascal’s wager is also an example of using expected value to think about the world. Humans all bet with their lives either that God exists or that he does not. Pascal argues that a rational person should live as though God exists and seek to believe in God. If God does actually exist, such a person will have only a finite loss (some pleasures, luxury, etc.), whereas they stand to receive infinite gains (as represented by eternity in Heaven) and avoid infinite losses (eternity in Hell).