What Is Expected Value?
Expected value (EV) is an anticipated average value for an investment at some point in the future. Investors use expected value to estimate the worth of investments, often relative to their risk.
Modern portfolio theory (MPT), for instance, attempts to solve for the optimal portfolio allocation based on investments' expected values and standard deviations (i.e., risk).
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 produce the outcome that they seek.
Key Takeaways
- Expected value describes the long-term average level of a random variable based on its probability distribution.
- In investing, the expected value of a stock or other investment is an important consideration and is used in scenario analyses.
- Modern portfolio theory uses expected value in conjunction with an investment's risk (standard deviation) to come up with optimized portfolios.
- Expected value can help investors size up whether an investment's risk is worth the potential reward.
Understanding Expected Value
The EV of a random variable gives a measure of the center of the distribution of the variable. Essentially, the EV is the long-term average value of the variable. Because of the law of large numbers, the average value of the variable converges to the EV as the number of repetitions approaches infinity. EV is also known as expectation, the mean or the first moment.
EV can be calculated for single discrete variables, single continuous variables, multiple discrete variables, and multiple continuous variables. For continuous variable situations, integrals must be used.
Scenario analysis is one technique for calculating the EV of an investment opportunity. It uses estimated probabilities with multivariate models to examine possible outcomes for a proposed investment. Scenario analysis also helps investors to determine whether they are taking on an appropriate level of risk given the likely outcome of the investment.
The difference between expected value and arithmetic mean is that the first involves a distribution of probability and the second involves a distribution of occurrence.
Formula for Expected Value
EV=∑P(Xi)×Xi
where:
- X is a random variable
- P(X) is the probability of the random variable
Thus, the EV of a random variable X is taken as each value of the random variable multiplied by its probability, and each of those products is summed.
Example of Expected Value
To calculate the EV for a single discrete random variable, you must multiply the value of the variable by the probability of that value occurring.
Take, for example, a normal six-sided die. Once you roll the die, it has an equal one-sixth chance of landing on one, two, three, four, five, or six. Given this information, the calculation is straightforward:
(61×1)+(61×2)+(61×3)+(61×4)+(61×5)+(61×6)=3.5
If you were to roll a six-sided die an infinite amount of times, you would find that the average value equals 3.5.
What Is a Dividend Stock's Expected Value?
The expected value of a stock is estimated as the net present value (NPV) of all future dividends that the stock pays. If you can estimate the growth rate of the dividends, you can predict how much investors should willingly pay for the stock using a dividend discount model such as the Gordon growth model (GGM).
How Do I Find the Expected Value of a Stock That Doesn't Pay Dividends?
For non-dividend stocks, analysts often use a multiples approach to come up with expected value. For example. the price-to-earnings (P/E) ratio is often used and compared to industry peers. So, if the tech industry has an average P/E of 25x, a tech stock's EV would be 25 times its earnings per share.
How Is the Expected Value of a Stock Used in Portfolio Theory?
Modern portfolio theory and related models use mean-variance optimization to come up with the best portfolio allocation on a risk-adjusted basis. Risk is measured as the portfolio's standard deviation, and the mean is the expected value (expected return) of the portfolio.
The Bottom Line
Comprehending the concept of expected value is important for investors. It can aid them in determining the level of return that they might expect from an investment.
Expected value and scenario analysis can provide insight into the risk of an investment versus its return and help an investor decide whether or not to include it in their portfolio.
