What Is Expected Value (EV)?
Expected value (EV) denotes the anticipated average value of an investment at some point in the future. Investors use expected value to estimate the worth of investments, often relative to their risk. By calculating expected values, investors can choose the scenario most likely to produce the outcome they seek. In statistics and probability analysis, the expected value is calculated by multiplying each of the possible outcomes by the likelihood that each outcome will occur and then summing all of those values.
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 create optimized portfolios.
- Expected value can help investors size up whether an investment’s risk is worth the potential reward.
Formula and Calculation of Expected Value (EV)
The formula for expected value is:
1EV = ∑ P(X_i) × X_i
where:
- X is a random variable
- X_i are specific values of X
- P(X_i) is the probability of X_i occurring
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.
Understanding Expected Value (EV)
As noted above, the term expected value is commonly used in the investment industry. It refers to the anticipated value of an asset in the future. The EV of a random variable gives a measure of the center of the distribution of the variable. The EV is essentially 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 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 occurrences.
Expected Value (EV) in Portfolio Construction
Investors need to understand several key factors when they want to construct their investment or financial portfolios. These include how assets work and their associated risks. They should also have a grasp of their financial situation, investment goals, and time horizon.
Modern portfolio theory (MPT) attempts to solve for the optimal portfolio allocation based on investments’ expected values and standard deviations like risk. Put simply, investors and their financial advisors employ the MPT to build a portfolio to maximize their returns while minimizing their risks.
As an investor, you can use EV to determine which assets to add to your portfolio based on your preferences. Different assets have different EVs, so a stock comes with a different expected value (and risk profile) than a bond or an exchange-traded fund (ETF).
You can also make changes to your portfolio by using EV. This includes selling assets to swap them out for others. For instance, selling an asset that may have plateaued with no expectation of a rise in value and replacing it with another (similar or different) with a higher EV.
Example of Expected Value (EV)
To calculate the EV for a single discrete random variable, you must multiply each 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:
1\left(\frac{1}{6} \times 1\right) + \left(\frac{1}{6} \times 2\right) + \left(\frac{1}{6} \times 3\right) + \left(\frac{1}{6} \times 4\right) + \left(\frac{1}{6} \times 5\right) + \left(\frac{1}{6} \times 6\right) = 3.5
If you were to roll a six-sided die an infinite number 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). However, it should be noted that this is a different formula than the statistical expected value presented in this article.
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. This is, again, different than the statistical expected value presented in this article. However, it is another commonly used method for examining a stock’s value.
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. This does utilize the concepts presented in this article.
The Bottom Line
Understanding 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.
Related Terms: Expected Return, Risk Management, Investment, Portfolio Theory, Modern Portfolio Theory.
