Unconditional expected value, variance and standard deviation

What is an expected value?

The unconditional mean of a random variable (also called the unconditional expectation, or simply the expected value) is the probability-weighted average of its possible outcomes. Take each outcome, multiply by its probability, and add. “Unconditional” means it is computed over all outcomes, without assuming that any particular scenario has happened. The contrast, a forecast given a scenario, comes in Lesson 6. The mean looks forward: it is a forecast built from a probability distribution , a list of outcomes with the probability of each, summing to 1. Laying out the outcomes as a short table of scenarios, such as recession, normal and boom, is scenario analysis .

$$E[X] = P(x_1)\,x_1 + P(x_2)\,x_2 + \cdots + P(x_n)\,x_n = \sum_{i=1}^{n} P(x_i)\,x_i$$

The 6% is a balance point: if the outcomes were weights on a seesaw, the board would balance at 6. It is not the return you should expect in any one year. No scenario returns exactly 6%.

Exam tip On the BA II Plus in chain mode, wrap each product in brackets after the first one. Otherwise 0.25 × 12 + 0.5 is evaluated before the 0.5 is multiplied.

How do you measure how uncertain a forecast is?

The unconditional variance is the probability-weighted average of the squared deviations from the expected value. The unconditional standard deviation is its square root, in the same units as the outcomes, so it is the figure to quote.

$$\mathrm{Var}(X) = \sigma_X^2 = \sum_{i=1}^{n} P(x_i)\,[x_i - E[X]]^2$$

There is a faster equivalent that needs no deviations. Square each outcome, weight it by its probability, add, and subtract the squared mean. The deviation form shows what variance means; the shortcut is quicker on a table with many scenarios.

$$\mathrm{Var}(X) = E\big[(X - E[X])^2\big] = E[X^2] - (E[X])^2$$

$$\sigma_X = \sqrt{\mathrm{Var}(X)}$$

Drag a probability up and watch the balance point move toward its outcome. Spread the outcomes apart and the shaded band widens even though the expected value does not change.

Why is there no n − 1 here?

In Learning Module 5 (Statistical Characteristics of Asset Returns) the data were past returns, so the mean was estimated from the sample and the deviations were averaged with n − 1. Here nobody has estimated anything from a sample. The probabilities are your forecast of how often each outcome occurs, and they already do the averaging.

	Historical data (Learning Module 5)	Forecast with probabilities (this lesson)
Center	Sample mean, equal weights	Expected value, weights = probabilities
Variance	Σ(X − X̄)² ÷ (n − 1)	Σ P × (X − E[X])²
Standard deviation	Square root of the variance	Square root of the variance

Notice the shape is the same: a weighted average of squared deviations. Equal weights of 1/n (or 1/(n − 1)) are replaced by the probabilities.

How far is an outcome from the forecast?

Divide an outcome’s distance from the expected value by the standard deviation. The answer counts how many standard deviations the outcome sits above (positive) or below (negative) the mean.

$$\dfrac{x - E[X]}{\sigma_X}$$

How do you use these in a decision?

Compare the expected value for reward and the standard deviation for risk. Suppose stock B returns 4% or 8%, each with probability 0.5. Its expected return is 6%, the same as the share above, but its standard deviation is only 2%. Two forecasts with the same reward can carry very different risk, and a risk-averse investor prefers the lower standard deviation. The same arithmetic applies to any forecast: earnings per share, revenue growth, project cash flows or credit losses. With two variables, the same table gives the unconditional covariance, E[XY] − E[X]E[Y]: Lesson 7.

Probability distribution of outcomeseach outcome with its probability; probabilities add to 1

• Unconditional mean (expected value) Memorize Σ P × x, the probability-weighted average
  • Reads as the forecast, not a promised outcome
• Unconditional variance Memorize E[X²] − (E[X])², units are squared
  • Weights probabilities, so no n − 1
• Unconditional standard deviation Memorize √variance, same units as the outcomes
  • Reads as the typical distance from the forecast

Go deeper Why a fair bet can still be a bad idea

A coin flip that wins 100 or loses 100 has an expected value of zero, so it is “fair”. It still has a standard deviation of 100. A risk-averse investor needs a positive expected return to accept that spread, which is why an investment’s reward has to be judged against its risk and not on its own. Portfolio risk in Learning Module 8 builds on exactly these two numbers.

Go deeper Why this variance is the one used in portfolios

Learning Module 8 combines the variances and the covariance of several assets into a portfolio variance. Each of those inputs is a probability-weighted quantity of this form when it comes from a forecast, so the formula in this lesson is the building block. Lesson 7 of this module computes the covariance from the same kind of table.