# Use of conditional expectation in investment applications

CFA level I / Quantitative Methods: Basic Concepts / Probability Concepts / Use of conditional expectation in investment applications

The conditional expectations are also used in the calculation of expected values just like conditional probabilities. To understand the use of conditional expectations in investment applications, we need to understand the concept of expected value first.

The expected value of a random variable is the probability-weighted average of the possible outcomes of the random variable.

Suppose a company's EPS (earnings per share) for the next year is expected to be \$4.50 if the inflation is greater than or equal to 5 percent. The EPS is expected to be \$6.00 if the inflation is less than 5 percent. The probability of inflation having a value of greater than or equal to 5 percent is 60 percent and the probability of inflation less than 5 percent is 40 percent. Note that the events have to be mutually exclusive and exhaustive to calculate the expected value.

E(X) = Expected value of EPS = ∑P(Xi)Xi (i from 1 to n)

E(X) = 0.60*4.50 + 0.40*6.00 = \$5.10

The unconditional expected value formula can also be written as below:

E(X) = E(X|E)P(E) + E(X|Ec)P(Ec)

where Ec is the complement of event E.

E(X) = E(X|E1)P(E1) + E(X|E2)P(E2) + .... + E(X|En)P(En)

Where E1, E2, ....., En are mutually exclusive and exhaustive events.

And E(X|En) is the expected value of X given that event En has occurred and is also called as conditional expected value.

 Example 3: Using conditional expectations A company's dividend policy is based on its earnings per share (EPS). The earnings per share for the company is based on the state of the economy. For a good economy, the conditional probabilities of EPS are as follows: P(Increase in EPS| good economy) =0.60 P(No change in EPS| good economy) = 0.30 P(Decrease in EPS| good economy) = 0.10 For a bad economy, the conditional probabilities of EPS are as follows: P(Increase in EPS| bad economy) = 0.10 P(No change in EPS| bad economy) = 0.20 P(Decrease in EPS| bad economy) = 0.70 The dividend payment is \$1.00 per share if the EPS increases, \$0.70 if there is no change in EPS and \$0.20 if the EPS decreases. Compute the conditional expected value of the dividend payment for each statement of the economy. The probabilities of occurrence of a good economy and a bad economy are 0.60 and 0.40 respectively. Solution: The expected value of dividend payment when the state of the economy is good = E(X| good economy) = 0.60*\$1.00 + 0.30*\$0.70 + 0.10*\$0.20 = \$0.83 The expected value of dividend payment when the statement of the economy is bad = E(X| bad economy) = 0.10*\$1.00 + 0.20*\$0.70 + 0.70*\$0.20 = \$0.38 Once we get the conditional expected values, we can easily calculate the unconditional expected value. Unconditional expected value of dividend payment = Conditional expected value for good economy*Probability of good economy + Conditional expected value for bad economy*Probability of bad economy = \$0.83*0.60 + \$0.38*0.40 = \$0.65

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