Conditional Expectation in Investments

In the context of investments, conditional expectation refers to the expected value of an investment, given a certain set of real-world events that are relevant to that particular investment. This means that in their calculation and prediction of the expected value of an investment, analysts take the events that are likely to occur in the future and the probability of their occurrence into account. Competitors, governments, and other financial institutions keep releasing new information. Such pieces of information may have a positive or a negative impact on investment. This means that a project’s expected value must be based on real-world dynamics.

It beats financial logic for an investor to stick to the initially predicted returns from an investment if they believe that the events occurring after the initial assessment of the investment can significantly impact earnings. For example, let us consider a sugar importer who calculates their expected return assuming that the government will not impose import tariff. If the government is planning to introduce a tariff in the coming months, then the investor will have to estimate the expected returns considering the possibility of a tariff.

The total probability rule is very useful when determining the unconditional expected value of an investment. The unconditional expected value, \(E(X)\), is the sum of conditional expected values. Therefore,

$$ E\left( X \right) =\sum { \left\{ E\left( X|{ S }_{ i } \right) P\left( { S }_{ i } \right) \right\} } $$

Example: Conditional Expectation

The probability of relaxed trade restrictions in a given country is 40%. Therefore, shareholders of XYZ Company Limited expect a 5% share return if trade restrictions are maintained and a loss of 8% if they are relaxed. The expected change in return is closest to:

Solution

We must take every possibility into account. We have a 40% chance of relaxed trade restrictions in this case. Intuitively, this means that there is a 60% chance that the current restrictions will be maintained. Therefore:

$$ \begin{align*}
E\left( X \right) & =\sum { \left\{ E\left( X|{ S }_{ i } \right) P\left( { S }_{ i } \right) \right\} } \\
& = 0.6(0.05) + 0.4(-0.08) \\
& = -0.002 \\
\end{align*} $$

Question

There is a 20% chance that the government will impose a tariff on imported cars. A company that assembles cars locally expects returns of 14% if the tariff is imposed and returns of 11% if the tariff is not imposed. The (unconditional) expected return is closest to:

A. 11.6%.

B. 12.8%.

C. 12.5%.

Solution

The correct answer is A.

The unconditional expected return will be the sum of:

1. The expected return given no tariff times the probability that a tariff will not be imposed.
2. The expected return given tariff times the probability that the tariff will be imposed. Therefore,

$$ \begin{align*}
E\left( X \right) & =\sum { \left\{ E\left( X|{ S }_{ i } \right) P\left( { S }_{ i } \right) \right\} } \\
& = 0.11(0.8) + 0.14(0.2) \\
& = 0.116 \text{ or } 11.6\% \\
\end{align*} $$