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# Application of Conditional Expectations in Investments

The conditional expectation, in the context of investments, 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 analysts calculate and predict the expected value of an investment, taking the events likely to occur in the future and their probabilities into account. Competitors, governments, and financial institutions keep on releasing new pieces of information. Such pieces of information may have a positive or negative impact on an investment. This means that a project’s expected value must be based on real world dynamics, events or scenarios.

It beats financial logic to stick to the initially predicted returns from an investment if an investor believes that the post-initial assessment events can significantly impact earnings. For example, let’s consider a sugar importer who calculates their expected return assuming the government won’t 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

The probability of relaxed trade restrictions in a given country is 40%. Therefore, shareholders of XYZ Company Limited expect their shares to return 5% if trade restrictions are maintained. On the other hand, they expect an 8% loss if trade restrictions are relaxed. Compute the expected change in return.

Solution

We must take every possibility into account. In this case, we have a 40% chance of relaxed trade restrictions. Intuitively, this means 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. Determine the (unconditional) expected return.

A. 11.6%

B. 12.8%

C. 12.5%

Solution

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, and

(2) The expected return given tariff times the probability that the tariff will be imposed. Thus,

\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*}

Note to Candidates: Focus should be on the calculations.

Explain the use of conditional expectation in investment applications.

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