 Decision trees and dealing with uncertainty

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# 3 Expected values

Allow approximately 1 hour to complete this section.

You can use probability to arrive at a weighted average of the value of an outcome, reflecting the various levels of likelihood. This weighted average can be called the expected value. Work through the following example to see how this idea is used in financial forecasting.

## Box 2 Worked example on forecasting interest rates

Economic forecasters are unsure what interest rates will be next year. However, combining the various contributing economic scenarios, they believe the outcomes shown in Table 1 are possible.

### Table 1 Probability of different possible interest rates

Possible interest rate % Probability
1.50 0.29
2.75 0.54
3.90 0.17

Note that there are only three possible outcomes.

Their probabilities must total one.

So, .

The ‘expected value’ of interest rates can be calculated as follows:

As a common sense test, note that the expected value is close to the overwhelmingly most probable (2.75%, with a probability of 0.54).

Also, the expected value, being an average, must lie within the range of possible outcomes (that is, between 1.5% and 3.9%).

In the next activity you will use the idea of expected value to estimate stock returns. Although this approach may seem simplistic, it lies at the heart of modern-day corporate finance.

## Activity 4 Calculating the expected value of stock returns

Timing: Allow approximately 15 minutes to complete this activity

In this activity, you will calculate the expected value of stock returns. You will assume that the rates of return on the stock market over the last 120 years are as summarised in Table 2. The frequency tells you how many of those years a particular return was made.

### Table 2 Frequency of stock returns

Return % Frequency
3.00 4
3.25 12
4.00 19
4.50 23
5.12 28
6.00 18
6.50 12
7.00 4
120

### Part 1 Calculating the probability of stock returns

Before you calculate the expected value of stock returns, you will first need to find the probabilities and record them in Table 3.

Hint: if something occurs 4 times out of 120, what is its probability?

#### Table 3 Calculating the probability of stock returns

Return % Frequency Probability
3.00 4
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3.25 12
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4.00 19
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4.50 23
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5.12 28
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6.00 18
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6.50 12
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7.00 4
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120
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Tip: do not forget to check that all of the probabilities add up to a total of one.

So, taking 3.00% as an example from Table 2, you can see that it has occurred four out of a possible total 120 times. This gives it a probability as follows: .

If you then calculate the probability for all of the stock returns, you should get the results shown in Table 4.

##### Table 4 The probability of stock returns
Return % Frequency Probability
3.00 4 0.0333
3.25 12 0.1000
4.00 19 0.1583
4.50 23 0.1917
5.12 28 0.2333
6.00 18 0.1500
6.50 12 0.1000
7.00 4 0.0333
120 1.0000

### Part 2 Calculating the expected value of stock returns

You can now calculate the expected value of stock returns and complete Table 5.

#### Table 5 Calculating the expected value of stock returns

Return % Frequency Probability Expected value
3.00 4 0.0333
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3.25 12 0.1000
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4.00 19 0.1583
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4.50 23 0.1917
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5.12 28 0.2333
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6.00 18 0.1500
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6.50 12 0.1000
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7.00 4 0.0333
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120 1.000
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To get the expected value of stock returns, you multiply each stock return by its probability and then sum the result, as shown in Table 6 below.

##### Table 6 Expected value of stock returns
Return % Frequency Probability Expected value
3.00 4 0.0333 0.100
3.25 12 0.1000 0.325
4.00 19 0.1583 0.633
4.50 23 0.1917 0.863
5.12 28 0.2333 1.195
6.00 18 0.1500 0.900
6.50 12 0.1000 0.650
7.00 4 0.0333 0.233
120 1.000 4.899

So the expected value of stock returns is 4.899%, which you could round up to 4.9%.

You will use expected value in the next section on decision trees.

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