To compute the expected return given the different economic states, their probabilities, and the potential returns, we follow these steps:
1. Identify the probabilities and returns for each state:
- Fast Growth: Probability = 0.2, Return = 23%
- Slow Growth: Probability = 0.6, Return = 14%
- Recession: Probability = 0.2, Return = -30%
2. Use the formula for the expected return:
[tex]\[
\text{Expected Return} = (\text{Probability of Fast Growth} \times \text{Return in Fast Growth}) + (\text{Probability of Slow Growth} \times \text{Return in Slow Growth}) + (\text{Probability of Recession} \times \text{Return in Recession})
\][/tex]
3. Plug in the values and compute the expected return:
- Contribution of Fast Growth: [tex]\(0.2 \times 23\% = 0.2 \times 0.23 = 0.046\)[/tex]
- Contribution of Slow Growth: [tex]\(0.6 \times 14\% = 0.6 \times 0.14 = 0.084\)[/tex]
- Contribution of Recession: [tex]\(0.2 \times -30\% = 0.2 \times -0.30 = -0.060\)[/tex]
4. Sum these contributions:
[tex]\[
\text{Expected Return} = 0.046 + 0.084 + (-0.060)
\][/tex]
5. Calculate the final expected return:
[tex]\[
\text{Expected Return} = 0.046 + 0.084 - 0.060 = 0.070
\][/tex]
6. Express the expected return as a percentage:
[tex]\[
\text{Expected Return as a percentage} = 0.070 \times 100\% = 7.0\%
\][/tex]
Therefore, the expected return given the economic states, probabilities, and returns is [tex]\( \boxed{7.0 \%} \)[/tex].
