Practice Question
You have done a thorough study of the economy and Stock X and concluded that:
(1) the probability of a boom next year is 20%, a stable economy is 55%, and a recession is 25%, and
(2) the price of Stock X will be $45 in a boom, $25 in a stable economy, and $15 in a recession.
If the stock is currently priced at $24, what is the expected return and the standard deviation of returns?
Standard Deviation: 42.24%
Step 1: Identify Known Inputs
- Current Price: $24
- Boom: 20% probability → Price = $45
- Stable: 55% probability → Price = $25
- Recession: 25% probability → Price = $15
Step 2: Calculate Returns
\[ R_{\text{boom}} = \frac{45 - 24}{24} = 0.875 \\ R_{\text{stable}} = \frac{25 - 24}{24} = 0.0417 \\ R_{\text{recession}} = \frac{15 - 24}{24} = -0.375 \]
Step 3: Compute Expected Return
\[ E(R) = (0.20 \times 0.875) + (0.55 \times 0.0417) + (0.25 \times -0.375) \\ E(R) = 0.175 + 0.02294 - 0.09375 = 0.10419 = \textbf{10.42%} \]
Step 4: Compute Variance and Standard Deviation
\[
\text{Variance} = \sum P_i \times (R_i - E(R))^2
\]
- Boom: \( (0.875 - 0.1042)^2 \times 0.20 \approx 0.1188 \)
- Stable: \( (0.0417 - 0.1042)^2 \times 0.55 \approx 0.0022 \)
- Recession: \( (-0.375 - 0.1042)^2 \times 0.25 \approx 0.0574 \)
\[
\text{Variance} \approx 0.1784 \quad \Rightarrow \quad \sigma = \sqrt{0.1784} = \textbf{0.4224 or 42.24%}
\]
Conclusion: The expected return is 10.42% and the standard deviation is 42.24%, indicating a relatively high-risk, high-return investment.