Calculate the excess volatility

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The excess volatility test is a method used to determine whether stock prices are more volatile than can be justified by changes in fundamentals, such as dividends or earnings. This test can be applied by comparing the actual volatility of stock prices with the volatility implied by a model based on fundamental values.

Here’s an example of how to perform an excess volatility test with simplified calculations:

### Step-by-Step Example

#### 1. **Collect Data**
- Obtain historical stock prices and dividends data for a specific stock over a period.
- For simplicity, let's assume we have the following data for a stock over 5 years:
- **Year 1**: Price = $100, Dividend = $2
- **Year 2**: Price = $110, Dividend = $2.2
- **Year 3**: Price = $105, Dividend = $2.1
- **Year 4**: Price = $115, Dividend = $2.3
- **Year 5**: Price = $120, Dividend = $2.4

#### 2. **Calculate the Implied Fundamental Value**
- Assume the fundamental value is based on discounted dividends.
- Use a simple constant discount rate (e.g., 5% or 0.05).

The fundamental value \( P_t \) at any time \( t \) can be approximated using the Gordon Growth Model for simplicity:
\[
P_t = \frac{D_t}{r - g}
\]
where:
- \( D_t \) = dividend at time \( t \)
- \( r \) = discount rate (assumed 5% or 0.05)
- \( g \) = growth rate of dividends (assumed 2% or 0.02)

#### 3. **Compute the Fundamental Value for Each Year**
- Year 1: \( P_1 = \frac{2}{0.05 - 0.02} = \frac{2}{0.03} = 66.67 \)
- Year 2: \( P_2 = \frac{2.2}{0.05 - 0.02} = \frac{2.2}{0.03} = 73.33 \)
- Year 3: \( P_3 = \frac{2.1}{0.05 - 0.02} = \frac{2.1}{0.03} = 70.00 \)
- Year 4: \( P_4 = \frac{2.3}{0.05 - 0.02} = \frac{2.3}{0.03} = 76.67 \)
- Year 5: \( P_5 = \frac{2.4}{0.05 - 0.02} = \frac{2.4}{0.03} = 80.00 \)

#### 4. **Calculate Actual Price Volatility**
- Compute the standard deviation of the actual stock prices:
\[
\sigma_{actual} = \text{Standard Deviation of } [100, 110, 105, 115, 120]
\]

#### 5. **Calculate Fundamental Value Volatility**
- Compute the standard deviation of the fundamental values:
\[
\sigma_{fundamental} = \text{Standard Deviation of } [66.67, 73.33, 70.00, 76.67, 80.00]
\]

#### 6. **Compare the Volatilities**
- If \(\sigma_{actual}\) is significantly greater than \(\sigma_{fundamental}\), it suggests that stock prices are excessively volatile compared to what would be justified by fundamentals.

Let's compute the standard deviations for clarity:

- **Actual Prices**: $100, $110, $105, $115, $120
- Mean = \( \frac{100 + 110 + 105 + 115 + 120}{5} = 110 \)
- Variance = \( \frac{(100-110)^2 + (110-110)^2 + (105-110)^2 + (115-110)^2 + (120-110)^2}{4} = 50 \)
- Standard Deviation (Actual) = \( \sqrt{50} \approx 7.07 \)

- **Fundamental Values**: $66.67, $73.33, $70.00, $76.67, $80.00
- Mean = \( \frac{66.67 + 73.33 + 70.00 + 76.67 + 80.00}{5} \approx 73.33 \)
- Variance = \( \frac{(66.67-73.33)^2 + (73.33-73.33)^2 + (70.00-73.33)^2 + (76.67-73.33)^2 + (80.00-73.33)^2}{4} \approx 26.67 \)
- Standard Deviation (Fundamental) = \( \sqrt{26.67} \approx 5.16 \)

### Conclusion

In this simplified example, the actual price volatility (\( \approx 7.07 \)) is greater than the fundamental value volatility (\( \approx 5.16 \)), suggesting that the stock prices exhibit excess volatility compared to what would be justified by changes in fundamental values.

In a real-world scenario, the calculations would be more complex, taking into account more data points and possibly more sophisticated models for fundamental value estimation.

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