Let's break down each part: ### Part a: Adjusted Beta The "adjusted beta" is commonly calculated using the formula: \[ \text{Adjusted Beta} = 0.67 \times (\text{Raw Beta}) + 0.33 \times 1.0 \] Given: Raw Beta = 1.25 \[ \text{Adjusted Beta} = 0.67 \times 1.25 + 0.33 \times 1.0 \] \[ = 0.8375 + 0.33 \] \[ = 1.1675 \] **Rounded to 2 decimal places:** \[ \boxed{1.17} \] --- ### Part b: Predicted Beta Next Year The regression equation is: \[ \beta_t = 0.2 + 0.8\beta_{t-1} \] Given: \(\beta_{t-1} = 1.25\) \[ \beta_t = 0.2 + 0.8 \times 1.25 \] \[ = 0.2 + 1.0 \] \[ = 1.2 \] **Rounded to 3 decimal places:** \[ \boxed{1.200} \] --- ### Summary of Answers: - **Adjusted beta:** `1.17` - **Predicted beta:** `1.200`

**Introduction** The concept of beta in finance measures a stock’s sensitivity to market movements, indicating its systematic risk relative to the overall market. A beta value of 1 implies that the stock tends to move in tandem with the market, whereas a beta greater than 1 indicates higher volatility, and a beta less than 1 suggests lower volatility compared to the market. Adjusted beta accounts for the tendency of beta estimates to regress toward the mean over time, often using a weighted average of the observed beta and 1.0, which is considered the market’s beta. Regression analysis, on the other hand, helps forecast future beta values based on historical data, capturing the dynamic evolution of the stock’s risk profile. **Explanation** Understanding beta and its adjustment is crucial for portfolio management and risk assessment. The adjusted beta provides a more realistic estimate that considers the tendency of beta to revert toward the market average, reducing potential overestimations or underestimations of risk. Regression analysis enables the prediction of future beta by modeling the relationship between past and future values, capturing trends and shifts in the stock’s systematic risk over time. These concepts are fundamental in making informed investment decisions and managing portfolio risk effectively. --- ### **Relevant Formulas and Their Explanation** 1. **Adjusted Beta Formula:** \[ \text{Adjusted Beta} = 0.67 \times \text{Raw Beta} + 0.33 \times 1.0 \] *This formula adjusts the raw beta estimate by giving more weight (67%) to the observed beta and a smaller weight (33%) to the market average (which is 1). This reflects the tendency of beta estimates to regress toward the mean over time.* 2. **Regression Forecasting Formula:** \[ \beta_{t+1} = \alpha + \beta \times \beta_t \] *This is a simple linear regression model where \(\alpha\) (intercept) and \(\beta\) (slope coefficient) are estimated parameters. The next period’s beta (\(\beta_{t+1}\)) is predicted based on the current beta (\(\beta_t\)).* *In this context, the regression equation is given as:* \[ \beta_e = 0.2 + 0.85 \times \beta_{t-1} \] *This models the evolution of beta over time, allowing the prediction of future beta based on the previous year’s beta.* --- ### **Step-by-Step Solution** **Part a: Calculating the Adjusted Beta** **Step 1:** Identify the raw beta value given. \[ \text{Raw Beta} = 1.25 \] **Step 2:** Apply the adjusted beta formula: \[ \text{Adjusted Beta} = 0.67 \times 1.25 + 0.33 \times 1.0 \] **Step 3:** Perform the calculations: \[ 0.67 \times 1.25 = 0.8375 \] \[ 0.33 \times 1.0 = 0.33 \] **Step 4:** Sum the results: \[ 0.8375 + 0.33 = 1.1675 \] **Step 5:** Round to two decimal places (as per standard practice): \[ \boxed{1.17} \] *This adjusted beta incorporates the tendency of beta to regress toward the mean market beta of 1, providing a more conservative estimate of the stock’s systematic risk.* --- **Part b: Predicting the Next Year’s Beta Using Regression** **Step 1:** Write down the regression equation: \[ \beta_e = 0.2 + 0.85 \times \beta_{t-1} \] **Step 2:** Substitute the previous year's beta: \[ \beta_{t-1} = 1.25 \] **Step 3:** Calculate the predicted beta: \[ \beta_e = 0.2 + 0.85 \times 1.25 \] \[ 0.85 \times 1.25 = 1.0625 \] \[ \beta_e = 0.2 + 1.0625 = 1.2625 \] **Step 4:** Round to three decimal places: \[ \boxed{1.263} \] *This prediction suggests that based on the historical trend modeled by regression, the beta of the stock for the upcoming year is approximately 1.263, indicating a slight increase in systematic risk.* --- ### **Final Answers:** - **Adjusted Beta:** **1.17** - **Predicted Beta for Next Year:** **1.263** These calculations provide a refined understanding of the stock’s risk profile, incorporating both the regression trend and the adjustment for mean reversion in beta estimates.