Week 12 part 2

Definition of Risks:
  - Firm-specific (unsystematic) risk:
    - Refers to risks that can be eliminated through diversification.
    - Investors are not compensated for bearing this type of risk due to its reducibility through portfolio diversification.
  - Economy-wide (systematic) risk:
    - Pertains to shocks that affect the entire economy, impacting all stocks simultaneously.
    - This type of risk cannot be diversified away and therefore investors are compensated for taking it on.
Beta:
- Each stock's sensitivity to systematic risk is measured by beta (β).
- In a world governed by the Capital Asset Pricing Model (CAPM), expected returns on assets depend solely on beta, as it reflects exposure to non-diversifiable (priced) risk.

Market/Systematic/Nondiversifiable Risk:
  - Characterized by factors common to the entire economy, including:
    - Recession
    - Inflation
    - Interest rates
Firm-Specific/Nonsystematic/Diversifiable Risk:
  - Examples include:
    - CEO turnover
    - New product announcements
    - Earnings announcements

Concept:
- We observe data points but need to draw a line that best fits this data through regression analysis.
- Beta is defined as the measure of a stock’s return sensitivity relative to market returns.
Method for estimating beta:
   - The regression approach asks: When does the market's excess return increase by 1%, how much does the stock's excess return change on average?
   - Example:
     - If a stock's beta is 1.2, according to CAPM, the stock's average excess return (denoted as $r_i - r_f$ ) is predicted to be 1.2 times the excess market return (denoted as $r_m - r_f$ ).
     - Thus, we have:
        $r_i - r_f ext{ approximately equals } 1.2 imes (r_M - r_f)$ .
   - There are various reasons observed returns might deviate from this predicted relationship.

Concept:
- OLS regression examines the relationship between two variables, here represented as $Y$ and $X$ (e.g., income and level of education).
OLS Regression Model:
  - The relationship is expressed mathematically as:
     $Y = \beta_0 + \beta_1X + \beta$ ,
    where:
    - $\beta_0$ is the intercept,
    - $\beta_1$ is the slope (indicating how $Y$ changes with respect to $X$ ),
    - $\beta$ represents other unexplained factors influencing $Y$ .
Objective:
- Choose parameters $\beta_0$ and $\beta_1$ to minimize the squared prediction errors as shown:
$ext{Minimize } ext{summation } (Y_i - \beta_0 - \beta_1X_i)^2$ .
Goodness of Fit:
- R-squared (R²):
- Represents the fraction of variation in $Y$ that can be explained by $X$ , with values ranging from 0 to 1.

CAPM Prediction:
- The return equation is defined as:
$E[r_i - r_f] = \beta_i imes E[r_m - r_f]$ .
Empirical Model (using historical data):
- Represented as:
$r_{i,t} - r_{f,t} = \beta_i + \beta_i (r_{m,t} - r_{f,t}) + ext{error}$ .
Findings:
  - The regression indicates a linear association between a stock's excess return and the market's excess return.
  - Interpretation of Results:
    - $\beta_i$ : Sensitivity to market risk.
    - $\beta_i$ analysis leads to:
      - If $\beta = 0$ : indicates no outperformance.
      - If eta > 0: signifies outperformance.
      - If eta < 0: indicates underperformance.

Objective:
- Use monthly returns of Walmart, the S&P (as a market portfolio proxy), and treasury bills (as a risk-free asset proxy) to estimate Walmart's beta.
Questions Posed:
- What is Walmart’s beta from the regression output?
- What do alpha and R-squared represent in the regression results?
Definitions:
- Alpha: Represents the average monthly excess return relative to the benchmark return predicted by CAPM during the observed timeframe.
- R-squared: Indicates the fraction of the total return variation explained by fluctuations in the market.

Regression Model:
- Standard model form:
$r_{i,t} - r_{f,t} = \beta_i + \beta_i (r_{m,t} - r_{f,t}) + ext{error}$ .
Resulting Variances:
  - Taking variance of both sides gives:
     $ext{Var}(r_i) = \beta_i^2 ext{Var}(r_m) + ext{Var}( ext{error})$ .
  - Where:
    - $ext{Var}(r_i)$ = total risk.
    - $\beta_i^2 ext{Var}(r_m)$ = systematic risk (due to market movements and the asset's sensitivity to the market).
    - $ext{Var}( ext{error})$ = unsystematic risk (firm-specific).
Key Concepts:
  - Firm-specific risks can be eliminated through diversification and thus are not 'priced'.
  - Only systematic risks affect expected returns and are priced.
  - The ratio $rac{\beta_i^2 ext{Var}(r_m)}{ ext{Var}(r_i)}$ is approximately R-squared from the regression output, reflecting the proportion of variance that cannot be diversified away.

Example of Stock i:
  - Given a standard deviation: ext{Std. Dev.}(r_i) = 30 ext{%} and $\beta = 1.3$ .
  - Market return’s standard deviation: ext{Std. Dev.}(r_m) = 20 ext{%} .
  - Thus, the variance of stock returns can be computed as:
     $ext{Var}(r_i) = 0.3^2 = 0.09$ .
  - Systematic Component of the Variance:
     $\beta^2 ext{Var}(r_m) = 1.3^2 imes (0.2^2) = 0.0676$ .
  - Unsystematic Component:
     $ext{Var}( ext{error}) = 0.09 - 0.0676 = 0.0224$ .
Diversifiable Risk Proportion:
- The fraction of risk that can be diversified away:
rac{0.0224}{0.09} = 24.9 ext{%} .
For Another Firm:
- If another firm has the same standard deviation of 30% but a beta of 0.8, the proportion of risk that can be diversified away would be:
rac{0.09 - 0.8^2 imes 0.2^2}{0.09} = 71.6 ext{%} .

Given Data:
  - Market portfolio:
    - Expected return: E[r_M] = 8 ext{%}
    - Standard deviation: ext{Std. Dev.}(r_M) = 15 ext{%}
  - Risk-free rate: r_f = 4 ext{%}
  - Stock A:
    - Standard deviation: ext{Std. Dev.}(r_A) = 20 ext{%}
    - Correlation with market: $<br>ho_{AM} = 0.8$
  - Stock B:
    - Standard deviation: ext{Std. Dev.}(r_B) = 30 ext{%}
    - Correlation with market: $<br>ho_{BM} = 0.2$

CAPM Expected Returns of Stocks:
   - For Stock A:
      $\beta_A = rac{ ext{Cov}<em>{AM}}{ ext{Std. Dev.}^2(r_M)} = rac{0.8 imes 0.2 imes 0.15}{0.15^2} = 1.07$      - Expected return:         E[r_A] = 4 ext{%} + 1.07 imes (8 ext{%} - 4 ext{%}) = 8.28 ext{%}    - For Stock B:       $\beta_B = rac{ ext{Cov}</em>{BM}}{ ext{Std. Dev.}^2(r_M)} = rac{0.2 imes 0.3 imes 0.15}{0.15^2} = 0.4$
     - Expected return:
        E[r_B] = 4 ext{%} + 0.4 imes (8 ext{%} - 4 ext{%}) = 5.6 ext{%}
Current Stock Prices Assuming Same Expected Cash Flows:
   - Expected cash flows of $100$ per share in one year with liquidation thereafter (devoid of additional payoffs).
   - Stock Price for A:
      P_A = rac{100}{1 + 8.28 ext{%}} = 92.4
   - Stock Price for B:
      P_B = rac{100}{1 + 5.6 ext{%}} = 94.7