CAPM - estimation

The points in the graph shown bellow represent IBM and S&P 500 risk premiums over time. The price series for IBM and S&P-500 were obtained from Yahoo Finance. The 1-month risk free rate is computed from the investment in a 3-month US Treasury-Bills. T-Bill data was collected from Federal Reserve Bank of St. Louis (FRED). The period considered is from January 1962 to February 2022 (722 observations).

The red line is given by the linear model:

\[ r_{IBM, t} - r_{f,t} = \hat{\alpha} + \hat{\beta}(r_{S\&P-500, t} - r_{f,t}) + \epsilon_{t} \qquad(1)\]

\[ \epsilon_{t} = (r_{IBM, t} - r_{f,t}) - \left[ \hat{\alpha} + \hat{\beta}(r_{S\&P-500, t} - r_{f,t}) \right] \]

The model parameters \(\alpha\) and the \(\beta\) can be manipulated using the sliders bellow.

Recall that the Ordinary Least Squares (OLS) regression models estimates are the ones that minimise the sum of the regression squared residuals (\(\epsilon_t\)):

\(\sum_{t=1}^N \epsilon_t^2 =\)

Below you find the OLS estimation results using a statistical software for the linear model presented in Equation 1.:

Residuals:
      Min        1Q    Median        3Q       Max 
-0.273658 -0.030311 -0.002053  0.029256  0.281918 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)      0.001825   0.002065   0.884    0.377    
I(sp500 - TB3MS) 0.975886   0.047808  20.412   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.05534 on 720 degrees of freedom
Multiple R-squared:  0.3666,    Adjusted R-squared:  0.3657 
F-statistic: 416.7 on 1 and 720 DF,  p-value: < 2.2e-16
      
Sum of squared residuals: 2.205391