Managerial economics guide 5

Gretl and CAPM

Equilibrium assumption

- Open market
- All the risky assets refer to all the tradable stocks available to all
- There is a risk-free asset (for borrowing and/or lending in unlimited quantities) with interest rate r_f
- All information is available to all, such as covariances, variances, mean rates of return of stocks, and so on.
- Everyone is a risk-averse rational investor who uses the same financial engineering mean-variance portfolio theory from Markowitz.

→ Everyone has a portfolio on the same efficient frontier and hence has a portfolio that is a mixture of the risk-free asset and a unique efficient fund F (of risky assets). In other words, everyone sets up the same optimization problem, does the same calculation, gets the same answer, and chooses a portfolio accordingly.

→ This efficient fund used by all is called the market portfolio and is denoted by M. The fact that it is the same for all leads us to conclude that it should be computable without using all the optimization methods from Markowitz: The market has already reached an equilibrium so that the weight for any asset in the market portfolio is given by its capital value (total worth of its shares) divided by the total capital value of the whole market (all assets together).

CAPM on Gretl

Time series data

Google_for_CAPM: The most important parameter is β. This parameter measures the systematic risk of the portfolio. Using the Google and Nasdaq data, we will perform a simple regression with the returns of one specific stock as the dependent variable, in this case, we have Google’s returns over the constant and the market returns.

What we will use is the OLS, Gretl: File > Open data > google_for_CAPM.csv
Model > Ordinary least squares > Dependent variable: Google return, independent: Nasdaq return
We have 52 close prices but only 51 observations.

β = 1.06419
Here we have β, you will find this value under coefficient, this is also the beta of Google’s return, it implies that Google’s stock is slightly riskier than the overall market (beta > 1).

Analysis

> Confidence intervals for coefficients
95% confidence interval means that 95% of the interval will contain the true parameter.
The confidence interval for beta goes from 0.69 to 1.43, this means that the estimation for beta is not precise because 0.69 means the asset’s aggregate risk is much smaller than that of the market and 1.43 means the asset’s aggregate risk is much higher than that of the market. The larger extreme of the interval also exaggerates the variance of the model and increases the overall riskiness of the market.

The standard error, square root of the variance of the estimates. We have low s