relationship between the expected rate of return and the value measured by the beta factor

Question

Assume that only two companies are listed on an effective capital market: companies A and company B. Capitalization (market value of all shares) of both companies is the same. Expected rate of return from shares of company A is 19%, and the expected rate of return on company shares B is 14%. Standard deviation of the rate of return on shares of company A is 30%, and the standard deviation of the rate of return on the company's shares B is 20%. Correlation coefficient between the rate of return on shares company A and the rate of return on shares of company B is 0.5. Rate of return of risk-free assets is 3%, with investors being able to grant loans at this rate, but it is not possible to borrow loans at a rate of return on risk-free assets. However, it is permissible to take both long and short positions in actions (short sale). I have to:

a) Calculate the equation of the straight line showing the relationship between the expected rate of return and the risk measured by the beta coefficient, on which lie only portfolios composed of shares A and B

b) Calculate the equation of the straight line showing the relationship between the expected rate of return and the risk measured by the beta coefficient, on which, under market equilibrium, there lie an segment containing effective portfolios composed partly of shares and partly of risk free assets.

I compute: $\mu_m=\frac{33}{200}$ (expected return from market porfolio), so i guess that in point b) the line will be: $$\mu_w=R+(\mu_m-R)\beta=\frac{3}{100}+\frac{27}{100}\beta$$ But what answer will be in point a)? I have also point c) to answer but first I need to know what is the line in point a). Can anyone help?

c) Present solutions to tasks in point (a) and (b) in the overview sketch in coordinate system: expected rate of return (axis ordinates), beta factor (abscissa axis). Enter coordinates points that you consider important and justify the solution selection of these points.

Answer 1

It's worth reiterating the assumptions underlying CAPM. The first set assumes individuals are rational, mean-variance optimizers, etc. The second set is a bit more interesting. The model assumes no taxes, no transaction costs, and that investors can borrow or lend at the risk-free rate and take short positions. You can't build an efficient frontier without leverage portfolios.

1) I think what your looking for is the Security Market Line, which plots the expected rate of return as a function of market-risk (Beta). The SML can be found by

$ \beta = cov(r_{i},r_{m})/\sigma_{m}^2$

You would need to be able to compute the variance between each security $i$ and the market index, which in your case is evenly cap-weighted

2) The slope is just the market risk premium (with $i$ representing the individual stock and $m$ representing the hypothetical 2-stock market).

$ SML = r_{f} + \beta[E(r_{m})-r_{f}] $

3) To answer question B, you'll need to look at the Capital Market Line (CML). Since investors have different risk preferences you'll see the creation of a market for loanable funds targeted to low risk-return individuals. CAPM is often described as the borrowing-and-lending line that runs tangent to the efficiency frontier we found earlier. In equilibrium, all investors will end up somewhere along the CML with a portfolio yielding an expected return and risk by investing in the market portfolio (defined as the best efficient portfolio) and by going long or short in a risk-free security.