Capital Market Line and the CAPM
We have been analyzing an optimization problem whose solution resolves the Markowitz problem, specifically, what combination of weights in a k-asset portfolio return a targeted portfolio mean return at minimum variance.
The Two-Fund theorem basically states that for any two efficient portfolios (hence, two sets of weights corresponding to two targeted returns and ), any other efficient portfolio can be constructed as a linear combination of these two.
Now, we derive the One Fund. Add a risk-free asset to our analysis earning a rate of return rf. Then, suppose we have a proportion α in this asset and the remainder in the risky asset(s). The mean return on this portfolio will be
where r is the mean return to the risky portfolio with variance
Var is a scalar because both the variance and covariance of returns on the riskless asset is zero by definition. The standard deviation on this new portfolio is then . What is the relationship between the mean (u) and ...