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 img and img), 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 img 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 ...

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