The risk-free asset, the market portfolio and the CML

In short

• Combining the risk-free asset with a risky portfolio gives a straight capital allocation line (CAL), because the risk-free asset has zero variance and zero correlation with the risky portfolio.
• The best CAL is the steepest one. Each investor picks a point on it with their indifference curves: risk-averse investors hold mostly the risk-free asset.
• With homogeneous expectations, everyone arrives at the same optimal risky portfolio: the market portfolio. With different expectations, optimal risky portfolios differ.
• The capital market line (CML) is the CAL whose risky portfolio is the market portfolio: $E(R_p) = R_f + \frac{E(R_m) - R_f}{\sigma_m}\sigma_p$.
• Passive investors rely on market prices and hold index funds on the CML; active investors trust their own valuations and over- or underweight assets.
• Points above the CML are unachievable; points below it are inferior (dominated).