The Efficient Frontier
The best risk-return trade-offs trace a hyperbola that a risk-free asset upgrades into a straight line.
The efficient frontier is the set of portfolios offering the highest expected return for each level of risk. With two assets it traces a hyperbola as you vary the mix: imperfect correlation bends the curve left as diversification cancels risk, and the minimum-variance portfolio sits at its leftmost point. Adding a risk-free asset straightens the picture into the capital allocation line, tangent to the risky frontier at the tangency portfolio. Tobin separation follows: every investor holds that same tangency portfolio, differing only in how much risk-free borrowing or lending wraps around it. This geometry sets up the CAPM pricing of the next node.