Indifference curves and the capital allocation line

In short

• An indifference curve links risk–return pairs that give one investor the same utility.
• Risk-averse curves slope upward and are convex; steeper curves mean more risk aversion. Utility rises toward the northwest. One investor's curves never cross.
• Risk-neutral curves are horizontal; risk-seeker curves slope downward.
• Combining a risk-free asset with a risky portfolio gives the capital allocation line (CAL): $E(R_p) = R_f + \frac{E(R_i) - R_f}{\sigma_i}\sigma_p$.
• Left of the risky portfolio the investor lends at $R_f$; right of it the investor borrows at $R_f$ to invest more than 100%.
• The optimal portfolio is the tangency point between the CAL and the highest indifference curve that touches it.