The frontier, revisited
Track 3 lesson 5 introduced the efficient frontier — the set of portfolios where, for any σ, you can't get a higher expected return without taking more σ. Now we extend it. The big addition: combining risky portfolios with the risk-free asset changes the picture entirely.
From Track 3 alone: the frontier is a curve. Add a risk-free asset (T-bill, σ ≈ 0, return = r_f) and something elegant happens. You can now form portfolios that are mixes of any point on the risky frontier and the risk-free asset.
The Capital Allocation Line (CAL)
Pick any risky portfolio P with expected return E(R_P) and σ_P. Mix it with the risk-free asset in proportion w (weight in P) and 1−w (weight in r_f). The combined portfolio has:
- Expected return:
w × E(R_P) + (1−w) × r_f— linear in w - σ:
w × σ_P— also linear (because r_f has σ = 0, no correlation effect)
Plot this on the risk-return chart. It's a straight line from (0, r_f) to (σ_P, E(R_P)) and beyond (if w > 1, meaning leverage). Same slope throughout.
That slope is the Sharpe ratio of P (lesson 9-5):
Slope = excess return per unit of σ. Higher slope = better CAL.
The tangency portfolio — picking the best risky portfolio
Among all CALs you could draw (from r_f to any point on the risky frontier), the one that touches the frontier at exactly one point — the tangency point — has the highest slope. That single tangency portfolio is the best risky portfolio for everyone, regardless of risk preference.
Why "for everyone"? Because mixing it with r_f can produce any σ you want. Want low σ? Hold mostly r_f, small portion in tangency portfolio. Want high σ? Hold all tangency portfolio (or borrow at r_f to lever up).
Risk preference doesn't change which risky portfolio is best; it only changes the mix with r_f. This is the two-fund theorem — a beautiful result of Markowitz/Tobin's portfolio theory.
What the tangency portfolio is, in theory
Under Markowitz's full assumptions (everyone has the same expected returns and risk estimates, can borrow/lend at r_f, no taxes/transaction costs), the tangency portfolio equals the market portfolio — the value-weighted basket of all risky assets. Lesson 9-2 covers why.
This is a remarkable claim: the best risky portfolio for everyone is just "the whole market." It motivates passive investing (lesson 9-6) — if the market portfolio is theoretically best, just buy a low-cost index fund of everything and call it done.
The takeaway
Adding a risk-free asset to the efficient frontier creates the Capital Allocation Line — a straight line from r_f to any risky portfolio. The CAL with the highest slope (highest Sharpe ratio) defines the tangency portfolio, which under Markowitz's assumptions equals the market portfolio. Risk tolerance picks the mix between the tangency portfolio and r_f. Lesson 9-2 expands the market portfolio idea; 9-3 builds CAPM on it.