Risk-free asset, CML, and tangency portfolio

Adding a risk-free asset to the Markowitz universe transforms the problem. The efficient frontier is no longer a hyperbola — it becomes a straight line from r_f through the tangency portfolio. This Capital Market Line is the geometric heart of CAPM and the foundation of all of single-factor asset pricing.

Setup with risk-free asset

Let r_f be a risk-free borrowing/lending rate. Now investors can hold any combination of risky portfolios with the risk-free asset. Allocation θ ∈ [0, 1] to a risky portfolio with (μ_p, σ_p) and (1-θ) to cash:

μ_combined = θ μ_p + (1 - θ) r_f
σ_combined = θ σ_p

As θ varies, the combined portfolio traces a straight line from (0, r_f) through (σ_p, μ_p). The slope is the Sharpe ratio of the risky portfolio. With borrowing (θ > 1), the line extends beyond the risky portfolio.

The tangency portfolio

The risky portfolio that gives the steepest such line is the one with the highest Sharpe ratio. Geometrically, it is the point of tangency between a ray from (0, r_f) and the original (risk-only) efficient frontier — hence 'tangency portfolio'. The line itself is the Capital Market Line (CML).

w_tangency = Σ⁻¹(μ - r_f 1) / (1ᵀ Σ⁻¹ (μ - r_f 1))
SR_tangency = √((μ - r_f 1)ᵀ Σ⁻¹ (μ - r_f 1))

Mutual fund theorem (one-fund theorem)

With a risk-free asset, every efficient portfolio is a combination of cash and a single risky portfolio (the tangency portfolio). Different investors with different risk tolerances choose different θ, but they all hold the same risky portfolio. The choice problem reduces to: how much cash, how much tangency? This is the deepest practical insight of mean-variance theory.

Capital Market Line equation

μ_p = r_f + [(μ_T - r_f) / σ_T] · σ_p

Linear relationship between expected return and risk, with intercept r_f and slope = Sharpe of tangency. Every efficient portfolio lies on this line.

When the borrowing rate exceeds the lending rate

Realistically, retail investors borrow at a higher rate than they earn on cash. This kinks the CML: one slope from r_f_lend through one tangency portfolio (the 'low-risk' efficient set), another slope from r_f_borrow through a different tangency portfolio (the 'high-risk' efficient set). The kink defines a range of risky portfolios suitable for unleveraged investors.

Implications for portfolio construction

• Risk-aversion separation: choose your risky portfolio (the tangency) without knowing your risk aversion. Your risk aversion only determines your cash allocation θ.
• Leverage as substitute for asset selection: a risk-averse investor who wants higher returns doesn't have to pick higher-risk individual assets — they can lever up the tangency.
• Cash is always an asset: even 'fully invested' portfolios implicitly bet against holding cash.

The empirical reality

In data: investors do not all hold the same risky portfolio. They differ by tax, regulation, view, mandate, transaction cost. The tangency portfolio is also estimated with huge error (μ is noisy). The CML is a normative ideal more than a description — and exactly because it's a normative ideal, deviations from it are the basis of relative-value strategies.