What is the one-fund theorem and the tangency portfolio?

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AcadiFi · Accepted Answer

Yes — introducing a risk-free asset collapses the two-fund theorem into the one-fund (mutual fund) theorem. All mean-variance investors hold combinations of the risk-free asset and a single risky portfolio: the tangency portfolio.

Tangency portfolio T is the risky portfolio with maximum Sharpe ratio. Geometrically, it's the point where a line from the risk-free rate (0, rf) is tangent to the efficient frontier of risky assets.

Construction: w_T proportional to Σ⁻¹(μ − rf × 1), normalized to sum to 1.

Example: μ = (0.08, 0.12, 0.15), rf = 0.03, Σ diagonal with variances (0.04, 0.09, 0.16).

Excess returns vector μ − rf = (0.05, 0.09, 0.12).
Σ⁻¹(μ − rf) = (1.25, 1.00, 0.75).
Normalize: sum = 3.00, so w_T = (0.417, 0.333, 0.250).

All investors, regardless of risk aversion, hold this same risky mix — they differ only in the fraction invested in T versus rf.

Conservative investor: 30% in T, 70% in rf → low-risk efficient portfolio.
Aggressive investor: 150% in T, borrow 50% at rf → leveraged portfolio above T on the CAL.

This insight revolutionizes asset management: instead of every investor building a custom risky portfolio, a single optimal risky fund (the "market portfolio" under CAPM assumptions) serves all investors. Risk tolerance merely dictates the cash mix.

```mermaid
flowchart LR
    A[Risk-Free Asset] --> C[CAL Line]
    B[Tangency T] --> C
    C --> D[All Efficient Portfolios]
```

CAPM extends this: if all investors hold T, in equilibrium T must equal the market portfolio. This yields the CAPM equation E[Rᵢ] = rf + βᵢ(E[Rm] − rf).

Exam tip: one-fund theorem requires unrestricted borrowing at rf — a strong assumption relaxed in Black's zero-beta CAPM.

What is the one-fund theorem and the tangency portfolio?

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