How does adding a risk-free asset change the efficient frontier?

Question

AcadiFi · Accepted Answer

Adding a risk-free asset transforms the efficient frontier from a curved hyperbola into a straight line — the Capital Market Line (CML). Understanding why reveals a deep insight about portfolio theory.

Before rf — only risky assets:

The frontier is derived by solving min w'Σw for each target return with w'1 = 1. Solution traces a parabola in (σ², E[R]) space or hyperbola in (σ, E[R]) space because variance is a quadratic form.

Marginal substitution: to move up the frontier, you replace low-return assets with high-return ones, but correlations ensure you can't escape the curved shape.

After rf — risky + risk-free:

Risk-free has σ = 0 and E[R] = rf. Any mix with a risky portfolio P yields:
- E[R_c] = rf + y × (E[R_P] − rf)
- σ_c = y × σ_P (linear in y)

Both expected return and standard deviation scale linearly with y. Hence the mix traces a straight line in (σ, E[R]) space.

For any y, you can always improve by choosing the risky portfolio P with the highest Sharpe ratio — the tangency portfolio. All other lines through rf lie below this one.

The final efficient frontier: a straight line from (0, rf) tangent to the old curved frontier at the tangency point. Above tangency point: you're levering up via borrowing at rf. Below: lending (investing in rf) while holding the same risky mix.

```mermaid
flowchart LR
    A[Old Frontier Curve] --> C[Add Risk-Free]
    B[Risk-Free Point] --> C
    C --> D[Straight CML]
```

Key consequences:

1. Every investor holds the same risky portfolio (the tangency) — they differ only in rf weight.
2. The concept of "risky portfolio choice" separates from "risk tolerance."
3. In equilibrium, tangency must be the market portfolio (CAPM).
4. Diversification's value is compressed — with rf, further diversification within risky assets matters only insofar as it raises the tangency Sharpe ratio.

Real-world frictions complicate this: borrowing rate > lending rate creates a kinked frontier. Short-sale constraints may eliminate portions of the CML above tangency.

How does adding a risk-free asset change the efficient frontier?

Master Level II with our CFA Course

Related Questions

Practice Questions