What is the Capital Allocation Line and how do I derive it?

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AcadiFi · Accepted Answer

The Capital Allocation Line (CAL) shows risk-return combinations from mixing any specific risky portfolio P with the risk-free asset. It's a straight line in (σ, E[R]) space.

Derivation: let y be the weight in risky portfolio P, 1−y in rf. Then:

E[R_complete] = y × E[R_P] + (1−y) × rf = rf + y × (E[R_P] − rf)

σ_complete = y × σ_P (since rf has zero variance)

Solving σ_complete = y × σ_P for y = σ_c / σ_P and substituting:

E[R_complete] = rf + [(E[R_P] − rf) / σ_P] × σ_complete

The slope (E[R_P] − rf) / σ_P is the Sharpe ratio of portfolio P. Every risky portfolio generates its own CAL — the higher the Sharpe ratio, the steeper the line.

Example: rf = 3%, P has E[R] = 10%, σ = 15%. Sharpe = 7/15 = 0.467.

CAL equation: E[R_c] = 3% + 0.467 × σ_c.

- 0% risky, 100% rf: E[R] = 3%, σ = 0%
- 50% each: E[R] = 6.5%, σ = 7.5%
- 100% risky: E[R] = 10%, σ = 15%
- 150% risky (borrow 50% at rf): E[R] = 13.5%, σ = 22.5% (above the risky-only point)

Investor's optimal complete portfolio on CAL: choose y to maximize utility U = E[R] − ½ A σ² where A is risk aversion. Optimal y* = (E[R_P] − rf) / (A × σ_P²).

Higher-A (more risk averse) investor chooses smaller y — sits closer to rf end.

CAL vs CML: CAL applies to any risky portfolio; CML is the specific CAL drawn to the tangency portfolio (maximum Sharpe). Since tangency has the highest Sharpe, the CML has the steepest slope — it dominates all other CALs.

```mermaid
flowchart LR
    A[rf Point] --> B[Slope = Sharpe]
    B --> C[Risky Portfolio P]
    C --> D[Leverage Beyond P]
```

What is the Capital Allocation Line and how do I derive it?

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