How do I use utility maximization to select a portfolio on the CAL?

Question

AcadiFi · Accepted Answer

Utility maximization identifies the unique optimal point on the CML for each investor based on their risk aversion parameter A.

Standard mean-variance utility: U = E[R] − ½ × A × σ²

Higher A means higher risk aversion — the investor demands more return per unit of variance.

Optimization problem on the CML: let y be the weight in the tangency portfolio M, (1−y) in rf.

E[R_c] = rf + y × (E[R_M] − rf)
σ²_c = y² × σ²_M

Utility: U(y) = rf + y(E[R_M] − rf) − ½ × A × y² × σ²_M

First-order condition: dU/dy = (E[R_M] − rf) − A × y × σ²_M = 0

Solving: y* = (E[R_M] − rf) / (A × σ²_M)

This is the optimal risky-portfolio weight for the investor.

Example: rf = 3%, E[R_M] = 10%, σ_M = 18%, A = 3.

y* = (0.10 − 0.03) / (3 × 0.18²) = 0.07 / 0.0972 = 0.720

Investor holds 72% in market, 28% in rf. Complete portfolio: E[R_c] = 3% + 0.72 × 7% = 8.04%, σ_c = 0.72 × 18% = 12.96%.

Comparative statics:

- Higher A → smaller y* → more conservative.
- Lower A → y* > 1 → leveraged position (borrow at rf to buy more market).
- Higher excess return (E[R_M] − rf) → larger y*.
- Higher σ_M → smaller y* (penalty for volatility grows quadratically).

Typical A values:
- A = 1-2: aggressive, willing to leverage.
- A = 3-5: moderate.
- A = 6-10: conservative, high cash allocation.

```mermaid
flowchart TD
    A[Investor Risk Aversion A] --> B[Compute y*]
    B --> C[Weight in Market]
    B --> D[Weight in rf]
    C --> E[Optimal Portfolio]
    D --> E
```

Caveats: quadratic utility assumes normal returns or small deviations. For fat-tailed distributions, CARA or HARA utilities give different allocations — typically less aggressive in the tails.

How do I use utility maximization to select a portfolio on the CAL?

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