What do investor utility functions look like for different risk aversion levels?

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AcadiFi · Accepted Answer

In mean-variance utility U = E[R] − ½ × A × σ², the parameter A captures risk aversion — how much return an investor demands to accept an additional unit of variance.

Interpretation of A:

- A = 0: risk-neutral. Investor maximizes expected return regardless of risk.
- A = 1-2: low risk aversion. Typical of hedge funds, leverage-tolerant family offices.
- A = 3-5: moderate. Most retail and balanced institutional portfolios.
- A = 6-10: high risk aversion. Conservative retirees, insurance general accounts.
- A = ∞: extreme — invests only in risk-free asset.

Indifference curves: sets of (σ, E[R]) pairs yielding constant utility U₀. Solving U₀ = E[R] − ½Aσ² gives:

E[R] = U₀ + ½ × A × σ²

For each U₀, this is a parabola in (σ, E[R]) space opening upward. Higher U₀ → parabola sits higher.

Shape by A:

- Low A: shallow parabolas — investor trades small return increases for large variance increases.
- High A: steep parabolas — investor requires steep compensation for even small variance increases.

Example curves at U₀ = 6%:

- A = 2: E[R] = 6% + σ² → at σ = 15%, E[R] = 6% + 0.0225 = 8.25%.
- A = 6: E[R] = 6% + 3σ² → at σ = 15%, E[R] = 6% + 0.0675 = 12.75%.

At σ = 15%, the A=6 investor needs 12.75% return to feel as good as they do at rf, while A=2 investor only needs 8.25%.

```mermaid
flowchart LR
    A[Low A: Flat Curves] --> C[Choose High y*]
    B[High A: Steep Curves] --> D[Choose Low y*]
```

Choosing portfolio: tangency between highest-utility indifference curve and the CML. Higher A → tangency at lower-σ point → smaller allocation to risky assets.

Measuring A empirically:

1. Questionnaires (flawed — stated preferences differ from actions).
2. Revealed preferences: observed allocation → backed-out A via y* = (E[R_M]−rf)/(Aσ²_M) formula.
3. Experimental lotteries.

Limitations: mean-variance utility assumes preferences depend only on first two moments. Prospect theory (Kahneman-Tversky) shows investors weight losses more than equivalent gains (loss aversion), violating A framework.

What do investor utility functions look like for different risk aversion levels?

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