How does prospect theory change optimal portfolio allocation compared to expected utility theory?

Question

AcadiFi · Accepted Answer

Prospect theory portfolio optimization replaces the expected utility framework with Kahneman and Tversky's value function and probability weighting function. This produces portfolios that look materially different from mean-variance optimal portfolios because investors evaluate gains and losses relative to a reference point, are loss-averse, and distort probabilities.

**Key Behavioral Parameters:**

1. **Reference dependence**: Returns are evaluated relative to a reference point (usually the purchase price or a benchmark), not in terms of total wealth.

2. **Loss aversion**: Losses hurt approximately 2-2.5x more than equivalent gains feel good. The value function is:
   v(x) = x^alpha for gains (alpha ~= 0.88)
   v(x) = -lambda x |x|^beta for losses (lambda ~= 2.25, beta ~= 0.88)

3. **Probability weighting**: Small probabilities are overweighted; moderate-to-large probabilities are underweighted.
   w(p) = p^gamma / [p^gamma + (1-p)^gamma]^{1/gamma} (gamma ~= 0.65)

**How This Changes Portfolio Choice:**

| Feature | Expected Utility (Mean-Variance) | Prospect Theory |
|---|---|---|
| Reference point | Total wealth | Status quo or benchmark |
| Risk attitude | Uniformly risk-averse | Risk-averse for gains, risk-seeking for losses |
| Probability treatment | Linear (as given) | Overweight small, underweight large |
| Optimal portfolio | On efficient frontier | Typically more conservative + lottery tail |

**Worked Example:**

Devereaux considers allocating between a risk-free asset (3% return) and an equity index with the following outcome distribution:

| Outcome | Probability | Return |
|---|---|---|
| Market crash | 5% | -35% |
| Bear market | 15% | -12% |
| Modest growth | 50% | +8% |
| Bull market | 25% | +22% |
| Euphoria | 5% | +45% |

**Mean-variance optimal** (assuming risk aversion gamma = 3):
Optimal equity allocation: **65%**

**Prospect theory optimal** (lambda = 2.25, reference = 3% risk-free):

The prospect theory investor evaluates each outcome relative to 3%:
- Crash: -38% (extreme loss, heavily penalized by loss aversion AND overweighted probability)
- Bear: -15% (loss, penalized)
- Modest: +5% (small gain, diminishing sensitivity)
- Bull: +19% (gain, diminishing sensitivity)
- Euphoria: +42% (lottery-like, overweighted probability)

The loss aversion effect dominates: the 20% probability of losses (crash + bear) generates disproportionate disutility. But the overweighting of the 5% euphoria probability adds some equity demand.

Result: Prospect theory optimal equity allocation: approximately **40-45%** — significantly lower than mean-variance due to loss aversion.

However, the prospect theory investor also adds a small allocation (3-5%) to lottery-like assets (out-of-the-money calls, speculative stocks) because probability weighting overvalues small chances of extreme gains.

**The Resulting Portfolio Shape:**
- Large safe allocation (bonds, cash) protecting against losses
- Moderate equity core
- Small speculative tail seeking asymmetric upside
- This matches the BPT pyramid structure empirically observed in investor behavior

**Practical Implication for Advisors:**
Understanding prospect theory helps explain why clients resist recommended equity allocations after losses (reference point has shifted) and why they demand lottery-like products despite their negative expected value.

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How does prospect theory change optimal portfolio allocation compared to expected utility theory?

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