8.5.1. Prospect Theory

Prospect Theory

Prospect theory (Kahneman & Tversky, 1979) is a model of decision-making under uncertainty that replaces expected utility with a formulation built around two functions: a value function $v(\cdot )$ defined over gains and losses relative to a reference point, and a probability weighting function $w(\cdot )$ that distorts objective probabilities into decision weights.

For a gamble paying $x$ with probability $p$ and $y$ with probability $q=1-p$, expected utility assigns value $pU(W+x)+qU(W+y)$ where $W$ is the individual’s wealth. Prospect theory assigns
$w(p)v(x-R)+w(q)v(y-R),$
where $R$ is a reference point against which outcomes are judged.

The value function has three features:

• Reference dependence: $v(\cdot )$ is defined over gains and losses relative to a reference point $R$, not over final wealth levels $W$. The reference point is implicitly 0 in the gamble.
• Loss aversion: $v(\cdot )$ has a kink at the origin where ${\lim }_{x\to 0^{+}}{v}^{\prime }(x)<{\lim }_{x\to 0^{-}}{v}^{\prime }(x)$, so losses are felt more acutely than gains of the same magnitude.
• Diminishing sensitivity: $v(\cdot )$ is concave over gains and convex over losses, producing risk aversion in the gain domain and risk seeking in the loss domain.

The weighting function $w(\cdot )$ overweights small probabilities and underweights moderate ones. Together these produce a four-fold pattern: risk aversion over moderate-probability gains, risk seeking over moderate-probability losses, risk seeking over low-probability gains, and risk aversion over low-probability losses.

Prospect theory is typically paired with narrow framing — evaluating each gamble in isolation rather than merging it with background wealth — and requires specifying a reference point against which outcomes are coded as gains or losses. Kőszegi and Rabin (2006, 2007, 2009) propose that the reference point is the agent’s expectations about outcomes. Cumulative prospect theory (Tversky & Kahneman, 1992) extends the framework to many-outcome lotteries. The three-component value function plus probability weighting plus narrow framing generates a family of testable predictions across asset pricing (the equity premium puzzle via Benartzi-Thaler myopic loss aversion; “lottery stock” overpricing via Barberis & Huang skewness pricing), insurance and annuity markets, the endowment effect in labor and consumption, downward wage rigidity, executive stock-option compensation, and even comparative cognition in non-human primates — see the # Applications section below.

Intuition

The value function captures three deviations from how rational agents “should” evaluate risk. First, people care about changes from a reference point, not final wealth — a $500 bonus feels different depending on whether you expected $0 or $1,000. Second, losses sting roughly twice as much as equivalent gains feel good — most people reject a 50:50 bet to win $110 or lose $100, despite positive expected value. Third, sensitivity diminishes as you move away from the reference point in either direction: the difference between losing $100 and $200 feels larger than between $1,100 and $1,200.

Probability weighting reflects the psychological reality that the jump from “impossible” to “possible” (0% → 1%) feels much larger than from 30% → 31%. People overweight rare events, which is why the same person buys lottery tickets (overweighting small chance of a big gain) and insurance (overweighting small chance of a big loss).

Narrow framing matters because loss aversion alone cannot explain aversion to small favorable gambles. Without narrow framing, an agent merges the new bet with her existing wealth risk — a small independent gamble actually diversifies her portfolio. Only when she evaluates it in isolation does the $100 loss loom larger than the $110 gain.

The reference point is what makes a given outcome feel like a gain or a loss. Kőszegi and Rabin argue it is set by expectations: a worker who expected a $5,000 bonus and receives $4,000 codes the outcome as a $1,000 loss, while one who expected nothing codes it as a $4,000 gain.

Properties

Narrow Framing

(Barberis, Huang & Thaler, 2006) Loss aversion without narrow framing cannot explain rejection of small, actuarially favorable gambles. When merged with pre-existing risk, a small independent bet diversifies the portfolio and reduces total loss probability — even a loss-averse agent should accept. Narrow framing is necessary to generate the observed aversion.

(Tversky & Kahneman, 1981) In a pair of concurrent decisions, combining choices BC stochastically dominates AD. But 60–73% of subjects choose A and D by evaluating each decision in isolation — a direct consequence of narrow bracketing combined with diminishing sensitivity.

Reference Points

(Expectations-Based Reference Points) Kőszegi and Rabin (2006, 2007, 2009): the reference point is the agent’s recent expectations about outcomes. This explains why professional traders (who expect to sell inventory) do not exhibit the endowment effect, while ordinary consumers (who expect to keep what they own) do.

(Tractability vs. Realism) Models using only gain-loss utility do so for tractability; consumption levels still matter, and Kőszegi-Rabin emphasize that prospect theory utility should supplement, not replace, standard consumption utility.

Precursor: Swalm (1966)

(Elicited Utility) Swalm elicited utility functions from corporate executives via indifference questions. The resulting curves displayed the S-shape later formalized by Kahneman and Tversky — concave over gains, convex over losses, steeper in the loss domain. Executives maximized their own executive value, not shareholder value.

Examples

Framing Effects (Reference Dependence)

“You have been given $1,000. Choose: $500 for sure, or ($1,000, 0.5).” vs. “You have been given $2,000. Choose: $-$$500 for sure, or ($-$$1,000, 0.5).” Both problems have identical final wealth distributions. Under EU, choices should be the same. Under prospect theory, the reference point shifts and most people choose the sure gain in the first frame and the gamble in the second.

Loss Aversion and Small Gambles

A 50:50 bet to win $110 or lose $100. Expected value is +$5. Under EU with reasonable risk aversion, this is accepted. Under prospect theory with $\lambda =2.25$: $v(110)=110^{0.88}\approx 72.3$ and $v(-100)=-2.25\times 100^{0.88}\approx -148.5$. The prospect theory value is $0.5(72.3)+0.5(-148.5)=-38.1<0$, so the bet is rejected.

Diminishing Sensitivity in the Loss Domain

Choose: lose $500 for sure, or a 50:50 chance of losing $1,000 or nothing. Under EU (concave utility), the sure loss is preferred (less risky). Under prospect theory, $v(\cdot )$ is convex over losses, so the gamble is preferred — the sure loss of $500 feels almost as bad as potentially losing $1,000. This is why people “double down” at casinos.

Lottery stocks and skewness pricing (Barberis & Huang, 2008)

An IPO stock has a small probability of becoming “the next Google” — a positively skewed return distribution with a long right tail. Under expected utility, only the mean and variance of returns should affect price. Under prospect theory, the probability weighting function $w(\cdot )$ overweights the small tail probability, so prospect-theory investors bid up the lottery-like stock and accept a low average return. Barberis & Huang ($2008$) formalize this in equilibrium and show that, in an economy populated by prospect-theory investors, idiosyncratic skewness is priced — more positively skewed stocks earn lower average returns. The prediction has been confirmed empirically: IPOs underperform long-run, distressed-firm stocks underperform, out-of-the-money options have low returns, and high-idiosyncratic-volatility stocks (which are typically positively skewed) earn low returns. Underdiversified household portfolios also tilt toward these “lottery stocks,” which is exactly what prospect theory predicts undiversified investors should do.

Myopic loss aversion and the equity premium (Benartzi & Thaler, 1995)

Stocks have historically returned about $6\%$ above bonds — a premium too large to rationalize with standard expected-utility risk aversion at any plausible parameter value (Mehra-Prescott $1985$). Benartzi & Thaler combine loss aversion with narrow framing over annual returns: an investor evaluates each year’s stock return in isolation rather than aggregating over the holding period. Using calibrated CPT parameters $\alpha =0.88$, $\lambda =2.25$, $\delta =0.65$, they compute the prospect-theory value of the historical distribution of annual U.S. stock returns and the historical distribution of annual U.S. bond returns. The two values come out approximately equal — meaning that the observed $6\%$ equity premium is exactly the compensation a loss-averse, narrow-framing investor needs to be indifferent between the two. Pull either ingredient out (no loss aversion, or no annual narrow framing) and the puzzle reappears. Barberis, Huang & Santos ($2001$) extend this into a general-equilibrium model where the agent gets utility from both consumption and gains/losses on financial wealth.

The endowment effect, and why traders don't have it (KKT 1990; List 2003)

Kahneman, Knetsch & Thaler ($1990$) gave half their experimental subjects a coffee mug and the other half nothing. They then asked owners the minimum price at which they would sell (willingness to accept, WTA) and non-owners the maximum price at which they would buy (willingness to pay, WTP). Under standard preferences these should be equal. They were not: median WTA was $$5.75$, median WTP was $$2.25$, a factor of $2.5\times$. Loss aversion explains the gap mechanically — once you own the mug, parting with it is coded as a loss against the new reference point, so it requires roughly $\lambda$ times the gain you would have demanded to acquire it from scratch. The cleanest test of the expectations-based refinement comes from List ($2003$): professional sports-card traders, who fully expect to sell their inventory, show no endowment effect. Their reference point, per Kőszegi-Rabin, has already been updated to “I will sell this,” so the sale is not coded as a loss. The endowment effect is therefore not a stable feature of human cognition but a consequence of how reference points form.

Capuchin monkeys exhibit prospect-theoretic biases (Santos & Chen, 2009)

Capuchin monkeys diverged from the human lineage roughly $35$ million years ago and have no exposure to markets, money, or human cultural learning about gambles. Santos & Chen trained capuchins to use small metal tokens as currency and ran experiments testing each component of prospect theory. The monkeys displayed reference dependence (they preferred a vendor who showed $1$ apple slice and added a bonus over a vendor who showed $2$ slices and removed one, even though both delivered $1.5$ slices on average), loss aversion (they avoided the loss-frame vendor more strongly than the gain-frame vendor was preferred), diminishing sensitivity (risk averse over gains, risk seeking over losses), and the endowment effect (they preferred to keep tokens or food they had been given). The biases prospect theory describes are therefore not artifacts of human culture or financial education — they appear to be deeply rooted in primate cognition, suggesting an evolutionary origin that predates anything resembling modern markets.

Applications

Beyond the four worked examples above, prospect theory has been applied across a wide range of economic settings. In insurance markets, Sydnor ($2010$) shows that households routinely pay roughly $$100$/year extra in premiums to avoid a $5\%$ chance of a $$500$ deductible — an actuarially poor trade that probability weighting and expectations-based reference points rationalize cleanly (the premium is expected and therefore neutral, while the deductible is unexpected and coded as a loss). In annuity markets, Hu & Scott ($2007$) argue that the under-annuitization puzzle reflects households framing annuity purchase as a gamble on longevity: dying early is coded as a “loss” amplified by loss aversion and probability overweighting of that low-probability event. In labor economics, downward nominal wage rigidity is partly driven by acute loss aversion to nominal pay cuts, a friction that shows up in business-cycle models. In executive compensation, firms grant stock options (positively skewed payoffs) rather than equivalent stock grants because employees overweight the small probability of a large upside, accepting a lower expected value than equity-equivalent compensation. In gambling markets, the favorite-longshot bias in horse racing — where market odds systematically overstate longshots’ chances of winning — is consistent with probability overweighting of small win probabilities. The breadth of applications is part of what makes prospect theory the dominant alternative to expected utility for descriptive decision modeling.

See Also

• 8.5. Prospect Theory — parent hub.
• 8.5.2. Cumulative Prospect Theory — sibling: extension to many-outcome lotteries with rank-dependent weighting.
• 8.3. Challenges to Rational Pricing — the equity premium puzzle that myopic loss aversion explains.
• 8.8. Momentum — a different cross-asset anomaly addressed by belief-based models rather than prospect theory.