Week 2: Prospect Theory and Loss Aversion

1. The need for a new approach

Week 1 showed that the traditional model (rational investors, Expected Utility Theory, efficient markets) does not explain everything we observe. The starting point of this week is simple: traditional models are not sufficient, and combining psychology and economics gives Behavioral Finance.

The pioneers

• Herbert Simon: bounded rationality.
• Daniel Kahneman and Amos Tversky: Prospect Theory (Kahneman & Tversky, 1979).
• Andrei Shleifer and Richard Thaler: behavioral anomalies, such as limits to arbitrage, mental accounting and the endowment effect.

Bounded rationality

In plain words:

• People do not have perfect information. Not every piece of information that matters for a decision is known to us.
• Cognitive abilities are limited. There is a ceiling on our intelligence and on how much information we can process.
• Time is limited. How much time we have affects how much information we can collect and process.

Intuition: a fully rational investor would read every annual report of every listed company before choosing a portfolio. Nobody does this. Instead people "satisfice": they choose an option that is good enough, using shortcuts. Bounded rationality does not mean people are stupid; it means their rationality works within limits, which creates predictable errors.

"Rise of the degenerates": GameStop and Roaring Kitty

The slides show the GameStop (GME) episode. In early 2021 retail investors, coordinated on social media (Reddit's WallStreetBets) and inspired by Keith Gill ("Roaring Kitty"), bought GameStop shares and call options. The price chart shows GME rising from very low levels in mid-2020 to a peak of about $350 in January 2021, crashing, and rising again; by June 2021 it stood at $212.94, a gain of about 4,272% over one year. In 2024 GameStop surged again when Roaring Kitty returned to social media (the Bloomberg headline "GameStop surges as 'Roaring Kitty' returns").

Why it matters: these price moves had little to do with changes in GameStop's fundamental value. They were driven by attention, social influence and emotion, and they hurt professional short sellers. This is hard to square with efficient markets and rational investors, and it is a good example of why a behavioral approach is needed.

Main idea

• Markets are not always efficient.
• Investors are subject to biases.
• They rely on heuristics (rules of thumb).

This week's first and most central bias is loss aversion: when deciding, investors give a possible loss more weight than a possible gain.

2. Loss aversion

Intuition: from an evolutionary point of view, losing food or shelter could be fatal, while extra resources were only nice to have. Our brain therefore treats threats as more urgent than opportunities. In finance this shows up as a strong reluctance to accept bets with a possible loss, even when the expected value is positive.

The value function

The slide graph shows the perceived value v(x) on the vertical axis and gains or losses on the horizontal axis, with the origin as reference point. The curve rises in the gain domain but flattens (concave), and falls in the loss domain more steeply (convex, and steeper than for gains). This is the famous S-shape.

The standard functional form (Tversky & Kahneman's parameterisation, from which λ = 2.25 comes) is:

"One picture, 1000 words"

The slide shows the S-curve with a gain of $50 and a loss of $50. The $50 gain gives "less joy" than the "more pain" from the $50 loss. Both points are measured from the reference point at the origin.

Empirical evidence on loss aversion

• Benartzi & Thaler (1995): short-term loss aversion accounts for the equity premium puzzle. Over history, stocks have returned far more than safe bonds, by a margin larger than ordinary risk aversion can explain. When loss-averse investors check their portfolio frequently, the many small losses on stocks hurt, so they require a big premium before holding them (see myopic loss aversion below).
• Shefrin & Statman (1985): the disposition effect: investors sell winning positions too early and hold losing positions too long.
• Thaler (1980): the house money effect: after gains, people become more willing to take risks. A casino gambler who is ahead treats the winnings as "the house's money" and bets more freely, because losing part of a gain is not felt as a real loss relative to the starting point.

Example of house money: an investor buys a crypto coin for €1,000 that rises to €3,000. He now treats the €2,000 profit as "free money" and puts it into a much riskier meme coin, while he would never risk €2,000 of his salary that way. Economically, money is money; psychologically, the reference point changed.

3. Core ideas of Prospect Theory

Prospect Theory (Kahneman & Tversky, 1979) is a descriptive theory of how people actually choose between risky options. Its core ideas:

1. Gains and losses are measured relative to a reference point, not as final wealth.
2. Each outcome is coded as either a gain or a loss.
3. A loss hurts more than an equally large gain pleases (loss aversion).
4. Neither outcomes nor probabilities receive linear weights.
5. The value function is therefore non-linear: concave in the gain domain (risk aversion) and convex in the loss domain (risk seeking).

The reference point

• Decisions are made relative to the status quo, not in absolute terms.
• The reference point can be set by previous expectations, initial prices (such as the purchase price of a share), or the presentation frame. Usually it is the current situation.

Example: two investors both end the year with €110,000. Anna started at €100,000; Ben started at €130,000. Under EUT they should be equally happy (same final wealth). Under Prospect Theory Anna feels a gain of €10,000 and Ben a loss of €20,000. Ben may now take more risk to "get back to even", Anna may play it safe.

4. Paradoxes that violate Expected Utility Theory

The Allais paradox

People break the rules of Expected Utility Theory in a systematic way, in particular when they have to pick between two uncertain options. Maurice Allais demonstrated this using two pairs of gambles.

Expected value: 1A = $1 million; 1B = 0.89 × 1 + 0.01 × 0 + 0.10 × 5 = $1.39 million. Still, most people choose 1A: the certainty of becoming a millionaire is very attractive, and the 1% chance of ending with nothing feels terrible.

Expected value: 2A = 0.11 × 1 = $0.11 million; 2B = 0.10 × 5 = $0.5 million. Here most people choose 2B: both chances are small and look similar (11% versus 10%), so people go for the larger prize.

The slide summary: the violation arises because the common 89% component (the same $1 million in 1A and 1B, the same zero in 2A and 2B) should not affect the relative preference. Similar outcomes should cancel each other out; this is exactly the independence axiom from Week 1. People instead react to the move from 99% to 100% certainty much more strongly than to the move from 10% to 11%. Prospect Theory captures this with the certainty effect in its probability weighting function.

The Ellsberg paradox

There are two urns, each holding 100 balls. Urn A contains 50 red and 50 black ones. Urn B contains red and black balls in a mix that is unknown. These are the bets:

• 1A: you get €1 when the ball drawn from A is red, otherwise €0.
• 2A: you get €1 when the ball drawn from A is black, otherwise €0.
• 1B: you get €1 when the ball drawn from B is red, otherwise €0.
• 2B: you get €1 when the ball drawn from B is black, otherwise €0.

Typical result: people are indifferent between red and black within each urn, but they clearly prefer to bet on urn A (for red as well as for black). This is inconsistent with any single belief about urn B: if you prefer 1A to 1B, you must believe urn B has fewer than 50 red balls, but then it has more than 50 black balls and you should prefer 2B to 2A.

Why it matters for investors: ambiguity aversion helps explain home bias (investors prefer familiar domestic stocks whose risks feel "known") and why markets sell off strongly when uncertainty rises, for example during new trade policies whose effects are hard to estimate.

5. How decisions are made: two stages

Kahneman and Tversky describe decision making in two phases:

1. Processing (editing): mental shortcuts, ranking of priorities and desired outcomes. The problem is simplified and framed.
2. Evaluation: the decision itself, using probability weighting, not necessarily rationally.

The processing (editing) phase

During this phase a heuristic rule is chosen, the reference point is fixed and any framing effects are removed. There are six operations:

Because editing depends on how a problem is presented, the same choice can be edited differently in different frames. This is the source of many preference reversals.

The evaluation phase

In the evaluation phase we mentally compute the value of each edited prospect from its outcomes and probabilities. Kahneman and Tversky defined:

The difference with Expected Utility: EU = Σ p_i U(W_i) uses objective probabilities and final wealth. Prospect Theory uses decision weights π(p) and changes x relative to a reference point.

6. The probability weighting function

• It measures desire (the decision weight people attach), not probability itself.
• People overweight small probabilities (for example, lottery tickets).
• People underweight large probabilities (almost certain events feel less than certain).
• This explains the popularity of both insurance (overweighting a small chance of disaster) and gambling (overweighting a small chance of a big prize).

A common textbook form (Tversky & Kahneman's later parameterisation) is π(p) = p^γ / (p^γ + (1 − p)^γ)^1/γ with γ ≈ 0.61 for gains. It gives an inverse-S shape:

Look at the jump from 0.99 (weight 0.912) to 1.00 (weight 1.000): a 1 percentage point increase in probability adds almost 0.09 in weight, while moving from 0.10 to 0.11 adds very little. This is the certainty effect that drives the Allais paradox.

Characteristics according to the slides

• We prefer certainty over probability.
• We underestimate or completely ignore some small probabilities (the editing phase may drop them).
• We overestimate the probabilities of more likely outcomes and ignore improbable scenarios, which leaves us exposed to Black Swans (rare, extreme events such as the 2008 crisis).

Note the tension: small probabilities are either overweighted (when they are noticed, such as a lottery jackpot) or ignored altogether (when they are not noticed, such as a crash scenario). Which one happens depends on attention and framing.

7. The disposition effect

The punchline on the slide tells those who picked A and C that it has some news for them. Most people would sell A, the largest winner, to lock in the gain, and hold on to B, the largest loser, since selling turns the loss into a definite one. Deciding what to sell based on whether a stock trades above or below the price you paid, rather than on what it is likely to do next, is precisely the bias discussed below.

A rational investor would ignore the purchase price, which is a sunk cost, and ask only: which stocks have the best prospects from today on? Taxes even favour the opposite of the disposition effect: selling losers creates a deductible loss, selling winners creates taxable gains.

Why it matters: the disposition effect lowers investor returns (winners that keep rising are sold, losers are kept) and it can affect prices: because holders of winners sell early, good news is absorbed slowly, which is one explanation for momentum (later in the course).

8. Myopic loss aversion

The theoretical logic:

• Investors reassess their portfolio over a short horizon.
• Frequent evaluation means they see many small losses, and each one is painful.
• Result: they choose safer assets, even when the expected return on those assets is lower.

The slide graph shows perceived risk falling as the investment horizon lengthens: with frequent evaluation perceived risk is high; with a long-term view it is low.

Empirical application: the equity premium puzzle

Benartzi & Thaler (1995) explain the equity premium puzzle with MLA. The puzzle: historically stocks have earned much more than bonds, so much more that standard models need implausibly high risk aversion to explain why people still hold bonds. If investors are loss-averse and evaluate their portfolios often (in their paper an evaluation period of about one year fits the data), stocks' frequent short-term losses make them feel very risky, so investors demand a large premium. The premium is "too high" from a rational long-run view but understandable for myopic, loss-averse investors.

Behavioral implications and evidence

• Portfolios get checked too frequently.
• Temporary losses trigger strong reactions.
• Stocks and other assets that fluctuate in the short run are avoided.
• Evidence: participants who see returns per month put on average 30% less into equities than participants shown long-horizon returns.
• Evidence: institutions that evaluate less frequently earn higher Sharpe ratios (excess return over the risk-free rate per unit of volatility).

Practical lesson: an investor saving for a pension in 30 years should not judge the portfolio daily in an app. Pension funds and advisers often limit reporting frequency or show long-term charts for this reason.

9. Criticism and conclusions

Criticism of Prospect Theory

• It does not explain all phenomena.
• The reference point is vague and difficult to define empirically. Is it the purchase price, the highest price since purchase, last year's value, or an expected return? Different choices give different predictions, which makes testing hard.
• It is more descriptive (how people do choose) than normative (how people should choose). It does not tell an investor what the optimal decision is.

Conclusions

• Kahneman and Tversky offered a decision theory that serves as an alternative to expected utility.
• People judge wealth against a reference point rather than in absolute terms.
• Losses weigh more heavily than gains of equal size.
• How a problem is framed shifts our preferences.
• Probabilities are weighted subjectively: people overestimate small ones and underestimate large ones.
• By linking psychology with finance, the theory explains anomalies that the EMH leaves aside.