Expected Utility vs Prospect Theory

1. Bernoulli’s Utility Theory (Expected Utility Theory)

• Assumptions: People make decisions to maximize their expected utility.
• Utility Function: It assumes a concave utility function for gains (e.g., diminishing marginal utility of wealth).
• Reference Point: It doesn’t explicitly consider reference points; decisions are based on the absolute value of wealth.
• Risk Preferences: Assumes consistent risk aversion across different levels of wealth.
• Key Idea: People evaluate the expected outcomes of a decision by multiplying probabilities and utilities of the outcomes and then choose the option with the highest utility.

Example: A person will prefer $10 for sure over a 50% chance of $20 because the utility of $10 is greater than the weighted utility of the risky gamble.

This chart demonstrates how the expected utility of the gamble is calculated under Bernoulli’s Utility Theory:

1. Blue Curve: The utility function U=log ⁡(Wealth), showing diminishing marginal utility.
2. Orange Points: The utility of the gamble outcomes, such as Wealth=20 (gain) and Wealth=0.1(loss).
3. Orange Dotted Line: The linear connection between the two gamble outcomes in utility space, illustrating the gamble’s range.
4. Green Point: The expected utility of the gamble, calculated as a weighted average of the utilities for the gamble outcomes (based on their probabilities).
5. Red Point: The utility of the certain gain, which lies above the green point, showing why a risk-averse investor prefers the certain gain over the gamble.

This approach visually explains how the dotted line represents the gamble and why risk-averse behavior results in avoiding risky options with equivalent expected values in wealth terms.

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2. Prospect Theory

• Developed by: Daniel Kahneman and Amos Tversky.
• Assumptions: People value gains and losses relative to a reference point, not absolute wealth levels.
• Loss Aversion: People dislike losses more than they like equivalent gains (e.g., losing $100 hurts more than gaining $100 feels good).
• Value Function:
  • S-shaped and asymmetrical:
    • Convex for losses (risk-seeking behavior in losses).
    • Concave for gains (risk-averse behavior in gains).
  • Steeper in the loss domain than in the gain domain (indicating loss aversion).
• Weighting of Probabilities: People tend to overweight small probabilities and underweight large probabilities.
• Reference Point: Decisions depend heavily on the framing of outcomes as gains or losses relative to a reference point.

Example:

• If framed as a gain: “You can gain $500 or take a 50% chance to gain $1000.” Most choose $500 (risk-averse in gains).
• If framed as a loss: “You can lose $500 or take a 50% chance to lose $1000.” Most choose the gamble (risk-seeking in losses).

“When Amos and I began our work on prospect theory, we quickly reached two conclusions: people attach values to gains and losses rather than to wealth, and the decision weights that they assign to outcomes are different from probabilities. Neither idea was completely new, but in combination they explained a distinctive pattern of preferences” – Daniel Kahneman – Thinking, Fast and Slow.

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Unlike Bernoulli’s utility theory, which assumes a concave utility function (for risk aversion) that only curves downward, Prospect Theory’s value function has an S-shaped curve. Here’s why:

1. Concave for Gains (Risk Aversion): In the domain of gains, people tend to avoid risks, so the curve flattens as gains increase, similar to Bernoulli’s function.
2. Convex for Losses (Risk Seeking): In the domain of losses, people tend to take risks to avoid sure losses, so the curve bends downward, reflecting risk-seeking behavior.
3. Loss Aversion (Steeper for Losses): The curve is steeper for losses than for gains, showing that losses feel more impactful than equivalent gains (loss aversion).

I will now plot the S-shaped curve to better illustrate Prospect Theory’s distinctiveness.

This chart illustrates the S-shaped value function of Prospect Theory:

1. Gains (Right of 0): The curve is concave, reflecting risk aversion. As gains increase, the additional subjective value diminishes (e.g., gaining $10 feels better than $5, but the difference between $1000 and $1010 is negligible).
2. Losses (Left of 0): The curve is convex, reflecting risk-seeking behavior. People are willing to take risks to avoid sure losses, as the pain of losing diminishes for larger losses.
3. Loss Aversion (Steepness near 0): The curve is steeper for losses than for gains, showing that losses feel more impactful than equivalent gains (e.g., losing $100 feels worse than gaining $100 feels good).

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Why Prospect Theory is Important

Prospect Theory addresses real-world behaviors that Expected Utility Theory cannot explain, such as:

• Framing Effects: The way a problem is presented affects choices.
• Endowment Effect: People value what they own more than identical things they don’t own.
• Certainty Effect: Preference for certain outcomes over probabilistic ones, even when expected utilities are similar.

In summary, Bernoulli’s Utility Theory models rational economic decision-making, while Prospect Theory better reflects actual human behavior by incorporating psychological biases and framing effects.

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Prospect Theory has significant implications for stock market traders, as it highlights how psychological biases, and framing can influence trading behavior. Here are the key implications.

Here’s a table summarizing the behaviors, reasons, consequences, and advice based on the implications of Prospect Theory for stock market traders:

Behavior	Reason	Consequences	Advice
Holding on to losing stocks	Loss aversion: Traders dislike realizing losses and prefer to wait for recovery, even when it’s unlikely.	Capital remains tied up in underperforming assets, leading to poor portfolio performance.	Use stop-loss orders to limit losses and reallocate capital to better opportunities.
Selling winning stocks too soon	Disposition effect: Traders lock in gains to avoid regret and secure profit.	Winners are sold prematurely, reducing potential portfolio growth.	Set clear profit targets and evaluate holdings based on future potential, not past performance.
Overinvesting in speculative stocks	Overweighting small probabilities: High-reward outcomes are overvalued despite low likelihood.	Increased risk and portfolio volatility, with potential for significant losses.	Diversify investments and evaluate speculative stocks objectively using probabilities and fundamentals.
Overreacting to framing effects	Reference points: Decisions depend on whether outcomes are framed as gains or losses.	Irrational trading decisions, such as overreacting to news or market events based on presentation.	Reframe decisions by focusing on long-term strategy rather than short-term gains or losses.
Overvaluing owned stocks	Endowment effect: Traders assign higher value to stocks they own than identical alternatives.	Missed opportunities due to inertia in trading decisions.	Regularly review holdings objectively, comparing them with new opportunities in the market.
Anchoring to past prices	Anchoring bias: Traders use the purchase price or recent highs as a reference point for decisions.	Reluctance to sell below purchase price, even when rational.	Base decisions on intrinsic value and market conditions, not past prices.
Following the herd	Social proof: Fear of missing out (FOMO) or fear of losses drives herd behavior.	Amplification of market trends, potentially leading to bubbles or crashes.	Maintain a contrarian approach when appropriate and rely on independent analysis rather than market noise.
Fluctuating risk preferences	Gains encourage risk aversion; losses encourage risk-seeking behavior.	Inconsistent strategies and poor long-term returns.	Stick to a pre-defined risk management strategy and avoid emotional decision-making based on recent results.

This table offers a concise overview of the challenges posed by Prospect Theory and practical steps traders can take to counteract its effects.

Adapted from:

Thinking, Fast and Slow by Daniel Kahneman