Lecture 3_no solution.pptx

# Utility functions and risk aversion This topic explores how individuals make decisions under uncertainty, focusing on the concept of utility functions and their relationship to wealth, which explains the phenomenon of risk aversion through diminishing marginal utility and concave functions. ## 1. Utility functions and risk aversion ### 1.1 The utility function over final wealth In Expected Utility Theory (EUT), utility is defined over an individual's final wealth. The core idea is that the value or satisfaction derived from an additional amount of money is not constant. For instance, an extra one hundred dollars is more valuable to someone with one hundred dollars than to someone with one million dollars. This implies that the utility of money increases at a slower rate than wealth itself. Mathematically, this is represented by a **concave function** of wealth. > **Tip:** Concavity of the utility function is the fundamental mathematical representation of risk aversion. The association between wealth and its utility varies among individuals. A concave function signifies **diminishing marginal utility of wealth**, meaning each additional unit of wealth provides less additional utility than the previous unit. This diminishing marginal utility is the direct explanation for risk aversion. ### 1.2 Risk aversion explained **Risk aversion** is the tendency for individuals to prefer a certain amount of money over a gamble that has an equal or higher expected monetary value. **Example of risk-averse behavior:** Consider two options: * **Option A (Gamble):** A 50% chance of having one million dollars or a 50% chance of having seven million dollars. * **Option B (Sure Amount):** Having four million dollars for sure. If an individual chooses Option B, they are exhibiting risk aversion. This choice is made because the utility gained from having four million dollars for sure is greater than the expected utility derived from the gamble. The utility of a gamble is calculated as the weighted average of the utilities of its possible outcomes. For Option A, if we denote the utility of wealth $W$ as $U(W)$: * Expected Utility of Option A: $E[U(A)] = 0.5 \times U(\text{one million dollars}) + 0.5 \times U(\text{seven million dollars})$ * Utility of Option B: $U(B) = U(\text{four million dollars})$ A risk-averse individual will choose Option B if $U(\text{four million dollars}) > E[U(A)]$. This preference arises because the utility function is concave. ### 1.3 Visualizing risk preferences The relationship between wealth and utility can be visualized on a graph, with wealth on the x-axis and utility on the y-axis. * **Risk-averse:** The utility function is **concave**. As wealth increases, the slope of the utility function (marginal utility) decreases. * **Risk-neutral:** The utility function is **linear**. Marginal utility is constant. * **Risk-loving:** The utility function is **convex**. As wealth increases, the slope of the utility function (marginal utility) increases. ### 1.4 Prospect Theory and changes in wealth While Expected Utility Theory (EUT) defines utility over final wealth, **Prospect Theory (PT)**, developed by Kahneman and Tversky, proposes that utility is defined over **changes in wealth** (gains and losses) relative to a reference point. This leads to several key implications: * **Reference dependence:** Individuals derive utility from gains and losses relative to a reference point, which can be an initial state of wealth or an expectation. * **Loss aversion:** People have a disproportionate dislike for losses compared to gains of the same magnitude. The utility function has a **kink** at the reference point, indicating that losses are more painful than equivalent gains are pleasurable. * **Diminishing sensitivity:** The value function is concave over gains (similar to EUT) and convex over losses. This means that the impact of a gain diminishes as it gets larger, and the impact of a loss also diminishes as the loss becomes larger (though still more impactful than equivalent gains). * **Nonlinear probabilities:** Individuals do not perceive probabilities linearly. Small probabilities are often **overweighted**, and large probabilities are often **underweighted**. This leads to the **fourfold pattern of risk attitudes**, where individuals can be risk-seeking for low-probability gains (e.g., playing the lottery) and risk-averse for high-probability gains, while being risk-seeking for low-probability losses and risk-averse for high-probability losses. The value function $v(\cdot)$ in Prospect Theory for changes in wealth $y$ can be formally expressed as: $$ v(y) = \begin{cases} y^\alpha & \text{if } y \geq 0 \\ -\lambda (-y)^\beta & \text{if } y < 0 \end{cases} $$ where $0 < \alpha < 1$ and $0 < \beta < 1$ (indicating concavity over gains and convexity over losses, respectively), and $\lambda > 1$ (indicating loss aversion, meaning people are more sensitive to losses than to gains). For instance, $\alpha \approx 0.88$, $\beta \approx 0.88$, and $\lambda \approx 2.25$ have been estimated, suggesting people are about twice as sensitive to losses as to gains. The probability weighting function $w(p)$ transforms objective probabilities $p$. A common form is: $$ w(p) = \frac{p^\gamma}{(p^\gamma + (1-p)^\gamma)^{1/\gamma}} $$ with $\gamma > 0$. For $\gamma = 0.65$, small probabilities are generally overweighted and large probabilities are underweighted. The value of a prospect $A(p, x, y)$ with probability $p$ of outcome $x$ and probability $1-p$ of outcome $y$ is calculated as: $$ V(p, x, y) = w(p) \times v(x) + w(1-p) \times v(y) $$ This formula applies to the mixed domain (prospects with both gains and losses). For pure domains (only gains or only losses), the probability weighting function is adjusted. > **Tip:** The key distinction between EUT and Prospect Theory lies in the domain of utility (final wealth vs. changes in wealth) and the treatment of probabilities (linear in EUT vs. non-linear in PT). ### 1.5 Examples of behavior explained by Prospect Theory Prospect Theory offers explanations for various observed economic and financial phenomena that are puzzling under traditional EUT: * **Equity Premium Puzzle:** The reluctance of investors to invest in stocks despite their historically higher returns compared to bonds can be explained by loss aversion and myopic (short-term) bracketing. Investors fear short-term losses more acutely. * **Disposition Effect:** The tendency to sell winning stocks too early and hold onto losing stocks for too long is explained by risk aversion over gains (leading to selling winners) and risk seeking over losses (leading to holding losers in the hope of breaking even). * **Cross-Section of Average Returns:** The fact that certain stock characteristics, like positive skewness, appear to command a price premium (leading to lower average returns) can be explained by investors' tendency to overweight small probabilities of large gains. ### 1.6 Limitations and challenges of Prospect Theory Despite its explanatory power, Prospect Theory faces challenges: * **Defining reference points:** The choice of the reference point for gains and losses can be ambiguous and subjective. * **Empirical reproduction of the fourfold pattern:** Some studies suggest the fourfold pattern might be more a feature of specific experimental designs (e.g., choice lists) than a universal behavioral tendency. Alternative elicitation methods, such as binary choices, sometimes fail to reproduce the full pattern. * **Overspecification:** The theory's flexibility in defining reference points and applying loss aversion can sometimes lead to explaining too many phenomena, raising concerns about overspecification. ### 1.7 Exercises and Applications Various exercises illustrate the application of utility functions and Prospect Theory: * **Plotting utility and probability weighting functions:** Using certainty equivalents from experimental tasks to non-parametrically estimate utility and probability weighting functions. * **Calculating expected utility and certainty equivalents:** For given utility functions (e.g., exponential $U(w) = \exp(w)$ or power $U(w) = w^{1/2}$) and prospect distributions, one can calculate expected wealth, expected utility, and the certainty equivalent. Comparing the certainty equivalent to the expected value reveals risk preferences (CE $<$ EV for risk-averse, CE $>$ EV for risk-seeking). * **Ordering prospects:** Using expected utility to rank different risky prospects. The understanding of utility functions and risk aversion is crucial for analyzing individual decision-making in financial markets and for understanding various economic puzzles and applications in fields like finance, litigation, and policy. --- # Prospect theory fundamentals Prospect theory offers an alternative framework to Expected Utility Theory (EUT) by defining utility over changes in wealth relative to a reference point, incorporating key psychological factors that influence decision-making under risk. ### 2.1 Core tenets of prospect theory Prospect theory posits that individuals evaluate outcomes not in absolute terms of final wealth, but in terms of gains and losses relative to a subjective reference point. This reframing leads to several distinct features that differentiate it from EUT. #### 2.1.1 Reference dependence Value is derived from changes in wealth, meaning that whether an outcome is perceived as a gain or a loss depends on an individual's reference point. This reference point is not necessarily zero and can be influenced by prior expectations or the current state. #### 2.1.2 Loss aversion A central tenet of prospect theory is that losses are felt more intensely than equivalent gains. This means the disutility of losing a certain amount of money is greater than the utility of gaining the same amount. Graphically, this is represented by a kink in the value function at the reference point, with the slope being steeper for losses than for gains. * **Tip:** Loss aversion explains why people are often willing to take more risks to avoid a loss than to achieve a gain of the same magnitude. #### 2.1.3 Diminishing sensitivity The value function in prospect theory exhibits diminishing sensitivity. This means that for gains, the marginal value of an additional dollar decreases as wealth increases (concavity), and for losses, the marginal disutility of an additional dollar also decreases as wealth decreases (convexity). * **Example:** An increase in wealth from 100 dollars to 200 dollars feels more significant than an increase from 1,000,000 dollars to 1,001,000 dollars. Similarly, a loss of 100 dollars from 1000 dollars is more impactful than a loss of 100 dollars from 1,000,000 dollars. #### 2.1.4 Nonlinear probability weighting Prospect theory suggests that individuals do not perceive probabilities linearly. Instead, they transform objective probabilities using a probability weighting function. This function typically overweights small probabilities and underweights large probabilities. * **Tip:** The overweighting of small probabilities contributes to the appeal of lotteries (small chance of a large gain), while the underweighting of large probabilities can explain risk-seeking behavior in some loss scenarios. ### 2.2 The value function The value function, denoted by $v(\cdot)$, replaces the utility function in EUT and is defined over changes in wealth. It is characterized by reference dependence, loss aversion, and diminishing sensitivity. * The function is typically defined as: $$ v(x) = \begin{cases} x^{\alpha} & \text{if } x \geq 0 \\ -\lambda (-x)^{\beta} & \text{if } x < 0 \end{cases} $$ where: * $x$ represents the change in wealth (gain or loss). * $\alpha$ and $\beta$ are exponents between 0 and 1, reflecting diminishing sensitivity for gains and losses, respectively. Typical estimates for $\alpha$ and $\beta$ are around $0.88$. * $\lambda$ is the loss aversion coefficient, which is greater than 1, indicating that losses loom larger than gains. A common estimate for $\lambda$ is $2.25$, suggesting people are about twice as sensitive to losses. * **Tip:** The concavity of $v(x)$ for gains ($\alpha < 1$) and convexity for losses ($\beta < 1$) captures diminishing sensitivity. The steeper slope for losses due to $\lambda > 1$ represents loss aversion. ### 2.3 The probability weighting function The probability weighting function, denoted by $w(\cdot)$, transforms objective probabilities $p$ into decision weights. This function captures how individuals subjectively perceive the likelihood of events. * A commonly used functional form for the probability weighting function is: $$ w(p) = \frac{p^{\gamma}}{(p^{\gamma} + (1-p)^{\gamma})^{1/\gamma}} $$ where $\gamma$ is a parameter greater than 0. * When $\gamma < 1$, small probabilities are overweighted, and large probabilities are underweighted. A typical estimate for $\gamma$ is $0.65$. * **Example:** For a probability $p=0.05$, $w(0.05)$ might be around $0.179$, indicating an overestimation of the likelihood. For a probability $p=0.90$, $w(0.90)$ might be around $0.745$, indicating an underestimation. * **Tip:** The S-shape of the probability weighting function (with $\gamma < 1$) is crucial for explaining the "fourfold pattern" of risk attitudes. ### 2.4 Calculating the value of a prospect The overall value of a prospect $A(p, x; y)$ (where $p$ is the probability of outcome $x$, and $1-p$ is the probability of outcome $y$) is calculated by combining the value function and the probability weighting function. * For prospects involving both gains and losses (mixed domain), the value $V$ is: $$ V(p, x; y) = w(p)v(x) + w(1-p)v(y) $$ where $v(x)$ and $v(y)$ are values from the value function and $w(p)$ and $w(1-p)$ are the decision weights. * For prospects involving only gains or only losses (pure domain), the formula simplifies, and it is often assumed that $w(1-p)$ is replaced by $1-w(p)$. * **Example:** Consider a prospect with a $0.001$ probability of gaining $5000$ dollars and a $0.999$ probability of receiving $0$ dollars. Using prospect theory parameters ($\alpha \approx 0.88$, $\gamma \approx 0.65$, and assuming $v(0)=0$), the value would be: * Calculate the decision weight for $p=0.001$: $w(0.001) \approx 0.011$. * Calculate the value of the gain: $v(5000) = 5000^{0.88} \approx 1799.26$. * The overall value is $V \approx 0.011 \times 1799.26 + (1-0.011) \times v(0) \approx 19.79$. * **Tip:** This calculation highlights how prospect theory can assign a higher perceived value to small-probability events than EUT would, due to the overweighting of small probabilities. ### 2.5 The fourfold pattern of risk attitudes Prospect theory explains the "fourfold pattern" of risk attitudes, which describes how people behave differently depending on whether they are facing gains or losses, and whether the probabilities are small or large. 1. **Small probability gains:** People are typically **risk-seeking** (e.g., buying lottery tickets), driven by the overweighting of small probabilities. 2. **Large probability gains:** People are typically **risk-averse** (e.g., preferring a sure gain over a gamble with a higher expected value), driven by diminishing sensitivity and underweighting of large probabilities. 3. **Small probability losses:** People are typically **risk-seeking** (e.g., gambling to avoid a certain small loss), driven by the overweighting of small probabilities, making the potential for avoiding the loss seem more appealing. 4. **Large probability losses:** People are typically **risk-averse** (e.g., buying insurance), driven by loss aversion and diminishing sensitivity. ### 2.6 Applications and implications of prospect theory Prospect theory has significant implications across various fields, particularly in finance, by offering explanations for anomalies that traditional EUT struggles to account for. #### 2.6.1 Equity premium puzzle The equity premium puzzle refers to the historically high average return of stocks over bonds, which is much larger than predicted by standard EUT models unless individuals are assumed to be extremely risk-averse. Prospect theory explains this by: * **Loss aversion:** Investors are sensitive to short-term losses in stock markets, which are more frequent than in bond markets. * **Myopic (narrow) framing:** Investors tend to evaluate investment performance over short periods in isolation, leading them to experience more frequent perceived losses. * **Implication:** This suggests that reducing the frequency of portfolio reports (e.g., from quarterly to yearly) might encourage more investment in stocks by mitigating the impact of short-term fluctuations. #### 2.6.2 Disposition effect The disposition effect describes the tendency for investors to sell winning assets too early and hold onto losing assets for too long. Prospect theory explains this through: * **Risk aversion over gains:** Investors want to lock in realized gains and avoid the potential of losing them. * **Risk seeking over losses:** Investors are willing to gamble on losing assets to avoid the pain of realizing a loss. * **Tip:** This behavior contributes to momentum effects in stock markets, where past winners continue to perform well and past losers continue to perform poorly. #### 2.6.3 Cross-section of average returns Prospect theory can help explain why certain stocks have different average returns, which is not always well-predicted by models like CAPM that solely rely on systematic risk. * **Skewness preference:** Investors may be attracted to assets with positive skewness (a small probability of very large gains), even if their average long-term return is low. This is because they overweight the small probability of a significant payoff. * **Implication:** This can lead to lower average returns on assets like Initial Public Offerings (IPOs) and under-diversification, as investors may concentrate on positively skewed assets. #### 2.6.4 Litigation The fourfold pattern of risk attitudes can also shed light on settlement decisions in legal contexts: * **High probability gains:** Individuals might accept unfavorable settlements due to risk aversion for likely gains. * **High probability losses:** Individuals might reject favorable settlements, gambling on the small chance of a complete win. * **Low probability gains:** Individuals might reject favorable settlements, similar to the lottery effect. * **Low probability losses:** Individuals might accept unfavorable settlements to avoid a small chance of a large loss. ### 2.7 Challenges and limitations of prospect theory Despite its explanatory power, prospect theory faces several challenges: * **Defining the reference point:** A fundamental difficulty is determining the appropriate reference point, which can be subjective and context-dependent. What constitutes a "gain" or "loss" can vary significantly. * **Overspecification:** The theory's ability to explain phenomena like the disposition effect through multiple mechanisms (e.g., loss aversion and risk seeking over losses) might indicate overspecification, where a single model is used to explain diverse behaviors. * **Experimental elicitation:** The reliance on certainty equivalents (CEs) as an elicitation method has been questioned. Some research suggests that behaviors observed in binary choice experiments may not always align perfectly with those derived from CE elicitation, potentially challenging the universality of the fourfold pattern in all choice settings. * **Narrow framing:** Prospect theory is often applied in conjunction with "narrow framing," where decisions are evaluated in isolation. However, in reality, individuals may integrate risks more broadly, which prospect theory does not fully capture. --- # Applications and limitations of prospect theory Prospect theory offers a more descriptive account of decision-making under risk than expected utility theory, particularly by incorporating psychological factors like reference dependence, loss aversion, and probability weighting. This section explores its applications in explaining financial anomalies and its limitations. ### 3.1 Prospect theory's core components Prospect theory, developed by Tversky and Kahneman, posits that individuals derive utility from changes in wealth (gains and losses) relative to a reference point, rather than from absolute wealth levels. It is characterized by: * **Reference dependence:** The value of an outcome is determined by its difference from a reference point, leading to the perception of gains and losses. * **Loss aversion:** Losses loom larger than equivalent gains. This is represented by a kink in the value function at the reference point, with the slope for losses being steeper than for gains. * **Diminishing sensitivity:** The value function is concave over gains (meaning the marginal value of an additional dollar decreases as wealth increases) and convex over losses (meaning the marginal disutility of an additional dollar lost decreases as losses become larger). * **Probability weighting:** Individuals do not perceive probabilities linearly. They tend to overweight small probabilities and underweight large probabilities. The value of a prospect $(p, x; 1-p, y)$ (a probability $p$ of outcome $x$ and a probability $1-p$ of outcome $y$) under prospect theory is given by: $$V(p, x; 1-p, y) = w(p)v(x) + w(1-p)v(y)$$ where $v(\cdot)$ is the value function and $w(\cdot)$ is the probability weighting function. For prospects with only gains or only losses (pure domains), the formula simplifies to $V(p, x) = w(p)v(x)$. #### 3.1.1 The value function The value function $v(\cdot)$ is defined as: $$ v(x) = \begin{cases} x^{\alpha} & \text{if } x \geq 0 \\ -\lambda(-x)^{\beta} & \text{if } x < 0 \end{cases} $$ where $0 < \alpha < 1$ and $0 < \beta < 1$. Typically, $\alpha$ and $\beta$ are estimated to be around $0.88$, indicating concavity for gains and convexity for losses. The parameter $\lambda > 1$ captures loss aversion, with an estimated value of approximately $2.25$, suggesting people are more than twice as sensitive to losses as to gains. #### 3.1.2 The probability weighting function The probability weighting function $w(\cdot)$ is suggested by Tversky and Kahneman (1992) to be: $$w(p) = \frac{p^{\gamma}}{(p^{\gamma} + (1-p)^{\gamma})^{1/\gamma}}$$ where $\gamma > 0$. An estimated value of $\gamma = 0.65$ suggests that small probabilities are often overweightened, and large probabilities are underweightened. This results in a probability weighting function that deviates from the straight line $w(p)=p$, which represents risk neutrality in probability perception. > **Tip:** The S-shape of the value function (concave for gains, convex for losses) and the inverse S-shape of the probability weighting function (overweighting small probabilities, underweighting large probabilities) are key to understanding the "fourfold pattern" of risk attitudes described by prospect theory. ### 3.2 Applications of prospect theory Prospect theory has been applied to explain several empirical observations and puzzles in finance and decision-making. #### 3.2.1 The equity premium puzzle The equity premium puzzle refers to the historically large difference between the average returns of stocks and risk-free bonds, suggesting that investors should have a much higher propensity to invest in stocks than observed. Traditional Expected Utility Theory (EUT) struggles to explain this reluctance without assuming extremely high levels of risk aversion. Prospect theory offers an explanation by considering: * **Loss aversion:** Stock markets exhibit significant short-term volatility, leading to frequent perceived losses. Loss-averse individuals may reduce or avoid stock investments due to the fear of these losses, even if the long-term trend is positive. * **Myopic or narrow framing:** Investors may focus on short reporting periods (e.g., quarterly) in isolation, rather than the overall long-term investment. This myopia exacerbates the impact of short-term losses due to loss aversion. Experiments show that increasing the delay in feedback on portfolio performance can increase stock market investments, consistent with the idea that less frequent reporting reduces the salience of short-term losses. #### 3.2.2 The disposition effect The disposition effect describes investors' tendency to sell assets that have appreciated (winners) too early and hold onto assets that have depreciated (losers) for too long. Prospect theory explains this by: * **Risk aversion over gains and risk seeking over losses:** When an asset is trading at a gain, investors are risk-averse and prefer to lock in the profit by selling. When an asset is trading at a loss, they become risk-seeking, holding onto it in the hope that it will recover to break even and avoid realizing a painful loss. * **Loss aversion and reference points:** Investors may hold onto losing assets to bring them back to their purchase price (a reference point), thereby avoiding the psychological pain of a realized loss. This effect has also been observed in the housing market, where sellers of properties at a loss tend to set higher asking prices than comparable houses. However, the existence of multiple explanations for the disposition effect within prospect theory can be seen as a limitation, suggesting potential overspecification of the model. #### 3.2.3 Cross-section of average returns Traditional finance models like the Capital Asset Pricing Model (CAPM) explain asset returns based on systematic risk. However, empirical data often shows that CAPM does not fully capture the cross-section of average returns, as high beta stocks do not consistently yield higher average returns. Prospect theory, particularly through probability weighting, can offer explanations: * **Overweighting low-probability, high-payoff events:** Investors may be attracted to assets with a small probability of very large returns, even if their long-term average return is low. This is because they overweight these small probabilities. * **Positively skewed securities:** Assets with positively skewed return distributions (a long tail on the right, indicating a small chance of a very large gain) may be overvalued. Investors are willing to pay a higher price (thus accepting a lower average return) for the prospect of these exceptionally large gains, due to overweighting the low probability of such an event. This can help explain phenomena such as the low long-run average returns on Initial Public Offerings (IPOs), which are often positively skewed, and under-diversification, where individuals may hold more positively skewed stocks to increase their chances of a large windfall. #### 3.2.4 Litigation decisions The fourfold pattern of risk attitudes in prospect theory can shed light on out-of-court settlement decisions in litigation. The theory suggests different behaviors based on the probability and valence (gain/loss) of outcomes: * **High probability gains:** Individuals may accept unfavorable settlements because the outcome is likely positive. * **High probability losses:** Individuals may reject favorable settlements and gamble further, as they are in the domain of high probability of losses and may seek to avoid them by gambling for a rare positive outcome. * **Low probability gains:** Individuals may reject favorable settlements, hoping for a large improbable gain. * **Low probability losses:** Individuals may accept unfavorable settlements, as the potential loss is perceived as unlikely. Understanding these patterns can inform policy interventions, such as framing effects, to influence litigation behavior and potentially combat a "compensation culture." ### 3.3 Limits and challenges of prospect theory Despite its explanatory power, prospect theory faces several challenges and limitations: * **Defining reference points:** A fundamental difficulty in applying prospect theory is defining the relevant reference point. Gains and losses can be relative to total wealth, financial wealth, specific assets, or even expected returns compared to a risk-free rate. The choice of reference point can significantly alter predictions. * **Frequency and type of gains/losses:** The frequency (e.g., annual, weekly, daily) and specific nature of gains and losses also need to be defined, adding complexity to its application. * **Reliance on Certainty Equivalents (CEs) as elicitation method:** Early empirical tests heavily relied on eliciting certainty equivalents. Some research suggests that when prospect theory parameters are measured using binary choices instead of certainty equivalents, the fourfold pattern may not hold as robustly, potentially requiring new explanations for the coexistence of gambling and insurance. * **Overspecification:** As seen with the disposition effect, prospect theory can sometimes offer multiple explanations for a single phenomenon, which can be a sign of overspecification and a lack of parsimony. * **Narrow framing:** Prospect theory is often implemented with "narrow framing," where risks are evaluated in isolation. In reality, individuals may integrate risks with their overall financial situation, which is more in line with traditional EUT. > **Tip:** Be mindful of how reference points and framing are defined when analyzing decisions through the lens of prospect theory. The "correct" definition can be subjective and context-dependent. In summary, prospect theory provides a powerful behavioral framework for understanding deviations from rational decision-making. Its key contributions lie in accounting for loss aversion, reference dependence, and probability weighting, which are crucial for explaining financial puzzles and other observed behaviors. However, challenges in defining core concepts and empirical validation using different elicitation methods highlight areas for further research and refinement. --- ## Common mistakes to avoid - Review all topics thoroughly before exams - Pay attention to formulas and key definitions - Practice with examples provided in each section - Don't memorize without understanding the underlying concepts

| Term | Definition | |------|------------| | Utility Function | A mathematical representation that describes an individual's preferences for different outcomes, typically relating wealth or consumption levels to a measure of satisfaction or utility. It is often expressed as $U(Wealth)$. | | Risk Aversion | The tendency of individuals to prefer a certain outcome over a gamble with an equal or higher expected value. This behavior is explained by the concept of diminishing marginal utility of wealth. | | Diminishing Marginal Utility | The principle that as an individual's wealth increases, the additional utility gained from each extra unit of wealth decreases. This means that the first €100 is worth more in utility than the next €100 when one already possesses substantial wealth. | | Concave Function | A function where the line segment connecting any two points on the function lies below or on the function itself. In the context of utility, a concave utility function signifies diminishing marginal utility and, consequently, risk aversion. | | Expected Utility Theory (EUT) | A framework that posits individuals make decisions under risk by choosing the option that maximizes their expected utility. Utility is defined over final wealth levels, and probabilities are treated as objective. | | Certainty Equivalent (CE) | The guaranteed amount of money that an individual would accept as an equivalent to a gamble or uncertain prospect. If the CE is less than the expected value of the gamble, the individual is risk-averse. | | Risk Neutral | An individual whose utility function is linear. They are indifferent between a certain amount and a gamble with the same expected monetary value, as their utility is directly proportional to the monetary amount. | | Risk Loving | An individual whose utility function is convex. They prefer a gamble with a certain expected value over a certain amount of money, indicating they are willing to take on more risk for potentially higher returns. | | Prospect Theory (PT) | A descriptive model of decision-making under risk that contrasts with Expected Utility Theory. It proposes that utility is defined over changes in wealth (gains and losses) relative to a reference point, and that probabilities are subjectively weighted. | | Reference Dependence | A key concept in Prospect Theory where the utility or value derived from an outcome depends on its difference from a reference point, rather than its absolute level. Gains and losses are evaluated relative to this point. | | Loss Aversion | The psychological phenomenon, central to Prospect Theory, where individuals experience a greater negative emotional impact from a loss than a positive emotional impact from an equivalent gain. This leads to a steeper value function for losses than for gains. | | Diminishing Sensitivity | In Prospect Theory, this refers to the concavity of the value function for gains (meaning additional gains are worth less) and the convexity of the value function for losses (meaning additional losses are less impactful). | | Nonlinear Probability Weighting | The phenomenon described in Prospect Theory where individuals do not perceive probabilities linearly. Small probabilities are often overweighted, while large probabilities are underweighted. | | Value Function | In Prospect Theory, this function assigns a subjective value to gains and losses relative to a reference point. It is typically concave for gains and convex for losses, with a kink at the origin reflecting loss aversion. | | Probability Weighting Function | In Prospect Theory, this function transforms objective probabilities into subjective decision weights, reflecting how individuals perceive and weigh different likelihoods of outcomes. | | Risk Seeking | The tendency of individuals to prefer a gamble over a certain outcome with an equal or lower expected value. This is often associated with a convex utility function. | | Fourfold Pattern of Risk Attitudes | A description of how individuals exhibit different risk attitudes (risk-seeking or risk-averse) depending on whether the outcomes are gains or losses and the probabilities associated with them. | | Probability Weighting | In Prospect Theory, individuals do not perceive probabilities linearly. Small probabilities are often overweighted, while large probabilities are underweighted, leading to a non-linear transformation of objective probabilities into subjective decision weights. | | Decision Weight | The subjective weight assigned to a probability in Prospect Theory, determined by the probability weighting function. These weights deviate from objective probabilities, especially for small and large probabilities. | | Equity Premium Puzzle | A long-standing puzzle in finance where historical data shows a significant difference between the average returns of stocks and bonds, suggesting investors should have a higher allocation to stocks than observed. Prospect Theory helps explain this by considering investors' loss aversion and myopic focus on short-term fluctuations. | | Disposition Effect | The tendency for investors to sell assets that have increased in value (winners) too early and hold onto assets that have decreased in value (losers) for too long. Prospect Theory explains this through risk aversion over gains and risk-seeking over losses. | | Narrow Framing | An assumption in some applications of Prospect Theory where individuals evaluate risks in isolation, separately from other concurrent risks or their overall wealth. This contrasts with traditional models where risks are integrated into a broader portfolio. |