Decision-making under risk and uncertainty

Outline The Standard Economic Model Prospect Theory Reference Points Loss Aversion Diminishing Marginal Sensitivity Weighted Probability Ambiguity Aversion The Endowment Effect Emotion

The standard economic model The standard economic model under risk is expected utility theory (EUT). Decision-making under risk can be considered as a process of choosing between prospects. In general terms a prospect can be described mathematically as (x1, p1 ;….. ; xn,pn ). EUT states that consumers will behave in such a way that they will maximize the preference function

Prospect theory Prospect theory was originally developed in the KT paper of 1979 and then extended in 1992. PT models choice as a two-phase process: the first phase involves editing, and the second involves evaluation. 1. Editing. This phase consists of a preliminary analysis of the offered prospects that aims to yield a simpler representation of these prospects. 2. Evaluation. The decision-maker evaluates each of the edited prospects, and is assumed to choose the prospect with the highest value.

Prospect theory The first scale v assigns to each outcome x a number v(x), which reflects the subjective value of that outcome. This scale entails an explanation of reference points, loss-aversion, and diminishing marginal sensitivity . The second scale π associates with each probability p a decision weight π(p), which reflects the impact of p on the overall value of the prospect. This scale entails an explanation of decision weighting or weighted probability function.

Prospect theory The mathematical exposition of the KT (1979) model Consider the simple prospects of the form (x,p;y,q) 1. Regular prospects V(x, p; y, q)= π (p)v(x) + π(q)v(y) 2. Strictly positive or strictly negative prospects In the editing phase, such prospects are segregated into two components: the riskless component and the risky component V(x, p; y, q)= v(y)+ π(p)[v(x)-v(y)]

Prospect theory - reference points In PT outcomes are defined relative to a reference point. Thus the scale v measures the value of deviations from that reference point, that is gains and losses. It is often assumed in analysis that the relevant point is the current status of wealth or welfare, but this need not be the case. The relevant reference point may be the expected status rather than the current status.

Prospect theory - reference points Reference points are also strongly influenced by the status of others. The reference point may not correspond to the current level of wealth when a person has not yet adapted to the current status. For example, imagine a person has already lost 2,000 and is now facing a choice between a sure gain of $1,000 and an even chance to win 2,000 or nothing. How will he code the problem?

Prospect theory - loss aversion In the words of KT(1979) the aggravation that one experiences in losing a sum of money appears to be greater than the pleasure associated with gaining the same amount v(x) < -v(-x) where x > 0 Some empirical evidence Asymmetric price elasticities of demand for consumer goods Disposition effect (Shefrin and Statman 1985)

Prospect theory - shape of the utility function In PT KT proposed a utility function that featured diminishing marginal sensitivities in the domains of both gains and losses. This type of function generally implies risk aversion in the domain of gains and risk-seeking in the domain of losses. v’’(x) < 0 for x > 0 and v’’(x) > 0 for x < 0 Reflection effect. The phenomenon that the preference between negative prospects is the mirror image of the preference between positive prospects.

Prospect theory - shape of the utility function Empirical evidence choose between (4000, 0.8) and (3000) choose between (-4000, 0.8) and (-3000) The important differences between PT function and the Markowitz function Markowitz proposed a utility function that has convex and concave regions in both the gain and the loss domains. Jullien and Salanie (1997) found that the utility function for small amount of money was convex. Besides people may be risk averse for very large losses.

Prospect theory - decision weighting As with some of the previous elements of PT, decision weighting also features in other theories prior to the original KT paper. There are two reasons why decision weights may be different form objective probabilities. 1. Estimation of probabilities (objective probabilities are unknown) Two examples of situations where people are often bad at estimating probabilities: rare events and conditional probabilities.

Prospect theory - decision weighting 2. Weighting of probabilities (objective probabilities are known) π(p) measures the impact of events on the desirability of prospects and not the perceived likelihood of these events. There are a number of important characteristics of the weighting function that were observed by KT. 1. π is an increasing function of p, with π(0)=0 and π(1)=1.

Prospect theory - decision weighting 2. Subadditivity. This characteristic relates to situations where p is small the prospect (6000, 0.001) versus (3000, 0.002) the prospect (-6000, 0.001) versus (-3000, 0.002) In general terms the subadditivity principle can be expressed as follows: π(rp) > r π(p) for 0<r<1

Prospect theory - decision weighting 3. Subcertainty. Allais (1953) noted that people tend to overweight outcomes that are considered certain, relative to outcomes that are merely probable. the prospect (2400) versus (2500, 0.33; 2400, 0.66) the prospect (2500, 0.33) versus (2400, 0.34) In general terms the subcertainty principle can be expressed as π(p) + π(1-p) <1

Prospect theory-decision weighting One main implication is that preferences are generally less sensitive to variations in probability than EUT would suggest. 4. Subproportionality. The decision weighting functions violate the axiom of EUT. (3000) to (4000, 0.8) and (3000, 0.25) to (4000, 0.2) In general terms the subproportionality principle can be expressed as π(pq)/ π(p) < π(pqr)/ π(pr) 0< p, q, r <1

Prospect theory - decision weighting For a fixed ratio of probabilities, the ratio of the corresponding decision weights is closer to unity when the probabilities are low than when they are high (0.25 is judged more similar to 0.2 than 1 is to 0.8). KT (1992) proposed an inverted S-shaped weighting functions for both gains and losses. It predicts risk- seeking for gains of low probability and risk-aversion for gains of high probability, with this pattern being reversed for losses.

Ambiguity aversion In reality, the probability of many events is not defined precisely and must usually be estimated subjectively when making a decision. When it comes to analyzing cases with undefined probability distribution, the equivalent of the classical utility hypothesis of Von Neumann and Morgenstern(1944) is the theory of subjective utility proposed by Savage (1954). Ambiguity aversion: empirical psychological studies suggest that most people try to avoid lotteries with undefined probability distribution.

Ambiguity aversion People often take actions that run counter to the theory of subjective utility. The most often cited example of such irrationality is the Ellsberg’s paradox. Ellsberg (1961). Subjects are presented with two urns. Urn 1 contains 100 balls, some of which are red, some blue. The ratio of blue and red balls is not known. Urn 2 contains the total of 100 balls—50 red and 50 blue. Respondents are asked to choose (1) a1 versus a2 (2) b1 versus b2

Ambiguity aversion In both choices most people prefer to draw from the urn with defined probability distribution choosing lottery a2 and b2 . Such behavior is irrational. Heath and Tversky (1991) suggest that the degree of ambiguity aversion may depend on how competent a decision maker feels in the field where he is supposed to estimate probability. The higher qualifications he thinks he has, the less concerned and more ready to accept the ambiguous situation he will be.

The endowment effect Status Quo bias: Samuelson and Zeckhauser (1988) documented that preferences may be heavily dependent on the status quo present when making a decision. We are incredibly often reluctant to take steps that would change the current situation. We are biased toward maintaining the status quo even though our preferences would be completely different if we were to make the same choices without any information on the current state of affairs.

The endowment effect Related to the status quo effect is the so-called endowment effect . It consists in people attaching more value to things they currently have than to identical objects that are not in their possession. Kahneman et al. (1990) demonstrated that for the owner of a thing the discrepancy between the sales price (he would be willing to sell) and the purchase prices (he would be prepared to pay) occurred not because the suggested purchase price was deflated but because owners inflated the sales price.

The endowment effect Loewenstein and Kahneman (1991) argue that the endowment effect does not result from assessing a given thing as particularly attractive, but first and foremost from the discomfort related to parting with something we already received.

emotion Mood and Weather Psychologists argue that the decisions we make may be heavily dependent on emotional states. In general, people in good mood are more optimistic in their judgments and more willing to take risks. The role of emotions in risk perception and decision making was systematically expanded on by Loewenstein et al. (2001). Dowling and Lucey (2005) also provide an overview of literature on the role of emotions and feelings in decision making.

emotion It is widely documented that mood depends on weather conditions. More daylight means less depression, less skepticism and more optimism, happiness, and well-being. Observations made by psychologists inspired studies into weather’s influence on stock market returns. Saunders (1993) proved that there is a statistical relationship between the level of cloud cover over New York City and changes of the Dow Jones Industrial index as well NYSE/AMEX indexes.

emotion With total cloud cover, returns for the indexes were usually below average. On sunny days (cloud cover up to 20%), indexes usually increased more than on average. Hirshleifer and Shumway (2003) carried out more comprehensive studies to analyze the influence of weather on index changes on 26 stock markets. The negative correlation between the level of could cover and rates of return was basically confirmed on 22 stock markets, although with relatively low statistical significance.

emotion One general remark about testing weather impact on stock returns relates to the implicit assumption that traders are geographically concentrated in the area where the exchange is located and the weather is observed. Regret: Regret is a psychological reaction to making a choice whose outcomes proved disadvantageous (Bell, 1982). The feeling of regret will be especially strong when it turns out that an alternative, previously rejected in favor of the wrong decision, would have brought desired results.

emotion We usually are less regretful of lost profits we have not earned because we did not decide to act than of the same losses suffered as a result of a wrong decision. The feeling of regret will also be much greater if the wrong decision is taken as an exception from rules or habits normally adhered to. Regret will also be exacerbated if the wrong decision was made individually. When this happens, one cannot put the blame on anyone else or identify external factors that could be made responsible for the failure.

emotion Disappointment. An emotion that is close to regret but usually has less of an impact on the decision- making process is disappointment. Disappointment is experienced when the results of a choice fall short of decision maker’s expectations. Greed and Fear Representatives of behavioral finance argue that investors experience two strong, contradictory feelings when making decisions in the capital market. Greed is related to the prospect of enrichment and it is the main causative factor for accepting risky investments.

emotion Fear is a negative feeling triggered by the possibility of suffering a loss and it works in an opposite direction, discouraging risky behavior. It is a sort of emergency brake preventing investors from taking excessive risk when pursuing profits.