# Prospect theory in decision making

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Prospect theory is an important theory for decision-making between alternatives that involve risk. The theory departs from the traditional expected utility theory because it attempts to explain how people really make decisions between risky alternatives, which attempts to model optimal decisions. This vital difference leads to the prospect theory departing from the traditional framework in important ways. Unlike the traditional approach, it attempts to incorporate psychology into the consideration process to provide a behavioural approach to portfolio selection (Barberis, Nichola, Huang, & Santos, 2001). During the course of this report, we will first look at how prospect theory differs from the traditional expected utility theory to gain a better understanding of the concept. Following this will be, a discussion of the key elements of prospect theory - the value function including a small reference to endowment effect and the status quo bias, reflection and framing effect, isolation effect and probabilistic insurance. Towards the end, we will have a precise look at the applications of prospect theory - equity premium puzzle and home bias.

The traditional finance theory assumes that investors try to maximize expected utility of wealth when they are making decisions under uncertainty. However, many studies have shown that the underlying assumptions of the traditional theory do not accurately describe how people actually behave when choosing among risky alternatives. This inadequacy leads to the weak correlation between the utility theory model and real decisions.

There are "four key features [that] distinguish prospect theory from mean-variance theory, which is the traditional approach to modelling decision-making." First, according to the traditional theory people choose among alternatives based on how the outcomes will affect their overall wealth. However, according to prospect theory people evaluate outcomes in terms of gains and losses relative to a reference point. So decisions are based on how the outcome changes their income, in relation to their reference point. (Han & Hsu, 2004).

Second, the mean variance analysis makes the assumption that people are risk averse in all their choices. In contrast, prospect theory agents are risk-averse in the domain of gains but are risk seeking when all changes in income are framed as losses.

The third feature of prospect theory is "loss aversion." An individual is loss averse if she or he dislikes symmetric 50-50 bets and their degree of aversion increases with the absolute size of the stakes. In other words, prospect agents don't perceive gains and losses of equal amounts evenly. For example, the loss of a particular amount is more painful then the pleasure received from the gain of an equal amount. This is also known as the endowment affect. People place a higher value on a good that they own than goods that they do not, and are willing to accept a higher risk if it means that they can avoid the loss.

Finally, in utility theory risk is treated objectively, by its probabilities. In contrast, the utility under prospect theory is not dependant on the original probability but rather on the transformed probability also known as decision weights. "They do not just measure the perceived likelihood of an event. Instead, they measure how events will impact the desirability of prospects." (Han & Hsu, 2004)

This feature of the prospect theory helps explain a number of violations of expected utility theory, including the famous Allais' paradox. People in prospect theory tend to overweight small probabilities. This overweighting explains why people buy lottery tickets offering a small chance of large gain, and insurance protecting against a small chance of a large loss (Kahneman & Tversky, 1979).

The four elements explained above and how risk is evaluated is usually explained by the 'value function'. The concept of the value function is based on gains and losses from a reference point, as explained in the first element of prospect theory above. Value function stresses the importance of the reference point (starting point) although changes and movement are observed more compared to the resting point, due to the concept of gains and loss. The following is the prospect theory value function:

Ï€= non-linear weighting function

V(x-r) = the value function

R= the reference point

PT = âˆ‘ Ï€ (pi) v(xi - r)

This function creates an S-shaped curve (Figure 1.1)

Figure 1.1

The curve clearly highlights the reference point, from where onwards gains and losses can be observed. It displays that as your gain increases the desire for it decreases demonstrating that people are risk averse when it comes to gains. On the contrary, as the loss increases the fear for more loss increases hence showing that people are risk seeking regarding losses. These two factors are highlighted in the graph by the steepness of the relevant sides. As gains increase the steepness decreases (indirectly proportional) and as losses increase the steepness increases (directly proportional) (Maher, 2010).

An example for this irrational behaviour is how a random sample would prefer to spend their $400.

Gain

Option A, where you will have a 100% chance of gaining $200

Option B, where you will have 50% chance of gaining $400 and a 50% chance of gaining $0

Loss

Option A, where you will have 100% chance of losing $500

Option B, where you will have 50% chance of losing $1000 and 50% of losing $0

In this scenario the vast majority of people would choose option A for gain and option B for loss confirming that people weight their losses more compared to their gains. As they would settle for a rational gain (even if it is small) but when it comes to losses they would prefer risk seeking to limit their loss. The determination of utility relating to the gain or loss mirrors the concept of psychophysical principle concerning the evaluation of outcomes.

This reflects loss aversion which then implicates two specific aspects. Firstly, the 'endowment effect' i.e. people would be willing to demand a higher value on product that they themselves own rather than a similar product that they do not. The second implication is 'status quo bias', in this case people like things to stay relatively in the same position they are in so they remain at the status quo they are in. In this scenario any sort of change either good or bad is taken to be a disadvantage.

Another key element of prospect theory is the reflection effect, which states that while investors are risk averse over prospects involving gains, they are risk seeking over prospects involving losses. This effect explains the observed preference for definite small gains over uncertain large gains and in opposition preference for uncertain large losses over small certain losses.

A remarkable interpretation of the reflection effect is that, a replacement of all positive payoffs by their negatives (reflection around zero) reverses the choice patterns. For example, a choice between a 90% choice of getting 2000 and a 45% chance of getting 4000 would be replaced by a choice between a 90% chance of losing 2000 and a 45% chance of losing 4000. This effect implies a risk-averse preference for high probability of the relatively safe 3000 gain, but a reversed preference for the risky option in the loss domain. Reflected choice patterns reported by Kahneman and Tversky (1979) were fairly high, i.e. 86% of subjects chose the safe lottery (90% chance of 3000) in the gain domain but only 8% chose the safe lottery when all payoffs were transformed into losses. (Laury & Holt, 2000).An important implication of this is the S shape of the value function in prospect theory - that is concave for gains and convex for losses.

It was also identified, that if the same decision problem was worded differently, the preferences of decision makers differed as well. This was referred to as the framing effect. Prospect theory implies a unique relationship of risk seeking to positive and negative framing- negatively framed problem encourage risk seeking.

For example: When a group of investors were faced with the following two propositions:

A gamble that offers a 10% chance of winning $95 and a 90% chance of losing $5 and another gamble B offering a 10% chance of winning $100 and a 90% chance of winning nothing. It was found that although the outcomes on both the gambles were the same, 74% of investors chose option B as paying $5(negative as compared to a loss) for the gamble than simply losing made the gamble more acceptable.

Von Restorff created the concept of the isolation effect, a way to make something that conforms within a similar a group 'stand out like a sore thumb'. An isolated item, in a list of otherwise similar items, is better remembered than an item in the same relative position in a list where all items are similar. This is a way of distracting attention from one event when the alternative holds exactly the same probability and can be of some help in explaining the prospect theory in decision making in relation to investments.

Kahneman and Tversky (1979) used the example of a two-stage test to better explain the use of the isolation effect practically in a behavioral finance situation. Isolation effect is important to show the irrationality of investors in situations that would normally produce a rational effect. This typifies the psychology of an investor having their attention diverted away from using a mean variance analysis of a situation.

The first step of the test is a .25 chance of progression to the second stage and a .75 chance of gaining nothing. The participant is asked to decide before the first stage whether, if successful, they would take 3000 or a 0.8 chance of taking 4000. It must be noted that in this game, the participant is choosing between 0.2 chance of 4000 or a 0.25 chance of 3000, the expected value of the former being greater (800 rather than 750). Of the 141 participants that Kahneman and Tversky (1979) tested, 78% chose the first option of the guaranteed 3000. The reasoning behind this is the greater appeal of the sequential certainty of the choice; most participants ignored the first stage of the experiment and just looked at the second test as a basis for their decision rather than weighing up the potential outcomes.

The concept is a strong indicator to suggest against all investors being mean variance optimising, there is clear evidence that given the right circumstances people will ignore the obvious rational choice and accept a decision based on the higher valuation of certain prospects. This evidence of irrational preference conforms to the reflection theory where the certainty of a small gain is valued higher than a chance of a large gain. Using this psychological weakness in peoples logic the Von Restorff effect distracted attention from the overall probability and coerced the decision maker into accept a decision based on something that stood out.

The rising popularity of insurance policies has been used by supporters of the utility function as strong evidence of the concavity of the utility curve for money. However Kahneman and Tversky (1979) demonstrated that not all insurance policies support this idea, basing their argument around the example of probabilistic insurance. Probabilistic insurance has also been used to highlight that decision weights tend to overweight small probabilities and large probabilities, but underweight moderate probabilities (Wakker, Thaler and Tversky, 1997).

Standard insurance provides the purchasers with a zero percent chance of any loss after a given incident, however a probabilistic insurance policy leaves the purchasers open to a small possibility that they will not be fully reimbursed. Following is an example of standard versus probabilistic insurance.

Suppose you want to insure iphone4 for the coming year, you can either insure your phone with Natwest bank for £10 a month or with ABC insurance who offer to insure the phone ever other day throughout the year for £4.50 per month. Most people would view the offer by ABC as unattractive and prefer to go with the deal offered by the bank of £10 per month. In this situation the purchaser is underweighting the fifty percent chance of damage to the phone occurring on a day that he or she is covered by the ABC insurance policy. This example demonstrates that reducing the probability of a loss from p to p over 2, is less valuable than reducing the probability of a loss from p over 2 to zero (Tomas and Viilar, 2002).

In contrast to the iphone4 insurance example given above, expected utility theory implies that probabilistic insurance is superior to regular insurance. This aversion towards probabilistic insurance is noteworthy because the most avid purchaser of insurance is still subjected to some level of risk. For example, most household contents insurance policies are void if the purchaser forgets to lock their front door.

This type of insurance represents many types of protective action, where the user pays a certain cost to reduce the probability of an undesirable event. For example, the purchase of a steering wheels lock or a carbon monoxide detector (Kahneman & Tversky, 1979).

Applications of Prospect theory

The underlying principles behind Prospect theory have been used on a number of occasions to explain a range of financial anomalies. The real world aspect of the model means it offers genuine explanations for some of the most prominent puzzles such as the Equity Premium Puzzle and Home Bias.

Equity Premium Puzzle

The equity premium puzzle refers to the empirical fact that stocks have outperformed bonds over the last century by a surprisingly large margin. Since 1926, the annual return on stocks has been around 7% while the return on bonds has been around 1% so, $1 invested in the S&P 500 on January 1, 1926 was worth $1100 by the end of 1995, while $1 invested in T-bills was worth $12.87. In 1985, Mehra and Prescott noted that under the assumptions of Expected Utility Theory, these abnormally high and low returns are difficult to explain. In 1995, Banartzi and Thaler offered an explanation to the puzzle based on key features of Prospect Theory. They claimed that the puzzle is caused mainly by two factors derived from the Prospect theory; loss aversion (investors being more sensitive to losses than gain) and a short evaluation period (investors checking their portfolio too often). This combination they termed Myopic Loss Aversion. They argue that the attractiveness, and therefore value of a stock depends on the time horizon of the investor and frequency of evaluation. The more frequently somebody evaluates their portfolio, the more likely they see their losses and suffer from loss aversion.

Putting this application into more contexts, a risky asset paying 7% per year with a standard deviation on 20%, like the average stock, has a probability of loss or gain of around 50%. For a loss averse investor who evaluates frequently, the stock market appears very risky. Considering this, an investor who is prepared to wait a long time between evaluating will find stocks much more appealing as there is an increased chance of them closing their position with a positive return. In turn, long-term investors will be willing to pay more for an identical stock than a short term, frequently evaluating investor.

Prospect theory has other various applications associated with it apart from the above mentioned equity premium puzzle. The Home bias phenomenon is another such example. This phenomenon contradicts the mean variance framework, which elucidates the benefits of international diversification helping in the minimization of risk of a given security's expected return. Home bias states that investors hold more domestic stocks and few foreign stocks than the optimal amounts actually predicted by the mean variance optimization (French and Poterba, 1991). Prospect theory explains this tendency of investors to choose domestic stocks. It says that one of the reasons for this could be a greater familiarity of investors with domestic assets and lower downside risk. This compels investors who may think globally to act locally (Campbell and Kraussl, 2006). Consider a foreign stock and a domestic stock with identical distribution payoffs. Since the foreign stock seem less familiar than the domestic one, investors may perceive it as having higher variance of payoff leading to low allocation to the foreign stock. However a direct implication of this is derived from the portfolio choice theory that home bias would decline as investors became more familiar with foreign stocks ( (Han & Hsu, 2004).

Thus, while the prospect theory can explain this behaviour of investors to concentrate risks on single assets rather than to hold a well diversified portfolio, it fails to explain why the single asset chosen by investors are domestic ones. In addition, the argument posed by Stracca (2002) says that if prospect theory is an accurate description of human attitude towards risk, the benefits of international diversification would be reduced to a significant extent.

Conclusion

We have looked over the principal elements behind the prospect theory proposed by Kahneman and Tversky in 1979. Prospect Theory is an alternative descriptive model of decision making under uncertainty, which incorporates real life choices and psychological analysis. Firstly, within prospect theory investors evaluate their outcomes in accordance with a reference point and make decisions based on how the outcome changes their wealth in relation to this unique reference. Within the expected utility theory, this relative level of wealth is not accounted for. Another key assumption behind prospect theory is the risk averse and seeking behaviour of investors under different circumstances. Investors are risk seeking in terms of losses and risk averse when it comes to profits. The assumptions of an endowment effect and decision weights are also included within the theory, where people place a higher value on a good that they already own and, in contrast to expected utility theory, risk is incorporated not by the original probability but by transformed decision weights. The S-shaped value function curve for prospect theory show this risk seeking and averse behaviour in investors, a reflection effect. The idea of framing is also a key element of prospect theory, where if the same decision problem is described in different words, it can lead to different preferences. Within the theory also is an isolation effect, where devices are used to draw additional attention to something that would otherwise conform, and probabilistic insurance, where decision weights tend to overweight small and large probabilities, but underweight moderate probabilities. The real world assumptions behind prospect theory have been used to explain a number of financial anomalies. We finally looked into prospect theory's applications to the equity premium puzzle and home bias which offer explanations to these anomalies.