The Prospect Theory Value Function

Published Date: 02 Nov 2017

Contrary to the classical financial theory, the efficient market hypothesis (EMH, Fama 1960) where investor trade on new information, Fischer Black (1986) introduces "noise" as a factor affecting investor decisions. Investors who are not trading on fundamental data rather on noise factors are known as the "Noise Trader". The concept of noise trader is modelled by Fischer Black as follow; "supposing that the signal s, has no information about future values of a security g. If an investor incorrectly believes that s is informative about g, and it is not, then the investor is a noise trader". The question which is controversial discussed is if noise traders should be considered for price formation or not. According to Friedman (1953) and Fama (1965) irrational investors are met in financial markets by rational arbitrageurs therefore both authors do not believe that noise traders should be considered for price formation.

-2σ

+2σ

Where it the price of the risky asset at time t. Arbitragers expectations is correct (blue) and noise traders’ expectations are biased (red).

DeLong, Shleifer, Summers & Waldmann (1990) have given empirical evidence that arbitrage does not remove the influence of noise because noise itself forms risk, the so called "Noise Trader Risk". Irrational traders can derive assets more down or up if they are more pessimistic or optimistic about an asset, which can lead arbitrageur so suffer a loss.

Price

-2σ

+2σ

+2σ

+2σ

(1)

The degree of noise traders misperceptions is measured by : . is the mean misperception degree of optimistic of noise traders. If equals to zero, than noise traders and rational traders have the same forecast about a security; if >0 than noise traders are on average more optimistic and if than noise traders are on average more pessimistic on a security than informed traders.

The classical finance theory implies that feelings do not impact the decision-making procedure and rational traders are able to ignore feelings.

Investment

Decision

(rational)

Outcome

Source: own source

Seasonal effects, such as the calendar effect, festival effect, weather effect and holiday effect represent an anomaly from the perspective of the classical pricing theory, because the fundamental information, which alone should determine the price changes do not include any seasonal patterns. Behavioral Finance debates that mood of humans are important factors in shaping investment decisions and preferences. The empirical study by Lo & Repin (2002) about professional trader’s decision-making process shows that in periods of high price-volatility decision are mood-driven.

Investment

Mood

Decision

(Irrational)

Outcome

Weather

Effect

Holiday

Effect

Festival Effect

Calendar

Effect

Source: own source

Forgas (1995) concluded in his empirical study that people in a bad (good) mood have a habit of to have more negative (positive) evaluations and make more pessimistic (optimistic) decisions. Saunders (1993) documented that weather has a significant effect on human behaviour and therefore on trading decision. Studies by Hirshleifer and Shumway (2003) examined the effect of sunlight on the stock market and concluded that the weather affects investor mood and decision-making.

Cross (1973) compared week days Standard and Poor’s 500 (S&P500) index returns and concluded that during 1953 to 1970 the S&P500 index closed to 62% higher on a Friday, while the probability of a higher return given on a Monday is 39.5%. This return anomaly is known as weekend effect and was the foundation of many further empirical studies on seasonal anomaly on financial markets.

The January effect is the main anomaly in the context of the so-called calendar effect. This effect is attributed to the observation that the monthly return of small size firms of January is above average. According to the empirical study of Rozeff and Kinney (1976) the average January return of shares of the NYSE is 3.5% between 1904 and 1974, while the average yield of the other months is only 0.5%. This means that a third of the annual return comes from the month of January alone.

The holiday effect was observed first in 1934 by Fields who investigated that stock prices in the USA rise one day before public holidays. Studies by Lakonishok and Smidt (1988) concluded that on days before public holidays returns are by a factor of over 23 higher than on normal trading days. Ariel (1990) concluded that a third of the stock return is done by the eight days before public holidays. Pettengill (1989) research on pre-holiday effect shows that returns differ by day of the week and month the holiday falls and firm size. The holiday effect can be seen not only in the USA, also in China, Singapore and Hong Kong on the Chinese (lunar) New Year and also in the Middle East during Ramadan Festival.

The saying "sell in may go away" is related to the "Gone Fishin effect" and is part of the research done by Hong and Yu (2004) who concludes that stock markets around the world is on average 8% lower during the summer. However the impact of the Gone Fishin effcet in in countries closes to the equator much lower than countries which are located far above or under the equator.

Many studies have focussed on the impact of non-economic phenomenon on stock markets, such as big sport events. Krueger and Kennedy (1990) analysed the impact of the world most watched sport event the U.S. National Football Legue´s Super Bowl ( more than 1 billion around the world), and found that the following five days after the Super Bowl, the New York Stock Exchange (NYSE) shows a higher return than the five days before the Super Bowl. Ashton et al. (2003) examined the impact of the performance of the England football team on the FTSE 100 index. The authors found that a good victory the FTSE 100 index closed higher at the next opening trading day.

The conclusion of this essay is that many investors tend to hubris as an impact of their moods and feelings and do not act rational. Even rational investors tend to behave irrational in times of high volatility on stock markets. Behavioural finance provides various explanations how investors mood impacts equity markets and weather seasonally anomaly can be explained by investors mood. Some seasonally anomalies such as Gone Finshin effect occurs on specific countries others such as the January effect can be seen on financial markets around the world. The way of thinking of "either/or" is detrimental to an objective scientific discussion on the behavioural finance. The proponents of the traditional theory often unconsciously assume the position that the market is either rational or irrational. Investors can be rational for instance for blue chips but irrational for small cap stocks, therefore this essay concludes that every investor could be rational and at the same time irrational. The only problem is that nobody wants to state that he might be irrational at some points.

The behaviour economics is recognized in many places of the literature as part of the behavioural theory of organization. The key question of behavioural economics is how individual decides given several possible outcomes. According to traditional finance and economic theories an individual is rational and always process and weights the right information before taking decision. If that would be the case, why did in the financial crisis 2007/08 so many Banks and individual investors faced huge losses? Human´s prefer the fast and immediate benefit and since the utility thereby plays a crucial role, the term utility will be explained at this point.

A utility is an arithmetic rating given to every possible outcome. In a choice among numerous different results, the one with the highest utility is recommended to choose. The utility theory and the expected utility method of lottery valuation were first introduced by Bernoulli (1738) for the St. Petersburg Paradox (Game). It is played with a fair coin, which is tossed until head appears. The player wins where n is the amount of tossed until head appears, for example after the third toss one would win (£8).

tosses

1

2

3

4

5

6

£ when head appears

2

4

8

16

32

64

The question which appears now is what is the price to pay for this game? Bernoulli's solution is based on the assumption that one does not consider the amount of expected profit rather the expected utility, playing the game. Most people response to that question with £3-5. Bernoulli´s proposal for a utility function under uncertainty is as follow:

(1)

This would give a lottery pay-out g of about £1.4.

The utility function u(x) transforms the several different choices into units. To do that Bernoulli assumed two properties, first, the utility function u(x) is an increasing function and secondly a concave function of x for risk averse or convex function of x for risk seeking.

u(1)

u(2)

u(x)

1

2

U(x)

£

Source: Krysztof Kontek (2010)

u(1)

u(2)

u(x)

1

2

U(x)

£

Source: Krysztof Kontek (2010)

The utility function is expressed as follow:

(2)

(3)

, (4)

where represents the Bernoulli utility function; given represent the gain, represents the probability with ≥ 0.

The equation (4) is an expectation utility function (or risk benefit function} often called Neumann-Morgenstern utility function. Different to previous assumption of the utility function, Neumann-Morgenstern preference is a cardinal utility function. The agent is indifferent between two choices with the same expected value, firstly a gamble or secondly receiving a given bundle. Consider an individual who is given two outcomes and given A than the Neumann-Morgenstern preference corresponds to the usual assumptions of rationality:

 Outcome is slightly preferred to outcome

>  Outcome is preferred to outcome

 Outcome and are equally preferred

The concept of a lottery is a normal tool which can be used to define the interaction of preferences with uncertainty about the outcome. Suppose A = then a lottery on A is a probability distribution (.

Neumann-Morgenstern utility function has six natural and desirable properties. These axioms are: transitivity, monotonicity, substitutability, completeness, decomposability and continuity.

The classical economics theory assumes that human mainly take decisions rational based on the basis of information. However empirical studies could not confirm that human always take decisions rational. Behavioural economics argues that psychological factors cause distortions of perception in decision taking.

The prospect theory of Kahneman and Tversky (1979) changed the way academicians study risk and how academicians link the important role of risk when decision needed to be taken given different choices. The prospect theory with certainty effect, reflection effect and isolation effect identified behaviour from experiments that was not rational and inconsistent with expected utility theory.

Kahnemann and Tyersky (1979) certainty effect states that people give less attention to events that are less possible to be happen and more attention to events that have a higher possibility to occur. This findings differs form the expected utility theory that states that utilities are weighted by probabilities.

External

Factors

Internal Factors

Individual

Choice A

100% chance to win £20

Choice B

80% chance to win £35 and 20% unsuccessful

Binomial Decision

A vs, B and C vs. D

Choice C

25% chance to win £20 and 75% unsuccessful

Choice D

20% chance to win £35 and 80% not unsuccessful

Certainty Effect

Source: own source

In the first case, probable choose mainly choice A (78%) which is risk-aversion behaviour. However in the second case more than the half of the probable choose D (58%) which has a lower probability than C. Kahneman and Tyersky (1974) reflection effect was developed based on the case when a negative outcomes are presented.

External

Factors

Internal Factors

Individual

Choice A

100% chance to win £2000

Choice B

80% chance to win £3000

Binomial Decision

A vs, B and C vs. D

Choice C

100% chance to loss £2000

Choice D

80% chance to los £3000

Reflection Effect

Source: own source

Individuals prefer -3,000 with 80% certainty versus -2,000 with 100% certainty. Again, this is inconsistent with expected utility theory.

An important aspect of that prospect theory model is that the theory divides the decision processes in two steps: editing and evaluation. Individuals evaluate differences and changes rather than absolute sizes. The start position called reference point (A) is the origin of the value function.

Value

Gains

Losses

Reference point (A)

The domain of gains: Risk averse

The domain of losses: Risk seeking

Concave

Convex

Source: Dr Jing-Ming Kuo (2013), Lecture Slide No 6

The prospect theory value function is usually concave for gains and convex for losses. The curve is steeper for losses than for gains which was noted by Mullainathan and Thaler (2000) and has a kink at the origin. The kink at the reference point (A) means that individuals behaviour at the domain of gains risk averse but show risk seeking behaviour in terms of potential loss.

Kahneman and Tversky express the evaluation phase as follow:

(5)

, (6)

where U represents the outcome of the expected utility, v is the value function, with i=1,2,3 … n are the potential outcomes and with i= 1,2,3 … n the individual probabilities of the outcomes.

When a stock has increased in value relative to the purchase price individual tent to sell assuming the stocks became risky (risk averse) and when a stock price falls individual tent to hold the stocks assuming they becoming relatively cheaper, risk averse. Ang, Chen and Xing (2006) findings extended the collected works on downside risk by giving evidence that stock returns show of reflecting a premium for downside risk. In simple words, individual investor’s losses outweigh gains. The value function shows a psychophysical principle; where individual consider a difference between 0 and 100 subjectively as greater than a difference between 1'000 and 1'100. The anchor heuristics affects the reference point of the value function. The turning point as a new reference point is anchored in a reversal of the performance.

Price Level

Time period

 

1 Reference point

2 Reference point

3 Reference point

£20

£15

£10

Source: own source

Let’s assume that an individual buys a stock at reference point (1) for 1300 pence if the stock price rallies to 1600 pence an individual investors takes 1600 pence as his new reference point and compares all movements regarding this the second reference point. If the stock price fluctuates and reaches after 2 month 1450 pence, than the individual investors would consider that as a loss of 150 pence and not as gain of 150 pence, considering the origin reference point (1) when the individual enters the position.

Individuals tend to associate their assessments with – quite irrelevant - "anchors" of memory. Tversky and Kahneman showed this with a famous study in 1974. This behaviour also explains why forecasts are often given relating to the last prices as a benchmark. The current price of shares is often seen as an anchor for the formation of expectations. However, the new price level of a stock should be determined based on profit estimates using a valuation model, like dividend-discount model.

Kahnemann and Tversky (1973. 74) found that individuals tend to be overconfident and therefore take more risk and they also tend to put lesser weight on base rates and more weight on new information and hence overreact to news. The findings of Kahnemann and Tyersky lead to the development of the overreaction hypothesis by DeBondt and Thaler (1985). The authors argued that investors overact to good and bad news. Therefore, the overreaction to news clues past losers to become under-priced and past winners become overpriced.

Over-

Reaction

Under- Reaction

Price

Time

Source: Dr Jing-Ming Kuo (2013), Lecture Slide No 6

Bremer & Hiraki (1999) and Da Costa (1994) analysed overreaction by comparing the impact of bad news versus good news on stock returns. Both studies concluded that bad news shows a higher overreaction than good news, which is known as asymmetric effect.

The conclusion of this assignment is, that individuals who recognize their own typical failing behaviour, can improve not only the process of taking their own decisions, but also can use this information to get advantage from the misconduct of others behaviour. Human beings are not perfect calculators show emotions and just a view people are able to move curves or juggle with formulas, before making a decision based on classical economics theory. The author of this assignment would use Camerer et al (2004) words to conclude how traditional financial theories can benefit from behavioural economics and finance "Behavioural economics increases the explanatory power of economics by providing it with more realistic psychological foundations".