FINC3023 Behavioral Finance TOPIC 1: Expected Utility Rational theories of finance tell us how people should behave and often do not reflect reality. A normative theory based on rational utility maximizers cannot be construed as a superior alternative to behavioral approaches, merely because it discusses how people should behave but a not well-explained empirically. If people do not behave this way, there are limitation to helping us understand the observed market behavior. Neoclassical economics 1. People have rational preferences across possible outcomes 2. People maximize utility and firms maximize profits 3. People make independent decisions based on all relevant information Expected Utility theory First proposed by Daniel Bernoulli in 1728 in response to solving what reasonable price an individual should pay to enter a gamble. A coin is flipped repeatedly until a head is produced; if you enter the game you receive a payoff of $2 n where n is the number of the throw producing the first head. E(V) = ½ (2) + ¼ (4) + ⅛(8) + = 1 + 1 + 1+ or in general E V = ( &)$ $ % & 2 & But even though the expected value of this gamble is infinite, most people would be unwilling to pay more than a few dollars St. Petersburg Paradox. Bernoulli chose a logarithmic utility function to explain that the expected utility of the game is indeed finite. E U V = ( &)$ Equivalent) of e $.789 $4.00 $ % & ln (2 & ) 1.386. This corresponds to a certain value (Certainty where U(x) = ln(x) Preferences are defined over prospects, where a prospect is a list of consequences or outcomes with associated probabilities. - Assume all consequences and probabilities are known to the investor.
In choosing among prospects, the investor can be said to confront a situation of risk (in contrast with situations of uncertainty in which as least some of the outcomes or probabilities are unknown) Any prospect q can be represented by a probability distribution q = (p 1, p 2,, p n ) over a fixed set of pure consequences X = (x 1, x 2,, x n ), where p i is the probability of x i, p i 0 and Σp i = 1. Reducing compound lottery All the lotteries involved in a compound lottery are always assumed to be independent of each other and so it is easy to reduce a compound lottery to a simple lottery Experimental evidence has suggested that people tend to prefer the compound form of the lottery (on the left, above), rather than its reduced form (on the right here). This is particularly likely when the probabilities of winning in the first part of the compound lottery are high.?? Axioms of EUT There are four axioms of the expected utility theory that define a rational decision maker. They are completeness, transitivity, independence and continuity. Completeness: For all q, r: either q r or r q or q r where represents the relation: q is (weakly) preferred to r. Transitivity: For all q, r, s: ifq r and r s then q s Continuity: requires that for all prospects q, r, s Where q r and r s, there exists some probability p such that q, p; s, 1 p r that is, there is some mixture of the prospects q and s for which the investor will be indifferent to choosing prospect r Independence: It assumes that two gambles mixed with an irrelevant third one will maintain the same order of preference as when the two are presented independently of the third one.
EUT provides one very simple way of combining probabilities and consequences into a single measure of value which has a number of appealing properties. One such property is monotonicity: Monotonicity is the property that stochastically dominating prospects are always preferred to prospects which they dominate. This is a fairly basic concept in rationality it says that if q stochastically dominates r, then the expected value of q is higher than r. The expected utility theory takes into account that individuals may be risk-averse, meaning that the individual would refuse a fair gamble (a fair gamble has an expected value of zero) The shape of the utility function also has a simple behavioural interpretation whereby concavity (convexity) of u(.) implies risk averse (seeking) behaviour. Someone with a concave utility function will always prefer a certain amount X to any risky prospect with expected value equal to X. E.g. A risk-averse individual will accept less than $50 rather than take a coin toss which would yield $100 if heads and $0 otherwise. If they would accept $40 for sure, rather than taking the gamble, this would be the certainty equivalent. We work this out by finding the certain value with the same utility as the risky gamble. The difference between the expected value and the certainty equivalent is the risk premium from taking the gamble.
Absolute risk aversion: Ra(x) = u (x)/u (x) More wealth is preferred to less, but utility grows at decreasing rate. This gives the concave shape we would normally expect in utility functions. Relative risk aversion: Rr(x) = x. u (x)/u (x) As wealth increases the level of the risk premium increases at a decreasing rate. Constant absolute risk aversion (CARA): E.g. u x = e STU level of risk aversion does not depend on wealth, risk aversion does not change with changes in wealth Constant relative risk aversion (CRRA): E.g. u(x) = x 1-β /(1-β) if β 1 and u(x) = log(x) is β = 1 As wealth increases the level of the risk premium increases proportionally Limitations of EUT It is precisely the simplicity and economy of EUT that has made it such a powerful and tractable modelling tool. The problem, however, is with the descriptive merits of the theory whether EUT provides a sufficiently accurate representation of actual choice behaviour. The evidence from a large number of empirical tests has raised some real doubts There is now a large body of evidence indicating that actual choice behaviour may systematically violate the independence axiom. Systematic violations, rather than random or idiosyncratic violations, suggest descriptive failings of Expected Utility Theory Two examples of such phenomena are Common Consequence Effect Common Ratio Effect The Allais paradox is a choice problem designed by Maurice Allais (1953) to show an inconsistency of actual observed choices with the predictions of expected utility theory. Example Problem 1: Choose between Prospect A and Prospect B. Prospect A: $2,500 with probability 0.33 $2,400 with probability 0.66 $0 with probability 0.01 Prospect B: $2,400 with certainty
Problem 2: Choose between Prospect C and Prospect D. Prospect C: $2,500 with probability 0.33 $0 with probability 0.67 Prospect D: $2,400 with probability 0.34 $0 with probability 0.66 It has been shown by Daniel Kahneman and Amos Tversky (1979, Prospect Theory: An analysis of decision under risk, Econometrica, 47 (2), 263-291) that more people choose B with Problem 1 and more people choose C when presented with Problem 2. These choices violate Expected Utility Theory. Why? Individuals fail to rationally value each gamble. In the first problem, there is an overvalue on certainty. In the second problem, there is more value placed on the higher potential upside, even though the probabilities are similar.
FINC3023 TOPIC 2: Expected Utility Theory Cont. The probability triangle By convention, horizontal axis measures the probability of the worst consequence ($0), increasing from left to right. The vertical axis measures the probability of the best consequence ($5M increase from bottom to top. Hypotenuse is the probability of outcome for the middle consequence. S1 - $1M for sure S2 0.89 probability of $0 and 0.11 probability of $1M R1 0.01 probability of $0, 0.1 probability of $5M and 0.89 probability of $1M. R2 0.9 probability of $0M and 0.1 probability of $5M The independence axiom of EUT restricts the indifference curves to being 1. Upward sloping 2. Linear 3. Parallel Slope = attitude to risk i.e. more risk averse individual, the steeper the slope of the indifference curves.
If individual 1 = blue, individual 2 = orange What we are shown is that relative to individual 1, we need to give individual 2 higher chances of winning the best possible outcome as we moving in the north west direction to generate indifference with the safe prospect of s. In relation to the Allais paradox problems (first triangle) for a given individual, EUT allows three possibilities. Indifference curves could have a steeper slope than the lines connecting prospects, in which case s $ r $ and s % r %. This is as in the figure. Alternatively, indifference curves could have a less steep slope, in which case r $ s $ and r % s % - Finally, the slope of indifference curves could correspond exactly with that of the lines joining pairs of prospects, in which case r $ s $ and r % s % But we know people often violate EUT revealing s $ r $ in the first lottery and r % s % in the second (right) lottery. So by choosing r2 over s2, we see people being more risk adverse than they should be given that they choose s1 over r1 i.e. common consequence effect.
There is a similar tendency in the common ratio effect. Assuming EUT preferences, an individual must prefer the safer option in both choices or the risker option in both choices. We instead see Many people choose s 1 ** over r 1 ** and r 2 ** over s 2 **. Therefore, any theory that seeks to explain this standard violation of EUT needs to have the following properties: Risk seeking (or at least less risk averse) for low probabilities: E.g. {s 2, r 2 } Risk Averse for high probabilities: E.g. {s 1, r 1 } Fanning out hypothesis There is where we get the fanning out hypothesis. Indifference curves are relatively steeper sloped in the neighborhood of prospect m, such that m lies on a higher indifference curve that p and r, and flatter in the bottom right-hand corner such that t lies on a higher indifference curve than s.
Hence, for an individual whose indifference curves fan out we can construct prospects over which we will observe: - A common consequence effect (m q and t s) - A common ratio effect (m r and t s). Loosely speaking, this means that agents tend to be more risk adverse as the prospects they face get better i.e. indifference curves become steeper or fan out as we move to the northwest corner of the triangle. Summary Individuals demonstrate predictable systematic violations of Expected Utility Theory The Allais paradox (i.e. common consequence/common ratio effects) demonstrated that people tended to be more risk averse in high probability gambles and less risk adverse in lower probability gambles. e.g. they selected $3,000 for sure over ($4,000, 0.8; $0, 0.2) but would typically select ($4,000, 0.2; $0, 0.8) over ($3,000, 0.25; $0, 0.75) This is inconsistent with EUT as: Indifference curves are parallel in the probability triangle the ratio of 0.8 / 1 (probability of getting $4000/ probability of getting $3000) is identical to 0.2 / 0.25 in the two lotteries. You must select either the safe option or the risky option in both lotteries according to EUT, which does not occur in empirical evidence. The failure of EUT to match observed decision making processes by individuals stems from the violation of the independence axiom More generalised decision making models allow for a relaxation of the independence axiom Independence requires that for all prospects q, r, s: if q r then (q, p; s, 1 p ) r, p; s, 1 p for all p