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Expected Utility Theory Introduction • we have talked about individual decision making in the absence of uncertainty • in reality, we usually make decision under uncertainty • example: uncertainty from product quality (second-hand vehicle) uncertainty in dealing with others -> often the outcome depends on what others purchase of financial assets (stocks and bonds) whose return is contingent on which state is realized This is the essence of Financial Economics Probability • probability of an event occurring is the relative frequency with which the event occurs • if αi = the probability of event i occurring and there are n possible events (states) then • αi > 0, i = 1…n n •  ‘i=1αi = • Lottery (X) with prizes (outcomes, (outcomes states, states events) X1, X2, X3, ,Xn with corresponding probabilities α1, α2, α3, ,αn , respectively (mutually exclusive and exhaustive) then the expected value of this lottery is E(X) = α1X1 + α2X2 + α3X3 + + αnXn E(X) =  αiXi n ‘i=1 Example • • • • Gamble (X) flip of a coin if heads, you receive $1 X1 = +1 if tails, you pay $1 X2 = -1 E(X) = (0.5) (1) + (0.5) (-1) = • if you play this game many times, it is likely that you break-even Example • • • Gamble (X) flip of a coin if heads, you receive $10 if tails, you pay $1 • E(X) = (0.5) (10) + (0.5) (-1) = 4.50 • • if you play this game many times, you will be a big winner How much would you pay to play this game: • perhaps as much as a $4.50 But of course the answer depends upon your preference to risk • X1 = +10 X2 = -1 Fair Gambles • if the cost to play these gambles = expected value of the outcome – then the gamble is said to be actuarially fair • Common empirical findings: individuals may agree to flip a coin for small amounts of money, but usually refuse to bet large sums of money people will pay small amounts of money to play actuarially unfair games (Lotto 649, where cost = $1, but E(X) < 1) - but will avoid paying a lot Why these empirical findings occur? Becoz’ it is not about E(W) St Petersburg Paradox • • Gamble (X): A coin is flipped until a head appears You receive $2n where n is the flip on which the head occurred • • states: prob: X1 = $2 α1 = 1/2 X2 = $4 α2 = 1/4 X3 = $8 α3 = 1/8 Xn = $2n αn = 1/2n  i 1  E(X) =   i x i   i   i 1 Paradox: no one would pay an “actuarially fair” price to play this game (no one would even pay close to the fair price) Explaining the St Petersburg Paradox • this paradox arises because individuals not make decisions based purely on wealth, but rather on the utility of their expected wealth • if we can show that the marginal utility of wealth declines as we get more wealth, then we can show that the expected value of a game is finite • Assume Ass me U(X) = ln(X) U'(X) U'(X)=1/x 1/ > MU positi positivee • U"(X)=-1/x2 < Diminishing MU Goals 1) Individual maximizes their expected Utility 0.4 100 0.6 0.3 10 60 0.7 30 Asset i Asset j E(W) = E[U(W)] = Prefer the one with higher E[U(W)] E(W) = E[U(W)] = 2) Individual preferences over risk and return y C2 x Return C1 Risk Expected Utility Theory • Objective: to develop a theory of rational decision-making under uncertainty with the minimum sets of reasonable assumptions possible • the following five axioms of expected utility provide the minimum set of conditions for consistent and rational behaviour 10 Axioms of Choice under uncertainty A1.Comparability (also known as completeness) For the entire set of uncertain alternatives, an individual can say either that either outcome x is preferred to outcome y (x > y) or y is preferred to x (y > x) or indifferent between x and y ((x ~ y) y) A2.Transitivity (also know as consistency) If an individual prefers x to y and y to z, then x is preferred to z If (x > y and y > z, then x > z) Similarly, if an individual is indifferent between x and y and is also indifferent between y and z, then the individual is indifferent between x and z If (x ~ y and y ~ z, then x ~ z) 11 Axioms of Choice under uncertainty A3.Strong Independence Suppose we construct a gamble where the individual has a probability α of receiving outcome x and a probability (1-α) of receiving outcome z This gamble is written as: G(x,z:α) Strong independence says that if the individual is indifferent to x and y, then he will also be indifferent as to a first gamble set up between x with probability α and a mutually exclusive outcome z, and a second gamble set up between y with probability α and the same mutually exclusive outcome z If x ~ y, then G(x,z:α) ~ G(y,z:α) NOTE: The mutual exclusiveness of the third outcome z is critical to the axiom of strong independence 12 Axioms of Choice under uncertainty A4.Measurability If outcome y is less preferred than x (y < x) but more than z (y > z), then there is a unique probability α such that: the individual will be indifferent between [1] y and [2] A gamble between x with probability α z with probability (1-α) In Maths, if x > y > z or x > y > z , then there exists a unique α such that y ~ G(x,z:α) 13 Axioms of Choice under uncertainty A5.Ranking If alternatives y and u both lie somewhere between x and z and we can establish gambles such that an individual is indifferent between y and a gamble between x (with probability α1) and z, while also indifferent between u and a second gamble, this time between x (with probability α2) and z, then if α1 is greater than α2, y is preferred to u If x > y > z and x>u> z then if y ~ G(x,z:α1) and u ~ G(x,z:α2), then it follows that if α1 > α2 then y > u, or if α1 = α2, then y ~ u 14 One more assumption • People are greedy, prefer more wealth than less • The axioms and this assumption is all we need in order to develop an expected utility theorem and actually apply the rule of max E[U(W)] = max ∑iαiU(Wi) 15 Utility Functions • Utility functions must have properties order preserving: if U(x) > U(y) => x > y Expected utility can be used to rank combinations of risky alternatives: [ ( ,y )] = αU(x) ( ) + ((1-α)) U(y) (y) U[G(x,y:α)] Deriving Expected utility theorem, one of the most elegant derivations in Economics, is tough Don’t worry about a formal derivation Just apply it • • Remark: Utility functions are unique to individuals 16 The Utility Function U(W) 3.40 3.00 Let U(W) = ln(W) 2.30 U'(W) > U''(W) < 61 1.61 U'(W) = 1/w U''(W) = - 1/W2 MU positive But diminishing 10 20 30 W 17 One more element: Risk Aversion • Consider the following gamble: • Prospect a prob = α • prospect b prob = 1-α G(a,b:α) • Question: Will we prefer the expected value of the gamble with certainty, or will we prefer the gamble itself? • • • ie consider the gamble with 10% chance of winning $100 90% chance of winning $0 • would you prefer the $10 for sure or would you prefer the gamble? E(gamble) = $10 if prefer the gamble, you are risk loving if indifferent to the options, risk neutral if prefer the expected value over the gamble, risk averse 18 Preferences to Risk U(W) U(W) U(W) U(b) U(b) U(a) U(b) U(a) U(a) a b W Risk Preferring a b Risk Neutral U'(W) > U''(W) > U'(W) > U''(W) = W a b W Risk Aversion U'(W) > U''(W) < 19 U[E(W)] and E[U(W)] • U[(E(W)] is the utility associated with the known level of expected wealth (although there is uncertainty around what the level of wealth will be, there is no such uncertainty about its expected value) • E[U(W)] is the expected utility of the utility associated with different levels of wealth that may obtain • The relationship between U[E(W)] and E[U(W)] is very important 20 Expected Utility • • • • • • Assume that the utility function is natural logs: U(W) = ln(W) Then MU(W) is decreasing U(W) = ln(W) U'(W)=1/W MU>0 => MU diminishing MU''(W) < Consider the following example: 80% change of winning $5 20% chance of winning $30 E(W) = (.80)*(5) + (0.2)*(30) = $10 U[E(W)] = U(10) = 2.30 E[U(W)] = (0.8)*[U(5)] + (0.2)*[U(30)] = (0.8)*(1.61) + (0.2)*(3.40) = 1.97 Therefore, U[(E(W)] > E[U(W)] uncertainty reduces utility 21 The Markowitz Premium U(W) 3.40 U(W) = ln(W) U[E(W)] = U(10) = 2.30 U[E(W)] = 2.30 E[U(W)] = 0.8*U(5) + 0.2*U(30) = 0.8*1.61 8*1 61 + 00.2*3.40 2*3 40 = 1.97 Therefore, U[E(W)] > E[U(W)] Uncertainty reduces utility E[U(W)] = 1.97 1.61 Certainty equivalent: 7.17 That is, this individual will take 7.17 with certainty rather than the uncertainty around the gamble 2.83 CE = 7.17 10 30 W 22 The Risk Premium • Risk Premium: – the amount that the individual is willing to give up in order to avoid the gamble • Recall the gamble 80% change of winning $5 20% chance of winning $30 E(W) = (.80)*(5) + (0.2)*(30) = $10 Suppose the individual has the choice now between the gamble and the expected value of the gamble E[U[W)] = 1.97 Certainty equivalent = $7.17 Investor would be willing to pay a maximum of $2.83 to avoid the gamble ($10 - $7.17) ie will pay an insurance premium of $2.83 THIS IS CALLED THE MARKOWITZ PREMIUM Ln(CE)=1.97, i.e U(CE)=E[U(W)], CE=7.17, RP=10-7.17=$2.83 23 The Risk Premium Risk Premium = an individual's expected wealth,given he plays the gamble In general, if U[E(W)] > E[U(W)] if U[E(W)] = E[U(W)] if U[E(W)] < E[U(W)] - level of wealth the individual would accept with certainty if the gamble were removed (ie the certainty equivalent) then risk averse individual then risk neutral individual then risk loving individual (RP > 0) (RP = 0) (RP < 0) risk aversion occurs when the utility function is strictly concave risk neutrality occurs when the utility function is linear risk loving occurs when the utility function is convex 24 The Arrow-Pratt Premium • • • • Risk Averse Investors assume that utility functions are strictly concave and increasing Individuals always prefer more to less (MU > 0) Marginal utility of wealth decreases as wealth increases A More Specific Definition of Risk Aversion W = current wealth gamble Z and the gamble has a zero expected value E(Z) = what risk premium (W,Z) must be added to the gamble to make the individual indifferent between the gamble and the expected value of the gamble? 25 Absolute Risk Aversion • Arrow-Pratt Measure of a Local Risk Premium (derived from (*) above) • define ARA as a measure of Absolute Risk Aversion • this is defined as a measure of absolute risk aversion because it measures risk aversion for a given level of wealth ARA > for all risk averse investors (U'>0, U'' U"(W) = -2W-3 < ARA = 2/W => dARA/dW < RRA = 2W/W = => dRRA/dW = This power utility function is consistent with the empirical evidence of Friend and Blume (1975) 29 An Example • U=ln(W) W = $20,000 • G(10,-10: 50) 50% will win 10, 50% will lose 10 premium associated with • What is the risk p this gamble? • Calculate this premium using both the Markowitz and Arrow-Pratt Approaches 30 10 Arrow-Pratt Measure • •  = -(1/2) 2z U''(W)/U'(W) 2z = 0.5*(20,010 – 20,000)2 + 0.5*(19,090 – 20,000)2 = 100 • U'(W) = (1/W) • U''(W)/U'(W) = -1/W = -1/(20,000) U''(W) = -1/W2 •  = -(1/2) 2z U''(W)/U'(W) = -(1/2)(100)(-1/20,000) = $0.0025 31 Markowitz Approach • • • • • E(U(W)) =  piU(Wi) E(U(W)) = (0.5)U(20,010) + 0.5*U(19,990) E(U(W)) = (0.5)ln(20,010) + 0.5 0.5*ln(19,990) ln(19,990) E(U(W)) = 9.903487428 ln(CE) = 9.903487428  CE = 19,999.9975 • The risk premium RP = $0.0025 • Therefore, the AP and Markowitz premia are the same 32 Markowitz Approach E(U(W)) = 9.903487 19,990 20,000 CE 20,010 33 11 Differences in two approaches • Markowitz premium is an exact measures whereas the AP measure is approximate • AP assumes symmetry payoffs across good or bad states as well as relatively small payoff changes states, changes • It is not always easy or even possible to invert a utility function, in which case it is easier to calculate the AP measure • The accuracy of the AP measures decreases in the size of the gamble and its asymmetry 34 12 ... -(1 /2) 2z U''(W)/U'(W) 2z = 0.5* (20 ,010 – 20 ,000 )2 + 0.5*(19,090 – 20 ,000 )2 = 100 • U'(W) = (1/W) • U''(W)/U'(W) = -1/W = -1/ (20 ,000) U''(W) = -1/W2 •  = -(1 /2) 2z U''(W)/U'(W) = -(1 /2) (100)(-1 /20 ,000)... a head appears You receive $2n where n is the flip on which the head occurred • • states: prob: X1 = $2 α1 = 1 /2 X2 = $4 2 = 1/4 X3 = $8 α3 = 1/8 Xn = $2n αn = 1/2n  i 1  E(X) =   i x... U (W) U (W) 27 Quadratic Utility Quadratic Utility - widely used in the academic literature U(W) = a W - b W2 U'(W) = a - 2bW ARA = U"(W) = -2b -U"(W) 2b - = U'(W) a -2bW quadratic
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