Topic 3 Utility theory and utility maximization for portfolio choices. 3.1 Optimal long-term investment criterion log utility criterion

Transcription

1 MATH362 Fundamentals of Mathematics Finance Topic 3 Utility theory and utility maximization for portfolio choices 3.1 Optimal long-term investment criterion log utility criterion 3.2 Axiomatic approach to the construction of utility functions 3.3 Maximum expected utility criterion 3.4 Characterization of utility functions 3.5 Two-asset portfolio analysis 3.6 Quadratic utility and mean-variance criteria 3.7 Stochastic dominance 1

2 3.1 Optimal long-term investment strategy log utility Suppose there is an investment opportunity that the investor will either double her investment or return nothing. The probability of the favorable outcome is p. Suppose the investor has an initial capital of X 0, and she can repeat this investment many times. How much should she invest at each time in order to maximize the longterm growth of capital? Statement of the problem Let α be the proportion of capital invested during each play. The investor would like to find the optimal value of α which maximizes the long-term growth. The possible proportional changes are given by { 1 + α if outcome is favorable 1 α if outcome is unfavorable, 0 α 1. 2

3 General formulation:- Let X k represent the capital after the k th trial, then X k = R k X k 1 where R k is the random return variable. We assume that all R k s have identical probability distribution and they are mutually independent. The capital at the end of n trials is X n = R n R n 1 R 2 R 1 X 0. 3

4 Taking logarithm on both sides or ln X n = ln X 0 + n k=1 ln R k ln ( Xn X 0 ) 1/n = 1 n n k=1 ln R k. Since the random variables ln R k are independent and have identical probability distribution, by the law of large numbers, we have 1 n n k=1 ln R k E[ln R 1 ]. 4

5 Remark Since the expected value of ln R k is independent of k, so we simply consider E[ln R 1 ]. Suppose we write m = E(ln R 1 ), we have ( Xn ) 1/n e m or X n X 0 e mn. X 0 For large n, the capital grows (roughly) exponentially with n at a rate m. Here, e m is the growth factor for each investment period. Log utility form m + ln X 0 = E[ln R 1 ] + ln X 0 = E[ln R 1 X 0 ] = E[ln X 1 ]. If we define the log utility form: U(x) = ln x, then the problem of maximizing the growth rate m is equivalent to maximizing the expected utility E[U(X 1 )]. 5

6 Essentially, we may treat the investment growth problem as a singleperiod model. The single-period maximization of log utility guarantees the maximum growth rate in the long run. Back to the investment strategy problem, how to find the optimal value of α such that the growth factor is maximized: m = E[ln R 1 ] = pln(1 + α) + (1 p)ln(1 α). Setting dm dα = 0, we obtain p(1 α) (1 p)(1 + α) = 0 giving α = 2p 1. 6

7 Suppose we require α 0, then the existence of the above solution implicitly requires p 0.5. What happen when p < 0.5, the value for α for optimal growth is given by α = 0? Lesson learnt If the game is unfavorable to the player, then he should stay away from the game. Example (volatility pumping) Stock: In each period, its value either doubles or reduces by half. riskless asset: just retain its value. How to use these two instruments in combination to achieve growth? Return vector R = { ( 12 1 ) if stock price goes down (2 1) if stock price goes up. 7

8 Strategy: Invest one half of the capital in each asset for every period. Do the rebalancing at the beginning of each period so that one half of the capital is invested in each asset. The expected growth rate m = 1 ( ) 1 2 ln prob of doubling + 1 ( 1 2 ln prob of halving ) We obtain e m , so the gain on the portfolio is about 6% per period. Remark This strategy follows the dictum of buy low and sell high by the process of rebalancing. 8

9 Combination of portfolio of risky stock and riskless asset gives an enhanced growth. 9

10 Example (pumping two stocks) Both assets either double or halve in value over each period with probability 1/2; and the price moves are independent. Suppose we invest one half of the capital in each asset, and rebalance at the end of each period. The expected growth rate of the portfolio is found to be 5 m = 1 4 ln ln ln 1 2 = 1 2 ln 5 4 = , so that e m = Remark 4 = This gives an 11.8% growth rate for each period. Advantage of the index tracking fund, say, Dow Jones Industrial Average. The index automatically (i) exercises some form of volatility pumping due to stock splitting, (ii) get rids of the weaker performers periodically. 10

11 Investment wheel The numbers shown are the payoffs for one-dollar investment on that sector. 1. Top sector: paying 3 to 1, though the area is 1/2 of the whole wheel (favorable odds). 2. Lower left sector: paying only 2 to 1 for an area of 1/3 of wheel (unfavorable odds). 3. Lower right sector: paying 6 to 1 for an area of 1/6 of the wheel (even odds). 11

12 Aggressive strategy Invest all money in the top sector. This produces the highest singleperiod expected return. This is too risky for long-term investment! Why? The investor goes broke half of the time and cannot continue with later spins. Fixed proportion strategy Prescribe wealth proportions to each sector; apportion current wealth among the sectors as bets at each spin. α 1 : top sector (α 1, α 2, α 3 ) where α i 0 and α 1 + α 2 + α 3 1. α 2 : lower left sector α 3 : lower right sector 12

13 If top occurs, R(ω 1 ) = 1 + 2α 1 α 2 α 3. If bottom left occurs, R(ω 2 ) = 1 α 1 + α 2 α 3. If bottom right occurs, R(ω 3 ) = 1 α 1 α 2 + 5α 3. We have m = 1 2 ln(1+2α 1 α 2 α 3 )+ 1 3 ln(1 α 1+α 2 α 3 )+ 1 6 ln(1 α 1 α 2 +5α 3 ). To maximize m, we compute m α i, i = 1,2,3 and set them be zero: 2 2(1 + 2α 1 α 2 α 3 ) 1 3(1 α 1 + α 2 α 3 ) 1 6(1 α 1 α 2 + 5α 3 ) = 0 1 2(1 + 2α 1 α 2 α 3 ) + 1 3(1 α 1 + α 2 α 3 ) 1 6(1 α 1 α 2 + 5α 3 ) = 0 1 2(1 + 2α 1 α 2 α 3 ) 1 3(1 α 1 + α 2 α 3 ) + 5 6(1 α 1 α 2 + 5α 3 ) = 0. 13

14 There is a whole family of optimal solutions, and it can be shown that they all give the same value for m. (i) α 1 = 1/2, α 2 = 1/3, α 3 = 1/6 One should invest in every sector of the wheel, and the bet proportions are equal to the probabilities of occurrence. m = 1 2 ln ln ln1 = 1 6 ln 3 2 so e m (a growth rate of about 7%). Remark: Betting on the unfavorable sector is like buying insurance. (ii) α 1 = 5/18, α 2 = 0 and α 3 = 1/18. Nothing is invested on the unfavorable sector. 14

15 Log utility and growth function Let w i = (w i1 w in ) T be the weight vector of holding n risky securities at the i th period, where weight is defined in terms of wealth. Write the random return vector at the i th period as R i = (R i1 R in ) T. Here, R ij is the random return of holding the j th security after the i th play. Write S n as the total return of the portfolio after n periods: S n = n i=1 w i R i. Define B = {w R n : 1 w = 1 and w 0}, where 1 = (1 1) T. This represents a trading strategy that does not allow short selling. When the successive games are identical, we may drop the dependence on i. 15

16 Single-period growth function Based on the log-utility criterion, we define the growth function by W(w; F) = E[ln(w R)], where F(R) is the distribution function of the stochastic return vector R. The growth function is seen to be a function of the trading strategy w together with dependence on F. The optimal growth function is defined by W (F) = max W(w; F). w B 16

17 Betting wheel revisited Let the payoff upon the occurrence of the i th event (denoted by ω i, which corresponds to the pointer landing on the i th sector) be (0 a i 0) T with probability p i. That is, R(ω i ) = (0 a i 0) T. Take the earlier example, the return vector is given by R(ω 1 ) = (3 0 0) T R(ω 2 ) = (0 2 0) T R(ω 3 ) = (0 0 6) T. ω 1 = top sector, ω 2 = bottom left sector, ω 3 = bottom right sector. For this betting wheel game, the gambler betting on the i th sector (equivalent to investment on security i) is paid a i if the pointer lands on the i th sector and loses the whole bet if otherwise. 17

18 Suppose the gambler chooses the weights w = (w 1 w n ) as the betting strategy with W(w; F) = = n i=1 n i=1 n i=1 w i = 1, then p i ln(w R(ω i )) = p i ln w i p i + n i=1 n i=1 p i ln p i + where the last two terms are known quantities. i=1 i=1 n p i ln w i a i n i=1 p i ln a i, Using the inequality: ln x x 1 for x 0, with equality holds when x = 1, we have n p i ln w ( ) i n wi n n p i 1 = w i p i = 0. p i p i The upper bound of i=1 i=1 i=1 p i ln w i p i is zero, and this maximum value is achieved when we choose w i = p i for all i. Hence, an optimal portfolio is w i = p i, for all i. 18

19 Remarks 1. Consider the following example p 1 = 0.5 a 1 = 1.01 p 2 = 0.2 a 2 = 10 6 p 3 = 0.3 a 3 = 0.8 Though the return of the second sector is highly favorable, we still apportion only w 2 = 0.2 to this sector, given that our goal is to achieve the long-term growth. However, if we would like to maximize the one-period return, we should place all bets in the second sector. 2. Normally, we should expect a i > 1 for all i. However, the above result remains valid even if 0 < a i 1. 19

20 3.2 Axiomatic approach to the construction of utility functions How to rank the following 4 investment choices? Investment A Investment B Investment C Investment D x p(x) x p(x) x p(x) x p(x) /4 10 1/5 0 1/2 10 1/5 40 1/4 20 2/5 30 1/5 When there is no risk, we choose the investment with the highest rate of return. Maximum Return Criterion. e.g. Investment B dominates Investment A, but this criterion fails to compare Investment B with Investment C. 20

21 Maximum expected return criterion (for risky investments) Identifies the investment with the highest expected return. E C (x) = 1 4 ( 5) (0) + 1 (40) = E D (x) = 1 5 ( 10) (10) (20) + 1 (30) = According to the maximum expected return criterion, D is preferred over C. However, some investors may prefer C on the ground that it has a smaller downside loss of 5 and a higher upside gain of 40. Is such procedure well justified? How to include the risk appetite of an individual investor into the decision procedure? 21

22 St Petersburg paradox (failure of Maximum Expected Return Criterion) Tossing of a fair coin until the first head shows up. The prize is 2 x 1 where x is the number of tosses until the first head shows up (the game is then ended). There is a very small chance to receive large sum of money. This occurs when x is large. Expected prize of the game = x=1 1 2 x2x 1 =. When people are faced with such a lottery in experimental trials, they refuse to pay more than a finite price (usually low). Alternative view In Mark Six, people are willing to pay a few dollars to bet on winning a reasonably large sum with infinitesimally small chance. 22

23 Certainty equivalent What is the certain amount that one would be willing to accept so that it is indifferent between playing the game for free or receiving this certain sum? This certain amount is called the certainty equivalent of the game. Let U(x) be the utility of the player, which measures the sense of satisfaction for a given wealth level x. Based on the expected utility criterion, the certainty equivalent c is given by U(c) = E[U(X)], where X is the random wealth at the end of the game. Certainty equivalent of the game of St. Peterburg paradox under log utility ln c = E[ln X] = x=1 1 2 x ln2x 1 = ln2 so that c = 2 is the certainty equivalent. x=1 x 1 2 x = ln2 23

24 Building block Pairwise comparison Consider the set of alternatives B, how to determine which element in the choice set B that is preferred? The individual first considers two arbitrary elements: x 1, x 2 B. He then picks the preferred element x 1 and discards the other. From the remaining elements, he picks the third one and compares with the winner. The process continues and the best choice among all alternatives is identified. 24

25 Choice set and preference relation Let the choice set B be a convex subset of the n-dimensional Euclidean space. The component x i of the n-dimensional vector x may represent x i units of commodity i. By convex, we mean that if x 1, x 2 B, then αx 1 + (1 α)x 2 B for any α [0,1]. Each individual is endowed with a preference relation,. Given any elements x 1 and x 2 B, x 1 x 2 means either that x 1 is preferred to x 2 or that x 1 is indifferent to x 2. 25

26 Three axioms for Reflexivity For any x 1 B, x 1 x 1. Comparability For any x 1, x 2 B, either x 1 x 2 or x 2 x 1. Transitivity For x 1, x 2, x 3 B, given x 1 x 2 and x 2 x 3, then x 1 x 3. Remarks 1. Without the comparability axiom, an individual could not determine an optimal choice. There would exist at least two elements of B between which the individual could not discriminate. 2. The transitivity axiom ensures that the choices are consistent. 26

27 Example 1 Total quantity Let B = {(x, y) : x [0, ) and y [0, )} represent the set of alternatives. Let x represent ounces of orange soda and y represent ounces of grape soda. It is easily seen that B is a convex subset of R 2. Suppose the individual is concerned only with the total quantity of soda available, the more the better, then the individual is endowed with the following preference relation: For (x 1, y 1 ),(x 2, y 2 ) B, (x 1, y 1 ) (x 2, y 2 ) if and only if x 1 + y 1 x 2 + y 2. 27

28 Example 2 Dictionary order Let the choice set B = {(x, y) : x [0, ), y [0, )}, the dictionary order is defined as follows: Suppose (x 1, y 1 ) B and (x 2, y 2 ) B, then (x 1, y 1 ) (x 2, y 2 ) if and only if [x 1 > x 2 ] or [x 1 = x 2 and y 1 y 2 ]. It is easy to check that the dictionary order satisfies the three basic axioms of a preference relation. 28

29 Definition Given x, y B and a preference relation satisfying the above three axioms. 1. x is indifferent to y, written as x y if and only if x y and y x. 2. x is strictly preferred to y, written as x y if and only if x y and not x y. 29

30 Axiom 4 Order Preserving For any x, y B where x y and α, β [0,1], [αx + (1 α)y] [βx + (1 β)y] if and only if α > β. Example 1 revisited checking the Order Preserving Axiom Recall the preference relation defined in Example 1, we take (x 1, y 1 ), (x 2, y 2 ) B such that (x 1, y 1 ) (x 2, y 2 ) so that x 1 +y 1 x 2 y 2 > 0. Take α, β [0,1] such that α > β, and observe α[(x 1 + y 1 ) (x 2 + y 2 )] > β[(x 1 + y 1 ) (x 2 + y 2 )]. Adding x 2 + y 2 to both sides, we obtain α(x 1 + y 1 ) + (1 α)(x 2 + y 2 ) > β(x 1 + y 1 ) + (1 β)(x 2 + y 2 ). 30

31 Axiom 5 Intermediate Value For any x, y, z B, if x y z, then there exists a unique α (0,1) such that αx + (1 α)z y. Remark Given 3 alternatives with rankings of x y z, there exists a fractional combination of x and z that is indifferent to y. Trade-offs between the alternatives exist. 31

32 Example 1 revisited checking the Intermediate Value Axiom Given x 1 + y 1 > x 2 + y 2 > x 3 + y 3, choose Rearranging gives α = (x 2 + y 2 ) (x 3 + y 3 ) (x 1 + y 1 ) (x 3 + y 3 ). α(x 1 + y 1 ) + (1 α)(x 3 + y 3 ) = x 2 + y 2 so that [α(x 1, y 1 ) + (1 α)(x 3, y 3 )] (x 2, y 2 ). 32

33 Dictionary order does not satisfy the intermediate value axiom Suppose (x 1, y 1 ),(x 2, y 2 ),(x 3, y 3 ) B such that (x 1, y 1 ) (x 2, y 2 ) (x 3, y 3 ) and x 1 > x 2 = x 3 and y 2 > y 3. For any α (0,1), we have α(x 1, y 1 ) + (1 α)(x 3, y 3 ) = α(x 1, y 1 ) + (1 α)(x 2, y 3 ) = (αx 1 + (1 α)x 2, αy 1 + (1 α)y 3 ). But for α > 0, we have αx 1 + (1 α)x 2 > x 2 so α(x 1, y 1 ) + (1 α)(x 3, y 3 ) (x 2, y 2 ) for all α (0,1). In other words, there is no α (0,1) such that αx + (1 α)z y. 33

34 Axiom 6 Boundedness There exist x, y B such that x z y for all z B. This Axiom ensures the existence of a most preferred element x B and a least preferred element y B. Example 1 revisited checking the Boundedness Axiom Recall B = {(x, y) : x [0, ) and y [0, )}. Given any (z 1, z 2 ) B, we have (z 1 + 1, z 2 ) (z 1, z 2 ) since z 1 + z > z 1 + z 2. Therefore, a maximum does not exist. 34

35 Motivation for defining utility Knowledge of the preference relation effectively requires a complete listing of preferences over all pairs of elements from the choice set B. We define a utility function that assigns a numeric value to each element of the choice set such that a larger numeric value implies a higher preference. We establish the theorem on the existence of utility function. The optimal criterion for ranking alternative investments is based on the ranking of the expected utility values of the various investments. 35

36 Theorem Existence of Utility Function Let B denote the set of payoffs from a finite number of securities, also being a convex subset of R n. Let denote a preference relation on B. Suppose satisfies the following axioms (i) x B, x x. (ii) x, y B, x y or y x. (iii) For any x, y, z B, if x y and y z, then x z. (iv) For any x, y B, x y and α, β [0,1], αx + (1 α)y βx + (1 β)y if and only if α > β. (v) For any x, y, z B, suppose x y z, then there exists a unique α (0,1) such that αx + (1 α)z y. (vi) There exist x, y B such that z B, x z y. Then there exists a utility function U : B R such that (a) x y iff U(x) > U(y). (b) x y iff U(x) = U(y). 36

37 To show the existence of U : B R, we write down one such function and show that it satisfies the stated conditions. Based on Axiom 6, we choose x, y B such that x z y for all z B. Without loss of generality, let x y. [Otherwise, x z y for all z B. In this case, U(z) = 0 for all z B, which is a trivial utility function that satisfies conditions (a) and (b).] Consider an arbitrary z B. There are 3 possibilities: 1. z x ; 2. x z y ; 3. z y. 37

38 We define U by giving its value under all 3 cases: 1. U(z) = 1 2. By Axiom 5, there exists a unique α (0,1) such that [αx + (1 α)y ] z. Define U(z) = α. 3. U(z) = 0. Such U satisfies properties (a) and (b). 38

39 Proof of property (a) Necessity Suppose z 1, z 2 B are such that z 1 z 2, we need to show U(z 1 ) > U(z 2 ). Consider the four possible cases. 1. z 1 x z 2 y 2. z 1 x z 2 y 3. x z 1 z 2 y 4. x z 1 z 2 y. Case 1 By definition, U(z 1 ) = 1 and U(z 2 ) = α, where α (0,1) uniquely satisfies αx + (1 α)y z 2. Now, U(z 1 ) = 1 > α = U(z 2 ). 39

40 Case 2 By definition, U(z 1 ) = 1 > 0 = U(z 2 ). Case 3 By defintion, U(z i ) = α i, where α i (0,1) uniquely satisfies so that α i x + (1 α i )y z i, z 1 [α 1 x + (1 α 1 )y ] and [α 2 x + (1 α 2 )y ] z 2. We claim α 1 > α 2. Assume not, then α 1 α 2. By Axiom 4, [α 2 x + (1 α 2 )y ] [α 1 x + (1 α 1 )y ]. This is a contradiction. Hence, α 1 > α 2 is true and U(z 1 ) = α 1 > U(z 2 ) = α 2. 40

41 Case 4 By definition, U(z 1 ) = α 1, where α 1 (0,1) uniquely satisfies α 1 x + (1 α 1 )y y 1 and U(z 2 ) = 0. We have U(z 1 ) = α 1 > 0 = U(z 2 ). 41

42 Sufficiency Suppose, given z 1, z 2 B, that U(z 1 ) > U(z 2 ), we would like to show z 1 z 2. Consider the following 4 cases 1. U(z 1 ) = 1 and U(z 2 ) = α 2, where α 2 (0,1) uniquely satisfies [α 2 x + (1 α 2 )y ] z U(z 1 ) = 1, where z 1 x and U(z 2 ) = 0, where z 2 y. 3. U(z i ) = α i, where α i (0,1) uniquely satisfies [α i x + (1 α i )y ] z i. 4. U(z 1 ) = α 1 and U(z 2 ) = 0, where z 2 y. 42

43 Case 1 z 1 x [1 x + 0 y ] and z 2 [α 2 x + (1 α 2 )y ]. By Axiom 4, 1 > α 2 so that z 1 z 2. Case 2 z 1 x y z 2. Case 3 z 1 [α 1 x + (1 α 1 )y ] z 2 [α 2 x + (1 α 2 )y ] Since α 1 > α 2, by Axiom 4, z 1 z 2. Case 4 z 1 [α 1 x + (1 α 1 )y ] and z 2 y [0x + (1 0)y ]. By Axiom 4 and Axiom 3, since α 1 > 0, z 1 z 2. 43

44 Proof of Property (b) Necessity Suppose z 1 z 2 but U(z 1 ) U(z 2 ), then U(z 1 ) > U(z 2 ) or U(z 2 ) > U(z 1 ). By property (a), this implies z 1 z 2 or z 2 z 1, a contradiction. Hence, Sufficiency U(z 1 ) = U(z 2 ). Suppose U(z 1 ) = U(z 2 ), but z 1 z 2 or z 1 z 2. By property (a), this implies U(z 1 ) > U(z 2 ) or U(z 2 ) > U(z 1 ), a contradiction. Hence, z 1 z 2. 44

45 3.3 Maximum expected utility criterion How to make a choice between the following two lotteries: L 1 = {p 1, A 1 ; p 2, A 2 ; ; p n, A n } L 2 = {q 1, A 1 ; q 2, A 2 ; ; q n, A n }? The outcomes are A 1,, A n ; p i and q i are the probabilities of occurrence of A i in L 1 and L 2, respectively. These outcomes are mutually exclusive and only one outcome can be realized under each investment. We are not limited to lotteries with the same set of outcomes. Suppose outcome A i will not occur in Lottery L 1, we can simply set p i = 0. Comparability When faced by two monetary outcomes A i and A j, the investor must say A i A j, A j A i or A i A j. 45

46 Continuity If A 3 A 2 and A 2 A 1, then there exists U(A 2 ) [0 U(A 2 ) 1] such that L = {[1 U(A 2 )], A 1 ; U(A 2 ), A 3 } A 2. For a given set of outcomes A 1, A 2 and A 3, these probabilities are a function of A 2, hence the notation U(A 2 ). Why is it called continuity axiom? When U(A 2 ) = 1, we obtain L = A 3 A 2 ; when U(A 2 ) = 0, we obtain L = A 1 A 2. If we increase U(A 2 ) continuously from 0 to 1, we hit a value U(A 2 ) such that L A 2. Remark Though U(A 2 ) is a probability value, we will see that it is also the investor s utility function. 46

47 Interchangeability Given L 1 = {p 1, A 1 ; p 2, A 2 ; p 3, A 3 } and A 2 A = {q, A 1 ;(1 q), A 3 }, the investor is indifferent between L 1 and L 2 = {p 1, A 1 ; p 2, A; p 3, A 3 }. Transitivity Given L 1 L 2 and L 2 L 3, then L 1 L 3. Also, if L 1 L 2 and L 2 L 3, then L 1 L 3. 47

48 Decomposability A complex lottery has lotteries as prizes. A simple lottery has monetary values A 1, A 2, etc as prizes. Consider a complex lottery L = (q, L 1 ;(1 q), L 2 ), where L 1 = {p 1, A 1 ;(1 p 1 ), A 2 } and L 2 = {p 2, A 1 ;(1 p 2 ), A 2 }, L can be decomposed into a simple lottery L = {p, A 1 ;(1 p ), A 2 }, with A 1 and A 2 as prizes where p = qp 1 + (1 q)p 2. 48

49 Monotonicity (a) For monetary outcomes, A 2 > A 1 = A 2 A 1. (b) For lotteries (i) Let L 1 = {p, A 1 ;(1 p), A 2 } and L 2 = {p, A 1 ;(1 p), A 3 }. If A 3 > A 2, then A 3 A 2 ; and L 2 L 1. (ii) Let L 1 = {p, A 1 ;(1 p), A 2 } and L 2 = {q, A 1 ;(1 q), A 2 }, also A 2 > A 1 (hence A 2 A 1 ). If p < q, then L 1 L 2. 49

50 Theorem The optimal criterion for ranking alternative investments is the expected utility of the various investments. Proof How to make a choice between L 1 and L 2 L 1 = {p 1, A 1 ; p 2, A 2 ; ; p n, A n } L 2 = {q 1, A 1 ; q 2, A 2 ; ; q n, A n } A 1 < A 2 < < A n, where A i are various monetary outcomes? By comparability axiom, we can compare A i. Further, by monotonicity axiom, we determine that A 1 < A 2 < < A n implies A 1 A 2 A n. Define A i = {[1 U(A i)], A 1 ; U(A i ), A n } where 0 U(A i ) 1. 50

51 By continuity axiom, for every A i, there exists U(A i ) such that A i A i. For A 1, U(A 1 ) = 0, hence A 1 A 1; for A n, U(A n ) = 1. For other A i,0 < U(A i ) < 1. By the monotonicity and transitivity axioms, U(A i ) increases from zero to one as A i increases from A 1 to A n. Substitute A i by A i in L 1 successively and by the interchangeability axiom, L 1 L 1 = {p 1, A 1 ; p 2, A 2 ; ; p n, A n }. 51

52 By the decomposability axiom, Similarly L 1 L 1 L 1 = {Σp i[1 U(A i )], A 1 ;Σp i U(A i ), A n }. L 2 L 2 = {Σq i[1 U(A i )], A 1 ;Σq i U(A i ), A n }. By the monotonicity axiom, L 1 L 2 if Σp i U(A i ) > Σq i U(A i ). This is precisely the expected utility criterion. The same conclusion applies to L 1 L 2, due to transitivity. 52

53 Remarks Recall A i A i = {[1 U(A i)], A 1 ; U(A i ), A n }, such a function U(A i ) always exists, though not all investors would agree on the specific value of U(A i ). By the monotonicity axiom, utility is non-decreasing. A utility function is determined up to a positive linear transformation, so its value is not limited to the range [0,1]. Determined means that the ranking of the projects by the MEUC does not change. The absolute difference or ratio of the utilities of two investment choices gives no indication of the degree of preference of one over the other since utility values can be expanded or suppressed by a linear transformation. 53

54 3.4 Characterization of utility functions 1. More is being preferred to less: u (w) > 0 2. Investors taste for risk averse to risk (certainty equivalent < mean) neutral toward risk (indifferent to a fair gamble) seek risk (certainty equivalent > mean) 3. Investors preference changes with a change in wealth. Percentage of wealth invested in risky asset changes as wealth changes. Jensen s inequality Suppose u (w) 0 and X is a random variable, then u(e[x]) E[u(X)]. 54

55 Write E[X] = µ; since u(w) is concave, we have u(w) u(µ) + u (µ)(w µ) for all values of w. 55

56 Replace w by X and take the expectation on each side Interpretation E[u(X)] u(µ) = u(e[x]). E[u(X)] represents the expected utility of the gamble associated with X. The investor prefers a sure wealth of µ = E[X] rather than playing the game, if u (w) 0. This indicates risk aversion. Recall that the certainty equivalent c is given by u(c) = E[u(X)] u(µ) so that c µ since u is an increasing function. The certainty equivalent may be visualized as the price of the game. The investor visualizes the price to be less than its mean value. 56

57 Insurance premium Individual s total initial wealth is w, and the wealth is subject to random loss Y during the period, 0 Y < w. Let π be the insurable premium payable at time 0 that fully reimburses the loss (neglecting the time value of money). 1. If the individual decides not to buy insurance, then the expected utility is E[u(w Y )]. The expectation is based on investor s own subjective assessment of the loss. 2. If he buys the insurance, the utility at the end of the period is u(w π). Note that w π is the sure wealth. 57

58 If the individual is risk averse [u (w) 0], then from Jensen s inequality (change X to w Y ), we obtain u(w E[Y ]) E[u(w Y )]. The fair value of insurance premium π is determined by u(w π) = E[u(w Y )] so that we can deduce that π E[Y ]. Suppose the higher moments of Y are negligible, it can be deduced that the maximum premium that a risk-averse individual with wealth w is willing to pay to avoid a possible loss of Y is approximately π µ Y + σ2 Y 2 R A(w µ), where R A (w) = u (w)/u (w), 0 Y < w and µ = E[Y ] < w. With higher R A (w), the individual is willing to pay a higher premium to avoid risk. 58

59 Proof We start from the governing equation for π u(w π) = E[u(w Y )]. Write Y = µ + zv, where V is a random variable with zero mean. Here, z is a small perturbation parameter. We then have u(w π) = E[u(w µ zv )]. (1) We are seeking the perturbation expansion of π in powers of z in the form π = a + bz + cz 2 + (i) Setting z = 0, u(w a) = E[u(w µ)] = u(w µ) so that a = µ. 59

60 (ii) Differentiating (1) with respect to z and setting z = 0, π (0)u (w π) = E[ V u (w µ)] (2) since E[V ] = 0 and π (0) = b, so b = 0. (iii) Differentiating (1) twice with respect to z and setting z = 0 π (0)u (w π) = E[V 2 u (w µ)] and observing var(v ) = E[V 2 ] since E[V ] = 0, we obtain c = var(v ) 2 u u. w µ 60

61 Define the absolute risk aversion coefficient: R A (w) = u (w) u (w), we have π µ + R A(w µ) z 2 var(v ) 2 = µ + σ2 Y 2 R A(w µ). π µ σ2 Y 2 R A(w µ) is called the risk premium. For low level of risks, π µ is approximately proportional to the product of variance of the loss distribution and individual s absolute risk aversion. 61

62 Relative risk aversion coefficient Let X be a fair game with E[X] = 0 and var(x) = σx 2. The whole wealth w is invested into the game. Choice A Choice B w + X w C (with certainty) The investor is indifferent to these two positions iff E[u(w + X)] = u(w C ). Note that w C = w (w w C ), indicating the payment of w w C for Choice B. The difference w w C represents the maximum amount the investor would be willing to pay in order to avoid the risk of the game. 62

63 Let q be the fraction of wealth an investor is giving up in order to avoid the gamble; then q = w w C or w C = w(1 q). Let Z be the w return per dollar invested so that for a fair gamble, E[Z] = 1. Write var(z) = σz 2. Suppose we invest w dollars, the return would be wz. Expand u(wz) around w: u(wz) = u(w) + u (w)(wz w) + u (w) 2 (wz w)2 + E[u(wZ)] = u(w) u (w) 2 w2 σ 2 Z + since σ 2 Z = E[(Z 1)2 ]. 63

64 On the other hand, u(w C ) = u(w(1 q)) = u(w) qwu (w) +. Equating u(w C ) with E[u(wZ)], we obtain so that u (w) 2 w2 σ 2 Z = u (w)qw q = σ2 Z 2 wu (w) u (w). Define R R (w) = coefficient of relative risk aversion = w u (w) u (w), then q = w w C w = percentage of risk premium = σ2 Z 2 R R(w). 64

65 Types of utility functions 1. Exponential utility u(x) = 1 e ax, x > 0 u (x) = ae ax u (x) = a 2 e ax < 0 so that R A (x) = a for all wealth level x. (risk aversion) 2. Power utility R A (x) = 1 α x u(x) = xα 1 α, α 1 u (x) = x α 1 u (x) = (α 1)x α 2 and R R (x) = 1 α. 65

66 3. Logarithmic utility (corresponds to α 0 in power utility) u(x) = aln x + b, a > 0 u (x) = a/x u (x) = a/x 2 R A (x) = 1 x and R R(x) = 1. Observe that lim α 0 x α 1 α = lim α 0 (ln x)x α 1 = ln x. 66

67 Properties of power utility functions: U(x) = x α /α, α 1 (i) α > 0, aggressive utility Consider α = 1, corresponding to U(x) = x. This is the expected value criterion. Recall that the strategy that maximizes the expected value bets all capital on the most favorable sector prone to early bankruptcy. For α = 1/2; consider two opportunities: (a) capital will double with a probability of 0.9 or it will go to zero with probability 0.10, (b) capital will increase by 25% with certainty. Since > 1.25, so opportunity (a) is preferred to (b). However, opportunity (a) is certain to go bankrupt. 67

68 (ii) α < 0, conservative utility For α = 1/2, consider two opportunities (a) capital quadruples in value with certainty (b) with probability 0.5 capital remains constant and with probability 0.5 capital is multiplied by 10 million. Since 4 1/2 > (10,000,000) 1/2, opportunity (a) is preferred to (b). Apparently, the best choice for α may be negative, but close to zero. This utility function is close to the logarithm function. 68

69 3.5 Two-asset portfolio analysis Example 1 Two-jump Model An investor has an initial wealth of w and can allocate funds between two assets: a risky asset and a riskless asset. m = expected rate of return on the risky security = pi u + (1 p)i d so that p = m i d i u i d, 1 p = i u m i u i d. 69

70 Impose the condition: m > i f, otherwise a rational investor will never invest any positive amount in the risky asset. The risk averse investor should command an expected rate of return higher than r f for bearing the risk. No arbitrage conditions: i u > i f > i d. If otherwise, suppose i u > i d > i f, then a riskfree profit can be secured by borrowing the riskfree asset as much as possible and use the proceeds to buy the risky asset. Also, m observes i u > m > i d given that 0 < p < 1. Let x be the fraction of initial wealth placed on the risky asset. Choose the power utility function: wα,0 < α < 1. α 70

71 Investor s expected utility: p {w[(1 + i f) + x(i u i f )]} α α + (1 p) {w[1 + i f) + x(i d i f )]} α. α To find the optimal strategy, we find x such that the expected utility is maximized. We differentiate the expected utility with respect to x and set it be zero. Since the utility function is concave, the second order condition for a maximum is automatically satisfied. Optimal proportion x = (1 + i f)(θ 1) (i u i f ) + θ(i f i d ) where 71

72 θ = [p(i u i f )] 1/(1 α) [(1 p)(i f i d )] 1/(1 α) = [( m id i f i d ) ( )] iu i 1/(1 α) f. i u m θ > 1 m i f i f i d > i u m i u i f m > i f (i) x > 0 θ > 1. (ii) As m i f, θ 1 and x 0. A risk-averse investor prefers the riskless asset if it has the same return as the expected return on the risky asset. (iii) θ is an increasing function of m and x < 1 as long as θ < 1 + i u 1 + i d. When x > 1, the investor short sells the riskfree asset to increase his leverage. 72

73 Example 2 Multi-state Model Relation between absolute risk aversion and demand function for risky asset Let a denote the number of units of risky asset b denote the number of units of riskfree asset r s = return from the risky asset in state s R = return from the riskless asset. 73

74 Assume a finite set of states S = {1,, s} with probability distribution p = (p 1,, p s ). Let the price of the risky asset be q and the price of the riskless asset be the numeraire (accounting unit). The price of the risky asset becomes r s q if the state s occurs. The investor s budget constraint is W 0 = qa + b, where W 0 is the initial wealth of the investor; b = W 0 qa. We assume no short selling so that a > 0. Return from the portfolio (a, b) if state s occurs W s (a, b) = aqr s + br = aqr s + (W 0 qa)r = RW 0 + (r s R)qa. 74

75 The optimization problem of an expected utility-maximizing investor: choose (a, b) to maximize the expected utility subject to qa + b = W 0. s S p s u(w s (a, b)) Determine the control variable a that maximizes the expected utility value s S p s u(rw 0 + (r s R)qa). The first order condition is s S p s u (RW 0 + (r s R)qa)(r s R)q = 0. 75

76 If the investor is risk-averse, u ( ) is strictly negative, then the second order condition is seen to be s S p s u (RW 0 + (r s R)qa) (r s R) 2 q 2 < 0. A solution to the first order condition must be a maximum if the investor is risk averse. Define a(w 0 ) = demand function for the risky asset at a given initial wealth level W 0, which is the optimal solution to the portfolio choice problem. Question Is the demand for the number of units of the risky asset increasing da or decreasing in initial wealth? That is, > 0 or otherwise! dw 0 76

77 Lemma Let R a (x) denote the absolute risk aversion coefficient as a function of wealth level x. We have a (W 0 ) > 0 if R a(x) < 0 a (W 0 ) = 0 if R a(x) = 0 a (W 0 ) < 0 if R a(x) > 0 Suppose the absolute risk aversion is a decreasing function of x, investors would invest more on the risky asset when the initial wealth level is higher. Proof Consider the derivative with respect to W 0 of the first order condition: s S p s u (RW 0 + (r s R)qa(W 0 )) (r s R)qR + s S p s u (RW 0 + (r s R)qa(W 0 )) (r s R) 2 q 2 a (W 0 ) = 0. 77

78 Solving for a (W 0 ): a (W 0 ) = p s u (RW 0 + (r s R)qa(W 0 ))(r s R) 2 q 2 s S s S p s u (RW 0 + (r s R)qa(W 0 ))(r s R)q. 1 R If the investor is risk-averse, u ( ) < 0. Hence, the sign of a (W 0 ) should be the same as the sign of s S p s u (RW 0 + (r s R)qa(W 0 )) (r s R)q. }{{} can be positive or negative Recall the definition: R a (x) = u (x) u ; the above term can be expressed (x) as s S p s u (RW 0 + (r s R)qa(W 0 )) (r s R)qR a (RW 0 + (r s R)qa(W 0 )). 78

79 For all s S (r s R)qR a (RW 0 ) (r s R)qR a (RW 0 + (r s R)qa(W 0 )) if and only if R a(x) 0. Easier to visualize if we write y = (r s R)q, x 0 = RW 0, λ = a(w 0 ) > 0. We have yr a (x 0 ) > yr a (x 0 + λy) iff R a(x) < 0. To show the claim, we consider the case R a(x) < 0: 79

80 (i) for y > 0 R a (x 0 ) > R a (x 0 + λy) (ii) for y < 0 R a (x 0 ) < R a (x 0 + λy). Consider R a(x) < 0, the sign of a (W 0 ) depends on the sign of p s u (RW 0 + (r s R)qa(W 0 )) (r s R)q s S R a (RW 0 + (r s R)qa(W 0 )) > R a (RW 0 ) p s u (RW 0 + (r s R)qa(W 0 )) (r s R)q s S = 0 [due to the first order condition] Hence, a (W 0 ) > 0. 80

81 Note that utility functions are only unique up to a strictly positive affine transformation. The second derivative alone cannot be used to characterize the intensity of risk averse behavior. The risk aversion coefficients are invariant to a strictly positive affine transformation of individual utility function, say Ũ(w) = au(w)+b, a > 0. We observe U (w)/u (w) = Ũ (w)/u (w). Absolute risk aversion A(w) = U (w) U (w) If A(w) has the same sign for all values of w, then the investor has the same risk preference (risk averse, neutral or seeker) for all values of w (global). Relative risk aversion R(w) = wu (w) U (w). 81

82 Changes in Absolute Risk Aversion with Wealth Condition Definition Property of A(w) Increasing absolute risk aversion Constant absolute risk aversion Decreasing absolute risk aversion As wealth increases, hold fewer dollars in risky assets As wealth increases, hold some dollar amount in risky assets As wealth increases, hold more dollars in risky assets A (w) > 0 A (w) = 0 A (w) < 0 Example w Cw2 e Cw ln w 82

83 Changes in Relative Risk Aversion with Wealth Condition Definition Property of Increasing relative risk aversion Constant relative risk aversion Decreasing relative risk aversion Percentage invested in risky assets declines as wealth increases Percentage invested in risky assets is unchanged as wealth increases Percentage invested in risky assets increases as wealth increases R (W) R (w) > 0 R (w) = 0 R (w) < 0 Examples of Utility Functions w bw 2 ln w e 2w 1/2 83

84 3.6 Quadratic utility and mean-variance criteria The mean-variance criterion can be reconciled with the expected utility approach in either of two ways: (1) using a quadratic utility function, or (2) making the assumption that the random returns are normal variables. Quadratic utility The quadratic utility function can be defined as u(x) = ax b 2 x2, where a > 0 and b > 0. This utility function is really meaningful only in the range x a/b, for it is in this range that the function is increasing. Note also that for b > 0 the function is strictly concave everywhere and thus exhibits risk aversion. 84

85 mean-variance analysis maximum expected utility criterion based on quadratic concave utility (risk averse) Suppose that a portfolio has a random wealth value of y. Using the expected utility criterion, we evaluate the portfolio using E[U(y)] = E [ay b2 ] y2 = ae[y] b 2 E[y2 ] = ae[y] b 2 (E[y])2 b var(y), a > 0, b > 0. 2 The expected utility value is dependent only on the mean and variance of the random wealth y. The optimal portfolio is the one that maximizes this value with respect to all feasible choices of the random wealth variable y. 85

86 For a given value of E[y], maximizing E[U(y)] minimizing var(y). For a given var(y), maximizing E[U(y)] maximizing E[y]. Note that U(x) = ax b 2 x2 is an increasing function of x in the range 0 x a/b, a > 0 and b > 0. U(x) a/b x 86

87 Normal Returns When all returns are normal random variables, the mean-variance criterion is also equivalent to the expected utility approach for any risk-averse utility function. To deduce this, select a utility function U. Consider a random wealth variable y that is a normal random variable with mean value M and standard deviation σ. Since the probability distribution is completely defined by M and σ, it follows that the expected utility is a function of M and σ. If U is risk averse, then E[U(y)] = U(y) 1 e (y M)2 /2σ 2 dy = f(m, σ), 2π with f M > 0 and f σ < 0. 87

88 1. Why f M > 0? Assume M high > M low and let Consider f(m high, σ) = = > ỹ = y (M high M low ). U(y) 1 2πσ 2 e (y M high ) 2 /σ 2 dy U(ỹ + M high M low ) U(ỹ) since U(y) is an increasing function of y. 1 2πσ 2 e (ỹ M low) 2 /σ 2 dỹ 1 2πσ 2 e (ỹ M low) 2 /σ 2 dỹ = f(m low, σ) 88

89 Portfolio selection problems Now suppose that the returns of all assets are normal random variables. Then the return of any linear combination of these asset is a normal random variable. Hence any portfolio problem is therefore equivalent to the selection of combination of assets that maximizes the function f(m, σ) with respect to all feasible combinations. For a risky-averse utility, this again implies that the variance should be minimized for any given value of the mean. Since f(m, σ) is a decreasing function of σ, a lower value of portfolio variance σ 2, the higher value of E[U(y)]. In other words, the solution must be mean-variance efficient. Portfolio problem is to find w such that f(m, σ) is maximized with respect to all feasible combinations. 89

90 3.7 Stochastic dominance Knowing the utility function, we have the full information on preference. Using the maximum expected utility criterion, we obtain a complete ordering of all the investments under consideration. What happens if we have only partial information on preferences (say, prefer more to less and/or risk aversion)? In the First Order Stochastic Dominance Rule, we only consider the class of utility functions, call U 1, such that u 0. This is a very general assumption and it does not assume any specific utility function. 90

91 Dominance in U 1 Investment A dominates investment B in U 1 if for all utility functions such that u U 1, E A u(x) E B u(x). Equivalently, U(F A ) U(F B ), where F A and F B are the distribution function of choices A and B, respectively, and for at least one utility function, there is a strict inequality. Choices among investments amount to choices on probability distributions. Efficient set in U 1 (not being dominated) An investment is included in the efficient set if there is no other investment that dominates it. 91

92 Inefficient set in U 1 (being dominated) The inefficient set includes all inefficient investments. An inefficient investment is that there is at least one investment in the efficient set that dominates it. Remarks The partition into efficient and inefficient sets depends on the choice of the class of utility functions. In general, the smaller the efficient set relative to the feasible set, the better for the decision maker. When we have only one utility function, we have complete ordering of all investment choices. The efficient set may likely contain one element (possibly more than one if we have investments whose expected utility values tie with each other). 92

93 First order stochastic dominance Can we argue that Investment A is better than Investment B? It is still possible that the return from investing in B is 11% but the return is only 8% from investing in A. 93

94 By looking at the cumulative probability distribution, we observe that for all returns and the odds of obtaining that return or less, B consistently has a higher or same value. 94

95 95

96 To compare two investment choices, we examine their corresponding probability distribution, where F X (x) = P r [X x]. Definition A probability distribution F dominates another probability distribution G according to the first-order stochastic dominance if and only if F(x) G(x) for all x C. Lemma F dominates G by FSD if and only if C u(x) df(x) C u(x) dg(x) for all strictly increasing utility functions u(x). 96

97 Proof Let a and b be the smallest and largest values that F and G can take on. Consider b b u(x) d[f(x) G(x)] = u(x)[f(x) a }{{ G(x)]b a } a u (x)[f(x) G(x)] dx zero since F(a) = G(a) = 0 and F(b) = G(b) = 1 b C C u(x) df(x) u(x) dg(x) a u (x)[f(x) G(x)] dx 0. Thus, for u (x) > 0, F(x) G(x) C u(x) df(x) C u(x) dg(x). 97

98 Second order stochastic dominance If both investments turn out the worst, the investor obtains 6% from A and only 5% from B. If the second worst return occurs, the investor obtains 8% from A rather than 9% from B. If he is risk averse, then he should be willing to lose 1% in return at a higher level of return in order to obtain an extra 1% at a lower return level. If risk aversion is assumed, then A is preferred to B. 98

99 Definition A probability distribution F dominates another probability distribution G according to the second order stochastic dominance if and only if for all x C x x F(y) dy G(y) dy. Theorem If F dominates G by SSD, then C u(x) df(x) C u(x) dg(x) for all increasing and concave utility functions u(x). 99

100 According to SSD, A is preferred to B since the sum of cumulative probability for A is always less than or equal to that for B. Write I A (x) = x F A(y) dy I A (8.6) = I A (8) + F A (8) 0.6 = = I A (13.5) = I A (12) + F A (12) 1.5 = =

101 Proof b a u(x) d[f(x) G(x)] = b = u (x) + = u (b) + a u (x)[f(x) G(x)] dx x a b x a u (x) a b a b x a u (x) a [F(y) G(y)] dy b a [F(y) G(y)] dydx [F(y) G(y)] dy [F(y) G(y)] dydx. Given that u (b) > 0 and u (x) < 0, x C C u(x) df(x) u(x) dg(x) if [F(y) G(y)] dy 0, x. a 101

102 Example F(x) = 0 if x < 1 x 1 if 1 x 2 1 if x 2 F dominates G by SSD since x F(y) dy x, G(x) = G(y) dy. 0 if x < 0 x/3 if 0 x 3 1 if x 3 F(x) is seen to be more concentrated (less dispersed).. In this example, F(x) G(x) is not valid for all x. 102

103 Sufficient rules and necessary rules for second order stochastic dominance Sufficient rule 1: FSD rule is sufficient for SSD Proof : If F dominates G by FSD, then F(x) G(x), x. This implies x a [G(y) F(y)] dy 0. Remark The inefficient set according to FSD is a subset of that of SSD. Proof : Suppose G lies in the inefficient set of FSD, say, it is dominated by F by FSD. Then F dominates G by SSD so that G must lie in the inefficient set of SSD. 103

104 Sufficient rule 2: Min F (x) > Max G (x) is a sufficient rule for SSD. Note that Min F (x) > Max G (x) is a very strong requirement. Example F G x p(x) x p(x) 5 1/2 2 3/4 10 1/2 4 1/4 Min F (x) = 5 Max G (x) = 4. Note that F(x) = 0 for x min F (x) while G(x) = 1 for x max G (x). Since F(x) and G(x) are nondecreasing functions in x, so F(x) G(x). Hence, F dominates G. Min F (x) Max G (x) FSD SSD E F u(x) E G u(x), u U

105 Necessary rule 1 (Geometric means) Given a risky project with the distribution (x i, p i ), i = 1,, n, the geometric mean, X geo, is defined as X geo = x p 1 1 xp n n = Taking logarithm on both sides n i=1 x p i i, x i 0. ln X geo = Σp i ln x i = E[ln X]. X geo (F) X geo (G) is a necessary condition for dominance of F over G by SSD. 105

106 Proof Suppose F dominates G by SSD, we have Since ln x = u(x) U 2, E F u(x) E G u(x), u U 2. E F ln x = ln X geo (F) E G ln x = ln X geo (G); we obtain ln X geo (F) ln X geo (G). Since the logarithm function is an increasing function, we deduce X geo (F) X geo (G). Therefore, F dominates G by SSD X geo (F) X geo (G). 106

107 Necessary rule 2 (left-tail rule) Suppose F dominates G by SSD, then Min F (x) Min G (x), that is, the left tail of G must be thicker. Proof by contradiction: Suppose Min F (x) < Min G (x), and write x k = Min F (x). At x k, G will still be zero but F will be positive. Observe that xk xk [G(y) F(y)] dy = [0 F(y)] dy < 0, implying that F is not dominated by G by SSD. Hence, if F dominates G, then Min F (x) Min G (x). 107