CHOICE THEORY, UTILITY FUNCTIONS AND RISK AVERSION

Transcription

1 CHOICE THEORY, UTILITY FUNCTIONS AND RISK AVERSION Szabolcs Sebestyén Master in Finance INVESTMENTS Sebestyén (ISCTE-IUL) Choice Theory Investments 1 / 66

2 Outline 1 An Introduction to Choice Theory Dominance Choice Theory Under Certainty Choice Theory Under Uncertainty How Restrictive Is Expected Utility Theory? The Allais Paradox 2 Risk Aversion How To Measure Risk Aversion? Interpreting Risk Aversion Risk Premium and Certainty Equivalent Constant Absolute Risk Aversion Constant Relative Risk Aversion Quadratic Utility Sebestyén (ISCTE-IUL) Choice Theory Investments 2 / 66

3 An Introduction to Choice Theory Outline 1 An Introduction to Choice Theory Dominance Choice Theory Under Certainty Choice Theory Under Uncertainty How Restrictive Is Expected Utility Theory? The Allais Paradox 2 Risk Aversion How To Measure Risk Aversion? Interpreting Risk Aversion Risk Premium and Certainty Equivalent Constant Absolute Risk Aversion Constant Relative Risk Aversion Quadratic Utility Sebestyén (ISCTE-IUL) Choice Theory Investments 3 / 66

4 An Introduction to Choice Theory Dominance Outline 1 An Introduction to Choice Theory Dominance Choice Theory Under Certainty Choice Theory Under Uncertainty How Restrictive Is Expected Utility Theory? The Allais Paradox 2 Risk Aversion How To Measure Risk Aversion? Interpreting Risk Aversion Risk Premium and Certainty Equivalent Constant Absolute Risk Aversion Constant Relative Risk Aversion Quadratic Utility Sebestyén (ISCTE-IUL) Choice Theory Investments 4 / 66

5 An Introduction to Choice Theory Dominance Preliminaries Risk means uncertainty in the future cash flow stream The cash flow of an asset in any future period is typically modelled as a random variable Example Consider the following asset pay-offs ($) with π 1 = π 2 = 1/2 for the two states θ: t = 0 t = 1 θ = 1 θ = 2 Investment 1 1, 000 1, 050 1, 200 Investment 2 1, , 600 Investment 3 1, 000 1, 050 1, 600 Sebestyén (ISCTE-IUL) Choice Theory Investments 5 / 66

6 An Introduction to Choice Theory Dominance Dominance State-by-state dominance: the strongest possible form of dominance We assume that the typical individual is non-satiated in consumption: she prefers more rather than less of goods the pay-offs allow her to buy Most of the cases are not so trivial and the concept of risk enters necessarily Risk is not the only consideration, the ranking between investment opportunities is typically preference dependent Sebestyén (ISCTE-IUL) Choice Theory Investments 6 / 66

7 An Introduction to Choice Theory Dominance Mean-Variance Dominance In our example, the mean returns and their standard deviations are E (r 1 ) = 12.5% σ 1 = 7.5% E (r 2 ) = 5% σ 2 = 55% E (r 3 ) = 32.5% σ 3 = 27.5% Mean-variance dominance: higher mean and lower variance It is neither as strong nor as general as state-by-state dominance, and it is not fully reliable A criterion for selecting investments of equal magnitude, which plays a prominent role in modern portfolio theory, is For investments of the same expected return, choose the one with the lowest variance For investments of the same variance, choose the one with the greatest expected return Sebestyén (ISCTE-IUL) Choice Theory Investments 7 / 66

8 An Introduction to Choice Theory Dominance Sharpe ratio criterion Example Consider the state-contingent rates of return with π 1 = π 2 = 1/2: θ = 1 θ = 2 E (r) σ Investment 4 3% 5% 4% 1% Investment 5 2% 8% 5% 3% No dominance in either state-by-state or mean-variance terms In the mean-variance framework, one would require specifying the terms at which the investor is willing to substitute expected return for a given risk reduction Sharpe ratio criterion: (E/σ) 4 = 4 > (E/σ) 5 = 5/3 However, decisions are again preference dependent Sebestyén (ISCTE-IUL) Choice Theory Investments 8 / 66

9 An Introduction to Choice Theory Choice Theory Under Certainty Outline 1 An Introduction to Choice Theory Dominance Choice Theory Under Certainty Choice Theory Under Uncertainty How Restrictive Is Expected Utility Theory? The Allais Paradox 2 Risk Aversion How To Measure Risk Aversion? Interpreting Risk Aversion Risk Premium and Certainty Equivalent Constant Absolute Risk Aversion Constant Relative Risk Aversion Quadratic Utility Sebestyén (ISCTE-IUL) Choice Theory Investments 9 / 66

10 An Introduction to Choice Theory Choice Theory Under Certainty Economic Rationality (1) Economic rationality: individuals behaviour is predictable in that it is systematic, i.e. they attempt to achieve a set objective Definition (Preference relation) There exists a preference relation, denoted by, that describes investors ability to compare various bundles of goods, services and money. For two bundles a and b, the expression a b means the following: the investor either strictly prefers bundle a to bundle b, or she is indifferent between them. Pure indifference is denoted by a b, whereas strict preference by a b. Sebestyén (ISCTE-IUL) Choice Theory Investments 10 / 66

11 An Introduction to Choice Theory Choice Theory Under Certainty Economic Rationality (2) Assumptions Economic rationality can be summarised by the following assumptions: A.1 Every investor possesses a preference relation and it is complete. Formally, for any two bundles a and b, either a b, or b a, or both. A.2 The preference relation satisfies the fundamental property of transitivity: for any bundles a, b and c, if a b and b c, then a c. A.3 The preference relation is continuous in the following sense: let {x n } and {y n } be two sequences of consumption bundles such that x n x and y n y. If x n y n for all n, then x y. Sebestyén (ISCTE-IUL) Choice Theory Investments 11 / 66

12 An Introduction to Choice Theory Choice Theory Under Certainty Existence of Utility Function Theorem Assumptions A.1 through A.3 are sufficient to guarantee the existence of a continuous, time-invariant, real-valued utility function u : R N R +, such that for any two bundles a and b, Remarks: a b u (a) u (b). Under certainty, utility functions are unique only up to monotone transformations The notion of a bundle is very general, different elements of a bundle may represent the consumption of the same good or service in different time periods Sebestyén (ISCTE-IUL) Choice Theory Investments 12 / 66

13 An Introduction to Choice Theory Choice Theory Under Uncertainty Outline 1 An Introduction to Choice Theory Dominance Choice Theory Under Certainty Choice Theory Under Uncertainty How Restrictive Is Expected Utility Theory? The Allais Paradox 2 Risk Aversion How To Measure Risk Aversion? Interpreting Risk Aversion Risk Premium and Certainty Equivalent Constant Absolute Risk Aversion Constant Relative Risk Aversion Quadratic Utility Sebestyén (ISCTE-IUL) Choice Theory Investments 13 / 66

14 An Introduction to Choice Theory Choice Theory Under Uncertainty Introduction Under uncertainty, the objects of choice are typically vectors of state-contingent money pay-offs Those vectors form an asset or an investment Assume that individuals have no intrinsic taste for the assets themselves (not really true in practice!), but rather they are interested in what pay-offs these assets will yield and with what likelihood Also assume that investors prefer a higher pay-off to a lower one Typically, no one investment prospect will strictly dominate the others There are two ingredients in the choice between two alternatives: The probability of the states The ex-post level of utility provided by the investment Sebestyén (ISCTE-IUL) Choice Theory Investments 14 / 66

15 An Introduction to Choice Theory Choice Theory Under Uncertainty Example Example The current price of all assets is $100. The forecasted prices per share in one period are the following: Which stock would you invest in? Asset State 1 State 2 A $100 $150 B $90 $160 C $90 $150 Sebestyén (ISCTE-IUL) Choice Theory Investments 15 / 66

16 An Introduction to Choice Theory Choice Theory Under Uncertainty Lotteries Definition (Lottery) A lottery is an object of choice, denoted by (x, y, π), that offers pay-off x with probability π and y with probability 1 π. Lottery (x, y, π): π x 1 π y Sebestyén (ISCTE-IUL) Choice Theory Investments 16 / 66

17 An Introduction to Choice Theory Choice Theory Under Uncertainty Example of a Compounded Lottery Example Lottery (x, y, π) = ( (x 1, x 2, τ 1 ), (y 1, y 2, τ 2 ), π ) : τ 1 x 1 π 1 τ 1 x 2 1 π τ2 1 τ 2 y 1 y 2 Sebestyén (ISCTE-IUL) Choice Theory Investments 17 / 66

18 An Introduction to Choice Theory Choice Theory Under Uncertainty Axioms and Conventions C.1 (a) (x, y, 1) = x (b) (x, y, π) = (y, x, 1 π) (c) (x, z, π) = ( x, y, π + (1 π) τ ) if z = (x, y, τ). C.2 There exists a preference relation, which is complete and transitive C.3 The preference relation is continuous C.4 Independence of irrelevant alternatives. Let (x, y, π) and (x, z, π) be any two lotteries; then, y z if and only if (x, y, π) (x, z, π) C.5 There exists a best (most preferred) lottery, b, as well as a worst lottery, w C.6 Let x, k, z be pay-offs for which x > k > z. Then there exists a π such that (x, z, π) k C.7 Let x y. Then (x, y, π 1 ) (x, y, π 2 ) if and only if π 1 > π 2 Sebestyén (ISCTE-IUL) Choice Theory Investments 18 / 66

19 An Introduction to Choice Theory Choice Theory Under Uncertainty The Expected Utility Theorem Theorem (The Expected Utility Theorem) If axioms C.1 through C.7 are satisfied, then there exists a utility function U defined on the lottery space such that U (x, y, π) = πu (x) + (1 π) u (y) where u (x) = U ( (x, y, 1) ). U is called the von Neumann Morgenstern (NM) utility function. Sebestyén (ISCTE-IUL) Choice Theory Investments 19 / 66

20 An Introduction to Choice Theory Choice Theory Under Uncertainty Remarks on the NM Utility Function Given the objective specification of probabilities, the NM utility function uniquely characterises an investor Different additional assumptions on u will identify the investor s tolerance for risk We require that u be increasing for all candidate utility functions The theorem confirms that investors are concerned only with an asset s final pay-off and the cumulative probabilities of achieving them, while the structure of uncertainty resolution is irrelevant Although the utility function is not defined over a rate of return, but on pay-off distribution, it can be generalised The NM representation is preserved under a certain class of monotone affine transformations: if U ( ) is a NM utility function, then V ( ) au ( ) + b with a > 0 is also such a function A non-linear transformation does not always respect the preference ordering Sebestyén (ISCTE-IUL) Choice Theory Investments 20 / 66

21 An Introduction to Choice Theory How Restrictive Is Expected Utility Theory? The Allais Paradox Outline 1 An Introduction to Choice Theory Dominance Choice Theory Under Certainty Choice Theory Under Uncertainty How Restrictive Is Expected Utility Theory? The Allais Paradox 2 Risk Aversion How To Measure Risk Aversion? Interpreting Risk Aversion Risk Premium and Certainty Equivalent Constant Absolute Risk Aversion Constant Relative Risk Aversion Quadratic Utility Sebestyén (ISCTE-IUL) Choice Theory Investments 21 / 66

22 An Introduction to Choice Theory How Restrictive Is Expected Utility Theory? The Allais Paradox The Allais Paradox (1) Example Consider the following four lotteries: L 1 = (10, 000; 0; 0.1) versus L 2 = (15, 000; 0; 0.09) L 3 = (10, 000; 0; 1) versus L 4 = (15, 000; 0; 0.9). Rank the pay-offs. Sebestyén (ISCTE-IUL) Choice Theory Investments 22 / 66

23 An Introduction to Choice Theory How Restrictive Is Expected Utility Theory? The Allais Paradox The Allais Paradox (2) Example (cont d) The following ranking is frequently observed in practice: L 2 L 1 and L 3 L 4. However, observe that ( ) L 1 = L 3, L 0, 0.1 ( ) L 2 = L 4, L 0, 0.1, where L 0 = (0, 0, 1). By the independence axiom, the ranking between L 1 and L 2, and between L 3 and L 4, should be identical. This is the Allais paradox (Allais, 1953). Sebestyén (ISCTE-IUL) Choice Theory Investments 23 / 66

24 An Introduction to Choice Theory How Restrictive Is Expected Utility Theory? The Allais Paradox How to Deal With the Allais Paradox? Possible reactions to the Allais paradox: 1 Yes, my choices were inconsistent; let me think again and revise them 2 No, I ll stick to my choices. The following things are missing from the theory: The pleasure of gambling; and/or The notion of regret The Allais paradox is but the first of many phenomena that appear to be inconsistent with standard preference theory Sebestyén (ISCTE-IUL) Choice Theory Investments 24 / 66

25 An Introduction to Choice Theory How Restrictive Is Expected Utility Theory? The Allais Paradox Example: Framing and Loss Aversion Example (Kahneman and Tversky, 1979) 1 In addition to whatever you own, you have been given $1, 000. You are now asked to choose between L A = (1, 000; 0; 0.5) versus L B = (500; 0; 1) 2 In addition to whatever you own, you have been given $2, 000. You are now asked to choose between L C = ( 1, 000; 0; 0.5) versus L D = ( 500; 0; 1) A majority of subjects chose B in case 1 and C in case 2, but this is inconsistent with any preference relation over wealth gambles. The difference between the two cases is that the outcomes are framed as gains relative to the reference wealth level in case 1, but as losses in case 2. Sebestyén (ISCTE-IUL) Choice Theory Investments 25 / 66

26 Outline 1 An Introduction to Choice Theory Dominance Choice Theory Under Certainty Choice Theory Under Uncertainty How Restrictive Is Expected Utility Theory? The Allais Paradox 2 Risk Aversion How To Measure Risk Aversion? Interpreting Risk Aversion Risk Premium and Certainty Equivalent Constant Absolute Risk Aversion Constant Relative Risk Aversion Quadratic Utility Sebestyén (ISCTE-IUL) Choice Theory Investments 26 / 66

27 Introduction Investors want to avoid risk, i.e., they want to smooth their consumption across states of nature Hence, restrictions on the NM expected utility representation must be imposed to guarantee such behaviour Since probabilities are objective and independent of investor preferences, we must restrict the utility function to capture risk aversion Sebestyén (ISCTE-IUL) Choice Theory Investments 27 / 66

28 How To Measure Risk Aversion? Outline 1 An Introduction to Choice Theory Dominance Choice Theory Under Certainty Choice Theory Under Uncertainty How Restrictive Is Expected Utility Theory? The Allais Paradox 2 Risk Aversion How To Measure Risk Aversion? Interpreting Risk Aversion Risk Premium and Certainty Equivalent Constant Absolute Risk Aversion Constant Relative Risk Aversion Quadratic Utility Sebestyén (ISCTE-IUL) Choice Theory Investments 28 / 66

29 How To Measure Risk Aversion? Definition of Risk Aversion Definition (Risk aversion) An investor is said to be (weakly) risk averse if u (w) E [ u ( w) ] for any random wealth w with mean w. Strict risk aversion is represented by strict inequality. An equivalent definition is: Definition (Risk aversion) An investor is said to be (weakly) risk averse if u (a) E [ u (a + ε) ], where ε is a zero-mean random variable and a is a constant. Sebestyén (ISCTE-IUL) Choice Theory Investments 29 / 66

30 How To Measure Risk Aversion? Concavity and Risk Aversion A risk-averse investor would prefer to avoid a fair bet The above inequalities are known as Jensen s inequality and are equivalent to concavity of the utility function See Appendix Concavity is preserved by monotone affine transformations, so, for given preferences, either all utility functions are concave or none are For a differentiable function u, concavity is equivalent to non-increasing marginal utility: u (w 1 ) u (w 0 ) if w 1 > w 0 Strict concavity is equivalent to decreasing marginal utility (strict inequality above) For a twice differentiable function u, concavity is equivalent to u (w) 0 for all w (strict concavity implies strict inequality) Sebestyén (ISCTE-IUL) Choice Theory Investments 30 / 66

31 How To Measure Risk Aversion? Concavity and Risk Aversion Intuition Consider a financial contract where the investor either receives an amount h with probability 1/2, or must pay an amount h with probability 1/2 (i.e., a lottery (h, h, 1/2)) Intuition tells us that a risk-averse investor would prefer to avoid such a security, for any level of personal income Y Formally, u (Y) > 1 2 u (Y + h) + 1 u (Y h) = E (u) = U (Y) 2 This inequality can only be satisfied for all income levels Y if the utility function is (strictly) concave Sebestyén (ISCTE-IUL) Choice Theory Investments 31 / 66

32 How To Measure Risk Aversion? A Strictly Concave Utility Function Source: Danthine and Donaldson (2005), Figure 4.1 Sebestyén (ISCTE-IUL) Choice Theory Investments 32 / 66

33 How To Measure Risk Aversion? Indifferent Curves Source: Danthine and Donaldson (2005), Figure 4.2 Sebestyén (ISCTE-IUL) Choice Theory Investments 33 / 66

34 How To Measure Risk Aversion? Measure Risk Aversion Given that u ( ) < 0, why not say that investor A is more risk averse than investor B iff u A (w) u B (w) for all wealth levels? Let u A (w) au A (w) + b with a > 0 Since the utility function is invariant to affine transformations, u A and u A must describe the identical ordering and must display identical risk aversion However, if a > 1, we have that u A (w) > u A (w) This a contradiction We need a measure of risk aversion that is invariant to affine transformations Sebestyén (ISCTE-IUL) Choice Theory Investments 34 / 66

35 How To Measure Risk Aversion? Coefficients of Risk Aversion Absolute risk aversion: α (w) = u (w) u (w) It depends on the preferences and not on the particular utility function representing the preferences For any risk-averse investor, α (w) 0 Relative risk aversion: ρ (w) = wα (w) = wu (w) u (w) Risk tolerance: τ (w) = 1 α (w) = u (w) u (w) Sebestyén (ISCTE-IUL) Choice Theory Investments 35 / 66

36 Interpreting Risk Aversion Outline 1 An Introduction to Choice Theory Dominance Choice Theory Under Certainty Choice Theory Under Uncertainty How Restrictive Is Expected Utility Theory? The Allais Paradox 2 Risk Aversion How To Measure Risk Aversion? Interpreting Risk Aversion Risk Premium and Certainty Equivalent Constant Absolute Risk Aversion Constant Relative Risk Aversion Quadratic Utility Sebestyén (ISCTE-IUL) Choice Theory Investments 36 / 66

37 Interpreting Risk Aversion Absolute Risk Aversion and the Odds of a Bet Consider an investor with wealth w who is offered the lottery L = (h, h, π) The investor will accept the bet if π is high enough and reject if π is low enough The willingness to accept the bet must also be related to her current level of wealth Proposition Let π = π (w, h) the probability at which the investor is indifferent between accepting and rejecting the bet. Then we have that π (w, h) = h α (w). Sebestyén (ISCTE-IUL) Choice Theory Investments 37 / 66

38 Interpreting Risk Aversion Proof. By definition, π (w, h) must satisfy u (w) = π (w, h) u (w + h) + [ 1 π (w, h) ] u (w h) Taylor approximation yields u (w + h) = u (w) + hu (w) h2 u (w) + R 1 u (w h) = u (w) hu (w) h2 u (w) + R 2 After substitution we have u (w) = π (w, h) [u (w) + hu (w) + 12 ] h2 u (w) + R [ 1 π (w, h) ] [ u (w) hu (w) + 1 ] 2 h2 u (w) + R 2 Sebestyén (ISCTE-IUL) Choice Theory Investments 38 / 66

39 Interpreting Risk Aversion Proof (cont d). Collecting terms yields u (w) = u (w) + [ 2π (w, h) 1 ] [ hu (w) + 1 ] 2 h2 u (w) + + π (w, h) R 1 + [ 1 π (w, h) ] R 2 }{{} R Solving for π (w, h) results in π (w, h) = h [ u (w) u (w) ] R 2hu (w) = h α (w) Sebestyén (ISCTE-IUL) Choice Theory Investments 39 / 66

40 Interpreting Risk Aversion Example: Exponential Utility Example Consider the utility function of the form It is easy to show that α (w) = ν. Hence, we have that u (w) = 1 ν e νw π (w, h) = h ν Now the odds demanded are independent of the level of initial wealth, but depend on the amount of wealth at risk. Sebestyén (ISCTE-IUL) Choice Theory Investments 40 / 66

41 Interpreting Risk Aversion Relative Risk Aversion and the Odds of a Bet Proposition Consider an investment opportunity similar to the one above, but now the amount at risk is proportional to the investor s wealth, i.e., h = θw. Then we have that π (w, θ) = θρ (w). 4 Proof. Very similar to the case of absolute risk aversion. Sebestyén (ISCTE-IUL) Choice Theory Investments 41 / 66

42 Interpreting Risk Aversion Example: Power Utility Example Consider the utility function of the form w 1 γ if γ > 0, γ = 1 u (w) = 1 γ ln w if γ = 1 It is easy to show that ρ (w) = γ. Hence, we have that π (w, h) = θγ Now the odds demanded are independent of the level of initial wealth. but depend on the fraction of wealth that is at risk. Sebestyén (ISCTE-IUL) Choice Theory Investments 42 / 66

43 Interpreting Risk Aversion Risk-Neutral Investors Risk-neutral investors are identified with an affine utility function with c > 0 u (w) = c w + d It is easy to see that both α (w) = 0 and ρ (w) = 0 Such investors do not demand better than even odds when considering risky investments (π (w, h) = π (w, θ) = 1/2) The are indifferent to risk and are concerned only with an asset s expected payoff Sebestyén (ISCTE-IUL) Choice Theory Investments 43 / 66

44 Risk Premium and Certainty Equivalent Outline 1 An Introduction to Choice Theory Dominance Choice Theory Under Certainty Choice Theory Under Uncertainty How Restrictive Is Expected Utility Theory? The Allais Paradox 2 Risk Aversion How To Measure Risk Aversion? Interpreting Risk Aversion Risk Premium and Certainty Equivalent Constant Absolute Risk Aversion Constant Relative Risk Aversion Quadratic Utility Sebestyén (ISCTE-IUL) Choice Theory Investments 44 / 66

45 Risk Premium and Certainty Equivalent Definitions Consider an investor, with current wealth w, evaluating an uncertain risky pay-off Z Relying on the definition of risk aversion, we have ( u w + E ( Z )) E [u ( w + Z )] If an uncertain pay-off is available for sale, a risk-averse agent will only be willing to buy it at a price less than its expected pay-off Certainty equivalent (CE): the maximal certain amount of money an investor is willing to pay for a lottery Risk premium (π): the maximum amount the agent is willing to pay to avoid the gamble The risk premium is the difference between the expected value of the bet and the certainty equivalent Sebestyén (ISCTE-IUL) Choice Theory Investments 45 / 66

46 Risk Premium and Certainty Equivalent Mathematical Formulation According to the above definition, the certainty equivalent and the risk premium are the solutions of the following equations: It follows that [ E u ( w + Z )] ( = u w + CE ( w, Z )) = = u (w + E ( Z ) π ( w, Z )) CE ( w, Z ) = E ( Z ) π ( w, Z ) or π ( w, Z ) = E ( Z ) CE ( w, Z ) Note that from the definition )) of CE we have that u (w + CE) < u (w + E ( Z Since u is increasing (recall that u > 0), we must have CE < E ( Z ) Sebestyén (ISCTE-IUL) Choice Theory Investments 46 / 66

47 Risk Premium and Certainty Equivalent CE and Risk Premium: An Illustration Source: Danthine and Donaldson (2005), Figure 4.3 Sebestyén (ISCTE-IUL) Choice Theory Investments 47 / 66

48 Risk Premium and Certainty Equivalent Example: Certainty Equivalent Example Consider an investor with log utility and initial wealth w 0 = 1, 000. She is offered the lottery Z = (200, 0, 0.5). What is the certainty equivalent of this gamble? Solution The equation that yields the CE is E [u ( w + Z )] = u ( w + CE ) Substitution results in 0.5 ln (1, 200) ln (1, 000) = ln (1, CE) from which CE = Sebestyén (ISCTE-IUL) Choice Theory Investments 48 / 66

49 Risk Premium and Certainty Equivalent Relation Between Risk Premium and Risk Aversion Assume that E ( Z ) = 0, u is twice continuously differentiable and the gamble is a bounded random variable It can be shown that π ( w, Z ) = 1 2 σ2 z α (w) The amount one would pay to avoid the gamble is approximately proportional to the coefficient of absolute risk aversion Let π = θw be the risk premium of w + w Z; then θ = 1 2 σ2 z ρ (w) The proportion θ of initial wealth that one would pay to avoid a gamble equal to the proportion Z of initial wealth depends on relative risk aversion and the variance of Z Sebestyén (ISCTE-IUL) Choice Theory Investments 49 / 66

50 Risk Premium and Certainty Equivalent Example: Power Utility Example Consider again the utility function u (w) = w1 γ 1 γ with γ = 3, w = $500, 000, and Z = ($100, 000; $100, 000; 0.5). Then the risk premium will be π ( w, Z ) = 1 2 σ2 z γ w = , 0002 To double-check the approximation, calculate 3 = $30, , 000 ( u w π ( w, Z )) = u (500, , 000) = [ E u ( w + Z )] = 1 2 u (600, 000) + 1 u (400, 000) = Sebestyén (ISCTE-IUL) Choice Theory Investments 50 / 66

51 Risk Premium and Certainty Equivalent Certainty Equivalent in Terms of Returns Let the equivalent risk-free return be defined as ( u w ( ) ) 1 + r f = u (w + CE ( w, Z )) The random pay-off Z can also be converted into a return distribution via Z = rw, or, r = Z/w Hence, r f is defined by the equation ( u w ( ) ) [ 1 + r f = E u (w ( 1 + r ))] The return risk premium, π r, is defined as π r = E ( r ) r f or E ( r ) = r f + π r Sebestyén (ISCTE-IUL) Choice Theory Investments 51 / 66

52 Risk Premium and Certainty Equivalent Example: Log Utility Example Consider the utility function u (w) = ln w, and assume w = $500, 000, and Z = ($100, 000; $50, 000; 0.5). Then the risky return is 20% with probability 0.5 and 10% with probability 0.5, and with an expected return of E ( r ) = 5%. The certainty equivalent must satisfy ln (500, CE) = 1 2 ln (600, 000) + 1 ln (450, 000) 2 which implies that CE = 19, 615 It follows then that the risk-free return is 1 + r f = 519, , 000 = The return risk premium is π r = 5% 3.92% = 1.08% Sebestyén (ISCTE-IUL) Choice Theory Investments 52 / 66

53 Constant Absolute Risk Aversion Outline 1 An Introduction to Choice Theory Dominance Choice Theory Under Certainty Choice Theory Under Uncertainty How Restrictive Is Expected Utility Theory? The Allais Paradox 2 Risk Aversion How To Measure Risk Aversion? Interpreting Risk Aversion Risk Premium and Certainty Equivalent Constant Absolute Risk Aversion Constant Relative Risk Aversion Quadratic Utility Sebestyén (ISCTE-IUL) Choice Theory Investments 53 / 66

54 Constant Absolute Risk Aversion CARA Utility Constant Absolute Risk Aversion (CARA): absolute risk aversion is the same at every wealth level Every CARA utility function is a monotone affine transform of the exponential utility function u (w) = e αw where α is a constant and equal to the absolute risk aversion Sebestyén (ISCTE-IUL) Choice Theory Investments 54 / 66

55 Constant Absolute Risk Aversion No Wealth Effects With CARA Utility With CARA utility there are no wealth effects Assuming that E ( Z ) = 0, we have that u (w π) = E [u ( w + Z )] Calculating the two sides yields u (w π) = e αw e απ and u ( w + Z ) = e αw e α Z Taking expectations and equating gives e απ = E ( e α Z ), from which it follows that π = 1 α ln E( e α Z ) Hence, an investor with CARA utility will pay the same to avoid a fair gamble no matter what her initial wealth might be Sebestyén (ISCTE-IUL) Choice Theory Investments 55 / 66

56 Constant Absolute Risk Aversion Normally Distributed Bet If the gamble Z is Gaussian, then the risk premium can be calculated more explicitly Recall that if x is Gaussian with mean µ and variance σ 2, then E ( e x) = e µ+ 1 2 σ2 Now we have x = α Z which has zero mean and variance α 2 σ 2 z ; thus the risk premium becomes π = 1 2 ασ2 z Sebestyén (ISCTE-IUL) Choice Theory Investments 56 / 66

57 Constant Relative Risk Aversion Outline 1 An Introduction to Choice Theory Dominance Choice Theory Under Certainty Choice Theory Under Uncertainty How Restrictive Is Expected Utility Theory? The Allais Paradox 2 Risk Aversion How To Measure Risk Aversion? Interpreting Risk Aversion Risk Premium and Certainty Equivalent Constant Absolute Risk Aversion Constant Relative Risk Aversion Quadratic Utility Sebestyén (ISCTE-IUL) Choice Theory Investments 57 / 66

58 Constant Relative Risk Aversion It is easy to show that ρ is the coefficient of relative risk aversion of the power utility function Sebestyén (ISCTE-IUL) Choice Theory Investments 58 / 66 CRRA Utility Constant Relative Risk Aversion (CRRA): relative risk aversion is the same at every wealth level Any CRRA utility function has decreasing absolute risk aversion as α (w) = ρ (w) /w Any monotone CRRA utility function is a monotone affine transform of one of the following functions: Log utility: u (w) = log w Power utility: u (w) = 1 γ wγ γ < 1, γ = 0 A more convenient formulation of power utility is u (w) = w1 ρ 1 ρ ρ = 1 γ > 0, ρ = 1

59 Constant Relative Risk Aversion Properties of CRRA Utility Functions Log utility has constant relative risk aversion equal to 1 An investor with power utility is said to be more risk averse than a log-utility investor if ρ > 1 and to be less risk averse if ρ < 1 The fraction of wealth a CRRA-utility investor would pay to avoid a gamble that is proportional to initial wealth is independent of her wealth Log utility is a limiting case of power utility obtained by taking ρ 1 (by l Hôpital s rule) Sebestyén (ISCTE-IUL) Choice Theory Investments 59 / 66

60 Quadratic Utility Outline 1 An Introduction to Choice Theory Dominance Choice Theory Under Certainty Choice Theory Under Uncertainty How Restrictive Is Expected Utility Theory? The Allais Paradox 2 Risk Aversion How To Measure Risk Aversion? Interpreting Risk Aversion Risk Premium and Certainty Equivalent Constant Absolute Risk Aversion Constant Relative Risk Aversion Quadratic Utility Sebestyén (ISCTE-IUL) Choice Theory Investments 60 / 66

61 Quadratic Utility Definition and Properties Quadratic utility takes the form u (w) = w b 2 w2 b > 0 The marginal utility of wealth is u (w) = 1 bw, and is positive only when b < 1/w Thus, this utility function makes sense only when w < 1/b, and the point of maximum utility, 1/b, is called the bliss point The absolute and relative risk aversion coefficients are α (w) = b 1 bw and ρ (w) = bw 1 bw It has increasing absolute risk aversion, an unattractive property Sebestyén (ISCTE-IUL) Choice Theory Investments 61 / 66

62 Quadratic Utility Quadratic Utility and Mean-Variance Preferences Quadratic utility has a special importance in finance as it implies mean-variance preferences Since all derivatives of order higher than 2 are equal to zero, the investor s expected utility becomes See Appendix E [ u ( w) ] = u ( E ( w) ) u ( E ( w) ) Var ( w) Specifically, E [ u ( w) ] = µ b 2 ( σ 2 + µ 2) For any probability distribution of wealth, the expected utility depends only on the mean and variance of wealth Sebestyén (ISCTE-IUL) Choice Theory Investments 62 / 66

63 APPENDIX Sebestyén (ISCTE-IUL) Choice Theory Investments 63 / 66

64 Some Important Definitions and Results Concavity and Jensen s Inequality Definition (Concavity) A function f is concave if, for any x and y and any λ (0, 1), f ( λx + (1 λ) y ) λf (x) + (1 λ) f (y). If the inequality is strict, the function is strictly concave. Theorem (Jensen s inequality) Let f be a concave function on the interval (a, b), and x be a random variable such that Pr [ x (a, b)] = 1. Provided that the expectations E ( x) and E [f ( x)] exist, E [ f ( x) ] f ( E ( x) ). Moreover, if f is strictly concave and Pr [ x = E ( x)] = 1, then the inequality is strict. Return Sebestyén (ISCTE-IUL) Choice Theory Investments 64 / 66

65 Some Important Definitions and Results Taylor Expansion of a Function Definition (Taylor Expansion of a Function) Let f (x) be a real- or complex-valued function, which is infinitely differentiable at a point x 0. Then the function can be written as a power series, called the Taylor expansion of the function, as f (x) = f (x 0 ) + f (x 0 ) 1! + f (x 0 ) 3! (x x 0 ) 3 + (x x 0 ) + f (x 0 ) 2! (x x 0 ) 2 + Sebestyén (ISCTE-IUL) Choice Theory Investments 65 / 66

66 Some Important Definitions and Results Taylor Expansion for the Mean of a Random Variable Theorem (Taylor Expansion for the Mean of a Random Variable) Let x be a random variable with mean µ and variance σ 2 (both finite), and let f be a twice differentiable function. Then the expected value of f ( x ) can be approximated as E [ f ( x) ] = f (µ) f (µ) σ 2. Proof. The Taylor expansion of the expected value of f ( x) is E [ f ( x) ] = E [ f (µ) + f (µ) ( x µ) f (µ) ( x µ) 2 ]. The expectation of the term containing the first derivative is zero, thus the expression simplifies to the one presented in the theorem. Return Sebestyén (ISCTE-IUL) Choice Theory Investments 66 / 66