Financial Economics: Making Choices in Risky Situations

Transcription

1 Financial Economics: Making Choices in Risky Situations Shuoxun Hellen Zhang WISE & SOE XIAMEN UNIVERSITY March, / 57

2 Questions to Answer How financial risk is defined and measured How an investor s attitude toward or tolerance for risk is to be conceptualized and then measured 2 / 57

3 Outline 3 / 57

4 State-by-state Dominance Mean-variance Dominance Sharpe Ratio Summary In the broadest sense, risk refers to uncertainty about the future cash flows provided by a financial asset. A more specific way of modeling risk is to think of those cash flows as varying across different states of the world in future periods... that is, to describe future cash flows as random variables. 4 / 57

5 State-by-state Dominance State-by-state Dominance Mean-variance Dominance Sharpe Ratio Summary Consider three investments where θ = 1 indicates the bad state and θ = 2 indicates good state. Which one do you prefer? 5 / 57

6 State-by-state Dominance State-by-state Dominance Mean-variance Dominance Sharpe Ratio Summary Investment 3 exhibits state-by-state dominance over investments 1 and 2, because it pays as much in all states and strictly more in at least one state. Any investor who prefers more to less (nonsatiated in consumption) would always choose investment 3 above the others. 6 / 57

7 State-by-state Dominance State-by-state Dominance Mean-variance Dominance Sharpe Ratio Summary But the choice between investments 1 and 2 is not as clear cut. Investment 2 provides a larger gain in the good state, but exposes the investor to a loss in the bad state. 7 / 57

8 Mean-variance Dominance State-by-state Dominance Mean-variance Dominance Sharpe Ratio Summary convert prices and payoffs to percentage returns: 8 / 57

9 Mean-variance Dominance State-by-state Dominance Mean-variance Dominance Sharpe Ratio Summary In probability theory, if a random variable X can take on n possible values, X 1 ; X 2 ;... ; X n, with probabilities p 1 ; p 2 ;...; p n, then the expected value of X is E(X ) = p 1 X 1 + p 2 X p n X n the variance of X is σ 2 (X ) = p 1 [X 1 E(X )] 2 +p 2 [X 2 E(X )] p n [X n E(X )] 2 and the standard deviation of X is σ(x ) = [σ 2 (X )] (1/2). 9 / 57

10 Mean-variance Dominance State-by-state Dominance Mean-variance Dominance Sharpe Ratio Summary E(r 1 ) = = 12.5 σ 1 = [0.5 ( ) (5 12.5) 2 ] ( 1/2) = / 57

11 Mean-variance Dominance State-by-state Dominance Mean-variance Dominance Sharpe Ratio Summary bad state good state E(r) σ Investment 1 5% 20% 12.5% 7.5% Investment 2-50% 60% 5% 55% Investment 3 5% 60% 32.5% 27.5% Investment 1 exhibits mean-variance dominance over investment 2, since it offers a higher expected return with lower variance. 11 / 57

12 Mean-variance Dominance State-by-state Dominance Mean-variance Dominance Sharpe Ratio Summary bad state good state E(r) σ Investment 1 5% 20% 12.5% 7.5% Investment 2-50% 60% 5% 55% Investment 3 5% 60% 32.5% 27.5% But notice that by the mean-variance criterion, investment 3 dominates investment 2 but not investment 1, even though on a state-by-state basis, investment 3 is clearly to be preferred. Mean-variance dominance neither implies nor is implied by state-by-state dominance. 12 / 57

13 Mean-variance Dominance State-by-state Dominance Mean-variance Dominance Sharpe Ratio Summary Mean-variance Dominance can be expressed in the form of a criterion for selecting investments of equal magnitude For investments of the same Er, choose the one with lower σ For investments of the same σ, choose the one with greatest Er 13 / 57

14 Sharpe Ratio Outline State-by-state Dominance Mean-variance Dominance Sharpe Ratio Summary Consider two more assets: Which one do you prefer? Neither exhibits state-by-state dominance, nor the mean-variance dominance. 14 / 57

15 Sharpe Ratio Outline State-by-state Dominance Mean-variance Dominance Sharpe Ratio Summary Consider two more assets: Which one do you prefer? Neither exhibits state-by-state dominance, nor the mean-variance dominance. 14 / 57

16 Sharpe Ratio Outline State-by-state Dominance Mean-variance Dominance Sharpe Ratio Summary Consider two more assets: William Sharpe (US, b.1934, Nobel Prize 1990) suggested that in these circumstances, it can help to compare the two assets Sharpe ratios, defined as E(r)/σ(r). Comparing Sharpe ratios, investment 4 is preferred to investment / 57

17 Sharpe Ratio Outline State-by-state Dominance Mean-variance Dominance Sharpe Ratio Summary Consider two more assets: William Sharpe (US, b.1934, Nobel Prize 1990) suggested that in these circumstances, it can help to compare the two assets Sharpe ratios, defined as E(r)/σ(r). Comparing Sharpe ratios, investment 4 is preferred to investment / 57

18 Sharpe Ratio Outline State-by-state Dominance Mean-variance Dominance Sharpe Ratio Summary But using the Sharpe ratio to choose between assets means assuming that investors weight the mean and standard deviation equally, in the sense that a doubling of σ(r) is adequately compensated by a doubling of E(r). Investors who are more or less averse to risk will disagree. 16 / 57

19 State-by-state Dominance Mean-variance Dominance Sharpe Ratio Summary State-by-state dominance is the most robust criterion, but often cannot be applied. Mean-variance dominance is more widely-applicable, but can sometimes be misleading and cannot always be applied. The Sharpe ratio can always be applied, but requires a very specific assumption about consumer attitudes towards risk. We need a more careful and comprehensive approach to comparing random cash flows. 17 / 57

20 Preferences Of course, economists face a more general problem of this kind. Even if we accept that more (of everything) is preferred to less, how do consumers compare different bundles of goods that may contain more of one good but less of another? Microeconomists have identified a set of conditions that allow a consumer s preferences to be described by a utility function. 18 / 57

21 Preferences Outline Preferences Let a, b, and c represent three bundles of goods. These may be arbitrarily long lists, or vectors (a R N ), indicating how much of each of an arbitrarily large number of goods is included in the bundle. A preference relation can be used to represent the consumer s preferences over different consumption bundles. 19 / 57

22 Preferences Outline Preferences a b, indicates that the consumer strictly prefers a to b a b indicates that the consumer is indifferent between a and b a b indicates that the consumer either strictly prefers a to b or is indifferent between a and b 20 / 57

23 Assumptions on Preferences Preferences A.1. The preference relation is assumed to be complete: For any two bundles a and b, either a b, b a, or both, and if both hold, a b. The consumer has to decide whether he or she prefers one bundle to another or is indifferent between the two. Ambiguous tastes are not allowed. A.2. The preference relation is assumed to be transitive: For any three bundles a, b and c, if a b, b c, then a c. The consumer s tastes must be consistent in this sense. Together, (A.1.) and (A.2.) require the consumer to be fully informed and rational. 21 / 57

24 Assumptions on Preferences Preferences A.1. The preference relation is assumed to be complete: For any two bundles a and b, either a b, b a, or both, and if both hold, a b. The consumer has to decide whether he or she prefers one bundle to another or is indifferent between the two. Ambiguous tastes are not allowed. A.2. The preference relation is assumed to be transitive: For any three bundles a, b and c, if a b, b c, then a c. The consumer s tastes must be consistent in this sense. Together, (A.1.) and (A.2.) require the consumer to be fully informed and rational. 21 / 57

25 Assumptions on Preferences Preferences A.3. The preference relation is assumed to be continuous: if a n and b n are two sequences of bundles such that a n a, b n b and a n b n for all n, then a b. Very small changes in consumption bundles cannot lead to large changes in preferences over those bundles. 22 / 57

26 Assumptions on Preferences Preferences Remark An two-good example that violates (A.3.) is the case of lexicographic preferences: a = (a 1, a 2 ) b = (b 1, b 2 ) if a 1 > b 1, or a 1 = b 1 and a 2 > b 2. It is not possible to represent these preferences with a utility function, since the preferences are fundamentally two-dimensional and the value of the utility function has to be one-dimensional. 23 / 57

27 Preferences Theorem If preferences are complete, transitive, and continuous, then they can be represented by a continuous, time-invariant, real-valued utility function. That is, if (A.1.)-(A.3.) hold, there is a continuous function u : R n R such that for any two consumption bundles a and b, a b if and only if u(a) u(b) 24 / 57

28 Preferences Note that if preferences are represented by the utility function u a b if and only if u(a) u(b) then they are also represented by the utility function v, where v( ) = F (u( )) where F : R R is any increasing function. The concept of utility as it is used in standard microeconomic theory is ordinal, as opposed to cardinal. 25 / 57

29 Cardinal V.S. Ordinal Preferences An ordinal utility function describing a consumer s preferences over two goods can be written as u(x, y), the same preferences could be expressed as another utility function that is an increasing transformation of u: g(x, y) = f (u(x, y)). Utility functions g and u give rise to identical indifference curve mappings. A cardinal utility function that preserves preference orderings uniquely up to positive affine transformations. Two utility indices are related by an affine transformation, i.e. u and v satisfies a relationship of the form v(x) = au(x) + b, where a and b are constants. 26 / 57

30 Under certainty, the goods are described by consumption baskets with known characteristics. Under uncertainty, the goods are random (state-contingent) payoffs. The problem of describing preferences over these state-contingent payoffs, and then summarizing these preferences with a utility function, is similar in overall spirit but somewhat different in its details to the problem of describing preferences and utility functions under certainty. 27 / 57

31 Consider shares of stock in two companies: bad state good state IBM RDS where the good state occurs with probability π and the bad state occurs with probability 1 π. We assume that if the two assets provide exactly the same state-contingent payoffs, then investors will be indifferent between them. 28 / 57

32 bad state good state IBM RDS We also assume that investors will prefer any asset that exhibits state-by-state dominance over another. Thus, if u(x) measures utility from the payoff x in any particular state, we will assume that u is increasing. 29 / 57

33 bad state good state IBM RDS Here, there is no state-by-state dominance, but it seems reasonable to assume that a higher probability π will make investors tend to prefer IBM, while a higher probability 1 π will make investors tend to prefer RDS. 30 / 57

34 A criterion that reflects both of these properties was suggested by Blaise Pascal (France, ): base decisions on the expected payoff, E(x) = πx G + (1 π)x B ; where x G and x B, with x G > x B, are the payoffs in the good and bad states. The expected payoff rises when either the payoff in either state rises or the probability of the good state goes up. 31 / 57

35 Nicolaus Bernoulli (Switzerland, ) pointed to a problem with basing investment decisions exclusively on expected payoffs: it ignores risk. To see this, specialize the previous example by setting π = 1 π = 0.5 but add, as well, a third asset: bad state good state IBM RDS T-bill The expected payoff of all three assets are E(x) = 125,but the T-bill is less risky than both stocks, and IBM stock is less risky than RDS stock. 32 / 57

36 Sony is dominated by both IBM and RDS. But the choice between the latter two can now be described in terms of an improvement of $10 over the Sony payoff, either in state 1 or in state 2. Which is better? The relevant feature is that IBM adds $10 when the payoff is low ($90), while RDS adds the same amount when the payoff is high ($150). Most people would think IBM more desirable, and with equal state probabilities, would prefer IBM. 33 / 57

37 Sony is dominated by both IBM and RDS. But the choice between the latter two can now be described in terms of an improvement of $10 over the Sony payoff, either in state 1 or in state 2. Which is better? The relevant feature is that IBM adds $10 when the payoff is low ($90), while RDS adds the same amount when the payoff is high ($150). Most people would think IBM more desirable, and with equal state probabilities, would prefer IBM. 33 / 57

38 Gabriel Cramer (Switzerland, ) and Daniel Bernoulli (Switzerland, ) suggested that more reliable comparisons could be made by assuming that the utility function u over payoffs in any given state is concave as well as increasing. This implies that investors prefer more to less, but have diminishing marginal utility as payoffs increase. 34 / 57

39 About two centuries later, John von Neumann (Hungary, ) and Oskar Morgenstern (Germany, ) worked out the conditions under which investors preferences over risky payoffs could be described by an expected utility function such as U(x) = E[u(x)] = πu(x G ) + (1 π)u(x B ); where the Bernoulli utility function u is increasing and concave and the von Neumann-Morgenstern utility function U is linear in the probabilities. 35 / 57

40 Simple lottery Outline Lottery Assumptions The simple lottery (x; y; π) offers payoff x with probability π and payoff y with probability 1 π. In this definition, x and y can be monetary payoffs, as in the stock and bond examples from before. Alternatively, they can be additional lotteries! 36 / 57

41 Compound lottery Lottery Assumptions The compound lottery (x; (y, z, τ); π) offers payoff x with probability π and lottery (y, z, τ) with probability 1 π. Notice that a simple lottery with more than two outcomes can always be reinterpreted as a compound lottery where each individual lottery has only two outcomes. 37 / 57

42 Compound lottery Lottery Assumptions Notice that a simple lottery with more than two outcomes can always be reinterpreted as a compound lottery where each individual lottery has only two outcomes. 38 / 57

43 Compound lottery Lottery Assumptions So restricting ourselves to lotteries with only two outcomes does not entail any loss of generality in terms of the number of future states that are possible. But to begin describing preferences over lotteries, we need to make additional assumptions. 39 / 57

44 Axioms Outline Lottery Assumptions C.1.a. A lottery that pays off x with probability one is the same as getting x for sure: (x, y, 1) = x. C.1.b. Investors care about payoffs and probabilities, but not the specific ordering of the states: (x, y, π) = (y, x, 1 π) C.1.c. In evaluating compound lotteries, investors care only about the probabilities of each final payoff: (x; z; π) = (x, y, π + (1 π)τ) if z = (x, y, τ) 40 / 57

45 Axioms Outline Lottery Assumptions C.2. There exists a preference relation defined on lotteries that is complete and transitive. Again, this amounts to requiring that investors are fully informed and rational. C.3. The preference relation defined on lotteries is continuous. Hence, very small changes in lotteries cannot lead to very large changes in preferences over those lotteries. 41 / 57

46 Axioms Outline Lottery Assumptions By the previous theorem, we already know that (C.2) and (C.3) are sufficient to guarantee the existence of a utility function over lotteries and, by (C.1a), payoffs received with certainty as well. What remains is to identify the extra assumptions that guarantee that this utility function is linear in the probabilities, that is, of the von Neumann-Morgenstern (vn-m) form. 42 / 57

47 Axioms Outline Lottery Assumptions C.4. Independence axiom: For any two lotteries (x; y; π) and (x; z; π), y z if and only if (x; y; π) (x; z; π). This assumption is controversial and unlike any made in traditional microeconomic theory: you would not necessarily want to assume that a consumer s preferences over sub-bundles of any two goods are independent of how much of a third good gets included in the overall bundle. But it is needed for the utility function to take the vn-m form. 43 / 57

48 Axioms Outline Lottery Assumptions C.5. There is a best lottery b and a worst lottery w. This assumption will automatically hold if there are only a finite number of possible payoffs and if the independence axiom holds. C.6. (implied by (C.3)) Let x, y, and z satisfy x y z. Then there exists a probability π such that (x; z; π) y. C.7. (implied by (C.4)) Let x y. Then (x; y; π 1 ) (x; y; π 2 ) if and only if π 1 > π / 57

49 Lottery Assumptions Theorem Expected Utility Theorem Consider a preference ordering, defined on the space of lotteries, that satisfies axioms (C.1) (C.7), then there exists a utility function U defined over lotteries, with Bernoulli utility function, such that U((x, y, π)) = πu(x) + (1 π)u(y) Note that we can prove the theorem simply by constructing the utility functions U and u with the desired properties. 45 / 57

50 Proof Outline Lottery Assumptions Begin by setting U(b) = 1; U(w) = 0 For any lottery z besides the best and worst, (C.6) implies that there exists a probability π z such that (b, w, π z ) z and (C.7) implies that this probability is unique. For this lottery, set U(z) = π z 46 / 57

51 Proof Outline Lottery Assumptions Condition (C.7) also implies that with U so constructed, z z implies and z z implies U(z) = π z > π z = U(z ) U(z) = π z = π z = U(z ) so that U is a utility function that represents the underlying preference relation. 47 / 57

52 Proof Outline Lottery Assumptions Now let x and y denote two payoffs. By (C.1a), each of these payoffs is equivalent to a lottery in which x or y is received with probability one. With this in mind, let u(x) = U(x) = π x u(y) = U(y) = π y 48 / 57

53 Proof Outline Lottery Assumptions Finally, let π denote a probability and consider the lottery z = (x, y, π). Condition (C.1c) implies, (x, y, π) ((b, w, π x ), (b, w, π y ), π) (b, w, ππ x +(1 π)π y ) this last expression is equivalent to U(z) = U((x, y, π)) = ππ x +(1 π)π y = πu(x)+(1 π)u(y) confirming that U has the vn-m form. 49 / 57

54 Remark Outline Lottery Assumptions Note that the key property of the vn-m utility function U(z) = U((x, y, π)) = πu(x) + (1 π)u(y). its linearity in the probabilities π and 1 π, is not preserved by all transformations of the form V (z) = F (U(z)) where F is an increasing function. In this sense, vn-m utility functions are cardinal, not ordinal. 50 / 57

55 Remark Outline Lottery Assumptions On the other hand, given a vn-m utility function U(z) = U((x, y, π)) = πu(x) + (1 π)u(y). consider an affine transformation and define V (z) = αu(z) + β v(x) = αu(x) + β v(y) = αu(y) + β 51 / 57

56 Remark Outline Lottery Assumptions U(z) = U((x, y, π)) = πu(x) + (1 π)u(y) V (z) = αu(z) + β v(x) = αu(x) + β; v(y) = αu(y) + β V ((x, y, π)) = αu((x, y, π)) + β = α[πu(x) + (1 π)u(y)] + β = π[αu(x) + β] + (1 π)[αu(y) + β] = πv(x) + (1 π)v(y) In this sense, the vn-m utility function that represents any given preference relation is not unique. 52 / 57

57 As mentioned previously, the independence axiom has been and continues to be a subject of controversy and debate. Maurice Allais (France, , Nobel Prize 1988) constructed a famous example that illustrates why the independence axiom might not hold in his paper Le Comportement de L Homme Rationnel Devant Le Risque: Critique Des Postulats et Axiomes De L Ecole Americaine, Econometrica Vol.21 (October 1953): pp / 57

58 Consider two lotteries: L 1 = { L 2 = { $10000 with probability 0.1 $0 with probability 0.9 $15000 with probability 0.09 $0 with probability 0.91 Which would you prefer?people tend to say L 2 L / 57

59 Consider two lotteries: L 1 = { L 2 = { $10000 with probability 0.1 $0 with probability 0.9 $15000 with probability 0.09 $0 with probability 0.91 Which would you prefer?people tend to say L 2 L / 57

60 But consider another two lotteries: L 3 = { L 4 = { $10000 with probability 1 $0 with probability 0 $15000 with probability 0.9 $0 with probability 0.1 Which would you prefer?the same people who say L 2 L 1 often say L 3 L 4 55 / 57

61 But consider another two lotteries: L 3 = { L 4 = { $10000 with probability 1 $0 with probability 0 $15000 with probability 0.9 $0 with probability 0.1 Which would you prefer?the same people who say L 2 L 1 often say L 3 L 4 55 / 57

62 L 1 = { L 2 = { L 3 = { L 4 = { $10000 with probability 0.1 $0 with probability 0.9 $15000 with probability 0.09 $0 with probability 0.91 $10000 with probability 1 $0 with probability 0 $15000 with probability 0.9 $0 with probability 0.1 But L 1 = (L 3 ; 0; 0.1) and L 2 = (L 4 ; 0; 0.1) so the independence axiom requires L 3 L 4 L 1 L 2 56 / 57

63 The Allais paradox suggests that feelings about probabilities may not always be linear, but linearity in the probabilities is precisely what defines vn-m utility functions. 57 / 57