ECON FINANCIAL ECONOMICS

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1 ECON FINANCIAL ECONOMICS Peter Ireland Boston College April 3, 2018 These lecture notes by Peter Ireland are licensed under a Creative Commons Attribution-NonCommerical-ShareAlike 4.0 International (CC BY-NC-SA 4.0) License.

2 Problem Set 7, Question 2 Consider an investor with initial wealth W 0 = 10, whose preferences over simple lotteries (x, y, π) can be described by the von Neumann-Morgenstern Expected Utility function U(x, y, π) = πu(w 0 + x) + (1 π)u(w 0 + y) where the Bernoulli utility function takes the special form with γ > 0. u(c) = c 1 γ 1 γ

3 Problem Set 7, Question 2 In the early 1700s, Daniel Bernoulli suggested that concavity of the utility function over payoffs could capture the element of risk aversion that Pascal s criterion base decisions on expected payoff alone missed. With u(c) = c 1 γ 1 γ u (c) = c γ > 0 and u (c) = γc γ 1 < 0 so that the Bernoulli utility function gets more concave as γ rises. Does this also make the investor more risk averse?

4 Problem Set 7, Question 2 To explore this possibility, consider three lotteries: L1) (x, y, π) = (5, 0, 1/2) = coin flip for 5 or 0 L2) (x, y, π) = (2.5, 0, 1) = 2.50 for sure L3) (x, y, π) = (2, 0, 1) = 2 for sure

5 Problem Set 7, Question 2 L1) (x, y, π) = (5, 0, 1/2) = coin flip for 5 or 0 L2) (x, y, π) = (2.5, 0, 1) = 2.50 for sure L3) (x, y, π) = (2, 0, 1) = 2 for sure Every risk averse investor will choose L2 over L1 and L3. But the choice between L1 and L3 will depend on risk aversion. Very risk averse investors will prefer L3 to L1 but less risk aversion investors may prefer L1 to L2.

6 Problem Set 7, Question 2 For each lottery (x, y, π), calculate expected utility [ ] [ ] (10 + x) 1 γ (10 + y) 1 γ U(x, y, π) = π + (1 π) 1 γ 1 γ for an investor with γ = 1/2. Rank the lotteries from most to least preferred, based on expected utility. Then repeat the exercise for γ = 2 and γ = 3.

7 Problem Set 7, Question 2 L1) (x, y, π) = (5, 0, 1/2) = coin flip for 5 or 0 L2) (x, y, π) = (2.5, 0, 1) = 2.50 for sure L3) (x, y, π) = (2, 0, 1) = 2 for sure All three investors get their highest expected utility from L2. But the choice between L1 and L3 depends on risk aversion. As γ rises, preferences shift from L1 to L3 as the second choice.

8 Problem Set 7, Question 2 Is it always the case that making u (c) more negative so that the Bernoulli utility function becomes more concave makes an investor more risk averse? Unfortunately, no.

9 Problem Set 7, Question 2 Suppose an investor s preferences are described by the expected utility function in the sense that U(x, y, π) = πu(x) + (1 π)u(y) (x, y, π) (x, y, π ) if and only if U(x, y, π) U(x, y, π ) Then those same preferences are described equally well by the expected utility function for any value of α > 0. V (x, y, π) = αu(x, y, π)

10 Problem Set 7, Question 2 Since where V (x, y, π) = αu(x, y, π) = α[πu(x) + (1 π)u(y)] = απu(x) + α(1 π)u(y) = παu(x) + (1 π)αu(y) = πv(x) + (1 π)v(y) v(c) = αu(c) = v (c) = αu (c) = v (c) = αu (c) we can make v (c) more or less concave by choosing larger or smaller values of α, without changing the underlying preference ordering.

11 The Allais Paradox 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

12 The Allais Paradox Consider two lotteries: { $10000 with probability 0.10 L 1 = $0 with probability 0.90 L 2 = { $15000 with probability 0.09 $0 with probability 0.91 Which would you prefer?

13 The Allais Paradox Consider two lotteries: { $10000 with probability 0.10 L 1 = $0 with probability 0.90 L 2 = { $15000 with probability 0.09 $0 with probability 0.91 People tend to say L 2 L 1.

14 The Allais Paradox But now consider other two lotteries: { $10000 with probability 1.00 L 3 = $0 with probability 0.00 L 4 = { $15000 with probability 0.90 $0 with probability 0.10 Which would you prefer?

15 The Allais Paradox But now consider other two lotteries: { $10000 with probability 1.00 L 3 = $0 with probability 0.00 L 4 = { $15000 with probability 0.90 $0 with probability 0.10 The same people who say L 2 L 1 often say L 3 L 4.

16 The Allais Paradox { $10000 with probability 0.10 L 1 = $0 with probability 0.90 { $15000 with probability 0.09 L 2 = $0 with probability 0.91 { $10000 with probability 1.00 L 3 = $0 with probability 0.00 { $15000 with probability 0.90 L 4 = $0 with probability 0.10 But L 1 = (L 3, 0, 0.10) and L 2 = (L 4, 0, 0.10) so the independence axiom requires L 3 L 4 L 1 L 2.

17 The Allais Paradox { $10000 with probability 0.10 L 1 = $0 with probability 0.90 { $15000 with probability 0.09 L 2 = $0 with probability 0.91 { $10000 with probability 1.00 L 3 = $0 with probability 0.00 { $15000 with probability 0.90 L 4 = $0 with probability 0.10 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.

18 Generalizations of Expected Utility Expected utility remains the dominant framework for analyzing economic decision-making under uncertainty. But a very active line of ongoing research continues to explore alternatives and generalizations.

19 4 Measuring Risk and Risk Aversion A Measuring Risk Aversion B Interpreting the Measures of Risk Aversion C Risk Premium and Certainty Equivalent D Assessing the Level of Risk Aversion

20 Measuring Risk Aversion We ve already seen that within the von Neumann-Morgenstern expected utility framework, risk aversion enters through the concavity of the Bernoulli utility function.

21 Expected Utility Functions When u is concave, a payoff of 5 for sure is preferred to a payoff of 8 with probability 1/2 and 2 with probability 1/2.

22 Measuring Risk Aversion We ve also seen previously that concavity of the utility function is related to convexity of indifference curves. In standard microeconomic theory, this feature of preferences represents a taste for diversity. Under uncertainty, it represents a desire to smooth consumption across future states of the world.

23 Expected Utility Functions A risk averse consumer prefers c A = (c G + c B )/2 in both states to c G in one state and c B in the other.

24 Measuring Risk Aversion Mathematically, u (p) > 0 means that an investor prefers higher payoffs to lower payoffs, and u (p) < 0 means that the investor is risk averse. But is there a way of quantifying an investor s degree of risk aversion? And is there a criterion according to which we might judge one investor to be more risk averse than another?

25 Measuring Risk Aversion Since u (p) < 0 makes an investor risk averse, one conjecture would be to say that an investor with Bernoulli utility function v(p) is more risk averse than another investor with Bernoulli utility function u(p) if v (p) < u (p) for all payoffs p.

26 Measuring Risk Aversion Recall, however, that the preference ordering of an investor with vn-m utility function U(z) = U(x, y, π) = πu(x) + (1 π)u(y) is also represented by the vn-m utility function V (z) = αu(z) = πv(x) + (1 π)v(y), where v(x) = αu(x) and v(y) = αu(y)

27 Measuring Risk Aversion And with for any payoff p, v(p) = αu(p), v (p) = αu (p) v (p) = αu (p), By making α larger or smaller, the Bernoulli utility function can be made more or less concave without changing the underlying preference ordering.

28 Measuring Risk Aversion Two alternative measures of risk aversion are R A (Y ) = u (Y ) u (Y ) R R (Y ) = Yu (Y ) u (Y ) = coefficient of absolute risk aversion = coefficient of relative risk aversion where Y measures the investor s income level. Since v(p) = αu(p) implies v (p) = αu (p) and v (p) = αu (p), these measures are invariant to affine transformations of the Bernoulli utility function.

29 Measuring Risk Aversion Two alternative measures of risk aversion are R A (Y ) = u (Y ) u (Y ) R R (Y ) = Yu (Y ) u (Y ) = coefficient of absolute risk aversion = coefficient of relative risk aversion where Y measures the investor s income level. And since both measures have a minus sign out in front, both are positive and increase when risk aversion rises.