Economic of Uncertainty

Transcription

1 Economic of Uncertainty Risk Aversion Based on ECO 317, Princeton UC3M April 2012 (UC3M) Economics of Uncertainty. April / 16

2 Introduction 1 Space of Lotteries (UC3M) Economics of Uncertainty. April / 16

3 Introduction 1 Space of Lotteries 2 L o er consequences (C 1, C 2,..., C n ) with probabilities (p 1, p 2,..., p n ) (UC3M) Economics of Uncertainty. April / 16

4 Introduction 1 Space of Lotteries 2 L o er consequences (C 1, C 2,..., C n ) with probabilities (p 1, p 2,..., p n ) 3 We established the existence of a utility function u() such that for any two lotteries L a and L b (UC3M) Economics of Uncertainty. April / 16

5 Introduction 1 Space of Lotteries 2 L o er consequences (C 1, C 2,..., C n ) with probabilities (p 1, p 2,..., p n ) 3 We established the existence of a utility function u() such that for any two lotteries L a and L b 1 EU(L a ) > EU(L b ) if and only if L a L b (UC3M) Economics of Uncertainty. April / 16

6 Introduction 1 Space of Lotteries 2 L o er consequences (C 1, C 2,..., C n ) with probabilities (p 1, p 2,..., p n ) 3 We established the existence of a utility function u() such that for any two lotteries L a and L b 1 EU(L a ) > EU(L b ) if and only if L a L b 2 has the expected utility property EU(L) n p i u(c i ) i =1 (UC3M) Economics of Uncertainty. April / 16

7 Risk aversion and concavity of utility Compare two situations (UC3M) Economics of Uncertainty. April / 16

8 Risk aversion and concavity of utility Compare two situations 1 L o which gives C 0 for sure (UC3M) Economics of Uncertainty. April / 16

9 Risk aversion and concavity of utility Compare two situations 1 L o which gives C 0 for sure 2 L which gives C 1 = C 0 + k and C 2 = C 0 k with probability 1/2 each. (UC3M) Economics of Uncertainty. April / 16

10 Risk aversion and concavity of utility Compare two situations 1 L o which gives C 0 for sure 2 L which gives C 1 = C 0 + k and C 2 = C 0 k with probability 1/2 each. Suppose the utility-of-consequences function u is concave. Then 1 2 u(c 1) u(c 2) < u(c 0 ) (UC3M) Economics of Uncertainty. April / 16

11 Risk aversion and concavity of utility Compare two situations 1 L o which gives C 0 for sure 2 L which gives C 1 = C 0 + k and C 2 = C 0 k with probability 1/2 each. Suppose the utility-of-consequences function u is concave. Then This is EU(L) < EU(L 0 ) 1 2 u(c 1) u(c 2) < u(c 0 ) (UC3M) Economics of Uncertainty. April / 16

12 Risk aversion and concavity of utility (UC3M) Economics of Uncertainty. April / 16

13 Risk aversion and concavity of utility There is a C CE < C 0 such that 1 2 u(c 1) u(c 2) = u(c CE ) (UC3M) Economics of Uncertainty. April / 16

14 Risk aversion and concavity of utility There is a C CE < C 0 such that 1 2 u(c 1) u(c 2) = u(c CE ) We call C CE the certainty-equivalent of L (UC3M) Economics of Uncertainty. April / 16

15 Risk aversion and concavity of utility There is a C CE < C 0 such that 1 2 u(c 1) u(c 2) = u(c CE ) We call C CE the certainty-equivalent of L Risk Premium= C 0 C CE = the highest insurance premium you are willing to pay to avoid the risk of the lottery (UC3M) Economics of Uncertainty. April / 16

16 Risk aversion and concavity of utility There are three di erent ways of characterizing a concave function: (UC3M) Economics of Uncertainty. April / 16

17 Risk aversion and concavity of utility There are three di erent ways of characterizing a concave function: 1 u 00 (C ) < 0 (UC3M) Economics of Uncertainty. April / 16

18 Risk aversion and concavity of utility There are three di erent ways of characterizing a concave function: 1 u 00 (C ) < 0 2 u 0 (C ) is a decreasing function of C (UC3M) Economics of Uncertainty. April / 16

19 Risk aversion and concavity of utility There are three di erent ways of characterizing a concave function: 1 u 00 (C ) < 0 2 u 0 (C ) is a decreasing function of C 3 For any C 1, C 2 and any p 2 (0, 1) pu(c 1 ) + (1 p)u(c 2 ) < u(pc 1 + (1 p)c 2 ) (UC3M) Economics of Uncertainty. April / 16

20 Jensen s Inequality We can develop more generally the idea that concavity of u implies risk-aversion. (UC3M) Economics of Uncertainty. April / 16

21 Jensen s Inequality We can develop more generally the idea that concavity of u implies risk-aversion. Consider a lottery with outcomes given by the rv c. Then u(c) is another random variable if u is concave, then E [u(c)] < u(e [c]) (UC3M) Economics of Uncertainty. April / 16

22 Jensen s Inequality We can develop more generally the idea that concavity of u implies risk-aversion. Consider a lottery with outcomes given by the rv c. Then u(c) is another random variable if u is concave, then E [u(c)] < u(e [c]) thta is, the expected utility of the lottery is less than the sure utility one would get from having the monetary expected outcome with certainty (UC3M) Economics of Uncertainty. April / 16

23 Quantitaive Measures of Risk Aversion Consider an agent with initial posiiton C 0, presented with a risk whose expected value is zero. (UC3M) Economics of Uncertainty. April / 16

24 Quantitaive Measures of Risk Aversion Consider an agent with initial posiiton C 0, presented with a risk whose expected value is zero. His nal position is a random variable C 0 + X where E [X ] = 0. (UC3M) Economics of Uncertainty. April / 16

25 Quantitaive Measures of Risk Aversion Consider an agent with initial posiiton C 0, presented with a risk whose expected value is zero. His nal position is a random variable C 0 + X where E [X ] = 0. For example, X take on values X i for i = 1, 2,..., n, with probabilities p i, and n p i X i = 0. i=1 (UC3M) Economics of Uncertainty. April / 16

26 Quantitaive Measures of Risk Aversion Consider an agent with initial posiiton C 0, presented with a risk whose expected value is zero. His nal position is a random variable C 0 + X where E [X ] = 0. For example, X take on values X i for i = 1, 2,..., n, with probabilities p i, and n p i X i = 0. i=1 The certainty equivalent is C CE u(c CE ) = n p i u(c 0 + X i ) i=1 (UC3M) Economics of Uncertainty. April / 16

27 Quantitaive Measures of Risk Aversion Consider an agent with initial posiiton C 0, presented with a risk whose expected value is zero. His nal position is a random variable C 0 + X where E [X ] = 0. For example, X take on values X i for i = 1, 2,..., n, with probabilities p i, and n p i X i = 0. i=1 The certainty equivalent is C CE u(c CE ) = n p i u(c 0 + X i ) i=1 The risk premium is Π C 0 C CE (UC3M) Economics of Uncertainty. April / 16

28 Quantitaive Measures of Risk Aversion In may cases we cannot solve that equation explicitly-> rely on numerical solution. (UC3M) Economics of Uncertainty. April / 16

29 Quantitaive Measures of Risk Aversion In may cases we cannot solve that equation explicitly-> rely on numerical solution. Alternative: for small risks we can use a Taylor approximation (UC3M) Economics of Uncertainty. April / 16

30 Quantitaive Measures of Risk Aversion In may cases we cannot solve that equation explicitly-> rely on numerical solution. Alternative: for small risks we can use a Taylor approximation Arrow-Pratt approximation Π = 1 2 Var[X ] u 00 (C 0 ) u 0 (C 0 ) (UC3M) Economics of Uncertainty. April / 16

31 Quantitaive Measures of Risk Aversion In may cases we cannot solve that equation explicitly-> rely on numerical solution. Alternative: for small risks we can use a Taylor approximation Arrow-Pratt approximation Π = 1 2 Var[X ] u 00 (C 0 ) u 0 (C 0 ) The risk premium is proportional to the variance of the random component C, (magnitude of the risk), and also proportional to a measure of the extent of concavity (curvature) of u(). (UC3M) Economics of Uncertainty. April / 16

32 Quantitaive Measures of Risk Aversion In may cases we cannot solve that equation explicitly-> rely on numerical solution. Alternative: for small risks we can use a Taylor approximation Arrow-Pratt approximation Π = 1 2 Var[X ] u 00 (C 0 ) u 0 (C 0 ) The risk premium is proportional to the variance of the random component C, (magnitude of the risk), and also proportional to a measure of the extent of concavity (curvature) of u(). Coe cient of absolute risk aversion A(C 0 ) = u00 (C 0 ) u 0 (C 0 ) (UC3M) Economics of Uncertainty. April / 16

33 Quantitaive Measures of Risk Aversion If the agent faces the risk C 0 (1 + bx ) where bx is a random variable with zero mean (UC3M) Economics of Uncertainty. April / 16

34 Quantitaive Measures of Risk Aversion If the agent faces the risk C 0 (1 + bx ) where bx is a random variable with zero mean Relative risk premium bπ u(c 0 (1 bπ)) = n p i u(c 0 (1 + bx i )) i=1 (UC3M) Economics of Uncertainty. April / 16

35 Quantitaive Measures of Risk Aversion If the agent faces the risk C 0 (1 + bx ) where bx is a random variable with zero mean Relative risk premium bπ u(c 0 (1 bπ)) = n p i u(c 0 (1 + bx i )) i=1 In a similar way to the result above we can obtain the Arrow-Pratt measure of relative risk aversion R(C 0 ) = C 0u 00 (C 0 ) u 0 (C 0 ) (UC3M) Economics of Uncertainty. April / 16

36 Example If u(c ) = 1 exp( ac ) where a is a positive constant, then A(C ) = u00 (C 0 ) u 0 (C 0 ) = a (UC3M) Economics of Uncertainty. April / 16

37 Example If u(c ) = 1 1 r C 1 r for r > 0, r 6= 1 ln(c ) for r = 1 then R(C ) = Cu00 (C 0 ) u 0 (C 0 ) = r (UC3M) Economics of Uncertainty. April / 16

38 Convex and Mixed Utility Functions A person with convex u() likes risk. (UC3M) Economics of Uncertainty. April / 16

39 Convex and Mixed Utility Functions A person with convex u() likes risk. In general u() does not have to be either concave over its whole domain or convex over its whole domain. (UC3M) Economics of Uncertainty. April / 16

40 Convex and Mixed Utility Functions A person with convex u() likes risk. In general u() does not have to be either concave over its whole domain or convex over its whole domain. A person with such a function will then be averse to gambles con ned to the concave part, and will like gambles con ned to the convex part. (UC3M) Economics of Uncertainty. April / 16

41 Convex and Mixed Utility Functions (UC3M) Economics of Uncertainty. April / 16

42 Convex and Mixed Utility Functions Friedmanand Savage argued that there would be risk aversion for very low and very high levels of wealth, and a middle range of risk preference. This is somewhat the opposite of the safety net idea described above. (UC3M) Economics of Uncertainty. April / 16

43 Convex and Mixed Utility Functions Friedmanand Savage argued that there would be risk aversion for very low and very high levels of wealth, and a middle range of risk preference. This is somewhat the opposite of the safety net idea described above. Kahneman and Tversky proposed the prospect theory of choice under risk, in which the status quo plays a special role. The utility-of-consequences function is generally concave for gains and convex for losses, with a discontinuity of slope (kink) at the status-quo point. Such people would be especially averse to small gambles around the status quo, but may like some larger gambles. (UC3M) Economics of Uncertainty. April / 16

44 Convex and Mixed Utility Functions (UC3M) Economics of Uncertainty. April / 16