Winter 2015/16. Insurance Economics. Prof. Dr. Jörg Schiller.

Transcription

1 Winter 15/16 Insrance Economics Prof. Dr. Jörg Schiller Yo ill find frther information on or ebpage: and on

2 Agenda I. Risk aversion and insrance demand nder symmetric information II. III. IV. Insrance demand: limited liability Insrance demand: adverse selection Insrance demand: moral hazard V. Empirical evidence for asymmetric information VI. Insrance Frad Winter 15/16 Insrance Economics

3 I. Risk Aversion and Insrance Demand nder Symmetric Information Choice nder ncertainty Risk Aversion Certainty eqivalent and risk premim Decreasing Absolte Risk Aversion and Prdence Empirical Evidence for Risk Aversion Measres of Risk Aversion to donside risk Insrance demand ith risk netral insrer Optimal Coinsrance Optimality of Dedctible Contracts Optimal risk sharing beteen to risk-averse parties Winter 15/16 Insrance Economics 3

4 The theory of expected tility The Bernolli principle: For every individal, there exists a tility fnction ( ) hich is defined on the otcome space X, invariant to positive linear transformations and hich permits to evalate different risky alternatives (lotteries tility. z~i ) according to their expected The lottery ith the highest expected tility shold be chosen. a) discrete case: E (z ~ n ) (x ) p max i j1 ij j b) continos case: E (z ~ ) (x) f (x)dx max i i Winter 15/16 Insrance Economics 4

5 Lotteries Basic lottery z~ x1,p, x p x 1 1 p x Simple lottery z~ x 1,...,x j,...,xn,p 1,...,p j,..., pn p 1 p p j x 1 x x j p n x n (Redction of) Compond lotteries p 1 x 1 z~ 1,, ~ z 1 1p 1 p x x 3 1 p, 1 p, 1 1 z~ x 1,x,x3,x4, p 1, 1 p 1 p x 4 Winter 15/16 Insrance Economics 5

6 Risk aversion Definition 1: An individal is risk-averse if, at any ealth level, she dislikes every lottery ith an expected payoff of zero: z~, z~ ith E ~ z, E Obviosly, e can generalize the definition to lotteries ith non-zero-means: z~ E ~ z E This definition implies that an risk-averse individal alays prefers the expected vale of a lottery to the lottery itself. With respect to insrance decisions, risk-aversion implies that individals alays prchase fll coverage hen insrance contracts are actarially fair priced (insrance premim = expected vale of indemnities). Winter 15/16 Insrance Economics 6

7 Risk aversion 4,; z~ ;8,;.5;.5 () (1,) E ~ z E ~ z C F E D () (4,) A 4, E ~ z 1, Winter 15/16 Insrance Economics 7

8 Risk aversion Proposition 1: An individal ith a tility fnction is risk-averse, if and only if is concave. This Proposition is in fact nothing more than reriting Jensen s ineqality. Jensen s ineqality: If and only if is a strictly concave fnction, E y~ E ~ y holds for any (non-degenerated) random y~ variable. There are to alternative definitions of concavity: A negative second-order derivative: (diminishing marginal tility) A convex-combination of to points on the crve mst lie belo this crve. Definition: A fnction f( ) is strictly concave on an interval H if and only if: f 1 ta tb 1 tf a tf b a bht,1 Winter 15/16 Insrance Economics 8

9 Risk premim Risk-averse individals dislike zero-mean risks. Bt, risk-averse individals may like risky lotteries if the expected payoff is large enogh. There is a trade-off beteen the expected payoff and risk aversion. No, e ant to qantify the degree of risk aversion. Ho mch are yo ready to pay to get rid of a zero-mean risk? Here, e determine the individal s risk premim for a zero-mean risk. E z~ De to Proposition 1, the risk premim is nonnegative if is strictly concave! Definition : The risk premim for any lottery ith a mean E ~ z is: z~ E ~ z E Winter 15/16 Insrance Economics 9

10 Certainty eqivalent When the risk z~ has an expectation that differs from zero, it is straight forard to se the concept of the certainty eqivalent. Definition 3: The certainty eqivalent e of risk z~ is the sre increase in ealth that has the same effect on elfare as having to bear the risk z~. z~ e E The general connection beteen certainty eqivalent and risk premim is: z~ e,,z E Therefore, if E z~ : e,,z ~ e,, ~ z Winter 15/16 Insrance Economics 1

11 Certainty eqivalent and risk premim 8,; z~ 4,;.5;4, () E (1,) z~ E z ~ () (4,) A z~ e,,z ~ E 4, 8, 1, 8, - Π Winter 15/16 Insrance Economics 11

12 Risk premim and risk aversion What is the amont that an individal is ready to pay for the elimination of a zero-mean risk hen considering small risks? z~ E Here, e can se the so-called Taylor approximation [see, e.g. Simon and Blme, Section 3.] The Taylor series is a representation of a fnction f( ) as an (infinite) sm of terms calclated from the vales of its derivatives at a single point a. It is common practice to se a finite nmber of terms of the series to approximate a fnction. The second-order Taylor approximation of f () at point a is: f 1 a h fa fah f ah Winter 15/16 Insrance Economics 1

13 Risk premim and risk aversion What is the amont that an individal is ready to pay for the elimination of a zero-mean risk hen considering small risks? First-order Taylor approximation of the RHS: z~ E Second-order Taylor approximation of the LHS: z~ E z~ z~ E 1 E ~ z E ~ z σ 1 1 Note: E ; E ~ z~ z Winter 15/16 Insrance Economics 13

14 Risk premim and risk aversion Using the Taylor approximations for the LHS and RHS leads to: z~ E 1 1 Conseqently, the risk premim for small risks is approximately: For small risks (ith σ ): 1 A ith A 1 lim A Winter 15/16 Insrance Economics 14

15 Risk premim and risk aversion When analyzing small risk A(), the Arro-Pratt measre of absolte local risk-aversion, measres the risk aversion of an individal at a specific ealth level. An individal ith a higher A() is more relctant to accept small risk. Proposition : The folloing conditions are eqivalent Individal v is more risk-averse than individal, i.e. the risk premim of any risk is larger for individal v than for individal. For all, A v () A (). For all z~, e,z ~ e,z ~. v Fnction v is a concave transformation of fnction : There exists a concave fnction sch that v. For large risks, e need to kno the degree of concavity of at all ealth levels in order to decide hether or not one individal is more risk averse than another. Winter 15/16 Insrance Economics 15

16 Risk premim and risk aversion 1 Example: Let s consider an individal ith =4. and z~ ;8; 1 ;. Let s compare the to tility fnctions 1 and ln. Comparison of certainty eqivalents: Comparison of absolte Arro-Pratt measres: Comparison of risk premims Winter 15/16 Insrance Economics 16

17 Decreasing Absolte Risk Aversion and Prdence It seems to be plasible that the risk premim and conseqently the risk aversion of an individal decreases in ealth. Can e characterize tility fnctions ith these properties? The risk premim can be defined as a fnction of : E ~ z, Flly differentiating the latter eqality ith respect to leads to E z~, 1,, E z~, The risk premim is (eakly) decreasing in ealth if and conseqently: E v z~ v, ith v Winter 15/16 Insrance Economics 17

18 Decreasing Absolte Risk Aversion and Prdence Becase v is increasing, e can interpret it as another tility fnction. Since E v z~ v,v, e get: In fact, e are comparing risk premims of to different individals. De to Proposition, the risk premim of the individal ith v is greater than that of the individal ith if the first individal is more risk-averse. Hence: Let s define: v,v v,,v, A v A The risk premim is only decreasing in ealth if and only if P P A A Winter 15/16 Insrance Economics 18

19 Decreasing Absolte Risk Aversion and Prdence Prdence can be interpreted as aversion against negative skeness. f() A B p() A B A prdent individal ill prefer A over B even thogh A and B have the same expectation and variance. Winter 15/16 Insrance Economics 19

20 Decreasing Absolte Risk Aversion and Prdence The condition P A is necessary and sfficient to garantee that an increase in ealth redces the risk premim. Alternatively, e can directly dedce the latter reslt from the Arro-Pratt coefficient. The risk premim is only decreasing in ealth if: Proposition 3: The risk premim associated to any risk z~ is decreasing in ealth, if and only if absolte risk aversion is decreasing; or eqivalently if and only if P () is for all eakly larger than A (). Decreasing Absolte Risk Aversion (DARA) reqires. Exercise: Sho that both conditions in Proposition 3 are eqivalent. A Winter 15/16 Insrance Economics

21 Relative Risk Aversion Sometimes one ants to measre ho risk aversion changes, hen both the risk and the ealth are increased by the same factor (e.g. dobled). Here, e can se the concept of relative risk aversion A rel. d () () d d () () d () () A What are the risk aversion measres for () = (1-r) /(1-r) ith r >? Winter 15/16 Insrance Economics 1

22 Empirical Evidence for Risk Aversion Holt and Lary () measre individals risk aversion by lottery choices: Lottery design Winter 15/16 Insrance Economics

23 Empirical Evidence for Risk Aversion Assming the tility fnction () = (1-r) /(1-r) one can infer r given the sitching from Option A to Option B. Winter 15/16 Insrance Economics 3

24 Measres of Risk So far, e have considered tility fnctions of individals and derived varios conclsions for a given lottery. No, e ant to compare different lotteries ithot considering a specific tility fnction. No, e ant to apply concepts of stochastic dominance! Winter 15/16 Insrance Economics 4

25 First-order stochastic dominance Definition 3: ~ is dominated by in the sense of the first-degree stochastic dominance (FSD) ~ 1 order if F for all. F 1 F() F () p() F 1 () f () f 1 () Proposition 4: The folloing conditions are eqivalent. All individals ith a nondecreasing tility fnction prefer ~ 1 to ~ : E ~ E ~ 1 for all nondecreasing fnctions. ~ is dominated by in the sense of FSD: is obtained from by a transfer of ~ ~ 1 ~ 1 probability mass from high ealth states to lo ealth states, or F. F 1 Winter 15/16 Insrance Economics 5

26 Adding noise Consider the basic lotteries: ~ 1 ~ 4,;.5;1, 4,;.5;1, ith 4,;.5;4, Becase E, in the compond lottery ~ there is an additional zero-mean noise. p 1/ ~ p 1 ~ 1/ 1/4 4, 1, 4, 8, 16, Winter 15/16 Insrance Economics 6

27 Adding noise Intition sggest that an risk-averse individal shold dislike the additional ncertainty of ~. Example : ~.5 4,.5 1, ~.5 4,.5.5 8,.5 E 1 16, E The reslt holds for any risk-averse individal. All risk-averse individals dislike adding zero-mean noises to the possible otcomes of their ealth! Winter 15/16 Insrance Economics 7

28 Mean-preserving spread Definition 4: ~ is a mean-preserving spread (MPS) of ~ 1 if: 1. E ~ E ~ 1, and. There exists an Interval M sch that f for all in M, and f1 f f1 for all otside M. Adding noise or constrcting a seqence of MPS s are obviosly to eqivalent ays to increase risk. f() F() f 1 () f () F 1 () F () M M Winter 15/16 Insrance Economics 8

29 Mean-preserving spread It is often sefl to translate the definition of a mean-preserving spread into a condition on the cmlative distribtion fnction of ~ and ~. 1 If ~ is an MPS of ~ 1 the folloing ineqality mst hold F s F s ds S a 1 F() S() F 1 () F () S() a b a b M Winter 15/16 Insrance Economics 9

30 Mean-preserving spread Adding noise or constrcting a seqence of MPS s are obviosly to eqivalent ays to increase risk. In some circmstances, it is easier to se one representation than another. ~ 1 Example: Compare 4,;.5;1, and ~,;.5;14,. 1. Adding some noise: 1/ 4, 1 5/6 1/6 -, + 1, E 1 1/ 1, 5/6 +, E 1/6-1, Winter 15/16 Insrance Economics 3

31 Mean-preserving spread Adding noise or constrcting a seqence of MPS s are obviosly to eqivalent ays to increase risk. In some circmstances, it is easier to se one representation than another. ~ 1 Example: Compare 4,;.5;1, and ~,;.5;14,.. MPS - Transfer of probability mass: p 1/, 4, 1, 14, Winter 15/16 Insrance Economics 31

32 Mean-preserving spread Adding noise or constrcting a seqence of MPS s are obviosly to eqivalent ays to increase risk. In some circmstances, it is easier to se one representation than another. ~ 1 Example: Compare 4,;.5;1, and ~,;.5;14,. 3. MPS Interval approach: F() F () F 1 () M 3,;13,, 4, 14, 1, M Winter 15/16 Insrance Economics 3

33 Mean-preserving spread One can sho that changes in risk redce the expected tility of risk-averse agents. See e.g. Eeckhodt et al. (5) for a proof. Proposition 5: Consider to random variables ~ 1and ~ ith the same mean. The folloing conditions are eqivalent. ~ All risk-averse agents prefer to ~ : for all concave fnctions. 1 E ~ E ~ 1 ~ is obtained from by adding zero-mean noise terms to the possible otcomes of. ~ ~ 1 1 ~ is obtained from by a seqence of mean-preserving spreads. ~ 1 S F s F1 s ds. a Winter 15/16 Insrance Economics 33