Finance: Risk Management Module II: Optimal Risk Sharing and Arrow-Lind Theorem

Transcription

1 Institte for Risk Management and Insrance Winter 00/0 Modle II: Optimal Risk Sharing and Arrow-Lind Theorem Part I:

2 Efficient risk-sharing between risk-averse individals Consider two risk-averse parties A and B. Institte for Risk Management and Insrance Assme that the risk of a random expense (e.g. a property loss) x ~ is present. How shold this risk be allocated between the two parties? Withot loss of generality assme that Q(x~ ) is the part of the risk borne by B, sch that x~ Q( x ~ ) is the portion borne by A 33

3 Institte for Risk Management and Insrance Efficient risk-sharing rles The model: The model: Risk averse individal A A with with (Bernolli) tility fnction Risk averse individal B B with (Bernolli) tility fnction No No transaction costs (Random) final wealth of of B: A: (Random) final wealth of of A: B: Notation: - W A, W B : A s and B s initial wealth - : random loss loss x ~ ~ ), W B Q(x x ~ Q( x~ ). Q( ) risk-sharing arrangement: For a given x, Q(x) represents the portion of - Q( ) : risk-sharing arrangement: For a given x, Q(x) represents the portion of the loss that is borne by B. x Q(x) wold ths be A s share. the loss that is borne by B. x Q(x) wold ths be A s share. In this framework, we will be looking at characteristics of In pareto-optimal this framework, risk-sharing we will be rles looking at characteristics of pareto-optimal risk-sharing rles W A 34

4 Institte for Risk Management and Insrance Efficient risk-sharing rles We are looking at Pareto-optima via max Q( ) L : E[ (W Q(x ~ Q(x ~ A ))] E[ (WB ))],, x ~ 0 (A precise jstification for the se of this approach for characterizing Pareto-optimal soltions can be fond, e.g. in Kreps (990), A Corse in Microeconomic Theory, pp. 60) The following slides will provide a motivation, which is sfficient for or prposes. 35

5 Soltion space and Pareto-efficient soltions Let U i denote expected tility, i.e. U =E[ ], U =E[ ] U Institte for Risk Management and Insrance Consider the soltion space for possible risk sharing fnctions Q It can be shown that this soltion space has a concave bondary soltion space Pareto-efficient soltions The bondary of the soltion space represents Pareto-optimal soltions U 36

6 Soltion space and Pareto-efficient soltions Now again look at Institte for Risk Management and Insrance max L : E[ ( ~ ( ~ WA x Q x))] E[ ( W Q( ) B Q( x ~ ))],, 0 Motivation of the approach Maximising L leads to a fnction * * * Q ( ) and a respective combination ( U, U) in the soltion space which is on the straight line U * L( Q ) U / This is exactly the straight line with the slope and the highest U axis intercept, sbject to the condition that there is a joint point with the soltion space (niqe becase of concavity) 37

7 Institte for Risk Management and Insrance Soltion space and Pareto-efficient soltions * L( Q ) U const. soltion space U 38

8 Soltion space and Pareto-efficient soltions Now again look at Institte for Risk Management and Insrance max L : E[ ( ~ ( ~ WA x Q x))] E[ ( W Q( ) B Q( x ~ ))],, 0 Motivation of the approach By determining the relation any possible Pareto-optimm can be attained. In the following, the analysis can be limited to the condition, 0. Setting one of these parameters eqal to zero implies trivial soltions. 39

9 Efficient risk-sharing rles Institte for Risk Management and Insrance For every x 0 ( point-wise ) we are looking for the vale of Q(x) that maximizes ( Y( x)) ( Z( x)) with Y( x) : WA x Q( x), Z( x) : W Q( x). B The so-derived indemnity fnction Q(x) also maximizes the objective fnction. Becase of dy/dq = and dz/dq = the vale can be obtained by solving (.) or (.) ( Y( x)) ( Z( x)) 0 ( Z( x)) ( Y( x)) 40

10 Institte for Risk Management and Insrance Efficient risk-sharing rles Consider dy(x)/dx= +dq(x)/dx and dz(x)/dx= dq(x)/dx and differentiate (.) with respect to x. We obtain dq( x) dq( x) ( Y( x)) [ ] ( Z( x)) [ ] dx dx 0 4

11 Institte for Risk Management and Insrance 4 can be written as (sing (.) ) dx dq, )) ( ( )) ( ( )) ( ( x Z x Y x Y dx dq (.3) Efficient risk-sharing rles (.3)*

12 Institte for Risk Management and Insrance Efficient risk-sharing rles What is the slope of a Pareto-efficient risk-sharing fnction, when both parties tility fnctions are of the type ( z) e az ( z) e bz dq dx What happens if a risk-netral party is involved? 43

13 The Arrow-Lind Theorem Preliminary remarks Institte for Risk Management and Insrance We have characterized optimal risk-sharing soltions between two parties, e.g. two individals. In the following, we will analyze the case of a large nmber (n) of risk-averse individals sharing a risk; in particlar we ask what happens for n? For this prpose we assme that these n individals form a syndicate. We are interested in the decisions made by the syndicate. More specifically we are looking at their willingness to pay for a risky project. As a starting point, please recall what yo know abot the willingness to pay of a risk averse individal for a project with risky otcomes. 44

14 Institte for Risk Management and Insrance Arrow-Lind Theorem Assmptions: Consider the case of the above-mentioned syndicate investing into the risky project with random otcome ~z. For the sake of simplicity, assme that each member of the syndicate is characterized by the same preferences, probability beliefs and incomes. An individal s (strictly concave) tility fnction is ( ). Her income from other sorces is ~y. Assme that Cov[ z~, y~ ] 0. (Note that this is certainly tre if y is deterministic) Let k denote the individal s share in the otcome (with k = /n). P denotes the maximm amont that the syndicate wold be willing to pay for the risky project (each individal wold therefore pay p := P/n). Under these assmptions, the individal s random (total) income is No transaction costs. y ~ kz~ p. 45

15 Institte for Risk Management and Insrance Arrow-Lind Theorem The willingness to pay for the risky project depends on the size of the syndicate. Therefore, denote P = P(k) and p=p(k) We know that p( k) E[ kz~ ] as individals are risk-averse. This implies P( k) n p( k) E[ z ~ ]. However, the Arrow-Lind Theorem tells s that nder the assmptions made [ ~ n ( E z] np( k)) 0. i.e., for large n : ~ ] P E[z In other words: For very large n, the syndicate acts as if it is risk-netral. 46

16 Institte for Risk Management and Insrance Arrow-Lind Theorem Proof (I) Obviosly: lim ( E[ z ~ ] np( k)) n E[ z ~ ] lim ( np( k)) n Ths, we only need to consider lim n np( k) lim np( k) lim k0 k0 p( k) k Applying the de l Hôpital Rle, we get lim k0 p( k) k dp( k)/ dk lim k0 dk / dk dp( k) lim k0 dk We know that p(k) is defined implicitly by ( y~ ) E ( y ~ kz~ p( k)) 0 E Applying the Implicit Fnction Theorem gives dp( k) E ( y ~ kz~ p( k)) z~ dk E ( y ~ kz~ p( k)) 47

17 Institte for Risk Management and Insrance Arrow-Lind Theorem Proof (II) This proves dp( k) lim k 0 dk E ( y ~ kz~ p( k)) z~ lim ( ~ k 0 E y kz~ p( k )) ( y ~ kz~ p) Ez~ cov ( y ~ kz~ p), z ~ E ( y ~ kz~ p E lim k 0 ) ~ E z cov ( y ~ ), z ~ E ( y ~ ) E z ~ 48

18 Arrow-Lind Theorem - Applications / implications (I) Institte for Risk Management and Insrance Consider a large and widely held pblic company and assme that its managers wish to act on behalf of the shareholders. According to the Arrow-Lind Theorem, this company wold act as if it was risk-netral (accepting projects with a non-negative expected vale of retrn less invested amont). Rationale for modelling the behavior of a firm as if it were owned by a single risk-netral owner. If, in contrast to or assmptions the individal shareholders other income is correlated with the project, what wold that mean for the decision? 49

19 Arrow-Lind Theorem - Applications / implications (II) Institte for Risk Management and Insrance Pblic sector projects: Under the assmptions of the Arrow-Lind Theorem the government shold be risk-netral in evalating pblic sector projects even if individals are risk-averse. Mtal insrance: If a large grop/syndicate of insred decides to share everybody s risk via splitting losses among all the members of the grop, and if these individal risks are stochastically independent and also independent from the insred s other income, premims shold eqal expected losses (assming zero transaction costs). 50