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  1. 2/22-3.........................1/11

  2. MA417Spring2021 Investmentand Insurance 2/22-3.........................1/11

  3. Consider AsetoflotteriesL(X) A setofactionsA AnagentwithBernoullifunctionu. Everyactiona∈Aofthis agentleadstoalotteryLa∈L(X). Theagentwantstochoosetheactiona∗thatwillleadtothe mostpreferredlotteryL∗.

  4. The problem of theagent; Maxa∈AELa(u) Examples:InsuranceandInvestmentinRiskyAssets.

  5. Investmentin Risky Asset AriskaverseagenthasaBernoullifunctionu(x) TheagenthaswealthWdollarsthatcanbeinvestedintwo assets Ariskyasset:returnsR1withprobabilityqandR0with probability1−q Asafeasset:withareturnrateofzero(e.g.keepthemoneyas “cash”) Theagentneedstodecidehowmuchtospendineachasset.

  6. Problem1 Letabethe amountinvestedintheriskyasset. Maxa∈[0,W]qu((1+R1)a+W−a)+(1−q)u((1+R0)a+W−a) To find the optimal amount a∗ that the agent invests in the risky asset, we take the derivative with respect to a, set it to zero and thensolvefora. Thisgiveusa∗afunctionofR1,R0,q,andW.

  7. Problem1 Letabethe amountinvestedintheriskyasset. Maxa∈[0,W]qu((1+R1)a+W−a)+(1−q)u((1+R0)a+W−a) To find the optimal amount a∗ that the agent invests in the risky asset, we take the derivative with respect to a, set it to zero and thensolvefora. Thisgiveusa∗afunctionofR1,R0,q,andW. 2/22-3.........................5/11

  8. Problem1 Letabethe amountinvestedintheriskyasset. Maxa∈[0,W]qu((1+R1)a+W−a)+(1−q)u((1+R0)a+W−a) To find the optimal amount a∗ that the agent invests in the risky asset, we take the derivative with respect to a, set it to zero and thensolvefora. Thisgiveusa∗afunctionofR1,R0,q,andW. 2/22-3.........................5/11

  9. Problem1 Letabethe amountinvestedintheriskyasset. Maxa∈[0,W]qu((1+R1)a+W−a)+(1−q)u((1+R0)a+W−a) To find the optimal amount a∗ that the agent invests in the risky asset, we take the derivative with respect to a, set it to zero and thensolvefora. Thisgiveusa∗afunctionofR1,R0,q,andW.

  10. Problem2 Letαbethe fractionofWthatyouinvestintheriskyasset. Thetochosetheoptimalα,theagentmustsolve MAXα∈[0,1]qu(W+αWR1)+(1−q)u(W+αWR0) Tofindtheoptimalα∗asafunctionofW,q,R1, andR0wehave totakethederivativewithrespecttoαandandsetittozero, andsolveforα. 2/22-3.........................6/11

  11. Problem2 Letαbethe fractionofWthatyouinvestintheriskyasset. Thetochosetheoptimalα,theagentmustsolve MAXα∈[0,1]qu(W+αWR1)+(1−q)u(W+αWR0) Tofindtheoptimalα∗asafunctionofW,q,R1, andR0wehave totakethederivativewithrespecttoαandandsetittozero, andsolveforα. 2/22-3.........................6/11

  12. Problem2 Letαbethe fractionofWthatyouinvestintheriskyasset. Thetochosetheoptimalα,theagentmustsolve MAXα∈[0,1]qu(W+αWR1)+(1−q)u(W+αWR0) Tofindtheoptimalα∗asafunctionofW,q,R1, andR0wehave totakethederivativewithrespecttoαandandsetittozero, andsolveforα. 2/22-3.........................6/11

  13. Theimpactofwealthonα∗anda∗ Wecompute∂α∗ and∂a∗ ∂W ∂W Foru(x)=ln(x),thefirstpartialis0andthesecondis>zero.

  14. InterpretationofAu(x): IfAu(x)isanincreasingfunctioninx,then∂a∗<0 ∂W IfAu(x)isandecreasingfunctioninx,then∂a∗>0. ∂W 2/22-3.........................8/11

  15. InterpretationofAu(x): IfAu(x)isanincreasingfunctioninx,then∂a∗<0 ∂W IfAu(x)isandecreasingfunctioninx,then∂a∗>0. ∂W 2/22-3.........................8/11

  16. InterpretationofRu(x) IfRu(x)isanincreasingfunctioninx,then∂α∗<0. ∂W IfRu(x)isandecreasingfunctioninx,then∂α∗>0 ∂W 2/22-3.........................9/11

  17. InterpretationofRu(x) IfRu(x)isanincreasingfunctioninx,then∂α∗<0. ∂W IfRu(x)isandecreasingfunctioninx,then∂α∗>0 ∂W 2/22-3.........................9/11

  18. RevisitingInsurance AriskaverseagentwithwealthlevelWandBernoullifunction u. Listheamountofpotentiallosswithprobabilityp Thecostofinsuranceiscper1perdollarinsured( Forexample,fullinsurancewillcostatotalofcL 0≤z≤Listheamountinsured.Theagentchoosesthevalue ofz. Fullinsurancemeansz=L 2/22-3.........................10/11

  19. EverychoicezleadstoalotteryLzthatgivestheagentwealth W− L + z − cz with probability p and W− cz with probability 1−p Tofindtheoptimalz,theagentsolves; Maxzpu(W−L+z−cz)+(1−p)u(W−cz) 2/22-3.........................11/11

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