Optimal Investment Strategy Insurance Company

Transcription

1 Opimal Invesmen Sraegy for a Non-Life Insurance Company Łukasz Delong Warsaw School of Economics Insiue of Economerics Division of Probabilisic Mehods

2 Probabiliy space Ω I P F I I I he filraion saisfies he usual hypoheses of compleeness righ coninuiy

3 Insurance risk process Collecive risk model J N i Y i N Poisson process wih inensiy λ Y i i N sequence of posiive iid random variables { } y 4 p dy < EY i µ he process J is F -adaped RCLL

4 Insurance risk process Poisson sochasic inegral M J A # ym ds dy { s J A } J J J M A random variable Poisson disribued wih λp A ~ he proces M A M A λp A is a maringale

5 Risk-free asse Financial marke db B rd B Risky asses i..n ds S i i n a i d σ ij dw j S i s i j > W... W W n sandard Brownian moion -adaped F

6 Insurer s wealh process i fracion of available wealh invesed in he risky asse fracion of available wealh invesed in he risk-free asse i d n ds db n i i i i i S i B dj d d Σ d W π rd dj

7 Admissible sraegies {... n < } respec o filraion F predicable process wih P i d < i... n P d < { } he process is an semimaringale RCLL F -adaped

8 Insurer Insurer s wealh wealh process process Levy Levy-ype ype sochasic sochasic inegral inegral < < < < Σ ~ y y y dy d ym dy d ym d d dy p y r d W λ π

9 Reserve Reserve Insurer Insurer s risk risk profile profile r e R Y J dj e R N i i s < I δ λ λ µ µ δ λ µ δ δ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ E

10 Wealh pah dependen disuiliy opimizaion Find an invesmen sraegy which minimizes he quadraic loss funcion E [ { } R s s α R s s ds β α ]

11 Dynamic Dynamic Programming Programming Principle Principle { } [ ]} { inf ] [ inf V d R d R V ds s s R s s R V n n < E E R R α α β α

12 Io Io s formula formula { } { } Σ ΣΣ dy d M V y V d V d V d r V d V dv W π

13 If here eiss a funcion V C Bellman equaion wih he boundary condiion and [ ]R is he opimal conrol for he problem. saisfying sfying Hamilon-Jacobi Jacobi- R α R V V r { V y V } inf V n R π V α V β ΣΣ { V s s y V s s } such ha he processes ~ M ds dy are maringales and here eiss an admissible conrol infimum is reached hen V V s s s i s dwj s i j... n λp dy for which he { R s s α R s s } ds β α inf E n R

14 Opimal Opimal invesmen invesmen sraegy sraegy π π ϕ π π φ αβ ϕ µλ α β φ π ' ' ΣΣ ΣΣ ΣΣ r r b b b a R a a a a b g g

15 Insurer Insurer s wealh wealh under under he he sraegy sraegy dy ds ym s Z ds r s g s g s Z g Z g } ' { Σ Σ ΣΣ ep r Z W π π π π ' m r m m g m ΣΣ λµ π π

16 Opimal invesmen sraegy Shor-selling he asse < < > g Borrownig from a bank accoun > < < g σ a r

17 Opimal invesmen sraegy he higher he reserve he higher he fracion of he wealh invesed in he risky asse given he same posiive level of available wealh and he higher he epeced value of he insurer's wealh he higher he value of alpha he higher he fracion of he wealh invesed in he risky asse given he same posiive level of available wealh and he higher he epeced value of he insurer's wealh

18 Simulaion sudy One year policy discreizaion one week Insurance risk process: : λ Gamma disribuion epeced value and variance 5 Financial marke r 4 % a % σ % ˆ δ 35% simulaions

19 Simulaion sudy he higher he risk allowance in he reserve he lower he fracion of he wealh invesed in he risky asse he higher he epeced erminal wealh and he lower he ruin probabiliy he higher he value of alpha he higher he fracion of he wealh invesed in he risky asse he higher he epeced erminal wealh and he higher he ruin probabiliy he value of bea has only marginal effec on he resuls

20 Simulaion sudy he ruin probabiliy is lower and he epeced erminal wealh is higher compared wih he siuaion when he insurer has only a bank accoun a is disposal he disribuion of he erminal wealh under he opimal sraegy has ligher lef ail 5% percenile and heavier righ ail 95% percenile compared wih he disribuion of he erminal wealh in he case of he risk free invesmen sraegy

21 Reserve5% alfa6 bea perceniles 5h5h85h

22 Reserve5% alfa6 bea Perceniles Risk-freerisky invesmen Risk-free invesmen 5% % % % % % % Mean Deviaion Ruin probabiliy 96 6

23 Reserve5% alfa6 bea perceniles 5h5h85h

24 Reserve5% alfa6 bea Perceniles Risk-freerisky invesmen Risk-free invesmen 5% % % % % % % Mean Deviaion Ruin probabiliy 4 6

25 Reserve5% alfa8 bea perceniles 5h5h85h

26 Perceniles Reserve5% alfa8 bea Risk-freerisky invesmen Risk-free invesmen 5% % % % % % % Mean Deviaion Ruin probabiliy 4 6

27 hank you for your aenion Łukasz Delong Warsaw School of Economics