Mean- Variance Portfolio Selection for a Non- life insurance CompanyŁukasz Delong, Russell Gerrard

Agata Kłeczek, Prague 8.03.20121

The insurance risk process• collective insurance risk model• C(t) denote aggregate claim amount paid up to time t• the process is a compound Cox process

Agata Kłeczek, Prague 8.03.2012 2

)0),((: TttCC

)(

1

)(tN

iiYtC

amounts of successive claims counts the number of claims

Agata Kłeczek, Prague 8.03.2012 3

iY

iY

)(tN

)(

1

)(tN

iiYtC

The Financial Market• Levy diffusion version of a Black- Scholes financial market• The price of a risk- free asset is described• A risky stock and the dynamics of its price is given by

1)0(,)(

)(0

0

0 SrdttS

tdS

)0),((: 00 TttSS

1)0(),()(

)(

StdLdt

tS

tdS

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)0),((: TttSS

THE MODEL•Financial market•Claim process•Claim intensity process

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Problem formulation• Portfolio selection for a general insurance company• Wealth process of the insurer • it’s dynamics are given by the stochastic differential equation

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)0),((: TttXX

Two optimization problems1) Classical mean-variance portfolio selection. Investment strategy should be chosen in the following way

where P is a specified target. Agata Kłeczek, Prague 8.03.2012 7

PTXE

TXVar

)]([

)]([inf

2) includes also a running cost penalizing deviations of the insurer’s wealth froma specified profit-solvency target which isa random processAgata Kłeczek, Prague 8.03.2012 8

0)]()([

)]()([]))()(([inf0

2

TRTXE

TRTXVardttRtXET

Solution of optimizationproblems

• Stochastic theory

• Verification theorem

• Levy diffusion financial market

Agata Kłeczek, Prague 8.03.2012 9