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M ean - Variance Portfolio Selection for a Non- life insurance Company

M ean - Variance Portfolio Selection for a Non- life insurance Company. Łukasz Delong, Russell Gerrard. Plan. Mathematical concepts Construction of the wealth process Formulation of the problem Solutions of the optimization problem. Stochastic process. Wealth Process X(t).

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M ean - Variance Portfolio Selection for a Non- life insurance Company

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  1. Mean- VariancePortfolioSelection for a Non- life insurance Company Łukasz Delong, Russell Gerrard Agata Kłeczek, Prague 29.03.2012

  2. Plan • Mathematicalconcepts • Construction of thewealthprocess • Formulation of the problem • Solutions of theoptimization problem Agata Kłeczek, Prague 29.03.2012

  3. Stochasticprocess Agata Kłeczek, Prague 29.03.2012

  4. WealthProcess X(t) • Amount of the wealth investedin the risky asset 2. Aggregated claim amount 3. Premium rate Agata Kłeczek, Prague 29.03.2012

  5. Amount of thewealthinvestedintheriskyasset • Amount of moneyinvested in the stockon therisky market = π • We canearnorlosemoneybuyingstocks Agata Kłeczek, Prague 29.03.2012

  6. Aggregated claim amountpaid upto time t • number of claims 1,2,3,…,N(t) • value of i-thclaim • Insurerisobliged to pay until time t Agata Kłeczek, Prague 29.03.2012

  7. Premium rate • How much must we pay for insurance if we buy: motor, property insurance? For example: • 1$ - insurer will go bankrupt • 1000$ - nobodybuys insurance Agata Kłeczek, Prague 29.03.2012

  8. Summarize Wealthprocess (t)= +moneyinvestedinriskyasset + allpremiumrate - Aggregatedclaimamount Agata Kłeczek, Prague 29.03.2012

  9. Formulation of the problem • Expectedvalue • Variance • Problem formulation Agata Kłeczek, Prague 29.03.2012

  10. Expectedvalue • Theweightedaverage of possiblevaluethatthis random variablecantake on • EX=100*0,1+200*0,3 300*0,2+500*0,3 1000*0,1 =380 Agata Kłeczek, Prague 29.03.2012

  11. Variance • Thesimplestriskmeasure • How far do valuesliefromtheexpectedvalue? • Var(X)=E (X-EX)^2=61600 • Squareroot of Var(X)= 248,19 Agata Kłeczek, Prague 29.03.2012

  12. For example Agata Kłeczek, Prague 29.03.2012

  13. Problem formulation • Minimalize variance at terminal time T • Expected value should be equal to the value which we assumed to get at terminal time T where P is a specified target Agata Kłeczek, Prague 29.03.2012

  14. Agata Kłeczek, Prague 29.03.2012

  15. Solution of optimizationproblems • We canfind an optimal strategy • Optimalstrategyexists and itisuniqe • Verificationtheorem Agata Kłeczek, Prague 8.03.2012

  16. Theend Agata Kłeczek, Prague 29.03.2012

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