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Mean-Variance Portfolio Selection for Non-life Insurance: Mathematical Concepts and Optimization Solutions

This study delves into the mean-variance portfolio selection for a non-life insurance company, focusing on mathematical concepts, construction of the wealth process, and formulating and solving optimization problems. It covers stochastic processes, wealth accumulation in risky assets, aggregated claim amounts, and premium rates for insurance. The text also discusses expected value, variance as a risk measure, and formulating the problem to optimize variance at a specified target. Solutions found through optimal strategies are highlighted, along with a verification theorem.

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Mean-Variance Portfolio Selection for Non-life Insurance: Mathematical Concepts and Optimization Solutions

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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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