m ean - variance portfolio selection for a non- life insurance company
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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). - PowerPoint PPT PresentationTRANSCRIPT
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Mean- Variance Portfolio Selection for a Non- life insurance CompanyŁukasz Delong, Russell Gerrard
Agata Kłeczek, Prague 29.03.2012
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Agata Kłeczek, Prague 29.03.2012 2
Plan
1. Mathematical concepts2. Construction of the wealth process3. Formulation of the problem4. Solutions of the optimization problem
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Agata Kłeczek, Prague 29.03.2012 3
Stochastic process
0 1 2 3 4 5 60
2
4
6
8
10
12
123
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Agata Kłeczek, Prague 29.03.2012 4
Wealth Process X(t)
1. Amount of the wealth investedin the risky asset
2. Aggregated claim amount
3. Premium rate
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Agata Kłeczek, Prague 29.03.2012 5
Amount of the wealth investedin the risky asset
• Amount of money invested in the stockon the risky market = π
• We can earn or lose money buying stocks
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Agata Kłeczek, Prague 29.03.2012 6
Aggregated claim amountpaid up to time t• number of claims 1,2,3,…,N(t)• value of i-th claim • Insurer is obliged to pay until time t
iY
)(
1
)(tN
iiYtC
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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$ - nobody buys insurance
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Agata Kłeczek, Prague 29.03.2012 8
Summarize
Wealth process (t)=
+money invested in risky asset
+ all premium rate
- Aggregated claim amount
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Agata Kłeczek, Prague 29.03.2012 9
Formulation of the problem
• Expected value
• Variance
• Problem formulation
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Agata Kłeczek, Prague 29.03.2012 10
Expected value
• The weighted average of possible value that this random variable can take on
• EX=100*0,1+200*0,3300*0,2+500*0,31000*0,1=380
Value Xprobabili
ty
1 100 0,1
2 200 0,3
3 300 0,2
4 500 0,3
5 1000 0,1
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Agata Kłeczek, Prague 29.03.2012 11
Variance
• The simplest risk measure
• How far do values lie from the expected value?
• Var(X)=E (X-EX)^2=61600
• Square root of Var(X)= 248,19
Value Xprobabili
ty
1 100 0,1
2 200 0,3
3 300 0,2
4 500 0,3
5 1000 0,1
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Agata Kłeczek, Prague 29.03.2012 12
For example
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.50
200
400
600
800
1000
1200
dateexpected valuestandard deviation
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Agata Kłeczek, Prague 29.03.2012 13
Problem formulation
1. 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
)]([inf TXVar
PTXE )]([
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0
200
400
600
800
1000
1200
1400
1600
1800
wealthexpected valuevariance
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Agata Kłeczek, Prague 8.03.2012 15
Solution of optimizationproblems
• We can find an optimal strategy
• Optimal strategy exists and it is uniqe
• Verification theorem
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The end