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

Agata Kłeczek, Prague 29.03.2012

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Plan

1. Mathematical concepts2. Construction of the wealth process3. Formulation of the problem4. Solutions of the optimization problem

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Stochastic process

0 1 2 3 4 5 60

2

4

6

8

10

12

123

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Wealth Process X(t)

1. Amount of the wealth investedin the risky asset

2. Aggregated claim amount

3. Premium rate

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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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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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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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Summarize

Wealth process (t)=

+money invested in risky asset

+ all premium rate

- Aggregated claim amount

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Formulation of the problem

• Expected value

• Variance

• Problem formulation

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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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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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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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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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Solution of optimizationproblems

• We can find an optimal strategy

• Optimal strategy exists and it is uniqe

• Verification theorem

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The end