OptimalOptimal InvestmentInvestment StrategyStrategyfor a for a NonNon--LifeLife InsuranceInsurance

CompanyCompany

ŁŁukasz Delongukasz DelongWarsawWarsaw SchoolSchool ofof EconomicsEconomics

InstituteInstitute ofof EconometricsEconometricsDivisionDivision ofof ProbabilisticProbabilistic MethodsMethods

ProbabilityProbability spacespace

( ) ( ) ℑ=ℑℑ=ℑΩ ≤≤ TTtt 0,, FP

TheThe filtrationfiltration satisfiessatisfies thethe usualusual hypotheseshypotheses ofofcompletenesscompletenessrightright continuitycontinuity

InsuranceInsurance riskrisk processprocess

CollectiveCollective riskrisk modelmodel

PoissonPoisson processprocess withwith intensityintensitysequencesequence ofof positivepositive iidiid randomrandom variablesvariables

TheThe processprocess isis --adaptedadapted, RCLL, RCLL

∑ ==

)(

1)( tN

i iYtJ

)( tN

N∈iYi ,

F)(tJ

λ

µ=∞<∫∞

iYdypy0

4 )( E

InsuranceInsurance riskrisk processprocess

PoissonPoisson stochasticstochastic integralintegral

randomrandom variablevariable, , PoissonPoisson distributeddistributed withwithTheThe proces proces isis a a martingalemartingale

)()()(

)(,0#),(

),()(0 0

−−=∆∈∆≤≤=

= ∫ ∫∞

tJtJtJAtJtsAtM

dydsyMtJt

),( AtM (Atp )λ)(),(),(~ AtpAtMAtM λ−=

FinancialFinancial marketmarketRiskRisk--freefree assetasset

RiskyRisky assetsassets

standard standard BrownianBrownian motionmotion,,--adaptedadapted

1)0(,)()(

== BrdttBtdB

0)0(,)()()(

1>=+= ∑ = ii

n

j jijii

i sStdWdtatStdS

σ

ni ,..,2,1=

( )T1 )(...,),()( tWtWt n=W

F

InsurerInsurer’’ss wealthwealth processprocess

fractionfraction ofof availableavailable wealthwealth investedinvested inin thethe riskyrisky assetassetfractionfraction ofof availableavailable wealthwealth investedinvested inin thethe riskrisk--freefree assetasset

iθ

( )∑ ∑= =−−−+−=

n

i

n

i ii

ii dJ

tBtdBtXt

tStdS

tXtt1 1 )(

)()()(1)()(

)()()( θθ

i

0

TT

)0(),()()()()()()()(

xXtdJrdttXtdttXdtttXt

=−−++Σ−+−= Wθπθ

0θ

tdX )(

dX

AdmissibleAdmissible strategiesstrategies

predictablepredictable processprocess withwithrespectrespect to to filtrationfiltration

TheThe processprocess isis anan --adaptedadaptedsemimartingalesemimartingale, RCLL, RCLL

Tttt n ≤<0),(...,),(1 θθF

1))()(( 0 0 =∞<−∫T dttXtθP

TttX ≤≤0),( F

nidttXtTi ...,,1,1))()(( 0

22 ==∞<−∫ θP

InsurerInsurer’’ss wealthwealth processprocess

LevyLevy--typetype stochasticstochastic integralintegral

∫∫

∫

<<

≥

<<

=−

+−Σ−+

+⎟⎠⎞⎜

⎝⎛ −−+−=

10 0

1

T

10

T

)0(),,(~),()()()(

)()()()()(

y

y

y

xXdydtMy

dydtyMtdttX

dtdypyrtXttXtdX

Wθ

λπθ

ReserveReserve

InsurerInsurer’’ss riskrisk profileprofile

( ) retR

YtJ

TttJdetR

tT

tN

i i

T

t tts

≤≥≥−=

=

<≤⎥⎦⎤

⎢⎣⎡ ℑ=

−−

=

−−

∑

∫

δλλµµδλµ δ

δ

ˆ,ˆ,ˆ,1ˆˆˆ

)(

ˆ)(ˆ

0,)(ˆ)(

)(ˆ

)(ˆ

1

)(ˆE

WealthWealth pathpath dependent dependent disutilitydisutilityoptimizationoptimization

FindFind anan investmentinvestment strategystrategy whichwhich minimizesminimizes thethequadraticquadratic lossloss functionfunction

( ) ( ) ( )])()(

)()()()([2

0

2

TXTX

dssXsRsXsRT

αβ

α

−+

+−+−∫E

DynamicDynamic ProgrammingProgramming PrinciplePrinciple

( ) ( )

( )

( ) ( )

[ ])())(,(

)()(inf),(

0],)()()(

)()()()([inf),(

2

)(

2

2

)(

xtXtXtV

dtxtRdtxtRxtV

TtxtXTXTX

dssXsRsXsRxtV

n

n

t

T

t

=−+

+−−+−−=−

<≤=−+

+−+−=

∈

∈⋅ ∫

E

E

R

R

α

αβ

α

θ

θ

ItoIto’’ss formulaformula

∫∞

−−−−+

+Σ−−∂∂

+ΣΣ−−∂∂

+

+−+−−∂∂

+−∂∂

=

0

TTT22

2

T

),())(,())(,(

)()()())(,()()()())(,(21

)()()())(,())(,())(,(

dydtMtXtVytXtV

tdttXtXtxVdttttXtXt

xV

dtrtXttXtXtxVdttXt

tVtXtdV

Wθθθ

πθ

IfIf therethere existsexists a a functionfunction satsatiisfyingsfying HamiltonHamilton--JacobiJacobi--BellmanBellman eequationquation

withwith thethe boundaryboundary conditioncondition , , suchsuch thatthat thethe processesprocesses

areare martingalesmartingales, , andand therethere existsexists anan admissibleadmissible controlcontrol for for whichwhich thetheinfimuminfimum isis reachedreached, , thenthen

andand isis thethe optimaloptimal controlcontrol for for thethe problem.problem.

( )R],,0[),( 2,1 TCxtV ∈

( ) ( )

⎥⎦⎤

⎢⎣⎡ ΣΣ++

+−−+++−+−=

∈

∞

∫θθπθ

λα

θ

TT2T

0

2

21inf

)(),(),()()(0

xVxV

dypxtVyxtVxrVVxtRxtR

xxx

xt

nR

( )xxxTV αβ −= 2),(

∫ ∫∞

−−−−t

dydsMsXsVysXsV0 0

),(~))(,())(,(

)(⋅∗θ

)(⋅∗θ

( ) ( ) ( ) ⎥⎦⎤

⎢⎣⎡ =−+−+−= ∫∈⋅

T

txtXTXTXdssXsRsXsRxtV

n)()()()()()()(inf),( 22

)(αβα

θE

R

njisdWssXsXsxVt

ji ,...,1,),()()())(,(0 =−−∂∂

∫ θ

OptimalOptimal investmentinvestment strategystrategy

( ) ( )

( )( ) ππϕ

ππφ

αβϕµλαβφ

πθ

1TT

1TT

1T

2

)(,0)()(')(2)(2)(,0)()('1

)(2)()(

)(1)()()(

−

−

−

∗∗∗

ΣΣ−=

ΣΣ−=

−==++−−−==++

−=

ΣΣ−

−−=

r

r

TbtbtbtatRTatata

tatbtg

tXtXtgt

InsurerInsurer’’ss wealthwealth underunder thethe strategystrategy

( )

∫ ∫

∫∞

∗

+

+−+−−=

t

t

dydsyMsZ

dsrsgsgsZxgtZ

tgtX

0 0

00

),()(

)()(')()0()(

1)()(

( )( ) ( )⎭⎬⎫

⎩⎨⎧ Σ+Σ+ΣΣ−−= −−− )(

21exp)( T1211TT tttrtZ Wππππ

( ) ( ) 01TT )0(,)()()()(' xmrtmtmtgtm =−+ΣΣ−=

−λµππ

OptimalOptimal investmentinvestment strategystrategy

ShortShort--sellingselling thethe assetasset

BorrownigBorrownig fromfrom a bank a bank accountaccount

)()(0)(0)( tgtXtXt >−∨<−⇔< ∗∗∗θ

ra

tgtXt

−+

<−<⇔> ∗∗2

1

)()(01)(σ

θ

OptimalOptimal investmentinvestment strategystrategy

thethe higher thehigher the reserve, the higher the fraction of the reserve, the higher the fraction of the wealth invested in the riskywealth invested in the risky asset (given the same asset (given the same positive level of available wealth) and thepositive level of available wealth) and the higher the higher the expected value of the insurer's wealthexpected value of the insurer's wealththethe higher the value of alpha, the higher the fractionhigher the value of alpha, the higher the fraction of of the wealth invested in the risky asset (given the same the wealth invested in the risky asset (given the same positivepositive level of available wealth) and the higher the level of available wealth) and the higher the expected value of theexpected value of the insurer's wealthinsurer's wealth

SimulationSimulation studystudy

One One yearyear policypolicy, , discretizationdiscretization one one weekweekInsuranceInsurance riskrisk processprocess: , Gamma : , Gamma distributiondistribution, , expectedexpected valuevalue 100 100 andand variancevariance50005000FinancialFinancial marketmarket1000 1000 simulationssimulations

%5,3ˆ%,20%,10%,4 ==== δσar

100=λ

SimulationSimulation studystudy

the higher the risk allowance in the reserve, the lower the higher the risk allowance in the reserve, the lower the fraction of the wealth invested in the riskythe fraction of the wealth invested in the risky asset, asset, the higher the expected terminal wealth and the lower the higher the expected terminal wealth and the lower the ruinthe ruin probabilityprobabilitythe higher the value of alpha, the higher the fraction of the higher the value of alpha, the higher the fraction of the wealth invested in the risky assetthe wealth invested in the risky asset,, the higher the the higher the expected terminal wealth and theexpected terminal wealth and the higher the ruin higher the ruin probabilityprobabilitythe value ofthe value of beta has only marginal effect on thebeta has only marginal effect on theresultsresults

SimulationSimulation studystudy

the ruin probability is lower and thethe ruin probability is lower and the expected expected terminalterminalwealth is higher compared with the situationwealth is higher compared with the situation when the when the insurer has only a bank account at its insurer has only a bank account at its disposaldisposalTheThe distribution of the terminal wealth under the distribution of the terminal wealth under the optimal strategyoptimal strategy has lighter left tail (5% percentile) and has lighter left tail (5% percentile) and heavier right tailheavier right tail (95% percentile) compared with the (95% percentile) compared with the distribution of the terminaldistribution of the terminal wealth in the case of the wealth in the case of the risk free investment strategyrisk free investment strategy

Reserve=15%, alfa=6000, beta=1Reserve=15%, alfa=6000, beta=1((percentilespercentiles 15th,50th,85th)15th,50th,85th)

-0,5

0

0,5

1

1,5

2

2,5

3

3,5

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52

Reserve=15%, alfa=6000, beta=1Reserve=15%, alfa=6000, beta=1

Percentiles Risk-free+riskyinvestment

Risk-free investment

5% -619,66 -644,46

15% 420,48 236,66

30% 1183,56 952,05

50% 1735,93 1593,28

70% 2361,60 2269,56

85% 2906,07 2904,24

95% 3723,49 3209,12

Mean 1688,13 1579,77

Deviation 1262,64 1297,11

Ruin probability 0,096 0,116

Reserve=25%, alfa=6000, beta=1Reserve=25%, alfa=6000, beta=1((percentilespercentiles 15th,50th,85th)15th,50th,85th)

-0,6

-0,4

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52

Reserve=25%, alfa=6000, beta=1Reserve=25%, alfa=6000, beta=1

Percentiles Risk-free+riskyinvestment

Risk-free investment

5% 632,18 427,53

15% 1364,94 1178,12

30% 1978,17 1817,23

50% 2694,00 2515,11

70% 3225,25 3181,54

85% 3850,07 3816,71

95% 4469,84 4166,16

Mean 2600,06 2498,84

Deviation 1162 1239,28

Ruin probability 0,014 0,016

Reserve=15%, alfa=8000, beta=1Reserve=15%, alfa=8000, beta=1((percentilespercentiles 15th,50th,85th)15th,50th,85th)

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52

Reserve=15%, alfa=8000, beta=1Reserve=15%, alfa=8000, beta=1

Percentiles Risk-free+riskyinvestment

Risk-free investment

5% -600,35 -644,46

15% 569,16 236,66

30% 1294,73 952,05

50% 2039,13 1593,28

70% 2646,94 2269,56

85% 3323,38 2904,24

95% 3931,54 3209,12

Mean 1886,31 1579,77

Deviation 1462,39 1297,11

Ruin probability 0,114 0,116

ThankThank youyou for for youryour attentionattention

ŁŁukasz Delongukasz DelongWarsawWarsaw SchoolSchool ofof EconomicsEconomicsEE--mailmail: : lukasz.delong@sgh.waw.pllukasz.delong@sgh.waw.pl