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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: : [email protected]@sgh.waw.pl