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November-December 2011 Centre for ICT Education
PROPORTIONAL REINSURANCE ON PROBABILITY OF RUIN IN A SURPLUS
PROCESS COMPOUNDED WITH A CONSTANT FORCE OF INTEREST
by
Christian Kasumo, MSc, MBA, BSc, Dip Ed
1
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OUTLINEINTRODUCTION
MODEL
RESULTS
CONCLUSION
OPEN PROBLEMS
ACKNOWLEDGEMENTS
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INTRODUCTIONStudy considers a diffusion-
perturbated insurance process compounded with a constant force of interest.
Overall purpose of the study is to assess impact of proportional reinsurance on the ruin probabilities in this model.
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INTRODUCTION (CONTD.)It is assumed in this study that the
insurance company invests some of its surplus in a risk-free asset (e.g., a bond) and that it buys proportional reinsurance from a reinsurer.
Proportional reinsurance is considered as opposed to other types of reinsurance as it is the easiest way of covering an insurance portfolio.
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MODELAll processes and r.v.’s are defined on a
filtered probability space (Ω,F,{F}tϵR+,P) satisfying the usual conditions.
The model considered is:
where: - is the insurer’s
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)()()()(0
sdRsYtPytYt bbb (1)
)(
1,)()(
tN
iiPPP
bP
bStWbbpttP
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MODEL (CONTD.)
surplus generating process,- is the investment generating process,- is the value of the insurer’s total surplus just before time t,- y=Y(0) is the initial surplus or capital of the insurance company,- bϵ(0,1] is the retention percentage for proportional reinsurance,- bp represents the premium rate net of reinsurance premiums. If there is no
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rttR )()( tY b
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MODEL (CONTD.)
reinsurance (i.e., when b=1), then the premium left to the insurer is simply p,
the premium rate paid by policyholders.It should be noted that (1) is but an extension
of the Cramér-Lundberg model, for when σP=r=0 and when b=1, then the model (1) becomes:
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)(
1
)(tN
iiSptytY
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MODEL (CONTD.)Definitions
Time of ruin: Ruin prob.:
where is the survival prob.
Infinitesimal generator of Y is given using Itô’s formula byA
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yYtYt bbb )0(|0)(:0inf yyYPy bb
bb 1)0(|
)(1 yy bb
)()()('''21
0
22 sdFygbsygygbpryygbygy
P
(2)
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MODEL (CONTD.)
from which we obtain the relevant Volterra integro-differential equation (VIDE):
The survival probability satisfies (3) only if it is strictly increasing, strictly con-cave and twice continuously differentiable, and if it satisfies for and
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y
P ysdFbsyybpryyb0
22 )('''21
(3) y
0y 0y
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MODEL (CONTD.)
(Paulsen and Gjessing, 1997).
Theorem: The VIDE (3) can be represented as a Volterra integral equation (VIE) of the second kind
where the kernel and forcing function are prescribed and the method of solution of (4) is the Block-by-Block method, which is considered as the best of the higher order methods for solving such equations (Press et al. 1992).
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1lim y
y
ydsssyKyy
0 , (4)
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RESULTSExponential claims:
Cramér-Lundberg model, p=6, λ=2, μ=0.5
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y0 0.66666667 0.66666667 0.000000005 0.28973213 0.28973214 0.0000034510 0.12591706 0.12591707 0.0000079415 0.05472333 0.05472333 0.0000000020 0.02378266 0.02378266 0.0000000025 0.01033590 0.01033590 0.0000000030 0.00449196 0.00449196 0.0000000035 0.00195220 0.00195220 0.0000000040 0.00084842 0.00084842 0.00000000
y y01.0 yD 01.0
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RESULTS (CONTD.)CLM: Exp(0.5) claims, p=6, λ=2
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0 5 10 15 20 25 30 35 400
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Initial surplus (y)
Rui
n pr
obab
ility
()
Exact (y)
B-by-B (y)
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CLM compounded with constant force of interest
MULUNGUSHI UNIVERSITYPursuing the Frontiers of
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y0 0.61991511 0.58965945 0.5662883
45 0.21190867 0.16970729 0.1415629
410 0.06587874 0.04216804 0.0296554
815 0.01883184 0.00931477 0.0054555
820 0.00499659 0.00186956 0.0009090
725 0.00124044 0.00034661 0.0001401
630 0.00029012 0.00006011 0.0000202
935 0.00006431 0.00000984 0.0000027
940 0.00001357 0.00000153 0.0000003
7
yr 1.0 yr 2.0 yr 3.0
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MULUNGUSHI UNIVERSITYPursuing the Frontiers of
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0 5 10 15 20 25 30 35 400
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Initial surplus (y)
Rui
n pr
obab
ility
()
r=0.1r=0.2r=0.3
CLM with constant force of interest
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RESULTS (CONTD.)
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y0 1.00000000 1.00000000 1.000000005 0.58319951 0.18131975 0.0964024310 0.37292758 0.02501663 0.0062338615 0.23846905 0.00233005 0.0002561420 0.15248880 0.00015788 0.00000693
Diffusion approximation to CLM: Exp(1) jumps, p=1.1, λ=1, =0.2 P
yr 0.0 yr 05.0 yr 1.0
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CLM with Exp(1) jumps, p=1.1, λ=1, =0.2
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0 2 4 6 8 10 12 14 16 18 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Initial surplus (y)
Rui
n pr
obab
ility
()
r=0.0(diffusion approx. to CLM)r=0.05r=0.1
P
RESULTS (CONTD.)
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CLM with proportional reinsurance (Exp(0.5), λ=2, μ=0.5 )
RESULTS (CONTD.)
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0 1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Initial surplus (y)
Rui
n pr
obab
ility
()
b=1.0b=0.9b=0.8
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Pareto claims: CLM with Pareto(3,2) claims, p=6, λ=2
MULUNGUSHI UNIVERSITYPursuing the Frontiers of
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y ψ0.01,0.0(y) ψ0.01,0.1(y) ψ0.01,0.2(y) ψ0.01,0.3(y)0 0.333324
780.320162
160.310201
990.301906
3210 0.018031
110.012190
390.009304
850.007521
1850 0.000797
700.000263
060.000158
650.000113
47100
0.00018822
0.00003971
0.00002229
0.00001548
200
0.00003726
0.00000494
0.00000265
0.00000181
300
0.00000941
0.00000098
0.00000052
0.00000035
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CLM with Pareto(3,2) claims, p=6, λ=2
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0 10 20 30 40 50 60 70 80 90 1000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Initial surplus (y)
Rui
n pr
obab
ility
( )
r=0.0r=0.1r=0.2r=0.3
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Diffusion approximation to CLM with Pareto(2,1) jumps, =0.2
MULUNGUSHI UNIVERSITYPursuing the Frontiers of
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y ψr=0.1(y) ψr=0.2(y) ψr=0.3(y)0 1.00000000 1.00000000 1.000000005 0.15667318 0.08777727 0.05971971
10 0.05571672 0.02678351 0.0172495520 0.01422636 0.00663769 0.0042907130 0.00604205 0.00286467 0.0018721950 0.00201238 0.00096172 0.00063124100 0.00045139 0.00021724 0.00015526150 0.00011574 0.00006561 0.00003406
P
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CLM with Pareto(2,1) jumps, =0.2
MULUNGUSHI UNIVERSITYPursuing the Frontiers of
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P
0 50 100 1500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Initial surplus (y)
Rui
n pr
obab
ility
( )
r=0.1r=0.2r=0.3
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Asymptotic ruin probabilities for large claim case
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y ψ1.0(y) ψ0.9(y) ψ0.8(y) ψ0.7(y) ψ0.6(y) ψ0.5(y) ψ0.4(y)0 0.200000
000.2068965
60.2162162
20.2295081
90.250000
000.2857143
00.3636363
45 0.033333
340.0315604
90.0298229
30.0281852
20.026737
970.0259740
30.0269360
310
0.01818182
0.01708320
0.01601602
0.01501455
0.01415094
0.01360544
0.01398601
15
0.01250000
0.01171113
0.01094766
0.01023285
0.00961538
0.00921659
0.00944510
20
0.00952381
0.00890942
0.00831601
0.00776115
0.00728155
0.00696864
0.00713012
25
0.00769231
0.00718946
0.00670438
0.00625120
0.00585938
0.00560224
0.00572656
30
0.00645161
0.00602611
0.00561601
0.00523309
0.00490196
0.00468384
0.00478469
35
0.00555556
0.00518682
0.00483165
0.00450016
0.00421348
0.00402414
0.00410889
40
0.00487805
0.00455274
0.00423953
0.00394732
0.00369458
0.00352734
0.00360036
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Ruin probabilities reduce with a reduction in b (that is, as the amount reinsured increases), then start rising again after a certain b.
MULUNGUSHI UNIVERSITYPursuing the Frontiers of
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We have numerically obtained the ruin probabilities for a surplus process compounded with a constant force of interest
Proportional reinsurance minimizes the probability of ruin for insurance companies
MULUNGUSHI UNIVERSITYPursuing the Frontiers of
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OPEN PROBLEMS
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Use other forms of reinsurance (e.g. Excess of Loss, Stop Loss)
Consider investments of Black-Scholes type in the investment model
Allow sudden changes (jumps) in the investment process
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Conference organisersNOMAMulungushi University
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ACKNOWLEDGEMENTS
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END OF PRESENTATION
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