Samsa proportional reinsurance_and_probability_of_ruin

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

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MULUNGUSHI UNIVERSITYPursuing the Frontiers of

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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


KnowledgeRESULTS (CONTD.)

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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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



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



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



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



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



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.




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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


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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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Samsa proportional reinsurance_and_probability_of_ruin