Samsa proportional reinsurance_and_probability_of_ruin

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
November-December 2011
Centre for ICT Education 1
MULUNGUSHI UNIVERSITY
Pursuing the Frontiers of Knowledge

OUTLINE
INTRODUCTION
MODEL
RESULTS
CONCLUSION
OPEN PROBLEMS
ACKNOWLEDGEMENTS

INTRODUCTION
Study 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.

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.

MODEL
All 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
)()()()(
0
sdRsYtPytY
t
bbb


 (1)


)(
1
,)()(
tN
i
iPPP
b
P
bStWbbpttP 

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
rttR )(
)( 
tY b

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:


)(
1
)(
tN
i
iSptytY

MODEL (CONTD.)
Definitions
Time of ruin:
Ruin prob.:
where is the survival prob.
Infinitesimal generator of Y is given using Itô’s
formula by
A
 yYtYt bb
b  )0(|0)(:0inf
     yyYPy bb
b
b
  1)0(|
  )(1 yy bb
 
          )()()('''
2
1
0
22
sdFygbsygygbpryygbyg
y
P   
(2)

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
           
y
P ysdFbsyybpryyb
0
22
)('''
2
1

(3)
 y
  0y 0y

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).
  1lim 

y
y

       ydsssyKy
y
  0
, (4)

RESULTS
 Exponential claims:
 Cramér-Lundberg model, p=6, λ=2, μ=0.5
y
0 0.66666667 0.66666667 0.00000000
5 0.28973213 0.28973214 0.00000345
10 0.12591706 0.12591707 0.00000794
15 0.05472333 0.05472333 0.00000000
20 0.02378266 0.02378266 0.00000000
25 0.01033590 0.01033590 0.00000000
30 0.00449196 0.00449196 0.00000000
35 0.00195220 0.00195220 0.00000000
40 0.00084842 0.00084842 0.00000000
 y  y01.0  yD 01.0

RESULTS (CONTD.)
CLM: Exp(0.5) claims, p=6, λ=2
0 5 10 15 20 25 30 35 40
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Initial surplus (y)
Ruinprobability()
Exact (y)
B-by-B (y)

 CLM compounded with constant force of interest
RESULTS (CONTD.)
y
0 0.61991511 0.58965945 0.56628834
5 0.21190867 0.16970729 0.14156294
10 0.06587874 0.04216804 0.02965548
15 0.01883184 0.00931477 0.00545558
20 0.00499659 0.00186956 0.00090907
25 0.00124044 0.00034661 0.00014016
30 0.00029012 0.00006011 0.00002029
35 0.00006431 0.00000984 0.00000279
40 0.00001357 0.00000153 0.00000037
 yr 1.0  yr 2.0  yr 3.0

RESULTS (CONTD.)
0 5 10 15 20 25 30 35 40
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Initial surplus (y)
r=0.1
r=0.2
r=0.3
CLM with constant force of interest

RESULTS (CONTD.)
y
0 1.00000000 1.00000000 1.00000000
5 0.58319951 0.18131975 0.09640243
10 0.37292758 0.02501663 0.00623386
15 0.23846905 0.00233005 0.00025614
20 0.15248880 0.00015788 0.00000693
 Diffusion approximation to CLM: Exp(1)
jumps, p=1.1, λ=1, =0.2P
 yr 0.0  yr 05.0  yr 1.0

CLM with Exp(1) jumps, p=1.1, λ=1, =0.2
0 2 4 6 8 10 12 14 16 18 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Initial surplus (y)
r=0.0(diffusion approx. to CLM)
r=0.05
r=0.1
P
RESULTS (CONTD.)

 CLM with proportional reinsurance (Exp(0.5),
λ=2, μ=0.5 )
RESULTS (CONTD.)
0 1 2 3 4 5 6 7 8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Initial surplus (y)
b=1.0
b=0.9
b=0.8

Pareto claims:
 CLM with Pareto(3,2) claims, p=6, λ=2
RESULTS (CONTD.)
y ψ0.01,0.0(y) ψ0.01,0.1(y) ψ0.01,0.2(y) ψ0.01,0.3(y)
0 0.33332478 0.32016216 0.31020199 0.30190632
10 0.01803111 0.01219039 0.00930485 0.00752118
50 0.00079770 0.00026306 0.00015865 0.00011347
100 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

CLM with Pareto(3,2) claims, p=6, λ=2
RESULTS (CONTD.)
0 10 20 30 40 50 60 70 80 90 100
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Initial surplus (y)
r=0.0
r=0.1
r=0.2
r=0.3

 Diffusion approximation to CLM with Pareto(2,1)
jumps, =0.2
RESULTS (CONTD.)
y ψr=0.1(y) ψr=0.2(y) ψr=0.3(y)
0 1.00000000 1.00000000 1.00000000
5 0.15667318 0.08777727 0.05971971
10 0.05571672 0.02678351 0.01724955
20 0.01422636 0.00663769 0.00429071
30 0.00604205 0.00286467 0.00187219
50 0.00201238 0.00096172 0.00063124
100 0.00045139 0.00021724 0.00015526
150 0.00011574 0.00006561 0.00003406
P

CLM with Pareto(2,1) jumps, =0.2
RESULTS (CONTD.)
P
0 50 100 150
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Initial surplus (y)
r=0.1
r=0.2
r=0.3

Asymptotic ruin probabilities for large claim case
RESULTS (CONTD.)
y ψ1.0(y) ψ0.9(y) ψ0.8(y) ψ0.7(y) ψ0.6(y) ψ0.5(y) ψ0.4(y)
0 0.20000000 0.20689656 0.21621622 0.22950819 0.25000000 0.28571430 0.36363634
5 0.03333334 0.03156049 0.02982293 0.02818522 0.02673797 0.02597403 0.02693603
10 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

Ruin probabilities reduce with a
reduction in b (that is, as the
amount reinsured increases), then
start rising again after a certain b.
RESULTS (CONTD.)

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
CONCLUSION

OPEN PROBLEMS
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

Conference organisers
NOMA
Mulungushi University
ACKNOWLEDGEMENTS

END OF PRESENTATION

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