Surplus process
and ruin theory
and ruin theory
Risk theory
Warsaw University of Technology
Summer semester 2009/2010
R. Łochowski
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Risk Theory and Surplus Process: Ruin Probability and Adjustment Coefficient, Assignments of Material Science and Technology
The theory of ruin and surplus process in the context of insurance risk management. It covers the concepts of compound Poisson distribution, moment of ruin, ruin probability, loading factor, adjustment coefficient, and Lundberg inequality. The document also includes formulas and equations for calculating ruin probability and adjustment coefficient, as well as a problem related to proving the validity of the equations.
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Surplus process and ruin theoryand ruin theory
Risk theory
Warsaw University of TechnologySummer semester 2009/
R. Łochowski
Compound Poisson distribution
as a distribution of total claim amou
nt
-^
- size of a (typical) claim in insurer’s portfolio -^ - i.i.d. random variables -^ -^ homogeneous Poisson process
with
1
2 ,^
,^
Y Y
Y
Y^ N
-^ -^ homogeneous Poisson process
with
intensity
, modelling claims’ arrival (number
of claims between 0 and t)
-^
Total claim amount on time interval [0, t] maybe written as N^ t
1
2
t
t^
N
S^
Y^
Y^
Y
=^
+^
+^
λ
Moment of a ruin and ruin
probability
-^
When surplus process attains negative value itmeans, that the claims exceed initial capitaland the income from premiums
-^
Moment of a ruin
is
defined
as
-^
Moment of a ruin
is
defined
as
-^
Ruin probability
before moment t is defined
as
{^
}
inf
t
T^
t^
U
=^
≥^
(^
)^
(^
)
,u t
T^
t
ψ^
=^
P
Surplus process in discrete time
-^
When surplus process is observed in discretemoments
t^
= 0,1,2,3,…, we call it a
surplus
process in discrete time
,^
U^
u^
c n
S^
n
=^
+^
−^
i
-^
For
from law of large numbers we get
that almost surely
-^
from which follows
,^
n^
n
U^
u^
c n
S^
n
=^
+^
−^
i
, 1
c^
S
≤^
E
(^
)^
1
lim
/^
n^
U^ n
n^
c^
S
→ ∞
=^
−^
E
(^
)^
(^
)^
(^
)
:^
,^
u^
u^
T
ψ
ψ =^
=^
P
Adjustment coefficient
-^
For a positive safety factor, there exists exactlyone positive solution (in R) of the equation
(^
)^
1
1
RS
Rc
S
e^
M
R^
e
=^
=^
E
which is called
adjustment coefficient
-^
It may be proved that
(^
) S^1
e^
M
R^
e
=^
=^
E
(^
)^
RY
Y
cR
M
R^
e
λ
λ
λ
+^
=^
=^
E
Adjustment coefficient, cont.
-^
Problem (theoretical one): to prove statementfrom the previous slide, i.e. that for thepositive loading factor, adjustment coefficient exists
and
is
uniquely
determined
exists
and
is
uniquely
determined
-^
Problem (computational): calculate theadjustment coefficient in the special casewhen claim sizes have exponential distributionwith parameter
β
Exact theoretical formula for ruin
probablility
-^
For every time moment t we have
(^
) (^ )
(^
)^
(^ )
(^
t 1
t^
t
Y^
Y
R u
ct
S
R U
R S
Ru
Rct
t
t M
R
Rc
M^
R
Ru
Rct
Ru
e^
e^
e^
e
e^
e^
e^
e^
e
λ
λ^
λ
−^
+^
−^
−^
−
−^
−^
−^
−^
−^
−
=^
=
=^
=
E^
E^
E
i^
i
-^
and the following exact formula holds
(^ )
(^
)^
(^ )
(^
1 Y^
Y
t M
R
Rc
M^
R
Ru
Rct
Ru
Ru e^
e^
e^
e^
e
e
λ
λ^
λ
−^
−^
−^
−^
−^
−
− =^
=
=
(^
T| RU
R u
e^
T^
u^
e
ψ
−^
−
^
< ∞
=
^
E^
i
i
Functional equation for ruin
probability
-^
For u<0 let us define
-^
Then we have the following formula
-^
From
the
above
formula
we
derive
(a bit
(^
)^
(^
)^
(^
)
0
0
t
u^
u^
ct
y^
dP
y
e^
λdt
ψ
ψ
λ
∞^
∞^
−
=^
+^
(^
)^
u
ψ^
-^
From
the
above
formula
we
derive
(a bit
complicated) equation Problem (theoretical): prove the above equation
(^
)^
(^
)^
(^
)^
(^
)
(^
)^
(^
)^
(^
)^
(^
)
0
0
'
u
c^
u^
u^
u^
y^
dP
y
u^
u^
y^
dP
y
Y^
u
ψ
ψ
ψ
λ ψ
ψ
∞