Optimal Control by Franchise and Deductible Amounts
in the Classical Risk Model
Olena Ragulina
Taras Shevchenko National University of Kyiv, Ukraine
7 - 9 July, 2014
Olena Ragulina (Kyiv)
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Overview
1
Preliminaries
2
Optimal Control by the Franchise Amount
3
Optimal Control by the Deductible Amount
Olena Ragulina (Kyiv)
Optimal Control
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Classical Risk Model
the insurance company has an initial surplus x > 0
Xt (x) is a surplus at time t ≥ 0 provided that the initial surplus
equals x
Premium arrivals
the insurance company receives premiums with constant intensity
c>0
Claim arrivals
the claim sizes form a sequence (Yi )i≥1 of nonnegative i.i.d. random
variables with c.d.f. F (y ) = P[Yi ≤ y ] and finite expectations µ; τi is
the time when the ith claim arrives
the number of claims on the time interval [0, t] is a Poisson process
(Nt )t≥0 with constant intensity λ > 0; the random variables Yi ,
i ≥ 1, and the process (Nt )t≥0 are independent
P t
P0
the total claims on [0, t] equal N
i=1 Yi ; we set
i=1 Yi = 0 if
Nt = 0
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Classical Risk Model
The surplus of the insurance company at time t equals
Xt (x) = x + ct −
Nt
X
Yi ,
t ≥ 0.
(1)
i=1
We assume that the net profit condition holds, i.e.
c > λµ.
The insurance company uses the expected value principle for premium
calculation, i.e.
c = λµ(1 + θ),
where θ > 0 is a safety loading.
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Optimal Control Problems
optimal control by investments: C. Hipp, M. Plum (2000);
C. Hipp, M. Plum (2003); C. S. Liu, H. Yang (2004); P. Azcue,
N. Muler (2009)
optimal control by reinsurance: H. Schmidli (2001); C. Hipp,
M. Vogt (2003)
optimal control by investments and reinsurance: H. Schmidli
(2002); M. I. Taksar, C. Markussen (2003); S. D. Promislow,
V. R. Young (2005); C. Hipp, M. Taksar (2010)
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Franchise and Deductible
Definitions
A franchise is a provision in the insurance policy whereby the insurer
does not pay unless damage exceeds the franchise amount.
A deductible is a provision in the insurance policy whereby the
insurer pays any amounts of damage that exceed the deductible
amount.
Example 1
The franchise/deductible amount is 10.
Case 1: the claim size is 5
If the franchise is used, then the insurance company pays nothing.
If the deductible is used, then the insurance company pays nothing.
Case 2: the claim size is 100
If the franchise is used, then the insurance company pays 100.
If the deductible is used, then the insurance company pays
100 − 10 = 90.
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Franchise and Deductible
Motivation
a franchise and a deductible are applied when the insured’s losses are
relatively small to deter a large number of trivial claims
a deductible encourages the insured to take more care of the insured
property
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Optimal Control by the Franchise Amount
Additional assumptions
the insurance company adjusts the franchise amount dt at every time
t ≥ 0 on the basis of the information available before time t, i.e.
every admissible strategy (dt )t≥0 ((dt ) for brevity) of the franchise
amount choice is a predictable process w.r.t. the natural filtration
generated by (Nt )t≥0 and (Yi )i≥1
0 ≤ dt ≤ dmax , where dmax is the maximum allowed franchise amount
such that 0 < F (dmax ) < 1; in particular, if dt = 0, then the franchise
is not used at time t
the safety loading θ > 0 is constant
The premium intensity at time t depends on the franchise amount at this
time and it is given by
Z +∞
y dF (y ).
c(dt ) = λ(1 + θ)
dt
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Optimal Control by the Franchise Amount
(d )
Let Xt t (x) be the surplus of the insurance company at time t provided
its initial surplus is x and the strategy (dt ) is used. Then
(d )
Xt t (x)
Z
t
c(ds ) ds −
=x+
0
Nt
X
Yi I{Yi >dτi } ,
t ≥ 0.
(2)
i=1
The ruin time under the admissible strategy (dt ) is defined as
(d )
τ (dt ) (x) = inf t ≥ 0 : Xt t (x) < 0 .
The corresponding infinite-horizon survival probability is given by
ϕ(dt ) (x) = P τ (dt ) (x) = ∞ .
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Optimal Control by the Franchise Amount
Our aim is to maximize the survival probability over all admissible
strategies (dt ), i.e. to find
ϕ∗ (x) = sup ϕ(dt ) (x),
(dt )
and show that there exists an optimal strategy (dt∗ ) such that
∗
ϕ∗ (x) = ϕ(dt ) (x) for all x ≥ 0.
The optimal strategy will be a function of the initial surplus only.
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Hamilton-Jacobi-Bellman Equation
Proposition 1
(d )
Let the surplus process Xt t (x) t≥0 follow (2) under the above
assumptions. If ϕ∗ (x) is differentiable on R+ , then it satisfies the
Hamilton-Jacobi-Bellman equation
Z +∞
0
sup
(1 + θ)
y dF (y ) ϕ∗ (x)
d
d∈[0, dmax ]
∗
Z
d∨x
+ F (d) − 1 ϕ (x) +
ϕ (x − y ) dF (y ) = 0,
(3)
∗
d
which is equivalent to
0
ϕ (x) =
∗
inf
d∈[0, dmax ]
Olena Ragulina (Kyiv)
R d∨x
1 − F (d) ϕ∗ (x) − d ϕ∗ (x − y ) dF (y )
. (4)
R +∞
(1 + θ) d y dF (y )
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Hamilton-Jacobi-Bellman Equation
Remark 1
Note that if there exists one solution to (3) or (4), then there exist
infinitely many solutions to these equations which differ with a
multiplicative constant.
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Existence Theorem
Theorem 1
If the random variables Yi , i ≥ 1, have a p.d.f. f (y ), then there exists the
solution G (x) to (4) with G (0) = θ/(1 + θ), which is nondecreasing and
continuously differentiable on R+ , and θ/(1 + θ) ≤ limx→+∞ G (x) ≤ 1.
The solution G (x) to (4) that satisfies the conditions of
Theorem 1 can be
found as the limit of the sequence of functions Gn (x) n≥0 on R+ , where
G0 (x) = ϕ(0) (x) is the survival probability provided that dt = 0 for all
t ≥ 0, and
!
R d∨x
1
−
F
(d)
G
(x)
−
G
(x
−
y
)
dF
(y
)
n−1
n−1
d
Gn0 (x) =
inf
,
R +∞
d∈[0, dmax ]
(1 + θ) d y dF (y )
Gn (0) = θ/(1 + θ),
n ≥ 1.
(5)
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Verification Theorem
Theorem 2
(d )
Let the surplus process Xt t (x) t≥0 follow (2) and G (x) be the solution
to (4) that satisfies the conditions of Theorem 1. Then for any x ≥ 0 and
arbitrary admissible strategy (dt ), we have
ϕ(dt ) (x) ≤
G (x)
,
limx→+∞ G (x)
(6)
(d ∗ )
and equality in (6) is attained under the strategy (dt∗ ) = dt∗ Xt−t (x) ,
where dt∗ (x) minimizes the right-hand side of (4), i.e.
∗
ϕ∗ (x) = ϕ(dt ) (x) =
Olena Ragulina (Kyiv)
G (x)
.
limx→+∞ G (x)
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Verification Theorem
Remark 2
In Theorem 2 we used any solution to (4) that satisfies the conditions of
Theorem 1. However, Theorem 2 also implies uniqueness of such a
solution. The corresponding strategy (dt∗ ) may not be unique in the
general case.
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Exponentially Distributed Claim Sizes
Theorem 3
(d )
Let the surplus process Xt t (x) t≥0 follow (2), the claim sizes be
exponentially distributed with mean µ, and dmax = µ. Then the strategy
(dt ) with dt = 0 for all t ≥ 0 is not optimal.
Remark 3
Theorem 3 implies that we can always increase the survival probability
adjusting the franchise amount if the claim sizes are exponentially
distributed.
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Exponentially Distributed Claim Sizes
Example 2
If the claim sizes are exponentially distributed with mean µ = 10,
dmax = µ, and θ = 0.1, then
ϕ(0) (x) ≈ 1 − 0.9090909 e−x/110 , x ≥ 0 ,
(
0.111048767 ex/22
if x ≤ 8.93258,
ϕ∗ (x) ≈
x/110
1 − 0.90382792 e
if x > 8.93258,
(
10 if x ≤ 8.93258,
dt∗ (x) =
0
if x > 8.93258.
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Constant Franchise Amount
Theorem 4
(d )
Let the surplus process Xt t (x) t≥0 follow (2) with dt ≡ d where d > 0
is constant, and the claim sizes be exponentially distributed with mean µ.
(d)
Then ϕ(d) (x) = ϕn+1 (x) for all x ∈ [nd, (n + 1)d), n ≥ 0, where
(d)
ϕ1 (x) = C1, 1 ex/γ ,
(d)
ϕn+1 (x)
(d)
ϕ2 (x) = C2, 1 + A2, 0 x ex/γ + C2, 2 e−x/µ ,
n−1
X
i+1
= Cn+1, 1 +
An+1, i x
ex/γ
i=0
n−2
X
i+1
+ Cn+1, 2 +
Bn+1, i x
e−x/µ ,
n ≥ 2.
i=0
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Constant Franchise Amount
Theorem 4
Here
γ = (1 + θ)(µ + d),
C1, 1 = θ/(1 + θ),
θ
e−d/γ ,
A2, 0 = −
(1 + θ)(γ + µ)
θ
γµ + d(γ + µ) −d/γ
C2, 1 =
1+
e
,
1+θ
(γ + µ)2
θγµ
C2, 2 = −
ed/µ ,
(1 + θ)(γ + µ)2
and other coefficients can be found by recurrent formulas.
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Constant Franchise Amount
Theorem 5
Let conditions of Theorem
4 hold.
µ(1+θ)
If d > 0 and x ∈ 0, min
ln 1 +
θ
θd
µ(1+θ)
, d , then the
correspoding survival probability is less than the classical one.
If d ∈ 0, µ(1+θ)θln(1+θ) and x is large enough, then the
correspoding survival probability is greater than the classical one.
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Optimal Control by the Deductible Amount
Additional assumptions
the insurance company adjusts the deductible amount d̄t at every
time t ≥ 0 on the basis of the information available before time t, i.e.
every admissible strategy (d̄t )t≥0 ((d̄t ) for brevity) of the deductible
amount choice is a predictable process w.r.t. the natural filtration
generated by (Nt )t≥0 and (Yi )i≥1
0 ≤ d̄t ≤ d̄max , where d̄max is the maximum allowed deductible
amount such that 0 < F (d̄max ) < 1; in particular, if d̄t = 0, then the
deductible is not used at time t
the safety loading θ > 0 is constant
The premium intensity at time t depends on the deductible amount at this
time and it is given by
Z +∞
(y − d̄t ) dF (y ).
c(d̄t ) = λ(1 + θ)
d̄t
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Optimal Control by the Deductible Amount
(d̄ )
Let Xt t (x) be the surplus of the insurance company at time t provided
its initial surplus is x and the strategy (d̄t ) is used. Then
(d̄ )
Xt t (x)
Z
=x+
t
Nt
X
c(d̄s ) ds −
(Yi − d̄τi )+ .
0
(7)
i=1
The ruin time under the admissible strategy (d̄t ) is defined as
(d̄ )
τ (d̄t ) (x) = inf t ≥ 0 : Xt t (x) < 0 .
The corresponding infinite-horizon survival probability is given by
ϕ(d̄t ) (x) = P τ (d̄t ) (x) = ∞ .
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Optimal Control by the Deductible Amount
Our aim is to maximize the survival probability over all admissible
strategies (d̄t ), i.e. to find
ϕ∗ (x) = sup ϕ(d̄t ) (x),
(d̄t )
and show that there exists an optimal strategy (d̄t∗ ) such that
∗
ϕ∗ (x) = ϕ(d̄t ) (x) for all x ≥ 0.
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Hamilton-Jacobi-Bellman Equation
Proposition 2
(d̄ )
Let the surplus process Xt t (x) t≥0 follow (7) under the above
assumptions. If ϕ∗ (x) is differentiable on R+ , then it satisfies the
Hamilton-Jacobi-Bellman equation
Z +∞
0
sup
(1 + θ)
(y − d̄) dF (y ) ϕ∗ (x)
d̄
d̄∈[0, d̄max ]
+ F (d̄) − 1 ϕ∗ (x) +
Z
(8)
x+d̄
∗
ϕ (x + d̄ − y ) dF (y )
= 0,
d̄
which is equivalent to
∗
0
ϕ (x) =
inf
d̄∈[0, d̄max ]
Olena Ragulina (Kyiv)
R x+d̄ ∗
1 − F (d̄) ϕ∗ (x) − d̄
ϕ (x + d̄ − y ) dF (y )
.
R +∞
(1 + θ) d̄ (y − d̄) dF (y )
(9)
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Existence Theorem
Theorem 6
If the random variables Yi , i ≥ 1, have a p.d.f. f (y ), then there exists the
solution G (x) to (9) with G (0) = θ/(1 + θ), which is nondecreasing and
continuously differentiable on R+ , and θ/(1 + θ) ≤ limx→+∞ G (x) ≤ 1.
The solution G (x) to (9) that satisfies the conditions of
Theorem 6 can be
found as the limit of the sequence of functions Gn (x) n≥0 on R+ , where
G0 (x) = ϕ(0) (x) is the survival probability provided that dt = 0 for all
t ≥ 0, and
Gn0 (x) =
inf
d̄∈[0, d̄max ]
Gn (0) = θ/(1 + θ),
!
R x+d̄
1 − F (d̄) Gn−1 (x) − d
Gn−1 (x + d̄ − y ) dF (y )
,
R +∞
(1 + θ) d̄ (y − d̄) dF (y )
n ≥ 1.
(10)
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Verification Theorem
Theorem 7
(d̄ )
Let the surplus process Xt t (x) t≥0 follow (7) and G (x) be the solution
to (9) that satisfies the conditions of Theorem 6. Then for any x ≥ 0 and
arbitrary admissible strategy (d̄t ), we have
ϕ(d̄t ) (x) ≤
G (x)
,
limx→+∞ G (x)
(11)
(d̄ ∗ )
and equality in (11) is attained under the strategy (d̄t∗ ) = d̄t∗ Xt−t (x) ,
where d̄t∗ (x) minimizes the right-hand side of (9), i.e.
∗
ϕ∗ (x) = ϕ(d̄t ) (x) =
Olena Ragulina (Kyiv)
G (x)
.
limx→+∞ G (x)
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Exponentially Distributed Claim Sizes
Theorem 8
(d̄t )
Let the surplus process Xt
(x) t≥0 follow (7) and the claim sizes be
exponentially distributed. Then ϕ∗ (x) = ϕ(d̄t ) (x) for every admissible
strategy (d̄t ), i.e. every admissible strategy is optimal.
Remark 4
Theorem 8 implies that we cannot increase the survival probability
adjusting the deductible amount for the exponentially distributed claim
sizes.
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References
P. Azcue, N. Muler (2009)
Optimal investment strategy to minimize the ruin probability of an insurance
company under borrowing constrains
Insurance: Mathematics and Economics 44(1), 26–34.
C. Hipp, M. Plum (2000)
Optimal investment for insurers
Insurance: Mathematics and Economics 27(2), 215–228.
C. Hipp, M. Plum (2003)
Optimal investment for investors with state dependent income, and for insurers
Finance and Stochastics 7(3), 299–321.
C. Hipp, M. Taksar (2010)
Optimal non-proportional reinsurance control
Insurance: Mathematics and Economics 47(2), 246–254.
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References
C. Hipp, M. Vogt (2003)
Optimal dynamic XL reinsurance
ASTIN Bulletin 33(2), 193–207.
C. S. Liu, H. Yang (2004)
Optimal investment for an insurer to minimize its probability of ruin
North American Actuarial Journal 8(2), 11–31.
S. D. Promislow, V. R. Young (2005)
Minimizing the probability of ruin when claims follow Brownian motion with drift
North American Actuarial Journal 9(3), 109–128.
O. Ragulina (2014)
Maximization of the survival probability by franchise and deductible amounts in the
classical risk model
Springer Optimization and Its Applications 90, 287–300.
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References
H. Schmidli (2001)
Optimal proportional reinsurance polisies in a dynamic setting
Scandinavian Actuarial Journal 2001(1), 55–68.
H. Schmidli (2002)
On minimizing the ruin probability by investment and reinsurance
The Annals of Applied Probability 12(3), 890–907.
M. I. Taksar, C. Markussen (2003)
Optimal dynamic reinsurance policies for large insurance portfolios
Finance and Stochastics 7(1), 97–121.
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Thank you very much for your attention!
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