Pure risk premiums under deductibles
K. Burnecki
J. Nowicka-Zagrajek
A. Wylomańska
Hugo Steinhaus Center
Wroclaw University of Technology
www.im.pwr.wroc.pl/˜hugo/
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Pure risk premiums under deductibles
Introduction
The idea of a deductible is, firstly, to reduce claim handling costs by
excluding coverage for the often numerous small claims and, secondly,
to provide some motivation to the insured to prevent claims,
throught a limited degree of participation in claim costs.
The reasons for introducing deductibles:
(i) loss prevention
(ii) loss reduction
(iii) avoidance of small claims where administration costs
are dominant
(iv) premium reduction.
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Pure risk premiums under deductibles
General formulae for premiums under
deductibles
Let X be a risk. A premium calculation principle is a rule saying
what premium should be assigned to a given risk. We consider the
simplest premium which is called pure risk premium, namely the
mean of X. It is often applied in life and some mass lines of business
in non-life insurance. The pure risk premium can be of practical use
because, for one thing, in practice the planning horizon is always
limited and for another, because there are indirect ways of loading a
premium.
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Let
• X – a non-negative random variable describing the size of claim
(assume EX exists),
• F (t) – distribution of X,
• f (t) – probability function,
• h(x) – the payment function corresponding to a deductible.
The pure risk premium P is then equal to the expectation (P =EX).
We will express formulae for premiums under deductibles in terms of
so-called limited expected value function
Z x
E(X, x) =
yf (y)dy + x(1 − F (x)), x > 0.
0
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Pure risk premiums under deductibles
Franchise deductible
One of the deductibles that can be incorporated in the contract is a
co-called franchise deductible. Under the franchise deductible of a, if
the loss is less than a amount the insurer pays nothing, but if the loss
equals or exceeds a amount claim is paid in full.The payment
function can be described by

 0, x < a,
hF D(a) (x) =
 x, otherwise.
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Pure risk premiums under deductibles
Franchise deductible cont.
The pure risk premium under the franchise deductible can be
expressed in terms of the premium in the case of no deductible and
the corresponding limited expected value function:
PF D(a) = P − E(X, a) + a(1 − F (a)).
This premium is a decreasing function of a, when a = 0 the premium
is equal to the one in the case of no deductible and if a tends to
infinity the premium tends to zero.
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Figure 1: The payment function under the franchise deductible (solid line)
and no deductible (dashed line).
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STFded01.xpl
Pure risk premiums under deductibles
Fixed amount deductible
An agreement between the insured and the insurer incorporating a
deductible b means that the insurer pays only the part of the claim
which exceeds the amount b; if the size of the claim falls below this
amount, then the claim is not covered by the contract and the insured
receives no indemnification. The payment function is thus given by
hF AD(b) (x) = max(0, x − b).
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Pure risk premiums under deductibles
Fixed amount deductible cont.
The premium in the case of fixed amount deductible has the following
form in terms of the premium under the franchise deductible.
PF AD(b) = P − E(X, b) = PF D(b) − b(1 − F (b)).
The premium is a decreasing function of b, for b = 0 it gives the
premium in the case of no deductible and if b tends to infinity, it
tends to zero.
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Figure 2: The payment function under the fixed amount deductible (solid
line) and no deductible (dashed line).
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STFded02.xpl
Pure risk premiums under deductibles
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Proportional deductible
In the case of the proportional deductible of c, where c ∈ (0, 1), each
payment is reduced by c · 100% (the insurer pays 100%(1 − c) of the
claim). The payment function is given by
hP D(c) (x) = (1 − c)x.
The relation between the premium under the proportional deductible
and the premium in the case of no deductible has the following form:
PP D(c) = (1 − c)EX = (1 − c)P.
The premium is a decreasing function of c, PP D(0) = P and
PP D(1) = 0.
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Figure 3: The payment function under the proportional deductible (solid
line) and no deductible (dashed line).
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Limited proportional deductible
The proportional deductible is usually combined with a minimum
amount deductible so the insurer does not need to handle small
claims and with maximum amount deductible to limit the retention
of the insured. For the limited proportional deductible of c with
minimum amount m1 and maximum amount m2 (0 ≤ m1 < m2 ) the
payment function is given by


0,
x ≤ m1 ,




 x − m , m < x ≤ m /c,
1
1
1
hLP D(c,m1 ,m2 ) (x) =

(1 − c)x, m1 /c < x ≤ m2 /c,




 x − m , otherwise.
2
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Pure risk premiums under deductibles
Limited proportional deductible cont.
The following formula expresses the premium under the limited
proportional deductible in terms of the premium in the case of no
deductible and the corresponding limited expected value function
m o
n m 2
1
PLP D(c,m1 ,m2 ) = P − E(X, m1 ) + c E X,
− E X,
.
c
c
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Figure 4: The payment function under the limited proportional deductible
(solid line) and no deductible (dashed line).
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Pure risk premiums under deductibles
Disappearing deductible
In the case of disappearing deductible the payment depends on the
loss in the following way: if the loss is less than an amount of d1 , the
insurer pays nothing, if the loss exceeds d2 (d2 > d1 ) amount, the
insurer pays the loss in full, if the loss is between d1 and d2 , then the
deductible is reduced linearly between d1 and d2 . The payment
function is thus given by:


x ≤ d1 ,

 0,
d2 (x−d1 )
hDD(d1 ,d2 ) (x) =
d2 −d1 , d1 < x ≤ d2 ,



x,
otherwise.
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Pure risk premiums under deductibles
Disappearing deductible cont.
The following formula shows the premium under the disappearing
deductible in terms of the premium in the case of no deductible and
the corresponding limited expected value function
PDD(d1 ,d2 )
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d2
d1
=P +
E(X, d2 ) −
E(X, d1 ).
d2 − d1
d2 − d1
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Figure 5: The payment function under the disappearing deductible (solid
line) and no deductible (dashed line).
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STFded05.xpl
Pure risk premiums under deductibles
Lognormal distribution of loss
Consider a random variable Z which has the normal distribution. Let
X = eZ . Then the distribution of X is called a lognormal
distribution. The distribution function is given by
(
Z t
2 )
ln t − µ
1
1 ln y − µ
√
F (t) = Φ
=
dy,
exp −
σ
2
σ
2πσy
0
where t, σ > 0, µ ∈ R and Φ(.) is the standard normal distribution
function.
We will illustrate the formulae for premium under deductibles using
the catastrophe data example. As the example we consider the
lognormal loss distribution with parameters µ = 18.4406 and
σ = 1.1348.
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Figure 6: The premium under the franchise deductible (thick line) and fixed
amount deductible (thin line).
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Pure risk premiums under deductibles
Figure 7: The thick solid line - the premium for c = 0.2 and m1 = 10
million, the solid - for c = 0.4 and m1 = 10 million, the dotted - for c = 0.2
and m1 = 50 million, and the dashed - for c = 0.4 and m1 = 50 million.
STFded07.xpl
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Figure 8: The thick line represents the premium for d1 = USD 10 million
and the thin line the premium for d1 = USD 50 million.
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Pure risk premiums under deductibles
Pareto distribution of loss
The Pareto distribution is defined by the formula:
α
λ
,
F (t) = 1 −
λ+t
where t, α, λ > 0. The expectation of the Pareto distribution exist
only for α > 1.
As the example we consider the PCS data. The analysis showed that
the catastrophe-linked losses can be well modelled by the Pareto
distribution with parameters α = 2.3872 and λ = 3.0320 · 108 .
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Figure 9: The premium under the franchise deductible (thick line) and fixed
amount deductible (thin line).
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STFded09.xpl
Pure risk premiums under deductibles
Figure 10: The thick solid line - for c = 0.2 and m1 = 10 million, the solid
- for c = 0.4 and m1 = 10 million, the dotted - for c = 0.2 and m1 = 50
million, and the dashed - for c = 0.4 and m1 = 50 million.
STFded10.xpl
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Figure 11: The thick line represents the premium for d1 = USD 10 million
and the thin line the premium for d1 = USD 50 million.
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STFded11.xpl
Pure risk premiums under deductibles
Burr distribution of loss
Experience has shown that the Pareto formula is often an
appropriate model for the claim size distribution, particularly where
exceptionally large claims may occur. However, there is sometimes a
need to find heavy tailed distributions which offer greater flexibility
than the Pareto law. Such flexibility is provided by the Burr
distribution which distribution function is given by
α
λ
F (t) = 1 −
,
τ
λ+t
where t, α, λ, τ > 0. Its mean exists only for ατ > 1.
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For the Burr distribution with ατ > 1 the following formulae hold:
(a) franchise deductible premium
1
τ
PF D(a)
τ
λ Γ (α − 1/τ ) Γ (1 + 1/τ )
1
1
a
1 − B 1 + ,α − ,
,
=
τ
Γ(α)
τ
τ λ+a
(b) fixed amount deductible premium
1
τ
PF AD(b)
=
∗
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λ Γ (α − 1/τ ) Γ (1 + 1/τ )
Γ(α)
α
τ
λ
1
1
b
1 − B 1 + ,α − ,
−
b
,
τ
τ
τ
τ λ+b
λ+b
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(c) proportional deductible premium
1
PP D(c)
λ τ Γ(α − 1/τ )Γ(1 + 1/τ )
,
= (1 − c)
Γ(α)
(d) limited proportional deductible premium
1
τ
λ Γ (α − 1/τ ) Γ (1 + 1/τ )
∗
Γ(α)
(
τ
τ
1
1
m1
1
1
(m1 /c)
1 − B 1 + ,α − ,
+ cB 1 + , α − ,
τ
τ λ + mτ1
τ
τ λ + (m1 /c)τ
)
τ
1
1
(m2 /c)
−cB 1 + , α − ,
τ
τ λ + (m2 /c)τ
α
α
α
λ
λ
λ
−m1
+
m
−
m
,
1
2
τ
τ
τ
λ + m1
λ + (m1 /c)
λ + (m2 /c)
PLP D(c,m1 ,m2 ) =
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(e) disappearing deductible premium
1
∗
−
λ τ Γ (α − 1/τ ) Γ (1 + 1/τ )
PDD(d1 ,d2 ) =
Γ(α)
n
o
τ
τ
d2 − d1 + d1 B 1 + 1/τ, α − 1/τ, d2 /(λ + d2 )
d2 − d1
o
n
τ
τ
d2 B 1 + 1/τ, α − 1/τ, d1 /(λ + d1 )
d2 − d1
1
+
λ τ Γ (α − 1/τ ) Γ (1 + 1/τ ) d2 d1
Γ(α)
d2 − d1
λ
λ + dτ2
α
−
λ
λ + dτ1
where the functions Γ(·) and B(·, ·, ·) are defined as:
R ∞ a−1 −y
Γ(a+b) R x a−1
Γ(a) = 0 y
e dy and B(a, b, x) = Γ(a)Γ(b) 0 y
(1 − y)b−1 dy.
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α ,
Pure risk premiums under deductibles
Weibull distribution of loss
Another frequently used analytic claim size distribution is the
Weibull distribution which is defined by
α t
F (t) = 1 − exp −
,
β
where t, α, β > 0. For the Weibull distribution the following
formulae hold:
(a) franchise deductible premium
α 1
a
1
1−Γ 1+ ,
,
PF D(a) = βΓ 1 +
α
α β
(b) fixed amount deductible premium
α α 1
b
1
b
PF AD(b) = βΓ 1 +
1−Γ 1+ ,
− b exp −
,
α
α β
β
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(c) proportional deductible premium
PP D(c) = (1 − c)βΓ 1 +
1
α
,
(d) limited proportional deductible premium
α 1 m1
1
1−Γ 1+ ,
PLP D(c,m1 ,m2 ) = βΓ 1 +
α
α
β
α α 1
1 m1
1 m2
+ βΓ 1 +
c Γ 1+ ,
−Γ 1+ ,
α
α
cβ
α
β
α α α m1
m1
m2
− m1 exp −
+ m1 exp −
− m2 exp −
,
β
cβ
cβ
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(e) disappearing deductible premium
"
α 1
βΓ 1 + α
1 d2
d2 − d1 + d1 Γ 1 + ,
PDD(d1 ,d2 ) =
d2 − d1
α
β
α #
α α 1
d1
d1 d2
d2
d1
−d2 Γ 1 + ,
+
exp −
− exp −
,
α cβ
d2 − d1
β
β
where the function Γ(·, ·) is defined as
Z x
1
y a−1 e−y dy.
Γ(a, x) =
Γ(a) 0
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Gamma distribution of loss
The gamma distribution given by
Z t
F (t) = F (t, α, β) =
0
y
1
α−1 − β
y
e dy,
α
Γ(α)β
for t, α, β > 0 does not have these drawbacks. For the gamma
distribution following formulae hold:
(a) franchise deductible premium
PF D(a) = αβ {1 − F (a, α + 1, β)} ,
(b) fixed amount deductible premium
PF AD(b) = αβ {1 − F (b, α + 1, β)} − b {1 − F (b, α, β)} ,
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(c) proportional deductible premium
PP D(c) = (1 − c)αβ,
(d) limited proportional deductible premium
PLP D(c,m1 ,m2 ) = αβ {1 − F (m1 , α + 1, β}
n m
m
o
1
2
, α + 1, β − F
, α + 1, β
+ cαβ F
c
c
n
m
o
n
m
o
1
2
+ m1 F (m1 , α, β) − F
, α, β
− m2 1 − F
, α, β ,
c
c
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(e) disappearing deductible premium
PDD(d1 ,d2 )
αβ d2 {1 − F (d1 , α + 1, β)}
=
d2 − d1
−d1 {1 − F (d2 , α + 1, β)} +
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d1 d2
{F (d1 , α, β) − F (d2 , α, β)} .
d2 − d1