Stochastic Models in Insurance

Stochastic Models in Insurance
Begleitmaterial zur Vorlesung von
o. Univ.Prof. Dr. Georg Pflug
Zeichnungen:
Gerald Kamhuber
Hinweise:
Dies ist nur ein Begleitmaterial und ersetzt nicht den Besuch der Vorlesung.
Dieses Werk ist urheberrechtlich geschützt. Jede Vervielfältigung ist verboten.
Institut für Statistik und Decision Support Systeme
Universitätsstraße 5/9, 1010 Wien
1
1
1.1
Life insurance
Life time distribution
Let L be a random variable describing the lifetime. Let G be the distribution function of
L, i.e.
G(u) = P{L ≤ u}
and let g be the pertaining density.
The residual lifetime distribution given that time t is survived is
P{L ≤ t + u|L > t} =
P{t < L ≤ t + u}
G(t + u) − G(t)
=
.
P{L > t}
1 − G(t)
The density of the residual lifetime is
∂ G(t + u) − G(t)
g(t + u)
=
.
∂u
1 − G(t)
1 − G(t)
The residual lifetime density at u = 0 is called the hazard function.
h(u) =
1.2
g(t)
.
1 − G(t)
Life insurance contracts
A life insurance policy consists of obligation of the insurance company to pay a predetermined amount of claims (sum insured), if a defined event (death, survival, disability etc.)
occurs in policy period against of payment of predetermined premium by the policyholder
(insured person). The premium is calculated by means of equivalence principle: at the
time of policy issue, the expected value of discounted future benefits payable by the insurer (plus administrative costs) equals the expected value of discounted future premiums
payable by the customer. The basis of premium calculation are (technical) interest rate,
probability distributions and cost shares. For simplicity, we will neglect the costs here.
We will use the following notations:
r
1+r
1
ν = 1+r
n
L̃x
Lx
interest rate,
interest factor,
discount factor,
integer duration of years of a contract,
the residual life time of a person aged x,
the number of completed years of a person aged x, i.e. Lx = bL̃x c
the integer part of L̃x .
2
An annuity is a sequence of payments of limited duration n. The present value of an
annuity with n annual payments of 1 starting at time 0 is
än| =
n−1
X
νj =
j=0
1 − νn
.
1−ν
The lifetime of a newborn person is L̃0 . Of course, the lifetime distributions are different
for males and females. L̃0 is a random variable with distribution function F and range
[0, ω], where ω is the end of the mortality table. We assume that F does not have jumps.
However in practical applications, we use the integer rounded version L0 = bL̃0 c. The
future lifetime of a person aged x is a random variable L̃x with probability distribution
function Fx , where
F (x + t) − F (x)
Fx (t) =
.
1 − F (x)
Again, we will typically use the rounded version Lx = bL̃x c for integer x. In a precise
formulation, Lx is the number of completed years lived after the x-th birthday.
We introduce the basic probabilities
px = 1 px = P{Lx > 0} = P{L̃x ≥ 1} = 1 − Fx (1)
(the one-year survival probability) and
qx = 1 qx P{Lx = 0} = P{0 ≤ L̃x < 1} = 1 − px = Fx (1) =
F (x + 1) − F (x)
1 − F (x)
(the one-year death probability, also called the hazard). Set pω = 0 and qω = 1. Using
these probabilities, we may form the derived probabilities
k px
= P{Lx ≥ k} = px · px+1 · · · px+k−1 .
The values of qx – of course for females and males separately – can be found in mortality
tables issued by statistical bureaus. If one assumes that these values do not change over
time, then the distribution of Lx can be calculated (however, a more refined model works
with estimations of cohort dependent mortalities):
k rx
:= P{Lx = k} = P{Lx ≥ k} − P{Lx ≥ k + 1} =
k px
− k px · px+k =
k px
· qx+k
denotes the probability that the person aged x will survive k years but not k + 1 years,
for 0 ≤ k ≤ ω − x.
3
1.2.1
Mortality tables
A typical mortality table looks like this
Männliches Geschlecht
Genaues
Alter
(am x-ten
Geburtstag)
in Jahren
x bis x +1
x
0
1
2
3
4
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
Sterbewahrscheinlichkeit
im Altersintervall
Überlebende
im
Alter x
q(x)
l(x)
0.0084686
0.0005477
0.0004539
0.0003739
0.0003086
0.0002571
0.0001495
0.0003961
0.0014673
0.0011736
0.0012464
0.0016246
0.0024129
0.0039948
0.0059701
0.0094913
0.0154529
0.0240385
0.0365059
0.0571156
0.0939633
0.1515216
0.2213000
0.3222411
1.0000000
Von den Überlebenden
im Alter x
bis x+1
noch zu durchlebende Jahre
Lebens-
L(x)
100000
99153.14
99098.83
99053.85
99016.81
98986.26
98888.87
98796.20
98319.38
97676.54
97104.01
96430.81
95545.29
94108.64
91898.74
88646.32
83585.58
76077.11
65781.37
52644.22
36536.51
19738.28
7419.76
1645.21
166.86
99291
99125.98
99076.34
99035.33
99001.53
98973.53
98881.48
98776.63
98247.25
97619.22
97043.49
96352.48
95430.02
93920.67
91624.42
88225.64
82939.76
75162.72
64580.67
51140.82
34819.96
18242.89
6598.76
1380.13
282.00
7248084.87
7148793.87
7049667.89
6950591.54
6851556.21
6752554.67
6257892.26
5763653.26
5270574.75
4780652.96
4293695.13
3809785.41
3329724.76
2855303.20
2389949.26
1938039.53
1506581.40
1
Weibliches Geschlecht
Genaues
Alter
(am x-ten
Geburtstag)
in Jahren
x bis x +1
x
0
1
2
3
4
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
Sterbewahrscheinlichkeit
im Altersintervall
q(x)
0.0067104
0.0005155
0.0004117
0.0003175
0.0002377
0.0001803
0.0000992
0.0002070
0.0004047
0.0003383
0.0004719
0.0007127
0.0012084
0.0020112
0.0029238
0.0043607
0.0065642
0.0107563
0.0186010
0.0334821
0.0628663
0.1195473
0.1935492
0.2935812
1.0000000
Überlebende
im
Alter x
l(x)
Von den Überlebenden
im Alter x
bis x+1
noch zu durchlebende Jahre
Lebens-
L(x)
100000
99328.96
99277.75
99236.88
99205.37
99181.79
99117.57
99060.42
98902.43
98718.26
98529.64
98254.19
97819.01
97077.36
95942.14
94301.41
91891.61
88267.86
82461.04
73116.37
58483.31
37922.64
17005.42
4552.07
569.32
99443
99303.35
99257.31
99221.12
99193.58
99172.85
99112.65
99050.17
98882.41
98701.56
98506.40
98219.17
97759.91
96979.74
95801.89
94095.80
91590.02
87793.14
81694.11
71892.33
56644.99
35655.86
15359.73
3883.87
1019.90
7904388.82
7804945.82
7705642.47
7606385.15
7507164.02
7407970.43
6912241.17
6416784.24
5921829.17
5427792.81
4934647.16
4442642.38
3952368.64
3464972.55
2982260.38
2506401.85
2040553.76
1589506.21
1
For understanding the process of estimation of the death and survival probabilities in a
population, we use the ”Becker’sche Darstellung”.
4
Becker'sche Darstellung
II.
I.
III.
Zeit
1 Jahr
Stichtag
Parametric models for the hazard function of the human lifetime were proposed by
(1) Gompertz (1825)
h(x) = aebx
(2) Mackeham (1860)
h(x) = aebx + c
(3) Perks (1932)
h(x) =
abx
+c
1 + αbx
The Gompertz-Mackeham distribution has density
a
f (x, a, b, c) = (aebu + c) exp(− (ebu − 1) − cu).
b
1.3
Simple life insurance contracts
For life insurance contracts, we review some typical structures. As a common notation,
we use Z b for the discounted benefits and Z c for the discounted contributions. Both are
random variables. The equivalence principle states that
E(Z c ) = E(Z b ) + costs.
To simplify, the costs are neglected. Moreover, again for simplification, we assume that
all contracts start at the x-th birthday of the insured person.
5
The benefit of a simple life insurance contract consists in at most one payment, the sum
insured. The basic types of life insurance are (a) term insurance and (a) pure endowment.
(a) The term insurance (Ablebensversicherung) of duration n consists of payment of 1 unit
at the end of the year of death, if it falls within the contract period. The sum insured is
fixed, only the time of payment Lx is random. Its net present value (NPV) is
Z b = ν Lx +1 1l{Lx ≤n−1} .
(1)
The distribution of Z b is:
P{Z b = ν k+1 } = P{Lx = k} =
k rx
= k px · qx+k
for k ≤ n − 1. Otherwise the value of Z b is zero.
Therefore the expected net present value (ENPV) is
|n Ax
b
= E(Z ) = E(ν
Lx +1
1l{Lx ≤n−1} ) =
n−1
X
ν k+1 · k rx .
k=0
In case of n = ∞ we have a so-called ”whole life insurance”, the net single premium is
denoted by Ax and given by
Ax = E(ν Lx +1 ) =
∞
X
ν k+1 · k rx .
k=0
The ENPV is also called ”net single premium”.
For a life insurance with benefits cj varying from year to year we have
Z b = cLx +1 · ν Lx +1 1l{Lx ≤n−1} ,
and the ENPV equals to
b
E(Z ) =
∞
X
ck+1 · ν k+1 · k rx .
k=0
For instance, the standard increasing term insurance has ck = k for k ≤ n and the present
value of this insurance is
Z b = (Lx + 1) · ν Lx +1 1l{Lx ≤n−1} .,
the ENPV is denoted by (IA)x,n| and is given by
(IA)x,n| =
n−1
X
(k + 1) · ν k+1 · k rx .
k=0
6
(b) Pure endowment (Erlebensversicherung) of duration n consists of payment of 1 unit
only if the insured is alive at the end the n-th year:
Z b = ν n 1l{Lx >n−1} ).
The ENPV is denoted by
n Ex
and is given by
n Ex
= νn ·
n px .
(c) The most common types of insurance are endowments (Er-und Ablebensversicherung).
An endowment consists of payment of 1 unit at the end of the year of death, if this occurs
within the first n years, otherwise at the end of the n-th year:
Z b = ν min(Lx +1,
n)
.
This contract is the combination of a term insurance and a pure endowment. As a
consequence, the net single premium for this insurance is Ax,n| = |n Ax + n Ex .
Periodic payments of premiums can be considered as a series of payments which are made
while the insured person is alive. For periodic premium the duration and the frequency
of premium payments must by specified in addition to the premium amount. The regular
premium can by understand as a life annuity payable in advance. The discounted net
present value of yearly payments of 1 is
c
Z =
n−1
X
ν k 1l{Lx >k−1} .
k=0
The expected net present value of these payments is
c
äx,n| = E(Z ) =
n−1
X
ν k · k px .
k=0
Based on this expression and the equivalence principle, the regular premium payable at
date of policy issue is
1
• Px:n|
=
|n Ax
äx,n|
• Px:n|1 =
n Ex
äx,n|
for term insurance,
for pure endowment,
1
• Px:n| = Px:n|
+ Px:n|1 =
Ax,n|
äx,n|
for endowment.
7
1.4
Spectra of life insurance
A life insurance contract is characterized by a triple
hC, S, Di,
where
C = (C1x , C2x , . . . , Cnx ), S = (S1x , S2x , . . . , Snx ), D = (D1x , D2x , . . . , Dnx )
are spectrum of premiums (contributions), spectrum benefits on survival and spectrum
x
of death benefits respectively. The net premium Cm
will be collected at the beginning of
x
year m, if the insured person survived this date, the benefit Sm
will be paid to insured
person at the end of year m ≤ n, if insured person is alive at this date and the payment
x
Dm
is due at the end of year m, if the insured person dies within year m ≤ n.
We have that
n−1
X
Z b = Z0b =
x
x
ν k+1 [Sk+1
1l{Lx >k} + Dk+1
1l{Lx =k} ]
k=0
c
Z =
Z0c
=
n−1
X
x
ν k Ck+1
1l{Lx ≥k} .
k=0
After m years, these two random variables discounted to the beginning of year m, given
that the insured person survived the year m − 1 are
b
Zm
=
n−1
X
x
x
ν k−m+1 [Sk+1
1l{Lx+m >k−m} + Dk+1
1l{Lx+m =k−m} ],
k=m
c
Zm
=
n−1
X
x
ν k−m Ck+1
1l{Lx+m ≥k−m} .
k=m
Calculating the expected values, one gets for m = 0, . . . , n − 1
b
E(Zm
)
=
=
n−1
X
x
ν k−m+1 [Sk+1
k=m
n−m−1
X
k−m+1 px+m
x
ν j+1 [Sj+m+1
j+1 px+m
x
+ Dk+1
k−m px+m qx+m ]
x
+ Dj+m+1
j rx+m ],
j=0
and
c
E(Zm
)
=
=
n−1
X
x
ν k−m Ck+1
·
k=m
n−m−1
X
k−m px+m
x
· j px+m .
ν j Cj+m+1
j=0
8
According to equivalence principle
E(Z0b ) = E(Z0c ),
i.e. at the date of policy issue, the expected present value of future premiums equals the
expected value of future benefit payments, making the expected loss of the insurer zero.
This equivalence between future payments and future benefits does not, in general, exist
at a later time. We define the needed reserve m Vx as the difference between the present
value of future benefits payments and the present value of future premium payments
calculated at the beginning of year m, given that the customer has survived this date, i.e.
m Vx
b
c
= E(Zm
) − E(Zm
).
A recursion formula for the reserve can be established (we omit the superscripts x here)
(
m−1 Vx
+ Cm )(1 + r) = px+m−1 ·
m Vx
+ px+m−1 · Sm + qx+m−1 · Dm
(2)
and this formula may be rewritten as
m Vx
=
1+r
(
px+m−1
m−1 Vx
+ C m ) − Sm −
qx+m−1
Dm .
px+m−1
The initial condition for this recursion is
0 Vx
= 0.
Alternatively, the formula (2) may be written as
Cm = ν ·
m Vx
−
m−1 Vx
+ ν · Sm + ν · qx+m−1 (Dm −
m Vx
− Sm ).
x
S
It makes sense to split the contributions Cm
into a savings part Cm
and a risk premium
R
Cm . The distinction is relevant for tax and regulatory reasons.
The risk premium is this part of the contribution, which is proportional to the death risk
qx+m−1 of the year under consideration: If this risk was zero (which is unrealistic, but just
an assumption for explanation), all of the contribution would go for covering the risks of
later years, i.e. everything would be saved. Thus
R
Cm
= ν · qx+m−1 (Dm −
m Vx
− Sm )
and
R
S
=ν·
= Cm − Cm
Cm
m Vx
9
−
m−1 Vx
+ νSm .
2
2.1
Non-life insurance and ruin probabilities
The Poisson process
In insurance mathematics, the most used stochastic process for describing random event
times is the Poisson process. It is defined as
Nt = max{n : Y1 + · · · + Yn ≤ t}
where Yi is a sequence of i.i.d. Exponential(λ) distributions. It is the renewal process for
exponential lifetimes.
We have that
E(Nt ) = λt
P {Nt ≥ k} = P {Y1 + · · · + Yk ≤ t}
Z t k k−1
λ u
e−λu du
=
(k
−
1)!
0
k
X
(λt)j −λt
=
e
j!
j=0
since Y1 + · · · + Yk is Erlang(k, λ) distributed with density
λk uk−1 −λu
e .
(k−1)!
Therefore
(λt)j −λt
P {Nt = k} = P {Nt ≥ k} − P {Nt ≥ (k + 1)} =
e ,
j!
which is a Poisson distribution. This justifies the name of the process.
If the damage sizes per event are Xi , i.i.d sequence, then the total claim sum St per period
t is a random sum
Nt
X
St =
Xi .
i=1
If the sequence (Xi ) is independent of the process (Nt ), we have that
Nt
X
E(St ) = E(
Xi ) = E(Nt ) · E(X1 ).
i=1
Introducing the generating function of a nonnegative random variable GX (u) = E(uX ),
one sees that
GNt (u) = eλt(u−1)
10
and
GSt (u) = GNt (GX (u)) = exp(λt(GY (u) − 1)).
2.2
The risk capital process and ruin probabilities
Let Zi identically and independently distributes random damage sizes, where Zi has distribution function
F (u) = P (Zi ≤ u)
and expectation µ. Let Nt be the (Poisson) event process with
E(Nt ) = λ · t.
Let c denote the continuous premium inflow per time and M the initial risk capital. Then
the risk capital process is M + V (t) with
V (t) = c · t −
Nt
X
Zk .
k=1
The total ruin probability is given by
Ψ(M ) = P {M + V (t) < 0 for some t > 0}.
We have that E(V (t)) = c · t − λtE(Zk ) = (c − λµ) · t.
We assume that c > λ · µ. Let ρ =
c−λµ
λµ
=
c
λµ
− 1 > 0 be the premium loading.
Z1
Z2
Z3
{
M
T1
T2
11
T3
T4
Setting now
Xi = −c(τi − τi−1 + Zi )
and
Sn =
n
X
τ0 = 0
Xi
τ =1
one sees that
Ψ(M ) = P {Sn > M for some n ≥ 0}.
Sometimes it is easier to work with the survival probability Φ(M ) = 1 − Ψ(M ).
2.2.1
The Lundberg exponent
An explicit expression for
Ψ(M ) = P {Sn > M for some n ≥ 0}.
is only available in exceptional cases. However, some limiting result is available.
Definition.
w = lim −
M →∞
1
log Ψ(M )
M
is called the Lundberg exponent of Xi .
It is equivalent to state that Ψ(M ) ∼ e−wM for large M .
Theorem. The Lundberg exponent w is the solution of the equation
E(exp(wXi )) = 1.
Proof:
Let
Yn = exp(−w(M − Sn )) n ≥ 1
Y0 = exp(−wM )
Then (Yn ) is a martingale, i.e.
E(Yn |X1 , ...., Xn − 1) = Yn−1 .
Let Un = M − Sn and let T = inf{n : Un > 0} be the stopping time at ruin. Notice that
ruin occurs if and only if T < ∞. By the martingale property
Y0 = e−wM = E(exp(−wUn∧T ))
= E(exp(−wUT )|T ≤ n) · P (T ≤ n) + E(exp(−wUT )|T > n) · P (T > n)
12
It might be shown that for n → ∞
E(exp(−wUT )|T > n) → 0
P (T > n) → P (T = ∞)
Therefore
e−wM = E(exp(−wUT )|T < ∞) · P (T < ∞)
i.e.
P (T < ∞) =
e−wM
E(exp(−wUT |T < ∞)
and for M → ∞
E(exp(−wUT )|T < ∞) → 1
leading to the assertion of the Theorem.
2.2.2
The integral equation approach
The ruin probability satisfies the integral equation
ZM
Ψ(M − v)g(v) dv
Ψ(M ) = 1 − G(M ) +
−∞
where G is the distribution and g is the density of Xi .
For the special case that Nt = P oisson(λ) and Zi ∼ Exponential( µ1 ), one gets
X ∼ Y1 − Y2
with Y1 ∼ Exponential( µ1 ) and Y2 ∼ Exponential( λc ).
The density of X is therefore
Z∞
g(s) =


u≥0
u≥s

1 − µ1 u λ − λ (u−s)
du
e
·e c
µ
c

Distinguishing the cases
(1) s < 0
λ +λs
e c
µc
Z∞
e− u(
0
1 λ
λ λ
1
+ ) du =
ecs 1
µ c
µc
+
µ
13
λ
λ
c
= ecs ·
λ
c + µλ
(2) s > 0
e
λ
s
c
x
c + µλ
Z∞
1
1
λ
1 x
λ
− e−u( µ + c ) |∞
( + ) e−µ( µ +λc) λµ = eλcs
s =
µ c
c + µλ
s
λ
ecs
λ
1
s
λ
λ
e−s( µ + c ) =
e− µ
c + µλ
c + µλ
one arrives at
(
g(s) =
λ
λ
es c
c+µλ
−s µs
λ
e
c+µλ
s<0
s>0
with distribution function
λ
G(t) =
·
c + µλ
Z0
s λc
e
λ
+
c + µλ
−∞
= −
Zb
c
λµ −s µs t
−
e |0
c + µλ c + µλ
s
e−s µ =
0
λµ − µt
c
λµ
λµ − µt
e +
+
=1−
e
c + µλ
c + µλ c + µλ
c + µλ
The solution of the integral equation in this special case is
Ψ(M ) =
ρM
1
λµ −M ( c−λµ
)
cµ
e− µ(1+ρ) =
·e
1+ρ
c
which can be seen by direct calculation:
ZM
1 − G(M ) +
Ψ(M − v)g(v) dv
−∞
λµ − Mµ
=
e
+
c + λµ
Z0
λµ −( c−λµ
λµ v λ
e cµ )(M −v) ·
e c dv +
c
c + λµ
−∞
ZM
λµ −( c−λµ
e cµ )(M −v)
c
0
λ
λµ −M c−λµ
λ
λµ − Mµ
λµ
cµ
e− µ d =
e
·
e
=
+
c + λµ
c + λµ
c + λµ c
Z0
λµ −M c−λµ
λµ
cµ
·
e
e dv +
c + λµ c
λ
µ
−∞
2 2
ZM
0
λ
λµ
λµ
λµ
e
e
e
+
−
· e−v c |M
0
c + λµ
(c + λµ) · c
(c + λµ)
c−λµ
c−λµ
c−λµ
λ
λµ − Mµ
λ 2 µ2
λµ
λµc
=
e
+
e−M ( cµ ) −
e−M ( cµ ) · e−M c +
e−M λµ
c + λµ
(c + λµ) · c
(c + λµ)
(c + λµ) · c
λµ(λµ + c) −M ( c−λµ
)
cµ
=
e
= Ψ(M )
c(c + λµ)
=
−M
µ
−M ( c−λµ
)
cµ
−M ( c−λµ
)
cµ
14
λ
e− µ dv
2.3
Premium calculation priciples
Let Y be the damage variable and let F (u) = P {Y ≤ u} be its distribution function. Let
g be a distortion function, i.e. a concave, monotonic function or [0, 1] satisfying g(0) = 0;
g(1) = 1. The distortion premium for Y is
Z∞
πg (Y ) =
g(1 − F (u)) du
0
This premium calculation principle was introduced by S. Wang. Notice that g(u) = u
leads to the expectation as premium. Frequently used distortion functions are
• The power distortion
g(u) = ur ,
0 < r < 1.
• The Wang distortion
g(u) = Φ(Φ−1 (u) + λ);
λ>0
where Φ is the distribution function of the standard normal.
• The function
g(u) = min(u/α, 1)
for 0 < α < 1.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 1. The power distortion function (r = 0.3, dotted) and the Wang distortion
function (λ = 1, dashed)
Setting F g (u) = 1 − g(1 − F (u)) one may equivalently say that πg is the expectation of
of F g .
The distortion premium may also be calculated as
15
πg (Y ) =
R∞
R∞
g(1 − F (u)) du = ug(1 − F (u))
0
R1
ug 0 (1 − F (u)) dF (u) =
0
R∞
0
−
R∞
udg(1 − F (u)) =
0
F −1 (p) · g 0 (1 − p) dp =
0
R1
F −1 (p)h(p) dp
0
where h(p) = g 0 (1 − p). Notice that h(p) is a monotonically increasing probably density,
since h(p) ≥ 0; h is monotonic, since g is concave, and
R1
h(p) dp =
R1
0
0
g | 0(1 − p) dp = −g(1 − p)|10 = 1
The previous examples give
• g(u) = ur
h(p) = r(1 − p)r−1
• g(u) = min( αu , 1)
h(p) =
1
α
· 1lp≥1−α
Thus we see that
R∞
0
min( 1−Fα(u) , 1) du =
1
α
R1
F −1 (p) dp =: U AV @Rα(Y )
1−α
Here U AV @Rα(Y ) is the upper average value at risk
U AV @Rα(Y ) =
1
α
R1
1−α
F −1 (p)dp =: −AV @Rα (−Y ) = min{b + α1 E([Y − b]+ }
with
AV @Rα(Y ) =
1
α
Rα
0
F −1 (p) dp = max{a − α1 E([Y − a]− ) : a ∈ R}
Moreover, we may show that
πg (Y ) = h(0) · E(Y ) +
R1
U AV @Rα(Y ) · k(α) dα
0
16
where k(α) is a probability density on [0, 1], namely k(α) = αh0 (1 − α).
Some insurances use functions H, which are ≥ 1 and calculate the premium according to
Z 1
πH (Y ) =
F −1 (p)H(p) dp.
0
If the damage is not fully covered, the calculation has to be modified:
By A(Y ) we denote the payment function of an insurance contract, and by Z = A(Y ) the
payment. Typical payment functions are:
1. The proportional insurance
A(y) = δ · y for a 0 < δ < 1
2. The deductible
A(y) = max(y − d, 0)
where d is the deductible.
3. The excess of loss insurance
A(y) = min(max(y − d, 0), m)
Here m is the maximal payment. We call d the attachment point and m + d the
exit point of the contract.
In all cases, the premium calculation follows the formula
Z 1
πH,A =
A(F −1 (p))H(p) dp.
0
References
[1] Gerber H. U. Life insurance. Third Edition. Springer, 1997.
[2] Haberman S. Advanced actuarial mathematics. Lecture Notes. London, 2001.
17