Optimal pricing of

a heterogeneous portfolio

for a given risk level

Yaniv Zaks1,2, Esther Frostig2 and Benny Levikson2

1Department of Mathematics, Bar-Ilan University, Israel2Department of Statistics, University of Haifa, Israel

In Memory of Prof. Benny Levikson

IME meeting, Piraeus, Greece, July 2007

2

The motivation

To determine the premiums based on the risk of ruin

according to a demand function.

In other words:

Find premiums based on the risk that the amount of

the insurer’s payments will exceed the amount of the

collected premiums where the population is not fixed.

3

The heterogeneous portfolio with a fixed populationConsider heterogeneous portfolio composed of k statistically risk

classes. Class j contains nj identically distributed risks, ,

each one is distributed with mean µj and variance σ2j j=1,…,k .

Let ,1

k

jj

n n

,1

jn

j j hh

S X

1

k

jj

S S

1

k

j jj

E S n

2 2

1

k

j jj

Var S n

,1 ,,jj j nX X

πj is the premium in class j .

4

21 1-

1 1 1

k k k

j j j j j jj j j

n q n q n

q1-α is the 1-α percentile of the distribution of .

DefinitionAn insurance company is revealed to a risk of size α if

1

n

i ii

P n S

S

5

An insurance company determines the premiums so as to minimize the

sum of the expected error in each class, corresponding to a risk .

Primal : Find the vector of premiums that minimizes the sum of the

expected distance functions under the constraint that the

probability for insolvency is less then a predetermined value α.

Dual : Find a vector of premiums that minimizes the insolvency

probability under the constraint that expected distance function is

below a predetermined value

6

The Primal problemThe Dual problem

21

1

1min

. .

k

i i iii

k

j jj

E S nr

s t P n S

1i

i ii j

rq

n r

1

2

1

min

1. .

k

j jj

k

i i iii

P n S

s t E S n Ar

i i iA

rr

2 2

1 1

where 1 , for a given 0 .k k

i i i i

i ii i

n nA t A t t

r r

7

The heterogeneous portfolio with dynamic population

Consider heterogeneous portfolio composed of k statistically risk classes.

Let π=(π1 ,…, πk) be a given premium vector.

The number of policyholders in class j is a function of the vector π .

nj = Dj(π) for j=1,…,k .

The function D(π) = (D1(π) ,…, Dk(π)) is called a demand function.

Let n = (n1 ,…, nk) be the population vector.

The premium in class j is a function of the vector n

πj = Pj(n) for j=1,…,k .

The function P(n) = (P1(n) ,…, Pk(n)) is called a pricing function.

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The existence of an equilibrium point

Starting with a premium vector π0=(π10 ,…, πk

0) .

The market reacts and sets the population vector as n1=D(π0) .

The insurer determines the premium vector according to the

pricing function π1=P(n1) .

Theorem

If P and D are bounded then the function Q(π) has an equilibrium point, i.e.

there exists a premium vector π* satisfying Q(π*) = π* .

π0

↓ n1=D(π0)

↓ π1=P(n1)

↓ n2=D(π1)

↓ π2=P(n2)

↓ . . .

Denote Q(π)=P(D(π))

Q(π1) = P(D(π1)) =P(n2) =π2

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The Primal problemThe Dual problem

1( ) ii i

i j

rP n q

n r

i i iA

P n rr

2

112 * *

1

k

j jii

ji i j j

q rq rD

r

2

* *1

k

jii

ji i j j

trD

r

A characterization of an equilibrium

pointA necessary and sufficient condition

Theorem: The premium vector π0=(π10 ,…, πk

0) is a fixed point of Q()

if, and only if, the following equality holds

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In the following examples we assume the following demand function:

.

ii i i i

i i

ii i i i

i i

ii i

i i

fm M

fD M

M

fm

m

1i

ir

f zr

2

*1 for each 1,...,i

i i z i kr

→

11

In the following examples we assume the following demand function:

1i

ir

f qr

2

*1 for each 1,...,i

i i q i kr

0.20

12

According to the demand function, the portfolio contains: 19836 policyholders who pays premium of 107.57, 444 policyholders who pays 1108.11, 113 policyholders who pays 3066.28.

1i

ir

f qr

2*

1i

i i qr

0.20

ClassE(X)n

1105107.5719835.44

210001108.13443.90

327003066.36112.97

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Thank You For

Your Attention