An insurance company classifies its policyholders into two groups: G (no claim i

Question

An insurance company classifies its policyholders into two groups: G (no claim in the last year) and A (made a claim or claims in the last year). In each year 20% of the G policyholders make a claim and therefore become A policyholders . As to the A policyholders, in each year 60% of the A policyholders make no claim and therefore become G policyholders.

Write xk for vector of the numbers of policyholders in each group at the start of year k. That is

xk =

number of policyholders in group G at the start of year k

number of policyholders in group A at the start of year k

(a) Fill in the matrix in the following formula:

xk+1 =

xk

(b) Suppose that at the start of the year 2013 the company had 110,000 policyholders, and that they were grouped as follows: 100,000 policyholders were in group G, and 10,000 policyholders were in group A.

How many of these 110,000 policyholders are expected to be in group G at the start of the year 2015? How many of the 110,000 policyholders are expected to be in group A at the start of year 2015? Assume that all of the 110,000 policyholders stay with the company until 2015.

You may use a calculator to perform the arithmetic operations.

Number in G =?

Number in A =?

xk =

number of policyholders in group G at the start of year k

number of policyholders in group A at the start of year k

(a) Fill in the matrix in the following formula:

xk+1 =

Explanation / Answer

a)
If you have n G-policy holders and m A-policy holders,
then the next year the numbeer of G-s will be 0.8*n + 0.6*m
and the number of A-s will be 0.2*n + 0.4*m

x^(k+1) =
[0.8 0.6] *x^k
[0.2 0.4]

b)
For b, just use the furmulas from a.
x^k = [100,000 10,000]
[0.8 0.6] *x^k = [86,000 24,000]
[0.2 0.4]

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