Etude des marchés d'assurance non-vie à l'aide d'équilibres de ...

tel-00703797, version 2 - 7 Jun 2012
2.2. A one-period model
where the sum is taken over the set {1, . . . , I} and fj is a price sensitivity function. In the
following, we consider two types of price functions
xj
¯fj(xj, xl) = µj + αj
xl
and ˜ fj(xj, xl) = ˜µj + ˜αj(xj − xl).
The first function ¯ fj assumes a price sensitivity with the ratio of the proposed premium xj and
competitor premium xl, whereas ˜ fj works with the premium difference xj − xl. Parameters
µj, αj represent a base lapse level and price sensitivity. We assume that insurance products
display positive price-elasiticity of demand αj > 0. One can check that
k lgk j (x) = 1.
The above expression can be rewritten as
lg k j (x) = lg j
j (x)
fj(xj,xk)
δjk + (1 − δjk)e ,
with δij denoting the Kronecker product. It is difficult to derive general properties of the
distribution of a sum of binomial variables with different probabilities, except when the size
parameters nj are reasonably large, in which case the normal approximation is appropriate.
With this insurer choice probability, the expected portfolio size of insurer j reduces to
ˆNj(x) = nj × lg j
j
(x) +
where nj denotes the last year portfolio size of insurer j.
2.2.2 Loss model
l=j
nl × lg j
l (x),
Let Yi be the aggregate loss of policy i during the coverage period. We assume no adverse
selection among insured of any insurers, i.e. Yi are independent and identically distributed
(i.i.d.) random variables, ∀i = 1, . . . , n. As already mentioned, we focus on short-tail business.
Thus, we assume a simple frequency – average severity loss model
Mi
Yi =
l=1
where the claim number Mi is independent from the claim severities Zi,l. Therefore, the
aggregate claim amount for insurer j is
Sj(x) =
Nj(x)
i=1
Yi =
Zi,l,
Nj(x)
i=1
Mi
Zi,l,
where Nj(x) is the portfolio size of insurer j given the price vector x. We consider two main
cases of the loss models: (i) Poisson-lognormal model: Mi i.i.d.
∼ P(λ) and Zi,l i.i.d.
∼ LN (µ1, σ 2 1 ),
(ii) negative binomial-lognormal model: Mi i.i.d.
∼ N B(r, p) and Zi,l i.i.d.
∼ LN (µ2, σ2 2 ). We choose
a different parameter set for the claim severity distribution, because if we want a significant
difference between the two loss models, changing only the claim number distribution does not
reveal sufficient. These two instances of the frequency-average severity model are such the
aggregate claim amount Sj(x) = Nj(x) i=1 Yi is still a compound distribution of the same kind,
since Yi are assumed i.i.d. random variables.
Hence, the insurer aggreggate claim amount Sj(x) is a compound distribution Mj(x)
l=1
Zl
such that the claim number Mj and claim severity Zl follow
l=1
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