0 INTRODUCTION 1
Chapter 6
Risk models (2)
0 Introduction
• In this chapter we will look at some of the practical
applications of risk models.
• The contents of this chapter are
– Aggregate claim distributions under proportional
and excess of loss reinsurance
∗ Proportional reinsurance
∗ Excess of loss reinsurance
– The individual risk model
0 INTRODUCTION 2
– Parameter variability/uncertainty
∗ Variability in a heterogeneous portfolio
e.g. Si has a compound Poisson distribution
with parameters λi.
∗ Variability in a homogeneous portfolio
e.g. S has a compound Poisson distribution
with a unknown parameters λ regarded as a
random variable with a known distribution.
∗ Variability in claim numbers and claim amounts
and parameter uncertainty
1 AGGREGATE CLAIM DISTRIBUTIONS UNDER REINSURANCE 3
1 Aggregate claim distributions under reinsurance
1.1 Proportional reinsurance
• In proportional reinsurance the distribution of the
number of claims involving the reinsurer is the
same as the distribution of the number of claims
involving the insurer.
• For a retention level α(0 ≤ α ≤ 1), the individual
claim amounts for the insurer are distributed as
αXi; and for the reinsurer as (1 − α)Xi.
• The aggregate claims amounts are distributed as
αS and (1 − α)S respectively.
1 AGGREGATE CLAIM DISTRIBUTIONS UNDER REINSURANCE 4
• Under a proportional reinsurance arrangement where
the direct writer retains a proportion k, if the
gross amount of an individual claim is X, the net
amount paid by the direct insurer will be Y = kX.
So, the MGF of Y will be:
MY (t) = E(etY ) = E(etkX ) = E[e(kt)X ] = MX (kt)
If the number of claims has a Poisson(λ) distribu-
tion, then the MGF of the aggregate claim amount
is:
MSnet (t) = MN [log MY (t)]
t
= eλ(e −1)|t=log MY (t)
= exp{λ[MY (t) − 1]}
= exp{λ[MX (kt) − 1]}
1 AGGREGATE CLAIM DISTRIBUTIONS UNDER REINSURANCE 5
1.2 Excess of loss reinsurance
• Under individual excess of loss reinsurance with
retention level M , the amount that an insurer pays
on the i-th claim is Yi = min(Xi, M ).
• The amount that the reinsurer pays is
Zi = max(0, Xi − M ).
• The insurerfs aggregate claims net of reinsurance
can be represented as:
S I = Y1 + Y2 + . . . + YN
• The reinsurerfs aggregate claims as:
SR = Z1 + Z2 + . . . + ZN
1 AGGREGATE CLAIM DISTRIBUTIONS UNDER REINSURANCE 6
• If N ∼ P (λ), we have:
E(SI ) = λ E(Y )
Var(SI ) = λ E(Y 2)
skew(SI ) = λ E(Y 3)
E(SR ) = λ E(Z)
Var(SR ) = λ E(Z 2)
skew(SR ) = λ E(Z 3)
1 AGGREGATE CLAIM DISTRIBUTIONS UNDER REINSURANCE 7
• Example on pages 3-6
The annual aggregate claim amount from a risk
has a compound Poisson distribution with Poisson
parameter 10. Individual claim amounts are uni-
formly distributed on (0,2000). The insurer of this
risk has effected excess of loss reinsurance with re-
tention level 1,600. Calculate the mean, variance
and coefficient of skewness of both the insurer’s
and reinsurer’s aggregate claims under this rein-
surance arrangement.
• The variance of S, the aggregate claim amount be-
fore reinsurance is not true that Var(SI )+Var(SR ) =
Var(S) because SI and SR are it not independent.
1 AGGREGATE CLAIM DISTRIBUTIONS UNDER REINSURANCE 8
• If FX (M ) > 0, then there is a non-zero probability
that Zi takes the value 0. From a practical point
of view, this definition of SR is rather artificial.
• The reinsurer’s aggregate claims can also be rep-
resented by:
SR = W 1 + W 2 + . . . + W N R
where the random variable N R denotes the ac-
tual number of (non-zero) payments made by the
reinsurer.
1 AGGREGATE CLAIM DISTRIBUTIONS UNDER REINSURANCE 9
• The density function of Wi is:
fX (w + M )
fW (w) = w>0
1 − FX (M )
because
P (W M )
w+M
fX (x)
= dx
M 1 − FX (M )
FX (w + M ) − FX (M )
=
1 − FX (M )
• To specify the distribution for SR , the distribution
of N R is needed.
1 AGGREGATE CLAIM DISTRIBUTIONS UNDER REINSURANCE 10
• Define:
N R = I1 + I2 + . . . + IN
where N denotes the number of claims from the
risk.
• Ij is an indicator random variable which takes the
value 1 if the reinsurer makes a (non-zero) pay-
ment on the jth claim, and takes the value 0 oth-
erwise.
• Thus N R gives the number of payments made by
the reinsurer.
1 AGGREGATE CLAIM DISTRIBUTIONS UNDER REINSURANCE 11
• Since Ij takes the value 1 only if Xj > M , then
P (Ij = 1) = P (Xj > M ) = π
P (Ij = 0) = 1 − π
• Ij has a B(1, π) distribution.
• Ij has MGF:
MI (t) = π exp(t) + 1 − π
• N R has MGF:
MN R(t) = MN (log MI (t))
1 AGGREGATE CLAIM DISTRIBUTIONS UNDER REINSURANCE 12
• If N has a Poisson(λ) distribution, and half of
the claims exceed the excess-of-loss retention limit,
then the MGF of N R is:
MN R(t) = MN [log MI (t)]
λ(et −1)
= e |t=log MI (t)
= exp{λ[MI (t) − 1]}
1 t 1
= exp{λ( e − )}
2 2
1
= exp{ λ(et − 1)}
2
which show that N R has a Poisson(1/2λ) distri-
bution.
• Question 6.5 on page 9.
2 THE INDIVIDUAL RISK MODEL 13
2 The individual risk model
• Under the individual risk model a portfolio con-
sisting of a fixed number of risks is considered.
• It will be assumed that:
– these risks are independent
– claim amounts from these risks are not (neces-
sarily) identically distributed random variables
– the number of risks does not change over the
period of insurance cover
2 THE INDIVIDUAL RISK MODEL 14
• The aggregate claims from this portfolio are de-
noted by S. So:
S = Y 1 + Y 2 + . . . + Yn
where Yj denotes the claim amount under the j-th
risk and n denotes the number of risks.
• This approach is referred to as an individual risk
model because it is considering the claims arising
from each individual policy.
2 THE INDIVIDUAL RISK MODEL 15
• For each risk, the following assumptions are made:
– the number of claims from the j-th risk, Nj , is
either 0 or 1
∗ This assumption is very restrictive. It means
that a maximum of one claim from each risk
is allowed for in the model. This includes
risks such as one-year term assurance, but
excludes many types of general insurance pol-
icy.
the probability of a claim from the j-th risk is
qj
• Under the assumptions, Nj ∼ b(1, qj ).
2 THE INDIVIDUAL RISK MODEL 16
• There are three important differences between this
model and the collective risk model:
1. The number of risks in the portfolio has been
specified. In the collective risk model, there
was no need to specify this number, nor to as-
sume that it remained fixed over the period of
insurance cover.
2. The number of claims from each individual risk
has been restricted. There was no such restric-
tion in the collective risk model.
3. It is assumed that individual risks are indepen-
dent. In the collective risk model it was indi-
vidual claim amounts that were independent.
2 THE INDIVIDUAL RISK MODEL 17
• If a claim occurs under the j-th risk, the claim
amount is denoted by the random variable Xj . Let
Fj (x), µj and σ 2 denote the distribution function,
mean and variance of Xj respectively.
• Thus, the distribution of Yj is compound binomial,
with individual claims distributed as Xj .
• It follows that:
E[Yj ] = qj µj
2
Var[Yj ] = qj σj + qj (1 − qj )µ2
j
2 THE INDIVIDUAL RISK MODEL 18
• S is the sum of n independent compound binomial
random variables. There is no general result about
the distribution of such a sum.
• This distribution can be stated only when the com-
pound binomial variables are identically distributed,
as well as independent.
• The mean of S is:
n n n
E[S] = E Yj = E[Yj ] = qj µj
j=1 j=1 j=1
2 THE INDIVIDUAL RISK MODEL 19
• If the individual risks are independent then the
variance of S is:
n n
Var[S] = Var Yj = Var[Yj ]
j=1 j=1
n
= 2
(qj σj + qj (1 − qj )µ2)
j
j=1
• In the special case when {Yj }n is a sequence
j=1
of identically distributed, as well as independent,
random variables, then:
E[S] = nqµ
Var[S] = nqσ 2 + nq(1 − q)µ2
2 THE INDIVIDUAL RISK MODEL 20
• Question 6.6 on page 12
Tho probability of a claim arising on any given
policy in a portfolio of 1,000 one year term as-
surance policies is 0.004, Claim amounts have a
Gamma(5, 0.002) distribution. Find the mean
and variance of the aggregate claim amount.
3 PARAMETER VARIABILITY / UNCERTAINTY 21
3 Parameter variability / uncertainty
3.1 Introduction
• So far risk models have been studied assuming
that the parameters, that is the moments and in
some cases even the distributions, of the number
of claims and of the amount of individual claims
are known with certainty.
• In general, these parameters would not be known
but would have to be estimated from appropriate
sets of data.
3 PARAMETER VARIABILITY / UNCERTAINTY 22
• In this section it will be seen how the models in-
troduced earlier can be extended to allow for pa-
rameter uncertainty / variability.
• This will be done by looking at a series of exam-
ples.
– Most, but not all, of these examples will con-
sider uncertainty in the claim number distri-
bution since this, rather than the individual
claim amount distribution, has received more
attention in the actuarial literature.
– All the examples will be based on claim num-
bers having a Poisson distribution.
3 PARAMETER VARIABILITY / UNCERTAINTY 23
3.2 Variability in a heterogeneous portfolio
• Consider a portfolio consisting of n independent
policies.
• The aggregate claims from the i-th policy are de-
noted by the random variable Si, where Si has
a compound Poisson distribution with parameters
λi; and F (x).
• Notice that, for simplicity, the distribution of in-
dividual claim amounts, F (x), is assumed to be
identical for all the policies.
• The distribution of individual claim amounts, F (x),
is assumed to be known but the values of the Pois-
son parameters, λis, are not known.
3 PARAMETER VARIABILITY / UNCERTAINTY 24
• In this subsection the λis are assumed to be inde-
pendent random variables with the same (known)
distribution.
• In other words {λi} is treated as a set of indepen-
dent and identically distributed random variables
with a known distribution.
• This means that if a policy is chosen at random
from the portfolio it is assumed that the Poisson
parameter for the policy is not known but that
probability statements can be made about it.
3 PARAMETER VARIABILITY / UNCERTAINTY 25
• It is important to understand that the Poisson pa-
rameter for a policy chosen from the portfolio is a
fixed number; the problem is that this number is
not known.
• Example on pages 14-15.
• Example on pages 15-16.
3 PARAMETER VARIABILITY / UNCERTAINTY 26
3.3 Variability in a homogeneous portfolio
• Suppose there is a portfolio of n policies.
• The aggregate claims from a single policy have a
compound Poisson distribution with parameters λ
and F (x).
• These parameters are the same for all policies in
the portfolio.
• It is assumed that the value of λ is not known,
possibly because it changes from year to year, but
that there is some indication of the probability
that λ will be in any given range of values.
3 PARAMETER VARIABILITY / UNCERTAINTY 27
• The uncertainty about the value of λ can be mod-
elled by regarding λ as a random variable (with a
known distribution).
• As before, it is assumed for simplicity that there is
no uncertainty about the moments or distribution
of the individual claim amounts, i.e. about F (x).
• Example on pages 17-18.
3 PARAMETER VARIABILITY / UNCERTAINTY 28
3.4 Variability in claim numbers and claim amounts and pa-
rameter uncertainty
• We now look at a complicated example involving
uncertainty over claim amounts as well as claim
numbers.
• Example on pages 19-21.
• Example on pages 21-24.