6 Empirical Bayesian credibility theory: Model 1 – the Bühlmann model
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4.6 Empirical Bayesian credibility theory: Model 1
177
Table 4.4. Structure of Bayesian and empirical Bayesian models
Prior
Conditional mean of Xi
Conditional variance of Xi
Normal/normal
Poisson/gamma
EBCT
θ ∼ N(μ, σ2 2 )
θ
σ1 2
λ ∼ gamma(α, β)
λ
λ
none
m(θ)
s2 (θ)
Since, given θ, the distribution of Xi does not depend on i, we can introduce
notation for the mean and variance of the conditional distribution using symbols which depend only on θ. We will adopt the symbols in general use, so we
define
m(θ) = E[Xi | θ]
and
s2 (θ) = Var[Xi | θ].
(4.29)
The pure premium/estimator for the risk is E[Xi | θ] = m(θ), and so we can
now state that our problem is to estimate m(θ), given data x = (x1 , x2 , . . . , xn ).
In Table 4.4 we give the earlier normal/normal and Poisson/gamma Bayesian
models alongside the new empirical structure. Using the table to compare the
normal/normal model (as in §4.5.2) and EBCT, we find that
E[m(θ)] in EBCT corresponds to E E[Xi | θ] = E[θ] = μ;
Var[m(θ)] in EBCT corresponds to Var E[Xi | θ] = Var[θ] = σ2 2 , the between
risk variance; and
E[s2 (θ)] in EBCT corresponds to E Var[Xi | θ] = E[σ1 2 ] = σ1 2 , the within
risk variance.
The credibility premium in the normal/normal model is Zx + (1 − Z)μ, where
Z=
n
n+
σ21
=
σ22
n
.
within risk variance
n+
between risk variance
A similar analogy between the Poisson/gamma model and EBCT can be
demonstrated, noting that Z can be expressed as
Z=
n
=
n+β
n
.
α/β
n+
α/β2
The above comparisons suggest that we tentatively adopt the formula
ZX + (1 − Z)E[m(θ)]
(4.30)
178
Model based pricing – setting premiums
for use in EBCT for the credibility premium/estimator, where the credibility
factor Z is given by
Z=
n
n
.
=
within risk variance
E[s2 (θ)]
n+
n+
between risk variance
Var[m(θ)]
(4.31)
Theorem 4.11 in the following shows that we can justify this adoption under
the criterion of using the “best” linear estimator of m(θ), that is the estimator
which is the linear function of the observations with minimum mean square
error.
As a preliminary to the proof of the theorem, we first establish some expectations that we will require. In doing so, we use E[Xg(θ) | θ] = g(θ)E[X | θ]
(see, for example, Sect. 7.7 of Grimmett and Stirzaker (2001)).
Lemma 4.10
With the set-up and notation of EBCT Model 1, we have
(i) E Xi = E X = E[m(θ)];
(ii) E Xi m(θ) = E Xm(θ) = E m2 (θ) ;
1
2
(iii) E X = E s2 (θ) + E m2 (θ) .
n
Proof For (i), note that the conditional expectation formula (1.3) implies that
E[Xi ] = E E[Xi | θ] = E[m(θ)], and it follows that
⎡ n
⎤
⎥⎥ 1
1 ⎢⎢⎢⎢
E X = E ⎢⎣
Xi ⎥⎥⎥⎦ = nE[m(θ)] = E[m(θ)].
n i=1
n
For (ii), using the conditional expectation formula (1.3) again, we have
E[Xi m(θ)] = E E[Xi m(θ) | θ] = E m(θ)E[Xi | θ] = E[m2 (θ)],
and hence E[Xm(θ)] = E[m2 (θ)].
For (iii), by conditional independence, note that, for i
j,
E[Xi X j ] = E E[Xi X j | θ] = E E[Xi | θ]E[X j | θ] ,
and so E[Xi X j ] = E[m2 (θ)]. Further, we have
E[Xi2 ] = E E[Xi2 | θ] = E Var[Xi | θ] + {E[Xi | θ]}2
= E[s2 (θ)] + E[m2 (θ)].
4.6 Empirical Bayesian credibility theory: Model 1
It follows that
EX
2
179
⎤
⎡ n
n
⎥⎥⎥
1 ⎢⎢⎢⎢
= 2 E ⎢⎢⎣
Xi
X j ⎥⎥⎦⎥
n
i=1
j=1
1
nE[Xi2 ] + n(n − 1)E[Xi X j ] for i j
n2
1
= 2 nE[s2 (θ)] + nE[m2 (θ)] + n(n − 1)E[m2 (θ)]
n
1
= E s2 (θ) + E m2 (θ) ,
n
=
as required. Alternatively, one can show Var[X | θ] = 1n s2 (θ), and use this to
2
find Var[X] and hence E[X ].
We now derive the credibility premium/estimator; that is, we verify that the
form of the optimum linear estimator a0 + a1 X1 + a2 X2 + · · · + an Xn of m(θ) is
indeed that of the credibility estimator tentatively adopted above.
Theorem 4.11 Let X1 , X2 , . . . , Xn be a sequence of random variables, each
of whose distribution depends on a parameter θ, and which, given θ, are iid,
with E[Xi | θ] = m(θ) and Var[Xi | θ] = s2 (θ). Let a0 , a1 , . . . , an be constants.
Then the estimator a0 + nj=1 a j X j of m(θ) for which
⎡⎧
⎢⎢⎢⎪
⎪
⎨
⎢⎪
E ⎢⎢⎢⎢⎪
m(θ) − a0 −
⎪
⎣⎪
⎩
n
j=1
⎫2 ⎤
⎪
⎥
⎪
⎪
⎬ ⎥⎥⎥⎥
⎥⎥
a j X j⎪
⎪
⎪
⎭ ⎥⎦
is minimised is given by
ZX + (1 − Z)E[m(θ)],
where
n
.
E[s2 (θ)]
n+
Var[m(θ)]
Z=
(4.32)
Proof The problem is symmetric in the Xi , and so a1 = a2 = · · · = an = a˜ ,
say, so that
n
a0 +
n
a j X j = ao + a˜
j=1
X j,
i=1
which means that the estimator is of the form a + bX. The problem is therefore
to find a and b such that
S = E m(θ) − a − bX
2
180
Model based pricing – setting premiums
is minimised. Taking the partial derivative of S with respect to a gives
∂S
= 0 ⇒ E[m(θ) − a − bX] = 0.
∂a
Using Lemma 4.10(i), we have a + bE[m(θ)] = E[m(θ)], and hence
a = (1 − b)E[m(θ)].
(4.33)
Taking the partial derivative of S with respect to b gives
∂S
= 0 ⇒ E[X{m(θ) − a − bX}] = 0,
∂b
so, using Lemma 4.10(i), (ii) and (iii), we have
aE[m(θ)] + b
1
E[s2 (θ)] + E[m2 (θ)] = E[m2 (θ)].
n
(4.34)
Solving (4.33) and (4.34) gives
b=
n
,
E[s2 (θ)]
n+
Var[m(θ)]
and so, denoting b by Z, we have that the best estimator is given by
ZX + (1 − Z)E[m(θ)],
and the result is proved.
The estimator involves three quantities, E[m(θ)], E[s2 (θ)] and Var[m(θ)],
which we have to estimate from collateral data. These quantities are sometimes
referred to as the three structural parameters.
We suppose now that the risk we are interested in is one of a collective of
a fixed number N of comparable risks. Our data now consist of values xi j of
random variables Xi j , where Xi j represents the aggregate amount (or number)
of claims for risk number i in year j, i = 1, 2, . . . , N, j = 1, 2, . . . , n , as in
Table 4.5. For convenience, we are making the (perhaps rash) assumption that
we have complete data – the same number of years data for each risk.
For each risk, say risk i, the distribution of each Xi j , j = 1, 2, . . . , n, depends
as before on the value of a risk parameter θi , which is fixed for that risk, but
unknown. We regard θi as a random variable with an unknown distribution
function. Each risk has its own risk parameter, which is fixed for that risk over
the period of years we are considering. It is very important to recognise that
the risks are heterogeneous – diﬀerent risks have diﬀerent risk parameters –
and we will set appropriate premiums which reflect this.
For each risk, say risk i, we make the following structural assumption.
4.6 Empirical Bayesian credibility theory: Model 1
181
Table 4.5. Collective of risks
Year
Risk
Assumption 1
1
2
·
·
N
1
2
·
·
n
X11
X21
·
·
XN1
X12
X22
·
·
XN2
·
·
·
·
·
·
·
·
·
·
X1n
X2n
·
·
XNn
Given θi , the Xi j , j = 1, 2, . . . , n, are iid.
This is exactly the same assumption we made earlier for a single risk, and gives
us the within risk structure we require.
We need to make an assumption to give us appropriate between risk
structure, and it is conveniently expressed as follows.
Assumption 2 For diﬀerent risks i, j (i
(θ j , X jk ), l, k = 1, 2, . . . , n, are iid.
j), the pairs of variables (θi , Xil ) and
It follows from assumption 2 that any two variables in diﬀerent rows in
Table 4.5 are iid. It also follows from assumption 2 that the risk parameters
θi , i = 1, 2, . . . , N, are iid. The “identicality” of the distributions is a formal
statement which firms up what we mean by the references to a collective of
“comparable risks”. We can think of the values of the risk parameters for the
diﬀerent risks as coming from some common underlying distribution.
For risk i define m(θi ) = E[Xi j | θi ] and s2 (θi ) = Var[Xi j | θi ]; these do not
depend on j (because of our assumptions). We identify m(θi ) and s2 (θi ) as the
mean and variance of the amounts (or numbers) of claims for risk i (row i in
Table 4.5).
Now, since θi , i = 1, 2, . . . , N, are identically distributed, it follows that none
of E[m(θi )], E[s2 (θi )] or Var[m(θi )] depend on i. So we write them as E[m(θ)],
E[s2 (θ)] and Var[m(θ)], respectively, bringing us back to the three structural
parameters we have already met:
E[m(θ)], the expected value (the average) of the risk means;
E[s2 (θ)], the expected value (the average) of the risk variances – it is is the
average variance within risks;
Var[m(θ)], the variance of the risk means – it is the variance between risks.
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Model based pricing – setting premiums
We now seek estimators of the three structural parameters. The first two, the
estimators of E[m(θ)] and E[s2 (θ)], are quite easy to identify, given the physical
nature of the parameters. The third, the estimator of Var[m(θ)], is less easy to
justify, but we can show that all three “usual” estimators (that is, the estimators
in everyday use) are unbiased for the parameters concerned.
We next define notation. Let
Xi =
1
n
n
Xi j
j=1
be the mean amount (or number) of claims for risk i over the n years for which
we have data (the mean for row i in Table 4.5), and let
X=
1
N
N
Xi =
i=1
1
Nn
N
n
Xi j
i=1 j=1
be the overall mean amount (or number) of claims for all years and all
risks involved. It is important to note our notation here: the mean amount
(or number) of claims for an individual risk (risk i) is now denoted X i ,
and X denotes the overall mean amount (or number) of claims for all risks
involved.
The usual estimators of the structural parameters are given in Table 4.6. The
estimator of E[m(θ)] is the overall mean of the claims data for all the risks
in the collective. The estimator of E[s2 (θ)] is the mean of the individual risk
sample variances. The estimator of Var[m(θ)] is the sample variance of the risk
means corrected for bias – the correction is a reduction given by the estimator of E[s2 (θ)] divided by n, the number of years data for each risk we have
available.
Table 4.6. Usual estimators of the structural parameters in
EBCT Model 1
Structural parameter
Estimator
E[m(θ)]
X
1
N
E[s2 (θ)]
Var[m(θ)]
N
i=1
1
N−1
1
n−1
n
(Xi j − X i )2
j=1
N
(X i − X)2 −
i=1
1
Nn
N
i=1
1
n−1
n
(Xi j − X i )2
j=1
4.6 Empirical Bayesian credibility theory: Model 1
183
It is easy to verify that X is unbiased for E[m(θ)]:
E[X] =
=
=
1
Nn
1
Nn
N
n
E[Xi j ] =
i=1 j=1
N
1
Nn
n
E[m(θi )] =
i=1 j=1
1
N
N
n
E[E(Xi j | θi )]
i=1 j=1
N
E[m(θi )]
i=1
1
NE[m(θ)] = E[m(θ)] .
N
The verification of the unbiasedness of the other two estimators is deferred to
Exercise 4.13 (some hints are given).
Comments on the calculation of the credibility premium for a risk
(i) The credibility factor
n
Z=
E[s2 (θ)]
n+
Var[m(θ)]
is the same for all risks in the collective – it only has to be calculated
once. Its value is between 0 and 1, and it is an increasing function of n.
(ii) A large value of E[s2 (θ)] implies large variability from year to year within
risks. This implies low credibility for the data from the particular risk
concerned, which implies a low value of the credibility factor Z.
(iii) A large value of Var[m(θ)] implies large variability between risks. This
implies that data from other risks are not very relevant/informative/reliable,
which implies high credibility for the data from the individual risk
concerned, and hence we have a high value of the credibility factor Z.
(iv) E[s2 (θ)] and Var[m(θ)] are positive parameters. While the estimator of the
former parameter is always positive, that of the latter can be negative. If
this occurs we take a pragmatic approach – we set Var[m(θ)] = 0; then
Z = 0 and the credibility estimate for risk i is just the overall mean X.
To sum up, the credibility premium for risk i in the collective is given by
Z × {mean for risk i} + (1 − Z) × {estimate of E[m(θ)]},
that is
ZX i + (1 − Z)X.
(4.35)
Example 4.12 Table 4.7 gives the aggregate claims in five successive years
from comparable policies covering the estate (buildings, vehicles, stock) of
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Model based pricing – setting premiums
Table 4.7. Aggregate claims for the
four risks in Example 4.12
Year
Risk
1
2
3
4
5
1
2
3
4
146
108
130
157
151
94
142
175
132
107
106
129
96
135
150
138
136
93
95
159
Table 4.8. Sample means and variances
for the four risks in Example 4.12
Risk
Risk mean
Risk variance
1
2
3
4
132.2
107.4
124.6
151.6
467.2
287.3
549.8
331.8
four medium-sized businesses. The claims are inflation-adjusted and are in
units of £1000.
We will calculate the credibility premium to be charged in the coming year
(year 6) for each risk, giving full details for risk 1. We are assuming that the
conditions which have held for the past five years justify our adoption of the
structural assumptions that underpin EBCT Model 1, and that these conditions
continue to hold in the coming year.
First we calculate the sample mean and variance for each risk. The values
are given in Table 4.8.
The estimate of E[m(θ)] is the mean of the four risk means (the overall
mean), namely x = (132.2 + 107.4 + 124.6 + 151.6)/4 = 128.95.
The estimate of E[s2 (θ)] is the mean of the four risk sample variances,
namely (467.2 + 287.3 + 549.8 + 331.8)/4 = 409.025.
The estimate of Var[m(θ)] is an adjusted version of the sample variance for
the four risk means, which is 335.637; the estimate is
335.637 − 409.025/5 = 253.83.
This gives the credibility factor as
Z=
5
= 0.756.
5 + 409.03/253.83
4.7 Empirical Bayesian credibility theory: Model 2
185
Table 4.9. Credibility premiums
for the four risks in Example 4.12
Risk
Premium (£)
1
2
3
4
131 410
112 650
125 660
146 080
The credibility premium for risk i is therefore given by
0.756xi + 0.244 × 128.95 = 0.756xi + 31.46.
The credibility premium for risk 1 is
0.756 × 132.2 + 31.46 = 131.40 (= £131 400).
The credibility premiums for all four risks (calculated in R with greater
accuracy throughout) are given in Table 4.9.
It is easy to check that the mean of the credibility premiums equals x, the
mean of the risk means (the overall mean of the claims data – the estimate
of E[m(θ)]). This will always be the case (see Exercise 4.14), and reflects the
fact that overall the insurer receives the appropriate total pure premium for the
group of risks. As emphasised earlier, the risks are heterogeneous – they have
diﬀerent risk parameters – and the credibility premiums for individual risks
vary, reflecting the claims experience of the risks. The higher the value of the
mean claims in the available history of the risk (xi ), the higher the credibility
premium for that risk. But over all the risks things average out as they should.
4.7 Empirical Bayesian credibility theory: Model 2 – the
Bühlmann–Straub model
This model was formulated by Bühlmann and Straub; see Bühlmann and
Straub (1970).
The model we have discussed in §4.6 (EBCT Model 1) clearly shows
similarities with a “pure” Bayesian approach and is a necessary and useful
introduction to “empirical” credibility methods. However, it involves rather
restrictive assumptions and is not very useful in practice.
EBCT Model 2 – the Bühlmann–Straub model – encompasses a major generalisation of Model 1 by allowing for changing levels of business (changing
risk volumes). It is easy to see why this is such an important and practical
186
Model based pricing – setting premiums
extension and improvement. The risk during one year may relate to cover for a
small business with four shops and three delivery vans on the road – the business may do well, expand, and next year have six shops and four vans on the
road. The increased estate (buildings, vans, stock) and general activity is not
taken account of by Model 1 but is taken account of by Model 2.
With this recognition of changing risk volumes, it is inappropriate now
to assume that, given the risk parameter, the claims variables are identically
distributed. The assumptions we do make for Model 2 are most conveniently
expressed in a manner which makes them less restrictive than was the case
for Model 1 – and these assumptions are made not about the claims variables
themselves, but about the variables representing claims per unit of risk volume.
So, let Y1 , Y2 , . . . , Yn represent the aggregate claims in n successive years
for a risk, and let P1 , P2 , . . . , Pn be corresponding risk volumes. These risk
volumes are known numbers (not random variables) and can be quantified in
various ways – for example, numbers of policies in a changing portfolio, numbers of shops in a chain, numbers of vehicles in a fleet, etc. A sensible general
measure which can be used – perhaps obvious once mentioned – is the annual
premium income the insurer has charged to cover the risk over recent years
(provided the premiums were set sensibly to reflect the risk).
We introduce Xi to represent the aggregate claims in year i scaled to take
account of the volume of business, that is
Xi = Yi /Pi ,
i = 1, 2, . . . , n,
(4.36)
so Xi is the aggregate claims per unit of risk volume in year i.
The basic structure of this model is that the distribution of each variable
Xi , i = 1, 2, . . . , n, depends on the value of a risk parameter θ, which is fixed
for that risk but unknown, and is regarded as a random variable with unknown
distribution function. It is not appropriate to assume that the Xi are identically
distributed, either conditionally (given θ), or unconditionally.
Assumptions
(1) Given θ, the Xi , i = 1, 2, . . . , n, are independent.
(2) E[Xi | θ] does not depend on i.
(3) Pi Var[Xi | θ] does not depend on i.
Under these assumptions we define
m(θ) = E[Xi |θ] and s2 (θ) = Pi Var[Xi |θ].
(4.37)
To motivate assumption (3), consider a risk which consists of a portfolio of
independent policies – suppose the number of policies in force in year i is Pi (a
known number). Suppose also that, for each policy, the aggregate claims in any
given year have mean m(θ) and variance s2 (θ), where θ is the risk parameter for