Download 1 Conditional Probability 2 Conditional Distributions 3 Conditional

1
Conditional Probability
Let A and B be two events such that P (B) > 0, then P (A|B) = P (A ∩ B)/P (B).
Bayes’ Theorem
Let A and B be two events such that P (B) > 0, then P (A|B) = P (B|A) · P (A)/P (B).
Theorem of total probability
Let A1 , A2 , · · · be a countable collection of mutually exclusive and exhaustive events, so that
Ai ∩ Aj = ∅ for i 6= j and ∪∞
i=1 Ai = Ω, then P (B) =
∞
X
P (B|Ai ) · P (Ai ).
i=1
2
Conditional Distributions
The conditional probability function of X given that Y = y is fX|Y (x|y) = fX,Y (x, y)/fY (y).
If X and Y are independent, then fX,Y (x, y) = fX (x)·fY (y) which implies fX|Y (x|y) = fX (x).
Rewrite fX,Y (x, y) = fZX|Y (x|y) · fY (y). Then fX (x) =
Application: fX (x) =
fX|Θ (x|θ) · fΘ (θ)dθ;
R
R
fX,Y (x, y)dy = fX|Y (x|y) · fY (y)dy.
FX (x) =
R
FX|Θ (x|θ) · fΘ (θ)dθ.
Note that fX|Y (x|y) · fY (y) = fX,Y (x, y) = fY |X (y|x) · fX (x), implying that
fX|Y (x|y) =
3
fY |X (y|x) · fX (x)
∝ fY |X (y|x) · fX (x).
fY (y)
Conditional Expectation
Let X be a discrete random variable such that x1 , x2 , . . . are the only values that X takes
on with positive probability. Define the conditional expectation of X given that event A has
occurred, denoted by E[X|A], as E[X|A] =
xi · P [X = x1 |A].
i=1
Z
Continuous case : E[X|Y = y] =
∞
X
x · fX|Y (x|y)dx
Let A1 , A2 , · · · be a countable collection of mutually exclusive and exhaustive events, and let
X be a discrete random variable for which E[X] exist. Then E[X] =
i=1
Z
Continuous case : E[X] =
∞
X
E[X|Y = y] · fY (y)dy
EY [EX (X|Y )] = E[X]
Z
E[h(X, Y )|Y = y] =
h(x, y) · fX|Y (x|y)dx
EY {EX [h(X, Y )|Y ]} = E[h(X, Y )]
V ar[X|Y ] = E{[X − E(X|Y )]2 |Y } = E[X 2 |Y ] − [E(X|Y )]2
E[V ar[X|Y ]] = E[E(X 2 |Y )] − E{[E(X|Y )]2 } = E[X 2 ] − E{[E(X|Y )]2 }
1
E[X|Ai ] · P [Ai ].
V ar[E(X|Y )] = E{[E(X|Y )]2 } − {E[E(X|Y )]}2 = E{[E(X|Y )]2 } − [E(X)]2
V ar[X] = E[V ar(X|Y )] + V ar[E(X|Y )]
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Nonparametric Unbiased Estimators
If X1 , X2 , . . . , Xn are independent but not necessarily identical with common mean µ =
E[Xj ] and common variance σ 2 = V ar[Xj ], then
n
1X
µ̂ = X̄ =
Xj is an unbiased estimator of µ, and
n j=1
n
1 X
σˆ2 =
(Xj − X̄)2 is an unbiased estimator of v
n − 1 j=1
5
Full and Partial Credibilities
The following Zs subject to Z ≤ 1; if Z ≥ 1 then a full credibility is assigned.
The standard for NO full credibility in terms of (λ0 = [yp /r]2 )
(1) (frequency case) the number of policies is (use observed number of policies for n)
µ
n ≤ nF = λ0
µ
⇒Z=
n
nF
µ
¶1/2
=
n
λ0
σX
θX
¶1/2
¶2
µ
X=N
= λ0
θ̂X X=N
=
σ̂X
µ
σN
θN
n
λ0
¶2
P oisson
=
¶1/2
θ̂N
σ̂N
λ0
λ
µ
P oisson
=
nλ̂
λ0
¶1/2
;
(2) (frequency case) the expected number of claims is (use observed number of claims
Pn
j=1 Nj = n · λ̂ for n · λ = n · E[Nj ] = n · θN )
µ
σX
nλ ≤ λ0 λ
θX
µ
⇒Z=
n
λ0
¶1/2
¶2
µ
σN
= λ0 λ
θN
X=N
θ̂X
=
σ̂X
½
¶2
=
λ0 2
σ
λ N
¾1/2
nλ̂
P oisson
P oisson
=
2
(λ0 /λ̂)σ̂N
=
λ0
µ
¶1/2
nλ̂
λ0
;
(3) Severity case (based on compound Poisson assumption) the number of policies is (use
observed number of policies for n)
µ
n ≤ λ0
σX
θX
¶2
σ 2 θ 2 + θN σ 2
= λ0 N Y 2 2 Y
θN θY
·
µ
λ0
σY
=⇒ n ≤
1+
λ
θY
P oisson
¶2 ¸
½
⇒Z=
n
¾1/2
(λ0 /λ̂)[1 + (σ̂Y /θ̂Y )2 ]
(4) Severity case (based on compound Poisson assumption) the expected number of
P
claims is (use observed number of claims nj=1 Nj = n · λ̂ for n · λ = n · E[Nj ])
·
µ
σY
n · E[Nj ] = n · λ ≤ λ0 1 +
θY
¶2 ¸
½
⇒Z=
2
nλ̂
λ0 [1 + (σ̂Y /θ̂Y )2 ]
¾1/2
;
;
(5) Severity case (based on compound Poisson assumption) the expected total dollars of
P
P i
Pn
claims is (use observed total dollars of claims ni=1 N
j=1 Yi,j =
i=1 Xi = n· θ̂X = n· λ̂· θ̂Y =
Pn
( j=1 Nj ) · θ̂Y for n · E[Xj ] = n · λ · θY )
·
¸
σ2
n · E[Xj ] = n · λ · θY ≤ λ0 θY + Y ⇒ Z =
θY
Pn
θ̂X =
Pure Premium=
6
i=1
n
Xi
Pn
=
j=1
n
Nj
Pn
Pn
·
i=1 Xi
Pn
j=1 Nj
j=1
=
½
Nj
n
¾1/2
nλ̂θ̂Y
λ0 [θ̂Y + (σ̂Y2 /θ̂Y )]
Pn
·
PNi
i=1
Pn
j=1
j=1
Yi,j
Nj
.
= θ̂N · θ̂Y .
Losses
# of Claims
Losses
=
·
=(Frequency)·(Severity).
Exposures
Exposures # of Claims
Predictive and Posterior Distributions
Assume we have observed X̃ = x̃, where X̃ = (X1 , . . . , Xn ) and x̃ = (x1 , . . . , xn ), and
want to set a rate to cover Xn+1 . Let θ be the associated risk parameter (θ is unknown
and comes from a r,v, Θ), and Xj have conditional probability density function fXj |Θ (xj |θ),
j = 1, . . . , n, n + 1. We are interested in
(1) fXn+1 |X̃ (xn+1 |x̃), the predictive probability density; and
(2) fΘ|X̃ (θ|x̃), the posterior probability density.
The joint probability density of X̃ and θ is
ind.
fX̃,Θ (x̃, θ) = f (x1 , . . . , xn |θ)π(θ) =
·Y
n
¸
fXj |Θ (xj |θ) ·π(θ),
j=1
(where π(θ) is called the prior probability density) and the probability density of X̃ is
fX̃ (x̃) =
Z ·Y
n
¸
fXj |Θ (xj |θ) ·π(θ)dθ,
j=1
implying that the posterior probability density is
·Y
n
f (x̃, θ)
=
πΘ|X̃ (θ|x̃) = X̃,Θ
fX̃ (x̃)
¸
fXj |Θ (xj |θ)
j=1
fX̃ (x̃)
· π(θ).
The predictive probability density is
fX̃,Xn+1 (x̃, xn+1 ) Z
= fXn+1 |Θ (xn+1 |θ) · πΘ|X̃ (θ|x̃)dθ.
fXn+1 |X̃ (xn+1 |x̃) =
fX̃ (x̃)
If θ is known, the hypothetical mean (the premium charged dependent on θ) is
Z
µn+1 (θ) = E[Xn+1 |Θ = θ] =
xn+1 · fXn+1 |Θ (xn+1 |θ)dxx+1 ;
3
if θ is unknown, the mean of the hypothetical means, or the pure premium (the premium
charged independent of θ) is
µn+1 = E[Xn+1 ] = E[E[Xn+1 |Θ]] = E[µn+1 (Θ)].
The mean of the predictive distribution (the Bayesian premium) is
Z
E[Xn+1 |X̃ = x̃] =
Z
xn+1 ·fXn+1 |X̃ (xn+1 |x̃)dxn+1 =
µn+1 (θ)·πΘ|X̃ (θ|x̃)dθ = E[µn+1 (Θ)|X̃ = x̃],
that is, (the mean of the predictive distribution) = (the mean of posterior distribution).
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The Credibility Premium
We would like to approximate µn+1 (Θ) by a linear function of the past data. That is, we
will choose α0 , α1 ,. . . ,αn to minimize squared error loss,
½·
Q=E
µn+1 (Θ) − α0 −
n
X
¸2 ¾
αj Xj
.
j=1
E[Xn+1 ] = E[µn+1 (Θ)] = α̂0 +
n
X
α̂j · E[Xj ],
(1)
j=1
For i = 1, 2, . . . , n, we have E[µn+1 (Θ) · Xi ] = E[Xn+1 · Xi ] and
Cov(Xi , Xn+1 ) =
n
X
α̂j · Cov(Xi , Xj ),
(2)
j=1
P
α̂0 + nj=1 α̂j · Xj is called the credibility premium. Equation (1) and the n equations
(2) together are called the normal equations which can be expressed in a matrix form as
follows:





µn+1
1 µ1 µ2 · · · µn
α̂0
 2




2
2
2
 σ1,n+1 
 0 σ1,1
  α̂1 
σ1,2
· · · σ1,n
 2





2
2
2
 σ




 2,n+1  =  0 σ2,1 σ2,2 · · · σ2,n   α̂2  ,


 .
 . 
.
.
.
.
.


 .
 . 
..
..
..
..
.. 


 .
 . 
2
σn,n+1
2
2
2
· · · σn,n
σn,2
0 σn,1
α̂n
2
= Cov(Xi , Xj ), i = 1, 2, . . . , n and j = 1, 2, . . . , n + 1.
where µj = E[Xj ] and σi,j
Note that the values α̂0 , α̂1 ,. . . , α̂n also minimize
½·
Q1 = E
E(Xn+1 |X̃) − α0 −
n
X
¸2 ¾
½·
and Q2 = E
αj Xj
Xn+1 − α0 −
n
X
¸2 ¾
αj Xj
.
j=1
j=1
P
That is, the credibility premium α̂0 + nj=1 α̂j · Xj is the best linear estimator of each the
hypothetical mean µn+1 (Θ)(=E[Xn+1 |Θ]), the Bayesian premium E[Xn+1 |X̃], and Xn+1 (all
µn+1 (Θ), E[Xn+1 |X̃] and Xn+1 have the same expectations).
4
2
2
= V ar(Xj ) = σ 2 , and σi,j
= Cov(Xi , Xj ) = ρ · σ 2 for
Special case: If µj = E[Xj ] = µ, σj,j
i 6= j, where the correlation coefficient ρ satisfies −1 < ρ < 1. Then
α̂0 =
(1 − ρ) · µ
ρ
and α̂j =
.
1 − ρ + nρ
1 − ρ + nρ
The credibility premium is then
n
X
n
X
ρ
1−ρ
α̂0 +
·µ+
·
α̂j · Xj =
Xj = (1 − Z) · µ + Z · X̄,
1 − ρ + nρ
1 − ρ + nρ j=1
j=1
where Z = nρ/(1 − ρ + nρ). Thus, if 0 < ρ < 1, then 0 < Z < 1 and the credibility premium
is a weighted average of the sample mean X̄ and the pure premium µn+1 = E[Xn+1 ] = µ.
8
(
9
The Loss Functions
L(Θ, Θ̂)
(Θ − Θ̂)2
|Θ − Θ̂|
c, if Θ̂ 6= Θ
0, if Θ̂ = Θ
Θ̂ minimizes
the mean,
the median,
the mode,
EΘ [L(Θ, Θ̂)],
EΘ [Θ],
Π−1
Θ [1/2],
EΘ [L(Θ, Θ̂)|X̃]
EΘ [Θ|X̃]
Π−1
[1/2]
Θ|X̃
Maxθ PΘ [Θ = θ], Maxθ PΘ|X̃ [Θ = θ]
The Parametric Buhlmann Model
Assume X1 |Θ, X2 |Θ, . . . , Xn |Θ are independent and identically distributed. Denote
the hypothetical mean µ(θ) = E[Xj |Θ = θ];
the process variance v(θ) = V ar[Xj |Θ = θ];
the expected value of the hypothetical mean (the collective premium) µ = E[µ(Θ)];
the expected value of the process variance v = E[v(Θ)] = E[V ar(Xj |Θ)]; and
the variance of the hypothetical mean a = V ar[µ(Θ)] = V ar[E(Xj |Θ)].
Then
E[Xj ] = E[E(Xj |Θ)] = E[µ(Θ)] = µ,
and
V ar[Xj ] = E[V ar(Xj |Θ)] + V ar[E(Xj |Θ)] = E[v(Θ)] + V ar[µ(Θ)] = v + a,
for j = 1, 2, . . . , n. Also, for i 6= j, Cov[Xi , Xj ] = V ar[µ(Θ)] = a. From the special case, we
have σ 2 = v + a and ρ = a/(v + a). The credibility premium is
α̂0 +
n
X
α̂j · Xj = Z · X̄ + (1 − Z) · µ,
j=1
a linear function of X̄ with the slope Z and the intercept (1 − Z) · µ, where
Z=
n
n
nρ
=
=
1 − ρ + nρ
n + v/a
n+k
5
is called the Buhlmann credibility factor, and
k=
v
E[v(Θ)]
E[V ar(Xj |Θ)]
=
=
.
a
V ar[µ(Θ)]
V ar[E(Xj |Θ)]
Note that Z is increasing in n and a = V ar[µ(Θ)] = V ar[E(Xj |Θ)], but decreasing in k and
v = E[v(Θ)] = E[V ar(Xj |Θ)].
Theorem: Suppose that (1) in a single period of observation there are n independent trials,
X1 , . . . , Xn , each with probability distribution function FX|Θ (x|θ), and (2) in each of n periods of observation there is a single trials, Xi , i = 1, 2, . . . , n, where the Xi ’s are independent
and identically distributed with probability distribution function FX|Θ (x|θ). Then Z1 = Z2
(the credibility factor of case (1) is equal to the credibility factor of case (2)) and k2 = n · k1 .
Z1 =
1
=
1 + k1
1
n
n
=
=
= Z2 .
EΘ [V ar(X1 + X2 + · · · + Xn |Θ)]
EΘ [V ar(X1 |Θ)]
n + k2
1+
n+
V arΘ [E(X1 + X2 + . . . + Xn |Θ)]
V arΘ [E(X1 |Θ)]
Note that the mode is not necessarily unique.
10
The Non-parametric Buhlmann Model
Given n policy years of experience data on r group policyholders, n ≥ 2 and r ≥ 2, let Xi,j
denote the random variable representing the aggregate loss amount of the ith policyholder
during the j th policy year for i = 1, . . . , r and j = 1, . . . , n, n + 1. We would like to estimate
E[Xi,n+1 |Xi,1 , . . . , Xi,n ] for i = 1, . . . , r. Let X̃i = (Xi,1 , . . . , Xi,n ) denote the random vector
of aggregate claim amount for the ith policyholder, i = 1, . . . , r. Furthermore, we assume
(1) X̃1 , . . . , X̃r are independent;
(2) For i = 1, . . . , r, the distribution of each element Xi,j (j = 1, . . . , n) of X̃i depends on
an (unknown) risk parameter Θi = θi ;
(3) Θ1 . . . Θn are independent and identically distributed random variables;
(4) Given i, Xi,1 |Θi , . . . , Xi,n |Θi are independent; and
(5) Each combination of policy year and policyholder has an equal number of underlying
exposure units.
For i = 1, . . . , r and j = 1, . . . , n, define
µ(θi ) = E[Xi,j |Θi = θi ],
v(θi ) = V ar[Xi,j |Θi = θi ],
µ = E[µ(Θi )], the expected value of the hypothetical means,
a = V ar[µ(Θi )] = V ar[E(Xi,j |Θi )], the variance of the hypothetical means, and
v = E[v(Θi )] = E[V ar(Xi,j |Θi )], the expected value of the process variances.
The Bühlmann estimate for the ith policyholder and the (n + 1)th policy year is
E[Xi,n+1 |Xi,1 , . . . , Xi,n ] = Ẑ X̄i + (1 − Ẑ)µ̂,
i = 1, . . . , r, where Ẑ = n/(n + k̂), k̂ = v̂/â,
6
n
1X
X1,j
n j=1
n
1X
X̃2 = (X2,1 X2,2 · · · X2,n ) ⇒ X̄2 =
X2,j
n j=1
..
..
..
..
..
..
.
.
.
.
.
.
n
1X
X̃r = (Xr,1 Xr,2 · · · Xr,n ) ⇒ X̄r =
Xr,j
n j=1
r
r
1 X
v̂
1X
â =
(X̄i − X̄)2 −
µ̂ = X̄ =
X̄i
r − 1 i=1
n
r i=1
X̃1 = (X1,1
X1,2
···
X1,n ) ⇒ X̄1 =
n
1 X
(X1,j − X̄1 )2
n − 1 j=1
n
1 X
v̂2 =
(X2,j − X̄2 )2
n − 1 j=1
..
.
n
1 X
v̂r =
(Xr,j − X̄r )2
n − 1 j=1
r
r X
n
X
1X
1
v̂ =
v̂i =
(Xi,j − X̄i )2
r i=1
r(n − 1) i=1 j=1
v̂1 =
Note that
(1) Ẑ and (1 − Ẑ)µ̂ are independent of i;
(2) it is possible that â could be negative due to the substraction. When that happens,
it is customary to set â = Ẑ=0, and the Bühlmann estimate becomes µ̂ = X̄.
11
The Die-Spinner Model
Xk |(Ai ∩ Bj ) = Ik · Sk |(Ai ∩ Bj ) = (Ik |Ai ) ∩ (Sk |Bj )
where the frequency of claims Ik |Ai ∼ Bernoulli(pi ), and Sk |Bj is the severity of claims.
P [Ai ∩ Bj ] = P [Ai ] · P [Bj ].
Bayesian Estimate
ind.
P [X̃ = x̃|(Ai ∩ Bj )] =
n
Y
P [Xk = xk |(Ai ∩ Bj )]
k=1
xk >0
P [Xk = xk |(Ai ∩ Bj )] = P [Ik = 1|Ai ] · P [Sk = xk |Bj ] = pi · P [Sk = xk |Bj ].
P [Xk = 0|(Ai ∩ Bj )] = P [Ik = 0|Ai ] = 1 − pi .
P [(Ai ∩ Bj )|X̃ = x̃] = P [X̃ = x̃|(Ai ∩ Bj )] · P [Ai ∩ Bj ]/P [X̃ = x̃].
P [X̃ = x̃] =
XX
P [X̃ = x̃|(Ai ∩ Bj )] · P [Ai ∩ Bj ].
i=1 j=1
P [Xn+1 = xn+1 |X̃ = x̃] =
XX
P [Xn+1 = xn+1 |(Ai ∩ Bj )] · P [(Ai ∩ Bj )|X̃ = x̃].
i=1 j=1
E[Xk |(Ai ∩ Bj )] = E[Ik |Ai ] · E[Sk |Bj ] = pi · E[Sk |Bj ].
E[Xn+1 |X̃ = x̃] =
=
XX
X
mk · P [Xn+1 = mk |X̃ = x̃]
k
E[Xn+1 |(Ai ∩ Bj )] · P [(Ai ∩ Bj )|X̃ = x̃].
i=1 j=1
7
Credibility Estimate (combined)
µ = E[Xk ] = E[E(Xk |Θ)] =
v = E{V ar[Xk |Θ]} =
XX
XX
E[Xk |(Ai ∩ Bj )] · P [Ai ∩ Bj ].
i=1 j=1
V ar[Xk |(Ai ∩ Bj )] · P [Ai ∩ Bj ].
i=1 j=1
V ar[Xk |(Ai ∩ Bj )] = V ar[Ik · Sk |(Ai ∩ Bj )] = E[Ik2 |Ai ] · E[Sk2 |Bj ] − {E[Ik |Ai ] · E[Sk |Bj ]}2
= E[Ik |Ai ]·V ar[Sk |Bj ]+V ar[Ik |Ai ]·{E[Sk |Bj ]}2 = pi ·V ar[Sk |Bj ]+pi ·(1−pi )·{E[Sk |Bj ]}2 .
a = V ar{E[Xk |Θ]} = E{[E(Xk |Θ)]2 } − {E[E(Xk |Θ)]}2 = E{[E(Xk |Θ)]2 } − {E[Xk ]}2
=
XX
{E[Xk |(Ai ∩ Bj )] − E[Xk ]}2 · P [Ai ∩ Bj ] =
i=1 j=1
XX
{E[Xk |(Ai ∩ Bj )] − µ}2 · P [Ai ∩ Bj ].
i=1 j=1
PC = Z · X̄ + (1 − Z) · µ where Z = n/(n + k) and k = v/a.
Credibility Estimate (separated)
Frequency:
µF = E[Ik ] = E[E(Ik |ΘA )] =
vF = E{V ar[Ik |ΘA ]} =
X
X
E[Ik |Ai ] · P [Ai ] =
i=1
V ar[Ik |Ai ] · P [Ai ] =
i=1
X
X
pi · P [Ai ].
i=1
pi · (1 − pi ) · P [Ai ].
i=1
aF = V ar{E[Ik |ΘA ]} = E{[E(Ik |ΘA )]2 } − {E[E(Ik |ΘA )]}2 = E{[E(Ik |ΘA )]2 } − {E[Ik ]}2
=
X
{E[Ik |Ai ] − E[Ik ]}2 · P [Ai ] =
i=1
X
{E[Ik |Ai ] − µF }2 · P [Ai ] =
i=1
X
[pi − µF ]2 · P [Ai ].
i=1
PF = ZF · I¯ + (1 − ZF ) · µF where ZF = n/(n + kF ) and kF = vF /aF .
Severity:
µS = E[Sk ] = E[E(Sk |ΘB )] =
vS = E{V ar[Sk |ΘB ]} =
X
X
E[Sk |Bj ] · P [Bj ].
j=1
V ar[Sk |Bj ] · P [Bj ].
j=1
aS = V ar{E[Sk |ΘB ]} = E{[E(Sk |ΘB )]2 } − {E[E(Sk |ΘB )]}2 = E{[E(Sk |ΘB )]2 } − {E[Sk ]}2
=
X
{E[Sk |Bj ] − E[Sk ]}2 · P [Bj ] =
j=1
X
{E[Sk |Bj ] − µS }2 · P [Bj ].
j=1
PS = ZS ·S̄+(1−ZS )·µS where ZS = nS /(nS +kS ), kS = vS /aS and nS is # of non-zero claims.
PC = PF · PS .
8
12
The Parametric Buhlmann-Straub Model
Assume X1 |Θ, X2 |Θ, . . . , Xn |Θ are independent distributed with common hypothetical mean
µ(θ) = E[Xj |Θ = θ]
but different process variances
v(θ)
mj
V ar[Xj |Θ = θ] =
where mj is a known constant measuring exposure, j = 1, . . . , n (mj could be the number of
months the policy was in force in past year j, or the number of individuals in the group in
past year j, or the amount of premium income for the policy in past year j). Let
the expected value of the hypothetical mean the collective premium µ = E[µ(Θ)],
the expected value of the process variance v = E[v(Θ)] (6= E[V ar(Xj |Θ)] = E[v(Θ)]/mj ),
and the variance of the hypothetical mean a = V ar[µ(Θ)] = V ar[E(Xj |Θ)]. Then
E[Xj ] = E[E(Xj |Θ = θ)] = E[µ(Θ)] = µ,
and
V ar[Xj ] = E[V ar(Xj |Θ)] + V ar[E(Xj |Θ)] =
E[v(Θ)]
v
+ V ar[µ(Θ)] =
+ a,
mj
mj
for j = 1, 2, . . . , n. Also, for i 6= j, Cov[Xi , Xj ] = a.
The credibility premium α̂0 +
Pn
j=1
α̂j · Xj minimizes
½·
Q=E
µn+1 (Θ) − α0 −
n
X
¸2 ¾
αj Xj
j=1
where
α̂0 =
µ
v/a
k
=
·µ=
·µ
1 + a · m/v
m + v/a
m+k
and
α̂j =
a α̂0
mj
m
mj
·
· mj =
=
·
v µ
m + v/a
m+k m
with k = v/a. The credibility premium is then
n
X
k
m
α̂0 +
α̂j · Xj =
·µ+
·
m+k
m+k
j=1
Pn
j=1
P
P
mj · Xj
= (1 − Z) · µ + Z · X̄,
m
where Z = m/(m+k) and X̄ = ( nj=1 mj ·Xj )/( nj=1 mj ). Note that if Xj is the average loss
per individual for the mj group members in year j, then mj · Xj is the total loss for the mj
group members in year j, and X̄ is the overall average loss per group over the n years. The
credibility premium to be charged to the group in year n+1 would be mn+1 ·[Z · X̄ +(1−Z)·µ]
for mn+1 members.
9
For the single observation X̄, the process variance is
· Pn
V ar[X̄|θ] = V ar
j=1
¯
n
n
X
mj · Xj ¯¯ ¸ X
m2j
m2j v(θ)
v(θ)
θ
=
·
V
ar[X
|θ]
=
·
=
,
j
¯
2
2
m
mj
m
j=1 m
j=1 m
and the expected process variance is
E[V ar(X̄|Θ)] =
E[v(Θ)]
v
= .
m
m
The hypothetical mean is
¯ ¸
·
n
n
X
¯
mj
1 X
E[X̄|θ] = E
mj · Xj ¯¯θ =
· E[Xj |θ] = µ(θ),
m j=1
j=1 m
the variance of the hypothetical means is
V ar[E(X̄|Θ)] = V ar[µ(Θ)] = a,
and the expected hypothetical means is
E[E(X̄|Θ)] = E[µ(Θ)] = µ.
Therefore, the credibility factor is
Z=
13
1
m
=
.
1 + v/(am)
m + v/a
The Non-parametric Buhlmann-Straub Model
Let r group policyholders be such that the ith policyholder has ni years of experience data,
i = 1, 2, . . . , r (r ≥ 2). Let mi,j denote the number of exposure units and Xi,j be the random
variable representing the average claim amount per exposure unit of the ith policyholder
during the j th policy year for j = 1, . . . , ni , ni + 1 (ni ≥ 2) and i = 1, . . . , r. We would
like to estimate E[Xi,ni +1 |Xi,1 , . . . , Xi,ni ] for i = 1, . . . , r. Let X̃i = (Xi,1 , . . . , Xi,ni ) denote
the random vector of average claim amount and m̃i = (mi,1 , . . . , mi,ni ) be the random vector
of the number of exposure units for the ith policyholder, i = 1, . . . , r. Furthermore, we assume
(1) X̃1 , . . . , X̃r are independent;
(2) For i = 1, . . . , r, the distribution of each element Xi,j (j = 1, . . . , ni ) of X̃i depends on
an (unknown) risk parameter Θi = θi ;
(3) Θ1 . . . Θn are independent and identically distributed random variables; and
(4) Given i, Xi,1 |Θi , . . . , Xi,ni |Θi are independent.
For i = 1, . . . , r and j = 1, . . . , ni (ni ≥ 2), define
E[Xi,j |Θi = θi ] = µ(θi ) and V ar[Xi,j |Θi = θi ] = v(θi )/mi,j .
10
Let
µ = E[µ(Θi )] denote the expected value of the hypothetical means,
a = V ar[µ(Θi )] = V ar[E(Xi,j |Θi )] denote the variance of the hypothetical means, and
v = E[v(Θi )] = mi,j · E[V ar(Xi,j |Θi )] denote the expected value of the process variances.
X̃1 =
(X1,1
X1,2
···
X1,n1 )
⇒ X̄1 =
m̃1 =
(m1,1
m1,2
···
m1,n1 )
⇒ m1 =
n1
1 X
m1,j · X1,j
m1 j=1
n1
X
v̂1 =
n1
1 X
m1,j · (X1,j − X̄1 )2
n1 − 1 j=1
v̂2 =
n2
1 X
m2,j · (X2,j − X̄2 )2
n2 − 1 j=1
m1,j
j=1
X̃2 =
(X2,1
X2,2
···
X2,n2 )
⇒ X̄2 =
m̃2 =
(m2,1
m2,2
···
m2,n2 )
⇒ m2 =
..
.
..
.
..
.
..
.
X̃r =
(Xr,1
Xr,2
···
Xr,nr )
⇒ X̄r =
m̃r =
(mr,1
mr,2
···
mr,nr )
⇒ mr =
â =
..
.
r
1 X
m−
m2
m i=1 i
m2,j
nr
1 X
mr,j · Xr,j
mr j=1
nr
X
µ̂ = X̄ =
..
.
v̂r =
nr
1 X
mr,j · (Xr,j − X̄r )2
nr − 1 j=1
mr,j
j=1
r
X
mi · (X̄i − X̄)2 − (r − 1) · v̂
i=1
n2
X
j=1
..
.
r
X
n2
1 X
m2,j · X2,j
m2 j=1
r
X
mi · X̄i
i=1
r
X
v̂ =
mi
i=1
In this case,
X̄i =
·(ni − 1) · v̂i
i=1
r
X
(ni − 1)
i=1
ni
1 X
mi,j · Xi,j
mi j=1
is the average claim amount per exposure unit and mi =
ni
X
mi,j is the total number of
j=1
exposure units for the ith policyholder during the first ni policy years for i = 1, . . . , r, and
ni
r
r X
1 X
1 X
X̄ =
mi · X̄i =
mi,j · Xi,j
m i=1
m i=1 j=1
is the overall past average claim amount per exposure unit of the r policyholders where
m=
r
X
i=1
mi =
ni
r X
X
mi,j is the total exposure units.
i=1 j=1
The Bühlmann estimate for the ith policyholder and the (ni + 1)th policy year is
E[Xi,ni +1 |Xi,1 , . . . , Xi,ni ] = Ẑi · X̄i + (1 − Ẑi ) · µ̂,
i = 1, . . . , r, where Ẑi = mi /(mi + k̂) and k̂ = v̂/â, and the credibility premium to
cover all mi,ni +1 exposure units for policyholder i in the (ni + 1)th policy year is mi,ni +1 ·
E[Xi,ni +1 |Xi,1 , . . . , Xi,ni ]. Note that
11
(1) Ẑi is dependent on i;
(2) it is possible that â could be negative due to the substraction. When that happens,
it is customary to set â = Ẑi =0, and the Bühlmann-Straub estimate becomes µ̂ = X̄;
(3) if mi,j = 1 and ni = n for all i and j, then mi = n, m = r · n, and the ordinary Bühlmann
estimators are recovered.
The method that preserves total losses (TP=TL): the total losses on all policyholders is
P
P
T L = ri=1 mi · X̄i , and the total premium is T P = ri=1 mi · [Ẑi · X̄i + (1 − Ẑi ) · µ̂]. Then
µ̂∗ =
r ·
X
i=1
¸
Ẑi
r
X
·X̄i
Ẑj
j=1
Ẑi
which is a credibility-factor-weighted average of X̄i s with weights wi = X
, i = 1, . . . , r.
r
Ẑj
Compare the alternative µ̂ with the original µ̂ =
r ·
X
i=1
mi
r
X
j=1
¸
·X̄i , which is a exposure-unit-
mj
j=1
mi
weighted average of X̄i s with weights wi = X
, i = 1, . . . , r. Note that
r
mj
r
X
(1) use X̄ =
j=1
mi · X̄i
i=1
r
X
for â of k̂ of µ̂∗ .
mj
j=1
(2) The difference of these two credibility premiums for policyholder i based on different
estimators of µ is (1 − Ẑi ) · (µ̂∗ − µ̂)
The above analysis assume that the parameters µ, v and a are all unknown and need
to be estimated. If µ is known, then v̂ given above can still be used to estimate v as it is
unbiased whether µ is known, and an alternative and simpler unbiased estimator for a is
ã =
r
X
mi
i=1
m
· (X̄i − µ)2 −
r
· v̂.
m
If there are data on only one policyholder (say policyholder i), ã with r = 1 becomes
ni
X
ãi =
mi
1
v̂i
· (X̄i − µ)2 −
· v̂i = (X̄i − µ)2 −
= (X̄i − µ)2 −
mi
mi
mi
12
mi,j · (Xi,j − X̄i )2
j=1
mi · (ni − 1)
.
14
Semi-parametric Estimation
Assume the number of claims Ni,j = mi,j · Xi,j for policyholder i in year j is Poisson distributed with mean mi,j · θi given Θi = θi , that is mi,j · Xi,j |Θi = θi ∼ Poisson(mi,j · θi ).
Then
E[mi,j · Xi,j |Θi ] = V ar[mi,j · Xi,j |Θi ] = mi,j · Θi
or
µ(Θi ) = E[Xi,j |Θi ] = Θi and v(Θi ) = mi,j · V ar[Xi,j |Θi ] = Θi .
Therefore,
µ = E[µ(Θi )] = E[Θi ] = E[v(Θi )] = v.
In this case, we could use µ̂ = X̄ to estimate v.
Example: Assume a (conditional) Poisson distribution for the number of claims per policyholder, estimate the Bühlmann credibility premium for the number of claims next year.
number of claims
number of insureds
0
r0
1
r1
2
r2
...
...
k
rk
total
r
Assume that we have r policyholders, ni =1 and mi,j =1 for i = 1, . . . , r. Since Xi,1 |Θi ∼
Poisson(Θi ), we have E[Xi,1 |Θi ] = V ar[Xi,1 |Θi ] = Θi , µ(Θi ) = v(Θi ) = Θi and µ =
E[µ(Θi )] = E[Θi ] = E[v(Θi )] = v. Moreover,
µ̂ = X̄ =
r
k
1X
1X
Xi,1 = [ j · rj ].
r i=1
r j=1
Since V ar[Xi,1 ] = V ar[E(Xi,1 |Θi )] + E[V ar(Xi,1 |Θi )] = V ar[µ(Θi ] + E[v(Θi )] = a + v =
a + u and E[Xi,1 ] = E[E(Xi,1 |Θi )] = E[µ(Θi ] = µ, implying that Xi,1 , Xi,2 , . . . , Xr,1 are
independent random variables with common mean µ and variance a + v = a + µ. Since the
sample mean of Xi,1 , Xi,2 , . . . , Xr,1 is an unbiased estimator of V ar[Xi,1 ] = a + v = a + µ, we
have
r
1 X
ˆ i,1 ] = â + v̂ = â + µ̂,
[Xi,1 − X̄]2 = V ar[X
r − 1 i=1
or
r
1 X
â =
[Xi,1 − X̄]2 − µ̂.
r − 1 i=1
Then k̂ = v̂/â = µ̂/â, Ẑ = 1/(1 + k̂) and the Bühlmann credibility premium for the number
of claim Xi,1 next year is
X̂i,2 = Ẑ · Xi,1 + (1 − Ẑ) · µ̂, for Xi,1 = 0, 1, . . . , k.
13
15
Parametric Estimator
Assume given i, Xi,j |Θi are identically and independently distributed with probability density function fXi,j |Θi (xi,j |θi ), and Θ1 , Θ2 , . . . , Θr are also identically and independently distributed with probability density function πΘ (θ). Let X̃i = (Xi,1 , Xi,2 , . . . , Xi,ni ); then the
unconditional joint density of X̃i is
Z
fX̃i (x̃i ) =
Z
ind.
fX̃i ,Θi (x̃i , θi )dθi =
fX̃i |Θi (x̃i |θi )πΘi (θi )dθi =
Z Y
ni
[
fXi,j |Θi (xi,j |θi )]πΘi (θi )dθi .
j=1
The likelihood function is given by
L=
r
Y
i=1
ni
r ½Z Y
Y
fX̃i (x̃i ) =
[
i=1
¾
fXi,j |Θi (xi,j |θi )]πΘi (θi )dθi .
j=1
Maximum likelihood estimator of the associated parameters are chosen to maximize L or
logL.
16
Exact Credibility
When ”the Bayesian premium = the credibility premium”, we say the credibility is ”exact”.
Recall that the solutions of the normal equations, α̃0 , α̃1 , . . . , α̃n yield the credibility premium
P
α̃0 + nj=1 α̃j · Xj which minimizes
½·
Q=E
µn+1 (Θ) − α0 −
n
X
¸2 ¾
αj Xj
,
j=1
½·
Q1 = E
E(Xn+1 |X̃) − α0 −
n
X
¸2 ¾
αj Xj
j=1
and
½·
Q2 = E
Xn+1 − α0 −
n
X
¸2 ¾
αj Xj
.
j=1
If the Bayesian premium, E(Xn+1 |X̃), is a linear function of X1 , X2 , . . . Xn (in general, it is
P
NOT), that is E(Xn+1 |X̃) = a0 + nj=1 aj · Xj , then α̃j = aj for j = 0, 1 . . . , n, and therefore
P
P
Q1 = 0. Thus, the credibility premium α̃0 + nj=1 α̃j · Xj = a0 + nj=1 aj · Xj = E(Xn+1 |X̃),
and the credibility is ”exact”.
P
In summary, the Bühlmann estimator α̃0 + nj=1 α̃j · Xj is the ”best linear” approximation
to the Bayesian estimate E(Xn+1 |X̃) under the squared error loss function Q1 .
Recall that in linear regression,
Yi = α 0 +
n
X
αj · Xj + ²i , where ²i ∼ N (0, σ 2 ), for i = 1, 2, . . . , m.
j=1
14
α̂0 , α̂1 , . . . , α̂n are chosen to minimize Q = E
the regression line Ŷi = α̂0 +
n
X
·X
m
¸
²2i = E
i=1
·X
m
(Yi −α0 −
i=1
n
X
¸
αj ·Xj )2 . Therefore,
j=1
α̂j ·Xj corresponds to the credibility premium α̃0 +
j=1
n
X
α̃j ·Xj ,
j=1
and observation Yi correspond to the Bayesian premium E[Xn+1 |X̃ = (x̃)i ], i = 1, 2, . . . , m.
Condition for exact credibility
Suppose that Xj |Θ = θ is independently distributed and is from the ”linear exponential family” (”linear” means the power of the exponential is a leaner function of xj ) with probability
function
p(xj ) · e−θxj
fXj |Θ (xj |θ) =
,
q(θ)
for j = 1, 2, . . . , n + 1, and Θ has probability function
πΘ (θ) =
[q(θ)]−k · eµkθ
c(µ, k)
where θ ∈ (θ0 , θ1 ) with πΘ (θ0 ) = πΘ (θ1 ) = 0 and −∞ ≤ θ0 < θ1 ≤ ∞. Then µ = EΘ [µ(Θ)] =
E[E(Xj |Θ)] = E[Xj ] and the posterior distribution is
∗
∗ k∗ θ
[q(θ)]−k · eµ
πΘ|X̃ (θ|x̃) =
c(µ∗ , k ∗ )
, θ ∈ (θ0 , θ1 ),
where k ∗ = n + k and
µ∗ =
n
k
n · X̄ + µ · k
=
· X̄ +
· µ.
n+k
n+k
n+k
The Bayesian premium is
E[Xn+1 |X̃] = EΘ|X̃ [µn+1 (Θ)] =
where Z = n/(n + k) and k =
Z θ1
θ0
µ(θ)πΘ|X̃ (θ|x̃)dθ = µ∗ = Z · X̄ + (1 − Z) · µ,
v
E[V ar(Xj |Θ)]
=
.
a
V ar[E(Xj |Θ)]
Note that the prior distribution πΘ (θ) is a conjugate prior distribution (a prior distribution
is a ”conjugate” prior distribution if the resulting posterior distribution is of the same type
as the prior one, but perhaps with different parameters).
Theorem: If fXj |Θ (xj |θ) is a member of a ”linear exponential family”, and the prior distribution πΘ (θ) is the ”conjugate” prior distribution, then the Bühlmann credibility estimator is
equal to the Bayesian estimator (i.e. exact credibility) assuming a squared error loss function.
15
conditional distribution Xj |Θ
prior distribution Θ
µ(Θ) = E[Xj |Θ]
µ = E[E(Xj |Θ)]
v(Θ) = V ar[Xj |Θ]
v = E[V ar(Xj |Θ)]
a = V ar[E(Xj |Θ)]
k = v/a
Z
credibility premium
posterior distribution Θ|X̃
Bernoulli(Θ)
Beta(α,β)
Θ
α
α+β
Θ(1 − Θ)
αβ
(α + β + 1)(α + β)
αβ
(α + β + 1)(α + β)2
α+β
n
n+α+β
α
Z · X̄ + (1 − Z) ·
α+β
µ
Beta
n
X
i=1
posterior mean E[Θ|X̃]
Xi + α, β + n −
n
X
i=1
Xi + α
Θ
αβ
αβ 2
1/β
nβ
nβ + 1
Z · X̄ + (1 − Z) · αβ
¶
Xi
µX
n
β
Gamma
Xi + α,
nβ + 1
i=1
β
µX
n
n+α+β
Bernoulli
n
X
¶
Xi + α
i=1
µ
predictive mean E[Xn+1 |X̃]
αβ
i=1
n
X
predictive distribution Xn+1 |X̃
n
X
Poisson(Θ)
Gamma(α,β)
Θ
Xi + α ¶
i=1
n+α+β
Xi + α
nβ + 1
µX
n
β
Xi + α,
NB
nβ + 1
i=1
β
µX
n
i=1
n+α+β
¶
Xi + α
i=1
nβ + 1
Note that
the predictive mean (the Bayesian premium), E[Xn+1 |X̃]
= the posterior mean, E[Θ|X̃] (= E[µn+1 (Θ)|X̃] since µn+1 (Θ) = E[Xn+1 |Θ] = Θ)
= the credibility premium, Z · X̄ + (1 − Z) · µ.
In some situations, we may want to work the logarithm of the data (Wj = logXj , j =
1, 2, . . . , n). Then µlog (Θ) = E[Wj |Θ] = E[logXj |Θ], vlog (Θ) = V ar[Wj |Θ] = V ar[logXj |Θ],
µlog = E[µlog (Θ)] = E[Wj ] = E[logXj ], vlog = E[vlog (Θ)] = E[V ar(Wj |Θ)] = E[V ar(logXj |Θ)],
alog = V ar[µlog (Θ)] = V ar[E(Wj |Θ)] = V ar[E(logXj |Θ)], and Zlog = n/(n + vlog /alog ).
Thus,
logC = Zlog · W̄ + (1 − Zlog ) · µlog , or C = eZlog ·W̄ +(1−Zlog )·µlog ,
which is denoted by Clog and is not unbiased because we use linear credibility to estimate the
mean of the distribution of logarithms. Recall Ĉ = Z · X̄ + (1 − Z) · µ where µ = E[µ(Θ)] =
E[E(Xj |Θ)] = E[Xj ], so E[Ĉ] = Z · E[X̄] + (1 − Z) · E[Xj ] = µ = E[Xj ]. To make Clog
unbiased, let
Clog = c · eZlog ·W̄ +(1−Zlog )·µlog
where c is determined by E[eWj ] = E[Xj ] = E[Clog ] = c · E[eZlog ·W̄ +(1−Zlog )·µlog ].
16
¶
¶