Let A and B be two events such that P (B) > 0, then P (A B) = P (B A) P (A)/P (B).

Transcription

1 1 Coditioal Probability Let A ad B be two evets such that P (B) > 0, the P (A B) P (A B)/P (B). Bayes Theorem Let A ad B be two evets such that P (B) > 0, the P (A B) P (B A) P (A)/P (B). Theorem of total probability Let A 1, A 2, be a coutable collectio of mutually exclusive ad exhaustive evets, so that A i A j for i j ad A i Ω, the P (B) P (B A i ) P (A i ). 2 Coditioal Distributios The coditioal probability fuctio of X give that Y y is f X Y (x y) f X,Y (x, y)/f Y (y). If X ad Y are idepedet, the f X,Y (x, y) f X (x) f Y (y) which implies f X Y (x y) f X (x). Rewrite f X,Y (x, y) f X Y (x y) f Y (y). The f X (x) f X,Y (x, y)dy f X Y (x y) f Y (y)dy. Applicatio: f X (x) f X Θ (x θ) f Θ (θ)dθ; F X (x) F X Θ (x θ) f Θ (θ)dθ. Note that f X Y (x y) f Y (y) f X,Y (x, y) f Y X (y x) f X (x), implyig that f X Y (x y) f Y X(y x) f X (x) f Y (y) f Y X (y x) f X (x). 3 Coditioal Expectatio Let X be a discrete radom variable such that x 1, x 2,... are the oly values that X takes o with positive probability. Defie the coditioal expectatio of X give that evet A has occurred, deoted by E[X A], as E[X A] x i P [X x 1 A]. Cotiuous case : E[X Y y] x f X Y (x y)dx Let A 1, A 2, be a coutable collectio of mutually exclusive ad exhaustive evets, ad let X be a discrete radom variable for which E[X] exist. The E[X] E[X A i ] P [A i ]. Cotiuous case : E[X] E[X Y y] f Y (y)dy E Y [E X (X Y )] E[X] E[h(X, Y ) Y y] h(x, y) f X Y (x y)dx E Y {E X [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

2 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 )] 4 Noparametric Ubiased Estimators If X 1, X 2,..., X are idepedet but ot ecessarily idetical with commo mea µ E[X j ] ad commo variace σ 2 V ar[x j ], the ˆµ X 1 X j is a ubiased estimator of µ, ad ˆσ (X j X) 2 is a ubiased estimator of v 5 Full ad Partial Credibilities The followig Zs subject to Z 1; if Z 1 the a full credibility is assiged. The stadard for NO full credibility i terms of (λ 0 [y p /r] 2 ) (1) (frequecy case) the umber of policies is (use observed umber of policies for ) ( ) σx 2XN ( ) σn 2P oisso F λ 0 λ 0 λ 0 θ X θ N λ ( )1/2 ( ) 1/2 ( ) ˆθX XN 1/2 ( ˆθN P oisso ˆλ Z F λ 0 ˆσ X λ0 ˆσ N λ 0 ) 1/2; (2) (frequecy case) the expected umber of claims is (use observed umber of claims N j ˆλ for λ E[N j ] θ N ) λ λ 0 λ ( σx θ X ) 2XN ( ) σn 2 λ 0 λ 0 λ θ N λ σ2 N ( ) 1/2 { ˆθX ˆλ Z λ0 ˆσ X (λ 0 /ˆλ)ˆσ N 2 } 1/2P oisso P oisso λ 0 ( ˆλ λ 0 ) 1/2; (3) Severity case (based o compoud Poisso assumptio) the umber of policies is (use observed umber of policies for ) ( ) σx 2 σ λ 0 λ0 Nθ 2 Y 2 + θ N σy 2 θ X θnθ 2 Y 2 λ [ ( ) 0 σy 2 ] { 1+ Z λ θ Y (λ 0 /ˆλ)[1 + (ˆσ Y /ˆθ Y ) 2 ] P oisso (4) Severity case (based o compoud Poisso assumptio) the expected umber of claims is (use observed umber of claims N j ˆλ for λ E[N j ]) E[N j ] λ λ 0 [1 + ( σy θ Y ) 2 ] { ˆλ Z λ 0 [1 + (ˆσ Y /ˆθ Y ) 2 ] 2 } 1/2; } 1/2;

3 (5) Severity case (based o compoud Poisso assumptio) the expected total dollars of claims is (use observed total dollars of claims Ni Y i,j X i ˆθ X ˆλ ˆθ Y ( N j ) ˆθ Y for E[X j ] λ θ Y ) [ ] { E[X j ] λ θ Y λ 0 θ Y + σ2 Y ˆλˆθ Y Z θ Y λ 0 [ˆθ Y + (ˆσ Y 2 /ˆθ Y )] ˆθ X X i Pure Premium Losses Exposures N j X i N j N j } 1/2. Ni Y i,j N j ˆθ N ˆθ Y. # of Claims Exposures Losses # of Claims (Frequecy) (Severity). 6 Predictive ad Posterior Distributios Assume we have observed X x, where X (X 1,..., X ) ad x (x 1,..., x ), ad wat to set a rate to cover X +1. Let θ be the associated risk parameter (θ is ukow ad comes from a r,v, Θ), ad X j have coditioal probability desity fuctio f Xj Θ(x j θ), j 1,...,, + 1. We are iterested i (1) f X+1 X (x +1 x), the predictive probability desity; ad (2) f Θ X(θ x), the posterior probability desity. The joit probability desity of X ad θ is f X,Θ ( x, θ) f(x 1,..., x θ)π(θ) id. [ ] f Xj Θ(x j θ) π(θ), (where π(θ) is called the prior probability desity) ad the probability desity of X is [ f X( x) implyig that the posterior probability desity is π Θ X(θ x) f X,Θ ( x, θ) f X( x) The predictive probability desity is ] f Xj Θ(x j θ) π(θ)dθ, [ ] f Xj Θ(x j θ) f X( x) π(θ). f X+1 X (x +1 x) f X,X+1 ( x, x +1 ) f X( x) f X+1 Θ(x +1 θ) π Θ X(θ x)dθ. If θ is kow, the hypothetical mea (the premium charged depedet o θ) is µ +1 (θ) E[X +1 Θ θ] x +1 f X+1 Θ(x +1 θ)dx x+1 ; 3

4 if θ is ukow, the mea of the hypothetical meas, or the pure premium (the premium charged idepedet of θ) is µ +1 E[X +1 ] E[E[X +1 Θ]] E[µ +1 (Θ)]. The mea of the predictive distributio (the Bayesia premium) is E[X +1 X x] x +1 f X+1 X (x +1 x)dx +1 µ +1 (θ) π Θ X(θ x)dθ E[µ +1 (Θ) X x], that is, (the mea of the predictive distributio) (the mea of posterior distributio). 7 The Credibility Premium We would like to approximate µ +1 (Θ) by a liear fuctio of the past data. That is, we will choose α 0, α 1,...,α to miimize squared error loss, {[ Q E µ +1 (Θ) α 0 ] 2 α j X j }. E[X +1 ] E[µ +1 (Θ)] ˆα 0 + ˆα j E[X j ], (1) For i 1, 2,...,, we have E[µ +1 (Θ) X i ] E[X +1 X i ] ad Cov(X i, X +1 ) ˆα j Cov(X i, X j ), (2) ˆα 0 + ˆα j X j is called the credibility premium. Equatio (1) ad the equatios (2) together are called the ormal equatios which ca be expressed i a matrix form as follows: µ +1 1 µ 1 µ 2 µ σ 1, σ σ2,+1 2 1,1 2 σ1,2 2 σ 2 1, 0 σ2,1 2 σ2,2 2 σ 2 2, σ, σ,1 2 σ,2 2 σ, 2 where µ j E[X j ] ad σ 2 i,j Cov(X i, X j ), i 1, 2,..., ad j 1, 2,..., + 1. Note that the values ˆα 0, ˆα 1,..., ˆα also miimize {[ Q 1 E E(X +1 X) α 0 ] 2 } α j X j ˆα 0 ˆα 1 ˆα 2. ˆα, {[ ad Q 2 E X +1 α 0 ] 2 α j X j }. That is, the credibility premium ˆα 0 + ˆα j X j is the best liear estimator of each the hypothetical mea µ +1 (Θ)(E[X +1 Θ]), the Bayesia premium E[X +1 X], ad X +1 (all µ +1 (Θ), E[X +1 X] ad X +1 have the same expectatios). 4

5 Special case: If µ j E[X j ] µ, σ 2 j,j V ar(x j ) σ 2, ad σ 2 i,j Cov(X i, X j ) ρ σ 2 for i j, where the correlatio coefficiet ρ satisfies 1 < ρ < 1. The ˆα 0 The credibility premium is the ˆα 0 + ˆα j X j (1 ρ) µ 1 ρ + ρ ad ˆα j 1 ρ 1 ρ + ρ µ + ρ 1 ρ + ρ ρ 1 ρ + ρ. X j (1 Z) µ + Z X, where Z ρ/(1 ρ + ρ). Thus, if 0 < ρ < 1, the 0 < Z < 1 ad the credibility premium is a weighted average of the sample mea X ad the pure premium µ +1 E[X +1 ] µ. 8 The Loss Fuctios L(Θ, ˆΘ) ˆΘ miimizes EΘ [L(Θ, ˆΘ)], E Θ [L(Θ, ˆΘ) X] (Θ ˆΘ) 2 the mea, E Θ [Θ], E Θ [Θ X] Θ ˆΘ the media, Π 1 Θ [1/2], Π 1 [1/2] { Θ X c, if ˆΘ Θ 0, if ˆΘ Θ the mode, Max θp Θ [Θ θ], Max θ P X[Θ Θ θ] 9 The Parametric Buhlma Model Assume X 1 Θ, X 2 Θ,..., X Θ are idepedet ad idetically distributed. Deote the hypothetical mea µ(θ) E[X j Θ θ]; the process variace v(θ) V ar[x j Θ θ]; the expected value of the hypothetical mea (the collective premium) µ E[µ(Θ)]; the expected value of the process variace v E[v(Θ)] E[V ar(x j Θ)]; ad the variace of the hypothetical mea a V ar[µ(θ)] V ar[e(x j Θ)]. The E[X j ] E[E(X j Θ)] E[µ(Θ)] µ, ad V ar[x j ] E[V ar(x j Θ)] + V ar[e(x j Θ)] E[v(Θ)] + V ar[µ(θ)] v + a, for j 1, 2,...,. Also, for i j, Cov[X i, X j ] V ar[µ(θ)] a. From the special case, we have σ 2 v + a ad ρ a/(v + a). The credibility premium is ˆα 0 + ˆα j X j Z X + (1 Z) µ, a liear fuctio of X with the slope Z ad the itercept (1 Z) µ, where Z ρ 1 ρ + ρ + v/a 5 + k

6 is called the Buhlma credibility factor, ad k v a E[v(Θ)] V ar[µ(θ)] E[V ar(x j Θ)] V ar[e(x j Θ)]. Note that Z is icreasig i ad a V ar[µ(θ)] V ar[e(x j Θ)], but decreasig i k ad v E[v(Θ)] E[V ar(x j Θ)]. Theorem: Suppose that (1) i a sigle period of observatio there are idepedet trials, X 1,..., X, each with probability distributio fuctio F X Θ (x θ), ad (2) i each of periods of observatio there is a sigle trials, X i, i 1, 2,...,, where the X i s are idepedet ad idetically distributed with probability distributio fuctio F X Θ (x θ). The Z 1 Z 2 (the credibility factor of case (1) is equal to the credibility factor of case (2)) ad k 2 k 1. Z k E Θ[V ar(x 1 + X X Θ)] V ar Θ [E(X 1 + X X Θ)] + E Θ[V ar(x 1 Θ)] V ar Θ [E(X 1 Θ)] + k 2 Z 2. Note that the mode is ot ecessarily uique. 10 The No-parametric Buhlma Model Give policy years of experiece data o r group policyholders, 2 ad r 2, let X i,j deote the radom variable represetig the aggregate loss amout of the i th policyholder durig the j th policy year for i 1,..., r ad j 1,...,, + 1. We would like to estimate E[X i,+1 X i,1,..., X i, ] for i 1,..., r. Let X i (X i,1,..., X i, ) deote the radom vector of aggregate claim amout for the i th policyholder, i 1,..., r. Furthermore, we assume (1) X 1,..., X r are idepedet; (2) For i 1,..., r, the distributio of each elemet X i,j (j 1,..., ) of Xi depeds o a (ukow) risk parameter Θ i θ i ; (3) Θ 1... Θ are idepedet ad idetically distributed radom variables; (4) Give i, X i,1 Θ i,..., X i, Θ i are idepedet; ad (5) Each combiatio of policy year ad policyholder has a equal umber of uderlyig exposure uits. For i 1,..., r ad j 1,...,, defie µ(θ i ) E[X i,j Θ i θ i ], v(θ i ) V ar[x i,j Θ i θ i ], µ E[µ(Θ i )], the expected value of the hypothetical meas, a V ar[µ(θ i )] V ar[e(x i,j Θ i )], the variace of the hypothetical meas, ad v E[v(Θ i )] E[V ar(x i,j Θ i )], the expected value of the process variaces. The Bühlma estimate for the i th policyholder ad the ( + 1) th policy year is i 1,..., r, where Ẑ /( + ˆk), ˆk ˆv/â, E[X i,+1 X i,1,..., X i, ] Ẑ X i + (1 Ẑ)ˆµ, 6

7 X 1 (X 1,1 X 1,2 X 1, ) X 1 1 X 2 (X 2,1 X 2,2 X 2, ) X 2 1 X 1,j ˆv X 2,j ˆv (X 1,j X 1 ) 2 (X 2,j X 2 ) X r (X r,1 X r,2 X r, ) X r 1 X r,j ˆv r 1 (X r,j 1 X r ) 2 â 1 r 1 ( X i X) 2 ˆv ˆµ X 1 X i ˆv 1 1 ˆv i (X i,j r r r( 1) X i ) 2 Note that (1) Ẑ ad (1 Ẑ)ˆµ are idepedet of i; (2) it is possible that â could be egative due to the substractio. Whe that happes, it is customary to set â Ẑ0, ad the Bühlma estimate becomes ˆµ X. 11 The Die-Spier Model X k (A i B j ) I k S k (A i B j ) (I k A i ) (S k B j ) where the frequecy of claims I k A i Beroulli(p i ), ad S k B j is the severity of claims. P [A i B j ] P [A i ] P [B j ]. Bayesia Estimate P [ X x (A i B j )] id. P [X k x k (A i B j )] k1 P [X k x k (A i B j )] x k>0 P [I k 1 A i ] P [S k x k B j ] p i P [S k x k B j ]. P [X k 0 (A i B j )] P [I k 0 A i ] 1 p i. P [(A i B j ) X x] P [ X x (A i B j )] P [A i B j ]/P [ X x]. P [ X x] P [ X x (A i B j )] P [A i B j ]. P [X +1 x +1 X x] P [X +1 x +1 (A i B j )] P [(A i B j ) X x]. E[X k (A i B j )] E[I k A i ] E[S k B j ] p i E[S k B j ]. E[X +1 X x] m k P [X +1 m k X x] k E[X +1 (A i B j )] P [(A i B j ) X x]. 7

8 Credibility Estimate (combied) µ E[X k ] E[E(X k Θ)] E[X k (A i B j )] P [A i B j ]. v E{V ar[x k Θ]} V ar[x k (A i B j )] P [A i B j ]. V ar[x k (A i B j )] V ar[i k S k (A i B j )] E[I 2 k A i ] E[S 2 k B j ] {E[I k A i ] E[S k B j ]} 2 E[I k A i ] V ar[s k B j ]+V ar[i k A i ] {E[S k B j ]} 2 p i V ar[s k B j ]+p i (1 p i ) {E[S k B j ]} 2. a V ar{e[x k Θ]} E{[E(X k Θ)] 2 } {E[E(X k Θ)]} 2 E{[E(X k Θ)] 2 } {E[X k ]} 2 k (A i B j )] E[X k ]} {E[X 2 P [A i B j ] {E[X k (A i B j )] µ} 2 P [A i B j ]. P C Z X + (1 Z) µ where Z /( + k) ad k v/a. Credibility Estimate (separated) Frequecy: µ F E[I k ] E[E(I k Θ A )] E[I k A i ] P [A i ] p i P [A i ]. v F E{V ar[i k Θ A ]} V ar[i k A i ] P [A i ] p i (1 p i ) P [A i ]. a F V ar{e[i k Θ A ]} E{[E(I k Θ A )] 2 } {E[E(I k Θ A )]} 2 E{[E(I k Θ A )] 2 } {E[I k ]} 2 {E[I k A i ] E[I k ]} 2 P [A i ] {E[I k A i ] µ F } 2 P [A i ] i µ F ] [p 2 P [A i ]. P F Z F Ī + (1 Z F ) µ F where Z F /( + k F ) ad k F v F /a F. Severity: µ S E[S k ] E[E(S k Θ B )] E[S k B j ] P [B j ]. v S E{V ar[s k Θ B ]} V ar[s k B j ] P [B j ]. a S V ar{e[s k Θ B ]} E{[E(S k Θ B )] 2 } {E[E(S k Θ B )]} 2 E{[E(S k Θ B )] 2 } {E[S k ]} 2 {E[S k B j ] E[S k ]} 2 P [B j ] k B j ] µ S } {E[S 2 P [B j ]. P S Z S S+(1 Z S ) µ S where Z S S /( S +k S ), k S v S /a S ad S is # of o-zero claims. P C P F P S. 8

9 12 The Parametric Buhlma-Straub Model Assume X 1 Θ, X 2 Θ,..., X Θ are idepedet distributed with commo hypothetical mea but differet process variaces µ(θ) E[X j Θ θ] V ar[x j Θ θ] v(θ) m j where m j is a kow costat measurig exposure, j 1,..., (m j could be the umber of moths the policy was i force i past year j, or the umber of idividuals i the group i past year j, or the amout of premium icome for the policy i past year j). Let the expected value of the hypothetical mea the collective premium µ E[µ(Θ)], the expected value of the process variace v E[v(Θ)] ( E[V ar(x j Θ)] E[v(Θ)]/m j ), ad the variace of the hypothetical mea a V ar[µ(θ)] V ar[e(x j Θ)]. The ad E[X j ] E[E(X j Θ θ)] E[µ(Θ)] µ, V ar[x j ] E[V ar(x j Θ)] + V ar[e(x j Θ)] E[v(Θ)] m j + V ar[µ(θ)] v m j + a, for j 1, 2,...,. Also, for i j, Cov[X i, X j ] a. The credibility premium ˆα 0 + ˆα j X j miimizes where ad ˆα 0 {[ Q E µ +1 (Θ) α 0 µ 1 + a m/v v/a m + v/a µ ˆα j a v ˆα 0 µ m j with k v/a. The credibility premium is the ˆα 0 + ˆα j X j k m + k µ + m j m + v/a ] 2 } α j X j m m + k m j X j m k m + k µ m m + k mj m (1 Z) µ + Z X, where Z m/(m+k) ad X ( m j X j )/( m j ). Note that if X j is the average loss per idividual for the m j group members i year j, the m j X j is the total loss for the m j group members i year j, ad X is the overall average loss per group over the years. The credibility premium to be charged to the group i year +1 would be m +1 [Z X +(1 Z) µ] for m +1 members. 9

10 For the sigle observatio X, the process variace is [ V ar[ X θ] m j X j V ar ] m θ ad the expected process variace is The hypothetical mea is E[ X θ] [ 1 E m m 2 j m 2 V ar[x j θ] E[V ar( X Θ)] E[v(Θ)] m v m. the variace of the hypothetical meas is ad the expected hypothetical meas is Therefore, the credibility factor is ] m j X j θ V ar[e( X Θ)] V ar[µ(θ)] a, E[E( X Θ)] E[µ(Θ)] µ. Z v/(am) m m + v/a. m 2 j m 2 v(θ) m j m j m E[X j θ] µ(θ), v(θ) m, 13 The No-parametric Buhlma-Straub Model Let r group policyholders be such that the i th policyholder has i years of experiece data, i 1, 2,..., r (r 2). Let m i,j deote the umber of exposure uits ad X i,j be the radom variable represetig the average claim amout per exposure uit of the i th policyholder durig the j th policy year for j 1,..., i, i + 1 ( i 2) ad i 1,..., r. We would like to estimate E[X i,i +1 X i,1,..., X i,i ] for i 1,..., r. Let X i (X i,1,..., X i,i ) deote the radom vector of average claim amout ad m i (m i,1,..., m i,i ) be the radom vector of the umber of exposure uits for the i th policyholder, i 1,..., r. Furthermore, we assume (1) X 1,..., X r are idepedet; (2) For i 1,..., r, the distributio of each elemet X i,j (j 1,..., i ) of Xi depeds o a (ukow) risk parameter Θ i θ i ; (3) Θ 1... Θ are idepedet ad idetically distributed radom variables; ad (4) Give i, X i,1 Θ i,..., X i,i Θ i are idepedet. For i 1,..., r ad j 1,..., i ( i 2), defie E[X i,j Θ i θ i ] µ(θ i ) ad V ar[x i,j Θ i θ i ] v(θ i )/m i,j. 10

11 Let µ E[µ(Θ i )] deote the expected value of the hypothetical meas, a V ar[µ(θ i )] V ar[e(x i,j Θ i )] deote the variace of the hypothetical meas, ad v E[v(Θ i )] m i,j E[V ar(x i,j Θ i )] deote the expected value of the process variaces. X 1 (X 1,1 X 1,2 X 1,1 ) X m 1,j X 1,j ˆv 1 1 m 1,j (X 1,j m X 1 ) 2 1 m 1 (m 1,1 m 1,2 m 1,1 ) m 1 m 1,j X 2 (X 2,1 X 2,2 X 2,2 ) X m 2,j X 2,j ˆv m 2,j (X 2,j m X 2 ) 2 2 m 2 (m 2,1 m 2,2 m 2,2 ) m 2 m 2,j X r (X r,1 X r,2 X r,r ) X r 1 r m r,j X r,j ˆv r 1 r m r,j (X r,j m r r 1 X r ) 2 r m r (m r,1 m r,2 m r,r ) m r m r,j m i ( X i X) 2 (r 1) ˆv m i X i ( i 1) ˆv i â m 1 ˆµ X ˆv m 2 i m i ( i 1) m I this case, X i 1 i m i,j X i,j m i i is the average claim amout per exposure uit ad m i m i,j is the total umber of exposure uits for the i th policyholder durig the first i policy years for i 1,..., r, ad X 1 m i m X i 1 m i m i,j X i,j is the overall past average claim amout per exposure uit of the r policyholders where i m m i m i,j is the total exposure uits. The Bühlma estimate for the i th policyholder ad the ( i + 1) th policy year is E[X i,i +1 X i,1,..., X i,i ] Ẑi X i + (1 Ẑi) ˆµ, i 1,..., r, where Ẑi m i /(m i + ˆk) ad ˆk ˆv/â, ad the credibility premium to cover all m i,i +1 exposure uits for policyholder i i the ( i + 1) th policy year is m i,i +1 E[X i,i +1 X i,1,..., X i,i ]. Note that 11 1

12 (1) Ẑi is depedet o i; (2) it is possible that â could be egative due to the substractio. Whe that happes, it is customary to set â Ẑi0, ad the Bühlma-Straub estimate becomes ˆµ X; (3) if m i,j 1 ad i for all i ad j, the m i, m r, ad the ordiary Bühlma estimators are recovered. The method that preserves total losses (TPTL): the total losses o all policyholders is T L r m i X i, ad the total premium is T P r m i [Ẑi X i + (1 Ẑi) ˆµ]. The [ ˆµ Ẑi Ẑ j ] X i which is a credibility-factor-weighted average of Xi s with weights w i Ẑi, i 1,..., r. Ẑ j [ ] mi Compare the alterative ˆµ with the origial ˆµ X i, which is a exposure-uitweighted average of Xi s with weights w i m i m j, i 1,..., r. Note that m j (1) use X m i X i for â of ˆk of ˆµ. m j (2) The differece of these two credibility premiums for policyholder i based o differet estimators of µ is (1 Ẑi) (ˆµ ˆµ) The above aalysis assume that the parameters µ, v ad a are all ukow ad eed to be estimated. If µ is kow, the ˆv give above ca still be used to estimate v as it is ubiased whether µ is kow, ad a alterative ad simpler ubiased estimator for a is ã m i m ( X i µ) 2 r m ˆv. If there are data o oly oe policyholder (say policyholder i), ã with r 1 becomes ã i m i m i ( X i µ) 2 1 m i ˆv i ( X i µ) 2 ˆv i m i ( X i µ) 2 i m i,j (X i,j X i ) 2. m i ( i 1) 12

13 14 Semi-parametric Estimatio Assume the umber of claims N i,j m i,j X i,j for policyholder i i year j is Poisso distributed with mea m i,j θ i give Θ i θ i, that is m i,j X i,j Θ i θ i Poisso(m i,j θ i ). The E[m i,j X i,j Θ i ] V ar[m i,j X i,j Θ i ] m i,j Θ i or Therefore, µ(θ i ) E[X i,j Θ i ] Θ i ad v(θ i ) m i,j V ar[x i,j Θ i ] Θ i. µ E[µ(Θ i )] E[Θ i ] E[v(Θ i )] v. I this case, we could use ˆµ X to estimate v. Example: Assume a (coditioal) Poisso distributio for the umber of claims per policyholder, estimate the Bühlma credibility premium for the umber of claims ext year. umber of claims k total umber of isureds r 0 r 1 r 2... r k r Assume that we have r policyholders, i 1 ad m i,j 1 for i 1,..., r. Sice X i,1 Θ i Poisso(Θ i ), we have E[X i,1 Θ i ] V ar[x i,1 Θ i ] Θ i, µ(θ i ) v(θ i ) Θ i ad µ E[µ(Θ i )] E[Θ i ] E[v(Θ i )] v. Moreover, ˆµ X 1 X i,1 1 k r r [ j r j ]. Sice V ar[x i,1 ] V ar[e(x i,1 Θ i )] + E[V ar(x i,1 Θ i )] V ar[µ(θ i ] + E[v(Θ i )] a + v a + u ad E[X i,1 ] E[E(X i,1 Θ i )] E[µ(Θ i ] µ, implyig that X i,1, X i,2,..., X r,1 are idepedet radom variables with commo mea µ ad variace a + v a + µ. Sice the sample mea of X i,1, X i,2,..., X r,1 is a ubiased estimator of V ar[x i,1 ] a + v a + µ, we have 1 [X i,1 r 1 X] 2 V ar[x ˆ i,1 ] â + ˆv â + ˆµ, or â 1 r 1 [X i,1 X] 2 ˆµ. The ˆk ˆv/â ˆµ/â, Ẑ 1/(1 + ˆk) ad the Bühlma credibility premium for the umber of claim X i,1 ext year is ˆX i,2 Ẑ X i,1 + (1 Ẑ) ˆµ, for X i,1 0, 1,..., k. 13

14 15 Parametric Estimator Assume give i, X i,j Θ i are idetically ad idepedetly distributed with probability desity fuctio f Xi,j Θ i (x i,j θ i ), ad Θ 1, Θ 2,..., Θ r are also idetically ad idepedetly distributed with probability desity fuctio π Θ (θ). Let X i (X i,1, X i,2,..., X i,i ); the the ucoditioal joit desity of Xi is f Xi ( x i ) f Xi,Θ i ( x i, θ i )dθ i f Xi Θ i ( x i θ i )π Θi (θ i )dθ i id. i [ f Xi,j Θ i (x i,j θ i )]π Θi (θ i )dθ i. The likelihood fuctio is give by r r { L f Xi ( x i ) i [ f Xi,j Θ i (x i,j θ i )]π Θi (θ i )dθ i }. Maximum likelihood estimator of the associated parameters are chose to maximize L or logl. 16 Exact Credibility Whe the Bayesia premium the credibility premium, we say the credibility is exact. Recall that the solutios of the ormal equatios, α 0, α 1,..., α yield the credibility premium α 0 + α j X j which miimizes ad {[ Q E µ +1 (Θ) α 0 {[ Q 1 E E(X +1 X) α 0 {[ Q 2 E X +1 α 0 ] 2 α j X j }, ] 2 } α j X j ] 2 α j X j }. If the Bayesia premium, E(X +1 X), is a liear fuctio of X 1, X 2,... X (i geeral, it is NOT), that is E(X +1 X) a 0 + a j X j, the α j a j for j 0, 1...,, ad therefore Q 1 0. Thus, the credibility premium α 0 + α j X j a 0 + a j X j E(X +1 X), ad the credibility is exact. I summary, the Bühlma estimator α 0 + α j X j is the best liear approximatio to the Bayesia estimate E(X +1 X) uder the squared error loss fuctio Q 1. Recall that i liear regressio, Y i α 0 + α j X j + ɛ i, where ɛ i N(0, σ 2 ), for i 1, 2,..., m. 14

15 [ m ] [ ˆα 0, ˆα 1,..., ˆα are chose to miimize Q E ɛ 2 m i E (Y i α 0 α j X j ) ]. 2 Therefore, the regressio lie Ŷi ˆα 0 + ˆα j X j correspods to the credibility premium α 0 + α j X j, ad observatio Y i correspod to the Bayesia premium E[X +1 X ( x) i ], i 1, 2,..., m. Coditio for exact credibility Suppose that X j Θ θ is idepedetly distributed ad is from the liear expoetial family ( liear meas the power of the expoetial is a leaer fuctio of x j ) with probability fuctio f Xj Θ(x j θ) p(x j) e θx j, q(θ) for j 1, 2,..., + 1, ad Θ has probability fuctio π Θ (θ) [q(θ)] k e µkθ c(µ, k) where θ (θ 0, θ 1 ) with π Θ (θ 0 ) π Θ (θ 1 ) 0 ad θ 0 < θ 1. The µ E Θ [µ(θ)] E[E(X j Θ)] E[X j ] ad the posterior distributio is where k + k ad π Θ X(θ x) [q(θ)] k e µ k θ, θ (θ c(µ, k 0, θ 1 ), ) The Bayesia premium is µ X + µ k + k + k X + k + k µ. E[X +1 X] E Θ X[µ +1 (Θ)] θ1 where Z /( + k) ad k v a E[V ar(x j Θ)] V ar[e(x j Θ)]. θ 0 µ(θ)π Θ X(θ x)dθ µ Z X + (1 Z) µ, Note that the prior distributio π Θ (θ) is a cojugate prior distributio (a prior distributio is a cojugate prior distributio if the resultig posterior distributio is of the same type as the prior oe, but perhaps with differet parameters). Theorem: If f Xj Θ(x j θ) is a member of a liear expoetial family, ad the prior distributio π Θ (θ) is the cojugate prior distributio, the the Bühlma credibility estimator is equal to the Bayesia estimator (i.e. exact credibility) assumig a squared error loss fuctio. 15

16 coditioal distributio X j Θ Beroulli(Θ) Poisso(Θ) prior distributio Θ Beta(α,β) Gamma(α,β) µ(θ) E[X j Θ] Θ Θ α µ E[E(X j Θ)] αβ α + β v(θ) V ar[x j Θ] Θ(1 Θ) Θ αβ v E[V ar(x j Θ)] αβ (α + β + 1)(α + β) αβ a V ar[e(x j Θ)] αβ 2 (α + β + 1)(α + β) 2 k v/a α + β 1/β β Z + α + β β + 1 credibility premium Z X α + (1 Z) Z α + β + (1 Z) αβ posterior distributio Θ X ( ) ( ) β Beta X i + α, β + X i Gamma X i + α, β + 1 ( ) X i + α β X i + α posterior mea E[Θ X] + α + β β + 1 predictive distributio X +1 X ( X i + α ) ( ) β Beroulli NB X i + α, + α + β β + 1 ( ) X i + α β X i + α predictive mea E[X +1 X] + α + β β + 1 Note that the predictive mea (the Bayesia premium), E[X +1 X] the posterior mea, E[Θ X] ( E[µ +1 (Θ) X] sice µ +1 (Θ) E[X +1 Θ] Θ) the credibility premium, Z X + (1 Z) µ. I some situatios, we may wat to work the logarithm of the data (W j logx j, j 1, 2,..., ). The µ log (Θ) E[W j Θ] E[logX j Θ], v log (Θ) V ar[w j Θ] V ar[logx j Θ], µ log E[µ log (Θ)] E[W j ] E[logX j ], v log E[v log (Θ)] E[V ar(w j Θ)] E[V ar(logx j Θ)], a log V ar[µ log (Θ)] V ar[e(w j Θ)] V ar[e(logx j Θ)], ad Z log /( + v log /a log ). Thus, logc Z log W + (1 Z log ) µ log, or C e Z log W +(1 Z log ) µ log, which is deoted by C log ad is ot ubiased because we use liear credibility to estimate the mea of the distributio of logarithms. Recall Ĉ Z X + (1 Z) µ where µ E[µ(Θ)] E[E(X j Θ)] E[X j ], so E[Ĉ] Z E[ X] + (1 Z) E[X j ] µ E[X j ]. To make C log ubiased, let C log c e Z log W +(1 Z log ) µ log where c is determied by E[e W j ] E[X j ] E[C log ] c E[e Z log W +(1 Z log ) µ log]. 16