Estimating Between Line Correlations in Insurance: Parameter Uncertainty, Lecture notes of Applied Statistics

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Estimating Between Line Correlations

Generated by Parameter Uncertainty

Glenn Meyers, FCAS, MAAA

Estimating Between Line Correlations

Generated by Parameter Uncertainty

Glenn Meyers

Insurance Services Office, Inc.

Abstract When appiymg the collect~ve nsk model to an analysis of insurer capttal needs, It is crucial to constder the effect of correlation betaeen lines of msurance. Recent^ tvork sponsored by the Comnuttee on the Theoty of Risk has sparked the development of methods that include correlation m the collecti\e risk model. One of these methods IS butlt around the view that correlation IS generated by parameter uncertamty affecting several lines of msurance simultaneously.

This paper uses simulation analyses to esplore the properties of both clawcal and Bayesmn methods of quantifymg parameter uncertainty. We conclude that m order to get suffícient accuracy to determine the necessary capnal, one must use the combined data of several insurers Using the combined data of several insurers forces us to constder a collecttve nsk model where parameter uncertamty affects several insurers - as well as several lines of tnsurance - simultaneously

RecogmLmg this problem. the CAS Committee on the TheoE of Risk commissioned Dr. Shaun Wang to debelop versions of the collectlve risk model that do not require one to ilssume Independence betiveen Iones of msurance Th~s work led lo a paper titled .‘Aggregatlon of Correlated Risk Portfolios Models & Algonthms” which 1s to appear m the ne-t Lolume of the ~ro,eedrngs c$rhe Curuaf~y.kruar~al S¿met

Insplrcd by Dr Wang‘s work, \e followed v,ith a dlscuwon of his paper, Mqers [ 19991, that focused on a verslon of the collective nsk model uhere the cltim count dlstribution for cach Ilne of insurance xras conditionally independent given a parameter a Treatmg^ CI as a random vanable leads to a particular kmd of dependence bet\veen lines of msurance.

In thls paper \ve propose a methodoloE for estnnatmg the variance of a and explore the data requirements necessary to provide reliable estimates of thls vanance.

2. The Collective Risk Model

For the hti Ime of msurance let ph = Espected claim severity, CJ,= Variance of the claim severity distributlon. hh = Espected claim count. and jih + ch.hi = Variance of the clam count distnbution Followng Heckman and Me):ers [ 19831. ne call ch the contaglon parameter Ifthe^ claim count dlstrlbutlon IS. Pmsson, then ch = 0: negative binomial. then ch > 0; and bmomial with n lrials, then ch = -I/n

A good way to wew the collecttve nsk model is by a Monte-Carlo simulation.

Simulation Algorithm #l

The Collective Risk Model Assuming Independence Between Lines of Insurance

1, For lmes of rnsurance 1 to n, select a random number of claims, I(h, for each line of msurance h.

2. For each line of insurance h, select random clarm amounts Zhk, for k = 1, (^) Kt, Each Z,,k has a common distribution (Zk}.

3. Set X, = EZt+.

k=l

4 SetX=EXh.

h=l The collective risk model describes the dtstribution of X

Meyers [ 1’9’991 sho\s that lf Kh is independent of Kd for d 3 h, and íT.+is mdependent of h we ha\ e:

and Cov[Xd,Xh]=Oford#h^ (2.2)

We now introduce parameter uncertainty that affects the clrum count distribution that affects several Imes of msurance simultaneously. We partition the Imes of msurance into cowrmnce groups {G,}. Our next version of the collecttve risk model is deftned as follows.

The ultimate purpose of thls papcr IS to dlscuss the estlmatlon of the 6,‘s from clami count data, so \e remove clalm sever+ from the aboye equatlons by settmg cach pi, = 1 and of> = 0 111s yves us:

C’ot[K<,>.K,,] = g, -L -h,, (2.6)

Co[K ,,,. K,,]=Var[K,]=h,“+(c,,+g,~c,, g,).h?,,. (^) (2 7) andforl#J Cov[K,,K,,] :- 0. (^) (2 w

3. The Impact of the Covariance Generator on Required Capital

The purpose of thls paper 15 IO y\e some estlmators of the co\ ariancc gcncrntor. g. To thls end, \e gve an esample on a hypothetlcal Insurer \vrttmg tòur Iones of~nsurance The Insurcr espects 1.000 clams In each lme. and the contayon parameter for each Imc 1s equal lo 0 ll2 The coianance generator 1s equal to 0 0-I The clalm scverlt! dlstrlbutlons are gl\en 111Meyrrs ~109~~~ Tablcs 3 1 and 3 2 gve various summar) statlstlcs of the Insurer’s aggregate loss dlstnbrmon

Table 3.

Aggregate Summary Statistics

Aggregate Mean 101.581, Aggregate Std De\ 23.270.

Table 3.

Claim Severity and Claim Count Statistics

Dlstributlon Name EICountJ StdlCount] E]Scvcr~t?] StdlSe\erityj ElTotal Loss)

GL-% I M 1000 248 60 36.966 16 124.853 59 36.X6,

CL-ã5M I 000 24x 60 40.3-M 87 160.21851 40.348.

AL-â I M IOOO 24X.60^ 11.456^65 76,434^03 1 I .456,

AL-FSM I no0 248.60 I2.809 55 99,730 27 12,809~

Table 3 3 and 3 4 give the correlations betaeen each of the lines of insurance for the clarrn counts. and for the total tosses.

Table 3.

Claim Count Correlation Matrix

CL-SIM GL-S5M AL-SlM AL-S5M CL-$ I M 1.ooo 0 647 0 647 0. CL-$SM 0.647 1.000 0 647 0. AL-$lM 0.647 0.647 1.ooo 0. AL-S5M 0.647 0.647 0 647 1.

Table 3.

Total Loss Cordation Matrix

GL-$1 M CL-$5M AL-$IM AL-SSM

GL-$1 M 1.000 0.53 1 0.453 0.

GL-S5M 0 531 1 000 0 440 0.

AL-S 1M 0.453 0 440 1.000 0351

AL-S5M 0.423 0410 0351 I.

We no\v consider some capttal requtrement formulas Let X be a random variable representmg the insurer’s aggregate loss. Let: F(s) = Pr(X < s) f(x) = F’(x) 0 = Standard Deviatton of X C = Required lnsurer Capnal Then the requtred capttal can be detined by one of the followmg equattons I Probabthty of Rutn Formula F(C^ + E[X])^ = 1^ -E

2. Expected Poltcyholder Delictt Formula:

E[Xl =q 3 Standard Devtation Formula C = T.a ### 4. The Likelihood Function for a Multivariate Claim Count Distribution From this point fonvard, \ve shall assume there is only one COI anance group and drop the subscnpts i and j m Simulatlon Algonthm # As we estnnate the g parameter across different hnes m a covanance group, we Wll be estlmatmg the parameters. xt,and ch. of each claim counl dlstnbution slmultaneously In effect. we \\~ll be estlmatmg the parameters of a muhivarlate dlslnbutlon on the random vector K = {Kh} At thls polnt. It IS helpful to adopt the vector notation C = {c,,}and i = {&} The negatlve bmomial claim count dlstnbutlon. condlConal on a. will be obtamed from the standard negatiue blnomlal distnbution by multlplymg ils mean, hh, by a Following Mqers 1ISUS]. we shall use the follo\vlng form of the negatike binomlal dlstributlon for the probablllty of kh condltlonal on a Pr{K, = k,la} = T^ r(lic,^ tk,,)^ tc,,a&)^ k. l71íc,,)-r(h,~ + 1) (I+c,,ah,,)““‘i” Grven g 2 0. define’: a,=I-&,az=l.anda,=l+fi. and Pr{a=a,}=l/6.Pr{u;az}=2/3.and Pr{a=a,}=l/G. One can easily venfy that E[a] = 1 and Varia] = g The condltlonal hkellhood of a clalmcount lector kla = {khla} is glven b) î(E,h,Cla)= nPr(K, = k,Iaj h (4 1) (4.2) #### (J 3) As I\C go about the cornputatlonal efforts descnbed belo\v. de ~111 \\ork \\lth the log- libchhood functlons ### 5. Masimum Likelihood Estimation Undrr thc assumptlon that clalms are generated b! the process descnbed m Slmulatlon Algonthm fil. .a” insurcr \\izhlng to csllmate the parnmeters 7,. C and gtight gaiher data hhe that III the follo\\~ng table from IIS OI\TI clams expenence ### Table 5. I ### Insurer Data for Estimating E and g ### Exposure by Line and Yeal Ycar Lene 1 Lene 2 Lene 3 Lene 4 I wx I OO^80 40 I 0’)7 100 x0 JO 20 1YK loo^ X0^40 1Cl')5 100 X0 40 20 1‘N4 100 x0 JO 20 ### Claim Count by Line and Year Lene 1 Lme 2 I.lne 3 L1ne 4 1‘)0X 153 131 53 31 I ‘90 7 96 17 JI 20 I ‘l’xí 53 x0 35 I 6 I 995 92 72 45 30 1994 02 ‘?O 43 I 0 Estlmated o CI72tj Frequenc‘ I^1475 1 1350^ 1 1300 ### Based on thls and other similar simulations we conclude that esttmatmg c and g in this ### manner can lead to bmsed and highty volatile results ### We now examine some other estimation methods ### The tirst alternauve is to combine the data of several “sinular” insurers Let A be the set ### of msurers and let a EA We created 40 nearly tdentical “coptes” of our insurer and ### simulated the MLE’s for c and g Table 5.3 belo\v shows the exposures and clatm counts ### for the tirst two msurers in a typical stmulation ### When combming the data of several msurers we maximrze the log-likelihood expression. (5 2) ### Table 5. ### Multi-lnsurer Data for Estimating c and g ### Lnsurer f# 1 Exposurr by Line and Year Year Lene I Lene 2 Line 3 Lme 4 1998 100 x0 40 20 19’17 100 80 40 20 1996 100 80 40 20 1995 100 80 40 20 1994 100 80 40 20 ### Claim Count by Line and Year Line I Lene 2 Lene 3 Lene 4 1998 69 69 53 20 1997 09 X0 51 17 1996 101 7x hX 18 1995 129 94 47. 17 1994 82 76 30 15 ### hurer #2 Exposurr by Line and Year Year Lene 1 Lene 2 Line 3 Lene 4 199x 20 100 x0 JO 1997 20 100 X0 40 I 9% 20 1OO no 40 199s 20 100 X0 40 I w-1 20 1OO no 40 ### Claim Count by Line and Year Line 1 Lene 2 Lme 3 Lene 4 1998 2S 10X 64 4s 1997 18 88 75 42 1996 22 x7 94 44 1995 22 130 69 47 1994 30 l-17 III 6X ### Insurer #3 Exposure by Line and Yea? Year Lene I Lme 2 Lene 3 Line 4 u ll u u u Estlmated Frequencq , ,oo88^ 1 0077^ 1 0088^ 0 9877 Then accordmg to Bayes’ Theorem, the posterior likehhood of each (c,,g,) will be proportlonal to’ As an Illustration, suppose that we choose a prior so that the pij’s are equally likely. For one simulated Ir<?} based on a single msurer’s esposure \re obtened the follo\\mg postcnor dlstnbutlon of (c,,gj), nhlch \ve show (part oT) graphically. ### Craph 6. Poslcrior Likclihood lora Sin& lnsunx with a UnifoormPnorDistribution As an esample, we construct a pnor dlstnbutlon so that P,, = flt(Q,c,.g,), 3.Y (6.2) where { i;:.} comes from the (simulated here, but m practlce real) data of the 40 “peer group” insurers gwen above. We obtamed the follo\\ing posterior distributlon for the same msurer that we show graphically. ### Belo\\. UC xx111shon how to use the posterior dlstributlon ZE input into the collectlve nsk model, as described m Simulation Algorithm HZ. ### Graph 6. Posten’or Likclihood fora Sin& Inm-ernith a We noi\ cnlcul3tc thc moments of the aggregate loss dlstnbutlon descrlbcd by Slmulallon Alg0nthn1 03 (7 1) ‘1‘0 c~~~lcula~ethe \ wanccs and COI annnces analogous to S~mulat~on Algorlthm #2. \\e s~mply rc+x the \ anance g, m tlquatlons 2 3.2 4.2 6 and 2 7 \\lth the ehpresslon g, ‘2,; ‘2, g,’ Let k>’ bc a 1ec~or of obscrwd cla~m counfs for thc “mdus~r) ” rn )‘ear J (^) An examplc of such a tector bnsed on l.able 5.3 IS c,:,,, = (69. 09. 53, 20. 2s. 108. 6-k 45. j’ Slnularl! let h: be a \ector of cxpccred clalm COUII~S for thc”mdustn“ III year > Thc Ilhellhood funct~on of L,’ condrtlonal nn a ” IS ylen b! 771~ assoclated lo;-llhcllhood functlon IS ylven b! Gven 9” 2 0 deEne a” = I-@.a; =l.anda;\ =l+m, and (^) (7.5) Pr{a~~=a~}=I/6,Pr{a”=a~}=2/3,andPr{aA=a;\}=1/ The uncondluonal log-llhehhood funcuon is then gven by: ### 8. Manimum Likelihood Estimation Revisited Conslder the follomng t\vo situations 1 g-r>Oandg.‘=O 2 g = 0 and 8,’ = r > 0. From the msurer’s pomt of vie\v, me two situations are identlcal Its espected clarm counls are muluphed by a random number each year But from the polnt of vle\v of one \\ho IS uymg IO esumate the vanance of the random multipller. the srtuations are different In the lirst situatlon. a nen- a IS pmked for each msurer for each year In the second situalion’ a” is p~ked owz each year for all rnsurers. The esumator should use ihe log-hhelihood function in Equauon 4.6 In the second sltuatlon the esumator should use the log-lrkellhood function In Equation 7 6 We dld 100 slmulations of our 40 insurers where the claim counts are generated b) Simulanon Ngonthm #3, \\lth c = 0 02, g = 0 and g* = 0.04. We hen estimated c and “g“ using maximum hkehhood on Equatron 4.6, wth the follo\\ing results