THE UNEARNED NO CLAIM BONUS. C. P. WELTEN Amsterdam

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1 THE UNEARNED NO CLAIM BONUS C. P. WELTEN Amsterdam I. The claims experience f a mtrcar insurance is assumed t give sme indicatin abut the risk (basic claim frequency) f that insurance. The experience rating systems in mtr insurance are based n this suppsitin. In these systems the premium t be paid in sme year is a functin f the individual claims experience f preceding years. That prcnliunl can be cnsidered as the difference between : -- a basic premium, being the expected value f the premium t be paid fr insurances with a basic claim frequency equal t tile mean value f the basic claim frequency f the prtfli r the tariff class. -- a psitive r negative bnus; althugh in the negative case the wrd bnus is misleading this wrd will nevertheless be used fr bth cases in this paper. The bnus usually cnsists f at least tile fllwing cmpnents: a. a cmpnent cncerning the individual claim frequency. b. an individual randm factr. c. a cllective randm factr. Other factrs, like the effect f the trend in the claim frequencies and the effect f the dependence f the claim frequency f age and experience f the driver arc nt taken int accunt in this paper. 2. The meaning f the three cmpnents mentined in sectin I can be den-tnstrated by the fllwing example. The structure functin U(x) f the prtfli, being the distributin functin f the individual basic claim frequencies x is assumed t be defined by d U (x) = I) q q SX ~ -1 e -- ~ q

2 26 THE UNEARNED NO CLAIM BONUS representing a gamma distributin with expected value q and variance qb. The prbability Pt fr an insurance t have i clailns in a certain year is suppsed t be p,=ff(xy),e-~v i! d u (~) d C (y) In this frmula the factr y has the same value fr all insurances in the prtfli in a certain year but varies frm 3,ear t ),ear accrding t the distributin functin G(y) with expected value i and variance a2v. The amunt f a claim is suppsed t be i. Fr a2v = several authrs have shwn that if an insurance has had n claims in the preceding t ),ears the risk premium Pt+~ ~,a fr the year t + I shuld be' q -4- b.u Pt+ll,,t -- z + b.t This premium can be explained as the difference between tile basic premium q and the bnus: b Bt+t,,,t = I + b.-t (ql-- n) If the claim frequency is x, and if the value attained by y in the 3rear sis Ys(S = 1,2...t), t with E Ys = Yt, tlle three cmpnents mentined in sectin I can be frmulated as fllws' a. cmpnent B(a) cncernh~g the individual claim frequency' b B(a) = i--+ ii (ql--xt) b. cmpnent B(b) cncerning the individual randm tactr" b e(b) -- I +bt (x Y, -- n) c. cmpnent B(c) cncerning the cllective randm factr" b B(c) -- (xt -- x Y~) i +bt S The term "unearned bnus" in the title f this paper is used fr the sum f B(b) and B(c), these cmpnents nt being due t the individual claim frequency. I

3 THE UNEARNED NO CLAIM BONUS B(a) tends asympttically t q- x fr increasing t; B(b) and B(c) bth tend t zer. Frm the pint f view f the insured a bnus system like this ne can therefre be said t be asympttically crrect. 4. Vrin the pint f view f the insurance cmpany, hwever, the situatin is mre cmplicated. The prfit f the cmpany in a certain year is due partly t the difference between the expected number and the actual number f claims and partly t the difference between the expected and the actual mean amunt f the claims. Accrding t the suppsitin that all claims are equal t I, the secnd part f the prfit is assumed t be zer; where in the fllwing the wrd prfit is used this wrd stands fr "part f the prfit due t the nulnber f claims". If in sme year y is attaining a lw value r if the individual randm effects are leading t a lw ttal number f claims, r if bth influences are wrking, tw cnsequences can be expected: a. a psitive prfit in that year. b. a tendency t psitive unearthed bnuses in future years, prbably leading t negative prfits in thse years. Fr this reasn it seems necessary t pay a part t the prfit t a "bnus reserve" t cver the future unearned bnuses cncerning that prfit. In this paper we will try t estimate the part f the prfit t be reserved fr a special case and t shw that this part may be t large t neglect in the practice f mtrcar insurance. 5. An insurance prtfli cnsists f Mt plicies, all being t years in existence, ut f which Meat (n =, I... " i =, I... ) with n claims in that perid f t years and with i claims in the year t n n I,, ~,, The structure functin and the distributin functin G(y) are assumed t be as mentined in sectin If the factr y attained the value Ys in year s (s = i, 2... t), with Yt = t E y,, the risk premium Pt+~ [,, fr an insurance in grup ]14~, fr the year t + I shuld be, as can be demnstrated easily:

4 28 THE UNEARNED NO CLAIM BONUS Ptt+ll,,t--I q + bn +byt The expected value f Pt+l ],,t x~dth respect t Y~ is: pt4 t q + bn,,,,.... d G(>) d C (y,)... d G (y~).. I+b ~ ys 8 l A useful apprximatin is given by: q + b'n I b et*2 3, In practice the crrectin term b 2 t *~3' (, + bt)~ prves t be less than,i and can therefre be neglected; then the wellknwn expressin mentined in sectin 2 results: q + bn Pt+l I,,t- I -~- bl This frmula will be used in the fllwing. 7. Assuming that the value attained by y in the year t + i is Yt+l we can write fr the prbability P,l,,t that an insurance with ~z claims in the preceding t years will have i claims in the year t + I: m j (x 3%+,) * e-xvt+t (x Yt)" e xvt d U (~) i~ n! Ptl f (x Yt)" e-xyt -- d U (~) which frmula may be reduced t n! i y,+l/ I

5 THE UNEARNED NO CLAIM BONUS 2 9 On accunt f Bayes' therem the a psteriri cnditinal distributin functin G(yt+~IM.. ) f Yt+> given Mnu (n, i =, I, 2... m) is defined by" da(y,+i Ira...) = This frmula can be reduced t 0 ~I (P") m,,.,.. M,at! t d G(y), fi t d ", M,,u! ) " yts+, d G(y) q {I + b (Yt + Y~.I) } ~M, +,Z nm m +s d g(yt+~lm...) =, s - f ~t+~ 0 d C(y) {~ + b (r~ + Y~+O } W, +.~"M"'+s 8. The ttal risk premium Pt+~ t be received in },ear t q- I is M,,t( q + bn) q Mt+ b ~ n Mnt.~ I + bl z + bl The prfit f the year l q--- z (in the sense defined in sectin 4) W is: W = Pt+I -- S The expected value Pt+u+t I.,,u f the premium that will be paid in year t + u + I fr an insuraalce belnging t M,, u is: ~ f q+b(n+i)+b(u--i) q + b(n+i) = I + b(t + Yt+t) dg(yt+l [M..). I + b (t + ) q ~- b ('1, -~ i) i I -~ b (t ~- u- I ~- Yt +1) -- Iq-b(t+u) J x+b(t+yt+o dg(yt+t[m...) The expected value ~t+,~+l I nu f the prenlium that shuld be received Jn year t + u + I fr an insurance belnging t m,,u is'

6 30 THE UNEARNED NO CLAIM BONUS et +tt +1 [ 't~{t SX Yt+u+, (X yt+)_) ~ (X Yt) n e--.~u,+, e -.~Y, d U(x) f (x yt +~)~ (x Yt) ~' e--.~v, +, e--~v, d U(x) d G(yt+u+t) d G(v,+, I M... ) This may be reduced t" fst+u+~ [,,u = {q + b (n + i)} ~{q + b (n + i)} f f dg(yt+tlm...) I + b (Yt, ~ '~'t;1) dg (y,+l [M I +b(t+ yt+t) The unearned part f the bnus in tim year t -~- u + I is" t+u+, I,,.- Pt+~,+~ I-- = q + b (n + i) ( b(i -- yt+t) d_g(yt+t [ 3g...) I + b (t + n) 3 I + b (t + yt ~) We assume that ut f the Mt insurances existing at the end f the 3,ear t + I a number ru. Mlt will still be existing during the year t + u + I (u = I, 2... ). Then the ttal amunt Du f unearned bnuses in the year t + u + I cncerning the prfit f the year t+ Iis: D,, lt ~t n i E M,,u q-+- b (n + I){ b_!i --yr.,)... dg(yt+l 1~[...) I + b (t --t- u) J I -t- b (t + yt+t) After additin ver u and further reductin this leads t: D = 2 Du= [(I + bt) W+ {I+ b(t+i)}s]. f b (I-- yt+t) dg(yt+t ] 31...) ~ ru I -q- b (t + Yt+O... I + b (t + u) 9. If Mt is large and if sme ther cnditins are fulfilled the a psteriri distributin functin G (Yt+~ I M...) tends t:

7 THE UNEARNED NO CLAIM BONUS 31 S S G (yt+, I M... ) ---- if Yt,-~ < Pt+l W + S S G(yt+~IM...) = z if yt+l>~ W-5 S Then the frmula fr D may be reduced t" D= b.w. ~2 ru..,i + b (l + u) S an apprximatin f the relative part f the prfit f any year that has t be paid t a bnus reserve t cver future unearned bnuses cncenung that prfit is given by: [. If we assume: b= O,I _b ~Z... W,, I 4- b (t -5 u) ru = I -- O,IU if u ~ I0; ru = i[.t > IO t = mean "age" f prtfli = 5 D then ~ = 0,24 This result seems t high t neglect in the balance sheet f the cmpany. ii. The aim f this paper is t demnstrate that the bnus reserve in mtrcar i.nsurance may be imprtant in practice, and t estimate this reserve in a special case. The methds described may, hwever, be used in ther cases and als fr ther bnus systems. In case Mt r ~v is s small that G (Yt+~ I M... ) has t be calculated the calculatin is relatively simple. T illustrate the effect f a mre precise calculatin in such a case we mentin that if in the example f sectin i, Mt = ; ~v g =,i; Pt+t = 3.800; S = (s that 1 = 200) the exact value f D i4--- ~ is,i 9 instead f 0, The abve mentined methds are cncerned with the prfit f ne year nly. In fact the bnus reserve has t be frmed as the

8 32 THE UNEARNED NO CLAIM BONUS sum f additins and reductins regarding psitive and negative prfits f successive years and regarding psitive and negative payments f unearned bnuses. The prblems cncerning this pint and cncerning the prcedure fr a negative reserve are nt discussed here. Amsterdam, September 1066.

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