Underwriting Risk. Glenn Meyers. Insurance Services Office, Inc.

Transcription

1 Underwrtng Rsk By Glenn Meyers Insurance Servces Offce, Inc. Abstract In a compettve nsurance market, nsurers have lmted nfluence on the premum charged for an nsurance contract. hey must decde whether or not to compete at the market prce. hs paper deals wth one factor n ths decson rsk. From polcyholder s standpont, the only rsk that matters s nsurer nsolvency. For the nsurer to stay n busness, t has to have suffcent captal to keep ths rsk below an acceptable level. Also, nvestors demand an acceptable return on ths captal. he problem s that the return comes from premums that are charged to ndvdual nsureds, each wth ther own rsk characterstcs. hs paper proposes a way to set standards for acceptng ndvdual nsureds based on the rsk each contrbutes to the nsurer s portfolo. hese standards wll assgn the margnal captal to the nsured. hese standards wll be expressed n terms of the acceptable return on allocated surplus. hey wll take nto account: (1) the varablty of the nsurer s loss; (2) the tme t takes untl all clams are pad; and (3) the correlaton of the nsured s losses wth the nsurer s other losses. We start by llustratng the basc concepts wth smple examples, and fnsh wth a comprehensve example that shows how we can put these standards nto practce.

2 1. Introducton Insurance companes are fnancal nsttutons wth fnancal objectves. hs paper s about lnkng the nsurer s fnancal objectves wth the myrad of ndvdual underwrtng and prcng decsons t makes as t goes about ts busness. From the fnancal pont of vew, the nsurer s msson s the followng. Under normal crcumstances, the nsurer expects ts premum ncome to generate enough money to pay for the nsured losses. On some occasons, nsured losses wll exceed premum ncome and the nsurer wll have to pay for losses out of ts own captal. he nsurer assumes the rsk of fnancal loss to ts customers,.e. the nsureds. Whle the nsurance contract covers losses arsng from accdents that occur n a predetermned perod, the losses themselves can be payable over a much longer perod of tme. he nsurer s owners provde the captal. In return for assumng the nsureds rsk of loss, the owners expect to receve a return on ther nvestment that s compettve wth other nvestments wth smlar rsk. hs return on the owner s captal nvestment s the nsurer s fnancal measure of success. In return for assumng ths rsk, the nsurer collects premum from the nsureds. hs premum s used to pay underwrtng expenses and set up the necessary reserves to pay future losses. he ncome that provdes the return on the owner s captal s derved from two prncpal sources: the underwrtng proft from ts nsurance operatons; and the nvestment ncome from the assets underlyng ts reserves and ts captal. Qute often, the underwrtng proft s negatve. hs s acceptable f the nvestment ncome generated from wrtng the busness s large enough to provde a compettve return on the owner s nvestment. More generally, the nsurer s ncome s the result of numerous underwrtng underwrtng and decsons made by employees of the nsurer. Each decson nvolves a consderaton of the Page 1

3 expected underwrtng proft, the length of tme that the reserve must be held, and the addtonal captal needed to protect the nsurer s solvency. From the owner s perspectve, the results of the ndvdual underwrtng decsons do not matter as long as the total ncome s large enough to provde a compettve return on the nvestment. But ultmately, the nsurer must make ndvdual underwrtng decsons that contrbute, n an actuaral sense, to ts overall fnancal objectves. hs paper descrbes some actuaral consderatons that an nsurer can make to lnk ts underwrtng decsons to ts fnancal objectves. 2. he Cost of Commttng Captal We begn wth a smple model that llustrates how the cost of captal nfluences the prce of an nsurance polcy. 1 he rsk s that an nsurer wll have to pay an amount of $1 at tme,, n nterval [0,t]. he dstrbuton of s unform throughout [0,t]. he probablty of makng a payment at some tme n ths nterval s q. Assume that the nsurer has to hold $1 of captal untl ether the clam s pad, or the lablty expres at tme t. he captal s nvested n a rsk free nterest bearng account wth force of nterest δ. Interest on ths nvestment s pad contnuously to the nsurer. In return for subjectng ts captal to the rsk, the nsurer requres a hgher rate of return wth force of nterest δ r. 1 hs example was motvated by a thought experment about rsk suggested to the author by Legh Hallwell. Page 2

4 Case 1. he clam occurs he nsurer receves contnuous nterest payments at an annual rate of δ untl tme, when the clam s pad. he nsurer s expected rate of return s δ r. he present value of the nsurer s nvestment s: hen: E PV wth clam 1 e PV wth clam = δ z e = δ 1 0 δ r δ r δ r q t d q δ e τ = 1 1 δ tδ δ t t rτ δ r r F HG r I KJ Case 2. he clam does not occur he nsurer receves contnuous nterest payments at an annual rate of δ untl tme t. At that tme the nsurer s captal of $1 s returned. he present value of the nsurer s nvestment s: 1 e PV wthout clam = δ δ r t δ r δ r t 1 e hen: E PV wthout clam = b1 qg δ + δ F HG r + e δ rt e δ rt I KJ If the nsurer s to expose ts captal to the rsk of loss, t must receve at least an amount, P, so that the expected rate of return on ts nvestment of $1 s at least δ r, hat s: 1 = P + E PV wth clam + E PV wthout clam. It s the nsured who must provde P, otherwse the nsurer would not accept the rsk. When prcng nsurance polces, actuares are accustomed to comparng P, the cost of the nsurance needed before they wll voluntarly wrte the nsurance, to q, the expected loss payments. Defne the rsk load R, as ths dfference. In ths example: R = P q Page 3

5 he followng table llustrates how the cost of nsurance can ncrease wth the length of tme that the supportng captal must be held. able 2.1 t δ 6% 6% 6% 6% 6% 6% δ r 10% 10% 10% 10% 10% 10% q E[PV wth clam] E[PV wthout clam] Cost of Insurance Rsk Load We leave t to the reader to nvestgate how the other factors δ, δ r and q affect the cost of nsurance. hs example represents a very smplfed vew of the nsurance busness. A more comprehensve example could nclude the followng consderatons. 1. δ should ncrease wth t. hs s the normal behavor for fxed-rate nvestments. 2. δ r depends upon the return of other nvestments wth comparable rsk and tme commtment, whch n turn depends upon the probablty and the tmng of the nsurer s loss. 3. he losses that are covered by a typcal collecton of nsurance polces are unlmted. he nsurance buyng publc appears wllng to accept the remote possblty that the nsurer won t be able to cover ts clams. here are a number of regulatory and ratng agences that take on the job of determnng the amount of captal that s necessary to assure that the probablty of nsolvency s ndeed remote. 4. An nsurer usually underwrtes several nsureds whose losses are of dfferent amounts, are pad at dfferent tmes and are, more or less, ndependent. he cost of provdng the total coverage depends upon the entre portfolo whle the premum that provdes for ths cost Page 4

6 comes pecemeal from ndvdual nsureds. Snce ndvdual nsureds may dffer n ther varablty of loss and payment tmes, ther effect on the nsurer s fnancal poston may dffer. Although an nsurer s management cannot deal wth these ssues separately, t s also true that a sngle paper cannot adequately cover all these ssues adequately. So n the remanng dscusson we wll restrct our consderatons by: (1) assumng a sngle fxed rsk-free rate of return; (2) assumng a sngle fxed rate of return to the nvestors for bearng the nsurer s rsk; and (3) usng three conventonal actuaral formulas for determnng the nsurer s requred captal. hs paper deals prmarly wth the ssues rased n (4) above. 3. Probablstc Captal Requrement Formulas In order to protect the polcyholders, the busness of nsurance s subject to solvency regulaton. he regulators have the authorty to revoke the nsurer s lcense. In addton, there are a number of prvate agences that rate nsurers on ther ablty to pay clams. hese ratngs are taken very serously by the nsurers because the ratngs have a strong nfluence on ther ablty to attract busness. hese nsttutons put a lower bound on an nsurer s captal. he nsurer s management wll often attempt to duplcate the regulator s and the ratng agences captal requrements. In addton, they may develop ther own probablstc captal requrement formulas that they use for plannng purposes. We now gve a descrpton of three such formulas. Let X be a random varable representng the nsurer s aggregate loss. Let: F( x) = Pr{ X x} f ( x) = F ( x) σ = Standard Devaton of X C = Requred Insurer Captal Page 5

7 hen the requred captal can be defned by one of the followng equatons 1. Probablty of Run Formula 2. Expected Polcyholder Defct Formula 3. Standard Devaton Formula z C+ E[ X] F( C + E[ X]) = 1 ε ( x C E[ X]) f ( x) dx = η E[ X] C = σ Each of these formulas depends upon a judgmental solvency threshold denoted by ether ε, η or. More often than not, the people makng these judgments also pay close attenton to the captal requrements of the regulatory and prvate ratng agences. We summarze the ratonale underlyng each of the formulas. 1. he probablty of run formula s the classc actuaral solvency formula. It represents nterests of the nsurer s stockholders who have lmted lablty. hat s, once the nsurer s nsolvent, nothng else matters. 2. he Expected polcyholder defct s a refnement of the probablty of run formula n that t takes the sze of nsolvency nto account. hs appeals to the nterests of the polcyholders. 3. he standard devaton prncple s equvalent to the probablty of run formula when the nsurer s dstrbuton of losses s normal. Whle the normal assumpton s not realstc, there s nothng to prevent one from usng the standard devaton formula on other loss dstrbutons. It s popular because t s easy to work wth. Page 6

8 We provde an llustratve example that can easly be programmed on a spreadsheet 2 wth formulas found n Klugman, Panjer and Wllmot [1998]. Let X be a random varable wth a gamma dstrbuton. hat s: Probablty Densty Functon f ( x) = α ( x / θ) e xγ( α) x/ θ (3.1) Cumulatve Dstrbuton Functon F( x) = Γ α; x / θ b g b g (3.2) GammaDst x, α, θ, RUE Expected Value b g Γb g b g θγ α + 1 E X = θ exp GammaLn( α + 1) GammLn( α) (3.3) α Lmted Expected Value Functon b g b g b g c b gh b g b g θγ α + 1 E X^ x = Γ α + 1; x / θ + x 1 Γ α + 1; x / θ Γ α θ exp GammLn( α + 1) GammaLn( α) GammaDst( x, α + 1, θ, RUE) + x 1 GammaDst( x, α, θ, RUE) Varance b g b g b g 2 2 θ Γ α E X = θ exp GammaLn( α + 2) GammLn( α) Γ α σ = Var X = E X E X (3.4) (3.5) 2 he spreadsheet formulas gven below are for McroSoft Excel 97. Page 7

9 z b g C+ E[ X] Usng the relatonshp ( x C E[ X]) f ( x) dx = E X E X^ C + E[ X] and Equatons , one can set up a spreadsheet to solve for the captal requred for nsurer loss dstrbutons descrbed by a gamma dstrbuton. he followng table shows the results for varous solvency thresholds. In ths table we set α = θ = 100. hs yelds a sze of loss dstrbuton wth mean, E[X] = 10,000 and standard devaton, Std[X] = 1,000. For reference, we have also ncluded a premum to surplus rato. 3 For comparson, the NAIC Early Warnng est penalzes any nsurer who has a premum to surplus rato that s hgher than 3.0 to 1. able 3.1 Illustratve Captal Requrements Probablty of Run hreshold Captal P/S 1.0% 2, to 1 0.5% 2, to 1 Expected Polcyholder Defct hreshold Captal P/S 0.10% 2, to % 2, to 1 Standard Devaton hreshold Captal P/S , to , to 1 3 In ths paper we wll use the term surplus and captal nterchangeably, gnorng the formal dstnctons there are between the two concepts. Also, we assume an expected loss rato of 2/3. Page 8

10 We now add parameter uncertanty to ths example. Let β be a random varable wth E[β] = 1 and Var[β] = b. We then make Equatons condtonal on β by replacng the parameter θ wth θ β. For example: f ( xb) = b g a x / ( qb) e -x/( qb) xg( a) (3.1 ) We then modfy our three probablstc captal requrement formulas to account for parameter uncertanty. 1. Probablty of Run Formula Eβ F( C + E[ X] β) = 1 ε 2. Expected Polcyholder Defct Formula 3. Standard Devaton Formula z C+ E[ X] E ( x C E[ X]) f ( xβ) dx β L NM E[ X] O QP = C = E Var Xβ + Var E Xβ We use a three-pont dstrbuton for β n ths example. Let: β β = 1 3b, β = 1, β = 1+ 3b, l 1q l 3q l 2q β Pr β = β = Pr β = β = 1/ 6 and Pr β = β = 2 / 3. (3.6) η We have that E[β] = 1 and Var[β] = b. Page 9

11 Let b = hen: θ β 1 = , θ β 2 = , and θ β 3 = Recall α = 100. If C = 4,443.25, then: E F( C + E[ X] β) β b 1 g b 2 g b 3 g = Γ α; / ( θβ ) / Γ α; / ( θβ ) / 3+ Γ α; / ( θβ ) = hs means that the requred surplus to make the probablty of run equal to 0.01 = 4, We smlarly solved for the requred surplus for the other formulas and parameters. he results are n the followng table. able 3.2 Illustratve Captal Requrements wth Parameter Uncertanty Probablty of Run hreshold Captal P/S 1.0% 4, to 1 0.5% 4, to 1 Expected Polcyholder Defct hreshold Captal P/S 0.10% 4, to % 4, to 1 Standard Devaton hreshold Captal P/S , to , to 1 Page 10

12 4. he Margnal Cost of Captal Consder the followng stuaton. A sngle nsured s up for renewal. An analyss of market condtons has determned the premum necessary to retan the nsured. he expected loss and all other expenses are known. You must decde whether or not to renew the nsured. o make ths decson, the nsurer performs the followng calculatons. C = the captal needed for ts current busness. R = the total rsk load (.e. the total premum suppled by all nsureds less the expected loss along wth all underwrtng and acquston expenses) that s needed to attract the captal C. C C = the total captal needed f the th nsured s not renewed. R R = the total rsk load needed f the th nsured s not renewed. he nsurer s decson to renew wll depend other nvestment opportuntes for C. Under stable condtons, the nsurer mght decde to renew f R C R C R < C. However, f the nsurer can fnd another prospect that requres the same margnal captal, C, and wll pay a premum that yelds a hgher proft, the nsurer may decde not to renew. Determnng C s complcated snce, as the followng examples wll show, C depends upon the characterstcs of the nsurer s total book of busness. In the followng example, we assume that the nsurer s dstrbuton of losses has a gamma dstrbuton wth θ = 100 and α = α. We also assume that the 1 st nsured s dstrbuton of losses has a gamma dstrbuton wth θ = 100 and α = 1. It s a property of the gamma dstrbuton that the parameters of the nsurer s dstrbuton of losses are gven by θ = 100 and α = α 1 when Page 11

13 the 1 st nsured s removed. Snce the nsurer s expected loss s gven by θ α, α can reasonably be vewed as an ndcator of the sze of the nsurer. able 4.1 Illustratve Margnal Captal Calculatons Probablty of 1.0% b α C C , , , , , , Expected Polcyholder 0.10% b α C C , , , , , , Standard 2.33 b α C C , , , , , , Page 12

14 We should note that parameter uncertanty generates correlaton between the nsured under consderaton for renewal and the rest of the nsurer s busness. If the parameters of the loss dstrbutons are mxed by the random varable β we have: Var X = E Var Xβ + Var E Xβ ; (4.1) β β Cov X, Y = E Cov X, Yβ + Cov E Xβ, E Yβ ; and (4.2) β β ρ = Cov X, Y Var X Var Y (4.3) In our example we have: Xβ ~ gamma 1, θ β Yβ ~ gamma α 1, θ β E Xβ = θ β b b E Yβ = θ β α Var Xβ = θ β 2 E Xβ Var Yβ = θ β α α 1 E Yβ g 2 2 g b g b g b g b g 3 c h c h l q Usng Eβ g Xβ = g Xβ Pr β = 1 (4.4) appled to Equatons for the g s n Equatons 4.4 and the Pr{β } s n Equaton 3.6, we obtan the followng coeffcents of correlaton for our example wth b = able 4.2 Illustratve Coeffcents of Correlaton α ρ Page 13

15 Note that the coeffcent of correlaton ncreases wth α. hs happens because the mean of the frst nsured s losses vares lnearly wth the mean of the nsurer s remanng losses. But as the sze of the nsurer s remanng busness ncreases, the nsurer s random devatons decrease as a proporton of the mean. hs leads to a hgher coeffcent of correlaton. hs phenomenon can be seen clearly n the graphs below. he graphs below were generated by a smulaton where θ was frst selected at random. hen two random numbers were selected from a gamma dstrbuton wth the same α. Each graph shows 100 smulatons. Graph 4.1 a = 25, E[q] = 100 6,000 5,000 4,000 3,000 2,000 1, ,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 Graph 4.2 a = 100, E[q] = ,000 16,000 14,000 12,000 10,000 8,000 6,000 4,000 2, ,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000 Page 14

16 hese examples llustrate that the margnal captal an nsurer must have to renew an nsured depends upon propertes of the nsurer s entre book of busness. In partcular, we observed dfferences due to the nsurer s sze 4 and the correlaton between the nsured beng consdered for renewal, wth the nsurer s exstng book of busness. hs means that an nsured, whch s acceptable to one nsurer may not be acceptable to another nsurer wth the same underwrtng standards and fnancal goals. We see ths happenng n the current market for property nsurance where there s an exposure to catastrophes. In property nsurance, geographc proxmty drves correlaton n much the same way that parameter uncertanty does above. One nsurer who s concentrated n an area wll reject new busness, whle another nsurer who s not concentrated n the same area wll readly accept new busness. 4 One mght expect that an ncrease n the coeffcent of correlaton wth the nsurer s book of busness mght lead to an ncrease n the margnal cost of captal. As ables 4.1 and 4.2 show, ths s not necessarly the case. Page 15

17 5. Allocatng Captal In the last secton we ndcated that an nsured mght decde to renew f: R C R C R C. (5.1) After all, renewng n ths case wll mantan the return on the nsurer s captal. Wth a lttle algebra, one can show that the above equaton s true f and only f: R R C. (5.2) C If the nsurer plans to contnue n ts busness, strct equalty n Equaton 5.2 for all nsureds presents a problem. If (as s usually the case for nsurers) C < C, then: R C = R < R. C Acceptng all nsureds at equalty wll not meet the nsurers fnancal objectves. herefore there must be a strct nequalty for at least some nsureds. We assume that a strct nequalty for some at the expense of others cannot exst n the long run. o solve ths problem we propose a formula of the form: R C K R > for all. (5.3) C he nsurer must have an expected return of R to keep ts nvestors captal. hs means that: R R. (5.4) Page 16

18 Combnng Equaton 5.3 and 5.4 we fnd that K must be no smaller than: R K = C. (5.5) Equatons 5.3 and 5.5 do not provde the only soluton to the problem posed by the strct equalty of Equaton 5.2. hs problem s smlar to what Mango [1998] refers to as the orderng problem. Mango s other solutons to the orderng problem could also be used here. he nsurer s management could nstruct ts underwrters to gve due consderaton to Equatons 5.3 and 5.5 when acceptng nsureds. But they often have another objectve to focus the underwrters attenton on mantanng an adequate return on captal. A common way to do ths s to assgn allocated captal, A, to ndvdual nsureds accordng to the followng formula. 5 6 R A R. (5.6) C We combne Equatons 5.3, 5.5 and 5.6 to arrve at a formula for allocatng captal. A K C C C = = C R C j j. (5.7) hat s, we allocate captal to ndvdual nsureds n proporton to ther margnal captal. We now contnue wth our llustratve example. We use our three captal requrements formulas on an nsurer wth α = 100. We populate the nsurer wth nsureds wth α = 1, 2, 3 and 4. We allow 10 nsureds for each α. We chose b = he results are n the followng table. 5 Allocatng captal has been controversal. Opponents to the dea say that the nsurers potental lablty to the nsured s lmted by ts entre captal, not the captal allocated to the nsured. We agree. Conventonal use of the term should be lmted to communcatng management fnancal goals to areas of underwrtng responsblty. 6 hs could be summarzed to hgher levels f desred, but for now we wll allocate captal to ndvdual nsureds. Page 17

19 able 5.1 Illustraton of Captal Allocated to Indvdual Insureds Probablty of 1.0% α Number of Insureds C per Insured % Allocated to Insured % % % % otal Margnal Captal Expected Polcyholder 0.10% α Number of Insureds C per Insured % Allocated to Insured % % % % otal Margnal Captal Standard 2.33 α Number of Insureds C per Insured % Allocated to Insured % % % % otal Margnal Captal It s nterestng to note that approxmately the same proporton of captal s allocated to the nsurer for each of the three captal requrement formulas. hs s no accdent, as we now demonstrate. Page 18

20 Express the captal as a functon of the mean, µ, and the varance, σ 2, of the nsurer s aggregate loss dstrbuton. he captal requrement s µ + σ 2 for the standard devaton formula. In our example above, the probablty of run and the expected polcyholder defct was a functon of the parameters of the gamma dstrbuton, whch one can calculate from the mean and varance of the gamma dstrbuton. Let: C = C µ, σ C C σ (For small σ ) 2 σ 2 A C 2 2 µ, σ c C = C C 2 σ C 2 σ σ j 2 j 2 h 2 µ, σ 2 µ, σ j 2 σ j = j σ 2 2 j σ (5.8) Equaton 5.8 says that we can allocate captal n proporton to the margnal varance f: 1. We calculate the captal requrement wth any dfferentable functon of the mean and varance of the nsurer s aggregate loss dstrbuton; and 2. he nsured s varance of loss s small compared to the nsurer s varance of loss. If these condtons are met, allocatng surplus becomes a smple task once one has the covarance matrx for all nsureds. One calculates the margnal varance of the nsured by summng all the covarances n the approprate row and column of the covarance matrx. But, as we shall see below, these condtons are not always met. Page 19

21 6. A Comprehensve Example So far, ths paper has developed the noton that there s a cost of nsurng rsk that depends on the nsurer s cost of captal. Secton 2 demonstrated that the cost of nsurng depends upon the length of tme that the captal must be held wth a smple stochastc model. Secton 3 ntroduced a more complex stochastc model but gnored the length of tme that the captal must be held. Secton 4 ntroduced the noton of margnal captal and Secton 5 showed how to use margnal captal to allocate the cost of captal to a sngle nsured. hs secton combnes both the tme and stochastc elements of rsk nto a sngle comprehensve example. he XYZ Insurance Company wrtes three lnes of nsurance: Property; General Lablty; and Auto. o lmt extraneous detals, we shall assume that: All polces go nto effect on January 1 and expre on December 31. he property losses are all pad by the end of the year. All Auto and General Lablty losses are pad wthn three years. he lnes of busness have been stable for the last three years and are expected to reman so for the foreseeable future. XYZ has a conservatve nvestment polcy, so asset rsk s not an ssue. Invested assets earn nterest at an annual rate of 6%. XYZ does not purchase rensurance. he expected loss rato s 2/3 for all lnes. he nvestors n XYZ demand a before-tax return on captal of 15%. XYZ s executves do not montor the prces on ndvdual nsureds but they do hold ther lne managers/underwrters Page 20

22 responsble for meetng proftablty targets. XYZ s actuary, Jane, has the job of allocatng surplus by lne of nsurance for use n evaluatng the underwrtng results of the year Pror to dong ths job, Jane projected XYZ s aggregate loss dstrbuton for the year Noteworthy features of the aggregate loss dstrbuton nclude: Losses for unpad clams from accdent years 1998 and 1999 are ncluded as well as losses for the accdent year he property clam severty dstrbuton and clam count dstrbutons are both very skewed. Auto and General Lablty losses are correlated, but Property losses are ndependent of the lablty losses. he correlaton s generated by smultaneously varyng the means of the clam count dstrbutons n a manner analogous to that explaned n Secton 4 above. he followng table provdes summary statstcs for XYZ s aggregate loss dstrbuton. A more detaled descrpton s gven n Appendx A. hs descrpton ncludes varous percentles of the aggregate loss dstrbuton as well as the covarance matrx. We calculated the aggregate loss dstrbutons wth the Heckman/Meyers [1983] algorthm. Page 21

23 able 6.2 Summary Statstcs for XYZ s Aggregate Loss Dstrbuton Aggregate Mean 348,737,619 Aggregate Std. Dev 51,143,663 Lne Statstcs Dstrbuton Name E[Count] Std[Count] E[Severty] Std[Severty] E[otal Loss] Property AY 2000 Lag 0 2, , , ,399,448 G.L. AY 2000 Lag 0 1, , , ,418,644 G.L. AY 2000 Lag , , ,878,980 G.L. AY 2000 Lag , , ,774,319 G.L. AY 1999 Lag , , ,878,980 G.L. AY 1999 Lag , , ,774,319 G.L. AY 1998 Lag , , ,774,319 A.L. AY 2000 Lag 0 1, , , ,717,080 A.L. AY 2000 Lag , , ,423,474 A.L. AY 2000 Lag , , ,758,194 A.L. AY 1999 Lag , , ,423,474 A.L. AY 1999 Lag , , ,758,194 A.L. AY 1998 Lag , , ,758,194 Usng ths aggregate loss dstrbuton, Jane calculated the needed captal under three dfferent crtera wth the followng results. able 6.2 Captal Requrements for XYZ Insurance Company Standard ,164,734 Probablty of 1.0% 120,538,640 Expected Polcyholder 0.05% 116,871,140 After consultaton wth the approprate ratng agences, XYZ s management concluded that a captal of 120,000,000 would lead to an acceptable ratng. Assumng an expected loss rato of 2/3, ths leads to a premum to surplus rato of 2.7 to he expected loss calculaton dd not nclude the reserves from pror years. Page 22

24 Now Jane went about her task of settng proftablty targets by lne of nsurance. She proceeded as follows. 1. Snce the agreed upon captal was close to her pror projectons, she worked wth the same captal requrements crtera as before. 2. For each captal requrement crtera, she calculated the margnal contrbuton to captal by, n turn, removng each of the lnes and settlement lags from XYZ s portfolo. 3. She allocated XYZ s captal n proporton to the margnal captal for each lne and settlement lag. he allocaton proportons are shown n ables 6.3a-c below. 4. Jane then calculated the captal, C 0 that was ntally needed to support each lne of nsurance wrtten n 2000 by multplyng the sum of the year 2000 allocaton factors for each lne by 120,000,000. Now as losses are pad, the captal needed to support the nsurance wrtten n 2000 can be released. Usng the allocaton factors, Jane smlarly calculated the amount of captal, C 1 that was stll needed at the begnnng of 2001 and the amount of captal, C 2 that was needed at the begnnng of hese amounts are shown ables 6.4a-c below. 5. he captal wll be nvested at a rate of = 6%. akng the nvestment earnng nto account, Jane then calculated the amount of captal that XYZ expects to release at the end of the frst, second and thrd years wth the formula: Re l t = C t 1 ( 1 + ) Ct (6.1) 6. Jane then calculated the rsk load, R, that must be collected from the nsureds at tme t = 0, to gve the nvestors a return of r = 15% on ther nvestment of C 0. She used the formula: C 0 = R + 3 l Re t= 1 1+ r b g t t he resultng proftablty targets, R, are gven n ables 6.4a-c. Page 23

25 able 6.3a Margnal Surplus Standard 2.33 Property ,431, General Lablty ,608, ,687,172 7,608, ,070,791 11,687,172 7,608,686 Auto Lablty ,252, ,436,987 3,252, ,358,640 7,436,987 3,252,286 otal Margnal Captal 106,692,300 Captal 119,164,734 Allocated Captal Property General Lablty Auto Lablty Page 24

26 able 6.3b Margnal Surplus Probablty of 1.0% Property ,954, General Lablty ,429, ,877,505 7,429, ,292,833 10,877,505 7,429,797 Auto Lablty ,794, ,486,603 2,794, ,753,638 6,486,603 2,794,383 otal Margnal Captal 101,402,172 Captal 120,538,640 Allocated Captal Property General Lablty Auto Lablty Page 25

27 able 6.3c Margnal Surplus Expected Polcyholder 0.05% Property ,170, General Lablty ,153, ,772,421 6,153, ,899,936 9,772,421 6,153,199 Auto Lablty ,389, ,582,538 2,389, ,964,669 5,582,538 2,389,980 otal Margnal Captal 90,374,524 Captal 116,892,764 Allocated Captal Property General Lablty Auto Lablty Page 26

28 able 6.4a Proftablty arget Calculaton Standard 2.33 t Property C t 4,984,395 Rel t 5,283,458 R 390,083 General Lablty C t 39,777,920 21,702,624 8,557,715 Rel t 20,461,971 14,447,067 9,071,178 R 5,096,397 Auto Lablty C t 29,296,861 12,022,543 3,657,942 Rel t 19,032,131 9,085,953 3,877,419 R 3,327,431 otal R 8,813,911 able 6.4b Proftablty arget Calculaton Probablty of 1.0% t Property C t 8,230,527 Rel t 8,724,359 R 644,128 General Lablty C t 39,762,622 21,664,982 8,792,471 Rel t 20,483,397 14,172,410 9,320,019 R 5,106,530 Auto Lablty C t 27,259,326 10,983,180 3,306,892 Rel t 17,911,705 8,335,279 3,505,305 R 3,076,466 otal R 8,827,125 Page 27

29 able 6.4c Proftablty arget Calculaton Expected Polcyholder 0.05% t Property C t 10,848,811 Rel t 11,499,740 R 849,037 General Lablty C t 39,602,606 21,146,162 8,170,265 Rel t 20,832,600 14,244,666 8,660,481 R 5,021,880 Auto Lablty C t 26,472,751 10,585,971 3,173,434 Rel t 17,475,145 8,047,696 3,363,840 R 2,979,979 otal R 8,850,897 Jane could have calculated proftablty targets for ndvdual nsureds usng the same methodology, but that was not her task. Nevertheless, she has a standng offer to calculate these targets, should she be asked. Note that the three methods allocate surplus to the lnes n dfferent proportons n contradcton to Equaton 5.8. hs s because the captal requrements crtera are not all smple functons of the mean and varance of the aggregate loss dstrbuton. When one does not derve the aggregate loss dstrbuton from the frst two moments, we should expect ths to happen. At XYZ, the underwrters bonuses depend upon how well ther lnes of nsurance perform relatve to the targeted returns. he fact that the three captal requrements crtera produce dfferent results has sparked a debate among XYZ s management. hey have yet to nform us of ther decson of whch crteron to accept. Page 28

30 7. Data and echnology Requrements You wll need the followng tems to perform a captal allocaton analyss lke the one above. 1. Clam severty dstrbutons by lne and settlement lag 2. Clam count dstrbutons by lne and settlement lag 3. A correlaton model between lnes of nsurance 4. An aggregate loss model A large nsurer mght be able to analyze ts own data to derve a clam severty dstrbuton. hose who have nether the necessary volume of data nor the nclnaton to analyze ther own data can obtan ths nformaton from an nsurance advsory organzaton. Clam count dstrbutons and correlatons between are harder to come by. he clam count depends upon exposure, whch vares by observaton. When one gets one observaton per year, t s dffcult to get a suffcent number of observatons to get a relable estmate of the clam count dstrbuton parameters. A smlar problem occurs when you are modelng the correlaton structure. However, f one accepts the dea that smlar clam count dstrbuton and correlaton structures apply to dfferent nsurers, more relable estmates can be made. We are n the process of makng such estmates and Meyers [1999b] outlnes our methodology. Wang [1998] and Meyers [1999a] have wrtten papers about aggregate loss models that allow one to account for correlaton. Between the two papers, there are a varety of correlaton structures and calculaton methods. Wang shows how to use the Fast Fourer ransform, and Meyers shows how to use the method of Heckman and Meyers [1983] to get the aggregate loss dstrbuton wth correlated lnes of nsurance. o summarze, the data and the technology are avalable to do these analyses. Page 29

31 References 1. Heckman, Phlp E. and Meyers, Glenn G., 1983, he Calculaton of Aggregate Loss Dstrbutons from Clam Severty Dstrbutons and Clam Count Dstrbutons PCAS LXX. 2. Klugman, Stuart A.; Panjer, Harry H.; and Wllmot, Gordon E., 1998, Loss Models: From Data to Decsons. John Wley & Sons. 3. Mango, Donald F., 1998, An Applcaton of Game heory: Property Catastrophe Rsk Load to appear n the Proceedngs of the Casualty Actuaral Socety. 4. Meyers, Glenn G., 1999a, A Dscusson of Aggregaton of Correlated Rsk Portfolos: Models & Algorthms, by Shaun S. Wang, to appear n the Proceedngs of the Casualty Actuaral Socety. 5. Meyers, Glenn G., 1999b, Estmatng Between Lne Correlatons Generated by Parameter Uncertanty, submtted for publcaton. 6. Shaun S. Wang, 1998, Aggregaton of Correlated Rsk Portfolos: Models & Algorthms, to appear n the 1998 Proceedngs of the Casualty Actuaral Socety. Page 30

32 Appendx A XYZ Insurance Company Aggregate Loss Dstrbuton for EPD Aggregate Mean 348,737,619 Aggregate Std. Dev 51,143,663 Entry Aggregate Excess Excess Excess Pure Excess Rato Loss Probablty Pure Premum Premum Rato Std. Devaton ,436, ,300, ,143, ,873, ,863, ,143, ,310, ,426, ,143, ,747, ,990, ,143, ,184, ,553, ,143, ,621, ,116, ,143, ,058, ,679, ,143, ,495, ,242, ,143, ,931, ,805, ,143, ,368, ,368, ,143, ,805, ,931, ,143, ,242, ,495, ,143, ,686, ,058, ,125, ,238, ,621, ,878, ,337, ,184, ,639, ,590, ,747, ,721, ,983, ,310, ,717, ,791, ,873, ,088, ,406, ,155, ,870, ,679, ,436, ,917, ,846, ,693, ,546, ,526, ,949, ,981, ,869, ,205, ,181, ,059, ,462, ,099, ,304, ,718, ,697, ,806, ,974, ,964, ,804, ,231, ,915, ,900, ,487, ,532, ,473, ,743, ,584, ,704, , ,048, ,608, , ,151, ,066, , ,782 Page 31

33 XYZ Insurance Company Aggregate Loss Dstrbuton for Probablty of Run Aggregate Mean 348,737,619 Aggregate Std. Dev 51,143,663 Entry Aggregate Cumulatve Lmted Lmted Pure Lmted Rato Loss Probablty Pure Premum Premum Rato Std. Devaton ,985, ,264, ,419, ,087, ,337, ,497, ,341, ,878, ,625, ,466, ,340, ,358, ,041, ,201, ,886, ,355, ,682, ,300, ,593, ,908, ,655, ,911, ,958, ,993, ,477, ,897, ,353, ,516, ,784, ,786, ,405, ,693, ,367, ,898, ,745, ,249, ,798, ,177, ,787, ,646, ,102, ,387, ,069, ,410, ,765, ,237, ,211, ,093, ,507, ,849, ,240, ,853, ,960, ,452, ,358, ,067, ,658, ,069, ,168, ,860, ,052, ,265, ,056, ,408, ,357, ,248, ,301, ,444, ,435, ,036, ,527, ,617, ,276, ,604, ,796, ,076, ,675, ,971, ,191, ,726, ,108, ,127, ,732, ,125, ,496, ,736, ,139,995 Page 32

34 L1 Yr 3 L2 Yr3 Lag0 XYZ Insurance Company Correlaton Matrx for Lnes of Insurance L2 Yr3 Lag1 L2 Yr3 Lag2 L2 Yr2 Lag1 L2 Yr2 Lag2 L1 Yr L2 Yr3 Lag L2 Yr3 Lag L2 Yr3 Lag L2 Yr2 Lag L2 Yr2 Lag L2 Yr1 Lag L3 Yr3 Lag L3 Yr3 Lag L3 Yr3 Lag L3 Yr2 Lag L3 Yr2 Lag L3 Yr1 Lag L2 Yr1 Lag2 L3 Yr3 Lag0 L3 Yr3 Lag1 L3 Yr3 Lag2 L3 Yr2 Lag1 L3 Yr2 Lag 2 L3 Yr1 Lag2 Page 33