An Examination of Insurance Pricing and Underwriting Cycles

Transcription

1 An Examinaion of Insurance Pricing and Underwriing Cycles Madsen, Chris K., ME, ASA, CFA, MAAA Pricing eader, GE Frankona Re Grønningen 5, 7 Copenhagen K, Denmark Phone: , Fax: chris.madsen@ercgroup.com Pedersen,Hal W., Warren Cenre for Acuarial Sudied and Research Faculy of Managemen Universiy of Manioba Winnipeg, ManiobaR3X J8 Canada Phone: pedersn@ms.umanioba.ca December, Absrac This paper lays ou a heory of a "price of risk" as defined in his paper and suggess ha his price links all risky financial ransacions. In paricular, he paper deails insurance pricing in erms of his price of risk and suppors his wih an analysis of he performance of he propery and casualy insurance indusry for he pas fify years. The performance of he propery and casualy insurance indusry is measured by he indusry's combined raio, essenially losses and underwriing expenses divided by premiums, he volailiy of which has long been a mysery o he indusry iself. The heory presened here is exended o also provide a model for he underwriing cycle, ha is, he apparen cyclical naure of he combined raio. While he heory in he paper is incomplee and is a mos a reduced-form perspecive, i is hoped ha he ideas can be used o develop insighs ino he insurance pricing process. Key Words Price of risk, risk-adjused value of insurance, insurance pricing, opion pricing, underwriing cycle, propery and casualy insurance, general insurance, marke cycle, implied volailiy

2 Table of Conens Absrac Table of Conens Inroducion 3 Risky Financial Transacions 3 Developing he Price of Risk 3 Wha is Insurance? 4 Adding he Marke Price of Risk o he Black-Scholes Model 4 Uiliy Theory 6 Adding he Marke Price of Risk o Uiliy Theory 7 Deermining he Marke Price of Risk 8 Insurance Puing I Togeher 4 Discussion 5 Assumpions 8 Pu-Call Pariy 9 Applying he Theory in Pracice 9 Economic Raionale Conclusion References

3 Inroducion Jus a few years ago, here was pleny of capial willing o ake seemingly enormous risks. The shif in invesor and speculaor menaliy since hen is simply asounding. Invesors' appeie for risk (or lack hereof) permeaes hrough he enire economy and affecs he price of all risky ransacions. Thus, any heory, ha can faciliae our undersanding of cycles in invesor risk-aversion, should furher our undersanding of how differen financial ransacions are linked and priced. Risky Financial Transacions Financial markes basically exis o faciliae risk ransfers. To generalize his concep, we define - for now - he price of risk o be he price for one sandard uni of risk. This is no precisely he same marke price of risk ha one normally alks abou in arbirage-free models. Equaion : Price of Risk Price of Risk Price for one sandard uni of risk In he simples of erms, here are wo ways o be a buyer of risk: Pay oday > Ge uncerain cash flows in he fuure Ge oday > Pay uncerain cash flows in he fuure Boh of hese are buyers of risk, bu he naure of he ransacions has dramaically differen effecs on how he price of risk influences he profiabiliy of he ransacion. The firs of he bulles is he radiional equiy or fixed income invesor i. The second is he radiional insurer. Boh are buyers of risk, bu he differences high-lighed above have significan consequences in erms of how hese players respond o changes in he marke price of risk. In paricular, when he riskiness of he cash flows increases, he equiy or fixed income price will drop if he price of risk is unchanged. Conversely, in insurance, when he riskiness of cash flows increases, he price increases. Furher, when he price of risk increases, equiy prices drop, bu he price of insurance rises. The sandard presen value of cash flow model shown below fails o accoun for his behavior. Equaion : Presen Value of Cash Flows P Ε[ CF + r * ] In his equaion, r is he discoun rae, P is he price paid or received for he risk ransfer, and CF is he uncerain cash flow a ime. * Developing he Price of Risk 3

4 I is relaively easy o ge a sense for he price of risk over ime. When equiy prices soar, for example, here may be good fundamenal reasons. Bu ofen, he price increase goes above he fundamenal jusificaion. In oher words, he price of risk begins o fall as invesors are willing o pay more and more for he same Dollar of risky ransacion. This senimen carries hrough o all financial ransacions. When he price of risk increases: Prices of equiies fall Prices of corporae bonds fall (bond spreads widen) Prices of opions rise Prices of insurance rise When he price of risk decreases: Prices of equiies rise Prices of corporae bonds rise (bond spreads narrow) Prices of opions fall Prices of insurance fall Wha is Insurance? Insurance is a financial ransacion on losses. A he ime a policy is wrien, he insurer has a cerain esimae of wha he expeced loss is (reserve). Over ime, his esimae develops as more informaion becomes available. Thus, he expeced loss develops over ime much like he price of a sock develops over ime. In insurance, using a log-normal disribuion o describe aggregae losses is nohing new. The log-normal is ofen used as a simplifying assumpion when deailed daa is no available. I is convenien, alhough no essenial, for us o assume ha losses have a log-normal disribuion. Under his assumpion we may hen use he usual formulas for limied expeced values. One way o view he buying of insurance is o consider i as buying a call opion on losses. Selling insurance is he equivalen of selling a call opion on losses. The insured has an opion on losses, ha pays only if losses exceed a cerain specified hreshold (in he case of a simple deducible). The insurer is generally shor naked calls (naked because he insurer does no have a cash flow sream o hedge he paymens hey mus make for he call hey have wrien). We sugges ha opion-pricing heory and an assumpion abou relaionships beween changes in risk aversion in he financial and insurance markes can be used o gain some insigh ino insurance pricing. We explore his concep in he following secions. Adding he Marke Price of Risk o he Black-Scholes Model Equaion 3: Black-Scholes Opion Pricing Model C S N r ( d ) K e N ( ) d, where d d S s ln + r K + s d s 4

5 C is he price of a call opion wih srike price K on an underlying securiy wih curren price S. The risk-free rae is denoed by r and he ime o opion expiraion is. s is he sandard deviaion of equiy price reurns over a ime period of. To use he Black-Scholes opion pricing model, we mus firs clarify how he various parameers are esimaed in an insurance environmen. C is he price of insurance wih deducible K on losses wih curren expeced discouned value of S. The risk-free rae is denoed by r and he insurance erm is. s is he sandard deviaion of he change in expeced discouned losses over a ime period of. We will use he coefficien of variaion (raio of he sandard deviaion o he mean) as a sandardized measure of riskiness. The riskier a cash flow, he higher is coefficien of variaion. Equaion 4: Coefficien of Variaion SdDev α Mean We now define he marke price of risk as he price invesors are willing o pay per uni of riskiness as measured by he coefficien of variaion,α. Thus, we relae he volailiy of reurns, s, o a sandardardized marke price of risk, λ. We can use his relaionship o deermine he marke price of risk from exising daa such as equiy opion prices. In paricular, we use he Chicago Board Opions Exchange (CBOE) volailiy index or VIX as a proxy for he s of equiy reurns as measured by he S&P 5 index ii. The coefficien of variaion, α, for equiies is based on long-erm volailiy divided by long-erm reurn. This is shown in Equaion 6. Equaion 5: The Marke Price of Risk In general, s λ α c s λ α S& P5 S& P5 We can use he S&P5 o find our price of risk a ime : λ s α S& P5 S& P5 s s S& P 5 S& P 5 µ S& P5 Once he marke price of risk, λ, is deermined from equiy opion prices, we need a mehod for deermining our loss volailiy parameer o use in he equaion. This work is laid ou in Equaion 7. 5

6 6 Equaion 6: Deerminaion of Implied Insurance Volailiy Parameer We wan o find he implied insurance volailiy parameer, which we can hen use o price he insurance policy. The volailiy we refer o is ha of he "loss reurn", or simply, he volailiy of he expeced change in he discouned loss over a given ime period s s s, where is he expeced discouned losses a ime (no variabiliy) iii. is he expeced discouned losses a ime (unknown a ime zero). The coefficien of variaion associaed wih his volailiy is ( ) r s r S S s oss + µ µ α, where r is he risk-free rae. This is because is he discouned esimae of losses a ime. From his, we can solve for he insurance volailiy parameer, oss s, a ime : oss oss s α λ The coefficien of variaion measures volailiy relaive o is expeced value. We can adjus he Black-Scholes formula o handle his (Equaion 8): Equaion 7: Black-Scholes Opion Pricing Model ( ) ( ) d N e K d N S C r, where r K S s s r K S d α λ α λ ) ( ln ln d d α λ λ is he marke price per sandardized uni of volailiy or, simply, he marke price of risk. α is he coefficien of variaion. Uiliy Theory Gerber and Pafumi iv show how various uiliy funcions can be applied o insurance pricing. They also exend heir research o show how uiliy heory can be exended o derive Black-

7 Scholes opion prices. Based on he deail we have described here so far, we should hus be able o coninue building our framework incorporaing some of Gerber s and Pafumi s work. This aracive for many reasons: I provides greaer suppor for our framework Due o he simpliciy and elegance of some of Gerber s and Pafumi s resuls, we can find more inuiive and simple ways o price I may be inuiively easier o explain an economic heory for insurance pricing on he basis of uiliy heory as opposed o opions heory. We will focus on one aspec of heir analysis, he exponenial uiliy funcion. Here, hey show ha he premium for an insurance policy can be deermined based on he following equaion: Equaion 8: Premium Deerminaion Based on Exponenial Uiliy Funcio n a P µ oss + σoss, where a is he risk aversion parameer. This equaion possesses he characerisics and behavior we se forh iniially for insurance pricing. Adding he Marke Price of Risk o Uiliy Theory We add he price of risk as a facor o risk aversion parameer. This is consisen wih our view ha invesor risk aversion changes over ime wih he price of risk. I may also be a funcion of oher hings, such as wealh. We will ignore he wealh effec for now, as we will coninue o use he exponenial uiliy funcion. Equaion 9: Premium Calculaion Using Exponenial Uiliy and Price of Risk a λ P µ oss + σ oss Assuming independence of losses over ime, we can exend his o muliple ime periods, which will be a useful resul. Equaion : Premium Calculaion from Equaion wih Muliple Time Periods P µ oss + a λ σ oss The above does no explicily consider a deducible, bu when we need o consider a deducible, he mean and variance parameers above can be esimaed based on ne losses. Alernaive, we can break Equaion ino wo policies: One covers all losses ( P ), and he oher covers losses below he deducible, K ( P, ). The price of a policy wih deducible K ( P, ) mus necessarily be he difference beween buying a policy wih no deducible and 7

8 selling one ha covers losses up o K. If his were no so, here would be an arbirage opporuniy. Equaion : Wih Deducible, K v P, a λ ( µ µ ) + ( σ σ COV ) P P, oss oss oss oss oss, oss Noe, ha we have subraced a covariance erm. This is essenial, since he losses are correlaed. They cover he same risk, hough differen aspecs of i. I is a firs ineresing o observe ha a posiive covariance decreases price, bu i makes sense, since we are buying one cover and selling he oher. The more correlaed hey are, he more hey hedge each oher and he lower he premium. Before we move on, le us consider he premium porfolio problem. Wha happens when a company wries more han one policy and losses are correlaed in some form? Coninuing wih our curren frame-work, we look a wo policies in Equaion 3 below: Equaion : Wriing Two Policies On Correlaed osses a λ ( µ + µ ) + ( σ + σ + COV ) P P + P oss oss oss oss oss, oss This, of course, suggess ha when one wries a policy in a correlaed porfolio, one mus charge as laid ou in Equaion 4. Equaion 3: Wriing A Policy in a Correlaed Porfolio vi P a λ ( µ ) + ( σ + COV ) oss oss oss, Oherosses A posiive covariance increases he premium and a negaive decreases i. Exending his o he company level, a company s premium is simply he following: Equaion 4: Insurance Porfolio P Company N a λ + µ ossi i i j N N COV ossi, ossj In he case where all losses are independen, his simplifies o Equaion. We can use Equaion 5 o find he opimal level of premium a company should wrie for a given level of risk, if losses are correlaed. In esablishing a frame-work for insurance pricing, his is ouside he scope of his paper, and we leave his paricular opic as a fascinaing one for fuure research. Deermining he Marke Price of Risk 8

9 In a February, 999 A.M. Bes repor, A.M. Bes saed ha "A.M. Bes believes he propery/casualy underwriing cycle has been replaced by a permanen 'down marke'". If his brough reminiscences of Irving Fisher's vii "Socks have been replaced by wha looks like a permanenly high plaeau", i was appropriae. The price of risk had been dropping for much of he 9's much like i did in he 's. During he 9's, his mean soaring equiy prices and poor insurance prices. Margins may erode and markes may become more efficien, reward new echnology and heory, bu i is exremely unlikely ha here is a hing such as a permanen down marke. I is equally unlikely ha a permanen up marke exiss. As underwriers are well aware, he financial markes are cyclical. Someimes, risk-akers ge compensaed well for aking risk and someimes hey do no. This is rue for all financial ransacions wheher in insurance or invesing. In he lae 9's, invesors bid many speculaive issues way in excess of heir fundamenal value. This essenially pushed he price of risk o nohing or perhaps ino negaive erriory. In oher words, no one was geing compensaed for aking risk, as he marke place did no perceive any real risk. Conversely, afer Sepember h, viii, he percepion of risk was grea and equiy markes fell and insurance prices rose as he price of risk rose. We will examine our model from Equaions 8 and o see if hey can help describe some of he cyclical behavior ha propery and casualy insurers have been exposed o. In order o do his, we mus firs deermine he marke price of risk. We know from equiy opion pricing ha opion prices ofen do no reflec he long-erm volailiy of equiy reurns. Thus, he Black-Scholes formula is ofen reversed and solved for he volailiy given prices. This volailiy is called he implied volailiy. Using he generalizaion from Equaion 8, he implied volailiy can also be separaed ino sandardized volailiy and marke price of risk. The following is an excerp from he Chicago Board Opions Exchange websie: "One measure of he level of implied volailiy in index opions is CBOE's volailiy index, known by is icker symbol VIX. VIX, inroduced by CBOE in 993, measures he volailiy of he U.S. equiy marke. I provides invesors wih up-o-he-minue marke esimaes of expeced volailiy by using real-ime OEX index opion bid/ask quoes. This index is calculaed by aking a weighed average of he implied volailiies of eigh OEX calls and pus. The chosen opions have an average ime o mauriy of 3 days. Consequenly, he VIX is inended o indicae he implied volailiy of 3-day index opions. I is used by some raders as a general indicaion of index opion implied volailiy. Implied volailiy levels in index opions change frequenly and subsanially." Using he VIX as implied volailiy for he equiy marke, we can - by assuming long-erm monhly expeced equiy price reurn and volailiy - solve for he marke price of risk. Equaion 5: Coefficien of Variaion for Equiies α σ µ S& P5 S& P 5.9%.%.66 This means we can develop he following graph for λ which, in effec, is jus a rescaled graph of he VIX index: 9

10 Graph : The Price of Risk Price of Risk.9%.7%.5% Price of Risk.3%.% Price of Risk One Year Moving Average ong Term Average.9%.7%.5% //986 //987 //988 //989 //99 //99 //99 //993 //994 //995 //996 //997 //998 //999 // // // Dae From his graph, we can clearly see ha for mos of he 9's, he price of risk was depressed. This would seem o sugges ha i was a good ime o buy opions and a bad ime o sell hem. Remember ha selling insurance may be inerpreed as selling opions. Of course, ineres raes also play a role in opion pricing. We already know wih he benefi of hindsigh ha he 9's were a ough ime for many propery and casualy insurers because of low prices. We now develop an insurance price index based on he above informaion and examine how i sacks up agains hisorical experience. Insurance Consider now an insurance policy. The ground-up losses have a sandard deviaion of hree imes he expeced loss (Example a). Example a: oss Parameers Expeced oss: $, Sandard Deviaion of osses: $3, Risk-free Rae: 5.75% This means 3,, α We insure losses ha exceed $5, (Example b).

11 Example b: Terms of Policy Duraion: Year Deducible: $5, Given ha we know he price of risk over ime, we can now use he Black Scholes model o deermine he price of he insurance policy. Since he price of risk flucuaes over ime, he price of insurance will oo. Graph looks much like Graph, alhough here are some suble differences. In paricular, he previous char only looked a he price of risk and did no consider he effec of ineres raes. Ineres raes are accouned for in he char below. A feaure of he model we are using is ha he model prices insurance by adjusing he volailiy of he losses used for pricing in response o changes in he VIX. Inuiively, as he VIX can be used as a proxy for changes in he aiude oward risk, we are allowing he pricing of insurance o reflec hese changing aiudes as well. Graph : The Price of Insurance "Fair" Price of Insurance, 95, 9, Price 85, Marke Price One Year Moving Average ong-term Average 8, 75, 7, /3/986 /3/987 /3/988 /3/989 /3/99 /3/99 /3/99 /3/993 /3/994 /3/995 /3/996 /3/997 /3/998 /3/999 /3/ /3/ Dae We can also use our uiliy heory model o mirror he resuls above. In fac, for cerain values of he risk aversion parameer, he models are very similar. Insurance prices are cyclical. We already know his and we need look no furher han combined raios o see his. The combined raio is defined in Equaion 7 below. Equaion 6: Combined Raio CR + E, where PE PW

12 CR is he combined raio a ime, is he incurred loss a ime, E is he incurred expenses a ime, PE is he earned premium a ime, and PW is he wrien premium a ime. Incurred losses are paid losses plus changes in loss reserves. In Graph 3 below, we show he hisorical values of he calendar year combined raio. Graph 3: Hisorical Combined Raios Calendar Year Combined Raios (%). Acual Year We can ry o model his ime series wih a simple auo-regressive model as proposed by Kaye D. James in her discussion of underwriing cycles ix from 98. She specifically proposes a model of he form: Equaion 7: K.D. James Model CR CR. 95 CR + CR is he combined raio a ime..93, where K.D. James used daa from 95 hrough 98 o fi her model. She cied he following saisics on he model:

13 SE.597 R 7.9% value CR 7.6 However, in rying o reproduce resuls using he same period of daa and he same daa as far as we can ell, we ge a differen bes-fi model: Equaion 8: Re -fied K.D. James Model CR CR CR SE.989 R 47.97% value CR 5.8 If we allow for a more general ime series model (wihou resricing he CR parameer o be one), second order auo-regressive, we can fi he following model. This ime we use all daa (95-) Equaion 9: Second Order Auo-Regressive Time Series Model CR. CR.98 CR +.69 ha has he following saisics SE.367 R 7.4% value 7. value CR CR.484 This model is graphed in Graph 4 below along wih he acual daa. Graph 4: Combined Raios - Acual and Modeled 3

14 Calendar Year Combined Raios (%). Acual Model Year While an AR() model is excellen a modeling he general characerisics of a ime series over ime, i has a major shor-coming in ha i is essenially a moving average of he ime series i is se up o model. This means i will end o overesimae he combined raios in a down-rend and underesimae hem in an up-rend. In he nex secion, we will explore if he insurance pricing index, ha we developed, can help explain beer he hisorical behavior of combined raios. Puing I Togeher Using our previously developed view of he insurance world as selling naked call opions on losses, we can fi a ime series model as before, bu include our insurance index or call opion price. We develop he following model for indusry calendar year combined raios: Equaion : Combined Raio Model CR CR CR C, where CR is he combined raio a ime, and C is he call opion price a ime divided by,. The daa used o fi his model was 988-, since he implied volailiy daa for previous periods was no readily available. The saisics for his model were: 4

15 SE.555 R 55.56% value value value CR CR C While a firs i may seem ha he R-squared here is lower han in he K.D. James model, i is no. They are fied on differen ime periods and i urns ou ha he period 988- is much more difficul o fi! The generalized K.D. James model fied on he same daa as above yields only an R-squared of 9.9%. Noe, ha our insurance index ( C ) is he mos significan variable and ha he parameer is posiive suggesing ha he indusry combined raio rises wih our pricing index. This is he exac opposie of wha should be happening if he indusry was properly reflecing he marke price of risk. Graph 5: Combined Raios - Acual and Modeled Calendar Year oss Raios. 5.. % 5. Acual Model Year Discussion Wha is paricularly ineresing and couner-inuiive abou Equaion is he fac ha when he call opion price rises, so does he combined raio. This seems o sugges ha when insurers can and should raise prices, hey do no or a leas no by enough. They are, in fac, selling under-priced opions. 5

16 Insurance is generally priced as discouned cash flows. This means ha he higher he ineres rae, he lower he price. Bu for opions he relaionship is opposie. The higher he ineres rae, he higher he price of he opion x. Insurers raionalize ha hey can charge less for insurance when ineres raes rise, since hey can inves a higher yields. Bu his fails o accoun for he fac ha when ineres raes rise, losses - in effec - are more likely o hi he aachmen poin (generally due o higher inflaion). In oher words, o move only he ineres rae wihou adjusing losses and he aachmen poin is like geing or giving a free lunch. When ineres raes rise, he price of insurance all oher hings remaining equal - should rise as well. As for he second componen of opion pricing, implied volailiy, insurers do raise prices when implied volailiy increases dramaically. This is in reacion o supply and demand, as implied volailiy is much easier o read in he marke place ha small changes in ineres raes. Afer Sepember h,, everyone was much more concerned wih insurance all of a sudden. The same was he case wih Hurricane Andrew in he early 9's. Insurers correcly use hese opporuniies o raise prices. I is quesionable wheher hey raise hem by enough. We know his from he marke place and no from he daa, so we mus conclude ha over a full year, he effecs of he poor decisions on pricing wih respec o ineres raes, have a greaer influence on he well-being of he insurance indusry. In shor, i appears he insurance indusry reacs properly o changes o implied volailiy bu may in fac do he exac opposie of wha hey should do when ineres raes change. If his asserion is rue, we should be able o "correc" for his behavior and model he combined raio cycles. In order o es his, we produce wo new daa series. The firs represens radiional insurance pricing and is a presen value of expeced cash flows. In his case, we assume an even payou ha occurs over five years. Thus, he preliminary radiional pricing index is Equaion : Tradiional Insurance Pricing Index TP r r N ( + r ) where N is he number of years of payou. N The opion pricing index is simply he price of he call opion a ime. Boh indexes are normalized by dividing by heir respecive values a //986. //986 has no special significance oher han ha i is he beginning of he ime series and i is, imporanly and convenienly, very close o he means of he ime series. Equaion : Insurance Opion Pricing TP TP * *, OP TP C C In order o ge he final pricing index for he year, we ake he average of he normalized indexes for he prior year. This makes sense, since mos renewals occur a he beginning of he year and hus, are based on he prior year's daa. 6

17 The indusry is over-pricing when he radiional pricing index is above he opion pricing index. The indusry is under-pricing when he radiional pricing index is below he opion pricing index. This leads o Graph 6 below. Graph 6: Combined Raios versus Pricing Indexes Sysemaic Over and Under Pricing Combined Raio (%) Pricing Index Combined Raio Sandard Pricing Opion Pricing Year.8 As can be seen, we have had hree cycles during he pas 4 years. Combined raios worsened up hrough 99, hen improved unil 997 and have since worsened again. The model presened here explains all hree rends. From 987 o 99, he indusry was under-pricing, so combined raios worsened. From 99 o 997, he indusry overpriced, so combined raios improved. Saring in 997, he indusry again under-priced and combined raios worsened. Noe, ha he green line (marked wih squares) maches he underwriing cycle almos perfecly and gives one year advance noice. If we bring he daa forward o he ime of wriing, we see he following: Graph 7: Same as Graph 6, bu wih and Forecas 7

18 Sysemaic Over and Under Pricing Combined Raio (%) Pricing Index Combined Raio Sandard Pricing Opion Pricing Year.8 The model properly accouns for he poor underwriing resuls of (ou of sample) and hough prices have increased in, i would no be surprising if also is a poor year for propery and casualy insurers. Indeed, all iniial indicaions are ha his will be he case. Using Equaion and a value for a of., we can generae an almos idenical graph. This value suggess ha afer he firs million USD, uiliy is prey consan. Of course, wih he exponenial uiliy funcion, he value of a ha roughly maches he values for Black Scholes depends on he loss parameers. When he expeced loss and sandard deviaion grow by a facor of, he a ha maches falls by a facor of. This is due o he characerisics of variance, and as such is an undesirable feaure. Ideally, we wan he risk aversion parameer, a, o be a consan for differen sizes of losses. Assumpions The assumpions made here are similar o hose made by Black and Scholes. No payous before he end of he erm. Markes are efficien (meaning cash flows can be replicaed wih oher securiies). No commissions are charged for ransacions. Ineres raes remain consan and are known. osses are log-normally disribued. I is rue ha hese assumpions - in paricular, he second and hird assumpions -are somewha less valid in an insurance marke which is, almos wihou excepion, over-hecouner. Mos ransacions are direc and negoiaed person o person. An efficien marke does no exis. Bu his was also rue of equiy opions prior o he Black Scholes model. While i will clearly ake some ime, here is no reason o believe ha he insurance marke will no evenually 8

19 blend in wih oher financial ransacions. We have winessed some of his wih caasrophe bond issues and oher securiizaions, bu hese conceps have no caugh on ye. This is mosly due o he "buyer's marke" ha has exised in insurance for much of he pas decade. When people are willing o sell goods a below fair marke value, here is lile incenive o pursue fair marke value. As prices increase and players charging oo lile are eliminaed, he incenive for fair prices will once again be esablished. Regulaors are a wild-card. Clearly, barriers o enry, mandaed pricing and similar regulaory issues work agains efficien insurance markes. arge fees and ransacion coss also work agains efficiency. Any ransiion is necessarily slow and incremenal, bu if our findings here have any relevance, i is jus a maer of ime. Pu-Call Pariy In insurance here are no pu opions per se and ha impacs he liquidiy and he efficiency of he marke. I is no easy o replicae an insurance payoff. In opion heory, being long a sock wih price S, shor a call on he sock wih srike price K, and long a pu wih srike price K, is equivalen o he discouned srike price. In oher words, holding a sock, a pu and being shor a call is equivalen o holding cash. This is because he opions offse he sock payoffs compleely. The equaion is shown in Equaion 7. Equaion 3: Pu-Call Pariy C P + dk S This also means ha we can replicae he sock reurn wih a call, a pu and some cash. In insurance i is no so easy. We do no really have insrumens ha we can replicae cash flows wih. An insurance pu opion means ha if losses came in less han expeced, here is a posiive payoff from he opion. We already know ha here is a marke for insurance call opions, since ha is wha insurance is. Bu who would be ineresed in insurance pu opions? The exac same eniies ha are buying call opions. Insurers should be ineresed in selling loss pu opions as i is a way o ge some exra cash ha can offse higher han expeced losses. Insured eniies should be ineresed in buying hem as i gives hem cash back when losses are less han expeced. In fac, his gives insurers anoher incenive o sell pu opions. A cash incenive below expeced losses may make smaller and adminisraively cosly claims less likely o occur. If such a marke develops, i would serve o increase efficiency and, as such, beer pricing. Applying he Theory in Pracice The examples laid ou above describe in deail how o use he pricing model. Even if a praciioner does no believe in he pricing mehod and heory laid ou here, he praciioner may sill benefi from he model of he underwriing cycle. Of course, o he exen ha praciioners use he pricing model and markes become more efficien, i changes he underwriing cycle, bu i will be many years. 9

20 If he coverage o be priced is a layer, hen simply calculae wo opions prices, as he price of he layer mus necessarily be he difference beween he wo opions prices. If we look a he propery and casualy marke in segmens, here may be pockes of opporuniy and perhaps greaer insighs can be gained from doing such as sudy. For he ime being, we leave ha as ouside he scope of his paper and sugges i as an area of fuure sudy. Economic Raionale The basic raionale is as laid ou in he following graphic. The Price of Risk, as defined in his paper, permeaes hrough all financial ransacions. Expeced Earnings Marke Price of Risk Ineres Raes Equiy Prices Bond Prices Opion Prices Insurance Prices In efficien markes, here mus be one price for idenical producs. Risk can be viewed as one such produc and is marke price is a componen of all risky financial ransacions. Conclusion There are several implicaions of he ideas we have presened here. Firsly, a fairly general heory of he price of risk was developed o creae anoher perspecive on he bridge beween he asse pricing world and he insurance world. Secondly, his heory was exended o creae an insurance pricing model, which in urn was used o model he underwriing cycle. Thirdly, he findings from he underwriing cycle model suggesed ha he insurance cycle could poenially be explained by sysemaic over and under pricing by insurers. Indeed, based on he daa available, his appears o be he case. The exension of his is ha if insurers price he opions hey are graning correcly, he underwriing cycle, as we know i oday, will largely disappear. The insurance world changes gradually. Indeed, pas prophecies of sudden changes in he insurance marke have no been realized. Such gradual change is naural, in par because companies are hesian o underake new pricing pracices if i means siing ou a whole renewal season. There are more ideas o explore here. The pricing model should be se on a full economic framework, which explains why changes in volailiy for pricing purposes are reasonable. The

21 idea of an inegraed financial framework has coninued o gain ground. I is our hope ha he ideas we have proposed will provide he basis for oher classes of insurance pricing models.

22 References A.M. Bes Company, "Discipline and Specialized Disribuion Are Keys o Program Underwriers' Success", February, 999 Special Repor Black, Fischer and Scholes, "The Pricing of Opions and Corporae iabiliies", Journal of Poliical Economy, 8:3, 973, pp Ciezadlo, Greg, "Marke Cycle Updae, Personal ines", CAS Spring Meeing, Commerce Deparmen's Naional Insiue of Sandards (NIST) and Inernaional Semaech (IST), "E-Handbook of Saisical Mehods" Gerber and Pafumi: "Uiliy Funcions: From Risk Theory o Finance", Norh American Acuarial Journal, 998, Volume, Number 3. Hull, John C., "Opions, Fuures, and Oher Derivaives", Prenice Hall, 997 Insurance Informaion Insiue Insurance Journal, "Combined Raio for on Track o Hi Record High, ISO Presiden Says", Propery and Casualy Magazine, April James, Kaye D. and Oakley, David, "Underwriing Cycles by Kaye D. James - Discussion by David J. Oakley" uenberger, David, "Invesmen Science", Oxford Universiy Press, 998 Moore, James, "Hard Markes", RiskIndusry.com, Augus 5, Skurnick, David, "The Underwriing Cycle", CAS Underwriing Cycle Seminar, April, 993 Swiss Re, "Profiabiliy of he non-life insurance indusry: i's back-o-basics ime", Sigma, No. 5/ Wang, Shaun: A Universal Framework for Pricing Financial and Insurance Risks i Fixed income is risky a some level. Though a coupon is specified, here is no cerain guarany ha he company, or he governmen for ha maer, will no defaul. ii VIX is acually based on eigh S&P opions. In general, when dealing wih implied volailiy, one mus consider he volailiy smile. We avoid his by using a marke based measure based on several opions. iii This is an area ha could be explored furher. In keeping he discouned expeced loss consan, we are possibly changing he rue loss hrough ime. The correc discouned loss o use should perhaps be a risk adjused discouned value. If we were o implemen his, we would also achieve he desirable feaure ha our opion price changes when here is no deducible. Bu i also complicaes he analysis grealy, and makes i more difficul o rack. We leave his as an area for furher sudy. iv See references v Since we are cuing he disribuion a K, i is no clear ha variance is he bes risk measure, bu we are aemping o specify a framework here. The framework can be expanded o more advanced measures of uiliy or opions pricing, bu ha is beyond he scope of his paper. vi Technically, here are an infinie number of ways o spli he covariance. Wihou going ino furher deail here, we borrow a concep from Game Theory known as he Shapley Value, which has many

23 desirable qualiies. Spliing wice he covariance by assigning one imes he covariance o each produc reflecs his propery. vii Irving Fisher, a Professor of Economics a Yale Universiy, made hose remarks in 99 a few weeks before he Ocober crash. viii On Sepember h,, he World Trade Cener and Penagon erroris aacks occurred. ix See references. x The Black-Scholes opion pricing models ells us ha he price of an opion is he expeced loss wih no insurance minus he presen value of he aachmen poin. The higher he risk-free ineres rae, he lower he presen value of he aachmen poin, and hus, he higher he price of he opion. 3