Pragmatic Insurance Option Pricing

Transcription

1 Paper to be presented at the XXXVth ASTIN Colloquum, Bergen, 6 9th June 004 Pragmatc Insurance Opton Prcng by Jon Holtan If P&C Insurance Company Ltd Oslo, Norway Emal: jon.holtan@f.no Telephone: Abstract Ths paper deals wth theoretcal and practcal prcng of non-lfe nsurance contracts wthn a fnancal opton prcng context. The market based assumpton approach of the opton context fts well nto the practcal nature of non-lfe nsurance prcng and valuaton. Basc facts n most nsurance markets lke the exstence of qute dfferent nsurer prce offers on the same clams rsk n the same market, support the need for ths approach. The paper outlnes nsurance and opton prcng n a parallel setup. Frst t takes a complete market approach, focusng dynamc hedgng, no-arbtrage and rsk-neutral martngale valuaton prncples wthn nsurance and optons. Secondly t takes an ncomplete market vew by ntroducng supply and demand effects va purchasng preferences n the market. Fnally the paper dscusses pragmatc nsurance prce models, parameter estmaton technques and nternatonal best practce of nsurance prcng. The overall am of the paper s to descrbe and unte the headlnes of the more or less common nsurance and opton prce theory, and hence ncrease the pragmatc understandng of ths theory from a busness pont of vew. Keywords Insurance and fnancal opton contracts; nsurance and opton prcng theory; complete and ncomplete markets; dynamc hedgng and no-arbtrage; rsk-neutral martngales; purchasng preferences; rsk, cost and market prce adjustments; parameter estmaton; pragmatc and best practce prcng. 1

2 1. Introducton There has been a consderable attenton on the relatonshp between nsurance and fnancal prcng theory the last two decades after the breakthrough establshment of the fnancal martngale theory made by Harrson and Kreps (1979) and Harrson and Plska (1981). There also seems to have been an ncreasngly development speed n the 1990-thes after the mportant works on dynamc prcng of arbtrage-free (re)nsurance markets made by Delbaen and Haezendonck (1989) and Sondermann (1991); see Embrechts (000) for an extng overvew descrpton. In ths paper we wll try to descrbe and unte the headlnes of ths common theory to a practcal understandng and practse n non-lfe nsurance prcng, and thus also try to address some future theoretcal challenges. A drect comparson of nsurance prcng versus opton prcng wll also gve deeper nsght to the relatonshp between the two prcng technques, and hence make the brdge between the fnancal and the actuaral approach to prcng a bt wder, - not only for academcs on the unverstes, but also for us practcal workng actuares n the nsurance busness ndustry. The paper s organzed as follows. Secton addresses some actual prcng questons and problems. Secton 3 outlnes characterstcs of nsurance contracts wthn an opton prcng context. Secton 4 outlnes and dscusses dynamc hedgng, no-arbtrage and put-call partes n a parallel opton and nsurance approach. Secton 5 puts the prcng challenges nto a rsk-neutral martngale prcng approach. Secton 6 dscusses effects generated by ncomplete markets and ntroduces a purchasng preference element to nsurance prcng. Secton 7 ntroduces some pragmatc nsurance prce models. Secton 8 dscusses relevant parameter estmaton technques wthn nsurance prcng. Secton 9 presents and dscusses nternatonal best nsurance prcng practces. Secton 10 gves some concludng remarks.. Prcng non-lfe nsurance contracts versus fnancal optons The common basc dea of both fnancal optons and nsurance contracts s to transfer an economcal rsk from one part to another aganst a specfc payment. To do so the buyer and the seller of the contract need to agree upon a prce before the contract begns to run. A common used market methodology approach s to splt ths prcng nto one techncal valuaton part and one part purely nfluenced by supply and demand mechansms. Hence the pont s frst to fnd the techncal far values of the contracts, and then adjust these values to real sales prces based on purchasng preferences or dspostons n the market. Fnn and Lane (1997) supports ths vew by sayng: There are no rght prce of nsurance; there s smply the transacted market prce whch s hgh enough to brng forth sellers, and low enough to nduce buyers. Ths statement defntely rules also n the fnancal dervate market. Hence, qute dstnct speakng, the prce of an opton or nsurance contract s the premum one pays for t, whle the value s what t s worth.

3 The dfference between value and prce/premum depends very much on the market condtons of the opton or nsurance contract. We have to behave some man market dfferences: Insurance market: - Most often standard one-year contracts - Most often no real effcent market place for buyng and sellng contracts - Not only prce, but also product and servce qualty are mportant - Prce per contract depends on and vares between the contract buyers Fnancal opton market: - Many dfferent types of contracts wth no standard tme condton - Most often an effcent market place for buyng and sellng contracts - Only prce s mportant to most customers - Prce per contract does not depend on the buyers of the contracts The more or less complete market condtons of fnancal optons generate a smaller dfference between value and prce/premum than the more ncomplete nsurance market. Hence, partcularly for nsurance contracts the theoretcal challenge s to nclude not only the valuaton part of prcng, but also the supply/demand part nto the prce models. Some nterestng key questons wthn ths approach are: Why do dfferent nsurance players offer dfferent prces for unque rsks? and under what condtons wll more unque market prces for unque rsks be generated? What s the optmal prce per rsk n a complete nsurance market? and n an ncomplete market? In ths paper we wll stress these questons by frst focusng the valuaton part of prcng n a complete market setup, and then nclude also ncomplete market based purchasng preferences. Equvalent questons are well known and handled wthn fnancal opton prcng theory. Hence the market model context of opton prcng fts well nto the practcal nature of non-lfe nsurance prcng. To get more hands on prcng smlartes and dfferences let us therefore outlne nsurance and opton prcng n a parallel approach. 3. Insurance contracts n a fnancal opton context An opton s a contract gvng the holder a rght to buy or sell an underlyng object at a predefned prce durng a predefned perod of tme. An opton condton s that the underlyng object has an uncertan stochastc future value. Fnancal stocks are the most common opton object. However, the clams rsk of an nsurance customer may be nterpreted as another opton object. Hence we may defne a non-lfe nsurance contract as an opton. We have the followng correspondngly defntons: 3

4 Call opton on stocks: The holder of a call opton gves the rght, but not the oblgaton, to buy a stock at a predetermned date (maturty tme) and prce (strke prce). Insurance contract: The holder of an nsurance contract gves the rght to get covered all ncurred nsurance clams wthn a predetermned date (maturty tme) and at a predetermned prce (the deductble or excess pon. Assume two dfferent underlyng rsk processes S( and X (, where S ( s the stock prce process up to tme t and X ( s the accumulated nsurance clam N process up to tme t. That s, assume X = = Y 1, where N( s the number of ncurred clams up to tme t and the Y -es are the clams severtes. Let C be a European call opton contract on the stock prce process S ( and Z and Z * two dfferent nsurance contracts on the clams rsk process X (. Assume tme t = 0 as the start tme of all contracts and tme t = T as the maturty tme. Hence let: C( T ) = max( S( T ) K,0), where C (T ) = call opton payment value at tme T, S (T ) = stock prce at tme T and K = strke prce. [ Y,0] N ( T ) 1 D Z ( T) = = max, where Z (T ) = sum nsurance payment value at tme T, N (T ) = number of clams up to tme T, Y = ncurred clam amount of clam number up to tme T and D = deductble for each clam occurrence. [ X ( T ) *,0] Z * ( T) = max D, where Z * ( T ) s the sum nsurance payment value at tme T, X (T ) = accumulated clam szes at tme T, and D * = deductble or excess pont of X (T ). Hence Z s an excess-of-loss nsurance contract and Z* s a stop-loss nsurance contract. We also observe that Z may be nterpreted as a stochastc sum of N sngle European call optons and Z* as an ordnary European call opton. Hence wthn ths context we may name these nsurance contracts as nsurance call contracts, and the prcng of them as nsurance opton prcng. The crucal dfference between stock opton contracts and nsurance contracts s generated by the underlyng stochastc processes of the contracts. The future values of a stock may occur up and down from a spot value at tme t = 0, whle the accumulated nsurance clam values only wll ncrease from a start value equal zero at tme t = 0. These dfferences are reflected n the classcal stochastc assumptons of the processes where the varaton of S( assumes to follow a geometrc Brownan moton process and X( most often assumes to be a compound homogeneous Posson process where the clam number process N( s a homogeneous Posson process (wth clams frequency parameter λ ) and the 4

5 clams severtes Y -es are ndependent and dentcally dstrbuted random varables and ndependent of the countng process N (. The assumpton of a geometrc Brownan moton process for S( where the runnng tme ncrements of the varaton of the process follow a normal dstrbuton functon s based on the dea that the spot prce of a stock at tme t already reflects the future expected value of the stock. Hence all relevant nformaton nfluencng the future expected return on the asset s assumed contnuously buld nto the runnng value (the spot prce) of the asset, that s, we assume a complete stock market where effcent market tradng of the assets generate these characterstcs. Even f the geometrc Brownan moton process tself s not very smple, we may say that the effects of the model assumptons of the stock prce process generate qute smple and nce prcng challenges wthn the fnancal opton markets. The compound homogeneous (or even more realstc, a mxed (heterogeneous)) Posson process assumpton of the clams rsk makes on the other hand more ntrcate problems to handle wthn a realstc nsurance market. The crucal purpose of prcng the opton and nsurance contracts s n any case to fnd explct expressons of the current value of the contracts at the contractual start tme, that s, n ths setup at tme t = 0. Or even more general we may want to fnd explct expressons of the current values of the contracts C, Z and Z* at any tme t (< T) before the maturty tme T. 4. Dynamc hedgng, no-arbtrage and put-call partes Followng the orgnal and now Nobel famous development lnes made by Black and Scholes (1973) and Merton (1973), the explct expressons of C, Z and Z* should be based on dynamc hedgng of the underlyng stochastc portfolos. The key bass of the Black-Scholes formula of fnancal opton values s to create rsk free synthetc portfolos by contnuously (dynamc) purchasng (hedgng) a certan share of the underlyng asset and a certan amount of a rsk less bond. The hedgng strategy has two key propertes: 1) It replcates the payoff of the opton, and ) It has a fxed and known total cost. Hence the value of the opton at any tme t may be explctly expressed by the combnaton of the shares of the asset and the bond. The most mportant assumpton behnd ths dynamc hedgng based prcng theory s the no-arbtrage assumpton. That s, we assume no opportuntes to make rsk less profts through buyng and sellng fnancal securty contracts. Ths assumpton generates the exstence of a put-call party, that s, a fundamental relatonshp between the values of a call opton and a put opton (the rght to sell a securty to a strke prce). Let C ~ be an European put opton wth the same strke K and maturty T as the call opton C : ~ ~ C( T ) = max[ K S( T ),0] where C ( T ) = put opton value at maturty tme T, S (T ) = stock prce at tme T and K = strke prce. 5

6 Then we have the basc put-call party known from every basc opton text book: ~ rt C C( = S( e K, (1) where r s the rsk free rate of nterest. In words: the value of the call mnus the value of the put s equal to the value of the stock mnus the present value of a rsk less bond K. Any devaton from the put-call party does consttute an arbtrage opportunty because smultaneously sellng and buyng the put-call portfolo (left hand sde of (1)) and the stock-bond portfolo (rght hand sde of (1)) yeld a rsk less proft equal the dfference between the values of the portfolos. Hence wthn opton prce theory the no-arbtrage assumpton s equvalent to the exstence of the put-call party. In fact, a more far-reachng consequence of the no-arbtrage assumpton s that two dentcal fnancal rsks must have the same value (the law of one prce). Turnng to the nsurance contracts, we may also construct put-call partes for the excess-of-loss and stop-loss contract. Ths depends however on the exstence of ~ ~ so-called nsurance put contracts. Hence let Z ( T ) and Z * ( T ) respectvely be the values at maturty tme T of an excess-of-loss put contract and a stop-loss put contract. We then have: N ( T ) ~ Z ( T) = max, () = 1 [ D,0] Y ~ Z *( T ) = max. (3) [ D * X ( T ),0] The practcal nterpretaton of the nsurance put contract s that the holder of the * contract gves the rght to a payment of D or D aganst a self-fnancng of the clams X (T ) durng the perod ( 0, T ). Ths s obvously a rght the holder of the * contract only wll use f the ncurred clams are less than D or D. Hence the owner of a clams rsk could by buyng an nsurance call contract and at the same tme sell an nsurance put contract on the same underlyng rsk gan a rsk-free * cash flow equal to the deductble D or D. The clams occurrence durng the contract perod wll not nfluence ths cash flow. By use of smple algebra we fnd straghtforward the general put-call party expressons of the nsurance contracts: ~ Z( T ) Z ( T ) = X ( T ) D( T ), where D ( T ) = N( T ) D (4) Z * ~ * ( T ) Z ( T ) = X ( T ) D * ( T ), where D *( T ) = D * (5) That s, the value of the call mnus the value of the put s equal to the value of the accumulated clam amounts mnus the (presen value of D (T ) of each contract. Hence the structure of the partes are dentcal to the fnancal opton party. 6

7 Followng the same arguments as for opton contracts, any devaton from the putcall party does consttute an arbtrage opportunty n the nsurance market because smultaneously buyng and sellng the nsurance put-call portfolo (left hand sde of (4) and (5)) and the clams-bond portfolo (rght hand sde of (4) and (5)) yeld a rsk less proft equal the dfference between the values of the portfolos. It s however mportant that these relatonshps depend on the exstence of an nsurance put contract Z ~ ~ or Z * and the exstence of an effcent market place of buyng and sellng nsurance calls and puts. But what about dynamc hedgng of nsurance contracts? The exstence of nsurance put-call partes and no-arbtrage n an nsurance market are mportant elements, but not suffcent to replcate a rsk free payoff of an nsurance rsk by a hedgng strategy. It s well known that the ordnary nsurance clams rsk process X ( s dffcult to drectly hedge away by dynamc tradng, and by ths follow the classcal fnancal prcng theory. Delbaen and Haezendonck (1989), however, assumed that an nsurer at any tme t can sell the remanng clam payments X ( T ) X over the perod ( t, T ) for some premum p( T. Hence the value of the portfolo of clams rsks at tme t s I = X + p( T, where I ( 0) = p( T ) and I ( T ) = X ( T ). Ths assumpton of dynamc buyng and sellng nsurance rsks durng the nsurance perod makes the crucal possblty for dynamc hedgng also n nsurance markets. By followng the Black-Scholes approach, a straghtforward descrpton of ths rsk less nsurance hedge at tme t s: The value of the nsurance contract at any tme t may be expressed by a combnaton of a share 1( of the underlyng clams rsk portfolo I ( and an amount of a rsk less zero-coupon bond B 1 maturng at tme T. Hence the value of the nsurance hedge portfolo at tme t s ( T I( e r B ( ). (6) 1( 1 t Hereby the dynamc hedgng of I ( am at contnuously rebalance (as for optons we assume that rebalancng costs do not nclude transacton costs) the hedge by holdng ( ) shares of I ( long and a bond maturng at tme T to B ( ) short. 1 t Correspondngly, the value of the opton hedge portfolo at tme t s ( T S( e r B ( ), (7) ( t where the share - the delta of the opton s the rate of change of the value of the opton wth respect to the underlyng value of the stock, that s, the relatve amount the opton value C ( wll change when S ( change. Hence, s close to 1 when the opton s deep n the money and near expraton (hgh probablty to expre n the money), and close to 0 f the opton s deep out of money and near expraton (hgh probablty to expre out of the money). 1 t 7

8 We may correspondngly nterpret 1( of I ( as the relatve amount the nsurance contract value Z( wll change when I( change. What ths nterpretaton really means s not obvous. The future values of a stock may ncrease and decrease wth no lmts (but not below zero) from a runnng spot value at tme t. These ncrease and decrease are also true for I(, but there s a lower lmt of I( equal X(. That s, at tme t the future value of I( can not be lower than X( because I = X + p( T and X ( can not decrease. Hence, n that moment X ( exceeds the deductble D or D * the nsurance contract wll be and stay n the money, that s, Z ( T ) > 0 or Z * ( T ) > 0. Because I ( conssts of both X ( and p( T there s n contrast to the opton and S ( - no one-to-one relatonshp between the values of I ( and Z ( even f Z ( wll certanly stay n the money the rest of the perod ( t, T ). Correspondngly to Black-Scholes, the value of the nsurance hedge portfolo s n fact equal to the value of the nsurance contract. Partcularly we have Z (0) = Value of nsurance contract at tme t = 0 = Value of nsurance hedge portfolo at tme t = 0 rt rt = 0) I (0) e (0) = 0) p( T ) e (0). (8) 1( B1 1( B1 That s, the value of the nsurance contract at tme t = 0 s equal to a lnear functonal 1 of the premum p (T ) of the rsk perod ( 0, T ) mnus the present value of the bond B 1 (whch should be nterpreted as the amount needed to pay deductbles n ( 0, T ) ). The most effcent mathematcal machnery to fnd more explct expressons of 1 and B 1, and hence value expressons of Z and Z *, s to use the now classcal martngale approach under the rsk-neutral valuaton prncple. 5. Rsk-neutral martngale prcng The dynamc hedgng strategy and the no-arbtrage assumpton are closely related to the rsk-neutral valuaton prncple. Ths prncple states that n a rsk-neutral world all assets have an expected rate of return equal to the rsk-free rate of return. In the real world we have to consder the relatonshp between rsk (volatlty) and reward (expected return) to prce an asset. In a rsk-neutral world, reward has been normalzed through the no-arbtrage mechansms, that s, all relevant market nformaton are assumed known and buld nto the prcng models so that no nvestors wll expect a hgher reward than the rsk-free rate of return. Ths nformaton transacton s techncally gven by tunng the classcal real world probablty measure P on the probablty space ( Ω, F, P) nto a fltered unquely equvalent probablty measure (by use of the so-called Grsanov transformaton) on the probablty space ( Ω, F,( Ft ) t 0, ) where F t can be vewed as the market nformaton avalable at tme t. Hence we make expected value calculatons based on nstead of P. Ths nformaton transacton s assumed vald at every tme t, and hence the underlyng rsk process (wth respect to 8

9 measure ) turns out to be a martngale process where at each tme t the expected next value drecton of the process s zero (corrected for the rsk-free rate of return). In practce ths mples that t s mpossble to earn (rsk less) money by studyng the hstory of a stock, that s, all relevant and avalable nformaton about the stock and the market are assumed fully ntegrated nto the runnng spot prce of the stock. The techncal lnk called the Fundamental Theorem of Asset Prcng - between the no-arbtrage assumpton and the martngale process was the key dscovery made by Harrson and Kreps (1979) and further developed by Harrson and Plska (1981) and generalzed by Delbaen and Schachermayer (1994). See Delbaen and Schachermayer (1997) for a rgorous mathematcal summary of relevant man results. The setup by Delbaen and Haezendonck (1989) of the nsurance clam rsk portfolo I = X + p( T wth effcent dynamc hedgng possbltes, generates an arbtrage-free nsurance market. Hence ths specfc market setup turns nto be rsk nformaton transformed, that s, from only assumng the orgnal probablty dstrbuton of the nsurance rsk process X( (the classcal statc actuaral approach) to nstead assume a fltered dstrbuton of X(, n ths case, gve more weght to unfavourable events of X(. Agan ths approach s techncally gven by tunng the classcal real world probablty measure P nto a equvalent fltered rsk neutral probablty measure. Hence we make expected value calculatons on the portfolo value process I( by usng the measure nstead of drectly on the rsk process X( by usng the measure P. The portfolo value process I( s hence called a -martngale,.e. at each tme t the expected next value drecton of the process I( s zero (corrected for the rsk-free rate of return). Wthn ths model a practcal nterpretaton s that t s mpossble to earn money by studyng the clams hstory of (0, of the nsurance portfolo I( because all relevant nformaton about the nsurance rsks are assumed fully reflected nto the runnng rsk premum p( T at tme t. To sum up: Whle the geometrcal Brownan moton assumpton of the varaton of the stock value S( drectly generates the martngale process of S(, we have to transform the accumulated Posson clams rsk process X( nto a clams rsk portfolo I( consstng of both hstorcal clams and premum for future clams to generate a martngale process ndrectly. Embrechts et al (1999) gves a thorough ntroducton to ths lnk between stochastc processes n nsurance and fnance. Stll followng the approach of Delbaen and Haezendonck (1989) and gven our clams rsk process assumptons n chapter 3, the far values at tme t of the excess-of-loss contract Z and the stop-loss contract Z * become r( T [ e Z( T )] Z( = E, where = adjusted rsk-free probablty measure, Z * r ( T * [ e Z ( )] = E, where * = adjusted rsk-free probablty measure, * T whle the correspondngly far value at tme t of the call contract becomes 9

10 r ( T [ e C( )] C( = E, where ** = adjusted rsk-free probablty measure. ** T Hence the far values of the contracts at tme t = 0 are rt [ e Z( )] Z( 0) = E T (9) Z * (0) rt * rt = E * [ e Z ( T )] = e [ E * X ( T ) E * D] rt = e [ E * N( T ) E * Y E * D] rt rt = E * [ e C( T )] = e [ E ** S( T ) E ** K ] C( 0) * Correspondngly, the straghtforward derved Black-Scholes formula of C ( turns out to be ( 1 t (10) (11) ( T C = N( d ) S( e r KN( d ( )), (1) where N ( ) = cumulatve normal dstrbuton functon and d 1( and d are gven n most basc text books of opton theory and of course n Black and Scholes (1973). Hence ( t ) = N( d1( ) and B = KN( d ). We also observe that the martngale fltered measure ** of (11) reflects the values of and B n (1), that s, E ** S( 1 t T ) = N( d ) S( = S( ), and E K = N d ) K = B ( ). ** ( t Hence N( d 1 ) s the rate of change at tme t of the value of the call wth respect to changes n stock prce S (, whle N ( d ) should be nterpreted as the probablty that C ( at the maturty tme T wll be n the money. When studyng (9), (10) and (11) we observe the structural smlartes between the far value expresson of the nsurances Z and Z* and the opton C. In fact the values are lnear functons defned on a set of marketable assets. ute general we have: Fnancal call opton Insurance contract The value at tme t ncreases The value at tme t ncreases - the longer tme to maturty, - the longer tme to maturty, - the lower the strke prce s, - the lower the deductble s, - the hgher the spot prce s*, - the hgher the clams have been*, - the hgher the rsk free rate s, - the hgher the rsk free rate s, - the hgher the stock volatlty s. - the hgher the clams rsk s. 10

11 Four out of fve parameters are easy to understand. However, one *-comment on the spot prce versus the clams value at tme t s needed: At any tme t the spot value of a stock s a postve value, whle the nsurance clams have a zero start value at tme t = 0 and possbly a postve value at any tme t > 0. Hence the expresson the hgher the clams have been s related to the clams occurred up to tme t. The great practcal usefulness of the Black-Scholes opton formula s due to that only the stock volatlty s unknown as valuaton parameter at the valuaton tme. Opton dealers are hence often just called volatlty dealers, where mpled volatlty or tght market knowledge s often used as the parameter estmaton technque of the volatlty. The fact that the value of an opton depends on the volatlty of the underlyng but not on ts expected return (whch s n accordance to rsk-neutralty ), numbers as one of the great nsghts from the Black-Scholes formula. Crss (1997) s hereby a pedagogcal reference as a wde and basc ntroducton to fnancal optons and opton prcng. As for optons the clams rsk s so far and wthn the above setup the most stochastc based parameter to estmate for nsurance contracts. However to generate more specfc expressons of (9) and (10) we have to deal wth the fltered probablty measure. Delbaen and Haezendonck (1989) see also Embrechts (000) outlnes that certan assumptons and condtons of the fltered -measure lead to well-known premum prncples dentcal to the expected value prncple, the varance prncple and the Esscher prncple, all generatng dfferent safety loadng factors n addton to the pure expected clams rsk up to tme T based on the classcal probablty measure P. Hence the choce of the techncal propertes of turns out to be the essental part of the valuaton of Z and Z*, or n other words, t all depends on what relevant extra nformaton we put nto the measure at tme t. The three premum prncples are for nstance based on the a pror choce of safety loadng as the (only) crtcal nformaton bas. The safety loadng s bascally mportant, and corresponds to the need for run safety for the nsurer and to the expected utlty wllngness of the customers to pay hgher premum than the pure rsk premum. However, other hghly relevant nformaton to nclude a pror n the prce models and hence affectng the -measure are e.g. nformaton about the admnstraton and reassurance expensve and the demand for nvestment return on the nsurance portfolo. Techncally ths expanded nformaton nput wll be handled by an expanded prce model combned wth an augment at tme t of the nformaton fltraton F t on. In a complete market setup, wth perfectly effcent buyng and sellng of nsurance contracts, we conclude that pure rsk and cost based premums are the suffcent prcng tasks to handle for an nsurance company. In such a market, where the market and prce equlbrum rules, the wnners of the market wll be those companes whch are able to delver the best unbased and ant selected rsk prces, the lowest admnstraton expensve and the most effcent reassurance and nvestment return demands. However, as we wll dscuss further on n chapter 6 and 7, true market nformaton lke purchasng preferences and the nsurers prce poston n the market wll be crucal nformaton to take nto account n an ncomplete market setup. 11

12 6. Incomplete market effects and purchasng preferences The trangle assumptons of dynamc hedgng, no-arbtrage and market completeness are mathematcally very pleasant to assume when developng asset prcng models and formulas. Even f there are many offence on these dealstc assumptons, the relatve realsm and robustness of them n the fnancal markets seems nevertheless to be one of the man reasons for the revoluton of fnancal mathematcs the last decades. The market place of fnancal dervatves and nsurance contracts are, however, normally qute dfferent. The nsurance markets are most often characterzed by hgher transacton costs, hgher and dfferentated rsk averson, less prce ratonalty and senstvty, less or no dynamc hedgng, less decson speed and fewer players than fnancal markets. In addton the nsurance market may be dvded nto a drect nsurance market and a rensurance market, where the above ncomplete characterstcs usually are more common n the drect market than n the rensurance market. However, the nsurance nternet market place, wth effectve and prce senstve shoppng of nsurance offers, may more and more generate less ncomplete effects on the drect markets. In a future settng wth an nsurance market as effcent and complete as the fnancal dervate market, we may keep the complete market assumpton approach of nsurance prcng. Untl so, f we take an average pragmatc approach on today s nsurance prcng, we defntely have to leave the dealstc assumpton of market completeness. However, the prcng mechansms and result outcomes of complete market models gve deep prcng nsght, and are mportant to have n mnd when dealng wth expanded ncomplete models. Mathematcal assumptons of ncomplete markets generate much heaver theoretcal models. A precse defnton gven n Harrson and Plska (1983) states that a market s complete f and only f there s only one equvalent martngale measure of the underlyng stochastc process S( (the stock marke, or n our case, I( (the nsurance marke. The practcal nterpretaton of ths defnton should be that n complete markets there exsts a unque prce for each unque contractual rsk - ndependent of the nsurance players - and that each contract can be perfectly hedged or replcated n the market. Market assumptons whch do not satsfy the prce unqueness per rsk contract generate ncomplete markets. Typcally assumptons see Embrechts and Mester (1997) - whch generate ncomplete markets are: Jumps n the underlyng stochastc process wth random sze occur: For nstance large jumps n stock markets because of bubble economy crash or n nsurance markets because of heavy large-clam nature/weather catastrophes, or even smpler, an unexpected strong ncrease n the underlyng market clams nflaton. Stochastc volatlty models, that s, assumptons of underlyng stochastc processes wth varable volatlty along the tme axs. For nstance the varablty of stocks or nsurance clams whch we assume to ncrease over the 1

13 perod of tme; ths may be realstc n e.g. some property nsurance markets whch have experenced ncreased whether varablty the last years. Markets wth transacton costs and/or nvestment constrants (so-called frcton): For nstance customers who loose some beneft buld ups or have a hgher product, servce and/or brand preferences than pure prce preferences, - or dfferent nvestment constrants between market players. Some techncal mathematcal technques have so far been developed to handle some problems of prcng dervatves n ncomplete markets. Super replcaton (fnd the cheapest self-fnancng hedgng strategy) and mean-varance hedgng (fnd a tradng strategy that reduces the actual rsk of the dervatve poston) - whch s one of the so-called quadratc methods are two examples; see e.g. Møller (000) for more rgorous and detaled nformaton and overvews on ths subject. The ntutve mathematcal reason for the ncompleteness of the above market assumptons s that n each case there are typcally many dfferent combnatons of model parameters whch nfluence the expanded model setup of the martngale process. Hence ths property generates many dfferent probablty measures that turn the underlyng process nto many dfferent so-called local martngales. A pragmatc consequence s that each fnancal or nsurance contract (where each one of them s based on one unque fnancal or nsurance rsk) turns out to be valuable unque for each combnaton of buyers and sellers of the contract. We may hence conclude that the more ncomplete an nsurance market s, the more the prces of equvalent rsks tend to vary n the market (the law of one prce turned upsde down). Remark that ths connected statement assumes a deregulated nsurance market wth no prce-cooperaton between the nsurer players n the market. Another assumpton whch generates ncomplete markets, at least wthn drect nsurance, s to assume postve product, servce and/or brand preferences of the buyers n addton to the prce preference. That s, the buyers of the contracts have other soft purchasng preferences n addton to prce when they act n the market. Ths assumpton s hghly realstc n most (or all) drect nsurance markets, and probably one of the man drvers for the ncompleteness of these markets. Ths market condton generates the ntroducton of purchasng rate as an mportant parameter wthn the prce models. A defnton of purchasng rate may be the lkeablty n the market to buy (or re-buy) a predefned nsurance contract. Ths parameter vares typcally ndvdually between the nsurance customers (just lke ther clams rsks) and should normally be a functon of the prce senstvty, the prce poston of the nsurer and the more soft preferences lke product-, servce- and brand preferences. Hence, we say that buyers wth less prce senstvty are more soft preferenced than other buyers, and vce versa. We conclude that ths approach of ndvdually purchasng rates or probabltes - matches the above local martngale nterpretaton of prce unqueness for each combnaton of nsurance buyer and seller. Moreover by ntroducng the nsurance purchasng preference the brdge to more sophstcated economcal equlbrum 13

14 prce models s qute close; see e.g. Aase (1993) for an equlbrum analyss of dynamc rensurance markets. To sum up the man answers to the questons we stressed n chapter : Dfferent nsurance players offer dfferent prces for unque rsks manly because each combnaton of buyers and sellers n ncomplete markets generates ndvdual unque contractual values. More unque market prces for unque rsks wll be generated n more complete markets wth hgher prce senstvty. Optmal prce per rsk n a complete nsurance market s pure rsk and cost prce based, whle the prce n addton should be market adjusted n ncomplete markets. 7. Pragmatc nsurance prce models Correspondng to the ndvdual martngale theory n chapter 6, we may put forward the followng pragmatc prce model approach of net nsurance sales prces n the ncomplete case: Net sales prce = pure rsk prce + nternal cost prce adjustments + external market prce adjustments Mathematcally we may express the model as: Net sales prce = ( ( Z ( T I ) I ) I ) p ( T) = M C (13) r c m where p ( = net premum of rsk contract for perod ( 0, T ) Z ( ) = pure rsk prce pr rsk contract C ( ) = the cost prce adjustment of the pure rsk prce Z M ( ) = the market prce adjustment of the cost prce C F r = gven clams rsk nformaton of ( 0, T ) F c = gven cost nformaton of ( 0, T ) F = gven market nformaton of ( 0, T ). m The nternal cost prce adjustment C ( ) n (13) s typcally generated by specfed goals of rsk rato (ncurred clams n percent of earned premums) and combned rato (rsk rato + expense rato (expenses n percent of earned premum)) for each relevant lne of busness. The rsk rato and combned rato goals are usually part of the fnancal plan of the nsurance company and based on opnons of necessary levels of rsk loadng, expenses, fnancal ncome, reassurance and demand for nvestment return of the company, all elements specfcally broken down to each lne of busness. The external market prce adjustment M ( ) n (13) s to some extent more dffcult and more sophstcated to deal wth than the pure rsk and cost prce elements. A 14

15 possble defnton of ths prce adjustments could be: All knds of prce devaton from the theoretcal best known rsk and cost based prce. Ths defnton supports the statement by Fnn and Lane (1997) that there s no rght prce of nsurance, but smply the transacted market prce whch s hgh enough to brng forth sellers, and low enough to nduce buyers, ref chapter. Ref also our dscusson n chapter 6 we may model the market prce adjustment by ncludng and modellng the purchasng rate of the customers/market n an compettve open-market setup. A well-known setup s hereby to structure p ( ) n (13) by usng the net present valuaton approach of the nsurance contracts. That s, to tune the premum p ( ) n a way so the net present values of the contracts and the portfolo are optmzed under some fnancal condtons (typcally predefned fnancal plan goals). Ths approach takes care of what nsurance busness steerng s all about, that s to fnd the optmal operatonal balance between runnng proftablty and market share. In addton to the clams rsk process Z (, we assume the stochastc process of the purchasng rate R ( at tme t. Let Z ( and R ( denote the clams rsk and the purchasng rate for contract number, and let V (T ) be the net present value of contract over the tme perod ( 0, T ). Hence we have: where T [ p Z c ] rt V ( T) = e R dt, (14) 0 R (0) = the ht sales rate of contract at tme t = 0, and ~ R t> 0 = 1 R t> 0, where R t> 0 = the renewal rate of contract at tme t and ~ R t> 0 = the cancellaton rate of contract at tme t. The practcal nterpretaton of these defntons s that nsurance customers are contnuously purchasers through ther contnuously decsons of contractual cancellaton or not. In (14) remark also the assumed known constant parameters of r = dscount rate, and c ( = the total nternal cost rate of contract at tme t. Hence the net present value V (T ) of the portfolo of all nsurance contracts over the tme perod ( 0, T ) s I [ p Z c ] rt V ( T ) = V ( T ) e R dtd. (15) = 1 I T 00 Algebrac - or more realstc, numercal - optmum solutons of (14) and (15) wth respect to the net premum p ( ) and preferable under some fnancal constrants, wll mplctly generate the structure of the prce model (13). The external market adjustments M n (13) s hereby materalzed through the stochastc purchasng varable R n (14) and (15) whch further on should be modelled by usng the customer explanaton parameters of for nstance prce senstvty, prce poston n 15

16 the market and soft preferences of product, servce and/or brandng, ref earler dscussons. Maybe also the dstrbuton power of the nsurer, whch really has purchasng effect n most markets, should be ncluded as an explanaton varable. Hence, dependent of the purchasng preferences n the market, too hgh prces or too low prces both wll generate lower net present values V (T ) and V (T ) than mddle path prce values. We should anyhow demand EV ( T ) > 0 and EV ( T ) > 0 for all = 1,..., I, that s, we should have net sales prces whch generate postve expected net present values of all nsurance customers. A challenge wthn ths model s, however, to lnk ths ncomplete market setup wth a local martngale approach based on market equlbrum, and hereby fnd explct closed form expressons of the net premum p ( ) n (13) as a functon of the expected values of EZ (T ) and ER (T ), ether under a real world probablty measure P or under an nformaton fltered measure. We leave ths challengng problem for future theoretcal research. One last comment before we fnally end the nsurance prcng approach by dealng wth the parameter estmaton: Sometmes the E N and E Y n (10) n the complete market case n chapter 5 are nterpreted as the market prce of clams frequency and market prce of clams severty. Ths nterpretaton may be correct n a specfc theoretcal sense of vew, but not necessarly n a general practcal one. Wthn practcal nsurance prcng t s most often operatonal sutable to treat the classcal pure rsk premum on the physcal rsk measure P apart from the other effects or elements of the prcng structure generated by some (ndvdual) fltered measure. Hence n an operatonal setup we should splt out, estmate and follow-up the real clams frequency and the real clams severty, nstead of envelope them n some knd of mprecse and msleadng market nterpretaton. Hence, all or most of the other effects of the nsurance premum wll turn out to be ether the cost adjustment or the market adjustment of the rsk prce n (13), and hence easer to focus and follow-up explctly. 8. Parameter estmaton = real nsurance prcng methodology The parameter estmaton of the prcng models represents the real heart of nsurance prcng. Compared to fnancal dervatves, the parameter estmaton wthn non-lfe nsurance s more complex. The man dfference between the dervatves and the nsurance contracts are hereby that dervatves are rather techncal n ther contractual consderaton but qute smple n ther parameter valuaton, whle the nsurance contracts have the opposte characterstcs. Hence whle the volatlty parameter of an opton s most often estmated by practcal market knowledge or mpled real market volatlty, the pure rsk prce pr rsk n (13) s usually splt nto expected clam frequency and expected clam severty and hence estmated separately by usng the data applcaton machnery of classcal statstcal models lke Generalzed Lnear (or Addtve) regresson (Mxed) Models (GLM, GAM or GLMM) or emprcal Bayes credblty models on large data sets of rsk exposure and reported clams. 16

17 In chapter 3 we assumed the underlyng clams number process N ( to follow a homogeneous Posson process wth constant frequency parameter λ, and hereby the clams rsk process X( to be a compound Posson process. A far more common and realstc model s however to allow stochastc varaton of the clams frequency from ts expected value λ. Hence f λ follows a Gamma dstrbuton, then N( becomes a Negatve Bnomal process. Ths setup matches very well our desre to make a rsk dfferentated premum tarff whch reflects rsk varances (both clams frequency and clams severtes) between customers/objects n the nsurance portfolo and market. Hence, ths matches also the standardzed GLM setup wth Posson/Gamma-assumptons. Important practcal elements to handle wthn rsk prcng are choce of explanaton rsk varables (tarff factors), estmaton levels (product, coverage or clams type level), categorzaton of varables, correlatons between varables, use of subjectve experence/knowledge, follow up and mantenance. There s a wde number of references to pure rsk prcng estmaton technques; an old classc s e.g. Brockman and Wrght (199), whle a newer reference of appled GLM s Fahrmer and Tutz (000). The purchasng rate R( n (15) may also be modelled by usng GLM. A well known and well used model s to assume the purchasng rate as a bnary varable (the customer ether purchase/renewal the contract at a dscrete tme, or he/she does no, and hence assume a Logstc Regresson model wth explanaton varables based on the ndvdual purchasng preferences of the customers,. e. lke pure customer characterstcs, premum ncrease(s) last year(s) and the nsurers prce and dstrbuton poston n the market. Understandng the purchasng rates (both sales ht rates and renewal rates) for dfferent customer segments preferable by use of some stochastc models should be a basc part of the model applcatons wthn pragmatc nsurance prcng n ncomplete markets. Fnally we stress the mportance of forecastng the clams rsk nflaton or trend level. A suffcent predcton of the future trend level of the clams rsk process X( s extremely mportant wthn fnancal goal steerng. The most pragmatc setup s to let the forecastng models be stand alone models apart from the GLM rsk dfferentaton models, and hence nfluencng only the premum levels of the premum tarffs. However, usng GLM-based leadng ndcators ntegrated wth autoregressve ntegrated or movng average (ARIMA) models gves often an effcent modellng of the clams rsk nflaton. 9. Internatonal best practce of nsurance prcng A standard term of the prcng qualty of the best nsurance players n a market s best market practce or when the prcng qualty s among the best n the world world class practce. The nternatonal trend s a move from an nternal cost based prce focus to a market-orented vew on prcng, just lke the wde approach n ths paper. An ncreased number of consultants offer ther knowledge and servce to help the companes to reach a best practce level of prcng. Key elements of a best practce market-orented vew on prcng are: 17

18 Outstandng qualty of the nsurance clams rsk and underwrtng processes and ther tme dependent rsk cycles. Ths qualty s fundamental for proftable market based prcng. Outstandng knowledge of customer behavour n terms of customer segments, products/servce standards and prce preferences, ncludng own prce and product/servce postons relatve to compettors n the market. Hence not only the tradtonal actuaral prcng approach of rsk and cost prcng, but also the busness elements of CRM (Customer Relatonshp Managemen ncludng product and servce standards should be mportant parts of the fnal premum tarff prces of nsurance products and contracts. An example of a mx of these elements s e.g. treated n Holtan (001), wth focus on customer purchasng ratonalty of ndvdual nsurance bonus-malus contracts. Ths wde busness approach demands defntely a dynamc prcng approach, that s, a process whch contnuously or regularly - adjust the prce levels to reach rsk rato and combned rato goals, - nnovate, update and re-estmate rsk prce models and parameters, - track compettors prce rates and product/servce value propostons, - test market prce adjustments to learn optmal prce adjustments, - mplement champon-challenger prce adjustments, - communcate wth dstrbuton departments to optmze the sales forces. Ths generates and assumes an actve role of the analysts (actuares) n the product and prce departments of the nsurance companes, wth focus on dynamc processes as an overall operatonal key factor to succeed. The methodology of rsk and cost prcng ref chapter 8 s qute famlar to most nsurance companes and markets. The methodology of market prcng s, however, far less famlar. It s mportant to recognze that there are no fxed rules or fnal answers to market prce challenges and questons. A dynamc data drven (and less statc theoretcal model drven) workng approach wth flexble and short tme-to-market IT prce calculaton systems seems, however, to be some knd of a common element of the best market prce practcng nsurance companes. Modellng the expected purchasng rate and the expected customer proftablty n a net present value context, are hereby some key ssues to handle. References to best practce nsurance prcng are n ts nature qute few and dffcult to put forward. The practce of Drect Lne Insurance Ltd n UK and ther explosve way to reach 15-0% market share n the UK motor nsurance market n the 1990-thes wth satsfactory proftablty s often mentoned as an example. Other best players - that s, players wth extremely good proftablty n combnaton wth stable or growng markets shares over a sgnfcant perod of tme are of course also good references. A readable journey n spte of ts age - to best busness focused rsk/cost prce and underwrtng practce s however McKnsey (1995). 18

19 10. Summary and concludng remarks A fundamental dfference between fnancal opton valuaton technques and the classcal actuaral valuaton prncples s that the opton valuaton s formulated wthn a market framework whch ncludes the possblty of tradng and hedgng. Ths a pror model approach whch very much s consstent wth the dervate market realtes generates n most cases complete market condtons wthout arbtrage possbltes. Ths proactve market model approach has tradtonally been absent wthn classcal actuaral prcng prncples, whereas most of the prncples have only been based on more or less ad hoc statstcal consderatons nvolvng the law of large numbers. There seems however to be a theoretcal trend to nvolve the fnancal market model approach nto nsurance prcng problems. Wthn such a framework there seems to be easer to expand the nsurance prce models by ncludng components lke purchasng preferences n ncomplete market setups, and by ths nclude not only the valuaton part of prcng, but also the supply/demand part nto the prce models. A short overvew of the prcng elements of nsurance versus fnance versus theory versus practce may be presented as follows: Practse Theory Rsk and cost dfferentated prces Data drven GLM parameter estmaton Incomplete and non-effcent markets Purchasng/prce senstve preferences Data drven market adjusted net prces Insurance clams as underlyng rsk Implct complete markets No hedgng or arbtrage focus Homo/heterogeneous Posson processes Rsk valuaton wth safety loadngs Insurance Stll complete and effcent markets Extreme contractual complexty Dfferent contractual objects Computer formula based valuaton Impled volatlty and market knowledge Stocks/securtes as underlyng rsk Explct complete and effcent markets Dynamc hedgng wth no arbtrage Geometrc Brownan moton processes Volatlty valuaton and prcng Fnance The whole area of brngng together elements from fnancal and actuaral valuaton and prcng technques s stll very much under constructon. We should stll expect a consderable progress wth effects for nsurance prcng n the years to come. Some future nsurance challenges n the 1st century wth opposte drectons are a) optmal prcng n ncomplete frcton markets wth less prce senstvty (whch s stll the case for many or most tradtonal markets/customer segments) and b) optmal prcng n effcent and complete nsurance markets wth contnuously nsurance tradng and hedgng possbltes (future effcent nternet and broker based markets; see Fnnegan and Moffat (000) for a pragmatc and problematc vew on ths subjec. In the latter case the brdge to opton prcng s very short. In both cases the prcng optmalty should have a balanced focus on proftablty versus market share of the (drec nsurance portfolo. In such an nnovatve framework actuares n all countres should know how to act and take leadng postons n ther nsurance companes to unte the future theoretcal and practcal parts of nsurance prcng. 19

20 References Aase, K. K. (1993) Equlbrum n a rensurance syndcate; Exstence, unqueness and characterzaton. ASTIN Bulletn 3, Black, F. and M. Scholes (1973) The prcng of optons and corporate labltes. Journal of Poltcal Economy 81 (3), Brockman, M. J. and T. S. Wrght (199) Statstcal Motor Ratng: Makng effectve use of your data. Journal of Insttute of Actuares 119, Crss, N. A. (1997) Black-Scholes and Beyond Opton Prcng Models. McGraw-Hll. Delbaen, F. and J. Haezendonck (1989) A martngale approach to premum calculaton prncples n an arbtrage free market. Insurance: Mathematcs and Economcs 8, Delbaen, F. and W. Schachermayer (1994) A general verson of the fundamental theorem of asset prcng. Matematsche Annalen 300, Delbaen, F. and W. Schachermayer (1997) Non-Arbtrage and the fundamental theorem of asset prcng: Summary and man results. Proceedngs of Symposa n Appled Mathematcs, Amercan Mathematcal Socety. Embrechts, P. and S. Mester (1997) Prcng nsurance dervatves, the case of CAT-future. Proceedngs of the 1995 Bowles Symposum on Securzaton of Rsk. Georga State Unversty, Atlanta. Socety of Actuares, Monograph M-FI97-1, Embrechts, P., R. Frey and H. Furrer (1999) Stochastc processes n nsurance and fnance. Workng Paper, Department of Mathematcs, ETH Zürch, Swtzerland. Embrechts, P. (000) Actuaral versus fnancal prcng of nsurance. Workng Paper, Department of Mathematcs, ETH Zürch, Swtzerland. Fahrmer, L. and G. Tutz (000) Multvarate Statstcal Modellng Based on Generalzed Lnear Models. Second Edton. Sprnger. Fnn, J. and M. Lane (1997) The perfume of premum or prcng nsurance dervatves. Proceedngs of the 1995 Bowles Symposum on Securzaton of Rsk, Georga State Unversty, Atlanta. Socety of Actuares, Monograph M- FI97-1, Fnnegan, D. and S. Moffat (000) Auto Insurance Prcng Crss. Research report, ualty Plannng Corporaton, San Francsco, USA. 0