Pragmatic Insurance Option Pricing

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "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: 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

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ). REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

More information

Analysis of Premium Liabilities for Australian Lines of Business

Analysis of Premium Liabilities for Australian Lines of Business Summary of Analyss of Premum Labltes for Australan Lnes of Busness Emly Tao Honours Research Paper, The Unversty of Melbourne Emly Tao Acknowledgements I am grateful to the Australan Prudental Regulaton

More information

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange

More information

The Application of Fractional Brownian Motion in Option Pricing

The Application of Fractional Brownian Motion in Option Pricing Vol. 0, No. (05), pp. 73-8 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qng-xn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com

More information

Simon Acomb NAG Financial Mathematics Day

Simon Acomb NAG Financial Mathematics Day 1 Why People Who Prce Dervatves Are Interested In Correlaton mon Acomb NAG Fnancal Mathematcs Day Correlaton Rsk What Is Correlaton No lnear relatonshp between ponts Co-movement between the ponts Postve

More information

An Alternative Way to Measure Private Equity Performance

An Alternative Way to Measure Private Equity Performance An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate

More information

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy 4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

More information

Capital asset pricing model, arbitrage pricing theory and portfolio management

Capital asset pricing model, arbitrage pricing theory and portfolio management Captal asset prcng model, arbtrage prcng theory and portfolo management Vnod Kothar The captal asset prcng model (CAPM) s great n terms of ts understandng of rsk decomposton of rsk nto securty-specfc rsk

More information

Stress test for measuring insurance risks in non-life insurance

Stress test for measuring insurance risks in non-life insurance PROMEMORIA Datum June 01 Fnansnspektonen Författare Bengt von Bahr, Younes Elonq and Erk Elvers Stress test for measurng nsurance rsks n non-lfe nsurance Summary Ths memo descrbes stress testng of nsurance

More information

Can Auto Liability Insurance Purchases Signal Risk Attitude?

Can Auto Liability Insurance Purchases Signal Risk Attitude? Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159-164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? Chu-Shu L Department of Internatonal Busness, Asa Unversty, Tawan Sheng-Chang

More information

Hedging Interest-Rate Risk with Duration

Hedging Interest-Rate Risk with Duration FIXED-INCOME SECURITIES Chapter 5 Hedgng Interest-Rate Rsk wth Duraton Outlne Prcng and Hedgng Prcng certan cash-flows Interest rate rsk Hedgng prncples Duraton-Based Hedgng Technques Defnton of duraton

More information

Traffic-light a stress test for life insurance provisions

Traffic-light a stress test for life insurance provisions MEMORANDUM Date 006-09-7 Authors Bengt von Bahr, Göran Ronge Traffc-lght a stress test for lfe nsurance provsons Fnansnspetonen P.O. Box 6750 SE-113 85 Stocholm [Sveavägen 167] Tel +46 8 787 80 00 Fax

More information

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo

More information

Multiple discount and forward curves

Multiple discount and forward curves Multple dscount and forward curves TopQuants presentaton 21 ovember 2012 Ton Broekhuzen, Head Market Rsk and Basel coordnator, IBC Ths presentaton reflects personal vews and not necessarly the vews of

More information

Recurrence. 1 Definitions and main statements

Recurrence. 1 Definitions and main statements Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

More information

Course outline. Financial Time Series Analysis. Overview. Data analysis. Predictive signal. Trading strategy

Course outline. Financial Time Series Analysis. Overview. Data analysis. Predictive signal. Trading strategy Fnancal Tme Seres Analyss Patrck McSharry patrck@mcsharry.net www.mcsharry.net Trnty Term 2014 Mathematcal Insttute Unversty of Oxford Course outlne 1. Data analyss, probablty, correlatons, vsualsaton

More information

Multiple-Period Attribution: Residuals and Compounding

Multiple-Period Attribution: Residuals and Compounding Multple-Perod Attrbuton: Resduals and Compoundng Our revewer gave these authors full marks for dealng wth an ssue that performance measurers and vendors often regard as propretary nformaton. In 1994, Dens

More information

Bond futures. Bond futures contracts are futures contracts that allow investor to buy in the

Bond futures. Bond futures contracts are futures contracts that allow investor to buy in the Bond futures INRODUCION Bond futures contracts are futures contracts that allow nvestor to buy n the future a theoretcal government notonal bond at a gven prce at a specfc date n a gven quantty. Compared

More information

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..

More information

The Short-term and Long-term Market

The Short-term and Long-term Market A Presentaton on Market Effcences to Northfeld Informaton Servces Annual Conference he Short-term and Long-term Market Effcences en Post Offce Square Boston, MA 0209 www.acadan-asset.com Charles H. Wang,

More information

On the pricing of illiquid options with Black-Scholes formula

On the pricing of illiquid options with Black-Scholes formula 7 th InternatonalScentfcConferenceManagngandModellngofFnancalRsks Ostrava VŠB-TU Ostrava, Faculty of Economcs, Department of Fnance 8 th 9 th September2014 On the prcng of llqud optons wth Black-Scholes

More information

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt. Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces

More information

Efficient Project Portfolio as a tool for Enterprise Risk Management

Efficient Project Portfolio as a tool for Enterprise Risk Management Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27 Effcent Proect Portfolo as a tool for Enterprse

More information

Stock Profit Patterns

Stock Profit Patterns Stock Proft Patterns Suppose a share of Farsta Shppng stock n January 004 s prce n the market to 56. Assume that a September call opton at exercse prce 50 costs 8. A September put opton at exercse prce

More information

Interest Rate Futures

Interest Rate Futures Interest Rate Futures Chapter 6 6.1 Day Count Conventons n the U.S. (Page 129) Treasury Bonds: Corporate Bonds: Money Market Instruments: Actual/Actual (n perod) 30/360 Actual/360 The day count conventon

More information

Communication Networks II Contents

Communication Networks II Contents 8 / 1 -- Communcaton Networs II (Görg) -- www.comnets.un-bremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP

More information

Portfolio Loss Distribution

Portfolio Loss Distribution Portfolo Loss Dstrbuton Rsky assets n loan ortfolo hghly llqud assets hold-to-maturty n the bank s balance sheet Outstandngs The orton of the bank asset that has already been extended to borrowers. Commtment

More information

The OC Curve of Attribute Acceptance Plans

The OC Curve of Attribute Acceptance Plans The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

More information

A Model of Private Equity Fund Compensation

A Model of Private Equity Fund Compensation A Model of Prvate Equty Fund Compensaton Wonho Wlson Cho Andrew Metrck Ayako Yasuda KAIST Yale School of Management Unversty of Calforna at Davs June 26, 2011 Abstract: Ths paper analyzes the economcs

More information

On the Optimal Control of a Cascade of Hydro-Electric Power Stations

On the Optimal Control of a Cascade of Hydro-Electric Power Stations On the Optmal Control of a Cascade of Hydro-Electrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;

More information

Implied (risk neutral) probabilities, betting odds and prediction markets

Implied (risk neutral) probabilities, betting odds and prediction markets Impled (rsk neutral) probabltes, bettng odds and predcton markets Fabrzo Caccafesta (Unversty of Rome "Tor Vergata") ABSTRACT - We show that the well known euvalence between the "fundamental theorem of

More information

A Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression

A Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy S-curve Regresson Cheng-Wu Chen, Morrs H. L. Wang and Tng-Ya Hseh Department of Cvl Engneerng, Natonal Central Unversty,

More information

Risk Model of Long-Term Production Scheduling in Open Pit Gold Mining

Risk Model of Long-Term Production Scheduling in Open Pit Gold Mining Rsk Model of Long-Term Producton Schedulng n Open Pt Gold Mnng R Halatchev 1 and P Lever 2 ABSTRACT Open pt gold mnng s an mportant sector of the Australan mnng ndustry. It uses large amounts of nvestments,

More information

Pricing index options in a multivariate Black & Scholes model

Pricing index options in a multivariate Black & Scholes model Prcng ndex optons n a multvarate Black & Scholes model Danël Lnders AFI_1383 Prcng ndex optons n a multvarate Black & Scholes model Danël Lnders Verson: October 2, 2013 1 Introducton In ths paper, we consder

More information

9.1 The Cumulative Sum Control Chart

9.1 The Cumulative Sum Control Chart Learnng Objectves 9.1 The Cumulatve Sum Control Chart 9.1.1 Basc Prncples: Cusum Control Chart for Montorng the Process Mean If s the target for the process mean, then the cumulatve sum control chart s

More information

Robust Design of Public Storage Warehouses. Yeming (Yale) Gong EMLYON Business School

Robust Design of Public Storage Warehouses. Yeming (Yale) Gong EMLYON Business School Robust Desgn of Publc Storage Warehouses Yemng (Yale) Gong EMLYON Busness School Rene de Koster Rotterdam school of management, Erasmus Unversty Abstract We apply robust optmzaton and revenue management

More information

The Cox-Ross-Rubinstein Option Pricing Model

The Cox-Ross-Rubinstein Option Pricing Model Fnance 400 A. Penat - G. Pennacc Te Cox-Ross-Rubnsten Opton Prcng Model Te prevous notes sowed tat te absence o arbtrage restrcts te prce o an opton n terms o ts underlyng asset. However, te no-arbtrage

More information

Time Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6 - The Time Value of Money. The Time Value of Money

Time Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6 - The Time Value of Money. The Time Value of Money Ch. 6 - The Tme Value of Money Tme Value of Money The Interest Rate Smple Interest Compound Interest Amortzng a Loan FIN21- Ahmed Y, Dasht TIME VALUE OF MONEY OR DISCOUNTED CASH FLOW ANALYSIS Very Important

More information

Financial Mathemetics

Financial Mathemetics Fnancal Mathemetcs 15 Mathematcs Grade 12 Teacher Gude Fnancal Maths Seres Overvew In ths seres we am to show how Mathematcs can be used to support personal fnancal decsons. In ths seres we jon Tebogo,

More information

DEFINING %COMPLETE IN MICROSOFT PROJECT

DEFINING %COMPLETE IN MICROSOFT PROJECT CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMI-SP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,

More information

Fixed income risk attribution

Fixed income risk attribution 5 Fxed ncome rsk attrbuton Chthra Krshnamurth RskMetrcs Group chthra.krshnamurth@rskmetrcs.com We compare the rsk of the actve portfolo wth that of the benchmark and segment the dfference between the two

More information

IN THE UNITED STATES THIS REPORT IS AVAILABLE ONLY TO PERSONS WHO HAVE RECEIVED THE PROPER OPTION RISK DISCLOSURE DOCUMENTS.

IN THE UNITED STATES THIS REPORT IS AVAILABLE ONLY TO PERSONS WHO HAVE RECEIVED THE PROPER OPTION RISK DISCLOSURE DOCUMENTS. http://mm.pmorgan.com European Equty Dervatves Strategy 4 May 005 N THE UNTED STATES THS REPORT S AVALABLE ONLY TO PERSONS WHO HAVE RECEVED THE PROPER OPTON RS DSCLOSURE DOCUMENTS. Correlaton Vehcles Technques

More information

Using Series to Analyze Financial Situations: Present Value

Using Series to Analyze Financial Situations: Present Value 2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated

More information

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

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

More information

Study on CET4 Marks in China s Graded English Teaching

Study on CET4 Marks in China s Graded English Teaching Study on CET4 Marks n Chna s Graded Englsh Teachng CHE We College of Foregn Studes, Shandong Insttute of Busness and Technology, P.R.Chna, 264005 Abstract: Ths paper deploys Logt model, and decomposes

More information

Vasicek s Model of Distribution of Losses in a Large, Homogeneous Portfolio

Vasicek s Model of Distribution of Losses in a Large, Homogeneous Portfolio Vascek s Model of Dstrbuton of Losses n a Large, Homogeneous Portfolo Stephen M Schaefer London Busness School Credt Rsk Electve Summer 2012 Vascek s Model Important method for calculatng dstrbuton of

More information

Intra-year Cash Flow Patterns: A Simple Solution for an Unnecessary Appraisal Error

Intra-year Cash Flow Patterns: A Simple Solution for an Unnecessary Appraisal Error Intra-year Cash Flow Patterns: A Smple Soluton for an Unnecessary Apprasal Error By C. Donald Wggns (Professor of Accountng and Fnance, the Unversty of North Florda), B. Perry Woodsde (Assocate Professor

More information

The impact of hard discount control mechanism on the discount volatility of UK closed-end funds

The impact of hard discount control mechanism on the discount volatility of UK closed-end funds Investment Management and Fnancal Innovatons, Volume 10, Issue 3, 2013 Ahmed F. Salhn (Egypt) The mpact of hard dscount control mechansm on the dscount volatlty of UK closed-end funds Abstract The mpact

More information

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001. Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.

More information

1 Approximation Algorithms

1 Approximation Algorithms CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons

More information

BERNSTEIN POLYNOMIALS

BERNSTEIN POLYNOMIALS On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful

More information

Prediction of Disability Frequencies in Life Insurance

Prediction of Disability Frequencies in Life Insurance Predcton of Dsablty Frequences n Lfe Insurance Bernhard Köng Fran Weber Maro V. Wüthrch October 28, 2011 Abstract For the predcton of dsablty frequences, not only the observed, but also the ncurred but

More information

Forecasting the Direction and Strength of Stock Market Movement

Forecasting the Direction and Strength of Stock Market Movement Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract - Stock market s one of the most complcated systems

More information

Lecture 3: Force of Interest, Real Interest Rate, Annuity

Lecture 3: Force of Interest, Real Interest Rate, Annuity Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annuty-mmedate, and ts present value Study annuty-due, and

More information

Statistical Methods to Develop Rating Models

Statistical Methods to Develop Rating Models Statstcal Methods to Develop Ratng Models [Evelyn Hayden and Danel Porath, Österrechsche Natonalbank and Unversty of Appled Scences at Manz] Source: The Basel II Rsk Parameters Estmaton, Valdaton, and

More information

Quality Adjustment of Second-hand Motor Vehicle Application of Hedonic Approach in Hong Kong s Consumer Price Index

Quality Adjustment of Second-hand Motor Vehicle Application of Hedonic Approach in Hong Kong s Consumer Price Index Qualty Adustment of Second-hand Motor Vehcle Applcaton of Hedonc Approach n Hong Kong s Consumer Prce Index Prepared for the 14 th Meetng of the Ottawa Group on Prce Indces 20 22 May 2015, Tokyo, Japan

More information

Joe Pimbley, unpublished, 2005. Yield Curve Calculations

Joe Pimbley, unpublished, 2005. Yield Curve Calculations Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward

More information

Prediction of Disability Frequencies in Life Insurance

Prediction of Disability Frequencies in Life Insurance 1 Predcton of Dsablty Frequences n Lfe Insurance Bernhard Köng 1, Fran Weber 1, Maro V. Wüthrch 2 Abstract: For the predcton of dsablty frequences, not only the observed, but also the ncurred but not yet

More information

Loss analysis of a life insurance company applying discrete-time risk-minimizing hedging strategies

Loss analysis of a life insurance company applying discrete-time risk-minimizing hedging strategies Insurance: Mathematcs and Economcs 42 2008 1035 1049 www.elsever.com/locate/me Loss analyss of a lfe nsurance company applyng dscrete-tme rsk-mnmzng hedgng strateges An Chen Netspar, he Netherlands Department

More information

Traffic-light extended with stress test for insurance and expense risks in life insurance

Traffic-light extended with stress test for insurance and expense risks in life insurance PROMEMORIA Datum 0 July 007 FI Dnr 07-1171-30 Fnansnspetonen Författare Bengt von Bahr, Göran Ronge Traffc-lght extended wth stress test for nsurance and expense rss n lfe nsurance Summary Ths memorandum

More information

ECONOMICS OF PLANT ENERGY SAVINGS PROJECTS IN A CHANGING MARKET Douglas C White Emerson Process Management

ECONOMICS OF PLANT ENERGY SAVINGS PROJECTS IN A CHANGING MARKET Douglas C White Emerson Process Management ECONOMICS OF PLANT ENERGY SAVINGS PROJECTS IN A CHANGING MARKET Douglas C Whte Emerson Process Management Abstract Energy prces have exhbted sgnfcant volatlty n recent years. For example, natural gas prces

More information

Kiel Institute for World Economics Duesternbrooker Weg 120 24105 Kiel (Germany) Kiel Working Paper No. 1120

Kiel Institute for World Economics Duesternbrooker Weg 120 24105 Kiel (Germany) Kiel Working Paper No. 1120 Kel Insttute for World Economcs Duesternbrooker Weg 45 Kel (Germany) Kel Workng Paper No. Path Dependences n enture Captal Markets by Andrea Schertler July The responsblty for the contents of the workng

More information

Risk-based Fatigue Estimate of Deep Water Risers -- Course Project for EM388F: Fracture Mechanics, Spring 2008

Risk-based Fatigue Estimate of Deep Water Risers -- Course Project for EM388F: Fracture Mechanics, Spring 2008 Rsk-based Fatgue Estmate of Deep Water Rsers -- Course Project for EM388F: Fracture Mechancs, Sprng 2008 Chen Sh Department of Cvl, Archtectural, and Envronmental Engneerng The Unversty of Texas at Austn

More information

HOUSEHOLDS DEBT BURDEN: AN ANALYSIS BASED ON MICROECONOMIC DATA*

HOUSEHOLDS DEBT BURDEN: AN ANALYSIS BASED ON MICROECONOMIC DATA* HOUSEHOLDS DEBT BURDEN: AN ANALYSIS BASED ON MICROECONOMIC DATA* Luísa Farnha** 1. INTRODUCTION The rapd growth n Portuguese households ndebtedness n the past few years ncreased the concerns that debt

More information

Questions that we may have about the variables

Questions that we may have about the variables Antono Olmos, 01 Multple Regresson Problem: we want to determne the effect of Desre for control, Famly support, Number of frends, and Score on the BDI test on Perceved Support of Latno women. Dependent

More information

Number of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000

Number of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000 Problem Set 5 Solutons 1 MIT s consderng buldng a new car park near Kendall Square. o unversty funds are avalable (overhead rates are under pressure and the new faclty would have to pay for tself from

More information

Chapter 7. Random-Variate Generation 7.1. Prof. Dr. Mesut Güneş Ch. 7 Random-Variate Generation

Chapter 7. Random-Variate Generation 7.1. Prof. Dr. Mesut Güneş Ch. 7 Random-Variate Generation Chapter 7 Random-Varate Generaton 7. Contents Inverse-transform Technque Acceptance-Rejecton Technque Specal Propertes 7. Purpose & Overvew Develop understandng of generatng samples from a specfed dstrbuton

More information

Analysis of the provisions for claims outstanding for non-life insurance based on the run-off triangles

Analysis of the provisions for claims outstanding for non-life insurance based on the run-off triangles OFFE OF THE NSURANE AND PENSON FUNDS SUPERVSORY OMMSSON Analyss of the provsons for clams outstandng for non-lfe nsurance based on the run-off trangles Ths Report has been prepared n the nformaton Systems

More information

State function: eigenfunctions of hermitian operators-> normalization, orthogonality completeness

State function: eigenfunctions of hermitian operators-> normalization, orthogonality completeness Schroednger equaton Basc postulates of quantum mechancs. Operators: Hermtan operators, commutators State functon: egenfunctons of hermtan operators-> normalzaton, orthogonalty completeness egenvalues and

More information

The Current Employment Statistics (CES) survey,

The Current Employment Statistics (CES) survey, Busness Brths and Deaths Impact of busness brths and deaths n the payroll survey The CES probablty-based sample redesgn accounts for most busness brth employment through the mputaton of busness deaths,

More information

A Critical Note on MCEV Calculations Used in the Life Insurance Industry

A Critical Note on MCEV Calculations Used in the Life Insurance Industry A Crtcal Note on MCEV Calculatons Used n the Lfe Insurance Industry Faban Suarez 1 and Steven Vanduffel 2 Abstract. Snce the begnnng of the development of the socalled embedded value methodology, actuares

More information

Portfolio Risk Decomposition (and Risk Budgeting)

Portfolio Risk Decomposition (and Risk Budgeting) ortfolo Rsk Decomposton (and Rsk Budgetng) Jason MacQueen R-Squared Rsk Management Introducton to Rsk Decomposton Actve managers take rsk n the expectaton of achevng outperformance of ther benchmark Mandates

More information

STAMP DUTY ON SHARES AND ITS EFFECT ON SHARE PRICES

STAMP DUTY ON SHARES AND ITS EFFECT ON SHARE PRICES STAMP UTY ON SHARES AN ITS EFFECT ON SHARE PRICES Steve Bond Mke Hawkns Alexander Klemm THE INSTITUTE FOR FISCAL STUIES WP04/11 STAMP UTY ON SHARES AN ITS EFFECT ON SHARE PRICES Steve Bond (IFS and Unversty

More information

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression. Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook

More information

Nasdaq Iceland Bond Indices 01 April 2015

Nasdaq Iceland Bond Indices 01 April 2015 Nasdaq Iceland Bond Indces 01 Aprl 2015 -Fxed duraton Indces Introducton Nasdaq Iceland (the Exchange) began calculatng ts current bond ndces n the begnnng of 2005. They were a response to recent changes

More information

x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60

x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60 BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true

More information

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of

More information

An Empirical Study of Search Engine Advertising Effectiveness

An Empirical Study of Search Engine Advertising Effectiveness An Emprcal Study of Search Engne Advertsng Effectveness Sanjog Msra, Smon School of Busness Unversty of Rochester Edeal Pnker, Smon School of Busness Unversty of Rochester Alan Rmm-Kaufman, Rmm-Kaufman

More information

THE USE OF RISK ADJUSTED CAPITAL TO SUPPORT BUSINESS DECISION-MAKING

THE USE OF RISK ADJUSTED CAPITAL TO SUPPORT BUSINESS DECISION-MAKING THE USE OF RISK ADJUSTED CAPITAL TO SUPPORT BUSINESS DECISION-MAKING By Gary Patrk Stefan Bernegger Marcel Beat Rüegg Swss Rensurance Company Casualty Actuaral Socety and Casualty Actuares n Rensurance

More information

Chapter 15: Debt and Taxes

Chapter 15: Debt and Taxes Chapter 15: Debt and Taxes-1 Chapter 15: Debt and Taxes I. Basc Ideas 1. Corporate Taxes => nterest expense s tax deductble => as debt ncreases, corporate taxes fall => ncentve to fund the frm wth debt

More information

ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING

ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING Matthew J. Lberatore, Department of Management and Operatons, Vllanova Unversty, Vllanova, PA 19085, 610-519-4390,

More information

ADVERSE SELECTION IN INSURANCE MARKETS: POLICYHOLDER EVIDENCE FROM THE U.K. ANNUITY MARKET

ADVERSE SELECTION IN INSURANCE MARKETS: POLICYHOLDER EVIDENCE FROM THE U.K. ANNUITY MARKET ADVERSE SELECTION IN INSURANCE MARKETS: POLICYHOLDER EVIDENCE FROM THE U.K. ANNUITY MARKET Amy Fnkelsten Harvard Unversty and NBER James Poterba MIT and NBER Revsed May 2003 ABSTRACT In ths paper, we nvestgate

More information

A Probabilistic Theory of Coherence

A Probabilistic Theory of Coherence A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want

More information

IMPACT ANALYSIS OF A CELLULAR PHONE

IMPACT ANALYSIS OF A CELLULAR PHONE 4 th ASA & μeta Internatonal Conference IMPACT AALYSIS OF A CELLULAR PHOE We Lu, 2 Hongy L Bejng FEAonlne Engneerng Co.,Ltd. Bejng, Chna ABSTRACT Drop test smulaton plays an mportant role n nvestgatng

More information

DI Fund Sufficiency Evaluation Methodological Recommendations and DIA Russia Practice

DI Fund Sufficiency Evaluation Methodological Recommendations and DIA Russia Practice DI Fund Suffcency Evaluaton Methodologcal Recommendatons and DIA Russa Practce Andre G. Melnkov Deputy General Drector DIA Russa THE DEPOSIT INSURANCE CONFERENCE IN THE MENA REGION AMMAN-JORDAN, 18 20

More information

Credit Limit Optimization (CLO) for Credit Cards

Credit Limit Optimization (CLO) for Credit Cards Credt Lmt Optmzaton (CLO) for Credt Cards Vay S. Desa CSCC IX, Ednburgh September 8, 2005 Copyrght 2003, SAS Insttute Inc. All rghts reserved. SAS Propretary Agenda Background Tradtonal approaches to credt

More information

Chapter 4 Financial Markets

Chapter 4 Financial Markets Chapter 4 Fnancal Markets ECON2123 (Sprng 2012) 14 & 15.3.2012 (Tutoral 5) The demand for money Assumptons: There are only two assets n the fnancal market: money and bonds Prce s fxed and s gven, that

More information

FAST SIMULATION OF EQUITY-LINKED LIFE INSURANCE CONTRACTS WITH A SURRENDER OPTION. Carole Bernard Christiane Lemieux

FAST SIMULATION OF EQUITY-LINKED LIFE INSURANCE CONTRACTS WITH A SURRENDER OPTION. Carole Bernard Christiane Lemieux Proceedngs of the 2008 Wnter Smulaton Conference S. J. Mason, R. R. Hll, L. Mönch, O. Rose, T. Jefferson, J. W. Fowler eds. FAST SIMULATION OF EQUITY-LINKED LIFE INSURANCE CONTRACTS WITH A SURRENDER OPTION

More information

The influence of supplementary health insurance on switching behaviour: evidence on Swiss data

The influence of supplementary health insurance on switching behaviour: evidence on Swiss data Workng Paper n 07-02 The nfluence of supplementary health nsurance on swtchng behavour: evdence on Swss data Brgtte Dormont, Perre- Yves Geoffard, Karne Lamraud The nfluence of supplementary health nsurance

More information

Pricing Multi-Asset Cross Currency Options

Pricing Multi-Asset Cross Currency Options CIRJE-F-844 Prcng Mult-Asset Cross Currency Optons Kenchro Shraya Graduate School of Economcs, Unversty of Tokyo Akhko Takahash Unversty of Tokyo March 212; Revsed n September, October and November 212

More information

SPECIALIZED DAY TRADING - A NEW VIEW ON AN OLD GAME

SPECIALIZED DAY TRADING - A NEW VIEW ON AN OLD GAME August 7 - August 12, 2006 n Baden-Baden, Germany SPECIALIZED DAY TRADING - A NEW VIEW ON AN OLD GAME Vladmr Šmovć 1, and Vladmr Šmovć 2, PhD 1 Faculty of Electrcal Engneerng and Computng, Unska 3, 10000

More information

Simple Interest Loans (Section 5.1) :

Simple Interest Loans (Section 5.1) : Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part

More information

Interest Rate Fundamentals

Interest Rate Fundamentals Lecture Part II Interest Rate Fundamentals Topcs n Quanttatve Fnance: Inflaton Dervatves Instructor: Iraj Kan Fundamentals of Interest Rates In part II of ths lecture we wll consder fundamental concepts

More information

Mathematical Option Pricing

Mathematical Option Pricing Mark H.A.Davs Mathematcal Opton Prcng MSc Course n Mathematcs and Fnance Imperal College London 11 January 26 Department of Mathematcs Imperal College London South Kensngton Campus London SW7 2AZ Contents

More information

Estimation of Dispersion Parameters in GLMs with and without Random Effects

Estimation of Dispersion Parameters in GLMs with and without Random Effects Mathematcal Statstcs Stockholm Unversty Estmaton of Dsperson Parameters n GLMs wth and wthout Random Effects Meng Ruoyan Examensarbete 2004:5 Postal address: Mathematcal Statstcs Dept. of Mathematcs Stockholm

More information

Cautiousness and Measuring An Investor s Tendency to Buy Options

Cautiousness and Measuring An Investor s Tendency to Buy Options Cautousness and Measurng An Investor s Tendency to Buy Optons James Huang October 18, 2005 Abstract As s well known, Arrow-Pratt measure of rsk averson explans a ratonal nvestor s behavor n stock markets

More information

Brigid Mullany, Ph.D University of North Carolina, Charlotte

Brigid Mullany, Ph.D University of North Carolina, Charlotte Evaluaton And Comparson Of The Dfferent Standards Used To Defne The Postonal Accuracy And Repeatablty Of Numercally Controlled Machnng Center Axes Brgd Mullany, Ph.D Unversty of North Carolna, Charlotte

More information

Section 5.4 Annuities, Present Value, and Amortization

Section 5.4 Annuities, Present Value, and Amortization Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today

More information

Valuing Customer Portfolios under Risk-Return-Aspects: A Model-based Approach and its Application in the Financial Services Industry

Valuing Customer Portfolios under Risk-Return-Aspects: A Model-based Approach and its Application in the Financial Services Industry Buhl and Henrch / Valung Customer Portfolos Valung Customer Portfolos under Rsk-Return-Aspects: A Model-based Approach and ts Applcaton n the Fnancal Servces Industry Hans Ulrch Buhl Unversty of Augsburg,

More information

Power-of-Two Policies for Single- Warehouse Multi-Retailer Inventory Systems with Order Frequency Discounts

Power-of-Two Policies for Single- Warehouse Multi-Retailer Inventory Systems with Order Frequency Discounts Power-of-wo Polces for Sngle- Warehouse Mult-Retaler Inventory Systems wth Order Frequency Dscounts José A. Ventura Pennsylvana State Unversty (USA) Yale. Herer echnon Israel Insttute of echnology (Israel)

More information