CHRISTOPH MÖHR ABSTRACT

Transcription

1 MARKET-CONSISTENT VALUATION OF INSURANCE LIABILITIES BY COST OF CAPITAL BY CHRISTOPH MÖHR ABSTRACT This paper invesigaes marke-consisen valuaion of insurance liabiliies in he conex of Solvency II among ohers and o some exen IFRS 4. We propose an explici and consisen framework for he valuaion of insurance liabiliies which incorporaes he Solvency II approach as a special case. The proposed framework is based on replicaion over muliple (one-year) ime periods by a periodically updaed porfolio of asses wih reliable marke prices, allowing for limied liabiliy in he sense ha he replicaion can in general no always be coninued. The asse porfolio consiss of wo pars: () asses whose marke price defines he value of he insurance liabiliies, and (2) capial funds used o cover risk which canno be replicaed. The capial funds give rise o capial coss; he main exogenous inpu in he framework is he condiion on when he invesmen of he capial funds is accepable. We invesigae exisence of he value and show ha he exac calculaion of he value has o be done recursively backwards in ime, saring a he end of he lifeime of he insurance liabiliies. We derive upper bounds on he value and, for he special case of replicaion by risk-free one-year zero-coupon bonds, explici recursive formulas for calculaing he value. In he paper, we only parially consider he quesion of he uniqueness of he value. Valuaion in Solvency II and IFRS 4 is based on represening he value as a sum of a bes esimae and a risk margin. In our framework, i urns ou ha his spli is no naural. Noneheless, we show ha a spli can be consruced as a simplificaion, and ha i provides an upper bound on he value under suiable condiions. We illusrae he general resuls by explicily calculaing he value for a simple example. KEYWORDS Marke-consisen valuaion of insurance liabiliies, echnical provisions, muli-period replicaion, Solvency II, Swiss Solvency Tes, cos of capial, risk margin, bes esimae. Asin Bullein 4(2), doi: 0.243/AST by Asin Bullein. All righs reserved.

2 36 C. MÖHR. INTRODUCTION Our saring poin is marke-consisen valuaion of insurance liabiliies ( echnical provisions ) under Solvency II. References o he approach include he Solvency II Framework Direcive DIRECTIVE 2009/38/EC [], he Solvency II draf Level 2 Implemenaion Measures Rules relaing o echnical provisions EIOPC/SEG/IM3/200 [], as well as relaed documens such as he CRO Forum posiion paper [9], he repor by he Risk Margin Working Group [5], and CEIOPS-DOC-36/09 (former CP 42) [4]. Many of he conceps used by Solvency II had earlier been inroduced in he Swiss Solvency Tes (SST), see for insance Federal Office of Privae Insurance [7]. In Solvency II, according o Aricle 77 in DIRECTIVE 2009/38/EC [], he marke-consisen value of an insurance liabiliy is deermined in one of wo ways: If he cash-flows of he liabiliy (or par of he cash-flows) can be replicaed reliably using financial insrumens for which a reliable marke value is observable, hen he value (of he par of he cash-flows) is deermined on he basis of he marke value of hese financial insrumens. Oherwise, he value is equal o he sum of bes esimae and risk margin, Marke-consisen value = bes esimae + risk margin. () In Aricle 77 of he DIRECTIVE 2009/38/EC [], he bes esimae is defined as he probabiliy-weighed average of fuure cash-flows, aking accoun of he ime value of money (expeced presen value of fuure cash-flows), using he relevan risk-free ineres rae erm srucure, and he risk margin is calculaed by deermining he cos of providing an amoun of eligible own funds equal o he Solvency Capial Requiremen necessary o suppor he insurance and reinsurance obligaions over he lifeime hereof. In EIOPC/SEG/IM3/200 [], he risk margin is expressed by a cos of capial approach as he sum of he coss of fuure required capial SCR by he expression SCR Risk margin = CoC / +. (2) ( + r + ) $ 0 where CoC denoes he cos of capial rae, which is assumed deerminisic and consan and, in EIOPC/SEG/IM3/200 [], is se o 6% above he riskfree rae. The sum is over all years, and r + denoes he risk-free discoun rae for + years, which means ha he coss of capial for year are discouned back from he end of year. The infinie sum above will be finie in pracice, limied by he lifeime of he corresponding liabiliies. In he formula (2), SCR denoes he Solvency Capial Requiremen from Solvency II for he year, i.e. he required capial, which is defined in Aricle 0 of DIRECTIVE 2009/38/EC [] o correspond o he Value-a-Risk of he basic own funds of an insurance or reinsurance underaking subjec o a

3 MARKET-CONSISTENT VALUATION OF INSURANCE LIABILITIES 37 confidence level of 99,5% over a one-year period. We consider SCR in more deail in Secion 2, bu noe here he following: for fuure years > 0, SCR depends on he fuure sae a he beginning of year, which is currenly (a ime = 0) no known. Consequenly, SCR for > 0 is a random variable as seen from ime = 0, implying ha he risk margin as defined by (2) is a random variable and no a number, as i ough o be. To avoid his problem, every SCR in (2) could be replaced wih he curren expeced value of he random variable SCR, so ha he risk margin would correspond o he expeced coss of fuure required capial. One migh hen hink ha his expeced risk margin is only sufficien in expecaion. However, as we show in Secion 5, i urns ou ha, under suiable assumpions, he expeced risk margin is sufficien no jus in expecaion bu always. As an addiional complicaion, according o EIOPC/SEG/IM3/200 [], he SCR used for calculaing he risk margin in Solvency II is no calculaed for he company (underaking) under consideraion, bu for a reference underaking o which he insurance liabiliies are hypoheically ransferred. The feaures of he ransfer and he properies of he reference underaking are specified in deail in EIOPC/SEG/IM3/200 []. The preceding commens aim o indicae ha valuaion of insurance liabiliies according o Solvency II is no obvious and ha a more explici heoreical framework migh be needed. The objecive of his paper is o propose such a framework, which incorporaes he Solvency II approach as a special case. The proposed framework expresses he value in erms of he marke price of a porfolio of asses. I is based on he replicaion over muliple ime periods of he cash-flows of he insurance liabiliies by porfolios of asses wih reliable marke prices. In his sense, i relies on he seminal idea of valuaion by replicaion underlying he (risk-neural) pricing of financial derivaives. Muli-period replicaion refers here o he fac ha he replicaion is dynamic in he sense ha he asse porfolio used for he replicaion is updaed a he end of every ime period. The framework needs o capure wo addiional aspecs. The firs addiional aspec is ha insurance liabiliies can ypically no be perfecly replicaed by asses wih reliable marke prices, so here remains a par of he cash-flows which canno be replicaed. According o Solvency II, he non-replicable par of he cash-flows is covered by capial funds, giving rise o capial coss. The second addiional aspec is ha he replicaion canno always be coninued. In Solvency II, his is because he required capial funds are given by he Solvency Capial Requiremen in erms of he Value-a-Risk (VaR) a 99.5%, which implies ha hey will be insufficien wih 0.5% probabiliy. The main exogenous assumpion in our proposed framework is wha we call he accepabiliy condiion in he remainder of his paper. The accepabiliy condiion is he condiion on when he sochasic reurn on he capial funds is accepable o he invesor of he capial funds. In oher words, i specifies he price of he capial invesmen. In his paper, we work wih he accepabiliy condiion implici in he definiion of he risk margin in Solvency II, which is ha he expeced excess reurn over he risk-free reurn is equal o he

4 38 C. MÖHR cos of capial rae CoC. We noe ha his accepabiliy condiion is formulaed independenly of he capial invesor and so does no ake ino accoun he specific risk profile and risk aversion of a given invesor. In general, he value of he insurance liabiliies can depend on he assumpions made abou fuure new business wrien, as fuure new business migh diversify wih he run-off of he curren business. In his paper, we consider a run-off siuaion in he sense ha we assume ha no fuure new business is wrien. Under he proposed framework, i urns ou ha a precise calculaion of he value needs o be done recursively backwards in ime, saring a he end of he lifeime of he insurance liabiliies. Moreover, we find ha here is no naural spli of he value ino a bes esimae and a risk margin ; he value is simply given as he marke price of a specific porfolio of asses. However, we show in Secion 5 ha, under cerain condiions, a spli can be inroduced, and ha he resuling sum of bes esimae and risk margin is no equal o he value bu provides an upper bound. The proposed framework can be siuaed in he conex of (marke-consisen) valuaion in incomplee markes. A presen, on he one hand, here is exensive academic lieraure on aspecs of valuaion by replicaion and in incomplee markes, while, on he oher hand, from a praciioner s perspecive, here are numerous aricles abou cerain aspecs of he Solvency II valuaion, such as simplified approaches, he risk-free rae, he cos of capial rae ec. This paper aims o bridge he wo areas, by formulaing Solvency II valuaion in he framework of replicaion in incomplee markes. The recen paper Salzmann-Wührich [0] provides a discussion of a mahemaically consisen muli-period risk measure approach for he calculaion of a risk margin o cover possible shorfalls in he liabiliy runoff of general (i.e. non-life) insurance companies. Moreover, explici calculaions are presened by means of a Bayes chain ladder model and a risk measure chosen o be a muliple of he sandard deviaion. Our approach is relaed o he Valuaion Porfolio (VaPo) according o Bühlmann [3] and Wührich e al. [2]: An insurance obligaion can be beer undersood no in erms of moneary values bu as a collecion of appropriaely chosen financial insrumens. In conras o he VaPo approach, we do no express he acual liabiliy as a porfolio of poenially synheic financial insrumens, bu consider replicaion of he liabiliy s cash-flows by a porfolio of asses wih reliable marke prices. The risk margin in he conex of he one-year risk is also invesigaed in Ohlsson-Lauzeningks [8]. We menion here also he classical paper Arzner e al. [2] on coheren risk measures or, equivalenly, accepable fuure ne worhs. While risk measures play a prominen par in wha follows, ha paper considers a one-period seing and does no consider replicaion over muliple ime periods. An alernaive approach o he accepabiliy condiion is given by uiliy indifference pricing similar o Møller [6].

5 MARKET-CONSISTENT VALUATION OF INSURANCE LIABILITIES 39 ORGANIZATION OF THE PAPER The paper is organized as follows: In Secion, we se up he mahemaical noaions and assumpions, including he filraion used o express available informaion, and risk-free discouning. In Secion 2, we invesigae he Solvency II approach o valuaion and solvency as a moivaion for our formulaion, in Secion 3, of he proposed framework for valuaion. In Secion 4, we hen invesigae valuaion over muliple one-year ime periods in he proposed framework. In Secion 5, we consider he risk margin and prove one of he main resuls of his paper: under suiable assumpions, he sum of bes esimae and risk margin is an upper bound for he value. Finally, in Secion 6, we explicily calculae he value for a simple example and show ha he upper bound in his case is sharp and, moreover, can reverse he ordering of he value beween differen liabiliies.. SET UP AND NOTATION We consider ime periods of one year, where year = 0, refers o he imeperiod [, + ). To be able o describe acions aken a he end of year, we denoe by ( + ) a poin in ime jus before ime +. We assume ha here exiss a filraion = (F ), wih F expressing he informaion available (known) a ime. To specify he filraion, we use he noaion idenical o Wührich e al. [2]. Tha is, we define a filered probabiliy space by choosing a probabiliy space (W, F, ) and an increasing sequence of s-fields = (F ) = 0,, n wih {0, W} = F 0 3 F 3 3 F n, where we assume F n = F for simpliciy. The main objecive of he paper is he valuaion of a given insurance liabiliy L wih sochasic cash-flows (X ) corresponding o claims paymens, expenses ec., where X denoes he cash-flow in year. For simpliciy, we assume ha he cash-flow X occurs and is known a ime ( + ), i.e. X is F + -measurable. In erms of he filered probabiliy space, his means ha he process (X ) is adaped o he filraion = ( F ). A ime, inuiively speaking, he value of X is no known, bu he disribuion of X is known. We assume hroughou he paper ha marke prices of a given se of financial insrumens are available a fuure poins in ime. Tha is, he informaion F available a ime includes he marke prices of financial insrumens a ime, i.e. he corresponding marke price processes are assumed o be adaped o he filraion. The fuure marke prices of hese financial insrumens are given by sochasic models. A reference marke (or replicaing marke) is defined o be a se of financial insrumens for which reliable marke prices are assumed o exis. As an

6 320 C. MÖHR idealizaion, marke prices of a financial insrumen are reliable if any quaniy of he financial insrumen can insananeously be raded (bough or sold) wihou affecing he marke price. Typically, i is assumed ha, if a financial insrumen is raded in a deep and liquid marke, hen is (unique and addiive) reliable marke price is an emergen propery of he corresponding marke. An asse porfolio consising of financial insrumens from he reference marke is called a reference porfolio (or replicaing porfolio). Deep and liquid (and ransparen) markes are defined in he Solvency II conex in EIOPC/SEG/IM3/200 [], which also specifies ha he model used for he projecion of marke parameers (or marke prices) needs o ensure ha no arbirage opporuniy exiss. In line wih his requiremen, we assume in he following ha he reference marke is arbirage-free. We assume: Risk-free zero-coupon bonds are par of he reference marke. We do no specify which oher financial insrumens migh be in he reference marke. As menioned above, we assume models for he sochasic fuure marke prices for he financial insrumens in he reference marke. To express risk-free discouning of a cash-flow x occurring a ime s discouned o ime # s, we wrie pv (s " ) (x), which is o be undersood as he value a ime of a risk-free zero-coupon bond in he appropriae currency wih face value x mauring a ime s. I is in his sense no possible o risk-free discoun sochasic (as opposed o deerminisic) cash-flows, because he cash-flow of a risk-free zero-coupon bond is deerminisic. We define he risk-free erminal value of an amoun x invesed a ime in a risk-free zero-coupon bond mauring a ime s $ by v ( " s) (x). Le R (m) denoe he annual rae for a risk-free zero-coupon bond a ime wih a erm of m =, 2 years, so R (m) is F -measurable, and pv ( + m " ) (x + m ) = ( + R (m) ) m x + m. (3) Consider a risk-free forward conrac se up a ime, which specifies ha, a ime +, for a price of B m + () fixed a ime, a risk-free zero-coupon bond is purchased wih a payoff of a ime + + m. Because of no-arbirage, we mus have ha ( + R () ) B m (m + + () = ( + R ) ) m. (4) I is common o idenify he forward price wih he expecaion a ime of he corresponding bond price, i.e.

7 MARKET-CONSISTENT VALUATION OF INSURANCE LIABILITIES 32 B m + () = {( + R (m) + ) m F }. In general, he price of a forward conrac migh conain an addiional premium for liquidiy, so Equaions (5) and (4) imply ha B m + () $ {( + R (m) + ) m F }. (5) ( + R () ) {( + R (m) + ) m (m + F } # ( + R ) ) m. (6) Given a se A, we denoe is complemen by A c and is characerisic funcion by A. The characerisic funcion akes he value on A and 0 on A c. We consider risk measures r, aking a random variable X o a real number r{x}. We define losses o be negaive numbers and he risk r of a loss o be a posiive number. A risk measure r is called ranslaion-invarian (or cashinvarian) if, for any random variable X and any real number b, r{x + b} = r{x } b. I is called monoone if, for any wo random variables wih X # X 2, we have r{x 2 } # r{x }. 2. MARKET-CONSISTENT VALUATION IN SOLVENCY II Because we are proposing a framework for valuaion which incorporaes Solvency II as a special case, we invesigae in he following he Solvency II approach o valuaion and solvency in more deail. The expressions we derive here are used o moivae he definiion of he general framework in Secions 3 and 4. To begin wih, we consider he Solvency Capial Requiremen SCR, which is defined o correspond o he Value-a-Risk of he basic own funds of an insurance or reinsurance underaking subjec o a confidence level of 99,5% over a one-year period. (DIRECTIVE 2009/38/EC[]) For he acual balance shee of he company (or underaking ) under consideraion, for simpliciy, we idenify in he following basic and eligible own funds as defined under Solvency II wih he available capial, denoed by AC a ime, which is defined as he difference beween he marke-consisen value V (A ) of he asses A and he marke-consisen value V (L ) of he liabiliies L in he company s balance shee a ime, AC := V (A ) V (L ). SCR can hen be wrien in erms of he one-year change of he available capial, SCR := pv ( + " ) (r{ac ( + ) v ( " + ) (AC ) F }), (7)

8 322 C. MÖHR where he risk measure r is prescribed o be he Value-a-Risk VaR a a he a = 99.5-percenile r{z} := VaR a { Z}. (8) Noe ha SCR is calculaed based on he informaion F available a ime. SCR is he capial requiremen under Solvency II in he assessmen of he solvency of a company. Solvency is effecively specified by he condiion ha, wih 99.5% probabiliy, a he end of year 0 (a ime = ), he marke-consisen value of he asses exceed he marke-consisen value of he liabiliies, V (A ) $ V (L ), which corresponds o he requiremen a ime = 0 ha he available capial exceed he required capial, AC 0 $ SCR 0, wih SCR 0 given by (7) for = 0. In order o assess he solvency condiion, we hus in paricular need o know he value of he insurance liabiliies. Regarding he value of he insurance liabiliies, we recall from he inroducion ha he risk margin as a componen of he value is defined in erms of he Solvency Capial Requiremen SCR. However, SCR is no calculaed for he company which currenly holds he insurance liabiliies, bu for a so-called reference underaking o which he insurance liabiliies are hypoheically ransferred for he purpose of valuaion. We denoe in he following he insurance liabiliies o be valued by L. In conras o he noaion L wih dependency on he poin in ime from above, we consider a fixed block of business L, whereas L migh change wih ime due o new business being wrien. For he valuaion of he insurance liabiliies L in his paper, we consider a run-off siuaion, i.e. we assume ha no new business is wrien. We come back o his assumpion below. The feaures of he ransfer and he properies of he reference underaking are specified in EIOPC/SEG/IM3/200 []. Afer he ransfer, he liabiliy side of he balance shee of he reference underaking is assumed o consis of he ransferred insurance liabiliies. The asses are assumed o consis of wo pars. The firs par is a reference porfolio of asses we denoe by RP, which is used o cover he value of he insurance liabiliies. Tha is, he value V (L) of he insurance liabiliies L a ime is given by he marke price of he reference porfolio V (L) = V (RP ). (9) The second par of he asses consiss of available capial AC, assumed invesed risk-free, equal o he Solvency Capial Requiremen SCR needed for he reference underaking.

9 MARKET-CONSISTENT VALUATION OF INSURANCE LIABILITIES 323 Under hese specificaions, SCR from (7) can be rewrien as follows. The available capial AC ( + ) a he end of year (before any poenial recapializaion) is given by he year-end value of he asses reduced by he cash-flow X in year and he year-end value of he insurance liabiliies V + (L), i.e. AC ( + ) = v ( " + ) (SCR ) + V + (RP ) X V + (L). Since AC = SCR, we ge from (7) he following formula for he SCR for he purpose of valuaion, SCR = pv ( + " ) ( r{v + (RP ) X V + (L) F }). (0) I becomes clear from his expression ha, in order o calculae he markeconsisen value V (L) a ime by (9), which hrough he risk margin (or hrough he accepabiliy condiion () below) depends on SCR, one firs needs o calculae he marke-consisen value V + (L) a ime + ec. This implies ha a precise calculaion of he marke-consisen value has o be recursively backwards in ime. The expression (0) also shows ha underlying marke-consisen valuaion of insurance liabiliies is replicaion wih a one-year ime period, i.e. he replicaion is updaed afer every one-year ime period. A ime, he porfolio RP, which defines he value V (L) hrough (9), is se up o replicae he random variable X + V + (L) a ime +. In he case of perfec replicaion, V + (RP ) is always equal o X + V + (L), so ha, a ime +, a new replicaing porfolio RP + can be consruced by a suiable reinvesmen of he asses RP reduced by he cash-flow X, and no capial funds are needed. For insurance liabiliies, perfec replicaion is ypically no possible. Hence, addiional capial funds are needed for he insances in which V + (RP ) is less han he sum X + V + (L), so capial funds accoun for he par of he liabiliy which canno be replicaed. This gives rise o capial requiremens SCR according o (0), which depend on he real-world probabiliies of differen amouns of he difference V + (RP ) X V + (L). In general, fuure new business migh be wrien and hus be added o he balance shee in he fuure, and he corresponding cash-flows migh diversify wih he cash-flows of he liabiliy L under consideraion. Since insurance liabiliies ypically run-off over several years, his means ha he curren value of an insurance liabiliy is poenially affeced by insurance obligaions which are added o he balance shee in he fuure, i.e. fuure new business, a leas unil he liabiliy is fully run-off. In Solvency II, he assumpions on fuure new business in he calculaion of he risk margin are currenly no really clear. In his paper, we consider a run-off siuaion in he sense ha we assume ha no fuure new business is wrien. The capial SCR comes wih a cos o make he capial invesmen accepable o he capial provider, which we express hrough he accepabiliy condiion. The accepabiliy condiion is encoded in he definiion of he risk margin, and

10 324 C. MÖHR requires ha he expeced reurn on he capial SCR a he end of year be equal o a cos of capial rae CoC in excess of he risk-free rae. The value of he capial invesmen a he end of he year is deermined from he available capial AC ( + ), considering ha is value is never negaive, since he capial provider has limied liabiliy. Hence, he accepabiliy condiion for year can be wrien as {max {0, AC ( + ) } F } = v ( " + ) (SCR ) + CoC SCR. () The lef hand side of equaion () is he expeced value a ime ( + ) of he invesmen of he capial funds, and he righ hand side is equal o he risk-free reurn plus he cos of capial rae on he capial funds SCR invesed a ime. We find in he following ha he accepabiliy condiion deermines he reference porfolio RP or allows o derive upper bounds. 3. FRAMEWORK FOR THE VALUATION OF INSURANCE LIABILITIES A a concepual level, he proposed framework for marke-consisen valuaion of an insurance liabiliy L is based on hree ideas:. Muli-period replicaion of he liabiliy cash-flows by asses given by financial insrumens wih reliable marke prices. 2. Covering he remaining non-replicable par of he cash-flows by capial funds provided by an invesor. 3. Limied liabiliy, i.e. he liabiliy cash-flows in general do no need o be provided for every sae of he world. The firs idea is analogous o no-arbirage or risk-neural pricing of financial insrumens in complee markes. The second idea accouns for he fac ha insurance liabiliies, in paricular, can usually no be perfecly replicaed by financial insrumens wih reliable marke prices, and relaes o he requiremens by he regulaory auhoriies, for insance in Solvency II, ha companies need o hold a required amoun of capial. The hird idea relaes o he fac ha he required regulaory capial ypically only needs o be large enough o ensure ha he insurance obligaions can be saisfied wih high probabiliy. In Solvency II, for insance, his is expressed by he 99.5% Value-a-Risk over a oneyear ime period. Valuing he liabiliy L hen means finding a replicaion procedure, which a a poin in ime consiss of a porfolio of asses composed of a reference porfolio RP and capial funds C. In a saic replicaion procedure, he porfolio RP is unchanged over he lifeime of he liabiliy L. In a muli-period replicaion procedure, he porfolio RP is adjused, in our case (a leas) over successive one-year ime periods, leading o a sequence of reference porfolios RP, RP +... The capial invesmen C for year is ied from ime o ime + and is used o cover cash-flow mismaches beween L and RP in year

11 MARKET-CONSISTENT VALUATION OF INSURANCE LIABILITIES 325 and o conver he asses a ime + o he nex reference porfolio RP +. A ime +, new capial funds poenially need o be raised for covering he nex ime period. The capial invesmen is assumed o have he following wo properies: As an obligaion, he capial invesmen has lowes senioriy (i.e. he capial funds are used for covering all oher obligaions). The capial invesmen comes wih limied liabiliy (i.e. is value is never negaive). The crucial assumpion abou he capial invesmen is he accepabiliy condiion: Under which condiions is he sochasic reurn from he capial invesmen accepable o he capial provider? The accepabiliy condiion specifies he risk-reurn preferences of he capial invesor and is he one inpu o he framework in addiion o he curren and fuure marke prices of he financial insrumens available for replicaion. If an accepabiliy condiion is specified and he reference porfolio RP is se up such ha he capial invesmen C fulfils he accepabiliy condiion, hen he value of L a ime is defined as in (9) o be he marke price of he reference porfolio RP, V (L) := V (RP ). (2) The implici assumpion is ha required capial funds can always be raised if an accepable (sochasic) reurn can be provided. In general, (2) only holds a he poin in ime a which he corresponding reference porfolio is se up and no in beween. A significan quesion which we only parially consider in his paper is he uniqueness of he value defined according o (2). In view of he hird idea underlying he proposed valuaion approach, here is he furher complicaion ha we allow for limied liabiliy in he replicaion procedure by limiing he required capial C. Tha is, he liabiliy L does no need o be replicaed for every sae of he world. In a dynamic muli-period replicaion procedure, limied liabiliy poenially applies boh backwards and forward in ime. Limied liabiliy applies backwards in ime because a any poin in ime we do no only reflec he defauls in he curren ime period, bu addiionally he defauls in any fuure ime period. Limied liabiliy also applies forward in ime, in he sense ha, a ime, here are saes of he world in which defaul has already occurred a a prior poin in ime. If he liabiliy L is considered o be a conrac wih a specific company, his means ha, in such a sae, he company has defauled on is Moreover, o specify accepabiliy of he sochasic fuure value of he capial invesmen, we have o specify a which ime he capial amoun C is deermined, as his is he dae a which accepabiliy of he reurn o he capial provider is decided. In he following, we assume ha C is deermined a ime and no before.

12 326 C. MÖHR obligaions prior o, and so he obligaions owards fuure cash-flows canno be fulfilled anymore o he exen required. We use a differen approach, which appears reasonable from he perspecive of an insurance regulaor, and consider he value a ime of he liabiliy as such, characerized by fuure cash-flows and fuure limied liabiliy, disregarding he replicaion hisory prior o ime. 4. VALUATION UNDER THE FRAMEWORK The valuaion of he liabiliy L according o (2) is achieved by calculaing recursively backwards in ime, saring a he end of he lifeime of he liabiliy. Le he number T denoe he final year of he lifeime of L, i.e. we assume ha here are no more ousanding liabiliies afer ime T. Tha is, T is he smalles whole number such ha X T +, X T + 2 = 0. Then, V T + (L) = 0. In he recursion sep, we assume ha he value V + (L) a ime + is known and equal o he marke price of a reference porfolio RP +, V + (L) = V + (RP + ). We hen have o calculae he value V (L) a ime as he marke price of a suiable reference porfolio RP. To his end, define he random variable Y + o be he sum of he cash-flow X in year and he value V + (L) a he end of he year, Y + := X + V + (L). (3) In paricular, Y T + = X T. For he replicaion in year, he random variable Y + needs o be mached by asses given by a reference porfolio RP ogeher wih capial funds C $ 0 provided for one year by a capial invesor. The capial funds C are assumed o be invesed a ime in a risk-free one-year zero-coupon bond. We allow for he fac ha he replicaion canno always be coninued pas ime +. To formalize hese assumpions, given a reference porfolio RP and capial funds C, he se A is defined o be he se of saes in which he cash-flow X can be provided and he replicaion can be coninued pas ime + by convering he asses available a ime + wih value V + (RP ) + v ( " + ) (C ) X o he new reference porfolio RP +. The se A and is probabiliy g are hus given by

13 MARKET-CONSISTENT VALUATION OF INSURANCE LIABILITIES 327 A := {Y + # v ( " + ) (C ) + V + (RP )}, (4) g := {A F } = { A F }. In view of he characerisics of he capial invesmen oulined in Secion 3, he value of he capial invesmen a ime + is given by he maximum of zero and he value of he asses lef afer all oher obligaions have been considered, so he value C o he capial provider a ime + of he capial invesmen C can be wrien as C := A (v ( " + ) (C ) + V + (RP ) Y + ). (5) The accepabiliy condiion is specified in he remainder of he paper as prescribed under Solvency II and, in paricular, expressed in erms of he expecaion of he value C of he capial invesmen. Corresponding o (), he accepabiliy condiion is defined o be he condiion ha he expeced excess reurn over risk-free of he capial invesmen be equal o a given F - measurable dividend D $ 0, {C F } v ( " + ) (C ) = D. (6) The accepabiliy condiion (6) ranslaes ino an equivalen condiion on he reference porfolio: if we inser he expression (5) for C ino (6), we ge he condiion on he reference porfolio RP ha { A V + (RP ) F } = { A Y + F } + ( g ) v ( " + ) (C ) + D. (7) Noe ha condiion (7) is complicaed in he sense ha i depends on RP, C, D, and A, all of which are in general inerlinked wih each oher. The value V (L) can hen be defined in he following way: Given Y + defined in (3), a reference porfolio RP, capial funds C, he se A and a dividend D such ha he accepabiliy condiion (6) or equivalenly (7) is saisfied, he value V (L) of he insurance liabiliy L a ime is defined o be he marke price of he reference porfolio, V (L) := V (RP ). (8) This immediaely enails wo quesions: Does here always exis a soluion o (6), i.e. can a value always be defined by (8)? If so, is such a soluion unique, i.e. is he value defined by (8) unique? We provide parial answers o hese quesions below, bu we do no invesigae he general quesion of he uniqueness of he value. In paricular, noe ha he value defined by (8) in general depends on he se A. In his respec, we sress ha we are no suggesing a new definiion of he marke-consisen value; all we claim o have done so far is provide a precise

14 328 C. MÖHR and more general formulaion of he valuaion approach for insurance liabiliies from Solvency II. The Solvency II approach follows from he general framework by he following hree assumpions:. The capial C is given in erms of he reference porfolio RP by SCR defined in (7) (compare wih (0)), i.e. for a ranslaion-invarian risk measure r, C := pv ( + " ) ( r{v + (RP ) Y + F }). (9) 2. r is given as in (8) by he Value-a-Risk VaR a he 99.5% percenile. 3. D is defined as a consan cos of capial rae j > 0 imes he capial C, i.e. D := j C. (20) In addiion, he curren prescripions from EIOPC/SEG/IM3/200 [] sugges ha he reference porfolio RP should be seleced o minimize he capial C. This can be hough of as a requiremen o ensure he uniqueness of he value. However, wih he Solvency II selecion of r as he 99.5% VaR, he capial C according o (9), he se A from (4), and he accepabiliy condiion (7) are no affeced by values of he difference Y + V + (RP ) beyond heir 99.5%-quanile. This suggess here migh no be uniqueness even if capial is minimized. As an informal example, assume ha he reference marke conains wo financial insrumens wih differen marke prices, bu wih he propery ha, if he firs financial insrumen is conained in RP, hen only he differences Y + V + (RP ) beyond heir 99.5%-quanile are affeced if he firs financial insrumen is replaced wih he second one. Then boh opions saisfy he accepabiliy condiion and lead o he same capial C and se A, bu resul in a differen value. Of course, an immediae way o ensure uniqueness would be o define he value as he minimum or infimum of he marke prices a ime of all reference porfolios RP saisfying he accepabiliy condiion (6) for he same amoun C and se A. In he following, we firs invesigae he exisence of soluions o condiion (6) under wo differen approaches. Nex, we derive in Lemma 2 an upper bound on soluions of (6) and hus on he value defined by (8). Finally, we show in Theorem 4 ha a unique soluion exiss and can be explicily calculaed if we assume ha he reference marke consiss only of risk-free zero-coupon bonds and ha he capial C is defined according o (9). For he following proposiion, we define an eligible dividend as follows: Definiion. An F -measurable dividend D $ 0 from (6) is called an eligible dividend if, given F, D is a coninuous and monoonously increasing funcion of C wih D = 0 for C = 0. Clearly, he dividend D defined by (20) is eligible.

15 MARKET-CONSISTENT VALUATION OF INSURANCE LIABILITIES 329 We now show ha soluions o he accepabiliy condiion (6) exis given a suiable form of he se A or he capial funds C. Proposiion. Assume ha he reference marke conains he one-year riskfree zero-coupon bonds. Le D be an eligible dividend, and le Y + from (3) be given. (a) Le RP (0) be a reference porfolio and define he se A (0) by A (0) := {Y + # V + (RP (0) )}. Then, here exiss a capial amoun C $ 0 and a reference porfolio RP such ha he corresponding se A defined by (4) is equal o A (0) and he accepabiliy condiion (6) is saisfied. (b) Le r be a ranslaion-invarian risk measure and le RP (0) be a reference porfolio. Assume ha he capial C (0) corresponding o RP (0) is given by (9) and he se A (0) by (4). Then, here exiss a reference porfolio RP wih he corresponding capial C $ 0 given by (9) and he se A given by (4) such (0) ha A = A and he accepabiliy condiion (6) is saisfied. (0) Proof. To prove (a), we spli up he porfolio RP ino a reference porfolio (0) RP and capial funds C $ 0 by removing from RP a one-year risk-free zerocoupon bond wih value C $ 0 o be deermined (or going shor in he bond). Then, V + (RP (0) ) = V + (RP ) + v ( " +) (C ), (2) and he accepabiliy condiion (7) for RP and C can be wrien as he condiion on v ( " +) (C ) + D ha v ( " +) (C ) + D = { A (0) (V + (RP (0) ) Y + ) F } $ 0, because (2) ensures ha he se A, if defined by (4) for RP and C, is equal o A (0), and he far righ inequaliy above holds by definiion of A (0). If equaliy holds in he far righ inequaliy, hen he accepabiliy condiion is saisfied for C := 0 and RP := RP (0). If no, hen he eligibiliy of he dividend ensures ha we find C > 0 such ha he accepabiliy condiion holds. To prove (b), we use a similar approach as for (a), removing a one-year risk-free zero-coupon bond wih value o be deermined from RP (0) o ge a new porfolio RP. The corresponding capial C given by (9) hen increases by he corresponding amoun because of ranslaion-invariance of he risk measure r, so V + (RP ) + v ( " +) (C ) = V + (RP (0) ) + v ( " +) (C (0) ),

16 330 C. MÖHR hence he se A defined by (4) for RP and C is equal o A (0), and (0) { A (V + (RP ) + v ( " +) (C (0) ) Y + ) F } = g v ( " +) (C ) + { A (V + (RP ) Y + ) F }, so using he accepabiliy condiion (7), we ge v ( " +) (C ) + D = { A (0) (V + (RP (0) ) + v ( " +) (C (0) ) Y + ) F } $ 0 by definiion of he se A (0). The argumen hen proceeds similarly o (a). Nex, we provide an upper bound on any soluion of (6). Lemma 2. Any soluion RP o he accepabiliy condiion (6) and equivalenly (7) saisfies {V + (RP ) F } # {Y + F } + D. Proof. By he definiion (4) of A, we have on he complemen A c of A, A c V + (RP ) < A c Y + A c v ( " +) (C ). Taking he expeced value condiional on F of his expression and adding he resul o (7), we ge he claimed inequaliy. If we assume in addiion ha he expeced reurn on RP over year is no less han he risk-free reurn, hen we ge from Lemma 2 a recursive upper bound on V (L): V (L) = V (RP ) # pv ( + " ) ( {X F } + {V + (L) F } + D ). (22) Under suiable assumpions, we can derive a closed formula upper bound from his recursive inequaliy. Proposiion 3. Assume ha (5) holds and ha, for any # s, {X s + D s F + } and R ( s- ) + are independen condiional on F. (23) Assume ha, for any year, he expeced reurn on any reference porfolio RP is larger han or equal o he risk-free reurn. Furher assume ha, for any, he se A is given by (4). Then, he value V (L) a ime of he liabiliy L is bounded from above by T V (L) # / pv ( s + " ) ( {X s + D s F }). (24) s=

17 MARKET-CONSISTENT VALUATION OF INSURANCE LIABILITIES 33 Proof. We proceed by inducion backwards in ime, saring from = T. For = T, he claim is given by (22), since V T + (L) = 0. Now assume ha (24) holds for +, i.e., using he noaion (3), T -(s- ) ( s- ) V ( L ) # / ` + j $ " X + D ; F,. (25) + R + s s= + s + The recursive upper bound (22) for can be wrien as V (L) # ( + R () ) {X + D F } + ( + R () ) {V + (L) F }. Insering (25) ino his inequaliy, using (23) and applying (6) proves (24). If we assume ha he reference marke consiss only of risk-free zero-coupon bonds, hen he accepabiliy condiion (7) on he reference porfolio RP explicily deermines he reference porfolio, given Y +, C and D, and provided ha g > 0. In fac, V + (RP ) = v ( " + ) (V (RP )) is hen F -measurable, so i can be aken ou of he expecaion in (7), and we ge g v ( " +) (V (RP )) = { A Y + F } + ( g ) v ( " +) (C ) + D. (26) This resul can be refined for he special case ha he capial C is defined in line wih Solvency II by (9) o derive an explici recursive expression for he value of L. Theorem 4. Assume ha he se A is given by (4) and he capial C by (9). Furher assume ha he reference marke consiss only of he risk-free zero-coupon bonds. Then, he value V (L) a ime of he liabiliy L is uniquely deermined by he recursive expression V (L) = pv ( + " ) ( { A Y + F } + ( g ) r{ Y + F } + D ), (27) where D is an eligible dividend from (6). The se A and he capial C can be wrien as A = {Y + # r{ Y + F }}, (28) C = pv ( + " ) ( r{ Y + F }) V (L). Proof. As he risk measure r is ranslaion-invarian, he capial from (9) is given by v ( " + ) (C ) = r{ Y + F } v ( " + ) (V (L)), (29) so he se A is given by (28). Insering (29) ino he expression (26) for V (RP ) hen proves (27) as, by definiion, V (L) = V (RP ).

18 332 C. MÖHR A more concise expression for (27) can be given if we define he random variable Z + as he cu-off of Y +, Z $ Y r{ - ; F }. = A + c + : + A $ Y + Then he recursion (27) can be wrien V L ) _ ( { } i. ( = pv + " ) Z+ ; F + D 5. THE RISK MARGIN Recall from he inroducion () he idea of defining he value of an insurance liabiliy L by he sum of a bes esimae, which we inerpre (maybe more generally han in Solvency II) as he marke price of a reference porfolio, and a risk margin corresponding o capial coss. The idea is ha he risk margin accouns for he non-hedgeable par of he cash-flows of he insurance liabiliy L o be valued. However, in he preceding par of he paper, a spli in bes esimae and risk margin was never required. Moreover, he definiion of he risk margin is ambiguous in he conex of muli-period replicaion, because here are wo conflicing inenions: on he one hand, he bes esimae is hough o capure only he cash-flows (X ) of he insurance liabiliy L and no capial coss. On he oher hand, he bes esimae should capure he hedgeable par over a one-year ime period, which in general includes also fuure capial coss. We show in he following ha i is possible o define a risk margin, and o use i o derive an upper bound on he value. However, he risk margin we define depends on cerain assumpions, and oher definiions of a risk margin would also be possible. To define he risk margin, he idea is o spli he reference porfolio RP ino a reference porfolio R RP, whose marke price is he bes esimae, and a porfolio we call dividend porfolio DP, such ha he dividend porfolio accouns for all capial coss, and is marke price corresponds o he risk margin. So RP consiss of he wo porfolios R RP and DP and, since marke prices are addiive, he value can be wrien as V ( L) = V ( = ( R ) + (. RP ) V RP V DP) We assume ha DP consiss of a risk-free one-year zero-coupon bond (compare o he formula (2) for he risk margin in Solvency II). Y + from (3) can be wrien as Y = X + V ( R RP ) + V ( DP ). (30)

19 MARKET-CONSISTENT VALUATION OF INSURANCE LIABILITIES 333 For deriving he spli, he reference porfolios R RP for every year are deermined firs; hey accoun for all fuure cash-flows (X s ) s $ of L and disregard limied liabiliy. Tha is, hey disregard he fac ha he replicaion canno always be coninued. The dividend porfolios DP are consruced aferwards. Define RM be he expeced risk margin a ime, RM T / : = pv s= )_ # D ; F -i, ( s+ " s where D s for s $ are eligible dividends from (6). Noe ha his expression is heoreical in he sense ha i is no necessarily he case ha risk-free zerocoupon bonds exis in he reference marke wih erms up o T +. We now prove one of our main resuls, which is ha he sum of he marke price of he reference porfolio R RP and he expeced risk margin RM is an upper bound for he value of L, see (33) as well as Corollary 6 below. Theorem 5. Le he se A on which he replicaion can be coninued be given by (4), A = {Y + # v ( " + ) (C ) + V + (RP )}. Assume ha he reference porfolio RP consiss of a dividend porfolio DP in he form of a one-year risk-free zero-coupon bond and a reference porfolio R RP, and ha he accepabiliy condiion (6) is saisfied for a given eligible dividend D $ 0. (a) Assume ha (5) holds, ha he porfolio R RP saisfies {V + ( R RP ) F } $ {X + V ( R RP + + ) F }, (3) and ha, for any # s, {D s F + } and R ( s- ) + Then, he value a ime of he liabiliy L saisfies are independen condiional on F. (32) V (L) = V ( R RP ) + V (DP ) wih V (DP ) # RM. (33) (b) Le he capial C be given by (9) for a ranslaion-invarian risk measure r, C = pv ( + " ) ( r{v + (RP ) Y + F }). Then DP is given by he recursive expression V (DP ) = pv ( + " ) ( { A (Y + V + ( R RP )) F }) + (34) + pv ( + " ) (( g ) r{v + ( R RP ) Y + F } + D ),

20 334 C. MÖHR where C and A can be wrien as C = pv ( + " ) (r{v + ( R RP ) Y + F }) V (DP ), (35) A = {Y + V + ( R RP ) # r{v + ( R RP ) Y + F }}. Proof. To prove (a), we need o prove (33), i.e. V (DP ) # RM. To his end, we firs derive a recursive expression for V (DP ). In fac, from Lemma 2, we have {V + (RP ) F } # {Y + F } + D. Insering R RP, DP, and Y + from (30) ino his inequaliy and using (3), we ge, since DP consiss of a one-year risk-free zero-coupon bond, V + (DP ) = {V + (DP ) F } # {V + (DP + ) F } + D. (36) This implies he recursive upper bound on V (DP ), V (DP ) # pv ( + " ) ( {V + (DP + ) F } + D ). Arguing as in he proof of Proposiion 3 and using (32), he upper bound (33) hen follows. To prove (b), he recursive expression (34) follows from he accepabiliy condiion (7) by similar argumens as in Theorem 4, using ha V + (DP ) is F -measurable and ha r is ranslaion-invarian. Remark. Noe ha he upper bound V (DP ) # RM from Theorem 5 (a) holds regardless of wheher equaliy holds in (3) or no. This means ha he upper bound RM on he value V (DP ) is no affeced by he selecion of R RP subjec o (3), alhough V (DP ) is. In order o obain he mos useful upper bound on he value, one should hus selec R RP as ha reference porfolio saisfying (3) which minimizes V ( R RP ). Remark 2. The upper bound from Theorem 5 (a) holds in paricular if he porfolio R RP is assumed o consis of a one-year risk-free zero-coupon bond and is defined o be he reference porfolio maching he expeced values of he cash-flows (X s ) s $ of he liabiliy L o be valued by risk-free zero-coupon bonds, i.e. if he value V ( R RP ) is given by T = pv( s+" ) s= V ( R RP ) / _ # X F -i. s ; This is also he reference porfolio which is opimal in he sense of minimizing V ( R RP ) as in Remark and for which equaliy holds in (3).

21 MARKET-CONSISTENT VALUATION OF INSURANCE LIABILITIES 335 If we assume ha he dividend D is given as in Solvency II by a consan cos of capial rae applied o he capial, see (20), hen we ge he following corollary. Corollary 6. Assume ha (5) holds. Assume ha he se A on which he replicaion can be coninued is given by (4), A = {Y + # v ( " + ) (C ) + V + (RP )}, wih he capial C given by (9) for a ranslaion-invarian risk measure r, C = pv ( + " ) ( r{v + (RP ) Y + F }). Assume ha he accepabiliy condiion (6) is saisfied for he dividend D given by (20), i.e. D = j C for some j > 0, and ha he reference porfolio RP consiss of a dividend porfolio DP in he form of a one-year risk-free zero-coupon bond and a reference porfolio R RP saisfying (3), {V + ( R RP ) F } $ {X + V ( R RP + + ) F }. Define Y + := X + V + ( R RP + ), DDP + := V + (DP + ) {V + (DP + ) F }, and assume ha, for any # s, # r{v R F - and R s + ( RPs )- Ys + - DDPs + ; Fs } ; + ( s-) + are independen condiional on F. (37) Then we have he upper bound V( L ) # V( R RP ) + R RM wih he adjused expeced risk margin T R j RM : = $ / pv ( ` { s + ( RPs )- Y s + DPs + F s - D ; s} ; F. + j + " ) $ r V R. j s= (38) Proof. We show ha V (DP ) # RMR for any. To his end, we noe ha he assumpions of Theorem 5 are saisfied, and so we can inser D = j C and he expression (35) for he capial C ino (36) o ge ( + R () + j) V + (DP ) # ( + R () ) {V + (DP + ) F } + j r{v + ( R RP ) Y + F }.

22 336 C. MÖHR Using ha r is ranslaion-invarian and in view of he definiions of Y + and DDP +, and wih he inequaliy + j # + R () + j, we can wrie his as he recursive upper bound j V + (DP ) # {V + (DP + ) F } + + j r{v + ( R RP ) Y + DDP + F }. Arguing as in he proof of Proposiion 3 and using (37), he upper bound hen follows. In pracice, i is ofen assumed as a simplificaion ha he risk margin does no conribue o he one-year volailiy, i.e. V + (DP ). V + (DP + ) + D. This assumpion is implici in he Swiss Solvency Tes (SST) and is ofen assumed in Solvency II when he Solvency Capial Requiremen needs o be calculaed. In our case, we can formulae he corresponding condiion as (for some small e > 0) DDP + = V + (DP + ) {V + (DP + ) F } # e r{v + ( R RP ) Y + F }. I.e. he possible increase in he esimae of V + (DP + ) from ime o + is small compared o he r-erm. If we assume his holds for any and, in addiion, ha he risk measure r is monoone, hen we ge r { V ( R + RP )-Y + -DDP + ; F } # r{ V ( R + RP) -Y+ -e$ r{ V ( R + RP) -Y+ ; F} ; F} = ( + e) $ r{ V ( RRP )-Y ; F } + + using ha r is ranslaion-invarian. We hen ge for he adjused expeced risk margin RMR from (38) he upper bound j $ ( e) T R + RM # $ / pv _ ( # { s+ ( RPs )-Xs - s+ ( s + ) Fs F. s ; ; + j + " ) r V R V RRP } -i s= This expression for he risk margin appears o be he one implicily used in mos acual calculaions in he conex of Solvency II (and he SST), bu wihou he j-erm in he denominaor of he fracion above and wih e se equal o zero. Noe ha he capial for any year is calculaed above in erms of he oneyear change of he bes esimae, and no in erms of he one-year change of he difference beween he value of he asses covering he bes esimae and he value of he liabiliy.

23 MARKET-CONSISTENT VALUATION OF INSURANCE LIABILITIES AN EXAMPLE FOR THE CALCULATION OF THE VALUE We now explicily calculae he value for a simple example wih wo insurance liabiliies L () and L (2). The example illusraes he upper bound in erms of he sum of bes esimae and risk margin from Theorem 5 and shows ha he ordering of he value of wo liabiliies can change: here are insances in which he value of one liabiliy is larger han he oher bu he inequaliy is reversed beween he upper bounds. We noe ha he example migh no be realisic as we assume ha he risk-free ineres rae is zero and ha successive cash-flows are independen. However, i migh be surprising ha he change of he ordering occurs even under hese assumpions. We assume ha only risk-free zero-coupon bonds are eligible for he replicaion and ha he risk-free ineres rae is zero. These wo condiions imply ha acually only cash is available. We furher assume ha he capial C is given by (9) wih he risk measure r as in (8) given by he Value-a-Risk a a confidence level 0 < a <, and ha he dividend D is given by (20) wih j > 0. Under hese assumpions, Theorem 4 implies ha he value V (L) of an insurance liabiliy L wih cash-flows (X ) is recursively given by V( L) = + ( g ). ` { A $ Y ; F + j - - ; F + j + } $ r{ Y + } j (39) The upper bound on he value according o Theorem 5 becomes V u ( L): = / { X ; F } + j $ / { C ; F } (40) wih C s { T T s s= s= = r -Y } V L), (4) s + ; Fs - s ( s where X denoes he claims paymen of he insurance liabiliy L in year. X is F + -measurable, and we assume in addiion ha is disribuion condiional on F s for any 0 # s < + is independen of s, i.e. no informaion abou X is revealed before ime +. In paricular, he X are independen of each oher. We consider he wo years = 0 and = wih cash-flows X 0 and X, respecively, and assume ha X = 0 for $ 2. We suppress condiioning on F 0 in he noaion, and hen ge from Theorem 4 and he above assumpion on X ha { $ Y ; F } = { $ X ; F } = { $ X } A 2 { X# r{ -X; F} } { X# r{ -X}} r{ - Y ; F } = r{ -X }. 2 Hence he value a ime = from (39) is V ( L ) = ` { $ X + ( ). j { X { X}} + j - + # r - } g $ r{ -X} j