Using DFA for Modelling the Impact of Foreign Exchange Risks on Reinsurance Decisions

Transcription

1 Casualy Acuarial Sociey 2001 Reinsurance Call for Papers Using Dynamic Financial Analysis o Opimize Ceded Reinsurance Programs and Reained Porfolios Published in Casualy Acuarial Socieey Forum, Summer 2001 Using DFA for Modelling he Impac of Foreign Exchange Risks on Reinsurance Decisions Peer Blum 1, 2, Michel Dacorogna 2, Paul Embrechs 1, Tony Neghaiwi 2, Huber Niggli 2 Absrac The flucuaions of he foreign exchange (FX) raes are a source of addiional risk, bu also an opporuniy for furher profis for inernaionally operaing reinsurers. A DFA model ha includes FX raes can be a means for measuring he poenial impac of FX rae flucuaions on porfolios of ceded reinsurance and inernaionally invesed asses. Moreover, such a DFA model can also be a means for esing differen sraegies for managing FX risk. We sar by surveying he empirical properies of FX rae daa and inroduce some simple approaches for generaing FX scenarios. We hen sudy he differen ways in which reinsurers are exposed o FX risk, and possible sraegies for managing hese risks. The emphasis is on so-called on-balance shee currency hedging echniques, i.e. on offseing FX risks on reinsurance conracs by invesing appropriae amouns of money in he respecive foreign currency. The heoreical reasonings are accompanied by a DFA sudy on an inernaional loss porfolio. This sudy shows ha FX flucuaions can have an impac on he reinsurer s business, and ha he opimal sraegy may no be a full hedge depending on he ype of business and on he economic circumsances. Acknowledgemens The auhors wish o acknowledge he useful advice of Sholom Feldblum a he onse of his work and he valuable commens and reviews of heir referee Gary Blumsohn on earlier versions of he manuscrip. 1 DEPARTMENT OF MATHEMATICS, ETH ZURICH, CH-8092 ZURICH, SWITZERLAND, addresses: {blum embrechs}@mah.ehz.ch 2 ZURICH INSURANCE COMPANY, REINSURANCE, PO BOX, CH-8022 ZURICH, SWITZERLAND, addresses: {peer.blum michel.dacorogna ony.neghaiwi huber.niggli}@zurichre.com 1/1

2 1. Inroducion Reinsurance business is becoming increasingly inernaional. Therefore, he impac of foreign exchange risk (FX risk) has o be deal wih by boh cedan and reinsurer when srucuring reinsurance programs. The sandard approach o dealing wih FX risks emanaing from (re)insurance conracs denominaed in foreign currencies is o ake sufficien asse posiions in he same currency and pay he liabiliies from hese posiions, hus avoiding cash flows across he currency border a he ime he losses occur (so-called currency hedging). Servicing losses a lowes possible cos is, however, only one par of he insurance business. The oher par is ha he company s asses should produce good invesmen reurns. While he approach of currency hedging definiely reduces cash flows ha are exposed o FX risk, i also leads o possibly subopimal invesmen reurns by binding asses in subopimal posiions. I should also be aken ino accoun ha FX risk has no only a downside, bu is iself an opporuniy for invesmen benefis which should no be lef unused given ha one has o ake exposure in foreign currencies. For inernaional financial asse managemen, corresponding resuls are given by Froo (1993) and Levich and Thomas (1993), and we will make some consideraions for he insurance-specific conex in Secion 5. Hence, in he conex of inegraed risk managemen, he managemen of FX risks should no only be considered from he poin of view of servicing inernaional liabiliies, bu also from he poin of view of maximizing he reurn of an inernaional invesmen porfolio. The big challenge is o find an opimal radeoff beween hese wo poins of view, and DFA can be a ool for achieving his goal. Therefore, a modern DFA sysem ha is able o model FX risks has many pracical applicaions for oday s global players. Taking ino accoun FX risks when doing DFA allows for more accurae and innovaive srucuring and pricing of ceded reinsurance arrangemens, which is in he ineres of boh cedan and reinsurer. This paper is accompanied by a pracical DFA sudy in order o underline he saemens being made. The seup of his sudy is explained in Secion 2, and he sudy will be revisied in each of he subsequen secions. Secion 3 invesigaes he properies of foreign exchange raes and relaed variables and proposes wo simple approaches for scenario generaion. Secion 4 presens he ways in which a (re)insurance company is exposed o FX risk and gives indicaions how hese exposures can be modelled in DFA, and Secion 5 surveys mehods for FX risk managemen. Secion 6 presens he applicaion of he previously inroduced mehods in a DFA sudy for srucuring reinsurance, and Secion 7 saes some conclusions and direcions for furher research. 2/2

3 2. The example We inroduce here he seup of a muli-currency (re)insurance deal ha we will use hroughou he whole paper o illusrae our reasoning. Noice ha his is in principle a real case, bu he number of involved currencies has been reduced (for simpliciy s sake), and he figures are modified such ha he cusomer canno be idenified. The insurance company (cedan) has is Head Office in he Unied Saes and consolidaes and repors is business in USD. The lines of business (LOB) involved in our deal are, however, siuaed in oher counries and denominaed in he respecive currency: LOB1 is in Swizerland (denominaed in CHF), LOB2 is in he Unied Kingdom (denominaed in GBP), and LOB3 is in Japan (denominaed in JPY). Therefore, each ransacion of funds beween he Head Office and he foreign LOBs is subjec o FX risk. Moreover, if he Head Office valuaes he hree foreign LOBs, all heir figures have o be ranslaed back ino USD. The hree LOBs are runoff business. Each LOB has an iniial loss reserve for ime 0 and a given sochasic payou paern, and all claims paymens are o be made in he currency of he respecive LOB. These loss reserves are ypical P/C loss reserves in ha hey bear risk in he iming as well as in he final amoun of he paymens. Therefore, from he poin of view of he Head Office, we are faced wih a superposiion of risks on iming and amoun (inheren o each LOB) and FX risk. The Head Office wans o proec is (USD-denominaed) resuls by aking ou a sop-loss conrac. There are differen degrees of freedom in he design of his conrac: One conrac for he aggregae porfolio or single conracs for each LOB. Differen reenions and limis. Currency in which o denominae he conrac(s) and parameers; use of fixed exchange raes in he conracs yes/no. Wha is he addiional exposure of he reinsurer in is layer due o FX volailiy? This, in urn, will influence he price of he conrac. Hedging issues for insurer and reinsurer: hedging yes /no, parial hedging, saic or dynamic hedging, single currency or cross currency hedging. The following figure shows he hree basic loss processes involved in he example (wihou influence of any FX raes): 3/3

4 3'500'000 Paid Losses in Local Currency 3'000'000 2'500'000 2'000'000 1'500'000 1'000'000 Losses from Swizerland in CHF Losses from GB in GBP Losses from Japan in JPY (00) 500' Year Figure 1: Losses per year and per LOB denominaed in local currency. The inervals around he observaions are plus/minus one sandard deviaion. 4/4

5 3. Scenario generaion 3.1 Requiremens Using DFA for assessing FX risks requires he generaion of FX scenarios. These scenarios mus no be generaed in isolaion, bu hey mus also reflec he relaions beween FX raes and oher economic variables used in DFA sudies, in paricular ineres raes and inflaions in he involved counries. Moreover, our aim is no o opimize our generaor so as o produce he mos accurae level or inerval forecass, bu credible and coheren scenarios for he ime up o he defined horizon. The generaor mus be such ha i can be calibraed wih low amouns of daa: The usual ime resoluion in DFA is one year, which means ha given he hisorical facs exhibied below - only abou 25 daa poins are available. Even if we use monhly daa, no more han abou 300 daa poins will be available. The generaor should be ransparen in ha each parameer has a pracical meaning, such ha manual adjusmens by he user are possible: eiher for correcing he calibraion or in order o bring in expecaions of he fuure behavior ha are no refleced by hisorical daa. A word of warning: I is no our ambiion o develop here a highly sophisicaed economeric model of he FX markes. We only have he aim o propose a simple model ha reflecs he mos imporan feaures of he daa while being racable for he average praciioner. 3.2 Definiion of Terms Unless oherwise saed, we will always use he spo exchange rae wih respec o he US Dollar XXX (USD), i.e. we denoe by S he USD price for one uni of currency XXX a ime. The cross rae, i.e. he price of one uni of currency YYY in unis of currency XXX, can hen be calculaed as S = S S. For he analyical reamen, we will use he logarihm of he spo XXX / YYY YYY XXX XXX XXX exchange rae, i.e. s = log( S ). By XXX I and XXX R we denoe he rae of inflaion and he risk free shor-erm ineres rae, respecively, for counry (currency) XXX. For analyical XXX XXX reamen we will also use some sor of logarihmic ransformaion i = log(1 + I /100) and r XXX XXX = log(1 + R /100) ha pu hese percenage raes on equal fooing wih he FX raes, which are essenially prices. By x we denoe he value of a mulivariae ime series a ime. Raher han in he level x, we may be ineresed in analyzing he changes of he ime series in a ime inerval, i.e. x = x x 1. For he analysis of one currency pair (always wih he USD as he reference), we will consider he ime series The exension o more currencies is obvious. x. 1 5 (,..., ) T ( USD, USD, XXX, XXX, XXX ) T = x x = r i r i s 3.3 Properies and sylized facs of FX daa FX raes can be governed by differen ypes of regimes: A currency is pegged if is exchange rae wih respec o some reference currency is fixed. A currency is semi-pegged if is exchange rae wih respec o a reference currency is only allowed o floa wihin a narrow band. Mainenance of 5/5

6 pegged and semi-pegged relaions requires suiable inervenions from he involved governmens. If no formal resricions are imposed on he exchange rae, hen i is called free-floaing. From he end of World War II up o 1971, he so-called Breon-Woods sysem (semi-) pegged all currencies o he US Dollar wihin a band of 1%. Free floaing of he mos raded currencies only emerged afer he breakdown of he Breon-Woods sysem in Therefore, FX daa from before 1972 should no be used for he calibraion of scenario generaors ha model free-floaing currencies. Pegging and semi-pegging relaions sill exis nowadays for some (mosly less raded) currencies. A special case is he European Moneary Union (EMU): Whereas he Euro is allowed o floa freely agains all ouside currencies, he domesic currencies of he members are pegged o he Euro. This sae will disappear by he end of 2001, when he Euro will fully replace all domesic currencies of EMU counries (e.g. he Deusche Mark). See Luca (2000) for more informaion on hisorical and poliical backgrounds, as well as for a horough reamen of he modern FX markes. There are wo basic ways o analyze and model he inerplay of FX raes wih he oher economic variables involved: fundamenal (economic) and echnical (saisical) analysis. Economiss have explored relaions of FX raes wih a high number of macro-economic variables (e.g. money supply, expor balance, inflaion, ec.); see MacDonald (2000) or Closermann and Schnaz (2000) for easy-o-read reamens. The ubiquious concep in hese reamens is he Purchasing Power Pariy (PPP), which saes ha under he assumpion of efficien markes and wihou ransporaion coss goods should have effecively he same price in he wo counries, i.e. YYY XXX / YYY XXX P = S P, where P denoes he prices in he respecive currencies. While his relaion is clearly oo sric o hold in pracice, here exis more elaborae versions. One ha akes ino accoun hose variables ha we need in DFA anyway is: s = α + α ( i i ) + α ( r r ) + I (1) XXX / YYY XXX YYY XXX YYY where α 0, α1 and α2 are consans, and I is a saionary sochasic process (e.g. auo-regressive). This equaion relaes he FX rae o he inflaion difference and o he ineres rae difference, which can be seen as wo key incenives for aking or leaving posiions in some currency. An alernaive is o relae he FX rae o he real raes of reurn in he wo counries, i.e. s = α + α ( r i ) + α ( r i ) + I. (2) XXX / YYY XXX XXX YYY YYY Clearly, possible invesmen benefis are no he only cause governing he demand (and hence he price) for a foreign currency. Therefore, we mus no expec he above relaions o explain all of he dynamics of he FX raes. Relaions of he above ype, which posulae ha some linear combinaion of componens of an oherwise non-saionary ime series are saionary, are called coinegraion relaions, and hey are very popular in saisics and economerics, see Clemens and Hendry (1998) for more informaion. 6/6

7 In saisical analysis we invesigae he mulivariae ime series ( x ) of variables of ineres by means of saisical daa analysis echniques, hus rying o explore properies and relaions in he daa wih he final aim of finding and calibraing a model ha reflecs he acual properies of he daa. Here is a shor survey of relevan properies. The resuls were obained from invesigaions of he auhors wih yearly and monhly observaions of several counries wih free-floaing exchange raes (USA, UK, Japan, Swizerland) over he pas wo decades, and hey go in line wih he findings of MacDonald (2000), Closermann and Schnaz (2000), and Clemens and Hendry (1998). See Figure 1, graphs (a) o (f), for some illusraive examples; ables and chars of all involved daa series can be found in Appendix A. The mulivariae series x as a whole shows clear signs of non-saionariy (i.e. drifs and non-consan volailiy, see (a)), whereas he differenced series x appears saionary. This is no fully obvious from Graph (b), bu saisical ess on monhly and higher-frequency observaions underpin he hypohesis of saionariy in differences. The values of ineres raes, FX raes and inflaion show clear dependence on heir own pasyear values (auocorrelaion) as well as on presen-year and pas-year values of oher variables (cross-correlaions). However, he cross-correlaion relaions are differen from counry o counry. As an example, Swiss inflaion is correlaed wih he USD/CHF exchange rae, whereas US inflaion is largely unaffeced by any single FX rae. Differences in one year are, moreover, ofen relaed o he levels of previous years, hus indicaing level-dependen volailiy and if he relaion has negaive sign mean reversion. The above-menioned coinegraion relaions (1) or (2) can be observed in he daa afer removing auoregressive componens. Wheher (1) or (2) is more significan depends on he currency pair. See (c) for he USD/JPY rae: The solid line is he predicion based on formula (2), he doed line represens he rue values; he residuals (doed line minus solid line) have zero mean, i.e. he model is unbiased. In monhly daa neiher (1) nor (2) are very significan. This goes in line wih he above-menioned economeric sources, which sae ha hese relaions only hold over he longer erm 3, bu large deviaions are possible in he shorer erm. Our observaions wih real daa sugges ha one year is already a sufficienly long ime span for he involved variables o rever o he equilibrium relaions, a leas for freely raded currency pairs and liquid markes. Moreover, on yearly daa, he inflaion could be shown o be very significanly correlaed XXX XXX wih he ineres rae of he respecive counry. The model i = β0 + β1r + I was found o fi o he daa very well. Graph (d) shows prediced and acual inflaion for he US in he same manner as described above. This does, however, no mean ha we posulae ha 3 Noice: In he economeric lieraure, ime horizons of one year and longer are usually referred o as long erm, whereas one day up o one quarer are referred o as shor erm. 7/7

8 he inflaion causally depends on he ineres rae; he real causaion is likely o be bidirecional. For modelling purposes, however, he above formula is useful, and is use is jusified by he fac ha i fis well o he daa. The converse modelling approach (ineres raes as a funcion of he inflaion) can also be aken, see e.g. Daykin e al. (1994). Mos of he ime, he log-ransformed variables show Gaussian behaviour. There exis, however, a few exreme moves in he sample ha canno be reconciled wih he hypohesis of Gaussianiy. Graphs (e) and (f) show wo QQ-normal plos 4 of residuals: The residuals of Swiss inflaion (Graph (e)) show clearly non-gaussian behaviour, whereas he residuals of he US inflaion (Graph (f)) look clearly Gaussian. Therefore, even on his very high level of aggregaion, we mus be aware of exreme, non-gaussian movemens. Figure 2: Some examples from daa analysis (Source: Daasream) 4 In a Quanile-Quanile-normal plo, he sample quaniles of he observed residuals are ploed agains he heoreical quaniles of he Normal disribuion. If he poins are ighly grouped around he diagonal line (which represens he heoreical Normal disribuion), hen he sample can be assumed o have Gaussian disribuion. See any exbook on saisical daa analysis for furher informaion. 8/8

9 If one looks a monhly insead of yearly daa, he following can be observed: Clearly he auoregressive relaions exend o several ime seps backwards. Moreover, paricularly he inflaion shows a srong inra-yearly seasonaliy, which can be overcome by using seasonally adjused daa (available for mos counries). FX raes and ineres raes have negligible seasonal effecs. Finally, here are relaively more exreme (non-gaussian) movemens due o he reduced smoohing caused by he lower ime aggregaion. Based on he summarized saisical analysis, i is possible o formulae some general rules, bu here are also differences beween he currencies. When seing up and calibraing a FX scenario generaor, i is essenial o do dedicaed saisical analysis on he involved currencies, and he above explanaions may serve as a guideline. 3.3 The generaor and is calibraion As one possibiliy for modelling FX raes ogeher wih heir consiuens relevan for DFA (i.e. ineres raes and inflaion) on a yearly basis we propose a boom up modular approach. This approach permis o reuse already-exising models for ineres raes and inflaion in one counry. The oupus of hese models are hen used o compue he exchange raes by using a coinegraion relaion, i.e. where: α XY XY 0 4 s = α + α r + α i + α r + α i + ε XXX / YYY XY XY XXX XY XXX XY YYY XY YYY XY s, ri, : logarihmic FX, ineres, and inflaion raes as inroduced above,, K, α : consan regression parameers specific o each currency pair XXX/YYY, ε : an iid series of random variables wih zero mean, XY eiher N σ if residuals are Gaussian, 2 (0, XY ) or wih Suden s disribuion 5 if residuals show non-gaussian exremes. XY XY XY XY If we le α1 = α3 and α2 = α4 his model is equivalen o he relaion (1). Similarly, one can also achieve equivalence wih relaion (2). In boh cases we have a reducion of he number of parameers o be esimaed, which is an advanage in view of he low amouns of daa available for esimaion. In he ligh of our findings in he previous secion, he inflaion for a given currency can be modelled by a simple linear regression on he respecive shor-erm ineres rae: where: X Y 0 1 i = β + β r + ε and i = β + β r + ε (3) XXX X X XXX ix YYY Y Y YYY iy ri, : logarihmic ineres and inflaion raes as above, β, K, β : consan regression parameers specific o currencies XXX and YYY, X Y ε, ε : iid series of random variables wih zero mean as above. 5 See Embrechs e al. (1997) for more informaion on he modelling of exremal evens. 9/9

10 For he shor-erm ineres raes, we ake here he well-known Cox-Ingersoll-Ross (CIR) model, which is described in Chan e al. (1992), and which can be exended o generae a full yield curve wihou exra effor for calibraion: where: r = a ( b r ) + s r ε and r = a ( b r ) + s r XXX X X XXX X XXX rx YYY Y Y YYY Y YYY ry r : logarihmic ineres raes as above abs,, : parameers of CIR model as specified in Chan e al. (1992) ε X Y ε, ε : iid series of Gaussian random variables wih zero mean and uni variance In order o accoun for cross-counry dependences, we have o inroduce some dependence srucure beween he innovaions of he wo ineres rae processes: eiher by leing ε r : = ( ε rx, ε ry ) T be an iid series of bivariae Gaussian random vecors wih zero mean and covariance marix Λ, or by linking he random variables ε rx and ε ry via a so-called copula. The concep of copulas is described in Embrechs e al. (1999). This is an advanced concep ha allows o accoun for sronger dependence in he ails of he involved random variables. The model inroduced here for he inflaion and ineres raes is he same as used in Dynamo, see D Arcy e al. (1997), bu any oher model for shor-erm ineres raes and inflaion can be used insead. I mus only be exended by formula (3) for he exchange rae and by a dependence srucure for he innovaions of he basis variables. Calibraion of his model is sraighforward: i. Calibrae he ineres rae model for each currency (for CIR: see Chan e al. (1992)). ii. Invesigae he join disribuion of he residuals of he ineres rae processes. If here are signs for sronger correlaion in he ails, use a copula, oherwise ake he esimaed covariance marix. iii. For each currency, esimae he regression coefficiens for he inflaion wih respec o he ineres rae. If he residuals show non-gaussian ouliers, fi a heavier-ailed disribuion. iv. For each currency pair, esimae he coefficiens of formula (3) by linear regression. If he residuals show non-gaussian behaviour, fi a heavier-ailed disribuion. The acual parameers esimaed for he seup of our example can be found in Appendix B. In he ligh of he findings of he daa analysis in he previous secion, his approach is well suied for modelling on a yearly basis. Moreover, i has he advanage ha already-used models for ineres raes and inflaion can be reused, and ha he parameers have clear pracical inerpreaions. On a quarerly or monhly basis, FX raes also depend on pas values of inflaion and ineres raes. A careful daa analysis along he lines presened in he previous secion should be done in his case before selecing and calibraing a model. A possible closed-form alernaive 10/10

11 would be he use of a purely saisical mulivariae ime series model for he process Vecor Auoregressive (VAR) model: x, e.g. he where: x : as inroduced above, p k k k = 1 x = τ + Γ x + ε τ : vecor of deerminisic drif erms, Γ : marices of auoregression coefficiens, k p : maximum order of auoregression, ε : iid series of Gaussian random vecors wih zero mean and covariance marix Λ. The main advanage of his closed-form model is ha i allows more easily o simulae condiional scenarios where he values of a par of he variables are fixed ( wha-if analysis ). I is also more open in ha i also allows dependences ha are no economically derived. Disadvanages include he (relaively) high number of parameers and he simple dependence srucure (only linear dependence on pas values) ha is no able o render cerain characerisics of he daa such as he level-dependen volailiy of ineres raes. When fied o yearly daa, he VAR(1) model was unbiased, bu he goodness-of-fi was no as good as for he modular model. The models were esed for qualiy by doing analysis of he residuals, i.e. he empirical values of he various ε s in he formulas. The model is unbiased if he mean of he residuals is zero, and if he ime series of he residuals shows no signs of serial correlaion. This means ha he model capures he behaviour of he hisorical daa wihou sysemaical deviaions. The model has a high goodness-of-fi if he variance of he residuals is considerably smaller han he oal variance of he hisorical daa, and if he empirical disribuion of he residuals can be reconciled wih he one given by he model (e.g. Gaussian). This means ha a relevan par of he variabiliy of he daa is explained by he model and no by he residual noise. If one uses a saisics sofware (in our case R, he freeware version of S-Plus), one can also obain es saisics ha give more elaborae indicaions for he goodness-of-fi. The ulimae means for measuring model qualiy would be o use a saisical informaion crierion, he mos classical one being AIC (Akaike s Informaion Crierion). The problem wih hese crieria is, however, ha hey are relaively difficul o compue in pracice. When fied o yearly daa, boh models (i.e. he boom-up modular and VAR(1)) urned ou o be significanly unbiased. The goodness-of-fi was beer for he boom-up modular model han for he VAR(1), due o he lower number of parameers o be esimaed wih he same amoun of daa. On he absolue scale he goodness-of-fi is no very high, bu we have a leas for all variables ha he major par of he variabiliy is explained by he model and no by he residual variance. The pracical meaning of his saemen is ha simulaed scenarios for he fuure will behave according o (roughly) he same sochasic discipline as he hisorical daa, i.e. our model 11/11

12 is a good predicion of he fuure provided ha he fuure behaviour of he modelled economies is fundamenally he same as in he preceding years. If expecaions for he fuure are differen, he esimaed coefficiens mus be modified accordingly, which is easy due o he simpliciy of boh models. We do no claim ha our proposed models are he bes possible generaors for FX scenarios in DFA. Bu we have chosen hem because hey are able o reflec he relevan saisical and economic properies of he respecive mulivariae ime series while being relaively simple, ransparen, and easy o calibrae. Anoher alernaive if available would be o use scenarios generaed exernally by one of he commercially available economic scenario generaors, such as LongRun from RiskMerics, see Kim e al. (1999), or mark-o-fuure from Algorihmics, see Dembo e al. (2000). For a general survey of models for sraegic long-erm financial risks see Kaufmann and Paie (2000) and he references herein. To conclude, we show here he resuls of a Mone Carlo evaluaion of he scenario generaor for he boom-up modular model. The sudy consised of generaing 1000 realizaions of he USD/CHF exchange rae and is consiuens for he ime inerval from 1991 up o 2000 by using he boom-up modular model calibraed wih daa from 1980 o This is called an in-sample evaluaion. In general, ou-of-sample evaluaions (i.e. wih disinc ime inervals for calibraion daa and reference daa) would be preferable, bu hey were no done here because of lack of daa. For each simulaed year, mean and sandard deviaion of he 1000 realizaions for each simulaed variable were generaed. The resuls are shown below: he poins represen he rue values of he respecive variable, he solid line is he ime series of he mean values of he forecass, he dashed lines represen he inerval mean +/- one sandard deviaion for each ime sep, and he doed lines represen he inerval mean +/- wo sandard deviaions. A word of warning: he power of his evaluaion is very limied; he graphs only give a hin on wheher he scenarios are able o capure he behaviour of he real pas series on he average. Wheher one wans o adjus he parameers or no depends on he expecaions for he fuure. The rue values of he ime series are in almos all cases comprised wihin one sandard deviaion from he mean, and always wihin wo sandard deviaions. Swiss ineres raes (and also inflaion) are sysemaically overprediced for he second half of he nineies. This is due o he fac ha hese raes were much higher in he eighies han in he nineies, wih he respecive influence on he parameers of he model. The average ineres rae predicions of he CIR model (and hence also inflaion and FX rae) show relaively srong mean reversion on he long run. This is no very realisic, bu maybe he bes guess for a disan fuure wih he accordingly high uncerainy. 12/12

13 Figure 3: Resuls of Mone Carlo evaluaion 3.4 Discussion of he limis of he modelling approach We conclude his secion by assessing wo imporan problems relaed o he simulaion of economic variables such as FX raes, ineres raes, or inflaion in he conex of DFA. The firs problem is he lack of sufficien amouns of daa for he calibraion of he model. Wih a ime resoluion of one year, we only have abou 25 years of observaions a hand. The model mus herefore be kep very simple, i.e. wih a low number of parameers o be esimaed. Even hen, he accuracy of he esimaed parameers is no very high, bu i was a leas possible o obain esimaes ha have no sysemaic bias wih respec o he hisorical daa. One way o achieve beer-deermined parameers is o use models and daa on higher frequencies, e.g. monhly or even weekly. This requires, however, a very good undersanding of he ime aggregaion properies of he involved models and he mehods for heir calibraion. This approach is quie well undersood for univariae models and on very high frequencies (inra-daily daa), see Dacorogna e al. (2001). For mulivariae models and long ime horizons he siuaion is more difficul and usable resuls are sparse, see e.g. Kaufmann and Paie (2000). If models are kep 13/13

14 simple and parameers have clear pracical meanings, hen here is also he possibiliy o adjus he parameers by hand, e.g. based on insighs or assumpions from oher sources han saisical analysis of hisorical daa. The approach of using simple bu ransparen models is ofen referred o as Simpliciy Posulae or Parsimony in he saisical and economeric lieraure, see e.g. Clemens and Hendry (1998). Anoher problem is of more fundamenal naure: When we have fied a model o hisorical daa, hen he scenarios generaed by his model have (roughly) he same sochasic behaviour as he hisorical daa, which is no necessarily he bes projecion for he fuure. This means in pracical erms: We implicily assume ha he fuure behaviour of he modelled economic variables is fundamenally he same as i used o be in ha par of he pas considered for calibraion. The generaed scenarios do, however, no accoun for fundamenal changes or drifs in he regime governing he modelled variables, nor do hey sufficienly accoun for hihero unexperienced exremal evens. The generaed scenarios simulae he same risk as i emanaed from he pas behaviour of he respecive variables. In paricular: Scenarios accoun for unusual or exreme evens o he same exen as such evens occurred in he pas. By using special models for he ails one can generae evens wih magniudes and probabiliies beyond wha has been experienced. Refer o Embrechs e al. (1997) for he fundamenals and he discussion in Müller e al. (1998) abou assessing exreme risks in he foreign exchange marke. See also Blumsohn (1999) for a presenaion of his problem and possible soluions in a differen conex. The severiy of his fundamenal uncerainy depends on he ime horizon of he sudy and also on non-saisical insighs and expecaions a he beginning of he period. Here are some possible ways o cope up wih he problem of fundamenal uncerainy. Their common characerisic is ha hey ry o explore sensiiviies raher han absolue levels of risk and reurn: Do several sochasic simulaions wih several differen parameer ses for he scenario generaor. These parameer ses can be based on saisical esimaions, bu also on manual adjusmens reflecing non-saisical insighs and assumpions. Idenify hose simulaed scenarios ha have led o exreme resuls (e.g. ruin) and explore heir common characerisics, c.f. Dembo e al. (2000) for an applicaion of his mehod in he conex of finance. Complemen he sochasic simulaions wih classical sress scenarios modelling paricularly adverse courses of evens. 14/14

15 4. Modelling FX risk exposure 4.1 FX risk in general Whenever an amoun of money A denominaed in one currency YYY mus be measured in erms of anoher currency XXX, his corresponds o muliplying he original amoun by he currenly XXX / YYY prevailing FX rae beween he wo currencies: S A, i.e. he original volailiy of A is superimposed by an addiional volailiy arising from he FX rae. In a ypical DFA seup, here will usually be a large number of variables o be convered from one currency o anoher, and several currencies may be involved. There are correlaions of differen amoun and sign beween he FX raes and oher sochasic variables: inflaion and ineres raes are correlaed wih he FX rae, inflaion may influence loss experience, and ineres raes influence invesmen reurns. Hence, even in a relaively simple real case, i is already very difficul if no impossible o make a valid quaniaive saemen on he impac of FX volailiy based only on analyical reasoning. See Loderer and Pichler (2000) for a survey of he difficulies ha firms have in deermining heir acual FX risk exposure. Sochasic simulaion, i.e. DFA, is one suiable means for resolving his issue. We can disinguish beween hree differen ypes of FX risk exposure, i.e. ranslaion exposure, ransacion exposure, and economic exposure; see Luca (2000) for more deails. Translaion exposure arises when foreign-denominaed asses or liabiliies are ranslaed ino he home currency for consolidaion purposes a prevailing FX raes, e.g. for he annual financial saemens of he company. If FX raes have changed since he las consolidaion, his can lead o considerable changes in consolidaed values of asses and liabiliies. Translaional changes are only nominal in he sense ha no gains or losses are acually realized, since no asses or liabiliies are liquidaed. If ranslaional flucuaions are high enough, hey can neverheless have an impac on he company s operaions, e.g. hrough changes in axaion, loss of credi raing, repuaional damage, or regulaory problems. Translaional gains and losses can be idenified by comparing financial saemens wih and wihou change in FX raes in he reporing period. They can also be a deecor for possible fuure ransacion exposures, in he case when he considered asses or liabiliies mus be liquidaed. Accouning rules, however, offer means for miigaing he effec of ranslaion exposure on he financial saemens. The basic principle is ha profis and losses from FX ranslaions are assigned o an equiy accoun and do no ener ino income figures unil he asse or liabiliy is liquidaed, bu he deails are raher complicaed and vary beween he differen accouning sandards; see FASB 52 (1981) for US Sauory and Chaper 22 of Bailey and Wild (2000) for IAS and GAAP. These rules make he implemenaion of ranslaion exposure measuremen in DFA highly complicaed, herefore a reamen of ranslaion exposure is beyond he scope of his paper. Transacion exposure arises when funds are acually convered from one currency ino anoher a prevailing FX raes. The respecive gains and losses are no longer nominal, bu hey are realized 15/15

16 gains and losses and, herefore, fully affec he income of he company. Transacional gains and losses mus be compued wih respec o some reference FX rae. For a foreign-denominaed invesmen, his would usually be he exchange rae a he ime when he invesmen was made. Esimaing ransacion exposure and sudying ways o anicipae i is he main aim of his paper. Economic exposure is he impac ha changes in FX raes can have on he compeiive posiion of he company in he marke. Economic exposure is very relevan for manufacuring companies, as he prices of heir producs are mainly made up by coss (labour, maerial, infrasrucure). For (re)insurance companies, economic exposure is no so imporan, since expenses represen only a small par of he price of he policies, and he funds for covering risk can a leas parly be moved from one counry o anoher if necessary. Besides he exposure o he FX raes hemselves, here arises also addiional exposure from he fac ha here are differen ineres and inflaion raes for each counry. This may, for insance, lead o beer invesmen reurns in some counry, bu hese reurns may in urn be compensaed by an adverse developmen of he respecive FX rae. As we have seen previously, here exis correlaions beween ineres raes, inflaion and FX raes, bu hese correlaions vary from case o case, and i is herefore no possible o make quaniaive saemens on he resuling overall exposure only based on analyical reasoning. Again, DFA is a good means o come o more insigh in a given case. 4.2 Specific issues in insurance and reinsurance Mos of he conceps for FX risk managemen mainly apply o banks or oher invesmen companies. An insurer or reinsurer is faced wih some specific issues. On he asse side (re)insurers are ofen subjec o regulaory consrains which preven hem from aking he posiions ha would bes cover heir needs (a Swiss insurer for insance is only allowed o hold a maximum of 20% of foreign-denominaed bonds). Given he usually high proporions of bonds in heir porfolios, (re)insurers are highly dependen on ineres raes. Since (re)insurers are ofen only profiable because of invesmen reurns, his issue mus no be negleced. The liabiliy side of an insurance company is much more complicaed han he one of a bank: There are ofen very large flucuaions in he loss developmen process, hink e.g. of a Ca X/L ha produces no claims in mos years and large claims in a few years. More uncerainy arises from he fac ha loss projecions are ofen difficul due o insufficien daa and due o he impossibiliy o foresee cerain facors ha affec loss developmen (e.g. changes in liabiliy legislaion). All hese effecs are increased by he fac ha ime horizons are ofen very long, e.g. up o 10 or 20 years. Hence, unlike banks, (re)insurers are faced wih a high uncerainy as o amoun and iming of heir liabiliies. 16/16

17 In a muli-currency seup, hese issues concerning asses and liabiliies become even more difficul o quanify since several inflaion and ineres rae regimes, as well as FX raes, and all he relaed correlaions are involved. An insurance or reinsurance company operaing in a foreign marke generally holds asses as well as liabiliies in he respecive currency, which gives he possibiliy of offseing liabiliies in one currency by asses in he same currency, hus reducing cross-currency cash flows and hence ransacion exposure. This is one form of currency hedging and will be he subjec of he nex secion. Noice however ha he offseing effec on he ranslaion exposure is limied due o special accouning rules (see Bailey and Wild (2000) for deails), even in case of foreign subsidiaries ha are legal eniies in he respecive counry. Moreover, in his case local regulaions applying o he subsidiary may force a company o allocae is capial sub-opimally. Le us now consider a ruly inernaional seup where a company runs business direcly in a foreign counry. Premiums are colleced, claims are paid, and invesmens are made eiher in he company s home currency or in a foreign currency. Depending on he seup of inernaional (re)insurance conracs, he risk emanaing from he FX volailiy is borne by differen paries. We illusrae our reasoning by he following lile example: Think of a UK cedan (calculaing in GBP) ha concludes reinsurance conracs wih a US reinsurer (calculaing in USD). A he presen ime = 0 when he conracs are concluded, he exchange rae is a 1.5 USD per GBP. A he ime = 1 when he losses will occur, he exchange rae may be eiher a 1.2 or 1.5 or 1.8 USD per GBP, each wih a cerain probabiliy. We consider he impac on he cash flows of cedan and reinsurer under differen conrac seups: i. Everyhing is quoed and seled in he currency of he cedan (here: GBP): ii. Rae: Claim Conrac: 10 XS 10 denominaed in GBP (GBP) Cedan receives (GBP) Reinsurer pays (USD) Whereas he cedan receives he same amouns irrespecive of he exchange rae prevailing a = 1, he liabiliy of he reinsurer changes, i.e. he reinsurer bears he full currency risk. I is easily verified ha he same applies also o a ground-up loss or a x% - quoa share. Everyhing is quoed and seled in he currency of he reinsurer (here: USD): Rae: Claim Conrac: 15 XS 15 denominaed in USD (GBP) Cedan receives (GBP) Reinsurer pays (USD) /17

18 iii. In his case he cedan as well as he reinsurer bear a currency risk. In paricular, wheher or no he conrac riggers is now also dependen on he exchange rae. I is easily verified ha for a quoa share, only he reinsurer would bear currency risk. Fixed raes can be agreed a which all losses will be valuaed or paid. As an example, loss amouns are ranslaed a 1.5 USD per GBP, paymens however are made a currenly prevailing raes, in which case he full exchange rae risk is borne by he cedan: Rae: Claim Conrac: 15 XS 15 denominaed in USD, valuaed a 1.5 USD/GBP (GBP) Cedan receives (GBP) Reinsurer pays (USD) In any case here exiss an addiional risk due o he addiional volailiy of he FX rae, and his risk mus be borne by someone a some price. A deailed consideraion of he pricing of FX risk or he implicaions of FX volailiy on calculaions of risk adjused capial is, however, beyond he scope of his paper. FX raes can also ener indirecly ino an insurance conrac. This is he case when a conrac is concluded in he common home currency of cedan and reinsurer, bu for a claims process ha depends on FX raes, e.g. a reinsurance conrac for a primary insurer ha wries business in a foreign currency. 4.3 Bringing FX risk exposure ino a DFA model A horough reamen of how exacly o model FX risk exposure in a DFA model is no possible a his poin, as DFA models are usually very complex and he aachmen poins for he FX raes differ from case o case. As an example, one may bear in mind he Dynamo model described by D Arcy e al. (1997). We resric ourselves here o sae some general rules and principles mainly for ransacion exposure. The implemenaion of a ranslaion exposure model is differen, depending on he accouning sandard in use, and highly complex due o he complicaed rules for redirecing gains and losses o special equiy accouns. The modelling of economic exposure, which would have o go in line wih he modelling of insurance and reinsurance business cycles, is no reaed here. The principle for modelling ransacion exposure is simple: Each cash flow crossing he currency border (and only he cash flows, bu no foreign-denominaed posiions ha are consolidaed ino some balance shee) mus be muliplied by he respecive FX rae prevailing a he ime he cash flow occurs. The concree implemenaion is highly dependen on he DFA model used and on he srucure of he modelled company. Applicaion of he above-saed rules corresponds o doing a DFA sudy ha simply akes ino accoun FX flucuaions. One migh, however, also be ineresed in measuring he porions of risk and reurn emanaing specifically from FX flucuaions in some given period. This can be 18/18

19 achieved by generaing wo ses of DFA resuls for ha period: one wih consan FX raes equal o he iniial values bu oherwise sochasic inpus, and one wih sochasic FX raes. A word of warning for his approach: If FX raes are kep consan, hen he variables correlaed wih hem (i.e. ineres raes and inflaion) mus be simulaed according o heir condiional probabiliy law given he fixed value of he FX rae. Oherwise, he generaed scenarios may become implausible. By doing several simulaions wih FX raes kep consan on some hypoheic levels and inflaion and ineres raes simulaed according o he condiional probabiliy given he FX raes, one can explore he sensiiviy of he resuls agains FX rae levels under oherwise equal circumsances. This is a dynamic version of he well-known scenario esing approach. Classical scenario esing would also be a possibiliy, bu given ha we have five dependen variables per currency pair, he number of scenarios may quickly become raher high. Finally, one migh also do saisical analysis of he oupu values of he simulaion agains he respecive values of he inpu scenarios in order o make inference abou sensiiviy o FX raes. This approach is largely dependen on he drill-down capabiliies of he DFA sofware used. 4.4 Simulaion resuls The following resuls from simulaions wih our example inroduced in Secion 2 show he impac ha FX raes and heir volailiy have on he claims as experienced by he Home Office in is reporing currency USD. The firs resul shows he hisorical impac of he FX raes on he consolidaed loss developmen: 18'000'000 16'000'000 14'000'000 12'000'000 10'000'000 8'000'000 6'000'000 4'000'000 2'000'000 - Average Losses Convered a 1983 FX Raes Average Losses Convered a Acual Hisorical FX Raes Figure 4: Hisorical impac of FX rae flucuaions (cumulaed) The (sochasic) loss amouns of he differen lines of business were convered ino USD according o wo differen deerminisic exchange rae regimes: In he firs case (doed line) he losses of each year were convered a he FX raes of 1983, hus excluding any kind of FX rae flucuaion. 19/19

20 In he second case (solid line) he same loss amouns of each year were convered a he hisorical FX raes ha prevailed in he respecive years. Hence, he difference beween he wo lines shows he acual hisorical impac ha he changing FX raes had on he cumulaed consolidaed losses as experienced by he head office in is reporing currency USD. I is easily seen ha he difference in ulimae loss amouns is quie considerable: USD 16.82M under he acual FX rae developmen agains USD 14.79M under consan rae. In he presen case, he resuls from he hisorical FX raes are higher han he ones under he consan rae, since he USD underwen a considerable depreciaion wih respec o GBP, JPY, and CHF during he considered period (e.g.: 1 CHF = 0.48 USD in 1983, bu 1 CHF = 0.67 USD in 1999). On he conrary, an appreciaing USD wih respec o he oher currencies would have had a favourable effec on he resuls of he company. Nex, we compare projecions of fuure losses (same sochasic simulaion for he losses as before) convered a deerminisic FX raes and convered a sochasically simulaed FX raes. 14'000'000 12'000'000 Ulimae Losses Paid in USD a Spo Raes Ulimae Losses Paid in USD a Todays FX Raes Ulimae Losses Paid in USD a 'Fair' FX Rae 10'000'000 8'000'000 6'000'000 4'000'000 2'000'000 - Swizerland UK Japan Figure 5: Ulimae losses under deerminisic and sochasic FX raes The figure shows he ulimae losses per line of business in USD, where he conversion of he yearly paymens ino USD was made according o hree differen FX rae regimes (he solid black bars indicaing a confidence inerval of plus/minus one sandard deviaion): FX raes sochasically simulaed according o he boom-up modular model inroduced in Secion 3 (referred o as Ulimae Losses Paid in USD a Spo Raes in he legend), wih he values of year 2000 as iniial values. FX raes deerminisic and consan over ime wih he acual values prevailing in he year 2000 ( Today s FX Raes ). 20/20

21 FX raes deerminisic and consan over ime wih arificial values such ha he mean ulimae losses become equal o he ones under he sochasic simulaion ( Fair FX raes ). This approach is unrealisic, bu allows bes o compare volailiies wih he case of sochasic FX raes. For UK and Swizerland, he esimaions for he means of he ulimae losses provided by he simulaion wih sochasic FX raes are quie differen from he ones provided by he simulaion wih consan FX raes equal o he values of he year This is due o he fac ha he acual raes in 2000 for CHF and GBP are below he long-erm mean of he raes generaed by he simulaor. For JPY, he year 2000 rae is close o his mean, and hence here is no big difference. The confidence inervals (indicaing he esimaed volailiy of he ulimae losses) are larger for he case of sochasic FX raes han for he case of deerminisic ones, bu he difference is no always very big, i.e. in some cases (UK, see below) he inheren variance of he claims process is considerably higher han he exra variance added by he FX rae flucuaions. To ge a clearer view of hese differences, we give here a diagram of he 99 h perceniles of he losses minus heir expecaion belonging o each of he above simulaion resuls: 3'500'000 99h Percenile in USD a Simulaed Spo Raes 99h Percenile in USD a Todays FX Raes 99h Percenile in USD a Unbiasing Consan FX Raes 3'000'000 2'500'000 USD 2'000'000 1'500'000 1'000' '000 - Swizerland UK Japan Figure 6: 99h Perceniles of (Losses - Expeced Losses) The picure here is also no uniform: In cerain cases, he added volailiy is quie considerable (Japan), whereas in oher cases (UK) i is lower (bu sill above 5%). From hese invesigaions one can conclude ha FX flucuaions definiely have an impac on amouns and volailiies of he losses, bu he impac can be quie differen in size. Comparing simulaions wih deerminisic and sochasic FX raes provides in any case a good feeling for he order of magniude of he impac. 21/21

22 5. FX risk managemen sraegies 5.1 General sraegies Underwriing liabiliies denominaed in a foreign currency is almos daily business for reinsurers. Ye, hey need o measure heir performance in erms of heir home currency and are hus exposed o FX risk. We sar here by inroducing general mehods for he managemen of FX risk ha apply o any company doing foreign business (see Luca (2000) and Loderer and Pichler (2000) for more deails), and pu hem ino he conex of (re)insurance in he nex secion. There are many ways o deal wih his ype of risks, here are brief descripions of some of hem: Avoid he occurrence of FX risk by no doing business abroad. This may be an opion if FX risks urn ou o be unconrollable, bu no in general. Accep he FX risk and do nohing. This may be an opion if exposure is sufficienly low or if he reay is wrien in a counry wih a weaker currency han he reinsurer s. Diversify risk by doing business in differen counries such ha he respecive FX risks are uncorrelaed or even negaively correlaed. Implemenaion of his approach will usually be difficul as business volume in differen counries canno be fully conrolled bu depends on demand and oher collaeral facors. Transfer risk o he cusomer by denominaing he conrac in home currency or by fixing a FX rae for all ransacions relaed wih he conrac. The former is a full ransfer of he FX risks o he cusomer, he laer a parial ransfer; check he examples in secion 4.2 o ge a feeling. Noice however ha his kind of risk ransfer is already an implici forward conrac ha will come a a price. Transfer risk o hird paries: buy currency derivaives (fuures, forwards, swaps, or opions), or buy insurance of FX risks. These mehods are called off-balance shee hedging. The effeciveness of his approach for large players sars, however, o be quesioned, see Lyon (2001) for he example of a large inernaional commodiy rader. Reduce likelihood and severiy of losses due o FX flucuaions by doing offseing ransacions. I.e. ake sufficien asse posiions in he foreign currency a he ime (and FX rae) he conrac is inceped and pay he liabiliies from hese posiions, hus avoiding or reducing cash flows across he currency border ha are exposed o FX risk. This approach is also called on-balance shee hedging, see Dacorogna e al. (2001) for more deails. The widely held wisdom in he (re)insurance indusry is ha foreign exchange exposure is no a problem as long as he reinsurer mainains asses in he currency in which he liabiliies are denominaed, i.e. does on-balance shee hedging. In principle, his is rue, alhough such a sraegy migh be difficul o follow in pracice and no opimal as i is equivalen o fully hedging he posiion. For invesmens, i has been shown by Froo (1993) ha a horizons of several years, complee hedging no only does no lower reurn variance, i acually increases i for many porfolios. In anoher conex, Levich and Thomas (1993) analyze he impac of acive currency 22/22

23 managemen on inernaionally diversified porfolios and show ha i can significanly improve reurns wihou overly affecing he risk. Dacorogna e al. (2001) show how i is possible o separae he differen aspecs of he invesmen sraegies and o rea he exchange exposure by iself. In he financial indusry he foreign exchange risk is no only an objec of sudy bu also a pracical problem leading o financial producs. There are firms like Pareo Parners ha offer specially designed currency overlay programs o proec foreign invesmen in an inernaionally diversified porfolio. 5.2 Special issues in insurance and reinsurance Unlike in inernaional banking and invesmen or commodiy rading, insurers and reinsurers are faced wih liabiliies (i.e. claims) ha are sochasic in amoun and iming, i.e. here may be considerable uncerainy on when o pay and how much (see also Secion 4.4 o ge a feel). Moreover, (re)insurance conracs may imply very long ime horizons for paying, e.g. en up o weny years. Finally, (re)insurance companies are subjec o regulaions ha may preven hem from placing heir asses in he way hey wan. These specific facors have an impac on some of he general FX risk managemen sraegies se forh in he previous secion: Taking ou currency derivaives (fuures, forwards, opions, swaps) becomes difficul due o he iming uncerainy as one does no know exacly for wha mauriy and for wha amoun o buy hem, which can resul in considerable mismaches. Exchange-raded FX derivaives are only available for ime horizons up o one or wo years, and i will also be difficul o ake ou overhe-couner (OTC) producs for longer imes o mauriy. Hence, he ime horizon of many (re)insurance liabiliies is oo long for proecing hem wih FX derivaives. On-balance shee currency hedging also becomes more difficul: One does no know how much o inves and a wha duraion in order o mach he liabiliies. One may inves oo much (a possibly sub-opimal reurns), or oo lile (such ha funds mus be neverheless ransferred across he currency border). If one sricly maches all liabiliies in foreign currencies by sufficien amouns of asses in he same currency, his may also lead o a silo effec and o he use of more risk based capial (which also has a cos) han would be necessary under a more flexible currency hedging regime. Moreover, regulaory consrains may preven he (re)insurer from aking cerain hedge posiions. Finally, an insurer is also an invesor ha mus inves is asses a an opimal rae of reurn, and invesmens in cerain currencies (dicaed by hedging requiremens) may lead o sub-opimal reurns in cerain ime periods. To ge a feeling for he working of on-balance shee hedging under sochasic liabiliies, we revisi our simple example from Secion 4.2 and consider he case of a 10 XS 10 conrac denominaed in GBP where he reinsurer bears he full currency risk. In order o (parly) hedge is possible liabiliy, he reinsurer opens a posiion of 5 GBP a = 0 and a 1.5 USD per GBP, hence he 23/23

24 value a = 0 of he hedge posiion is 7.5 USD. A = 1 when he loss occurs, he USD/GBP rae is a 1.2, 1.5, or 1.8. The balance of he reinsurer afer seling he claim looks as follows: Rae: Claim Paymen Hedge Accoun (GBP) (GBP) Balance (GBP) Reinsurer s Balance (USD) If he conrac does no rigger (claims of 5 and 10 GBP), hen he reinsurer has a gain or loss on is unused deposi depending on he FX rae prevailing a = 1. If he claim amoun is 15 GBP (i.e. he paymen is 5 GBP), he hedge accoun exacly covers he loss and no funds ransacion over he currency border akes place. If he full loss occurs (i.e. 10 GBP paymen), hen he money on he hedge accoun is no sufficien and anoher 5 GBP mus be ransferred a prevailing FX raes, leading o ransacion gains or losses wih respec o he original rae of 1.5. The example can easily be adaped for oher hedge amouns (e.g. full hedge or no hedge). In every case, due o he uncerainy on he acual amoun of he claims paymen, here is a poenial for gains or losses due o FX rae changes. 5.3 On-balance shee hedging conceps On-balance shee currency hedging means ha a (re)insurer who has o cover liabiliies of value s in a foreign currency invess a hedge amoun of s h in he respecive currency in order o cover he liabiliy, i.e. in order o be able o pay (a leas a par of) he claims wihou funds ransacions over he currency border. The conceps presened in his secion come from Dacorogna e al. (2001). The hedging raio is defined as h= s / s and denoes he porion of he oal liabiliy h covered by he hedge amoun. In a muli-period seup, s denoes he oal liabiliies from now up o some fixed ime horizon. The deerminaion of he liabiliy amoun s can already be a problem in he conex of (re)insurance: If he maximum liabiliy amoun is known (e.g. in case of an a XL b conrac), here is no problem and s would be se o his maximum liabiliy amoun (e.g. s = a ). In many cases, however, he maximum liabiliy amoun is unknown, as in our simulaion example. Possibiliies o deal wih his siuaion include: Se s equal o he x% (e.g. 99%) quanile of he loss disribuion and express s h in erms of his s. The advanage here is ha he hedge amoun s h can be expressed as a percenage of a fixed amoun s. The disadvanage is ha here exiss a residual risk ha is no covered by he hedging consideraions, i.e. he liabiliy amouns above he x% quanile. Direcly ake sh as he y% quanile of he loss disribuion and do no rea s explicily. The disadvanage here is ha he hedge amoun is no expressed wih respec o some fixed maximum, bu he advanage is ha here is no residual o be reaed separaely. 24/24

25 Oher conceps, e.g. based on expeced loss plus some securiy loading, or on PML would also be possible. We find however ha he conceps based on he quaniles of he loss disribuion are he mos objecive ones. The simples sraegy is saic hedging. This sraegy consiss of aking posiions in he foreign currency a incepion of he conrac and keeping hem unil he end wihou adding or wihdrawing funds (excep for acually paying liabiliies and unless he liabiliies exceed he funds and he negaive balance mus be covered by ransfer paymens). In pracice his could be o place he necessary reserves in he currency in which he claims have o be paid. There are wo ways of implemening such a hedge: eiher by covering fully he liabiliies as hey are expeced o occur (i.e. sh = s ), his is called full hedging, or by covering only a par of he liabiliies wih an invesmen in he respecive currency, his is called parial hedging (i.e. s h < s ). A porion of h 100% of each claim would hen be paid from he hedge amoun, and he res from funds ransferred across he currency border. In a muli-period seup, i is also possible o readjus he hedge amoun s h for each ime inerval depending on he acual course of evens and changed expecaions of he fuure. This approach is called dynamic hedging. Funds can be added o or removed from he foreign invesmen, or he porion of he liabiliies of a cerain ime inerval ha is paid from local funds can be adjused dynamically depending on he acual developmen of he FX rae. As can be seen from he simple example in he previous secion, i can be worhwhile for our US reinsurer o pay is GBPdenominaed liabiliies wih USD funds (alhough a sufficien GBP amoun would be in place), if he GBP has depreciaed wih respec o he USD. Besides hese forms of hedging, i is possible o combine a parial saic hedging wih a dynamic hedging sraegy designed o ake profi of he FX movemens. The ideal siuaion would be o only hedge he FX risk when he rae moves in an unfavorable direcion and o fully profi from he movemen in he oher direcion by no hedging. Of course, he problem remains o deermine in advance when he movemen will occur in he righ direcion. In all cases, i is essenial o find mehods o deermine an opimal value for he hedge amoun: once and for all imes in he case of saic hedging, and for each period afresh in he case of dynamic hedging. These mehods mus ake ino accoun he possible developmen of he liabiliies and of he drivers of he exposure: inflaion and ineres raes in boh counries. This can be done by using a DFA model as will be shown in his paper. The hedge raio will be deermined by considering he reurn and he risk of he enire porfolio and choosing he soluion which is he closes o he efficien fronier. The opimizaion of he hedge raio can be made for each currency separaely (his is single-currency hedging), or for all currencies simulaneously (muli-currency hedging), hus rying o achieve an overall opimum. Opimaliy can be defined in differen ways: In he mos basic case, he only goal is o service he liabiliies 25/25

26 reliably, in oher cases, his may be superimposed by profiabiliy requiremens for he invesed funds. When opimizing hedge amouns (saic and dynamic case) consrains mus also be aken ino accoun. This can be posiion limis for foreign invesmens, or in more advanced applicaions duraion maching issues beween liabiliies and invesed asses. Paricularly when designing dynamic hedging sraegies, one should also be aware of ransacion coss. These are usually low for FX ransacions, bu may neverheless consume a par of possible benefis when ransacions occur very frequenly. 5.4 Modelling issues As a general prerequisie for modelling currency hedging, he DFA model mus be enabled for inernaional modelling, i.e. a scenario generaor for FX raes and relaed variables (as explained in Secion 3), a model for invesmens in differen currencies, and a model for a leas FX ransacion exposure (as explained in Secion 4) mus be in place. Modelling of saic (full or parial) hedging is simple. The liabiliy and hedge amouns can be deermined prior o he simulaion. A preliminary DFA simulaion wih only he loss model can be used o obain he disribuions of he liabiliy amoun and oher variables ha may be used o deermine he hedge amoun. The deerminaion of he hedge amoun can be done ouside he DFA model (along he lines presened in he previous secion), hence no specific exensions of he DFA model are necessary. Modelling of cash flows from asses posiions o pay claims is basic DFA funcionaliy. Modelling of dynamic hedging is simple as long as he hedge amouns are no consan over ime, bu deermined prior o he DFA simulaion (and hence independen from he acual developmen of he respecive variables in some given scenario). This approach makes sense if one models clear rends in some of he underlying inpu variables. Think e.g. of a US reinsurer who expecs he GBP o depreciae in he subsequen years. Then i would make sense o decrease he hedge raio accordingly over ime, no based on he concree scenario bu only on he drif in he expecaions of he scenario generaor. Modelling of dynamic hedging becomes difficul if he hedge amouns (or hedge raios) are adapively adjused depending on he acual developmen of he losses or FX raes. In his case, he basic inernaional DFA model mus be exended wih respecive dynamic adjusmen funcions. A simple example would be a hedge raio ha is made proporional o he changes in he underlying FX rae. In more complex cases, his can also include he compuaion of condiional expecaions and embedded opimizaion funcions, in paricular in he case of mulicurrency hedging. In his conex, if one considers foreign currencies also as a mean for obaining rading gains, hen a model for he revenues of a rading sysem mus be in place (see Dacorogna e al. (2001) for more informaion). 26/26

27 5.5 Simulaions In his secion we invesigae saic parial on-balance shee hedging sraegies wih differen hedge raios for our example. To his end we consider our simulaion seup as a full loss porfolio ransfer ha mus be funded appropriaely. The disribuions of he loss amouns in local CHF GBP JPY currency ( L, L, L ) and of he consolidaed oal loss amoun in USD ( L USD ) were obained by simulaions as shown in Secion 4.4. Conversion of losses from local currency ino USD was done a sochasically simulaed FX raes. The reference liabiliy amouns s for hedging (as inroduced in Secion 5.3) were defined o be he expeced loss plus sandard deviaion for each LOB (in local currency), i.e. CHF CHF CHF GBP GBP GBP JPY JPY JPY s = E L + σ L, s = E L + σ L, s = E L + σ L. The oal funding amoun in USD ha is available a ime 0 for covering he loss porfolio was deermined in he same way: A ime 0, he amouns CHF h s, USD USD USD s = E L + σ L. GBP h s, JPY h s were invesed in CHF, GBP, and JPY USD respecively (convered a he iniial FX rae). The remainder of he oal capial s was kep in USD. h denoes he hedge raio as inroduced in Secion 5.3. All invesmens were simulaed as porfolios of a cash accoun and bonds wih differen imes o mauriy. The cash flows of he invesmens were mached wih he expeced cash flows of he liabiliies. During he simulaion, all ransfer paymens from USD ino he oher currencies were convered a curren (simulaed) FX raes, and a he end all foreign-denominaed invesmens were convered back ino USD a he final FX raes. The following figure shows he ne presen value in USD of he loss porfolio hedged in he described way for various hedge raios: Mean and Sandard Deviaion of NPV Disribuion Mean / SDev in USD (000) Mean Sigma Hedge Raio Figure 7: NPV of hedged loss porfolio The figure shows very clearly ha neiher full hedging nor no hedging is he opimal sraegy for he given seup. A hedge raio of abou 0.7 yields he highes reurn wih a relaively low risk. There are differen facors ha may have conribued o his resul: 27/27

28 US ineres raes are considerably higher han he ones in Japan and Swizerland and abou equal o he Briish ones. Hence, invesmens kep in USD generae higher reurns han invesmens in CHF and JPY, and invesmens in GBP do no yield much more han US ones. None of he hree foreign currencies have a clear upward or downward rend wih respec o he USD. Therefore, in some cases i is favourable o keep he money in USD, in oher cases i is favourable o have he money in foreign currencies. On he average, a mixed porfolio wih invesmens in USD as well as in foreign currencies (i.e. parial hedging) seems he mos reasonable. This is, of course, only one example, and resuls may look differen for oher loss porfolios and oher involved currencies. Bu he resuls here show ha i is definiely worhwhile o consider parial currency hedging insead of full currency hedging. Reurns from invesmens in foreign currencies could be furher opimized by using dynamic currency overlay sraegies as described in Dacorogna e al. (2001). The implemenaion of such sraegies in he simulaion model was, however, well beyond he scope of his sudy. 28/28

29 6. Reinsurance cover Three reinsurance sraegies for our example have already been reaed implicily by he simulaions in Secions 4.4 and 5.5: Reenion of he whole porfolio by he cedan, loss porfolio ransfer (an ART ransacion), and a quoa share where he cedan reains only a par of he whole loss porfolio (and where all consideraions apply o he cedan and o he reinsurer for heir respecive share of he loss porfolio). We have seen for hese cases ha, indeed, FX rae flucuaions bring addiional volailiy ha has o be deal wih, and ha careful selecion of he hedge raio can have a considerable impac on risk and reurn of he loss porfolio. We complemen hese cases here by he consideraion of a non-proporional reay. We inroduce a USD 20M XS 12.5M Sop Loss reay on he consolidaed losses of he porfolio in USD, cumulaed over ime. I.e. cumulaed losses above USD 12.5 Million and up o USD 32.5 Million would be paid by he reinsurer. For his case we have a look a expeced claims and heir volailiy in he layer of he reinsurer: Expeced Paymens in USD as % of Ulimae 30.00% 25.00% Sochasic Model for FX Raes FX Raes fixed a odays Spo Rae % of Ulimae 20.00% 15.00% 10.00% % of Expeced Paymen in ha Year 5.00% 0.00% 14.00% 12.00% 10.00% 8.00% 6.00% 4.00% 2.00% 0.00% / Sandard Deviaion of Paymens Sochasic Model for FX Raes FX Raes fixed a odays Spo Rae Figure 8: Mean and sandard deviaion for paymens in R/I layer The paymen figures here are given as percenages of he ulimae loss of he porfolio, knowing ha he laer is differen for fixed and sochasic FX raes. In he firs years, he conrac does no

30 rigger a all under boh FX rae regimes. The differences in he levels and in he volailiy of he paymens are very considerable in he hird up o he sixh year, because here is no only he added volailiy of he FX rae iself, bu FX flucuaions have also an impac on wheher or no he conrac riggers a all. Moreover, in his ime span he absolue size of claims is sill quie high; see also Secion 4.4. For he res of he ime inerval he differences beween fixed and sochasic FX raes are no very high, since here is no longer uncerainy as o wheher or no he conrac riggers, and since he absolue size of addiional claims arising in hese years is lower. Looking a he ulimae cumulaed paymen amoun of he reinsurer, he difference in volailiy is quie low: he risk raio (i.e. sandard deviaion divided by mean) of he reinsurer s paymens is 26.2% a fixed FX rae as opposed o 27.8% under sochasic FX raes. Hence, because of he srucure of he claims and because of he long duraion of he conrac, he difference in volailiy for he reinsurer is no very high, bu he changes in he FX rae add considerable uncerainy o he iming of he paymens in he ime period when he FX rae has an impac on wheher or no he conrac riggers a all (a phenomenon ha is well known for X/L reaies in he case of claims inflaion). This fac may be imporan for he reinsurer when srucuring he hedge accoun from which o pay he claims. The added volailiy due o uncerainy wheher or no he conrac riggers would be higher in he case of a series of individual X/L reaies for each year, because in ha case conrac riggers could arise each year and no only once for he whole period. On he absolue scale, he mean ulimae cumulaed paymens of he reinsurer amoun o USD 4.46 Million if he FX raes are kep consan a iniial values. Under sochasic scenarios for he FX raes his increases o USD 5.46 Million. We now apply exacly he same currency hedging approach as in Secion 5.5 o he claims of he reinsurance layer. Be aware ha he siuaion for he reinsurance company is now slighly differen. The presen reinsurance cover is fully denominaed and payable in USD (which we assume o be he reinsurer s home currency); we only have a dependence of he claims process on he FX raes. The same approach could, however, also be worked ou for a foreign-denominaed reinsurance conrac where he FX raes ener more direcly. The figure below shows ha a low hedge raio (20%), or even no hedging, is clearly beer han a high hedge raio or full hedging. Recall ha for he whole loss porfolio he opimal hedge raio was found o be a 70%. 30/30

31 Mean / SDev in USD (000) Mean and Sandard Deviaion of NPV Disribuion Mean Sigma 100% 90.0% 80.0% 70.0% 60.0% 50.0% 40.0% 30.0% 20.0% 10.0% 0.0% Hedge Raio Figure 9: Hedge resuls of reinsurer The reason for his lower hedge raio may be ha he Sop Loss conrac does no cause any claims a all during he firs wo or hree years (see above), i.e. he full hedge amoun can be invesed during his ime period. As we have menioned earlier, he US ineres raes are considerably higher on average han he Japanese and Swiss ones, and no much lower han he Briish ones. Therefore, high US invesmen reurns over he firs years may ouweigh he FX risks for he laer years. 31/31

32 7. Assessmen and oulook Measuremen, modelling, and managemen of FX risks is a vas and difficul field, for inernaionally operaing companies in general and for insurers and reinsurers in paricular. This paper can only scrach he surface of he issue and, hopefully, give some inpu for furher research and pracical applicaion. The syle of his paper was kep informal, and many opics had o be simplified or omied in order no o overburden he presenaion. Here is a survey of he mos imporan findings: In scenario generaion, he modelling of dependences beween he FX raes and oher economic variables, ha could be significanly measured in daa analysis, is very imporan. In yearly daa here are simple and also economically defensible equilibrium relaions ha can be used for scenario generaion. On finer ime scales (quarerly and monhly) he siuaion is more complicaed, as dependences on pas values a differen lags become highly significan. There is no silver bulle for scenario generaion. On he yearly level, he presened boom-up modular approach provides a simple and easy-o-use soluion, which comes, however, a he price of decreased flexibiliy if one wans o incorporae deerminisic rends or simulae according o condiional probabiliies. The alernaives are purely saisical mulivariae ime series models, e.g. he VAR model as presened in his paper. Finding an inernaional economic model ha capures he behaviour and dependences of he relevan economic variables on he one hand, bu ha is also undersandable and fully racable by he acuarial and financial praciioner on he oher hand is clearly an imporan issue for fuure applied research. The approach o be chosen is likely o be raher saisical han economical, as he economic relaions beween he involved economic variables mainly apply on very long ime horizons. Afer all, even he bes scenario generaor leaves us wih some fundamenal uncerainy on he adequacy of he simulaed scenarios for he fuure. I is herefore imporan o complemen all simulaions wih sensiiviy analyses ha give a leas a hin for he exposure of he company o evens or developmens no covered by he generaed scenarios. FX flucuaions affec he business of (re)insurers in differen ways: hrough ransacion, ranslaion and economic exposure. Translaion exposure is difficul o model in DFA as i largely depends on he accouning sandard in use and was herefore no subsanially reaed in his paper. Exploring he deails of modelling ranslaion exposure in DFA under some imporan accouning sandards (in paricular IAS and GAAP) would be an ineresing subjec for more accouning-oriened researchers. Economic exposure was also only menioned here. A horough reamen of economic exposure due o FX rae changes would have o go in line wih research on economic cycles in insurance. Modelling of ransacion exposure in DFA was found o be relaively sraighforward, alhough dependen on he concree model in use. Our invesigaions show ha, indeed, FX rae changes can have significan effecs on he business of an insurance or reinsurance company, paricularly over long ime horizons and here is also an addiional 32/32

33 volailiy due o he flucuaions of he FX raes. Wheher or no hese addiional uncerainies mus be aken ino accoun will usually depend on heir quaniaive relaion o he genuine volailiy in he claims process. The message here is ha, whenever FX raes influence he business, one should no neglec hem a priori, because hey poenially have a quie dramaic impac on he business resuls of he (re)insurer. DFA-syle sochasic simulaion is one way of deermining he impac of FX rae flucuaions on he business resuls. An alernaive would be scenario esing wih differen fixed FX scenarios bu oherwise sochasic inpu variables. This can very well show he sensiiviy of he resuls agains FX rae changes, bu i does no yield informaion abou addiional volailiy due o FX rae uncerainy. There exiss a number of approaches for FX risk managemen. The respecive conceps mainly come from he area of banking and invesmen. In (re)insurance, some imporan specialies mus be aken ino accoun: liabiliies are sochasic in amoun and iming, ime horizons may be very long, and regulaory consrains may preven (re)insurers from implemening cerain sraegies. On-balance shee currency hedging in is various forms was idenified as he mos imporan universally applicable sraegy for FX risk managemen. Simulaions showed ha here exis siuaions where full currency hedging is clearly dominaed by parial hedging sraegies. Currency hedging in an insurance conex is more complex han in a banking conex because here is addiional uncerainy as o amoun and iming of he liabiliies o be hedged. Hedging consideraions can be based on differen approaches, e.g. mean plus sandard deviaions or also perceniles. Only he mos basic saic hedging sraegies were invesigaed here. Dynamic mulicurrency hedging sraegies (so-called currency overlay, see Dacorogna e al. (2001)) would bear a poenial for even more opimizaion, bu adapaion of hese mehods for he specific needs of insurance liabiliies is an open poin for furher research. Our findings for primary insurance apply o reinsurance as well. For he cases of loss porfolio ransfer and proporional reinsurance, everyhing is equivalen. In he case of non-proporional reaies (X/L and Sop Loss) FX rae flucuaions creae an addiional uncerainy as o wheher or no a layer riggers a all, wih he respecive impac on expecaion and volailiy of he claims and, hence, on he price for he cover. This paper has shown very informally ha FX flucuaions can bear dangers as well as opporuniies for an inernaionally operaing insurer or reinsurer, and i is undoubedly worhwhile o explore he downside as well as he upside of his risk. Furher, probably more analyical, research on he following opics could add considerable value: impac of FX rae volailiy on he pricing of inernaional reinsurance conracs and on he mehodology for risk based capial, in paricular: deerminaion of possible exra amouns of risk based capial o cover FX risks (in paricular: deerminaion of he sign of his amoun), general mehodology for he managemen of a porfolio of asses and (insurance) liabiliies in he presence of volaile FX raes. 33/33

34 Bibliography BAILEY G.T., K. WILD (2000): Inernaional Accouning Sandards: A Guide To Preparing Accouns. Second Ediion, abg Professional Informaion. BLUMSOHN G. (1999): Levels of deerminism in workers compensaion reinsurance commuaions, CAS Proceedings, Vol. LXXXVI, Par 1. CHAN K.C., G.A. KAROLYI, F.A. LONGSTAFF, A.B. SANDERS (1992): An empirical comparison of alernaive models for he shor-erm ineres rae, Journal of Finance, 47, CLEMENTS M., D. HENDRY (1998): Forecasing economic ime series. Cambridge Universiy Press. CLOSTERMANN J., B. SCHNATZ (2000): The deerminans of he euro-dollar exchange rae, Discussion paper 2/00, Economic Research Group of he Deusche Bundesbank ( DACOROGNA M. M., R. GENÇAY, U. A. MÜLLER, R. B. OLSEN, O. V. PICTET (2001): Inroducion o High Frequency Finance. Academic Press. D ARCY S.P., GORVETT R.W., HERBERS J.A., HETTINGER T.E., LEHMANN S.G., MILLER M.J. (1997): Building a public access PC-based DFA model, Casualy Acuarial Sociey. DAYKIN C.D., T. PENTIKÄINEN, M. PESONEN (1994): Pracical Risk Theory for Acuaries, Chapman and Hall, Dembo R., A. Aziz, D. Rosen, M. Zerbs (2000): mark-o-fuure, A Framework for Measuring Risk and Reward. Algorihmics Publicaions, ( EMBRECHTS P., C. KLÜPPELBERG, T. MIKOSCH (1997): Modelling Exremal Evens for Insurance and Finance. Springer, New York. EMBRECHTS P., A. MCNEIL, D. STRAUMANN (1999): Correlaion: pifalls and alernaives, RISK Magazine, May, FASB 52 (1981): Saemen No. 52: Foreign currency ranslaion, Financial Accouning Sandards Board ( FROOT K. A. (1993), Currency hedging over long horizons, NBER Working Paper No KAUFMANN R. AND PATIE P. (2000), Sraegic long-erm financial risks, Inermediae research repor, Risklab / ETH Zürich ( KIM J., MALZ A.M., MINA J. (1999): LongRun Technical Documen. RiskMerics Group ( LEVICH, R. M. AND L.R. THOMAS, L. R. (1993): The meris of acive currency risk managemen: evidence from inernaional bond porfolios, NBER Working Paper No LODERER C., K. PICHLER (2000): Firms, do you know your currency risk exposure? Survey resuls, Journal of Empirical Finance, 7, LUCA C. (2000): Trading in he Global Currency Markes. New York Insiue of Finance (Prenice Hall). LYON (2001): BHP drops derivaives, RISK Magazine, February, 13. MACDONALD R. (2000): Conceps o calculae equilibrium exchange raes: an overview, Discussion paper 3/00, Economic Research Group of he Deusche Bundesbank ( MÜLLER U. A, DACOROGNA M. M. AND PICTET O. V. (1998): Heavy ails in high-frequency financial daa in A pracical Guide o Heavy Tails, ed. by R. J. Adler, R. E. Feldman and M. S. Taqqu, Birkhäuser, Boson. 34/34

35 Appendix A: Economic daa The following able shows he yearly daa for he economic variables ha were used hroughou our sudy. All daa were obained from Daasream. R denoes he shor-erm ineres rae of he respecive counry in percen (annualized). The acual figures are inerbank money marke raes. I denoes he inflaion rae of he respecive counry in percen (annualized). The acual figures are consumer price inflaion raes. S denoes he spo exchange rae of he respecive currency, i.e. he spo price in USD for one uni of foreign currency (excep for Japan: USD price of 100 Yen). USA Unied Kingdom Swizerland Japan Year R [%] I [%] 6 R [%] I [%] S [$] R [%] I [%] S [$] R [%] I [%] S [0.01$] Figures for ineres raes and spo exchange raes were obained by aggregaing daily daa: In a firs sep, a monhly aggregae was generaed by aking he median of he daily quoes, hen yearly aggregaes were obained by aking he average of he monhly aggregaes. Inflaion was aggregaed by averaging annualized monhly inflaion figures. Below we give a series of chars ha presen he above-saed ime series in differen combinaions so as o give a feeling no only for he behaviour of he single ime series, bu also for heir relaions wih one anoher. The solid lines always denoe he FX rae, he dashed lines denoe he ineres raes, and he doed lines denoe he FX raes. The difference beween he dashed and he doed line is he real rae of reurn, which urned ou o be he mos significanly explanaory facor for he FX raes in he boom-up modular model, see also Appendix B. These chars also allow o check more sysemaically he findings of he saisical analysis presened in Secion Noice: The inflaion figures given here do no correspond o he ones usually used by US praciioners. We used here a CPI All Iems series from Daasream and averaged annualized monhly inflaions. Usual pracice is he US is o use he CPI Urban series from he Bureau of Labor Saisics and o compue he inflaion for a given year based only on he CPI values a he end of ha year and he previous year. Given he purely illusraive naure of our example, his deviaion from common pracice is, however, irrelevan. 35/35

36 Figure A 1: USD/GBP rae and is consiuens Remarks: The srong correlaion beween ineres raes and inflaion can be clearly observed for all counries. The real rae of reurn (ineres minus inflaion) is almos always posiive, bu by far no consan over ime, and here are also considerable differences beween he counries. Also, i can be observed ha he ineres raes of he differen counries are quie srongly correlaed wih each oher. Periods of high and low ineres raes occur quie simulaneously in all counries, alhough he absolue levels of ineres are raher differen from counry o counry, and also he relaive increases and decreases differ. The ineres raes of he US and he UK are generally higher han he ones of Swizerland and Japan, which may have an impac on invesmen decisions when invesmen reurns maer (see he example in Secion 6). All hree FX raes agains he USD show a similar paern in erms of increases and decreases, which shows he role of he USD as a world reference currency. Changes in FX raes from one year o he nex can be quie drasical: 20% and more. All series show very clear signs of auocorrelaion. 36/36

37 Figure A 2: USD/CHF rae and is consiuens Remarks (con d): In spie of is repuaion as a save haven currency, also he CHF makes raher srong moves wih respec o he USD, see also Luca (2000). The high degree of simulaneiy in all series suggess ha he relevan economic variables of he involved counries are driven by some common economic cycle. 37/37

38 Figure A 3: USD/JPY rae and is consiuens Remarks (con d): Ineres raes in Japan and also inflaion can aain remarkably low levels (close o zero), which can have a srong impac when hedge raios are compued and invesmen reurns play a role. 38/38

39 Appendix B: Model parameers We now presen he esimaed parameers for he boom-up modular model as we inroduced i in Secion 3.3. Refer he laer secion for a deailed explanaion of he formulae and parameers. i.) Ineres rae generaor r = r + a ( b r ) + s r ε XXX XXX XXX XXX XXX XXX XXX r, XXX USD CHF GBP JPY a b s r Correlaion marix for residual erms ε : USD CHF GBP JPY USD CHF GBP JPY ii.) Inflaion generaor i = β + β r + ε XXX XXX XXX XXX i, XXX 0 1 USD CHF GBP JPY β β σ(ε) iii.) FX rae generaor s = α + α r + α i + α r + α i + ε USD / XXX XXX XXX USD XXX USD XXX XXX XXX XXX s, XXX CHF GBP JPY α α α α α σ(ε) Noice ha for CHF and GBP we have applied he simplificaions α1 = α2 and α3 = α4 as hey led o significanly beer fis, i.e. he FX rae is modelled by he real raes of reurn of he involved counries, and he effecive number of parameers o be esimaed is reduced. For JPY, his simplificaion urned ou o be applicable only for US inflaion and ineres rae. 39/39