ON THE MULTI-PERIOD ECONOMIC RISK CAPITAL OF UNIT-LINKED LIFE PORTFOLIOS

Transcription

1 ON H MULI-PRIOD CONOMIC RIK CAPIAL OF UNI-LINKD LIF PORFOLIO Werner Hürlimann alue and Ris Managemen Winerhur Life and Pensions Posfach CH-8401 Winerhur wizerland el e mail werner.huerlimann@winerhur.ch Absrac. A mehod of evaluaion of he muli-period economic ris capial of a life porfolio defined as he value-a-ris or condiional value-a-ris measure associaed o he muli-period aggregae surplus is presened. I is applied o a porfolio of uni-lined endowmen policies wih guaranee. Using a simple bu reliable gamma approimaion of he urance ris i is shown ha he guaraneed uni-lined endowmen urance performs beer han he radiional endowmen urance on a ris-adjused reurn scale. he ris-adjused reurn on capial or RAROC ha is he epeced gain per uni of economic ris capial is greaer for he guaraneed uni-lined conrac han for he radiional one. Key words : aggregae surplus value-a-ris condiional value-a-ris RAROC uni-lined urance endowmen urance 1. Inroducion. he presen paper shows how he muli-period economic ris capial of a life porfolio can be evaluaed. Defining he economic ris capial as he value-a-ris or condiional value-aris associaed o he muli-period aggregae surplus of a life porfolio he las quaniy mus be firs deermined. For his he muli-period aggregae surplus is divided ino hree componens namely he invesmen ris he urance ris and he urance margin where each componen is modeled separaely. Our applicaion concerns he evaluaion of he economic ris capial of a porfolio of uni-lined endowmen policies wih guaranee and is value based comparison wih a porfolio of radiional endowmen policies. Using a simple bu reliable gamma approimaion of he urance ris componen i is shown ha he guaraneed uni-lined conrac performs beer han he radiional one on a ris-adjused reurn scale. Indeed he ris-adjused reurn on capial abbreviaed RAROC ha is he epeced gain per uni of economic ris capial is greaer for he guaraneed uni-lined produc han for he radiional one. A more deailed ouline follows. In ecion 2 we consider a simple sochasic model of a life porfolio in a muli-period seing which allows one o deermine is accumulaed aggregae surplus over a finie number of urance periods as well as is associaed economic ris capial. he aggregae surplus is decomposed in hree summands. he firs erm represens he possible random invesmen gain or profi in ecess of he guaraneed echnical ineres and is negaive value is called invesmen ris. he second erm represens he possible random urance loss as deviaion of he random accumulaed value of he aggregae claims in each period from is epeced value and is called urance ris. Finally he hird erm represens he epeced accumulaed urance surplus induced by he ris premium marg and is called urance margin. In

2 2 pracical evaluaions i is necessary o specify a model of he accumulaed aggregae claims and a disribuion for he random accumulaed reurn on invesmen. ecion 3 recalls he noion of economic ris capial (RC) derived from he well esablished value-a-ris (ar) and condiional value-a-ris (CaR) capial requiremens. We poin ou diversificaion effecs beween he urance and invesmen riss and show how he overall RC can be allocaed o he urance and invesmen componens using a simple covariance principle. ecion 4 surveys he analysis and pricing of uni-lined conracs wih guaranees whose main feaure is he inclusion of random benefis as opposed o deerminisic benefis in radiional life urance. Inspired by he developmens in Aase and Persson(1994) we offer an analyical reamen close o he radiional viewpoin which can be easily applied o he evaluaion of economic ris capial along he presened approach. ecion 5 illusraes he acuarial use of our resuls. In ecion 5.1 we show by eample ha he disribuion of he random muli-period aggregae loss of a life porfolio can be well approimaed using a gamma disribuion assumpion for he urance ris. Maing his approimaion we compare in ecion 5.2 he radiional endowmen urance wih he guaraneed uni-lined endowmen urance and obain an improved ris-adjused performance for he laer. 2. he aggregae surplus of a life porfolio. We consider a simple sochasic model of a life porfolio in a muli-period seing which allows one o deermine is accumulaed aggregae surplus over a finie number of urance periods. his quaniy is essenial for a life urance company because i is he main ingredien required o evaluae he economic ris capial associaed o life urance porfolios. Our simplified urance business model aes ino accoun he urance ris associaed o random urance claims and he invesmen ris associaed o random reurns on invesed capial. I is assumed hroughou ha he random claims and reurns are sochasically independen. he managemen ris associaed o adminisraion epenses and he aiude o ris are no aen ino accoun here. he echnical values of a radiional life porfolio can be summarized as follows (e.g. Bowers e al.(1986) p. 209) : R : he ris premiums in period [ 1 : he saving premiums in period [ 1 R + : he ne premiums : he acuarial reserves required a ime due a ime 1 due a ime 1 : he random aggregae claims in period [ 1 I : he random reurn on invesmen in period [ 1 i : he one-period consan echnical ineres rae r 1 + i : he one-period accumulaed echnical ineres rae 1 v r : he discoun rae due a ime Recall some well-nown relaionships. he ris premiums are equal o he discouned value of he epeced aggregae claims and he loading on aggregae claims (o fi ideas hin of he loading as a muliple of he sandard deviaion of aggregae claims). In formulas one has R ( [ + l[ ) v. (2.1)

3 3 he discouned reserves a ime consiss of he reserves a ime 1 and he saving premiums ha is v 1 +. (2.2) One is ineresed in he aggregae surplus of he life porfolio a a fuure ime ha is afer a number of urance periods. he random value of he liabiliy L a ime of he porfolio consiss of he acuarial reserves a ime and he random accumulaed value of he ha is aggregae claims in each period [ L + 1 r. (2.3) Discouned a he echnical ineres rae he presen value of he asses A 0 a ime 0 of he life porfolio consiss of he acuarial reserves and he discouned value of all fuure premiums ha is A v. (2.4) 1 he aggregae surplus a ime defined as difference beween asses and liabiliies is denoed and equal o A 0 R L where 1 accumulaed reurn on invesmen over he period [ inducion he acuarial reserves a ime R ( 1+ I ) represens he random 0. Using (2.2) one oba hrough r r. (2.5) 1 Using (2.1)-(2.5) he aggregae surplus can be rewrien as A0 ( R r ) r ( + 1 [ + 1 ) + r l [ + 1. (2.6) 0 0 inv he firs erm abbreviaed represens he possible random invesmen gain in ecess of he guaraneed echnical ineres and is negaive value is called invesmen ris. he second erm represens he possible random urance loss as deviaion of he random accumulaed value of he aggregae claims in each period from is epeced value and is called urance ris abbreviaed. Finally he hird erm represens he epeced accumulaed urance surplus a ime provided by he ris premium marg and is called urance margin abbreviaed M. For (2.6) one uses he following shor hand noaion + M. (2.7) inv In pracice i is necessary o specify a model of he accumulaed aggregae claims (e.g. Hürlimann(2001a) for an individual muli-period life model wih one cause of decremen) and a disribuion for he random accumulaed reurn on invesmen (e.g. a normal ellipical lognormal disribuion ec.).

4 4 3. Muli-period economic ris capial. One of he major problems faced by urance companies is he deerminaion of capial requiremen associaed o he urance and invesmen riss of porfolios of urance policies. he basic approach consiss o apply an appropriae ris measure ha aes ino accoun he shape of he profi and loss disribuion especially is righ ail. he well esablished value-a-ris (ar) capial requiremen (quanile reserve defined as percenile of he loss disribuion) may fail o be sudaddiive (and hus fail o simulae diversificaion) and does no ae ino accoun he severiy of an incurred adverse loss even. he encounered deficiencies are capured by he noion of coheren ris measure largely discussed in he recen lieraure (e.g. Arzner e al.(1997/99) Arzner(1999) Wirch(1999) Wirch and Hardy(1999) Delbaen(2000) esuri and Uryasev(2000) Acerbi(2001) Acerbi and asche(2001a/b)). Besides ar our coheren ris measure will be he condiional value-aris measure (CaR). he economic ris capial (RC) associaed o a loss random variable X wih disribuion funcion F X () using hese wo approaches is defined as follows : alue-a-ris (ar) ar [ X Q ( ) inf { F ( ) } X X his value is he maimum possible loss which is no eceeded wih he probabiliy (in pracice 95%99% ). ar is in general no subaddiive and no a coheren ris measure. Condiional alue-a-ris (CaR) CaR [ [ X X X ar [ X > he condiional epeced loss given he loss eceeds is value-a-ris represens he average of he % 100 wors losses in a random sample of losses. One has he useful formula CaR 1 [ X ar [ X + m [ ar [ X ar [ X + [ ar [ X ) ε where m X ) : [ X X > X ( is he mean ecess funcion X ( ) X ( ) mx ( ) is he sop-loss ransform X ( ) 1 FX ( ) is he survival funcion and ε 1 is inerpreed as loss probabiliy. ome properies of CaR which is a coheren ris measure are discussed in several recen papers (e.g. esuri and Uryasev(2000) Acerbi(2001) Acerbi and asche(2001a/b) Hürlimann(2001b) Kusuoa(2001) Rocafellar and Uryasev(2001)). According o (2.7) he random aggregae loss of a life porfolio a ime is equal o X + M (3.1) inv inv inv where A0 ( r R ) represens he possible random invesmen loss which is posiive in case he echnical ineres canno be guaraneed. o be able o cover a possible loss > 0 wih a high probabiliy an urance company borrows a ime 0 he amoun RC 0 [ called economic ris capial. A ime ineres is due a he periodic rae i C. o

5 5 guaranee wih cerainy he value of he borrowed capial a ime he amoun RC 0 [ is invesed a he periodic risless rae capial a ime is herewih RC i < i f C. he accumulaed value of he economic ris [ RC [ ( 1+ r r ) f C 0 (3.2) where one ses be evaluaed : r + f 1 i f r C + ic 1. Applying he above approaches wo funcionals mus RC RC [ ar [ [ CaR [ (value-a-ris approach) (3.3) (condiional value-a-ris approach). (3.4) o eamine and poin ou diversificaion effecs beween he urance and invesmen riss i is of grea imporance o loo separaely a he RC of he urance loss called urance economic ris capial and abbreviaed I-RC and a he RC of he invesmen loss called mare economic ris capial and abbreviaed M-RC. imilarly o (3.3) and (3.4) one has herewih he alernaive funcionals I I inv inv RC [ ar [ M RC [ ar [ inv inv RC [ CaR [ M RC [ CaR [ In general one posulaes he subaddiive propery RC inv inv [ I RC [ + M RC [ (3.5). (3.6) +. (3.7) he non-negaive difference in (3.7) represens a diversificaion effec in form of an ecess ris capial which is no more required in case of separae evaluaions of I-RC and M-RC. here eiss several ris allocaion principles which apporion he non-negaive diversificaion effec o he urance and invesmen ris componens (e.g. asche(2000) Delbaen and Denaul(2000) Hürlimann(2001c)). According o a simple covariance principle he allocaed ris conribuions I-RC* and M-RC* such ha RC * inv [ I RC [ + M RC [ M * (3.8) are deermined by he formulas I RC M RC Cov [ [ [ ( [ [ ) + RC * ar * ar [ inv inv inv Cov [ [ [ ( [ [ ) + RC [ (3.9). (3.10) he ineresed reader should noe ha (3.9)-(3.10) can be inerpreed as a CAPM lie principle. his general principle has been firs advocaed in Borch(1982/90) and can be generalized applying a leas four differen approaches as shown in Hürlimann(1998).

6 6 4. Analysis and pricing of uni-lined life urance wih guaranee. he analysis and pricing of uni-lined conracs wih guaranees whose main feaure is he inclusion of random benefis as opposed o deerminisic benefis in radiional life urance has been discussed in he lieraure since abou 25 years by many researchers in paricular Brennan and chwarz(1976/79a/b) Boyle and chwarz(1977) Corby(1977) Delbaen(1986) Delvau and Magnée(1991) Bacinello and Oru(1993) Aase and Persson(1994) Nielsen and andmann(1995) Kurz(1996). Inspired by he developmens in Aase and Persson(1994) we offer an analyical reamen of uni-lined conracs wih guaranee which is very close o he radiional viewpoin and hus can be easily applied o he evaluaion of economic ris capial following he approach in ecions 2 and 3. Our aenion focuses on combinaions of pure endowmen and erm urance conracs especially uni-lined endowmen conracs wih guaranee aga consan periodic premiums. For clearness he presenaion is divided ino four ubsecions Uni-lined conracs. he descripion is based on he following quaniies : : age a enry of a policy-holder s : age a epiraion of a conrac s : erm of a conrac : curren ime of valuaion () : guaraneed deah benefi a ime < ( ) : guaraneed benefi a epiraion of a conrac v : echnical discoun rae () : random mare value a ime of one share of he uni-lined reference porfolio N () : prescribed number of shares a ime of he reference porfolio included in he benefi j : epeced reurn of he reference porfolio over he ime horizon [ 0 σ : volailiy of he reference porfolio One observes ha he guaraneed benefi a ime is covered hrough shares of he reference porfolio provided one has N ( ) ( ) ( ). (4.1) However a ime 0 of underwriing he fuure value () of he reference porfolio is unnown. herefore for he purpose of pricing we assume ha he epeced value of he shares in he reference porfolio covers he guaraneed benefi ha is N ( ) [ ( ) ( ). I follows ha he prescribed number of shares is deermined by N( ) ( ) w w (1 + j) 1. (4.2) If he mare value of he shares in he reference porfolio eceeds he guaraneed benefi a deah or epiraion hen he difference is paid as bonus o he beneficiary of he conrac. he financial payoff a ime saisfies he relaionship

7 7 ma { ( ) ( ) ( ) } ( ) + ( N( ) ( ) ( ) ) + N. (4.3) Wih (4.3) he payoff is decomposed in a deerminisic paymen () and a sochasic paymen from a call-opion on he mare value of he shares wih eercise price () Pricing of he uni-lined endowmen conrac wih guaranee. he benefi decomposiion (4.3) suggess o represen echnical values of uni-lined conracs as sums of echnical values of radiional conracs wih deerminisic payoff () and of echnical values of conracs wih sochasic call-opion payoff ( N ( ) ( ) ( ) ) +. he endowmen urance is he superposiion of a erm urance and a pure endowmen urance. herefore each echnical value can be addiionally decomposed in a erm and pure endowmen componen. Following his procedure he wo-fold decomposiion of he relevan echnical values wih heir noaions are summarized in he ables 4.1 and 4.2. able 4.1 : Decomposiion of single premiums benefi radiional uni-lined call-opion guaraneed uni-lined urance erm urance pure endowmen endowmen I : P IC : PC I : P C able 4.2 : Decomposiion of acuarial reserves benefi radiional uni-lined call-opion guaraneed uni-lined urance erm urance pure endowmen endowmen I IC I P PC P C he remaining required echnical values for each componen are derived from he single premium wrien in general noaion as he acuarial reserves + 1 and he discoun rae v using he following formulas : : periodic ne premium (4.4) a&& : v 1 : saving premium in period [ 1 (4.5) R : ris premium in period [ 1 (4.6) R vq 1 (4.7) ( ) 1 : sum a ris in period [ +

8 Analyical formulas. he valuaion of he radiional urance conracs wih deerminisic payoff is classical and well-nown. Presen values of he uni-lined call-opion conracs are deermined using Blac-choles formula where he risless rae is se equal o he echnical ineres rae. I follows ha he value of he call-opion a ime wih payoff ( N ( ( ( ) + a ime u > abbreviaed C ( is given by (use he relaionship (4.2)) : u [ d1( ( v Φ[ d ( ) ( δ + σ )( u ) C( N( ( ) Φ 2 N( ( ln 2 u d u ( ) 1( ) σ u d2( d1( σ u δ ln( v). (4.8) A he ime 0 of underwriing he fuure value () is unnown. As in ecion 4.1 one assumes in formula (4.8) ha he fuure value a ime of a share of he reference porfolio coincides wih is epeced value ha is ( ) (1 + j). Under his assumpion he required echnical values are calculaed eplicily as follows : ingle premiums 1 : ( + 1) 0 p v I p q (4.9) + P ( ) (4.10) 1 : C(0 + 1) 0 IC p q (4.11) + PC p C( 0 ) (4.12) I : + P (4.13) : I : + C IC : + PC (4.14) I IC: (4.15) P P + PC (4.16) + C I : + P (4.17) Acuarial reserves I ( + 1) v p+ q P I 0 p v ( ) + P 1 ( ) P v 1 p I + v p + (4.18) (4.19)

9 9 IC 1 C( + 1) p+ q PC PC p I + IC 0 C( ) 1 PC v 1 IC v p [ 1 1 [( N( ) ( ) ( )) ( ) 2Φ( σ ) P + p + 2 (4.20) (4.21) + (4.22) + (4.23) C I IC I PC + (4.24) P P IC + (4.25) PC + + (4.26) I P 4.4. A numerical illusraion. o illusrae numerically he obained formulas consider a uni-lined endowmen policy wih guaraneed paymen ( ) for a man aged 50 a enry wih a erm of 10 years. o calculae single premiums and acuarial reserves we use second order probabiliies of deah (here 60% of he wiss Life able KM80) and a echnical ineres rae i 2.5%. he epeced reurn and he volailiy of he reference porfolio are assumed o ae he values j 7% and σ 15%. Numerical resuls for he various quaniies of ineres are summarized in he ables below. able 4.3 : ingle premiums produc erm urance Pure endowmen ndowmen radiional Uni-lined call Uni-lined guaranee able 4.4 : Annual ne premiums produc erm urance Pure endowmen ndowmen radiional Uni-lined call Uni-lined guaranee able 4.5 : Acuarial reserves I P IC PC C

10 able 4.6 : Ris premiums R I R P R R IC R PC able 4.7 : aving premiums I P IC PC able 4.8 : ums a ris I P IC PC R C C C R

11 11 able 4.9 : Number of shares and epeced call-opion paymen () ( N( ) ( ) ( )) N [ 5. radiional versus guaraneed uni-lined endowmen urance. he purpose of he presen ecion is wo-fold. Firs in ecion 5.1 we show by eample how he disribuion of he random muli-period aggregae loss of a life porfolio and he associaed ar and CaR measures can be eacly evaluaed and well approimaed by he quaniies obained from a gamma disribuion assumpion for he urance ris. hen ecion 5.2 is devoed o a comparison of he radiional endowmen urance wih he guaraneed uni-lined endowmen urance. Wihin a value based managemen cone i is shown ha he guaraneed uni-lined conrac performs beer han he radiional one on a ris-adjused reurn scale. Indeed he epeced gain per uni of sandard deviaion ar or CaR of a porfolio of idenical policies is greaer for he guaraneed uni-lined produc han for he radiional one amma approimaion of he urance ris. o deermine he ar and CaR measures of a life porfolio i is necessary o evaluae he disribuion of he aggregae loss in (3.1). We follow he mehod applied in Hürlimann(2001a). he random accumulaed reurn R on invesmen is modeled by a 1 2 lognormal disribuion wih parameers [ ln( 1+ j) σ and σ wih j and σ as in ecion 4.1. he random accumulaed aggregae claims 2 1 r follows a muliperiod individual life model whose disribuion can be evaluaed using he wo sage recursive formulas of De Pril(1989) (see Hürlimann(2001a) for deails). As a simple bu reliable approimaion of one firs replaces 1... by independen copies 1... and hus replace by r 1. By Hürlimann(2001a) ecion 4 his implies he sop-loss order relaion sl which means ha has been replaced by he slighly more dangerous for which one has in paricular [ [ and ar [ ar[. hen one approimaes by a gamma disribued random variable wih mean [ and variance ar [. As he following illusraion shows he obained very racable approimaion yields enough accurae upper bounds for ar and CaR calculaions.

12 12 Consider a porfolio of N 100 radiional endowmen policies wih he characerisics of ecion 4.4. he parameers of he gamma approimaion are deermined by he formulas [ N ( r ) ar 1 1 p q (5.1) [ N ( r ) 1pq+ 1( 1pq+ 1) 1 1 (5.2) where second order probabiliies of deah are used as in ecion 4.4 and he sum a ris is found in able 4.8. By definiion of he urance margin in ecion 2 one has M [ 2 (recall ha one assumes ha 60% of he probabiliies in he ariff suffice o 3 cover he claims). Comparisons of eac and approimaed values of ar and CaR for he confidence level are summarized in he able 5.1. able 5.1 : Comparison of ar and CaR measures model ar CaR eac disribuion gamma approimaion Ris-adjused performance measuremen. he use of capial requiremens using RC ris measures lie ar and CaR is of grea imporance in ris-adjusmen performance measuremen and has implicaions for he value based managemen of urance producs. Consider he random reurn per uni of RC o a fied confidence level of a life porfolio a ime called RC gain raio which is defined by R :. (5.3) RC [ he epeced value of he RC gain raion measures he ris-adjused reurn on capial. his way of compuing he reurn is commonly called RAROC (e.g. Maen(1996) p.59) and is here defined by RAROC R. (5.4) [ [ A relaed recen discussion of RAROC is found in Hürlimann(2001d). he use of RAROC as a value based managemen ool is sraighforward. Indeed if a produc manager has o decide upon he more profiable of wo life porfolios wih accumulaed aggregae surplus and a decision in favor of he firs porfolio is aen if and only if one has 2 RAROC 1 2 [ RAROC [ a given confidence levels. his preference crierion ells us ha a porfolio is preferred o anoher if is epeced gain per uni of economic ris capial is greaer. As an illusraion we compare porfolios of N idenical policies wih radiional endowmen and guaraneed uni-lined endowmen conracs under he assumpions of ecion 4.4. We use a gamma approimaion for he urance ris as in ecion 5.1 and noe ha has o be assumed gamma disribued when [ < 0 as for our guaraneed uni- 1

13 13 lined policy. Our resuls for he confidence level by varying porfolio size N are summarized in he ables 5.2 and 5.3. I is remarable ha he guaraneed uni-lined conrac performs beer han he radiional conrac on he RAROC scale. One observes ha he same propery holds if he economic ris capial measure is replaced by he simpler sandard deviaion ris measure. able 5.2 : Measuremen of radiional and guaraneed uni-lined endowmen life urance radiional endowmen uaraneed endowmen N µ σ ar CaR µ σ ar CaR able 5.3 : Comparison of ris-adjused performance measures radiional endowmen uaraneed endowmen N µ µ µ µ µ µ σ ar CaR σ ar CaR References. Aase K.K. and.-a. Persson (1994). Pricing of uni-lined life urance policies. candinavian Acuarial Journal no Acerbi C. (2001). Ris aversion and coheren ris measures : a specral represenaion heorem. Woring paper hp:// Acerbi C. and D. asche (2001a). On he coherence of epeced shorfall. Woring paper hp:// Acerbi C. and D. asche (2001b). peced shorfall : a naural coheren alernaive o valuea-ris. Woring paper hp:// Arzner P. (1999). Applicaion of coheren ris measures o capial requiremens in urance. Norh American Acuarial Journal 3(2) Arzner P. Delbaen F. ber J.-M. and D. Heah (1997). hining coherenly. RIK 10(11) Arzner P. Delbaen F. ber J.-M. and D. Heah (1999). Coheren measures of ris. Mahemaical Finance 9(3) Bacinello A.R. and F. Oru (1993). Pricing equiy-lined life urance wih endogenous minimum guaranees. Insurance : Mahemaics and conomics Corrigendum. Insurance : Mahemaics and conomics Borch K. (1982). Addiive urance premiums : a noe. he Journal of Finance Reprined in Borch(1990) Borch K. (1990). conomics of Insurance. Advanced eboos in conomics vol. 29. Norh-Holland. Bowers N.L. erber H.U. Hicman J.C. Jones D.A. and C.J. Nesbi (1986). Acuarial

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