IMPLICIT LOADINGS IN LIFE INSURANCE RATEMAKING AND COHERENT RISK MEASURES

Anales del Insiuo de Acuarios Españoles, 3ª época, 9, 23/85- IMPLICIT LOADINGS IN LIFE INSURANCE RATEMAKING AND COHERENT RISK MEASURES Monserra Hernández Solís, Crisina Lozano Colomer 2 y José Luis Vilar Zanón 3 ABSTRACT In his paper we sudy a premium calculaion principle applied o life insurance based on a coheren risk measure called Proporional Hazard Transform. This is based on a probabiliy disorion by means of Wang s disorion power funcion (Wang S. (995)). We will see how his heoreically suppors he pracical use of implici loadings in life insurance raemaking. Eiher life insurance conracs wih deah coverage or life annuiies are considered as exemplificaions. KEY WORDS: Implici loading, disorion funcion, hazards risk ransform, coheren risk measure, life insurance. RESUMEN En ese arículo se propone un principio de cálculo de primas para seguros de vida, basado en la medida de riesgo coherene, esperanza disorsionada con la función de disorsión de Wang (995) en forma de poencia, la denominada Proporional Hazards Transforms (PH). Ese principio propone una jusificación eórica a la prácica habiual de recargar de manera implícia las probabilidades de fallecimieno y supervivencia. Se considera como ejemplo la modalidad de seguro de vida con coberura de fallecimieno, el seguro vida monserrah@cee.uned.es Deparmen of Business Economy and Accounancy Universidad Nacional de Educación a Disancia (UNED) 2 clozano@cee.upcomillas.es Deparmen of Quaniaive Mehods Universidad Ponificia de Comillas (ICADE) 3 jlvilarz@ccee.ucm.es Deparmen of Financial and Acuarial Economics. Universidad Compluense de Madrid * This sudy has been graned by he ECO2-2265-C3- (MICIN) projec. **A previous version of his research was presened as a communicaion o he congress Risk 2 held a Carmona (Spain) in Ocober 2h and 2s, see Hernández Solis M. Lozano-Colomer C., Vilar-Zanón J.L. (2). Ese arículo se ha recibido en versión revisada el 3 de mayo de 23. 85

Implici loadings in life insurance raemaking and coheren - Anales 23/8- enera, y a la modalidad de seguro de vida con coberura de supervivencia, el seguro de renas vialicio. PALABRAS CLAVE: Recargo implício, función de disorsión, ransformada del ano insanáneo, medida de riesgo coherene, seguro de vida.. INTRODUCTION In order o proec hemselves agains he risk arising from flucuaions in claims severiies, insurance companies calculae securiy loadings. These may be explicily calculaed or implicily assumed in he raemaking, he laer being he mos common pracice in life insurance. When considering life insurance wih deah coverage, an adverse claim severiy consiss in lives lasing shorer han expeced. Thus a common pracice when calculaing premiums is adding an implici loading eiher defined as a safey margin conceived as a percenage of he deah probabiliies q x, or using a moraliy able wih higher probabiliies han hose of he human group aken ino accoun. The las is equivalen o consider a higher moraliy insananeous rae (for insance considering oudaed moraliy ables). The opposie siuaion is found in life insurance wih survival benefis, where an adverse claim severiy occurs when life las longer han expeced. In his case he insurance company can implicily load he premium using moraliy ables wih lower deah probabiliies ha decreases he insananeous moraliy rae. Wang (995) proposes a loaded premium calculaion for non-life insurance using disored probabiliies. I is called proporional hazard ransform (PH), and is in fac a coheren risk measure (Arzner, P. (999)) wih a disorion / funcion of he form g(u) = u, where i is shown ha he necessary condiion for such a risk measure o be coheren is. This aricle applies his resul in he wo following ways. The more immediae one consiss in applying he resul from Wang (996) o he case of survival insurance (i.e. life annuiy insurance) showing ha he common pracice of implicily loading he premium considering a lower insananeous moraliy rae can be considered in fac as a proper coheren risk 86

Monserra Hernández, Crisina Lozano y José Luis Vilar Anales 23/85- measure. This is worked ou expressing he ne premium wih respec o he survival funcion, hen applying he proporional hazard ransform (which sands in he case and is coheren) and checking ha his corresponds o a new insananeous moraliy rae proporional o he former by a facor of /. This sep is in fac done in a second sage (see secion 5). Bu he firs sage and main way of he sudy (see secion 4), is he case of deah coverage producs (i.e. whole life insurance conracs) which is in fac ou of Wang s resul. This is moivaed by he fac ha considering an implici loading would be equivalen o he calculaion of a ne premium corresponding o a higher insananeous moraliy rae. If we ried hen o follow he same scheme as in he survival producs case, we would find he obsacle ha rying o express he ne premium wih respec o he survival funcion would no longer reward us wih an expression ha could be considered as a proporional hazard ransform. A careful inspecion of he expression obained hrough his mehod will ell us ha i is in fac a funcion of he proporional hazard ransform his ime wih an exponen (,). Our goal consiss in showing ha his funcion sill is a coheren risk measure. Therefore his aricle seeks o jusify a sandard pracice in life insurance showing ha i may be viewed as he applicaion of a coheren risk measure. Secion 2 is recalls some basic conceps on risk measures and premium calculaion principles, while he proporional hazard ransform is inroduced along secion 3. We will consider whole life insurance and life annuiy conracs as exemplificaions. 2. RISK MEASURES BASED ON PREMIUM CALCULATION PRINCIPLES Premium calculaion principles can be considered candidaes for being risk measures. Alhough here is a consensus abou how o calculae he ne premium, here are many ways o aggregae he loading in order o obain a premium reflecing he risk acceped by he insurer. On he oher hand, risk measures are very ineresing because hey allow o quanify he way he premium compensaes he insurer for he risk associaed o he loss. 87

Implici loadings in life insurance raemaking and coheren - Anales 23/8- Before inroducing a premium based on a risk measure, le us remember a formal definiion of he laer and he properies a risk measure mus fulfill o be coheren (Arzner, P. (999)). Definiion. Given a loss, modeled by means of a non-negaive random variable X, a risk measure H(X) is a funcional H:X. Arzner (999) suggess he following four axioms for a risk measure o be considered coheren: I. Translaion Invariance. For any random variable X and any nonnegaive consan b, H(X + b) = H(X) + b. This axiom saes ha if he loss X were increased by a fixed amoun b, he risk would increase by he same amoun. II. Subaddiiviy. For all pairs of random variables X and Y, H(X + Y) H(X) + H(Y). This axiom saes ha an insurance company canno reduce is risk by dividing is business in smaller blocks. III. Posiive homogeneiy. For any random variable X and any nonnegaive consan a> H(aX) = a H(X). This axiom saes ha a change in he moneary uni does no change he risk measure. IV. Monooniciy. For every pair of random variables X and Y, such ha P(X r Y) = hen H(X) H(Y). This axiom saes ha if one risk loss is no greaer han anoher for all saes of naure, he risk measure of he former canno be greaer han he risk measure of he laer. For insance i is very well known ha he risk measure based on he expeced value principle is H(X) = ( + α)μ X (α ), 88

Monserra Hernández, Crisina Lozano y José Luis Vilar Anales 23/85- where α is he risk loading and μ X sand for he mean of he loss X. When α =, he risk measure is called pure or ne premium. Oher imporan premium principles are for insance, he variance and he sandard deviaion: 2 H(X) = μx + βσx β H(X) = μ + γσ γ. I is known (Yiu-Kuen Tse (29) p.8) ha hese premium calculaion principles are no coheren risk measures since he expeced value principle does no verify he axiom of invariance under ranslaions, and he sandard deviaion principle does no verify he monoony axiom. On he oher hand, he ne premium principle ( α = ) is a coheren risk measure. X X 3-THE PROPORTIONAL TRANSFORM OF THE INSTANTANEOUS RATE AS A RISK MEASURE. Consider a risk X wih disribuion funcion and survival funcion: ( ) F(x) = Pr X x S(x) = F(x). () The ne premium based on he expeced value risk measure expressed by means of he survival funcion is: ( ) E X = xdf(x) = S(x)dx. We mus now define a loaded premium likely o be beer adjused o he risk and based on he disorion funcion (Wang(995)): Definiion 2: Given a risk X wih survival funcion S(x), and a non-decreasing funcion g : [,] [,] wih g() =, g() = called he disorion funcion, he risk premium adjused o he disored probabiliy risk measure is: E X = g S(x) dx. (2) g ( ) ( ) 89

Implici loadings in life insurance raemaking and coheren - Anales 23/8- Supposing ha g and S have firs derivaives, he disorion funcion verifies he following properies (Wang 996):. g(s(x)) is non-decreasing. 2. g(s(x)) for any x [, + ). 3. If g and S are coninuous funcions, g(s(x)) can be considered as he survival funcion of anoher random variable Y wih densiy funcion given by dg(s(x)) f Y (x) = = g'(s(x))s'(x) = g'(s(x))f(x). dx Therefore g (S(x)) is a weighing funcion of he densiy funcion f(x). Moreover, if g is concave hen dg[s(x)] = g[s(x)]s(x). dx Therefore he disorion funcion allows us o define a new random variable Y, since g(s(x)) has he properies of a survival funcion. We now consider a power funcion as a special case of disorion (Yiu-Kuen Tse. (29), p.29). This case is well known in non-life insurance because i saisfies he properies of a coheren risk measure. I is also well known ha considering a power of he survival funcion resuls in a proporional insananeous moraliy rae. As we are going o see soon, his has an ineresing inerpreaion in life insurance. Definiion 3: A proporional ransform of he insananeous rae is a Wang measure wih he following disorion funcion: / g(u) = u, >. (3) In his case a new random variable Y is defined from he original X, wih survival funcion and premium adjused o he risk given by: / S Y (x) = ( S(x) ), > + (4) / Π ( X) = E( Y) = ( S(x) ) dx. We can deduce from definiion 3 he following consequences: 9

Monserra Hernández, Crisina Lozano y José Luis Vilar Anales 23/85-. E(Y) is an increasing funcion wih respec o. The higher is, he higher will be he risk-adjused premium. Thus can be considered as a risk aversion parameer (see Yiu-Kuen Tse (29) p.29). 2. The insananeous raes of he random variables X and Y are proporional. Recalling ha: / / Y = ( ) = x = x μ μ S () S() exp (u) du exp (u) du. We can wrie: μ Y() = μ x(), >,. (5) Therefore he X and Y insananeous raes of moraliy are proporional. The new random variable Y is called he proporional insananeous rae ransform of X wih parameer (Wang (996)). This ransform only requires ha >, hough in he conex of general insurance is considered in order o give more weigh o he ail of he risk disribuion. Assuming ha Y ( ) / S (x) = S(x),, he Y survival funcion decreases slower han he X one, wih greaer probabiliies for larger values of he variable, so he risk-adjused premium or he loaded premium verifies Π (Y) = E( Y) E( X), he difference being he securiy loading. As indicaed in (Wang (995)), he risk-adjused premium reflecs he risk of he original loss, and he decision maker s risk aversion is adjused by means of he parameer values. As shown in (Wang (995)), for he disored risk probabiliies are a coheren risk measure. In fac, he consrain is mainly necessary for he measure o fulfill he subaddiiviy propery, he oher axioms requiring only >. 9

Implici loadings in life insurance raemaking and coheren - Anales 23/8-4. NET PREMIUM CALCULATION FOR A WHOLE LIFE INSURANCE CONTRACT. Whole life insurance is a deah coverage insurance such ha he insurer underakes o pay he guaraneed policy benefi o he beneficiaries, whaever he momen of he insured s deah could be (see for example Bowers e al. (997) p.94). I is an insurance policy wih a fixed amoun and random mauriy. Le us consider a whole life insurance conrac for an individual (x) in coninuous ime, wih one moneary uni as he insured capial. In his case he risk can be modeled by he random variable T(x), residual life or ime remaining unil he insured s deah. The following assumpions hold:. A he ime of deah a moneary uni is paid. 2. i is he echnical rae of ineres. 3. A newborn s deah ime is a coninuous random variable X, wih survival funcion S(x). Then he random variable T(x) has a disribuion funcion G x () and a survival funcion S x () whose expressions depending on S(x) are given by (Bowers e al (997) p.52) : S(x) S(x + ) S(x + ) G x () = = S(x) S(x) S(x + ) S x() = G x() =. S(x) (6) 4. Defining v= ( + i), he loss associaed o he policy is hen defined by means of he random variable: Tx Z= v (7) Applying he acuarial equivalence principle, he following expression is obained (Bowers e al. (997) p.95) for he pure premium: + Π( Z) x ( ) = v dg () (8) 92

Monserra Hernández, Crisina Lozano y José Luis Vilar Anales 23/85- In order o adap he acuarial equivalence principle o premium calculaion based on he disorion funcion, his is now expressed depending on S x (). Changing he variable o + + x x (9) Π( ) = = Z v dg () v ds (). v = z : Inegraing by pars: Π( ) ln z = x lnv Z z ds. z = u dz = du Ln z Ln z dsx = dv v = Sx Ln v Ln v Lnz Lnz Lnz Lnz x x x x x Ln v Ln v Ln v Lnv Π( Z) = z S S dz = ( S ( ) ) S dz = S dz We finally obain an expression for he pure premium based on S x () ( ) ln z = x Π Z S dz. () Now a loaded premium is obained subsiuing he survival funcion ino he power disorion funcion. In () we find is expression, where he subscrip emphasizes he dependency of he premium on he previous choice of he power funcion: / ln z Π ( Z) = Sx dz. () I is clear ha he exponen should be for he loaded premium o be greaer han he pure premium. 93

Implici loadings in life insurance raemaking and coheren - Anales 23/8- We are now going o show how his loaded premium can be deduced considering he proporional ransformaion of he insananeous rae, and also ha i is a coheren measure of risk (remember ha in his case ). Theorem. The loaded premium () equals he pure premium of anoher random variable wih he same survival law model, bu wih insananeous moraliy rae proporional o he one of he random variable X, by a proporionaliy facor of /. Proof: In fac, if we inegrae by pars expression () and wriing r = : r r ln z ln z ln z u = Sx du = r S x Sx dz z log v dv = dz v = z ln z Π ( Z) = Sx dz= r r r ln z ln z ln z = z S + zr S S dz= zd S x x x zlog v x r ln z. Now changing variable z= v : r / ( ) = ( x ) = ( x ) (2) Π Z vds() vds(). Comparing (2) wih (9), we see ha i corresponds o he pure premium of an insurance of he same kind, hough for a new random variable Y wih survival funcion S ( ) / Y() = S x(). Therefore we are in he same siuaion described in (5): μ Y() = μ X(), (3) where μ X () is now he insananeous rae of he original variable X. (Q.E.D.) 94

Monserra Hernández, Crisina Lozano y José Luis Vilar Anales 23/85- Thus we can conclude ha for any survival law he loaded premium coincides wih he pure premium ha would be obained for ha law, hough wih a proporional rae by a facor. Therefore he new insananeous rae is higher and his represens an adverse claim experience for he insurer. Going now ino de firs sep menioned during he inroducion, he following heorem shows ha he subaddiiviy propery is also verified by he premium defined in (): Theorem 2: For every pair of non-negaive random variables U and V, and he risk measure given by: / H (U) = ( SU ( z) ) dz, <, z= v, v =, + i he subaddiiviy propery holds: H (U+ V) H (U) + H (V). (4) We will proceed quie similarly as is done in Wang (996). Le us firsly show a previous lemma which plays a similar role as he one ha can be found in Wang (996). Lemma. If < a < b and hen x he following holds: Proof: Calling ( ( ) ) ( ) ( ) ( ) ( ) / / / / x+ b x+ a b a. g(x) = (b + x) (a + x) g (x) = (a + x) (b + x) < because >, and a + x < b + x. 95

Implici loadings in life insurance raemaking and coheren - Anales 23/8- Knowing ha g(x) is a decreasing funcion, i will ge is maximum value in x=, in which case we will have (b + x) (a + x) < b a. (Q.E.D.) Proof of Theorem 2: The proof uses he mehod of complee or srong inducion. Firsly he resul is shown o be rue for a random variables V and U, his las being discree such ha U {,, K,n}. Then applying he ranslaion invariance and he posiive homogeneiy properies, i will also be proved for any discree random variable U { kh, K,(n+ k)h }, (h>), wih h >. Finally, since any random variable can be arbirarily closely approximaed by a discree random variable U wih adequae h, k, he resul will have been proven for any random variable. Looking o (4) and reasoning by means of complee inducion: - If n= hen U= and he resul is rivial - Assuming (4) o be rue in he n-h case, le us examine he (n+) case: U,...n U,V U >. Given (U, V) wih { } Assuming U* {,,n + } + define (U*, V*) as ( ) K he inducion hypohesis saes ha: H (U* + V*) H (U*) + H (V*). Wriing ω Pr( U ), S () Pr ( V U ) = = = > =, for any > V This is he same han: On he oher hand Pr (U > ) = Pr ( U* > U > ) Pr( U > ), S U() = ( ω )S U* (). 96

Monserra Hernández, Crisina Lozano y José Luis Vilar Anales 23/85- Also: ( ) ( ) ( ) ( ) ( ) ( ) S V () = Pr V > = = Pr V > U = Pr U = + Pr V > U > Pr U > = = S () ω + S () ω. V V* ( ) ( ) ( ) ( ) ( ) () ( ) S U+ V() = Pr U+ V> = = Pr U + V > U = Pr U = + Pr U + V > U > Pr U > = = ω S + ω S (). Now according o Lemma : V U* + V* / / / U+ V U V / / ω S ω S U* V* () ω S U* () S () ω S V* () ω S () ( S () ) ( S () ) = / = ( V + ( ) + ) ( ( ( ) ) ) ( V + ( )) / / / / ( ω) ( S U* + V* () ( S U* () ) ( S V* () )) / / / U* + V* U* V* S () ( S () ) ( S () ). Nex we reconsruc he corresponding expressions by adding one o each inegral in boh sides (he reader will check ha in fac his changes nohing). Changing he ln z variable o v = z =, and inegraing wih respec o z we finally obain: S ( ) dz S ( ) dz+ S ( ) dz ln z / ln z / ln z / U+ V U V ln z / ln z / ln z / U* + V* U* V* (5) S ( ) dz S ( ) dz + S ( ) dz. The expression (4) can be wrien as * * * * ( ) ( ) H(U+ V) H(U) + H(V) H(U + V) H(U) + H(V). The las inequaliy is rue by virue of he inducion hypohesis, so we conclude ha he subaddiiviy axiom is also fulfilled by U and V. Therefore his premium calculaion principle is a coheren risk measure. (Q.E.D) 97

Implici loadings in life insurance raemaking and coheren - Anales 23/8- Observaion: Theorem 2 shows ha he premium calculaion principle for a Tx ( ) whole life insurance conrac wih loss Z= v / ln z H (Z) = Sx dz, <,z= v, v =, + i is a coheren risk measure. I canno be considered as a generalizaion of Wang s resul, bu i heoreically suppors he pracice of modifying he insananeous moraliy rae in he case of deah coverage producs. 5. CALCULATION OF THE SINGLE PREMIUM FOR A LIFE INSURANCE WITH SURVIVOR S COVERAGE (LIFE ANNUITY INSURANCE) A coninuous-ime life annuiy insurance (Bowers e al. (997) p.34) is characerized by annuiies payable coninuously. Paymens are made by he insurance company o an insured (x) as long as he is alive. In exchange he insured mus pay he amoun of he premiums o he company, eiher periodically or in he form of a single premium. In his case he risk can be modeled by he random variable T(x), residual life or ime remaining unil he insured s deah. The following assumpions hold:. We have annuiies payable coninuously a he rae of m.u. per year (Bowers e al. (997) p.34) unil deah. 2. i is he echnical rae of ineres. 3. Then he random variable T(x) has a disribuion funcion G x () and a survival funcion S x () whose expressions depending on S(x) are given in (6). 4. Defining v= ( + i) he loss associaed o he policy is hen defined by means of he random variable he presen value of paymens Z= a, where: Tx ( ) u a = v du (v ) = 98

Monserra Hernández, Crisina Lozano y José Luis Vilar Anales 23/85- Applying he acuarial equivalence principle o ge he pure premium (Bowers e al. (997) p.35) we find: where v = <, and p x = S x (). + i + Π(Z) = E(Z) = v p x d, Changing he variable on S() x v = z we obain an expression for he pure prime based + lnz x x (6) Π(Z) = v S ()d = S dz. The loaded premium using he disorion funcion proporional ransformaion of he hazard funcion given by (3) has he following form: lnz x Π (Z) = S dz wih, Π Π. (7) Now for he loaded premium o be greaer or equal han he pure premium, he exponen mus be less or equal han, so. Therefore hese disored probabiliies saisfy he properies of a coheren risk measure (Wang (996)). ln z For each value of = given in (7) he disored survival funcion is greaer han he iniial survival funcion, which means ha he insured is deemed o have a lower risk of deah. In his way, an annuiy loss rae increases if he insured lives longer han expeced. Thus he disorion funcion gives more weigh o he ail of he residual ime variable, resuling in a loaded premium. Theorem 3. The loaded premium (7) coincides wih he pure premium of anoher random variable, wih he same survival model law bu an insananeous moraliy rae 99

Implici loadings in life insurance raemaking and coheren - Anales 23/8- proporional o he insananeous rae of variable X, wih a proporionaliy facor equal o. Proof of Theorem 3: In fac, if in expression (7) we inroduce he change of variable obain: z =, = z =, = dz = v d / / = ( x ) ( x ) = Π (Z) S () v lnvd v S () d. z = v, we Calling S Y() = (S x()) he expression of he insananeous rae of variable Y equals (5): μ Y() = μ X(). Thus he insananeous rae of he new variable is proporional o he original variable. (Q.E.D.) REFERENCES: Arzner, P. (999): Applicaion of coheren risk measures o capial requiremens in insurance. Norh American Acuarial Journal. Vol. 3, N.2, pp.-25. Bowers, N.L. e al. (997): Acuarial Mahemaics. Sociey of Acuaries. Denui M., Dahene J., Goovaers M., Kaas R. (25): Acuarial Theory for dependen risks. John Wiley & Sons Ld. Hernández Solis M. Lozano-Colomer C., Vilar-Zanón J.L. (2): Tarificación en Seguros de Vida con la medida de riesgo esperanza disorsionada, Invesigaciones en Seguros y Gesión del Riesgo: Riesgo 2, Cuadernos de la Fundación 7, pp. 93-8. Roar, V. (26): Acuarial Models. Chapman and Hall Wang, S. (995): Insurance pricing and increased limis raemaking by proporional hazards ransforms. Insurance, Mahemaics and Economic, Vol.7, pp. 42-54. Wang, S. (996): Premium calculaion by ransforming he layer premium densiy. Asin Bullein, Vol.26, pp.7-92. Wang, S., Young, V. Panjer, H. (997): Axiomaic characerizaion of insurance prices. Insurance Mahemaics and Economics, Vol. 2, pp.73-83. Yiu-Kuen Tse (29): Nonlife Acuarial Models. Theory, mehods and evaluaion. Cambridge Universiy Press.