Error in Joint Mortality Formulas

Transcription

1 ausralian a c u a r i a l journal 2012 Volume 18 Issue 1 pp Error in Join Moraliy Formulas M Boggess & M Moyer* Absrac Life Coningencies is he sudy of probabiliy and he ime value of money whose objecive is he valuaion of quaniies associaed wih life insurance and annuiies Since policies on a husband and wife are common, acuarial eams covering life coningencies include he siuaion where he lifeimes of he husband and wife are dependen and he policy payou depends on he ime of he firs or second deah We show ha some well-known formulas are incorrec, and whose applicaion by he Sociey of Acuaries in heir soluions o eam quesions has led o incorrec answers These formulas can be recified by he inclusion of noaion ha makes specific he necessary condiioning evens Keywords: life coningencies; join moraliy formula error * Conac: May M Boggess, BMah(Hons I) MSc(Sa) PhD, Associae Professor, School of Mahemaical and Saisical Sciences, Arizona Sae Universiy, mboggess@asuedu Michelle Moyer, BSc(Applied Mah) MSc(Mah), Acuarial Analys II, USAA, San Anonio TX, USA, MichelleMoyer@usaacom Acuaries Insiue of Ausralia

2 ausralian a c u a r i a l journal 1 Inroducion In 2002 Youn, Shemyakin and Herman [12] claimed ha sandard acuarial formulas, Equaions (1), (3) and (4) below, are no always correc James Hickman and Donald Jones [4] responded ha, while here are some issues, i is jus a noaional deficiency In his response, S David Promislow [9] acknowledged here is a problem arising when modelling join lives wih dependency oher han common shock, including models on real daa [3] However, raher han supporing Youn, Shemyakin and Herman's call for a noaional fi o he problem, Promislow assered ha anyone sophisicaed enough o use he copula model in he firs place would be aware of he need o change he marginal [9, p115] In he even ha one inadverenly applied he formulas, he claimed (wihou supporing deail) ha he size of pricing errors caused would be negligible in pracice [12, p 167] Bu Youn, Shemyakin and Herman had used a copula model esimaed on daa of join and las-survivor annuiy conracs conaining 11,457 married couples from a large Canadian company [11] and showed ha he formulas would resul in an error in he las-survivor acuarial presen value of approimaely 10% I is ime for his debae o be resolved Acuaries Insiue of Ausralia 2012 Here we confirm ha errors nearing 10% are possible in pracice and show ha he use of Equaions (1) o (5) has led o incorrec answers on he US Sociey of Acuaries (SOA) Life Coningencies (MLC) Eam [10] All members of he acuarial profession have reason o be concerned when i is shown ha an eam quesion did no have an unambiguous correc answer, and when misleading formulas may resul in errors large enough o cause financial loss We begin wih definiions of sandard acuarial noaion from Bowers e al [1, Chaper 3] Le X and Y be random variables represening age a deah Le T() be he fuure lifeime for () a person aged, T() be he fuure lifeime for () he minimum of wo fuure lifeimes, and T( ) be he fuure lifeime for ( ) he maimum: 68

3 Error in Join Moraliy Formulas T X X, T y Y y Y y, T( ) min{ T( ), T( y)} min{ X Y, y} X Y, y, and T( ) ma{ T( ), T( y)} ma{ X, Y y} X, Y y The probabiliy ha a person aged () will live more years is denoed by p P( T( ) ) P( X X ), so ha p P T P X or Y yx, Y y, and p P T( ) P X or Y yx, Y y The following join moraliy funcions are derived in Bowers e al: p p, p py (1) e e e e, (2) y A A, A Ay (3) a a, a ay (4) a + a a a, (5) y he firs of which is equivalen o F F F F () T( ) T( ) T( ) T( y) [1, pages 268, 272, 282], [2, pages 265, 266, 286] To ge o he roo of he problem, he condiioning evens need o be carefully followed: T( ) T( ) F F P min X, Y y X, Y y P(ma X, Y y X, Y y),,, P X Y y X Y y P X Y y X Y y P ( X y, Y y ) P ( X Y, y) PX ( Y, y) PX ( Y, y) Acuaries Insiue of Ausralia

4 ausralian a c u a r i a l journal Py ( Y y X, ) P ( X y, Y y ) PX ( Y, y) PX ( Y, y) P ( X Y, y) Py ( Y y X, ) PX ( Y, y) PX ( Y, y),, P X X Y y P Y y X Y y, which may differ from: F F P X X P Y y Y y T y T if X and Y are no independen Hickman, Jones and Promislow's responses o Youn, Shemyakin and Herman included he asserion ha he size of he error obained when using Equaion (3) is negligible for pracical purposes This asserion was based on calculaions for which no supporing deail was provided Thus we begin wih he calculaions involved in heir eample, which we performed in Maple TM [7] (code available from corresponding auhor) 2 Youn, Shemyakin and Herman, Eample 2 Acuaries Insiue of Ausralia 2012 Youn, Shemyakin and Herman chose Weibull marginal ( / ) disribuions, ( m 1 / m F e ) and Hougaard's copula 1/ [8, page 96] Cu, vep [( ln u) ( ln v) ], wih 1638 [12, Eample 2] Thus, he join cumulaive disribuion funcion is H, y ep [ ln(1 e )] [ ln(1 e )], wih m 08951, 899, my 08598, and y 1124 (where m,, my and y have been scaled by 100) Le Fa () be he condiional cumulaive disribuion funcion for he coninuous lifeime random y 164 variable X a X a : Ha (, ) Ha (, ) Fa ( ) 1 Ha (, ) Following Youn, Shemyakin and Herman, = 065 and y = 07: 70

5 061 F65 ln(1 e 0042, 302(065 ) e p [ )] 061 F ep ln(1 e (07 ) [ )] Error in Join Moraliy Formulas The acuarial presen value of a paymen of one dollar made o a person currenly aged () a he ime of deah is r r 0 0 A e f () d re F () d [1, 426, page 96] Using r = 005, A 5e F ( ) d y and A The condiional cumulaive disribuion funcion of he maimum is 1H065, H,07 H(065,07) H 065,07 H 065,07 H 065,07 H(065,07) F () Since H065,07 = ,, , H 065, and H he denominaor is The numeraor has hree pars ha are funcions of : (07 ) 164 H(065,07 ) ep [ ln(1 e )] , (065 ) 164 H(065,07) ep [ ln(1 e )] 2 328, H 065, ep[ ln(1 e )] [ ln (1 e )] Acuaries Insiue of Ausralia

6 ausralian a c u a r i a l journal The acuarial presen value is hen A The inegral was evaluaed in Maple TM using Simpson's rule wih 10,000 pariions beween = 0 and = 30, and by comparison wih larger and smaller domains wih more and less pariions, his answer is accurae o a leas seven decimal places The condiional cumulaive disribuion funcion of he minimum is 1H065, H,07 H(065, 07) 1H 065, H,07 H 065,07 F () 1 The numeraor has he following componens ha are funcions of : (07 ) 164 H(,07 ) ep [ ln(1 e )], H 065, ep[ ln(1 e )] The acuarial presen value is hen A Using Equaion (3) A A A A 65:70 y : , Acuaries Insiue of Ausralia 2012 whereas he correc value is A , a percenage error of 88%, which is in close agreemen wih Youn, Shemyakin and Herman's original calculaions 3 Errors in SOA Sample Eam Quesions 31 Course 3, May 2001, Quesion 9 This quesion is also in he Sociey of Acuaries MLC Eam sample quesions [10, Q104]: () and (y) are wo lives wih idenical epeced moraliy You are given: P Py 01, P 006, P is he annual benefi premium for a fully discree insurance of 1 on () and 72

7 Error in Join Moraliy Formulas d = 006 Calculae he premium P, he annual benefi premium for a fully discree insurance of 1 on () For his eample we choose geomeric marginal disribuions for 1 boh X and Y, k F k 1 (1 p), where k 0,1,2,3, and p Frank's copula is used [6, 345, page 112], u v 1 ( e 1)( e 1) Cu, v ln1, e 1 wih Then he join cumulaive disribuion funcion is y H, y 05267ln 11177( e 1)( e 1) Take () = 1 and (y) = 1, meaning, he given even is X>0 and Y>0 The condiional marginal cumulaive disribuion funcion is P F ( k) 0 P X X 0 k 084, so ha ( ) F k 1 Since d = 006, he 094 annual discouning facor is v = 094 and he acuarial presen value of a paymen of one dollar made o a person currenly aged () a sar of each year prior o deah is k k 1 a v 1 F( k) k 0 [1, 524, page 135] The level annual benefi premium is valued a P 1 a d 01 [1, 632, page 180], and similarly for Y The condiional cumulaive disribuion funcion of he maimum is H, H,0 H 0, H(0,0) F ( ) 1 H 0, H,0 H(0, 0) The denominaor is 1-2( ) = The numeraor has he following componens ha are funcions of : Acuaries Insiue of Ausralia

8 ausralian a c u a r i a l journal H 0, 05267ln 11176( e 1)( e 1), H, 05267ln(1 1176( e 1 ) ) The condiional cumulaive disribuion funcion of he minimum is F 1 H, H, H(, ) 1 1 H 0, H,0 H(0,0) The numeraor has he following componens ha are funcions of : e H, 05267ln e Then a 8333, P 006, a and P A differen answer is obained by using Equaion (5): a a a a 4167 and P y The SOA answer is (C) 018 using Equaion (4), whereas by direc calculaion, we see ha he correc answer is (B) 016 in his insance 32 Course 3, May 2001, Quesion 23 Acuaries Insiue of Ausralia 2012 This quesion is also in he Sociey of Acuaries MLC Eam sample quesions [10, Q112]: A coninuous wo-life annuiy pays: 100 while boh (30) and (40) are alive; 70 while (30) is alive bu (40) is dead; and 50 while (40) is alive bu (30) is dead The acuarial presen value of his annuiy is 1180 Coninuous single life annuiies paying 100 per year are available for (30) and (40) wih acuarial presen values of 1200 and 1000, respecively Calculae he acuarial presen value of a wo-life coninuous annuiy ha pays 100 while a leas one of hem is alive Consider he following bivariae eponenial disribuion [5, page 352] wih join cumulaive disribuion funcion y y ( )( y ) y y y H, y 1e e e 74

9 Error in Join Moraliy Formulas The marginal disribuions are eponenials wih force of moraliies and y, and when 0 hey are independen Now le , 0072, and 0028, so ha y y y H, y 1 e e e y Then he acuarial presen values of one dollar paid coninuously for life are a 1( ) 12 [1, page 136], and ay 1( y ) 10, as required in he quesion The cumulaive disribuion funcion of he maimum is F 1H30, H,40 H(30,40) H 30,40 H 30,40 H 30,40 H(30,40) The denominaor is = The numeraor has he following componens ha are funcions of : ,40 1 H e e e (30 )(40 ), H 30, 40 1 e e e, and H 30,40 1 e e e The condiional cumulaive disribuion funcion of he minimum is F 1H 30, H,40 H(30,40 ) 1 1H 30, H,40 H(30,40) The numeraor has he following componens ha are funcions of : , 1, H e H e a , so ha,40 1 Then a 50a 20a a, y as required in he quesion The SOA soluion uses Equaion (4) o obain a a ay a , answer choice (A) Bu, by direc calculaion, a , which is less han all answer choices provided Acuaries Insiue of Ausralia

10 ausralian a c u a r i a l journal 33 Course 3, May 2005, Quesion 23 This quesion is also in he Sociey of Acuaries MLC Eam sample quesions [10, Q163]: You are given: T() and T(y) are no independen; q q 005, k 0,1,2, ; p 102 p p, k yk k k k y k 1, 2,3, Ino which of he following ranges does e, he curae epecaion of life of he las survivor saus, fall? Le =1/102, a = 0009, p = 095 and define, for k, l = 0, 1, 2,, he funcion 1, < 0, < 0, ( +), = 0, > 0, ( +), = 0, > 0,, =( >,>) = +, = 0, < 0, +, = 0, < 0,, = 0, = 0,, >0,>0 Le (,y) = 1, for real numbers 0< < 1, 0 < y < 1, and zero oherwise Define he join survivor funcion, for real numbers and y, by (, ) =, (,), join survivor so ha i is monoonic decreasing, wih consan value beween ineger laice poins Acuaries Insiue of Ausralia 2012 Then, for k 0, Sk (, ) ( ap ) PX kx p S(0, ) ( a) k 1 k 1, meaning he fuure lifeimes X X and Y y Y yare geomeric, so ha 1 ( ) 1 ( ) 1 pk P X k X k P X k X k k 2 PX ( k1) PX ( k1)/ PX ( ) p p, k 1 PX ( k ) PX ( k)/ PX ( ) p as required in he quesion Then, 76

11 k p P X k, Y y kx, Y y k1 l1 P X k, Y y k Skk (, ) p p P X, Y y S(0,0) Error in Join Moraliy Formulas k1 l1 102 p p, for k 1, as required in he quesion The epeced fuure lifeime k 1 of a person aged currenly () is e p 19 [1, page 73], e y 19 k1 k1 and e 102 p p The epeced las survivor ime k 0 is calculaed direcly by p k 0 k 1,0 0, (, ) 2( 1) S k S k S k k a p S(0,0) so ha e , which is answer choice (E) Using Equaion (2) as in he SOA soluions, e , which is answer choice (D) 4 Conclusion, Join moraliy formulas used in he sudy of life coningencies have been demonsraed here o be incorrec, resuling in incorrec answers o SOA eam quesions The wo responses by James Hickman and Donald Jones and S David Promislow did lile o refue he claims made by Youn, Shemyakin and Herman in 2002 Hickman and Jones agree ha Youn, Shemyakin and Herman are correc, bu asser ha i is a noaional inadequacy and have an insincively negaive reacion o more noaion [4, page 114] However, wih he amoun of noaion used in acuarial sudies, especially life coningencies, one more piece of noaion is a small price o pay o ge he correc answer Promislow agrees ha Equaion (1) is no rue in general [9, page 115] However, he couners ha he usual way of deriving muliple life premiums is o assume independence, which gives negligible errors in pracical applicaions Acuaries Insiue of Ausralia

12 ausralian a c u a r i a l journal We believe i is fundamenally wrong o assume independence when i is no rue, and his is an inadequae way o deal wih his problem, paricularly for SOA eams We pu forh wo suggesions based on his work: 1 Since P X X, Y y P( X X ) in common shock models, SOA eaminers could uilise hese join disribuions and hereby avoid any possible ambiguiy in he answer choices 2 Youn, Shemyakin and Herman sugges ha noaion be included on p o indicae he addiional condiioning on y as follows: p y P X yx, Y y Then formulas such as p p p y py, can be applied accuraely Based on he inaccuracies in he Sociey of Acuaries eams shown here, we sugges ha Youn, Shemyakin and Herman's noaion be adoped by acuarial sudens and praciioners alike Acuaries Insiue of Ausralia

13 Error in Join Moraliy Formulas References [1] Bowers, N, Gerber, H, Hickman, J, Jones, D, & Nesbi, C (1986) Acuarial Mahemaics Sociey of Acuaries, Iasca, IL [2] Dickson, D, Hardy, M & Waers, H (2009) Acuarial Mahemaics for Life Coningencies Cambridge Universiy Press, Cambridge, UK [3] Frees, E, Carriere, J & Valdez, E (1996) Annuiy evaluaion wih dependen moraliy Journal of Risk and Insurance, 63(2): [4] Hickman, J & Jones, D (2002) Response o A reeaminaion of he join moraliy funcions Norh American Acuarial Journal, 6(4): [5] Koz, S, Johnson, NL & Balakrishnan, N (2000) Coninuous Mulivariae Disribuions: Models and applicaions John Wiley and Sons, New York, NY [6] Malevergne, Y & Sornee, D (2005) Ereme Financial Risks: From Dependence o Risk Managemen Springer, New York, NY [7] Maplesof (2008) Maple 12 Waerloo ON, Canada [8] Nelsen, R (2009) An Inroducion o Copulas Springer, New York, NY [9] David Promislow, S (2002) Response o A re-eaminaion of he join moraliy funcions Norh American Acuarial Journal, 6(4): Acuaries Insiue of Ausralia

14 ausralian a c u a r i a l journal [10] Sociey of Acuaries (April 2010) Eam M, Acuarial Models, Life Coningencies Segmen (MLC) hp://wwwsoaorg/files/pdf/edu-2008-spring-mlcquesionspdf [11] Youn, H & Shemyakin A (Augus 1999) Saisical aspecs of join life insurance pricing In 1998 Proceedings of he Business and Economic Saisics Secion, American Saisical Associaion Meeing, Dallas, TX, pages [12] Youn, H & Shemyakin A & Herman, E (2002) A reeaminaion of he join moraliy funcions Norh American Acuarial Journal, 6(1): Acuaries Insiue of Ausralia