LIDSTONE IN THE CONTINUOUS CASE by Ragnar Norberg Absrac A generalized version of he classical Lidsone heorem, which deals wih he dependency of reserves on echnical basis and conrac erms, is proved in he case where premiums are payable coninuously and insurances are payable a he momen of deah. Key words: Ne level reserves, changes in echnical basis, coninuous paymens, Thiele's differenial equaion. 1 Inroducion A useful ool for deermining he effec on reserves of changes in echnical basis and conrac erms is Lidsone's (1905) heorem wih exensions due o Baillie (1951) and Gershenson (1951). These auhors reaed he "discree" case where premiums and benefis are payable a cerain erms, viz. annually for premiums and annuiies and a he end of he year of deah for insurances. An accoun of his heory is offered in Jordan's (1967) exbook. I urns ou ha he resuls which have been esablished for he discree case, easily carry over o he "coninuous" case where premiums and annuiies are payable coninuously and insurances are payable a he momen of deah. Since he srucure of he proof is virually he same in he wo cases - hey differ only wih respec o echnicaliies - we urn direcly o he coninuous case. Anicipaing evens, 'vie poin ou ha, in accordance wih Wha is usually observed also elsewhere in life insurance mahemaics, argumens and resuls are acually simpler in he coninuous case.
- 4 - for E ( 0, 0 ) ( 3) and (4) hen fore (O,n). (ii) The resul in {i) remains valid if all inequaliies are reversed. (iii) The resuls in (i) and (ii) remain valid if all inequaliies are made sric. Proof: Upon subracing (1) from he corresponding relaion for special basis quaniies and rearranging erms, we ge he differenial equaion d r - (o'+"' )r d ~x+ O<<n, ( 5) wih c defined by (2). The assumed equaliy of V and V' he limi as 4-0 or n yields he boundary condiions in and r = 0. n- (6) Muliplying in (5) by and forming a complee differenial on he lef hand side, we ge he equivalen equaion d - r exp{-f(o'+~' )ds}] = c exp{-f(o'+~'+ )ds}, O<<n. {7) dl 0 x+s 0 x s Inegraing (7) from 0 o and employing he firs condiion in (6), we obain he "rerospecive" formula = f 0 exp{f(o'+~' )ds}d~, x+s ~ (8) Similarly, inegraing from o n and employing he second condiion in (6), we obain he "prospecive" formula
- 5 - r = n ' f c exp{-f(o'+~' )ds}d', ' x+s O~~ n. ( 9) Now, applying (8) for ~ 0 and (9) for > 0, we arrive a he conclusions of he heorem. I I Before urning o a closer sudy of special cases, we noe ha, rivially, he condiions of he heorem could be furher specified. If, for insance, c = 0 for each in some inerval (O, 1 ) or ( 2,o), hen i is readily seen from he proof ha r = 0 in his inerval. If, in paricular, c = 0 for all E (O,n), hen r = 0 for all. The possibiliy of 0 being 0 or n is no reaed in he heorem since i would imply he case jus menioned, wih vanishing c. 3. Applicaions of he basic heorem We immediaely obain "coninuous" analogies of he wo corollaries presened in Jordan (1967) for he discree case: Corollary 1 Assume ha premiums are payable coninuously a a consan rae hroughou he duraion of he policy. Then, provided ha he sandard reserve increases wih duraion, we have: (i) An increase in he force of ineres produces a decrease in reserves. (ii) A decrease in he force of ineres produces an increase in reserves. Proof: Under he assumpions of he corollary we have ~~+ = ~x+, consan premiums n = n and n~ = n', say, and h~ = h, which makes (2) assume he form
- 6 - As V is assumed o be increasing, c is increasing or decreasing according as 6' > 6 or 6' < 6. Thus iems {i) and (ii) in he corollary follow from he corresponding iems in he heorem. I I As noed by Jordan, he requiremen ha V be increasing is normally saisfied in he case of pure endowmen and whole lifeor endowmen insurance, bu no in he case of erm insurance. The corollary in Jordan ha corresponds o our Corollary sudies, wihin he discree se-up, he effec of a change in he annual ineres rae, Which is equivalen o he change in he force of ineres considered here. The pracical relevance of sudying he effecs of such changes is obvious. The second of Jordan's corollaries deals wih changes in he annual raes of moraliy, 1 q x = 1 - exp {-f ~ ds }, 0 x+s x = 0,1,.., w-1. Resuls are obained for uniform changes in he moraliy rae, ha is, q' = q +k X X for all x = 0,1,...,w-1. Such changes are rarely, if ever, encounered in pracice, and he value of he menioned corollary herefore is mainly due o he indicaion i gives of wha can be expeced by more realisic variaions in moraliy assur~ions. The coninuous se-up, however, allows for resuls of immediae pracical significance since he criical funcion in his case depends on he moraliy laws hrough he forces of moraliy insead of he annual moraliy raes. He firs make some definiions: Saring from a sandard ~ y, we shall say ha he special ~ represens a progressive increase (decrease) in moraliy if y ~I ) ~ (~ ~~ ) for all y and is a non-vanishing and y y y y non-decreasing funcion of y. A degressive increase (decrease)
- 7 - in moraliy is defined by replacing "non-decreasing" by "nonincreasing" in he above definiion. A change in moraliy which is a he same ime boh progressive and degressive, ha is, ~ = y ~ +k for all y, will be called uniform. TI1ese noions of change y in moraliy comprise any changes in one of he hree parameers of he Gomperz-Makeham inensiy ~y = a+ ~cy; a change in a is uniform, and any change eiher in ~ or in c is progressive. We shall daaonsrae he following resuls: Corollary 2. Consider an n-year endowmen life insurance wih sum s and premiums payable coninuously a a consan rae hroughou he duraion of he policy. If he sandard reserve increases wih duraion, hen we have: (i) A uniform or degressive increase in moraliy produces a decrease in reserves. (ii) A uniform or degressive decrease in moraliy produces an increase in reserves. Proof: We now have 0 I = 0, 1 = 1 and n' = 1 (consans), h = s~x+, and h~ = s~~+" Subsiuing his in (2), we ge C = 1 I - 1 + ( ~ - ~ I ) ( S -v ) x+ x+ The conclusions of he corollary resul from he basic heorem by noing ha s-v is non-increasing and non-negaive. I I A nmnber of furher special resuls may be derived from Lidsone's heor~1. Corollary 2 is easily modified so as o yield a resul for an n-year pure endowmen wih consan, coninuous premium hroughou he duraion of he policy. In his case a change in moraliy gives c = n' - n + ( ~ ~+ - ~ x+ ) V '
- 8 - and i can be concluded ha a uniform or progressive increase in ). moraliy produces a decrease in reserves, provided ha he sandard reserve increases wih duraion (as will normally be he case He close our presen discussion wih an ineresing applicaion ha has been oulined wihin he discree framework by Sverdrup (1982). Consider a sandard basis wih h = 11: = 0 n 0 consan premium inensiy n = n, O<<n. Le he special basis differ from he sandard one only by a change o naural premium payn~n, ha is n' = 0 and n' = h for O<<n. Then (1) 0 applied o V, wih he iniial condiion v0+ = 0, implies ha V~ = 0 for all, and so he basic heorem above provides a crierion for deciding wheher he sandard reserve saisfies he and requiremen of being non-negaive for all. The criical funcion in (2) n~v reduces o and we conclude from par {i) of he main heorem ha a sufficien condiion for V o be non-negaive is ha h is a nondecreasing funcion of. On he oher hand, if h is sricly decreasing, as i may be for insance for decreasing life insurances, we can conclude ha v is negaive in (O,n). References Baillie, D.C. (1951): Acuaries 3, The equai9n of equilibrium. Trans. Soc. of 7 4-81. Gershenson, H. (1951): Reserves by differen moraliy ables. Trans. Soc. of Acuaries 3, 68-73. Jordan, c.w. (1967): Acuaries, Life Coningencies, 2nd ed. Chicago. Soc. of Lidsone, G.J. (1905): Changes in pure premium values consequen upon variaions in he rae of ineres or rae of moraliy. Journ. Insiue of Acuaries 3. Sverdrup, E. (1982): Forsikringsfonde i livsforsikring. Saisical memoirs No. 1, 1982. Insiue of Mahemaics, Universiy of Oslo. (In Norwegian.)