Mortality Variance of the Present Value (PV) of Future Annuity Payments

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1 Morali Variance of he Presen Value (PV) of Fuure Annui Pamens Frank Y. Kang, Ph.D. Research Anals a Frank Russell Compan Absrac The variance of he presen value of fuure annui pamens plas an imporan role in he annui business. No onl do we wan o know how dispersed a disribuion of PV's is, we also ma use he normal approimaion o approimae he disribuion of PV's of a specific annui produc. Based on he recen research b Kang [3], in a paou annui business, he morali provision for adverse deviaion, a requiremen in FASB 6 [], can be quanified b using he normal approimaion. Therefore, i is essenial o derive he variance formula for an kinds of annui producs. I. Inroducion This paper was moivaed in 998 b he Appoined Acuar of Aena Reiremen Services (now Aena Financial Services), Dr. Ja Vadiveloo, during m inernship in his deparmen and a he same ime being his advisee for m graduae sud a Universi of Connecicu. I began when Aena decided o implemen a new reserve calculaion compuer daabase ssem, Trion. To ensure a successful implemenaion and ransformaion from he old ssem, i was needed o have a reliable verificaion vehicle using spreadshees o calculae sauor reserves for a sample of annui conracs. Laer on, Dr. Vadiveloo was ineresed in he variance of benefi reserves for muliple lives paou annuiies. No unil his paper, he available variance formulas for annui producs are limied o hose of consan or increasing/decreasing (a a consan rae) benefi pamens. This paper aemps o provide a general variance formula for annuiies (single life and muliple lives) wih an paern of pamens. The variance of he presen value of fuure annui pamens plas an imporan role in life business. No onl do we wan o know how dispersed a disribuion of PV's is, we also ma use he normal approimaion o approimae he disribuion of PV's of a specific annui produc. The formulas for he variance of PV of classic annuiies such as annui due, firs-o-die and second-o-die are given in Bowers []. For more complicaed pes of annuiies wih nonlevel annui pamens such as 5% J&S or variable annuiies, he formulas for variance of he PV of fuure pamens have no been developed prior o his paper. In his paper, we generalize he variance o all annui producs b deriving an ieraion formula for he variance of he PV of fuure pamens for an paerns of benefi pamens. One of he advanages of using his variance is ha i can be easil implemened in spreadshee sofware such as MS Ecel.

No onl do we wan o know how dispersed a disribuion of PV's is, we also ma use he normal approimaion o approimae he disribuion of PV's of a specific annui produc.

2 II. Single Life Annui Recall ha in Bowers [] for a single life annui, le T be curae survival ime and Z be he presen value of fuure annui pamens. Denoe v = /(+i) and le B be he annui amoun o be paid a he end of ear. B so ha definiion = ω N where ω is he larges age in he morali Table. Similarl, Then Var (Z) = E (Z ) - (E (Z)). Z T = = E(Z) ( v B ) N q N = = = ω N v B E (Z ) [( v B )] q N N = =,,. III. Muliple Lives (): Recall he following formulas described in Bowers []. A. For second-o-die J&S conracs: Le T = curae survival ime for a second-o-die pair. Then Z T = v = B, where B equals annui pamens a he end of ear so long as a leas one of he annuians lives. Then E N ( Z ) = ω ( v B ) N q, N = = - N ( Z ) = ω [( v B )] N q, N = = E Again Var (Z) = E (Z ) (E (Z)).

Z T = = E(Z) ( v B ) N q N = = = ω N v B E (Z ) [( v B )] q N N = =,,. III. Muliple Lives (): Recall he following formulas described in Bowers []. A.

3 3 B. For firs-o-die J&S conracs: Le T be curae survival ime for a firs-o-die pair. Then Z T = v = B N ( Z ) = ω ( v B ) N q N = = E where B equals annui pamens a he end of ear when boh of he annuians live., E (Z ) [( v = ω N = N = B )] q N Then Var (Z) = E (Z ) (E (Z)). The above formulas assume ha he B 's are consans for each period. We also need o invesigae some annui producs where B varies b he saus of he pair. For eample, wha is he variance of a J&S conrac ha pas an annual amoun of $(.3), where is he numbers of ears afer issue, if boh survive; $.5 if onl survives; and $.5 if onl survives? Or wha is he variance of a J&S conrac ha pas an annual amoun of $.6 if boh are alive; $.5, if onl survives; and (.) s if onl survives, where s is numbers of ears since died. These quesions are no eas o answer. In he following secion, we provide an analical wa o find he variance of he PV of annui for an kind of annui produc. IV. Generalized Variance Formula for Non-Level Annui Pamens We would like o find he mean and he variance of Z: he PV of a J&S annui conrac wih non-level annual annui pamens. Firs, le us define he annui pamen amoun a ime (end of ear ) for an J&S conrac. Le B be he annual annui pamen amoun a ime (he end of ear ), given b one of he following: b b b : Benefi pamen a ime : Benefi pamen a ime when boh and arealive; when onl lives; :Benefi pamen a ime when onl lives. Wrie Z = Σ Z, where Z, he PV of annui pamens a ime, is equal o one of he following: v v v b b b ( PV of Benefi Pamens a ime when boh and are alive); (PV of Benefi Pamens a ime when (PV of Benefi Pamens a ime when onl onl lives); lives).

The above formulas assume ha he B 's are consans for each period. We also need o invesigae some annui producs where B varies b he saus of he pair.

4 4 and v is equal o /(+i), where i is he assumed fied ineres rae. We ma visualize he annui amoun a ime b he following ree: Year Year Year Year 3. b b b b b b b b b

5 5 Le α = Prob (Boh and survive a ime ) = p There is an obvious recursion formula α + = α p + p +. (.) For he probabili onl survives a end of ear, here are wo possibiliies from he viewpoin of end of ear (-): Boh and survive a end of ear (-), and survives for he following ear bu dies in he following ear; or Onl lives a ear (-), and survives for he following ear. Since hese wo cases are muuall eclusive, we sum hem up o obain a recursion formula for he probabili ha onl survives a ear. Le β = Prob(Onl survives a ime ) β + = (α ) p + q + + β p +. (.) The formula for onl survives is similar. Le γ = Prob (Onl survives a ime ) γ + = (α ) p + q + + γ p +. (.3) Ne, b using (.), (.), and (.3), we can consruc ables for each annui amoun Z, and he corresponding probabiliies. We can hen use hese ables o find he firs and second momens of Z. Table.. The PV of annui pamens and he corresponding probabiliies a he end of he firs ear. Z v b v b v b Prob(Z ) α = p β = p q γ = p q - α - β - γ Table. shows he PV of annual annui pamens and heir corresponding probabiliies a he end of firs ear. Table. gives he annui pamens and heir corresponding probabiliies a he end of he -h ear. In he hird row of Table., we calculae he desired probabiliies using he previous able and he recursion formulas (.), (.), and (.3).

) For he probabili onl survives a end of ear, here are wo possibiliies from he viewpoin of end of ear (-): Boh and survive a end of ear (-), and survives for he following ear bu dies in he following

6 6 Table.. The PV of annual annui pamens and he corresponding probabiliies a he end of he h ear. Z v b v b v b Prob(Z ) -p p + q + -p p + q + P + p q + p q - α - β - γ Prob(Z ) b using α, β, and γ α β = α - p + q + + β - p + γ = α - p + q + + γ - p + - α - β - γ We can calculae he firs momen of Z b definiion for all ears. Tha is, E (Z ) = (v b ) (α ) + (v b ) (β ) + (v b ) (γ ) + (- α - β - γ ); and E (Z ) = (v b ) (α ) + (v b ) (β ) + (v b ) (γ ) (.4). The reserve for his J&S conrac hen equals he sum of all E (Z ). We can also calculae he second momen of Z : E (Z ) = (v b ) (α ) + (v b ) (β ) + (v b ) (γ ) + (- α - β - γ ) (.5) Afer he firs momen and second momen for ear are obained b (.4) and (.5), we can find he variance for ear b Var (Z ) = E (Z ) - (E (Z )). Since Z and Z s are dependen, we also have o capure he covariance of all pairs Z and Z s, s in order o compue Var(Z) = Var( Z ). Var (Z) = Var ( Z ) = Var(Z ) + = = < s Cov(Z, Z ) s (.6) We can obain Σ Var (Z ) in (.6) b combining (.4) and (.5). In order o find he covariance of all pairs of (Z, Z s ), s, we consruc a able o find E(Z Z s ). Table.3 provides he join probabili of Z and Z s. B convenion, we le s = +k. In Table.3, he firs row and firs column represen he PV of annui amoun of Z and Z +k, respecivel. The middle bo in Table.3 consiss of he join probabiliies of PV of annui amoun.

Z b definiion for all ears. Tha is, E (Z ) = (v b ) (α ) + (v b ) (β ) + (v b ) (γ ) + (- α - β - γ ); and E (Z ) = (v b ) (α ) + (v b ) (β ) + (v b ) (γ ) (.4).

7 7 Table.3 Join densi for Z and Z +k Z v b v b v b Z +k v +k b +k α +k (,) (,) (,3) α +k v +k b +k β +k - β k p + (,) β k p + (,) (,3) β +k v +k b +k γ +k - γ k p + (3,) (3,) γ k p + (3,3) γ +k α k q + β kq k q + γ - α k q - β γ - α +k - β +k - γ +k α β γ - α - β - γ The probabili in cell (,) of he middle bo of Table.3 represens he probabili ha boh and survive a he end of ear and onl survives a he end of ear (+k). =Prob (Onl survives a he end of ear (+k)) - Prob (Onl survives a he end of ear and onl survives a he end of ear (+k)) = β +k - β k p + (.7) Similarl, we can find he probabiliies in he oher cells in he middle bo of Table.3. If we sum all he elemens column-wise from he second column o he fourh column, i is no surprise ha we have he marginal probabili of Z ; if we sum all he elemens row-wise from he second row o he fourh row, we have he marginal probabili of Z +k. Having done all he calculaion in Table.3, we can find he epeced values of all producs (Z Z k ). The covariance is hen obained b: Cov (Z, Z k ) = E (Z Z k ) - E (Z ) E (Z k ), for all k. (.8) Finall, Var (Z) can be calculaed b plugging (.4), (.5), and (.8) ino (.6).

q + γ - α k q - β - + + γ - α +k - β +k - γ +k α β γ - α - β - γ The probabili in cell (,) of he middle bo of Table.

8 8 References. AICPA. Life Insurance Accouning. American Insiue of Cerified Public Accounans, 996. BOWERS e al, Acuarial Mahemaics. ed. Socie of Acuaries, Frank Y Kang, Analsis and Implemenaion of Provision for Adverse Deviaion for Paou Annuiies: A Sochasic Approach, Docoral Thesis, Universi of Connecicu, 999

BOWERS e al, Acuarial Mahemaics. ed. Socie of Acuaries, 986 3.

A Re-examination of the Joint Mortality Functions

A Re-examination of the Joint Mortality Functions Norh merican cuarial Journal Volume 6, Number 1, p.166-170 (2002) Re-eaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali