pimal Annui Purchasing Virginia R. Young and Moshe A. Milevsk Version: 6 Januar 3 Young is an Associae Professor a he chool of Business, Universi of Wisconsin-Madison, Madison, Wisconsin, 5376, UA. he can be reached a Tel: (68) 65-3494, Fax: (68) 63-34, E-mail: voung@bus.wisc.edu. Milevsk is an Associae Professor of Finance a he chulich chool of Business, York Universi, Torono, nario, M3J P3, Canada, and he Direcor of he Individual Finance and Insurance Decisions (IFID) Cenre a he Fields Insiue. He can be reached a Tel: (46) 736- ex 664, Fax: (46) 763-5487, E-mail: milevsk@orku.ca.
pimal Annui Purchasing Absrac: We find he opimal annui-purchasing scheme for an individual who seeks o maximize her expeced uili of lifeime consumpion and beques. Milevsk and Young () found he opimal ime for an individual, who has no pre-exising annuiies, o annuiize all her wealh. We now allow he individual o possess pre-exising annuiies, o annuiize onl a porion of her wealh a a given ime, o bu annuiies more han once -- even coninuousl, if ha is opimal -- and o consume somehing oher han he annui income afer annuiizaion. JEL Classificaion: J6; G Kewords: Insurance; Life Annuiies; Asse Allocaion; Reiremen; pimal Consumpion; pimal Invesmen.. Inroducion We find he opimal annui-purchasing scheme for an individual who seeks o maximize her expeced uili of lifeime consumpion and beques. Milevsk and Young () found he opimal ime for an individual, who has no pre-exising annuiies, o annuiize all her wealh. Afer annuiizaion, Milevsk and Young () assumed ha he individual consumes exacl he annui income. B conras, we allow he individual o possess pre-exising annuiies, o annuiize onl a porion of her wealh a a given ime, o bu annuiies more han once (even coninuousl, if ha is opimal), and o consume somehing oher han he annui income afer annuiizaion. In ecion, we presen our model and argue ha if he marginal uili of annui income is larger han he adjused marginal uili of wealh, hen he individual will annuiize a lump sum. Thereafer, she will bu annuiies a a coninuous rae (possibl zero) in order o keep he marginal uili of annui income no larger han he adjused marginal uili of wealh. The marginal uili of wealh is adjused b mulipling b he price of an annui a a given age. Thus, he annui-purchasing problem is qualiaivel similar o he problem of opimal consumpion and invesmen in he presence of proporional ransacion coss. ee ecion for deails.
3 In ecion 3, we examine he annui-purchasing problem for he specific case of an individual wih preferences ha exhibi consan relaive risk aversion (CRRA). We use he homogenei proper of he value funcion o reduce he problem s dimension b one. We sud properies of he opimal consumpion, invesmen, and annuipurchasing policies. We show ha if he wealh-o-income raio is larger han a given number a ime, hen he individual will purchase a lump sum annui so ha he wealho-income raio equals ha specific number. Conversel, if he wealh-o-income raio is less han ha number, hen he individual will bu no annui a ha ime. ecion 4 concludes he paper.. pimal Annui Purchasing wih General Preferences In his secion, we consider he opimal annui-purchasing problem for an individual who seeks o maximize her expeced uili of lifeime consumpion and beques. We assume ha her preferences are raher general, and in ecion 3, we specialize o he case for which preferences exhibi CRRA. We allow he individual o bu annuiies in lump sums or coninuousl, whichever is opimal. ur resuls are similar o hose of Dixi and Pindck (994, pp 359ff). The consider he problem of a firm s (irreversible) capaci expansion. For our individual, annui purchases are also irreversible, and his leads o he similari in resuls. We assume ha an individual can inves in a riskless asse whose price a ime, X, follows he process dx s = rx s ds, X = x >, for some fixed r. Also, he individual can inves in a risk asse whose price a ime s, s, follows geomeric Brownian moion given b ds = µ sds + σsdbs, = >, in which µ > r, σ >, and B s is a sandard Brownian moion wih respec o a filraion {F s } of he probabili space (Ω,F, Pr). Le W s- be he wealh a ime s- of he individual (before purchasing annuiies a ha ime), and le π s be he amoun ha he decision maker invess in he risk asse a ime s. The noaion W s- denoes he lef-hand limi of wealh before he individual bus an annui a ha ime; we allow annui purchasing o
4 occur in lump sums, if ha is opimal. I follows ha he amoun invesed in he riskless asse is W s- - π s. Also, he decision maker consumes a a rae of c s a ime s. As for he acuarial assumpions, le p denoe he subjecive condiional probabili ha an individual aged (x) believes he or she will survive o age (x + ). I is defined via he subjecive hazard funcion,, x λ x + s b he formula px = ( λ x + s ds ) exp. ee Bowers e al. (997) for furher deails on his noaion. We have a similar formula for he objecive condiional probabili of survival, p, in erms of he objecive hazard funcion, λ +. The acuarial presen value of a life annui ha pas $ per ear x s coninuousl o (x) is wrien a x. I is defined b x r x x a = e p d. If we use he subjecive hazard rae o calculae he survival probabiliies, hen we wrie a, while if we use he objecive (pricing) hazard rae o calculae he survival probabiliies, hen we wrie a x. Jus o clarif, b objecive a x, we mean he acual marke prices of he annui, whereas a denoes wha he marke price would be were he insurance x compan o use he individual s subjecive assessmen of her morali. The individual has a non-negaive annui income rae a ime s of A s- before an annui purchases a ha ime. We assume ha she can purchase an annui a he (unloaded) price of x + s. Thus, wealh follows he process a x + s per dollar of annui income a ime s, or equivalenl, a age [ ] dws = rws + ( µ r) πs + As cs ds + σπsdbs ax+ sdas, (.) W = w. Again, he negaive sign for he subscrip on wealh and annuiies denoes he lef-hand limi of hose quaniies before an (lump-sum) annui purchases. We assume ha he decision maker seeks o maximize, over admissible {c s, π s, A s }, her expeced uili of lifeime consumpion and beques. Admissible {c s, π s, A s } are hose ha are measurable wih respec o he informaion available a ime s, namel F s, ha resric consumpion and wealh o be non-negaive, ha resric he annui-income process o be non-negaive and non-decreasing (i.e., annui purchases are irreversible), and ha resul in (.) having a unique soluion; see Karazas and x
5 hreve (998), for example. We also allow he individual o value expeced uili via her subjecive hazard rae (or force of morali), while he annui is priced b using he objecive hazard rae. Denoe he random ime of deah of our individual b τ. Thus, he value funcion of he individual a ime, or age x +, defined on D= {( w, A, ) : w, A, } is given b U( w, A, ) = sup E e p u ( c ) ds+ e u ( W ) W = w, A = A r( s ) r( τ ) s x s τ { cs, π s, As} + r( s ) sup E e s px { u( cs) λx su( Ws) } ds W w, A A + +, { cs, π s, As} = + = = (.) in which u and u are sricl increasing, concave uili funcions of consumpion and beques, respecivel. Noe ha we assume ha he individual discouns consumpion a he riskless rae r. If we were o model wih a subjecive discoun rae of sa ρ, hen his is equivalen o using r as in (.) and adding ρ - r o he subjecive hazard rae. Thus, here is no effecive loss of generali in seing he subjecive discoun rae equal o he riskless rae r. In he following proposiion, we presen some properies of he value funcion. Proposiion.: (i) The value funcion U is joinl concave in w and A and sricl increasing wih respec o boh w and A. (ii) The value funcion U is coninuous on D. Proof: The concavi of U follows from he concavi of u and u and from he lineari of (.) wih respec o he conrols. U is sricl increasing wih respec o wealh because if he individual has wealh w > w, hen she can bu an annui using w w (w w ) and consume an addiional for he res of her life. Therefore, a x + she is beer off han if she had onl wealh w. imilarl, if he individual has annui income A > A, hen she can consume an addiional (A A ) for he res of her life and be beer off han if she had income A. The proof of he coninui of U follows from hreve and oner (994).
6 We coninue wih a formal discussion on he derivaion of he associaed Hamilon-Jacobi-Bellman equaion. If da s = A is a lump-sum purchase, hen he HJB equaion for he value funcion U is ( r+ λ ) U = U + ( rw+ AU ) + max [ u ( c) cu ] + max ( µ r) πu + σ π U x+ w w w ww c π + max U( w ax+ A, A+ A, ) U( w, A, ) + λx+ u( w). w A a x+ (.3) ee Björk (998) for clear derivaions of such HJB equaions. The firs-order necessar condiion for A gives us ha A solves he following equaion: U ( w a A, A+ A, ) = a U ( w a A, A+ A, ). (.4) A x+ x+ w x+ pecificall, he lump-sum purchase is such ha he marginal uili of annui income equals he adjused marginal uili of wealh. This is parallel o man resuls in economics. Indeed, he marginal uili of annui income can be hough of as he marginal uili of he benefi, while he adjused marginal uili of wealh can be hough of as he marginal uili of he cos. Thus, he lump-sum purchase forces he marginal uiliies of benefi and cos o equal. From Proposiion., we know ha he value funcion U is increasing and concave wih respec o wealh w and annui income A. Thus, if U ( w, A, ) > a U ( w, A, ), (.5) A x+ w hen b decreasing wealh and increasing annui income, we can achieve equali as in equaion (.4). B following he argumens in Dixi and Pindck (994, pp 359ff) or in Zariphopoulou (99), we discover ha he opimal annui-purchasing scheme is a pe of barrier conrol. pecificall, if inequali (.5) holds a ime, hen he individual will spend a lump-sum amoun in order o reach equali as in (.4). n he oher hand, if a ime, we have U ( w, A, ) a U ( w, A, ), (.6) A x+ w hen he individual will bu annuiies a a coninuous rae (possibl zero) in order o mainain inequali (.6).
7 Thus, he curve in wealh-annui income space (w, A) a ime ha is defined b he equali U ( w, A, ) = a U ( w, A, ) (.7) A x+ w can be hough of as a barrier. If wealh and annui income lie o he righ of he barrier a ime, hen he individual will immediael spend a lump sum of wealh o move diagonall o he barrier. The move is diagonal because as wealh decreases o purchase more annuiies, annui income increases. Thereafer, annui income is eiher consan if wealh is low enough o keep o he lef of he barrier, or annui income responds coninuousl o infiniesimall small changes of wealh a he barrier. In he region of no annui purchasing, or inacion, namel where inequali (.6) holds sricl, we have ha he value funcion U saisfies he following HJB equaion: ( r+ λ x+ ) U = U+ ( rw+ AU ) w+ max [ u( c) cuw] + max ( µ r) πuw σ π Uww λx+ u( w). c π + + (.8) There exis verificaion heorems ha ell us if he value funcion U is smooh and if U % is a smooh soluion of he associaed HJB equaion, hen under cerain regulari condiions, U% = U (Fleming and oner, 993). However, in general, we can onl asser ha he value funcion is solves he HJB equaion in he sense of viscosi soluions. Indeed, b following he argumens of Zariphopoulou (99) or of Duffie and Zariphopoulou (993), one can show ha he value funcion is a consrained viscosi soluion of he HJB equaion under suiable regulari condiions. pecificall, we have he following proposiion. Proposiion.: The value funcion U is a consrained viscosi soluion on D of he Hamilon-Jacobi-Bellman equaion min ( r+ λ x+ ) U U ( rw+ AU ) w max [ u( c) cuw] max ( µ r) πuw σ π Uww λx+ u( w), c π + ax+ Uw U A =. In he nex secion, we analze he HJB equaion in (.8) for he case of CRRA preferences.
8 3. pimal Annui Purchasing wih CRRA Preferences In his secion, we specialize he resuls of ecion o he case for which he individual s preferences exhibi CRRA. pecificall, le = and u() c = k c, >,, k. (3.) () c, u c The parameer k > weighs he uili of beques relaive o he uili of consumpion. Davis and Norman (99) and hreve and oner (994) show ha for CRRA preferences in he problem of consumpion and invesmen in he presence of ransacion coss, he value funcion U is a soluion of is HJB equaion in he classical sense, no jus in he viscosi sense. Generall, if he force of morali is evenuall large enough o make he value funcion well-defined, hen his resul holds for our problem, oo. For he uili funcions in (3.), i urns ou ha he value funcion U is homogeneous of degree wih respec o wealh w and annui income A. Tha is, U(bw, ba, ) = b - U(w, A, ) for b >. Thus, if we define V b V(z, ) = U(z,, ), hen we can recover U from V b U( w, A, ) = A V( w A, ), for A >. I follows ha he HJB equaion for U from Proposiion. becomes he following equaion for V: cˆ z min ( r+ λ ) ( ) max ˆ max ( ) ˆ ˆ x+ V V rz+ Vz cvz r Vz Vzz k x, cˆ µ π σ π λ + ˆ π + ( z+ ax+ ) Vz ( ) V =, (3.) c π in which cˆ =, and ˆ π =. Davis and Norman (99) and hreve and oner (994) use A A he same ransformaion in he problem of consumpion and invesmen in he presence of ransacion coss. Also, Duffie e al. (997) and Koo (998) use his ransformaion o sud opimal consumpion and invesmen wih sochasic income. We nex sud properies of he opimal consumpion and invesmen policies. Throughou we assume ha he value funcion U is coninuousl wice differeniable and saisfies he HJB equaion given in Proposiion.. Equivalenl, we assume ha he value funcion V is coninuousl wice differeniable and saisfies he HJB equaion (3.)
9 for z = w/a. Thus, he opimal consumpion and invesmen policies are given b he firsorder necessar condiions. We follow he work of Davis and Norman (99) and Koo (998) b expressing he value funcion U in an inuiivel pleasing form. Lemma 3.: (i) Define he funcion p in he region of inacion b UA( w, A, ) p U ( w, A, ) for w >, A >,. w Then, p is a funcion of he wealh-o-income raio z = w/a and. Define he funcion q in he region of inacion b ( ) U( w, A, ) q ( + (, ) ) w p z A for w >, A >,. (ii) (iii) Then, q is also a funcion of z and. The value funcion U in he region of inacion can be wrien as qz (,) = + U( w, A, ) ( w p( z, ) A) for w >, A >,. The value funcion V in he region of inacion can be wrien as qz (,) V(,) z = ( z+ p(,)) z for z >,. Proof: The fac ha p and q are funcions of z and follows from he homogenei of U. The res of he lemma is sraighforward o show. I follows from (.6), (.7) and Lemma 3.(ii) ha p(,) z < a + if (w, A, ) is in he region of inacion, and p(,) z = a x + if (w, A, ) is on he barrier, in which z = w/a. Nex, we give some properies of he funcions p and q and use he firs-order necessar condiions from (.8) o derive he opimal consumpion and invesmen policies in he region of inacion. Proposiion 3.: (i) In he region of inacion, he funcion p(z, ) is a non-decreasing funcion wih respec o z, and he funcion q(z, ) saisfies qz(,) z ( ) pz(,) z = for z >,. qz (,) pz (,) x (ii) In he region of inacion, he opimal consumpion polic is given b
= + c* q( Z*, ) ( W* p( Z*, ) A* ), (iii) (iv) in which W* and A* are he opimall-conrolled wealh and annui income processes. Also, Z* = W* /A*. In he region of inacion, he opimal invesmen polic in he risk sock is given b µ r π * = ( W* + p( Z*, ) A* ). Define he accouning oal wealh b σ + pz( Z*, ) ATW ( w, A, ) = w + A a +. x ne can hink of he accouning oal wealh as he wealh required o have liquid wealh of w and an annui income of A. Then, he raio of consumpion o accouning oal wealh is increasing wih respec o he wealh-o-income raio. Proof: For he sake of brevi, we omi he proof of his proposiion. Please refer o Koo (998) for deails. From he fac ha p is non-decreasing wih respec o z, we have he following form for he barrier. Proposiion 3.3: For each value of, here exiss a value of he wealh-o-income raio z () such ha (i) If z > z (), hen he individual immediael bus an annui so ha w Aa x + A+ A = z( ); (ii) If z < z (), hen he individual bus no annui; i.e., she is in he region of inacion. I follows ha a each ime poin, he barrier is a ra emanaing from he origin and ling in he firs quadran of (w, A) space. Proof: If (w, A, ) is in he region of inacion, hen for z z = w /A, we have ha p z pz < a + (,) (,) x. Thus, an (w, A, ) such ha z < w/a is in he region of inacion. Le z () solve he equaion
p( z ( ), ) = a +. Then, z () has he properies claimed in he proposiion. Davis and Norman (99) and hreve and oner (994) find a similar resul for he problem of opimal consumpion and invesmen in he presence of proporional ransacion coss. We are now read o give a more complee formulaion of he value funcion U. Proposiion 3.4: In he case of CRRA preferences for consumpion and beques (3.), he value funcion U in (.) is given b qz U( w, A, ) = qz + = x (,) ( (, ) ) w+ p z A if z = w A< z ( ), ( ( ), ) ( ) w ax+ A if z w A z ( ), in which p and q are given in Lemma 3., and z () is given in Proposiion 3.3. Proof: The formulaion of U for z < z () follows from Lemma 3.(ii) because we are in he region of inacion. If z = z (), hen we are on he barrier, so b definiion of z (), we have p( z( ), ) = a +. The form for U follows b coninui (Davis and x Norman, 99; hreve and oner, 994). Finall, if z > z (), hen he individual immediael bus an annui so ha from which i follows ha w Aa x + A+ A = z( ), U( w, A, ) = U( w Aa, A+ A, ) x+ ( ( ), ) ( ) ( ) w Aax+ A A ax+ qz = + + qz = + ( ( ), ) ( ) w Aax+. 4. Linearizaion of he HJB Equaion wih No Beques Moive In his secion, we linearize he nonlinear parial differenial equaion for V in he region of inacion given b equaion (3.) wih no beques moive (k = ). To do his, we consider he convex dual of V defined b
z> [ ] V% (, ) = max V( z, ) z. (4.) The criical value z* solves he equaion = V ( z, ) ; hus, z* = I(, ), in which I is he inverse of V z wih respec o z. I follows ha V% (, ) = V[ I(, ), ] I(, ). (4.) z Noe ha V% (, ) = V [ I(, )] I (, ) I(, ) I (, ) z = I (, ) I(, ) I (, ) = I(, ). (4.3) We can rerieve he funcion V from V % b he relaionship V(,) z = min V% (,) + z. (4.4) > Indeed, he criical value * solves he equaion = V % (, ) + z = I(, ) + z; hus, * = V ( z, ), and z V% ( *, ) + z* = V% [ V ( z, ), ] + zv ( z, ) = V[ I( V ( z, ), ), ] V ( z, ) I( V ( z, ), ) + zv ( z, ) = V(,) z zv (,) z + zv (,) z = V(,), z in which we use equaion (4.) for he second equali. Nex, noe ha and z z z z z z z z V% (, ) = I(, ) = V[ I (, ), ], (4.5) zz V% (, ) = V [ I(, ), ] I (, ) + V[ I(, ), ] I (, ) z = I(, ) + VI [ (, ), ] I(, ) = V[ I(, ), ]. (4.6) In he parial differenial equaion for V wih no beques moive (k = ), le z = I(, ) o obain ( r+ λ ) V[ I(, ), ] = V[ I(, ), ] + ( ri(, ) + ) V [ I(, ), ] x+ z r ( Vz[ I(, ), ]) µ + ( Vz[ I(, ), ]). σ V [ I(, ), ] zz
3 Rewrie his equaion in erms of V % o ge ( r+ λx ) { V% + (, ) + I (, )} = V% (, ) + ( rv% (, ) + ) + m, V% (, ) in which µ r m =, σ or equivalenl, = V% ( r+ λx ) V% + + λx+ V % + m V%, (4.7) wih boundar condiions given implicil b V(, ) = and V( a +, ) =. Noe ha (4.7) is a linear parial differenial equaion. Nex, consider he boundar condiions p( z( ), ) = a + and from smooh pasing a he free-boundar, pz ( z( ), ) =. We can wrie hese in erms of V as and ( ) V( z ( ), ) + ( z ( ) + a ) V ( z ( ), ) =, (4.8) x+ z z x+ zz V ( z ( ), ) + ( z ( ) + a ) V ( z ( ), ) =. (4.9) We also have a boundar condiion a z = because a ha poin, he individual has no wealh o inves in he risk asse; hus, V zz (, ) = - for all. Because V z > is sricl decreasing wih respec o z, we have a () > () for all, in which a () and () are defined b () = V (, ), (4.) a z x x and () = V ( z (), ). (4.) z Thus, in erms of V %, he boundar condiions become and V% ( ( ), ) =, for V% ( ( ), ) =, (4.) a a ( ) V% ( (), ) + () V% ( (), ) = ax+ (), (4.3a) % ( ( ), ) ( ) % ( ( ), ). (4.3b) for V + V = ax+ To solve he second-order parial differenial equaion (4.7) wih free boundaries deermined b (4.) and (4.3), we propose he following algorihm. Firs, suppose we
4 have esimaes of he funcions and b. Use hese esimaes o solve he parial differenial equaion (4.7) wih normal condiions a and b given b he second equaions of (4.) and (4.3). Re-esimae and b b using he firs equaions of (4.) and (4.3), and repea he process unil i converges. Implemening his algorihm is he subjec of fuure research. In he nex secion, we solve he ssem in he case for which he forces of morali are consan. 5. Consan Forces of Morali 5. oluion of he Boundar-Value Problem λ If we assume ha he forces of morali are consan, ha is, λ λ + and x+ λ for all, hen we can obain an implici analical soluion of he value funcion V via he boundar-value problem given b (4.7), (4.), and (4.3). In his case, V, V %, a, and are independen of ime, so (4.7) becomes he ordinar differenial equaion: = ( r+ λ ) V% ( ) + λ V% ( ) + m V% ( ), (5.) wih boundar condiions V% ( a ) =, for V% ( a ) =, (5.) x and The general soluion of (5.) is V% + V% = (5.3a) r + λ ( ) ( ) ( ), % ( ) + % ( ) =. (5.3b) r + λ for V V B B V% ( ) = D + D + + C, (5.4) r wih D and D consans o be deermined b he boundar condiions, wih C given b and wih B and B given b λ C = r+ m, (5.5)
5 and and B = m λ + m λ + m r+ λ > m ( ) ( ) 4 ( ), B = m λ m λ + m r+ λ < m ( ) ( ) 4 ( ). (5.6) (5.7) The boundar condiions a give us B B D{ + ( B ) } + D{ + ( B ) } + =, r r + λ (5.8) B B DB { + ( B ) } + DB { + ( B ) } + =. r r + λ (5.9) olve equaions (5.8) and (5.9) o ge D and D in erms of : and B λ B D =, rr ( + λ ) B B + ( B ) D B λ B =. rr ( + λ ) B B + ( B ) (5.) (5.) Nex, subsiue for D and D in V% ( ) + V% ( ) = from (5.), specificall a a a B { } B DB { } + ( B ) a + DB + ( B ) a + =, o ge r B B ( ) a λ ( ) a λ λ B B B B + =. r+ λ B B r+ B B (5.) (5.) gives us an equaion for he raio a >. To check ha (5.) has a unique soluion greaer han, noe ha he lef-hand side () equals λ ( r + λ ) < when we se a =, () goes o infini as a goes o infini, and (3) is sricl increasing wih respec o a. Nex, subsiue for D and D in V% ( a ) = from (5.), specificall B B a a a DB + DB + + C =, o ge r
6 B ( ) λ ( ) B ( ) a ( ) a C a λ B B B B + + =. rr ( + λ ) B B + ( B ) rr ( + λ ) B B + ( B ) r (5.3) ubsiue for a in equaion (5.3), and solve for a. Finall, we can ge from and D and D from equaions (5.) and (5.), respecivel. nce we have he soluion for V %, we can recover V from V( z) = max V% ( ) + z > = a, (5.4) B B (5.5) = max D + D + + C + z, > r in which he criical value * solves a B B DB + DB + + C + z=. r (5.6) Thus, for a given value of z = w/a, solve (5.6) for and plug ha value of ino (5.5) o ge U(w, A) = V(z). Perhaps more imporanl, we are ineresed in he criical value z above which an individual spends a lump-sum o purchase more annui income. 5. Numerical Examples In his secion, we presen numerical examples o demonsrae he resuls of ecion 5.. Example 5.: uppose we have he following values of he parameers: λ = λ =.4; he force of morali is consan such ha he expeced fuure lifeime is 5 ears. r =.4; he riskless rae of reurn is 4%. µ =.8; he risk rae of reurn is 8%. σ =.; he risk asse s volaili is %. In Table, for various values of, we give he criical value of he raio of wealh o annui income z = w/a above which individual will spend a lump-sum of wealh o increase her annui income. We also include he amoun ha he individual will spend on annuiies for a given annui income of A = $5,: ( w za) ( + ( r+ λ ) z).
7 Table. Amoun of Mone pen on Annuiies for Various Levels of Wealh and Risk Aversion Wealh =.5 (z = 3.73) =. (z =.354) =.5 (z =.837) = 3. (z =.56) $,, $77,6 $79, $83,85 $858,9 $5, $33,384 $37,5 $395,99 $4,653 $5, $33,66 $6,866 $77,937 $89,59 $, $4,395 $34,635 $47,54 $55,655 $5, $ $ $3559 $,3 Noice from Table ha he amoun spen on annuiies increases, for a given level of wealh, as he individual becomes more risk averse, an inuiivel pleasing resul. Also, for a given level of risk aversion, he amoun spen on annuiies decreases as wealh decreases. Example 5.: In his example, we examine how he criical raio z changes as he parameers change. We ake given he values of he parameers as in Example 5. wih =. and deermine z b varing one parameer a a ime and keeping all he parameers fixed. In a sense, we did his for he coefficien of relaive risk aversion in Example 5.. In Tables hrough 7 we calculae z for changes in λ, λ = λ, λ, r, µ, and σ, respecivel. From Table, we see ha z increases monoonicall wih respec o λ because as he individual becomes healhier relaive o he objecive morali used in pricing he annuiies, annuiies become less aracive. Therefore, he individual will no annuiize as much of her wealh. Table. Criical Raio z as a Funcion of λ λ z..637..768..95.3.7.4.354.5.644.6.999.7 3.43.8 3.99.9 4.489. 5.33. 6.639.5 9.47
8. 5.95.5 5.5.3 38.5.4 84.38.5 78.7. 7,779 From Table 3, we see ha he criical raio z monoonicall decreases wih respec o λ = λ because annuiies become less aracive o he invesor as morali raes increase. Table 3. Criical Raio z as a Funcion of λ = λ λ = λ z..83. 5.664.3 3.45.4.354.5.79.6.35.7.4.8.845.9.7..59..437.5.99..8.5..3.87.4.5.5.34..9 From Table 4, we see ha he criical raio z decreases wih respec o λ for he same reason as in Table 3. However, he decrease is more marked here because he individual s subjecive morali is consan. Table 4. Criical Raio z as a Funcion of λ λ z..93. 7.79.3 3.93.4.354.5.56.6.
9.7.83.8.647.9.59..45..3.5.97..3.5.74.3.5.4.3.5.9..5 From Table 5, we see ha he criical raio z decreases as a funcion of r < µ because as r increases, annuiies become more aracive relaive o he risk asse. Table 5. Criical Raio z as a Funcion of r r z. 3.65. 7.63.3 4.97.4.354.5.88.6.497.7.4.75.3.78.5.79..8. From Table 6, we see ha he criical raio z increases as a funcion of µ because as µ increases, he risk asse becomes more aracive relaive o annuiies. Table 6. Criical Raio z as a Funcion of µ µ z.....3..4..5.74.6.638.7.36.8.354.9 3.65. 5.38
. 7.439..4.3 3.6.4 8.9.5 3.93 From Table 7, we see ha he criical raio z decreases as a funcion of σ because as σ increases, he risk asse becomes more volaile and hereb he risk asse becomes less aracive relaive o annuiies. Table 7. Criical Raio z as a Funcion of σ σ z.3 7,99.4 44..5.4.6 46.7.7 6.97.8 8.9.9 3.8..4. 6.674.5 4.63..354.5.537.3.9.4.638.5.4..4 References Bowers, N. L., H. U. Gerber, J. C. Hickman, D. A. Jones, and C. J. Nesbi (997), Acuarial Mahemaics, second ediion, ocie of Acuaries, chaumburg, Illinois. Björk, T. (998), Arbirage Theor in Coninuous Time, xford Universi Press, New York. Davis, M. H. A. and A. R. Norman (99) Porfolio selecion wih ransacion coss, Mahemaics of peraions Research, 5: 676-73.
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