Verification Theorems for Models of Optimal Consumption and Investment with Retirement and Constrained Borrowing


 Aron Stuart Rogers
 1 years ago
 Views:
Transcription
1 MATHEMATICS OF OPERATIONS RESEARCH Vol. 36, No. 4, November 2, pp issn X eissn hp://dx.doi.org/.287/moor INFORMS Verificaion Theorems for Models of Opimal Consumpion Invesmen wih Reiremen Consrained Borrowing INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. Philip H. Dybvig, Hong Liu Olin Business School, Washingon Universiy in S. Louis, S. Louis, Missouri 633 hp://dybfin.wusl.edu/, hp://apps.olin.wusl.edu/faculy/liuh/} Proving verificaion heorems can be ricky for models wih boh opimal sopping sae consrains. We pose solve wo alernaive models of opimal consumpion invesmen wih an opimal reiremen dae (opimal sopping various wealh consrains (sae consrains. The soluions are parameric in closed form up o a mos a consan. We prove he verificaion heorem for he main case wih a nonnegaive wealh consrain by combining he dynamic programming Slaer condiion approaches. One unique feaure of he proof is he applicaion of he comparison principle o he differenial equaion solved by he proposed value funcion. In addiion, we also obain analyical comparaive saics. Key words: volunary reiremen; invesmen; consumpion; free boundary problem; opimal sopping MSC2 subjec classificaion: Primary: 93, 49; secondary: 93E2, 49K, 49K2, 49L2 OR/MS subjec classificaion: Primary: finance/porfolio, finance/invesmen; secondary: finance/asse pricing Hisory: Received April 8, 2; revised November 5, 2, April 7, 2, June 3, 2. Published online in Aricles in Advance Ocober 4, 2.. Inroducion. Reiremen is one of he mos imporan economic evens in a worker s life. This paper conains a rigorous formulaion analysis of several models of life cycle consumpion invesmen wih volunary or maory reiremen wih or wihou a borrowing consrain agains fuure labor income. In hese models, opimal consumpion jumps a reiremen, if reiremen is volunary, he opimal porfolio choice also jumps a reiremen. If reiremen is volunary, he opimal reiremen rule gives human capial a negaive bea if wages are uncorrelaed wih he sock marke because reiremen comes laer when he marke is down. This leads o aggressive invesmen in he marke, a resul ha is dampened when borrowing agains fuure labor income is prohibied may be reversed when wages are posiively correlaed wih marke reurns. In he companion paper, Dybvig Liu [] focus on he economic inuiions for hese resuls. In his paper, we provide rigorous proofs. The main resuls in his paper are explici parameric soluions (up o some consans wih verificaion heorems analyical comparaive saics. In paricular, we combine he dual approach of Pliska [8], He Pagès [3], Karazas Shreve [5] Karazas Wang [6] wih an analysis of he boundary o obain a problem we can solve in a parameric form even if no known explici soluion exiss in he primal problem. Having an explici dual soluion allows us o derive analyically he impac of parameer changes, more imporanly, allows us o prove a verificaion heorem showing ha he firsorder (Bellman equaion soluion is a rue soluion o he choice problem. Compared o he exising lieraure (e.g., Pliska [8], Karazas Wang [6], he noborrowing consrain agains fuure labor income significanly complicaes he derivaions he proof of he verificaion heorem. The proof is suble because of ( he nonconvexiy inroduced by he reiremen decision, (2 he marke incompleeness (from he agen s view caused by he nonnegaive wealh consrain, (3 he echnical problems caused by uiliy unbounded above or below. Two common approaches o proving a verificaion heorem are he dynamic programming (FlemingRichel approach he separaing hyperplane (Slaer condiion approach. Boh approaches encouner difficulies in our seing so we use a hybrid of he wo (a separaing hyperplane afer reiremen dynamic programming before reiremen. The wo are combined wih opional sampling, where he coninuaion afer reiremen is replaced by he known value of he opimal coninuaion. One of he mos challenging asks for proving a verificaion heorem for opimal sopping problems is o show ha he proposed value funcion saisfies cerain inequaliy condiions so ha i is indeed opimal o sop a he proposed boundaries. One unique feaure of our proof is ha his is shown indirecly by applying he comparison principle o he differenial equaions solved by he proposed value funcion. This indirec mehod makes he proof simple elegan. So far as we know, we are he firs o use his approach o prove a verificaion heorem for his ype of conrol problem involving an opimal sopping ime. The res of he paper is organized as follows. Secion 2 presens he formal choice problems used in he paper. Secion 3 presens analyical soluions, comparaive saics, proofs. Secion 4 closes he paper. 62
2 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen Mahemaics of Operaions Research 36(4, pp , 2 INFORMS 62 INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. 2. Choice problems. We consider he opimal consumpion invesmen problem of an invesor who can coninuously rade a riskfree asse n risky asses. The riskfree asse pays a consan ineres rae of r. The risky asse price vecor S evolves as ds = d + dz S where Z is a sard n dimensional Wiener process; is an n consan vecor is an n n inverible consan marix so ha marke is complee wih no redundan asses; he division is elemen by elemen. The invesor also earns labor income y : [( y y exp y y ] y + 2 y Z ( where y is he iniial income from working y y are consans of appropriae dimensions. The invesor can choose o irreversibly reire a any poin in ime. The arrival ime d of he invesor s moraliy follows an independen Poisson process wih consan inensiy. The invesor can purchase insurance coverage of B W agains moraliy, where W is he financial wealh of he invesor a ime so ha, if deah occurs a, he invesor has a beques of W +B W = B. To receive he insurance coverage B W a he ime of moraliy, he invesor pays he insurer a a rae of B W, i.e., insurance is assumed o be fairly priced a he moraliy rae per uni of coverage. The invesor derives uiliy from ineremporal consumpion beques. The invesor has a consan relaive risk aversion (CRRA, ime addiive uiliy funcion (2 wih a subjecive ime discoun rae : [ d ( E e R c + R Kc d + e d kb d ] (2 where > is he relaive risk aversion coefficien, he consan K > indicaes preferences for no working in he sense ha he marginal uiliy of consumpion is greaer afer reiremen han before reiremen, he consan k > measures he inensiy of preference for leaving a large beques, he limi k implemens he special case wih no preference for beques, R is he righconinuous nondecreasing indicaor of he reiremen saus a ime (which is afer reiremen before reiremen. The sae variable R is he reiremen saus a he beginning of he invesmen horizon. Define { if saemen S is rue S = oherwise ( e T + if g (3 T + if = where r + y + y > (4 is he effecive discoun rae for labor income (assumed o be posiive r (5 is he price of risk. Below are he wo choice problems we focus on in his paper. Problem. Given iniial wealh W, iniial income from working y, he deerminisic ime o reiremen T wih associaed reiremen indicaor funcion R = T, choose adaped nonnegaive consumpion c, adaped porfolio, adaped nonnegaive beques B o maximize expeced uiliy of lifeime consumpion beques [ d ( E e R c + R Kc kb d + e d ] d = corresponds o he log uiliy case, which can be examined similarly. Mos of our resuls in he paper apply o he log case by aking.
3 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen 622 Mahemaics of Operaions Research 36(4, pp , 2 INFORMS subjec o he budge consrain W = W + rw s ds + s r ds + dz s + W s B s ds c s ds + R s y s ds (6 he labor income process (, he limied borrowing consrain W gy (7 INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. where gy is he marke value a of he fuure labor income. Le denoe he ype of borrowing consrains, wih = for he limied borrowing ype = for he noborrowing ype. Problem corresponds o Problem of Dybvig Liu [], Problem 2 wih = corresponds o Problem 2 of Dybvig Liu [], Problem 2 wih = corresponds o Problem 3 of Dybvig Liu []. Problem 2. Given iniial wealh W, iniial income from working y, iniial reiremen saus R, borrowing consrain ype, choose adaped nonnegaive consumpion c, adaped porfolio, adaped nonnegaive beques B, adaped nondecreasing reiremen indicaor R (i.e., a righconinuous nondecreasing process aking values o maximize he expeced uiliy of lifeime consumpion beques (2 subjec o he budge consrain (6, he labor income process before reiremen (, he borrowing consrain W R y (8 where R y / is he marke value a of he subsequen labor income. To summarize he differences across he problems, moving from Problem o Problem 2, he fixed reiremen dae T (R = T is replaced by free choice of reiremen dae (R, a choice variable along wih a echnical change in he calculaion of he marke value of fuure labor income gy o R y / when =. When =, hen he invesor faces a noborrowing consrain W Remark. Because he ime of moraliy d is independen of he Brownian moion, we have ha he objecive funcion [ d ( E e R c + R Kc [ ( = E e + R c d + e d [ ( K R c = E e + + kb kb d ] + R Kc + kb ] d ] d We denoe he value funcions for Problems 2 by vw y V W y R, respecively. Noe ha, because boh problems become he same afer reiremen, we have vw y T = V W y = V W y (9 3. The analyical soluion comparaive saics. The general idea of solving Problems 2 is o perform a change of variables ino a dual variable, he marginal uiliy of consumpion. The general advanage of his dual approach (especially for Problem 2 is ha i linearizes he nonlinear HamilonJacobiBellman (HJB equaion in he primal problem. This is consisen wih he mehod of Pliska [8] of convering a dynamic budge consrain ino a saic budge consrain. Le + r + + /2 ( For our soluions, we will assume >, which is also he condiion for he corresponding Meron problem (Meron [7] o have a soluion because, if <, hen an invesor can achieve infinie uiliy by delaying consumpion. Define he sae price densiy process by e r++ 2 Z (
4 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen Mahemaics of Operaions Research 36(4, pp , 2 INFORMS 623 This is he usual sae price densiy adjused o be condiional on living, given he moraliy rae fair pricing of long shor posiions in erm life insurance. Also, define b / INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. The soluion o Problem can be saed as follows. Theorem 3.. he iniial values ( f ˆ exp + k b T k b ˆ K b + k b Suppose > ha he limied borrowing consrain is saisfied wih sric inequaliy a W > gy (2 The soluion o an invesor s Problem can be wrien in erms of a dual variable ˆx (a normalized marginal uiliy of consumpion, where ( W + gy ˆx e + f y (3 Then, he opimal wealh process is he opimal consumpion policy is he opimal rading sraegy is = y he opimal beques policy is W = f y ˆx / gy (4 [ r Furhermore, he value funcion for he problem is c = K br y ˆx / (5 f ˆx / ] y g B = k b y ˆx / (6 W + gy vw y = f Proof. Because Problem can be ransformed ino a sard dual problem wih minor modificaions (e.g., Pliska [8], we omi he proof here. See Dybvig Liu [] for a skech of he proof using a separaing hyperplane o separae preferred consumpions from he feasible consumpions. Unlike Problem, Problem 2 also requires one o solve for he opimal reiremen decision. We conjecure ha i is opimal o reire when he wealhoincome raio is high enough, which corresponds o when a new dual variable x his a lower bound x. Then, as in a ypical opimal sopping problem, one imposes C condiion for he dual value funcion across x. In he presence of he noborrowing consrain (i.e., =, one also imposes he noriskyinvesmen condiion (i.e., he second derivaive of he dual value funcion is when wealh W his (or, equivalenly, when x his an upper bound x. Afer obaining he soluions, we verify ha all of our conjecures are indeed correc. Recall he definiions of in ( in (4. Define y y y 3 y y (7 3
5 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen 624 Mahemaics of Operaions Research 36(4, pp , 2 INFORMS INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/ (8 3 (9 ( b A + + xb + x ( A + x + b + b + xb + x (2 + ( ˆ/b b / x + b (2 / + ( ˆb x + x (22 b where is he unique soluion o q = 2 hus where ( K b q b + k b b ( + + ( K b b + k b b + ( + Then, he soluion o Problem 2 can be saed as follows. + b b + (23 Theorem 3.2. Suppose >, >, 2 >, 3 > ha he borrowing consrain holds wih sric inequaliy a he iniial condiion: 3 W > R y (24 The soluion o an invesor s Problem 2 can be wrien in erms of a dual variable x, where ( x x e + y x + e + y /y y max sup s min x e +s s y s /y / x where x solves y x x R = W (25 { ( = R inf x e + y x} y (26 ˆ xb b x R = A + x + A x + xb b + x Then, he opimal consumpion policy is he opimal rading sraegy is c = K br y x / if R = or x x oherwise (27 = y rx xx x R y x xx x R + xx R he opimal beques policy is B = k b y x / 2 The exisence uniqueness of he soluion o q = is shown in Lemma > is o ensure he finieness of he labor income; 2 > is o ensure he finieness of he expeced uiliy from working consuming labor income forever; 3 > is o avoid he degeneraion of he dual process x o a deerminisic funcion of ime. The razor edge case where 3 = is less ineresing needs a separae reamen.
6 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen Mahemaics of Operaions Research 36(4, pp , 2 INFORMS 625 he opimal reiremen policy is R = he corresponding reiremen wealh hreshold is W = y x x INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. he opimal wealh is Furhermore, he value funcion is where x solves W = y x x R (28 V W y R = y x R x x x R (29 y x x R = W (3 The invesor s problem can be associaed wih he dual opimal sopping problem, where he invesor solves [ ( ( ] x y max E e +s y s + k b xb s b + x s ds + e + y ˆ xb b subjec o (, he borrowing consrain where dx x = x d + x dz y xx y y x r 2 y y + y y (3 x y (32 The ransformed dual value funcion x y x/y y hen saisfies a variaional inequaliy 4 wih boundary condiions where L = x < x < x L < < x < x x > ˆ xb b x < x < x x = ˆ xb b < x x x x < < x < x xx x > < x < x x x x x = xx x = L 2 3x 2 xx 2 x x 2 + k b xb b + x 4 We do no prove ha he dualiy gap is zero or even ha he firsorder soluion of he dual problem is an acual soluion. However, we do no need hese resuls because our verificaion heorem works wih he primal objecive funcion.
7 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen 626 Mahemaics of Operaions Research 36(4, pp , 2 INFORMS INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. When =, Problem 2 can be ransformed ino one ha has been sudied by Karazas Wang (2 wih minor modificaions. 5 Therefore, we will only presen he proof for he case wih he noborrowing consrain, i.e., =. For noaion simpliciy, we use V W y R x R o represen V W y R x R, respecively. Le Then, by (27, (33, (34, we have x A + x + A x + xb b + x (33 x R = ˆx ˆ xb b (34 { ˆx if R = or x x x oherwise The following lemmas are useful for he proof of Theorem 3.2. Lemma 3.. Suppose =, >, >, 2 >, 3 >. Suppose here exiss a soluion o Equaion (23 (o be shown in Lemma 3.4. Then, (i ˆx is sricly convex sricly decreasing for x (ii x x, we have x ˆx; x x x, we have x x ˆ x x ( K b x < (35 b (iii ( + k b A < A + > x > b (iv x is sricly convex sricly decreasing for x < x. (v Given W >, here exiss a unique soluion x > o (25. In addiion, W defined in (28 saisfies he borrowing consrain (8. Proof of Lemma 3.. (i > implies ha b = / <. Then, because >, direc differeniaion shows ha ˆx is sricly convex sricly decreasing for x. (ii Le hx x ˆx I can be easily verified ha wih 2 3x 2 ˆ xx x 2 x ˆ x x 2 ˆx K b + k b xb b = (36 2 3x 2 xx x 2 x x x 2 x + k b xb + x = (37 b x = ˆx (38 x x = ˆ x x (39 x x = (4 xx x = 5 One such ransformaion is W W y y W y y We hank an anonymous referee for poining his ou. B B y c c y Ṽ W y R y V y W y R
8 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen Mahemaics of Operaions Research 36(4, pp , 2 INFORMS 627 INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. Then, by (36 (37, hx mus saisfy 2 3x 2 h 2 xh 2 h = K b x b x (4 b By (38 (4 he fac ha ˆx is monoonically decreasing for x >, we have Differeniaing (4 once, we obain hx = h x = h x > (42 2 3x 2 h xh h = K b x b (43 We consider wo possible cases. Case. K b x b <. In his case, he righh side of Equaion (43 is negaive. Because >, h x canno have any inerior nonposiive minimum. To see his, suppose ˆx x x achieves an inerior minimum wih h ˆx. Then, we would have h ˆx h ˆx =, which implies ha he lefh side is posiive. This is a conradicion. Because h x =, h x >, we mus have h x > for any x x x; oherwise, here would be an inerior nonposiive minimum. Then, he fac ha hx = implies ha hx > for any x x x. Because h x > for any x x x h x =, we mus have h x. In addiion, if h x were equal o, hen we would have h x < by (42 (43 because K b x b <. However, his would conradic he fac ha h x > for any x x x h x =. Therefore, we mus have h x >. Then, (4, (42, h x > imply ha ( K b x < b Case 2. K b x b. In his case, we mus have < b < because K >. Therefore, x K b < K b /b. This implies ha h x > by (4 (42. There exiss > such ha h x > for any x x x + because h x =. The righh side of Equaion (43 is monoonically decreasing in x. Le x be such ha he righh side of (43 is. Then, for any x x, he righh side is nonnegaive hus h x canno have any inerior nonnegaive (local maximum in x x for similar reasons o hose in Case. There canno exis any ˆx x + x such ha h ˆx. If x < x, hen, for any x x x, he righh side is nonposiive hus h x canno have any inerior nonposiive (local minimum in x x. There canno exis any ˆx x x such ha h ˆx. Therefore, here canno exis any ˆx x x such ha h ˆx hus we have h x > hx > for any x x x. Now, we show, for boh cases, ha hx > for any x < x. Equaion (35 implies ha he righh side of (4 is posiive for x < x h canno achieve an inerior posiive maximum for x < x. On he oher h, h x >, h x is coninuous a x h x =, which imply ha here exiss an > such ha x x x h x < Thus, x x x, hx >, herefore, x < x, hx > ; oherwise, h would achieve an inerior posiive maximum in x. (iii Recall ha =. I can be shown ha A + = ˆ + b x b + x b + + = + + k b + bb Equaion (35 hen implies ha A + >. Because we also have (2, x mus saisfy x > ( + b + Because + b + > b we have ( b x > (44 which (by he definiion (9 implies ha A <.
9 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen 628 Mahemaics of Operaions Research 36(4, pp , 2 INFORMS (iv Differeniaing (33 wice, we have, for x < x, xx x = ( A + + x + b + A + x b b x b 2 > xx xx/ x b 2 = (45 INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. where he inequaliy follows from he fac ha d dx A + + x + b + A + x b < This is implied by A <, A + >, + > > b >, <, he las equaliy in (45 follows from xx x =. Thus, x is sricly convex x < x. Because x x = x < x, xx x >, we mus also have x < x, x x <. (v By par (i, par (iv, x x = ˆ x x, x x R is coninuous sricly increasing in x x. By inspecion of (33 (34, x x R akes on all nonposiive values. Because y >, here exiss a unique soluion x > o (25 for each W >. Also, because x x R, (28 implies ha W Though he dual approach yields almos explici soluions, i is simpler o show he opimaliy of he cidae policies in he primal for his combined opimal sopping opimal conrol problem. Define ( ] c M = e [ +s s R s + kb s ds + V W s y s dr s + R e + V W y (46 The following lemma is a generalized dominaed convergence heorem ha is required for he proof of Lemma 3.3. Lemma 3.2. Suppose ha a.s. convergen sequences of rom variables X n X Y n Y saisfy X n Y n EY n EY <. Then, EX n EX. Proof of Lemma 3.2. Because X n Y n, by Faou s lemma, liminf EX n EX liminf EY n X n EY X. These inequaliies imply ha boh limsupex n EX liminf EX n EX because EY n EY <. Therefore, we mus have EX n EX. Lemma 3.3. Suppose =. Given he definiions for Theorem 3.2, (i M as defined in (46 is a supermaringale for any feasible policy a maringale for he claimed opimal policy. (ii For any feasible policy, lim E R e + V W y (47 wih equaliy for he claimed opimal policy. Proof of Lemma Define W = y x x. Then, for any W, V W y V W y (48 wih equaliy for W W. This can be shown as follows. Le x x R be such ha y x x=w y x x R =W. Then, we have x x R x x R x x R x x R =x x x x R x x R where he firs inequaliy follows from x x by Lemma 3. he second inequaliy follows from he convexiy of x. Afer rearranging, we obain (48. Applying he generalized Iô s lemma o M (see, e.g., Harrison [2, 4.7], we have M = M ( c R s {e +s e +s V W s y s V W s y s dr s s +kb s +LV W sy s } ds R s e +s V W W s y s s +y s V y W s y s y dz s (49
10 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen Mahemaics of Operaions Research 36(4, pp , 2 INFORMS 629 where LV = 2 V W W + y y V Wy + 2 y yy 2 V yy +rw + r+w B c + R y V W + y V y +V INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. By he definiions of V,, c B R W, he fac ha x saisfies (36 (39, we obain ha he firs inegral is always nonposiive for any feasible policy cbr is equal o zero for he claimed opimal policy c B R. By (48, he hird erm in (49 is always nonposiive for every feasible reiremen policy R equal o zero for he claimed opimal policy R. In addiion, using he expressions for he claimed opimal, V, B, W, we have ha, under he claimed opimal policy, he sochasic inegral is a maringale because y is a geomeric Brownian moion; wih =, x is bounded beween x x before reiremen; x is also a geomeric Brownian moion afer reiremen. This shows ha M is a local supermaringale for all feasible policies a maringale for he claimed opimal policy. Nex, we show ha M is acually a supermaringale for all feasible policies. Firs, we show ha V W y for every feasible policy. By (29 (3, V W W y=y x > hus V W y increases in W. If <, hen V W y because V W y V W y. If >, hen V W y< because V W y V W y=v W y< Therefore, V W y for every feasible policy. If <, hen V W y > he local supermaringale M is hen always nonnegaive hus a supermaringale. Suppose >. By (49, here exiss an increasing sequence of sopping imes n such ha M EM n, i.e., n ( ] c V W y E e [ R +s s s s +kb s ds +V W s y s dr s +E R n e + n V W n y n (5 Because he inegr in he inegral of (5 is always negaive, his inegral is monoonically decreasing in ime. In addiion, where R e + V W y e + V y = V e + y V N (5 N e y + yz (52 is a maringale wih EN =, he second inequaliy follows from V being negaive increasing in W W >, he equaliy follows from he form of V as defined by (29 (3, he las inequaliy follows from V < 2 >. In addiion, V >. Therefore, aking n in (5, by he monoone convergence heorem for he firs erm Lemma 3.2 for he second erm, we have ( ] c V W y E e [ R +s s s +kb s ds +V W s y s dr s +E R e + V W y Tha is, M EM for any. Because his argumen applies o any ime s, we have M s E s M for any s. Thus, M is a supermaringale for all feasible policies. 2. Because V W y for every feasible policy, we have lim E R e + V W y = lim E R e + y x x x x lim EL e + y + ˆe +/ b = (53
11 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen 63 Mahemaics of Operaions Research 36(4, pp , 2 INFORMS for some consan L, where he second inequaliy follows from he fac ha when =,x, x, x x are all bounded R = for >. The las equaliy in (53 follows from he condiions ha > 2 >. Therefore, for he claimed opimal policy, we obain lim E R e + V W y = INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. For any feasible policy, if <, hen V W y> herefore (47 holds. If >, because 2 >, we have lim Ee + y =. Therefore, aking he limi as in (5, we have ha (47 also holds. This complees he proof. Lemma 3.4. Suppose >, >, 2 >, 3 >. Then, here exiss a unique soluion o Equaion (23 (( K b < =min b+k b Proof of Lemma 3.4. Because >, >, 2 >, 3 >, + >>b > < Nex, because b + dominaes + as, we have If =, i is easy o verify ha limq= lim If = K b /b+k b <, hen we have I follows ha K b b+k b b + b + =+ q = + + K b +k b < + +k b K b b+k b b + + = + + q = K b b+k b b = + >> + ( + ( + + < Then, by coninuiy of q, here exiss a soluion such ha q =. Suppose here exiss anoher soluion ˆ such ha q ˆ=. Le V W y W be he value funcion boundary, respecively, corresponding o le ˆV W y ˆ W be he value funcion boundary, respecively, corresponding o ˆ. Wihou loss of generaliy, suppose W > ˆ W. Because ˆ W is he reiremen boundary, he value funcion corresponding o ˆ for W >W > ˆ W is equal o V W y. However, Lemma 3. implies ha V W y> V W y for any W < W. This implies ha ˆ W canno be he opimal reiremen boundary, which conradics Theorem 3.2. Therefore, he soluion o Equaion (23 is unique. We are now ready o prove Theorem 3.2. Proof of Theorem 3.2. If R =, hen Problem 2 is idenical o Problem. Therefore, he opimaliy of he claimed opimal sraegy follows from Theorem 3.. In addiion, as noed in (9, we have V W y=vw yt where vw yt (independen of T is he value funcion afer reiremen for Problem. From now on, we assume w.l.o.g. ha R =. I is edious bu sraighforward o use he generalized Iô s lemma Equaions (7 (23 (27 (28 o verify ha he claimed opimal sraegy W, c,, R in Theorems saisfy he budge consrain (6. In addiion, by Lemma 3., x exiss is unique W saisfies he borrowing consrain in each problem. Furhermore, by Lemma 3.4, here is a unique soluion o (23.
12 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen Mahemaics of Operaions Research 36(4, pp , 2 INFORMS 63 INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. By Doob s opional sampling heorem, we can resric aenion w.l.o.g. o he se of feasible policies ha implemen he opimal policy saed in Theorem afer reiremen. The uiliy funcion for such a sraegy can be wrien as ( ] c E e [ R +s s s +kb s ds +V W s y s dr s By Lemma 3.3, M is a supermaringale for any feasible policy cbr a maringale for he claimed opimal policy c B R, which implies ha M EM, i.e., ( ] c V W y E e [ R +s s s +kb s ds +V W s y s dr s +E R e + V W y (54 wih equaliy for he claimed opimal policy. In addiion, by Lemma 3.3, we also have ha lim E R e + V W y wih equaliy for he claimed opimal policy. Therefore, aking he limi as in (54, we have ( ] c V W y E e [ R +s s s +kb s ds +V W s y s dr s wih equaliy for he claimed opimal policy c B R This complees he proof. Theorem 2 provides essenially complee soluions because he soluion for x, given W, requires only a onedimensional monoone search o solve Equaion (25, for, one only needs o solve Equaion (23. We nex provide resuls on compuing he marke value of human capial a any poin in ime, which is useful for undersing much of he economics in he paper. Proposiion 3.. Consider he opimal policies saed in Theorems Afer reiremen, he marke value of he human capial (i.e., he capialized labor income is zero. Before reiremen, in Theorem 3., he marke value of he human capial is Hy =gy where y g are given in ( (3. In Theorem 3.2, if =, hen he marke value of he human capial is Hx y = y x x + if =, hen he marke value of he human capial is where Proof. Hx y = y Ax +Bx + + A= + x x + + x + x + B = x + x + x + For Problem, by Iô s lemma, (, (3, (4, (, simple algebra, Furhermore, d gy = R y d + gy y dz E s gsy s 2 y y ds < because s y s is a sard lognormal diffusion he oher facors are bounded for any zero for >T. Therefore, he local maringale gy + s R s y s ds =gy + s gsy s y dz s (55 s=
13 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen 632 Mahemaics of Operaions Research 36(4, pp , 2 INFORMS is a maringale ha is consan for >T. Picking any >maxt, he definiion of a maringale implies ha ] gy + s R s y s ds =E [ gy + s R s y s ds INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. Now, >maxt implies ha g=, he inegral on he lefh side is known a. Therefore, we can subrac he inegral from boh sides divide boh sides by o conclude gy = [ ] E s R s y s ds = E [ ] s R s y s ds where he second equaliy follows from he fac ha R s for s >. For he cases wih volunary reiremen, because here is no more labor income afer reiremen, he marke value of human capial afer reiremen is zero. We nex prove he claims for afer reiremen. Using he expressions of H he dynamics of x y, i can be verified ha, for x >x when = for x <x < x when =, we have ha he change in he marke value of human capial plus he flow of labor income will be given by d ( Hx y + y d = ( 2 3x 2 H xx 2 3 x H x H +y d + ( x H x x +H y dz (56 The drif erm in (56 is equal o zero afer plugging in he expressions for H (if =, he addiional local ime erm a x from applying he generalized Iô s lemma is also equal o zero because i can be verified ha H x xy =. This implies ha M Hx y + s y s ds is a local maringale. In addiion, here exiss a consan <L< such ha x H x x +H y <L y because < x >x if = x <x x if =. Because boh y are geomeric Brownian moions, we have ha M is acually a maringale. Recall he definiion (26 of he opimal reiremen ime. For all we have ] Hx y + s y s ds =E [ Hxy + s y s ds which implies ha Hx y = E [ ] s y s ds because i can be easily verified ha Hxy= Therefore, H as specified in Proposiion 3. is indeed he marke value of he fuure labor income. The following resul shows ha because of reiremen flexibiliy, human capial may have a negaive bea even when labor income correlaes posiively wih marke risk. Proposiion 3.2. As an invesor s financial wealh W increases, he invesor s human capial H decreases in Problem 2. Furhermore, if y </, hen human capial has a negaive bea measured relaive o any locally meanvariance efficien risky porfolio. The following lemma is useful for he proof of Proposiions Lemma 3.5. Suppose =, >, >, 2 >, 3 >. Then, (i ˆx is sricly decreasing sricly convex. (ii x is sricly convex x x /. (iii x, we have x ˆx x x, we have x x ˆ x x (iv Given (24, here exiss a unique soluion x > o (25. In addiion, W borrowing consrain (8. defined in (28 saisfies he
14 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen Mahemaics of Operaions Research 36(4, pp , 2 INFORMS 633 Proof of Lemma 3.5. (i This follows from direc differeniaion because ˆ > b <. (ii Firs, because >, >, 2 >, 3 >, i is sraighforward o use he definiions of + o show ha + >>b > < A + > INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. Then, he claimed resuls also follow from direc differeniaion. (iii This follows from a similar argumen o ha of par (ii of Lemma 3.. (iv By par (i, par (ii, x x= ˆ x x, xr is coninuous sricly increasing in x. By an inspecion of (33 (34, x xr akes on all values ha are less han or equal o /. Because y >, here exiss a unique soluion x > o (25 for each W y /. Also, because x xr /, (28 implies ha W > R y / Proof of Proposiion 3.2. Firs, as shown in Lemmas , he dual value funcion defined in Theorem 3.2 is convex hus he wealh level W defined in Theorem 2 decreases wih he dual variable x. By Proposiion 3., differeniaing he expression for human capial wih respec o x for he case = yields ha human capial is increasing in x because <. Therefore, human capial decreases wih financial wealh W for = in Problem 2. For = in Problem 2, differeniaing human capial H wih respec o x, we have ha, before reiremen, Hxy x = y A x +B + x + = y + x x x + x + (( x + > (57 x where he second equaliy follows from he expressions of A B in Proposiion 3. he inequaliy follows from he fac ha + >> x < x. Thus, human capial decreases wih financial wealh W also for =. Furhermore, because x = y, if y </, hen, as he marke risk Z increases, x decreases herefore human capial decreases by (57, i.e., human capial has a negaive bea. The following resul shows ha reiremen flexibiliy ends o increase sock invesmen. Proposiion 3.3. Suppose y = >r. Then, he fracion of oal wealh W +H invesed in he risky asse in Problem 2 when = is greaer han ha in Problem. Proof. By Theorem 3., he fracion of oal wealh W +H invesed in he risky asse in Problem is consanly equal o r/. Wih =, by Theorem 3.2 Proposiion 3., we have W +H = r A + x +x b A + x +x b / x x Plugging in he expressions for A + x using he fac ha <b, we have A + x +x b A + x +x b / x x > A reiremen decision is criical for an invesor s consumpion invesmen policies. The following proposiion shows ha he presence of he noborrowing consrain ends o make an invesor reire earlier. Proposiion 3.4. In Problem 2, he reiremen wealh hreshold for = is higher han for =. Proof. Le A, A +, x be defined as in Theorem 3.2. To make he exposiion clear, we now make heir dependence on explici, i.e., using A, A +, x insead: hx=a + x +A x + xb b + x+ ˆ xb b We prove by conradicion. Suppose x x. By Lemmas , we have hx=h x x =, hx=h x x=. From he proof of Lemma 3., we have h x x> for all x x x. By (22 (44, we have x< x. Therefore, hx =hx h x x =h x x (58
15 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen 634 Mahemaics of Operaions Research 36(4, pp , 2 INFORMS INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. The firs equaion of (58 implies ha which, in urn, implies A + x +A x + A + x A + >A + (59 because A < as shown in Lemma 3.. On he oher h, he second equaion of (58 implies ha which, in urn, implies A + x +A + x + A + x A + <A + (6 because A < + >. Resul (6 conradics (59. This shows ha we mus have x>x. Because, a reiremen, he financial wealh is equal o y x x for Problem 2 because y x x= ˆx /, we mus have ha he financial wealh level W a reiremen for = is higher han for =. One measure ha is useful for examining he life cycle invesmen policy is he expeced ime o reiremen. The following proposiion shows how o compue his measure. Proposiion 3.5. In Problem 2, suppose ha an invesor is no reired ha x 2 2 x <. Then, he expeced ime o reiremen for he opimal policy is for = for =, where Proof of Proposiion 3.5. which implies ha E x =x= logx/x 2 x 2 x >x x E x m x m x =x= 2 x 2 xm x + logx/x m 2 x 2 x x x (6 x m= 2 x 2 x Firs, we prove he resul for = in Theorem 3.2. Recall ha dx s =x s x ds + x dzs x s =x exp [ x 2 2 x s + xz s Z ] Because x >x x 2 2 x <, we have < almos surely (see, for example, Karazas Shreve [4, p. 349]. Le f x logx/x 2 x x x Then, by Iô s lemma, for any sopping ime, we have ( f x + ds =f x + 2 x xx 2 s f x xx + x x s f x + ds + 2 x dz s (62 x x which implies ha f x + ds is a maringale because i can be easily verified ha he drif erm is zero, given he definiion of f x, he sochasic inegral is a scaled Brownian moion hus a maringale. Thus, aking = + aking he expecaion in (62, we ge f x=e x =x because x + =x f x=. A similar argumen applies o he case =; noe ha whenevaluaed a x = x, he firs derivaive of he righh side of (6 wih respec o x is zero x is bounded. This complees he proof. The following proposiion shows how he expeced ime o reiremen is relaed o human capial financial wealh, which can help explain he graphical soluions presened in he previous secion. Proposiion 3.6. Suppose ha an invesor is no reired ha 2 2 x x >. Then, as he expeced ime o reiremen increases, financial wealh decreases human capial increases. Proof. By Proposiion 3.5, i can be easily verified ha he expeced ime o reiremen is increasing in x. By (57, we have ha human capial is increasing in x Proposiion 3.2 hen implies ha financial wealh is decreasing in x. Therefore, he claim holds.
16 Dybvig Liu: Verificaion Theorems for Models of Opimal Consumpion Invesmen Mahemaics of Operaions Research 36(4, pp , 2 INFORMS Conclusion. We examine he impac of reiremen flexibiliy borrowing consrains agains fuure labor income on opimal consumpion invesmen policies. We solve wo alernaive models almos explicily (a leas paramerically up o a mos a consan provide verificaion heorems ha are proved using a combinaion of he dual approach an analysis of he boundary. In addiion, we also obain prove some ineresing comparaive saics. INFORMS holds copyrigh o his aricle disribued his copy as a couresy o he auhor(s. Addiional informaion, including righs permission policies, is available a hp://journals.informs.org/. Acknowledgmens. The auhors hank Ron Kaniel, Masakiyo Miyazawa (an area edior of Mahemaics of Operaions Research, Anna Pavlova, Cosis Skiadas, wo anonymous referees an associae edior a Mahemaics of Operaions Research, he seminar paricipans a he 25 American Finance Associaion (AFA conference, Duke Universiy, MIT, Washingon Universiy in S. Louis for helpful suggesions. References [] Dybvig, P. H., H. Liu. 2. Lifeime consumpion invesmen: Reiremen consrained borrowing. J. Econom. Theory 45( [2] Harrison, J. M Brownian Moion Sochasic Flow Sysems. Wiley, New York. [3] He, H., H. F. Pagès Labor income, borrowing consrains, equilibrium asse prices: A dualiy approach. Econom. Theory 3( [4] Karazas, I., S. E. Shreve. 99. Brownian Moion Sochasic Calculus. SpringerVerlag, New York. [5] Karazas, I., S. E. Shreve Mehods of Mahemaical Finance. SpringerVerlag, New York. [6] Karazas, I., H. Wang. 2. Uiliy maximizaion wih discreionary sopping. SIAM J. Conrol Opim. 39( [7] Meron, R. C Lifeime porfolio selecion under uncerainy: The coninuous ime case. Rev. Econom. Sais. 5( [8] Pliska, S. R A sochasic calculus model of coninuous rading: Opimal porfolio. Mah. Oper. Res. (
Stochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationRenewal processes and Poisson process
CHAPTER 3 Renewal processes and Poisson process 31 Definiion of renewal processes and limi heorems Le ξ 1, ξ 2, be independen and idenically disribued random variables wih P[ξ k > 0] = 1 Define heir parial
More informationFair games, and the Martingale (or "Random walk") model of stock prices
Economics 236 Spring 2000 Professor Craine Problem Se 2: Fair games, and he Maringale (or "Random walk") model of sock prices Sephen F LeRoy, 989. Efficien Capial Markes and Maringales, J of Economic Lieraure,27,
More informationNewton's second law in action
Newon's second law in acion In many cases, he naure of he force acing on a body is known I migh depend on ime, posiion, velociy, or some combinaion of hese, bu is dependence is known from experimen In
More informationA Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)
A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More informationOptimal Stock Selling/Buying Strategy with reference to the Ultimate Average
Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July
More informationOptimal Life Insurance, Consumption and Portfolio: A Dynamic Programming Approach
28 American Conrol Conference Wesin Seale Hoel, Seale, Washingon, USA June 1113, 28 WeA1.5 Opimal Life Insurance, Consumpion and Porfolio: A Dynamic Programming Approach Jinchun Ye (Pin: 584) Absrac A
More informationFourier Series Solution of the Heat Equation
Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,
More informationGraduate Macro Theory II: Notes on Neoclassical Growth Model
Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.
More informationTanaka formula and Levy process
Tanaka formula and Levy process Simply speaking he Tanaka formula is an exension of he Iô formula while Lévy process is an exension of Brownian moion. Because he Tanaka formula and Lévy process are wo
More informationWorking Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationTime Consistency in Portfolio Management
1 Time Consisency in Porfolio Managemen Traian A Pirvu Deparmen of Mahemaics and Saisics McMaser Universiy Torono, June 2010 The alk is based on join work wih Ivar Ekeland Time Consisency in Porfolio Managemen
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationDebt Portfolio Optimization Eric Ralaimiadana
Deb Porfolio Opimizaion Eric Ralaimiadana 27 May 2016 Inroducion CADES remi consiss in he defeasance of he Social Securiy and Healh public deb To achieve is ask, he insiuion has been assigned a number
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationA NOTE ON THE ALMOST EVERYWHERE CONVERGENCE OF ALTERNATING SEQUENCES WITH DUNFORD SCHWARTZ OPERATORS
C O L L O Q U I U M M A T H E M A T I C U M VOL. LXII 1991 FASC. I A OTE O THE ALMOST EVERYWHERE COVERGECE OF ALTERATIG SEQUECES WITH DUFORD SCHWARTZ OPERATORS BY RYOTARO S A T O (OKAYAMA) 1. Inroducion.
More informationResearch Article Optimal Geometric Mean Returns of Stocks and Their Options
Inernaional Journal of Sochasic Analysis Volume 2012, Aricle ID 498050, 8 pages doi:10.1155/2012/498050 Research Aricle Opimal Geomeric Mean Reurns of Socks and Their Opions Guoyi Zhang Deparmen of Mahemaics
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationA Note on Renewal Theory for T iid Random Fuzzy Variables
Applied Mahemaical Sciences, Vol, 6, no 6, 97979 HIKARI Ld, wwwmhikaricom hp://dxdoiorg/988/ams6686 A Noe on Renewal Theory for T iid Rom Fuzzy Variables Dug Hun Hong Deparmen of Mahemaics, Myongji
More informationOptimal Life Insurance Purchase, Consumption and Investment
Opimal Life Insurance Purchase, Consumpion and Invesmen Jinchun Ye a, Sanley R. Pliska b, a Dep. of Mahemaics, Saisics and Compuer Science, Universiy of Illinois a Chicago, Chicago, IL 667, USA b Dep.
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO
Profi Tes Modelling in Life Assurance Using Spreadshees, par wo PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO Erik Alm Peer Millingon Profi Tes Modelling in Life Assurance Using Spreadshees,
More informationTechnical Appendix to Risk, Return, and Dividends
Technical Appendix o Risk, Reurn, and Dividends Andrew Ang Columbia Universiy and NBER Jun Liu UC San Diego This Version: 28 Augus, 2006 Columbia Business School, 3022 Broadway 805 Uris, New York NY 10027,
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationON THE PRICING OF EQUITYLINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT
Teor Imov r.amaem.sais. Theor. Probabiliy and Mah. Sais. Vip. 7, 24 No. 7, 25, Pages 15 111 S 949(5)6344 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITYLINKED LIFE INSURANCE
More informationarxiv: v1 [qfin.mf] 8 Mar 2015
Purchasing Term Life Insurance o Reach a Beques Goal: TimeDependen Case Erhan Bayrakar Deparmen of Mahemaics, Universiy of Michigan Ann Arbor, Michigan, USA, 48109 arxiv:1503.02237v1 [qfin.mf] 8 Mar
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More informationPresent Value Methodology
Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More informationA Generalized Bivariate OrnsteinUhlenbeck Model for Financial Assets
A Generalized Bivariae OrnseinUhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical
More informationOptimal Life Insurance Purchase and Consumption/Investment under Uncertain Lifetime
Opimal Life Insurance Purchase and Consumpion/Invesmen under Uncerain Lifeime Sanley R. Pliska a,, a Dep. of Finance, Universiy of Illinois a Chicago, Chicago, IL 667, USA Jinchun Ye b b Dep. of Mahemaics,
More informationOn the Role of the Growth Optimal Portfolio in Finance
QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 144 January 2005 On he Role of he Growh Opimal Porfolio in Finance Eckhard Plaen ISSN 14418010 www.qfrc.us.edu.au
More informationStochastic Calculus, Week 10. Definitions and Notation. TermStructure Models & Interest Rate Derivatives
Sochasic Calculus, Week 10 TermSrucure Models & Ineres Rae Derivaives Topics: 1. Definiions and noaion for he ineres rae marke 2. Termsrucure models 3. Ineres rae derivaives Definiions and Noaion Zerocoupon
More informationRepresenting Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More informationOption Pricing Under Stochastic Interest Rates
I.J. Engineering and Manufacuring, 0,3, 889 ublished Online June 0 in MECS (hp://www.mecspress.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecspress.ne/ijem Opion ricing Under Sochasic Ineres
More informationnonlocal conditions.
ISSN 17493889 prin, 17493897 online Inernaional Journal of Nonlinear Science Vol.11211 No.1,pp.39 Boundary Value Problem for Some Fracional Inegrodifferenial Equaions wih Nonlocal Condiions Mohammed
More informationOPTIMAL LIFE INSURANCE PURCHASE, CONSUMPTION AND INVESTMENT ON A FINANCIAL MARKET WITH MULTIDIMENSIONAL DIFFUSIVE TERMS
OPTIMAL LIFE INSURANCE PURCHASE, CONSUMPTION AND INVESTMENT ON A FINANCIAL MARKET WITH MULTIDIMENSIONAL DIFFUSIVE TERMS I. DUARTE, D. PINHEIRO, A. A. PINTO, AND S. R. PLISKA Absrac. We inroduce an exension
More informationA general decomposition formula for derivative prices in stochastic volatility models
A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 57 85 Barcelona Absrac We see ha he price of an european call opion
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationA martingale approach applied to the management of life insurances.
A maringale approach applied o he managemen of life insurances. Donaien Hainau Pierre Devolder 19h June 2007 Insiu des sciences acuarielles. Universié Caholique de Louvain UCL. 1348 LouvainLaNeuve, Belgium.
More informationDynamic Information. Albina Danilova Department of Mathematical Sciences Carnegie Mellon University. September 16, 2008. Abstract
Sock Marke Insider Trading in Coninuous Time wih Imperfec Dynamic Informaion Albina Danilova Deparmen of Mahemaical Sciences Carnegie Mellon Universiy Sepember 6, 28 Absrac This paper sudies he equilibrium
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationRelative velocity in one dimension
Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies
More informationMorningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
More informationPricing FixedIncome Derivaives wih he ForwardRisk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK8 Aarhus V, Denmark Email: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs
More informationOptimal Investment, Consumption and Life Insurance under MeanReverting Returns: The Complete Market Solution
Opimal Invesmen, Consumpion and Life Insurance under MeanRevering Reurns: The Complee Marke Soluion Traian A. Pirvu Dep of Mahemaics & Saisics McMaser Universiy 180 Main Sree Wes Hamilon, ON, L8S 4K1
More informationCointegration: The Engle and Granger approach
Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be nonsaionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require
More informationA MARTINGALE APPROACH APPLIED TO THE MANAGEMENT OF LIFE INSURANCES.
A MARTINGALE APPROACH APPLIED TO THE MANAGEMENT OF LIFE INSURANCES. DONATIEN HAINAUT, PIERRE DEVOLDER. Universié Caholique de Louvain. Insiue of acuarial sciences. Rue des Wallons, 6 B1348, LouvainLaNeuve
More informationYTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.
. Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure
More informationFollow links Class Use and other Permissions. For more information, send to:
COPYRIGHT NOTICE: David A. Kendrick, P. Ruben Mercado, and Hans M. Amman: Compuaional Economics is published by Princeon Universiy Press and copyrighed, 2006, by Princeon Universiy Press. All righs reserved.
More informationDependent Interest and Transition Rates in Life Insurance
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
More informationLecture III: Finish Discounted Value Formulation
Lecure III: Finish Discouned Value Formulaion I. Inernal Rae of Reurn A. Formally defined: Inernal Rae of Reurn is ha ineres rae which reduces he ne presen value of an invesmen o zero.. Finding he inernal
More informationWorking Paper No. 482. Net Intergenerational Transfers from an Increase in Social Security Benefits
Working Paper No. 482 Ne Inergeneraional Transfers from an Increase in Social Securiy Benefis By Li Gan Texas A&M and NBER Guan Gong Shanghai Universiy of Finance and Economics Michael Hurd RAND Corporaion
More informationA Probability Density Function for Google s stocks
A Probabiliy Densiy Funcion for Google s socks V.Dorobanu Physics Deparmen, Poliehnica Universiy of Timisoara, Romania Absrac. I is an approach o inroduce he Fokker Planck equaion as an ineresing naural
More informationHANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed.
Inverse Funcions Reference Angles Inverse Trig Problems Trig Indeniies HANDOUT 4 INVERSE FUNCTIONS KEY POINTS A.) Inroducion: Many acions in life are reversible. * Examples: Simple One: a closed door can
More informationStochastic Calculus and Option Pricing
Sochasic Calculus and Opion Pricing Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus 15.450, Fall 2010 1 / 74 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 BlackScholes
More informationAn Optimal Selling Strategy for Stock Trading Based on Predicting the Maximum Price
An Opimal Selling Sraegy for Sock Trading Based on Predicing he Maximum Price Jesper Lund Pedersen Universiy of Copenhagen An opimal selling sraegy for sock rading is presened in his paper. An invesor
More informationEconomics 140A Hypothesis Testing in Regression Models
Economics 140A Hypohesis Tesing in Regression Models While i is algebraically simple o work wih a populaion model wih a single varying regressor, mos populaion models have muliple varying regressors 1
More informationA Reexamination of the Joint Mortality Functions
Norh merican cuarial Journal Volume 6, Number 1, p.166170 (2002) Reeaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali
More informationDIFFERENTIAL EQUATIONS with TI89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations
DIFFERENTIAL EQUATIONS wih TI89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows
More informationLectures # 5 and 6: The Prime Number Theorem.
Lecures # 5 and 6: The Prime Number Theorem Noah Snyder July 8, 22 Riemann s Argumen Riemann used his analyically coninued ζfuncion o skech an argumen which would give an acual formula for π( and sugges
More informationModeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling
Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se
More informationOptimal Annuity Purchasing
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 WisconsinMadison, Madison, Wisconsin, 5376, UA. he
More informationOn the paper Is Itô calculus oversold? by A. Izmailov and B. Shay
On he paper Is Iô calculus oversold? by A. Izmailov and B. Shay M. Rukowski and W. Szazschneider March, 1999 The main message of he paper Is Iô calculus oversold? by A. Izmailov and B. Shay is, we quoe:
More informationOptimalCompensationwithHiddenAction and LumpSum Payment in a ContinuousTime Model
Appl Mah Opim (9) 59: 99 46 DOI.7/s45895 OpimalCompensaionwihHiddenAcion and LumpSum Paymen in a ConinuousTime Model Jakša Cvianić Xuhu Wan Jianfeng Zhang Published online: 6 June 8 Springer Science+Business
More informationINDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES
Journal of Applied Analysis 1, 1 (1995), pp. 39 45 INDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES A. LUCZAK Absrac. We find exponens of independen marginals
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationThe option pricing framework
Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.
More information4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay
324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find
More informationFourier series. Learning outcomes
Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Halfrange series 6. The complex form 7. Applicaion of Fourier series
More information4 Convolution. Recommended Problems. x2[n] 1 2[n]
4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discreeime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.11.
More information5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance.
5.8 Resonance 231 5.8 Resonance The sudy of vibraing mechanical sysems ends here wih he heory of pure and pracical resonance. Pure Resonance The noion of pure resonance in he differenial equaion (1) ()
More informationCBOE VIX PREMIUM STRATEGY INDEX (VPD SM ) CAPPED VIX PREMIUM STRATEGY INDEX (VPN SM )
CBOE VIX PREIU STRATEGY INDEX (VPD S ) CAPPED VIX PREIU STRATEGY INDEX (VPN S ) The seady growh of CBOE s volailiy complex provides a unique opporuniy for invesors inen on capuring he volailiy premium.
More informationCredit Index Options: the noarmageddon pricing measure and the role of correlation after the subprime crisis
Second Conference on The Mahemaics of Credi Risk, Princeon May 2324, 2008 Credi Index Opions: he noarmageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo  Join work
More informationI. Basic Concepts (Ch. 14)
(Ch. 14) A. Real vs. Financial Asses (Ch 1.2) Real asses (buildings, machinery, ec.) appear on he asse side of he balance shee. Financial asses (bonds, socks) appear on boh sides of he balance shee. Creaing
More informationDETERMINISTIC INVENTORY MODEL FOR ITEMS WITH TIME VARYING DEMAND, WEIBULL DISTRIBUTION DETERIORATION AND SHORTAGES KUNSHAN WU
Yugoslav Journal of Operaions Research 2 (22), Number, 67 DEERMINISIC INVENORY MODEL FOR IEMS WIH IME VARYING DEMAND, WEIBULL DISRIBUION DEERIORAION AND SHORAGES KUNSHAN WU Deparmen of Bussines Adminisraion
More informationChapter 1.6 Financial Management
Chaper 1.6 Financial Managemen Par I: Objecive ype quesions and answers 1. Simple pay back period is equal o: a) Raio of Firs cos/ne yearly savings b) Raio of Annual gross cash flow/capial cos n c) = (1
More informationEfficient Risk Sharing with Limited Commitment and Hidden Storage
Efficien Risk Sharing wih Limied Commimen and Hidden Sorage Árpád Ábrahám and Sarola Laczó March 30, 2012 Absrac We exend he model of risk sharing wih limied commimen e.g. Kocherlakoa, 1996) by inroducing
More informationINVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS
INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,
More informationDifferential Equations in Finance and Life Insurance
Differenial Equaions in Finance and Life Insurance Mogens Seffensen 1 Inroducion The mahemaics of finance and he mahemaics of life insurance were always inersecing. Life insurance conracs specify an exchange
More informationOptionPricing in Incomplete Markets: The Hedging Portfolio plus a Risk PremiumBased Recursive Approach
Working Paper 581 Business Economics Series 21 January 25 Deparameno de Economía de la Empresa Universidad Carlos III de Madrid Calle Madrid, 126 2893 Geafe (Spain) Fax (34) 91 624 968 OpionPricing in
More informationReturn Calculation of U.S. Treasury Constant Maturity Indices
Reurn Calculaion of US Treasur Consan Mauri Indices Morningsar Mehodolog Paper Sepeber 30 008 008 Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion
More informationApplied Intertemporal Optimization
. Applied Ineremporal Opimizaion Klaus Wälde Universiy of Mainz CESifo, Universiy of Brisol, UCL Louvain la Neuve www.waelde.com These lecure noes can freely be downloaded from www.waelde.com/aio. A prin
More informationDifferential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.
Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given
More informationABSTRACT KEYWORDS. Markov chain, Regulation of payments, Linear regulator, Bellman equations, Constraints. 1. INTRODUCTION
QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS BY MOGENS STEFFENSEN ABSTRACT Quadraic opimizaion is he classical approach o opimal conrol of pension funds. Usually he paymen sream is approximaed
More informationChapter 6. First Order PDEs. 6.1 Characteristics The Simplest Case. u(x,t) t=1 t=2. t=0. Suppose u(x, t) satisfies the PDE.
Chaper 6 Firs Order PDEs 6.1 Characerisics 6.1.1 The Simples Case Suppose u(, ) saisfies he PDE where b, c are consan. au + bu = 0 If a = 0, he PDE is rivial (i says ha u = 0 and so u = f(). If a = 0,
More informationINTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchangeraded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationMath 201 Lecture 12: CauchyEuler Equations
Mah 20 Lecure 2: CauchyEuler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem
More informationAP Calculus BC 2010 Scoring Guidelines
AP Calculus BC Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board
More informationOptimal Time to Sell in Real Estate Portfolio Management
Opimal ime o Sell in Real Esae Porfolio Managemen Fabrice Barhélémy and JeanLuc Prigen hema, Universiy of CergyPonoise, CergyPonoise, France Emails: fabricebarhelemy@ucergyfr; jeanlucprigen@ucergyfr
More informationUse SeDuMi to Solve LP, SDP and SCOP Problems: Remarks and Examples*
Use SeDuMi o Solve LP, SDP and SCOP Problems: Remarks and Examples* * his file was prepared by WuSheng Lu, Dep. of Elecrical and Compuer Engineering, Universiy of Vicoria, and i was revised on December,
More informationJumpDiffusion Option Valuation Without a Representative Investor: a Stochastic Dominance Approach
umpdiffusion Opion Valuaion Wihou a Represenaive Invesor: a Sochasic Doance Approach By Ioan Mihai Oancea and Sylianos Perrakis This version February 00 Naional Bank of Canada, 30 King Sree Wes, Torono,
More information