Optimal Investment, Consumption and Retirement Decision with Disutility and Borrowing Constraints

Transcription

1 Opimal Invesmen, Consumpion and Reiremen Decision wih Disuiliy and Borrowing Consrains Yong Hyun Shin Join Work wih Byung Hwa Lim(KAIST) June 29 July 3, 29 Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, 29 1 / 36

2 Conens 1 Hisorical Remarks Porfolio Selecion Borrowing Consrains 2 The Model Main Problem Dualiy Approaches and Variaional Inequaliy 3 Soluions 4 Examples: CRRA Uiliy Classes CRRA Uiliy Class Numerical Resuls 5 Conclusion Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, 29 2 / 36

3 Porfolio Selecion (1) Hisorical Remarks Porfolio Selecion Meron (1969 RESTAT, 1971 JET) Formulae a coninuous ime consumpion/invesemen problem. (dynamic programming mehod) [ max E e β u(c )d c,π subjec o dx = [rx + π (µ r) c d + π σdb. Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, 29 3 / 36

4 Porfolio Selecion (2) Hisorical Remarks Porfolio Selecion Karazas and Wang (2 SICON) Consider he mixure problem (c, π, τ). (maringale mehod) max E e β u 1 (c )d + e βτ u 2 (X τ ) c,π,τ subjec o dx = [rx + π (µ r) c d + π σdb. Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, 29 4 / 36

5 Porfolio Selecion (3) Hisorical Remarks Porfolio Selecion Choi and Shim (26 MF), Benchmark Model Labor income, disuiliy and opimal reiremen ime. (dynamic programming wihou considering borrowing consrains) [ max E ( ) u(c ) l1 { <τ} d c,π,τ subjec o dx = [rx + π (µ r) c + ɛ1 { <τ} d + π σdb. Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, 29 5 / 36

6 Hisorical Remarks Borrowing Consrains Borrowing Consrains Borrowing Consrain (Liquidiy Consrain) The economic agen has limied opporuniies o borrow agains fuure labor income and canno oally insure he risk of income flucuaion. So he borrowing consrain resrics he agen s choice in a non-rivial way. Lieraures He and Pagés (1993 JET), Duffie, Fleming, Soner, and Zariphopoulou (1997 JEDC), Koo (1998 MF), El Karoui and Jeanblanc (1998 FS), Dybvig and Liu (25 WP), Farhi and Panageas (27 JFE), Choi, Shim, and Shin (28 MF), Saha (28 WP) Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, 29 6 / 36

7 Hisorical Remarks Borrowing Consrains Borrowing Consrains Borrowing Consrain (Liquidiy Consrain) The economic agen has limied opporuniies o borrow agains fuure labor income and canno oally insure he risk of income flucuaion. So he borrowing consrain resrics he agen s choice in a non-rivial way. Lieraures He and Pagés (1993 JET), Duffie, Fleming, Soner, and Zariphopoulou (1997 JEDC), Koo (1998 MF), El Karoui and Jeanblanc (1998 FS), Dybvig and Liu (25 WP), Farhi and Panageas (27 JFE), Choi, Shim, and Shin (28 MF), Saha (28 WP) Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, 29 6 / 36

8 The Model The Basic Financial Marke Seup (1) riskless asse S ( ): ds () S () = rd risky asse S : ds S = µd + σdb r, µ, σ: consans B is a sandard Brownian moion on a probabiliy space (Ω, F, P) marke-price-of-risk θ µ r σ discoun process ζ exp{ r} exponenial maringale Z exp { θb 1 2 θ2 } pricing kernel(or sae-price-densiy) H ζ Z equivalen maringale measure P T (A) E[Z T 1 A, for any fixed T [, ) and for A F T B T B + θ : a sandard Brownian moion under he new measure P T by Girsanov heorem P on F, which agrees wih P T on F T for any T [, ). Furhermore B is a sandard Brownian moion under P. Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, 29 7 / 36

9 The Model The Basic Financial Marke Seup (2) labor wage: ɛ >, disuiliy: l > reiremen ime: a sopping ime τ consumpion process : c wih c sds <, a.s. porfolio process : π wih π2 s ds <, a.s. wealh process X wih an iniial wealh X = x dx = [rx + π (µ r) c + ɛ1 { <τ} d + π σdb = [rx c + ɛ1 { <τ} d + π σd B budge consrain [ E H τ X τ + τ τ H s c s ds H s ɛds x, for all τ Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, 29 8 / 36

10 The Model The Basic Financial Marke Seup (2) labor wage: ɛ >, disuiliy: l > reiremen ime: a sopping ime τ consumpion process : c wih c sds <, a.s. porfolio process : π wih π2 s ds <, a.s. wealh process X wih an iniial wealh X = x dx = [rx + π (µ r) c + ɛ1 { <τ} d + π σdb = [rx c + ɛ1 { <τ} d + π σd B budge consrain [ E H τ X τ + τ τ H s c s ds H s ɛds x, for all τ Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, 29 8 / 36

11 Borrowing Consrains The Model The borrowing consrain means ha he invesor canno borrow agains her/his fuure labor income. So he wealh of he invesor should always be nonnegaive. i.e. X, > Borrowing Consrain [ E H τ X τ + τ H s (c s ɛ)ds F, for all < τ. Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, 29 9 / 36

12 Borrowing Consrains The Model The borrowing consrain means ha he invesor canno borrow agains her/his fuure labor income. So he wealh of he invesor should always be nonnegaive. i.e. X, > Borrowing Consrain [ E H τ X τ + τ H s (c s ɛ)ds F, for all < τ. Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, 29 9 / 36

13 The Model Definiion 1 (General Uiliy Funcion) A funcion u : (, ) R is a uiliy funcion if i is sricly increasing, sricly concave, coninuously differeniable and saisfies u (+) lim c u (c) =, u ( ) lim c u (c) =. Labor Income ɛ: he agen receives he income coninuously Disuiliy l: disuiliy comes from labor Reiremen Time τ : he immoral agen can choose when o reire Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, 29 1 / 36

14 The Model Definiion 1 (General Uiliy Funcion) A funcion u : (, ) R is a uiliy funcion if i is sricly increasing, sricly concave, coninuously differeniable and saisfies u (+) lim c u (c) =, u ( ) lim c u (c) =. Labor Income ɛ: he agen receives he income coninuously Disuiliy l: disuiliy comes from labor Reiremen Time τ : he immoral agen can choose when o reire Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, 29 1 / 36

15 The Model Definiion 1 (General Uiliy Funcion) A funcion u : (, ) R is a uiliy funcion if i is sricly increasing, sricly concave, coninuously differeniable and saisfies u (+) lim c u (c) =, u ( ) lim c u (c) =. Labor Income ɛ: he agen receives he income coninuously Disuiliy l: disuiliy comes from labor Reiremen Time τ : he immoral agen can choose when o reire Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, 29 1 / 36

16 The Model Definiion 1 (General Uiliy Funcion) A funcion u : (, ) R is a uiliy funcion if i is sricly increasing, sricly concave, coninuously differeniable and saisfies u (+) lim c u (c) =, u ( ) lim c u (c) =. Labor Income ɛ: he agen receives he income coninuously Disuiliy l: disuiliy comes from labor Reiremen Time τ : he immoral agen can choose when o reire Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, 29 1 / 36

17 Main Problem The Model Main Problem Expeced Uiliy Maximizaion Problem The immoral invesor wans o maximize her/his expeced uiliy: [ J(x; c, π, τ) E e β ( ) u(c ) l1 { <τ} d i.e., V (x) = sup J(x; c, π, τ) (c,π,τ) A(x) subjec o he budge consrain and he borrowing consrain, where A(x) is he se of an admissible riple (c, π, τ) and β > is he subjecive discoun rae. Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

18 The Model Main Problem Expeced Uiliy Maximizaion Problem The invesor wans o maximize her/his expeced uiliy [ sup J(x; c, π, τ) sup E e β ( ) u(c ) l1 { <τ} d (c,π,τ) A(x) (c,π,τ) A(x) = sup E e β (u(c ) l) d + e βτ U(X τ ), (c,π,τ) A(x) subjec o he budge consrain E and he borrowing consrain, E τ H sc sds + H τ X τ H sɛds x, [ τ H τ X τ + H s(c s ɛ)ds F. Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

19 The Model Main Problem Lemma 2 (Value Funcion Afer Reiremen) The value funcion U( ) is given by U(x) = 2(λ ) n+ λ zi 1 (z) u(i 1 (z)) θ 2 (n + n ) ŷ z n++1 dz 2(λ ) n θ 2 (n + n ) λ ŷ zi 1 (z) u(i 1 (z)) z n dz + (λ )x, +1 where ŷ > is an arbirary consan, I 1 ( ) is he inverse funcion of u ( ) and λ is deermined by he algebraic equaion 2n+(λ ) n+ 1 θ 2 (n + n ) λ ŷ zi 1 (z) u(i 1 (z)) z n++1 dz + 2n (λ ) n 1 θ 2 (n + n ) λ ŷ zi 1 (z) u(i 1 (z)) z n dz = x. +1 Here n + > 1 and n < are wo roos of he following quadraic equaion 1 2 θ2 n 2 + (β r 12 ) θ2 n β =. (1) Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

20 The Value Funcion The Model Main Problem The Value Funcion The value funcion of our problem is given by V (x) = sup J(x; c, π, τ) (c,π,τ) A(x) = sup τ sup (c,π) Π τ (x) sup V τ (x) τ J(x; c, π, τ) where A(x) is he se of an admissible riple (c, π, τ) and Π τ (x) is he se of τ-fixed consumpion-porfolio plan (c, π) for which (c, π, τ) A(x) Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

21 Dualiy Approaches (1) The Model Dualiy Approaches and Variaional Inequaliy Individual s Shadow Prices Problem (He and Pagés (1993)) inf D > J(λ, D ; τ) inf E D > { } e β ũ(y λ ) + y λ ɛ l d + e βτ Ũ(y τ λ ), where D is he non-negaive, decreasing, and progressively measurable process, y λ = λd e β H ũ(y) sup {u(c) cy} = u(i 1 (y)) yi 1 (y) c Ũ(y) sup {U(x) xy} = U(I 2 (y)) yi 2 (y), x where I 1 ( ) u ( ) 1 and I 2 ( ) U ( ) 1. Moreover ũ( ) and Ũ( ) are sricly decreasing, sricly convex. Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

22 Dualiy Approaches (2) The Model Dualiy Approaches and Variaional Inequaliy J(x; c, π, τ) = E e β {u(c ) l λd e β H c }d + e βτ {U(X τ ) λd τ e βτ H τ X τ } E + λe E + E = E + λe D H c d + D τ H τ X τ τ e β ũ(λd e β H )d + e βτ Ũ(λDτ eβτ H τ ) D H c d + D τ H τ X τ τ e β ũ(λd e β H )d + e βτ Ũ(λDτ eβτ H τ ) λd H ɛd + λx e β { ũ(λd e β H )d + λd e β H ɛ l + e βτ Ũ(λDτ eβτ H τ ) + λx. } d e β ld e β ld Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

23 Dualiy Approaches (3) The Model Dualiy Approaches and Variaional Inequaliy E = E = E D H c d + D τ H τ X τ τ D H (c ɛ)d + D τ H τ X τ + τ D H ɛd + + E E E D H ɛd + x. τ H c d H s c s ds + H τ X τ D H ɛd H ɛd + H τ X τ τ H s ɛds F dd Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

24 Dualiy Approaches (4) The Model Dualiy Approaches and Variaional Inequaliy For any fixed τ S, previous inequaliies hold as equaliy if c = I 1 (λd e β H ), X τ = I 2 (λd τ e βτ H τ ), for all τ, τ E H c d + H τ X τ H ɛd = x, and τ E H s c s ds + H τ X τ H s ɛds F =. (2) Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

25 Dualiy Approaches (5) The Model Dualiy Approaches and Variaional Inequaliy (Based on Karazas and Wang (2)) V (x) = sup τ S V τ (x) = sup = inf {λ>,d >} sup τ S inf τ S {λ>,d >} [ J(λ, D ; τ) + λx, [ J(λ, D ; τ) + λx Proposiion (Value Funcion) Define Ṽ (λ) sup inf J(λ, D ; τ) = inf τ S D > sup D > τ S hen if Ṽ (λ) exiss and is differeniable for λ >, hen V (x) = inf [Ṽ (λ) + λx, λ> for any x (, ). J(λ, D ; τ), Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

26 Dualiy Approaches (5) The Model Dualiy Approaches and Variaional Inequaliy (Based on Karazas and Wang (2)) V (x) = sup τ S V τ (x) = sup = inf {λ>,d >} sup τ S inf τ S {λ>,d >} [ J(λ, D ; τ) + λx, [ J(λ, D ; τ) + λx Proposiion (Value Funcion) Define Ṽ (λ) sup inf J(λ, D ; τ) = inf τ S D > sup D > τ S hen if Ṽ (λ) exiss and is differeniable for λ >, hen V (x) = inf [Ṽ (λ) + λx, λ> for any x (, ). J(λ, D ; τ), Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

27 Variaional Inequaliy (1) The Model Dualiy Approaches and Variaional Inequaliy To find Ṽ (λ), define φ(, y) sup τ> =y inf Ey D > where y = λd e β H, y = λ >. Then e βs { ũ(y s ) + ɛy s l } ds + e βτ Ũ(y τ ), dy y = dd D + (β r)d θdb. φ(, λ) = Ṽ (λ). This opimal sopping problem can be solved by he variaional inequaliy. Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, 29 2 / 36

28 Variaional Inequaliy (2) The Model Dualiy Approaches and Variaional Inequaliy Suppose D has a following differenial form dd = ψ()d d for some ψ(). The Bellman equaion is given by { min Lφ(, y) + e β {ũ(y) + ɛy l}, φ } = y wih he differenial operaor L = + (β r)y y θ2 y 2 2 y 2. (He and Pagés (1993)) Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

29 The Model Dualiy Approaches and Variaional Inequaliy Variaional Inequaliy (3) Variaional Inequaliy 2.1 Find he free boundary ȳ, ŷ which makes zero wealh level and a funcion φ(, ) C 1 ((, ) R + ) C 2 ((, ) (R + \ {ȳ})) saisfying (1) L φ + e β {ũ(y) + ɛy l} =, ȳ < y ŷ (2) L φ + e β {ũ(y) + ɛy l}, < y ȳ (3) φ(, y) > e β Ũ(y), y > ȳ (4) φ(, y) = e β Ũ(y), < y ȳ, (5) φ (, y), < y ŷ y (6) φ (, y) =, y ŷ y for all >, wih boundary condiions φ y (, ŷ) = and 2 φ (, ŷ) =. y 2 Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

30 The Model Dualiy Approaches and Variaional Inequaliy one-o-one correspondence beween y and x. y ȳ ŷ Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

31 The Model Dualiy Approaches and Variaional Inequaliy one-o-one correspondence beween y and x. y ȳ ŷ e βũ(y) L φ + e β {ũ(y) + ɛy l} = φ y (, y) = Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

32 The Model Dualiy Approaches and Variaional Inequaliy one-o-one correspondence beween y and x. y ȳ ŷ e βũ(y) L φ + e β {ũ(y) + ɛy l} = φ y (, y) = x x Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

33 Variaional Inequaliy (4) The Model Dualiy Approaches and Variaional Inequaliy Proposiion 2 Consider he funcion C 1 y n+ + C 2 y n v(y) = + 2yn + y l+z(i 1 (z) ɛ) u(i 1 (z)) θ 2 (n + n ) ŷ z n yn y l+z(i 1 (z) ɛ) u(i 1 (z)) θ 2 (n + n ) ŷ 2y n + y zi 1 (z) u(i 1 (z)) dz θ 2 (n + n ) ŷ z n yn y θ 2 (n + n ) ŷ dz z n +1 dz, if ȳ < y ŷ, zi 1 (z) u(i 1 (z)) z n +1 dz, if < y ȳ, hen φ(, y) = e β v(y) is a soluion o Variaional Inequaliy. And he coefficiens C 1, C 2, ŷ and he free boundary value ȳ are deermined implicily. Ṽ (λ) = φ(, λ) = v(λ) Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

34 Value Funcion Soluions Theorem 3 The value funcion V (x) is given by C 1 (λ ) n+ + C 2 (λ ) n + (λ )x + 2(λ ) n + λ θ V (x) = 2 (n + n ) ŷ where 2(λ ) n θ 2 (n + n ) U(x), l+z(i 1 (z) ɛ) u(i 1 (z)) z n + +1 λ l+z(i 1 (z) ɛ) u(i 1 (z)) ŷ dz dz, if x < x, z n +1 if x x x = I 2 (ȳ), where λ is deermined from he following algebraic equaion n + C 1 (λ ) n + 1 n C 2 (λ ) n 1 2n + (λ ) n + 1 λ θ 2 (n + n ) ŷ + 2n (λ ) n 1 θ 2 (n + n ) l + z(i 1 (z) ɛ) u(i 1 (z)) z n + +1 dz λ l + z(i 1 (z) ɛ) u(i 1 (z)) ŷ z n +1 dz = x Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

35 Soluions Opimal Wealh Processes Before Reiremen X () = n + C 1 (y λ 2n + (yλ ) n + 1 θ 2 (n + n ) + 2n (yλ ) n 1 θ 2 (n + n ) ) n + 1 n C 2 (y λ ) n 1 y λ ŷ y λ ŷ l + z(i 1 (z) ɛ) u(i 1 (z)) z n + +1 dz l + z(i 1 (z) ɛ) u(i 1 (z)) z n +1 dz Afer Reiremen X () = 2n + (yλ ) n + 1 θ 2 (n + n ) + 2n (yλ ) n 1 θ 2 (n + n ) y λ ŷ y λ ŷ zi 1 (z) u(i 1 (z)) z n + +1 dz zi 1 (z) u(i 1 (z)) z n +1 dz Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

36 Opimal Policies (1) Soluions Theorem 4 The opimal policies (c, π, τ ) are given by { c I1 (y = λ ), if X < x I 1 (y λ ), if X x, Wih he opimal wealh process X (), he opimal sopping ime τ is deermined by τ = inf { > X () x}. Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

37 Opimal Policies (2) Soluions Theorem 5 (Coninued) π = { θ σ 2 σθ n + (n + 1)C 1 (y λ ) n n (n 1)C 2 (y λ ) n l+y λ θ 2 (I 1 (y λ ) ɛ) u(i 1 (y λ )) y λ + 2n + (n + 1)(yλ ) n + 1 y θ 2 λ l+z(i 1 (z) ɛ) u(i 1 (z)) (n + n ) ŷ z n + +1 dz 2n (n 1)(yλ ) n 1 y λ θ 2 l+z(i 1 (z) ɛ) u(i 1 (z)) (n + n ) ŷ z n +1 dz, y λ I 1 (y λ ) u(i 1 (y λ )) y λ + n + (n + 1)(yλ ) n + 1 y λ zi 1 (z) u(i 1 (z)) n + n ŷ z n + +1 dz n (n 1)(yλ ) n 1 y λ zi 1 (z) u(i 1 (z)) n + n ŷ z n +1 dz, if X < x if X x. Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

38 Examples: CRRA Uiliy Classes CRRA Uiliy Class Examples: CRRA Uiliy Class Definiion 6 (CRRA Uiliy Funcion) A CRRA uiliy funcion is defined by { 1 u(c) 1 γ c1 γ, if γ > and γ 1, log c, if γ = 1. Here γ is an invesor s coefficien of relaive risk aversion. Meron s Consan K r + β r γ + γ 1 2γ 2 θ2 >. Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

39 Examples: CRRA Uiliy Classes CRRA Uiliy Class CRRA Uiliy Class - Required Funcions Expeced Uiliy Maximizaion Problem (Power-Type) J(x; c, π, τ) = E = E ( ) 1 e β 1 γ c1 γ l d + e β ( 1 1 γ c1 γ l e β 1 ) τ d + e βτ U(X τ ) 1 γ c1 γ d Required Funcions U(x) = 1 K γ 1 1 γ x 1 γ ũ(y) = Ũ(y) = γ 1 γ y 1 γ γ γ K (1 γ) y 1 γ γ Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, 29 3 / 36

40 Examples: CRRA Uiliy Classes CRRA Uiliy Class CRRA Uiliy Class - Required Funcions Expeced Uiliy Maximizaion Problem (Power-Type) J(x; c, π, τ) = E = E ( ) 1 e β 1 γ c1 γ l d + e β ( 1 1 γ c1 γ l e β 1 ) τ d + e βτ U(X τ ) 1 γ c1 γ d Required Funcions U(x) = 1 K γ 1 1 γ x 1 γ ũ(y) = Ũ(y) = γ 1 γ y 1 γ γ γ K (1 γ) y 1 γ γ Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, 29 3 / 36

41 Examples: CRRA Uiliy Classes CRRA Uiliy Class CRRA Uiliy Class - Value Funcion Proposiion (Value Funcion) From Theorem 3, V (x) = c 1 (λ ) n+ + c 2 (λ ) n + γ K (1 γ) (λ ) 1 γ γ + ( x + ɛ ) r (λ ) l, if x < x β 1 1 K γ 1 γ x 1 γ, if x x where λ is deermined from he algebraic equaion n + c 1 (λ ) n+ 1 n c 2 (λ ) n K (λ ) 1 γ ɛ r = x, for x < x and he criical wealh level is given by x = 1 K ȳ 1 γ. Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

42 Examples: CRRA Uiliy Classes CRRA Uiliy Class CRRA Uiliy Class - Opimal Policies Opimal Policies c = { (y λ ) γ 1, if X < x KX, if X x, π = { θ σ θ σγ X, n +(n + 1)c 1 (y λ ) n+ 1 +n (n 1)c 2 (y λ ) n K γ λ (y ) γ 1 }, if X < x if X x, τ = inf { > X () x}, where he opimal wealh process before reiremen is given by X () = n +c 1 (y λ ) n+ 1 n c 2 (y λ ) n λ (y ) γ 1 ɛ K r. Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

43 Examples: CRRA Uiliy Classes Numerical Resuls Numerical Resuls for a CRRA Uiliy Funcion (1) Figure 1: Comparison of amoun of wealh invesed in he risky asse (β =.7, r =.1, µ =.5, σ =.2, γ = 2, ɛ =.2 and l =.5) Porfolio Wealh Level Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

44 Examples: CRRA Uiliy Classes Numerical Resuls Numerical Resuls for a CRRA Uiliy Funcion (2) Figure 2: Comparison of consumpion raio (β =.7, r =.1, µ =.5, σ =.2, γ = 2, ɛ =.2 and l =.5) Consumpion Wealh Level Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

45 Conclusion Conclusion We exended he opimal consumpion-porfolio selecion problem of an infiniely-lived working invesor whose wealh is subjec o borrowing consrain o he general uiliy funcion case We figured ou ha he criical wealh level wih borrowing consrain is lower han he level wih no consrain for he CRRA uiliy case The amoun of invesing o risky asse wih borrowing consrain is lower han he amoun wih no consrain for he CRRA uiliy case Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36

46 Conclusion Thank you! Yong Hyun Shin (KIAS) Workshop on Sochasic Analysis & Finance June 29 July 3, / 36