Optimal Investment, Consumption and Retirement Decision with Disutility and Borrowing Constraints Yong Hyun Shin Joint Work with Byung Hwa Lim(KAIST) June 29 – July 3, 2009 Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 1 / 36
Contents Historical Remarks 1 Portfolio Selection Borrowing Constraints The Model 2 Main Problem Duality Approaches and Variational Inequality Solutions 3 Examples: CRRA Utility Classes 4 CRRA Utility Class Numerical Results Conclusion 5 Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 2 / 36
Historical Remarks Portfolio Selection Portfolio Selection (1) Merton (1969 RESTAT, 1971 JET) Formulate a continuous time consumption/investement problem. (dynamic programming method) �� ∞ � e − β t u ( c t ) dt max c ,π E 0 subject to dX t = [ rX t + π t ( µ − r ) − c t ] dt + π t σ dB t . Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 3 / 36
Historical Remarks Portfolio Selection Portfolio Selection (2) Karatzas and Wang (2000 SICON) Consider the mixture problem ( c , π, τ ) . (martingale method) �� τ � e − β t u 1 ( c t ) dt + e − βτ u 2 ( X τ ) max c ,π,τ E 0 subject to dX t = [ rX t + π t ( µ − r ) − c t ] dt + π t σ dB t . Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 4 / 36
Historical Remarks Portfolio Selection Portfolio Selection (3) Choi and Shim (2006 MF), Benchmark Model Labor income, disutility and optimal retirement time. (dynamic programming without considering borrowing constraints) �� ∞ � � � c ,π,τ E max u ( c t ) − l 1 { 0 ≤ t <τ } dt 0 subject to dX t = [ rX t + π t ( µ − r ) − c t + ǫ 1 { 0 ≤ t <τ } ] dt + π t σ dB t . Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 5 / 36
Historical Remarks Borrowing Constraints Borrowing Constraints Borrowing Constraint (Liquidity Constraint) The economic agent has limited opportunities to borrow against future labor income and cannot totally insure the risk of income fluctuation. So the borrowing constraint restricts the agent’s choice in a non-trivial way. Literatures He and Pag´ es (1993 JET), Duffie, Fleming, Soner, and Zariphopoulou (1997 JEDC), Koo (1998 MF), El Karoui and Jeanblanc (1998 FS), Dybvig and Liu (2005 WP), Farhi and Panageas (2007 JFE), Choi, Shim, and Shin (2008 MF), Saha (2008 WP) Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 6 / 36
Historical Remarks Borrowing Constraints Borrowing Constraints Borrowing Constraint (Liquidity Constraint) The economic agent has limited opportunities to borrow against future labor income and cannot totally insure the risk of income fluctuation. So the borrowing constraint restricts the agent’s choice in a non-trivial way. Literatures He and Pag´ es (1993 JET), Duffie, Fleming, Soner, and Zariphopoulou (1997 JEDC), Koo (1998 MF), El Karoui and Jeanblanc (1998 FS), Dybvig and Liu (2005 WP), Farhi and Panageas (2007 JFE), Choi, Shim, and Shin (2008 MF), Saha (2008 WP) Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 6 / 36
The Model The Basic Financial Market Setup (1) riskless asset S 0 ( · ) : dS 0 ( t ) S 0 ( t ) = rdt risky asset S t : dS t S t = µ dt + σ dB t r , µ, σ : constants B t is a standard Brownian motion on a probability space (Ω , F , P ) market-price-of-risk θ � µ − r σ discount process ζ t � exp {− rt } � � − θ B t − 1 exponential martingale Z t � exp 2 θ 2 t pricing kernel(or state-price-density) H t � ζ t Z t equivalent martingale measure � P T ( A ) � E [ Z T 1 A ] , for any fixed T ∈ [ 0 , ∞ ) and for A ∈ F T P T by � t � B t + θ t : a standard Brownian motion under the new measure � B T Girsanov theorem P T on F T for any T ∈ [ 0 , ∞ ) . Furthermore ∃ � P on F ∞ , which agrees with � B t is a standard Brownian motion under � � P . Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 7 / 36
The Model The Basic Financial Market Setup (2) labor wage: ǫ > 0, disutility: l > 0 retirement time: a stopping time τ � t consumption process : c t with 0 c s ds < ∞ , a.s. � t 0 π 2 s ds < ∞ , a.s. portfolio process : π t with wealth process X t with an initial wealth X 0 = x ≥ 0 dX t = [ rX t + π t ( µ − r ) − c t + ǫ 1 { 0 ≤ t <τ } ] dt + π t σ dB t = [ rX t − c t + ǫ 1 { 0 ≤ t <τ } ] dt + π t σ d � B t budget constraint � � � τ � τ E H τ X τ + H s c s ds − H s ǫ ds ≤ x , for all τ 0 0 Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 8 / 36
The Model The Basic Financial Market Setup (2) labor wage: ǫ > 0, disutility: l > 0 retirement time: a stopping time τ � t consumption process : c t with 0 c s ds < ∞ , a.s. � t 0 π 2 s ds < ∞ , a.s. portfolio process : π t with wealth process X t with an initial wealth X 0 = x ≥ 0 dX t = [ rX t + π t ( µ − r ) − c t + ǫ 1 { 0 ≤ t <τ } ] dt + π t σ dB t = [ rX t − c t + ǫ 1 { 0 ≤ t <τ } ] dt + π t σ d � B t budget constraint � � � τ � τ E H τ X τ + H s c s ds − H s ǫ ds ≤ x , for all τ 0 0 Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 8 / 36
The Model Borrowing Constraints The borrowing constraint means that the investor cannot borrow against her/his future labor income. So the wealth of the investor should always be nonnegative. i.e. X t ≥ 0 , ∀ t > 0 Borrowing Constraint � � � � τ � � H τ X τ + H s ( c s − ǫ ) ds � F t ≥ 0 , for all 0 ≤ t < τ. E t Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 9 / 36
The Model Borrowing Constraints The borrowing constraint means that the investor cannot borrow against her/his future labor income. So the wealth of the investor should always be nonnegative. i.e. X t ≥ 0 , ∀ t > 0 Borrowing Constraint � � � � τ � � H τ X τ + H s ( c s − ǫ ) ds � F t ≥ 0 , for all 0 ≤ t < τ. E t Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 9 / 36
The Model Definition 1 (General Utility Function) A function u : ( 0 , ∞ ) → R is a utility function if it is strictly increasing, strictly concave, continuously differentiable and satisfies u ′ ( 0 +) � lim c ↓ 0 u ′ ( c ) = ∞ , u ′ ( ∞ ) � lim c ↑∞ u ′ ( c ) = 0 . Labor Income ǫ : the agent receives the income continuously Disutility l : disutility comes from labor Retirement Time τ : the immortal agent can choose when to retire Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 10 / 36
The Model Definition 1 (General Utility Function) A function u : ( 0 , ∞ ) → R is a utility function if it is strictly increasing, strictly concave, continuously differentiable and satisfies u ′ ( 0 +) � lim c ↓ 0 u ′ ( c ) = ∞ , u ′ ( ∞ ) � lim c ↑∞ u ′ ( c ) = 0 . Labor Income ǫ : the agent receives the income continuously Disutility l : disutility comes from labor Retirement Time τ : the immortal agent can choose when to retire Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 10 / 36
The Model Definition 1 (General Utility Function) A function u : ( 0 , ∞ ) → R is a utility function if it is strictly increasing, strictly concave, continuously differentiable and satisfies u ′ ( 0 +) � lim c ↓ 0 u ′ ( c ) = ∞ , u ′ ( ∞ ) � lim c ↑∞ u ′ ( c ) = 0 . Labor Income ǫ : the agent receives the income continuously Disutility l : disutility comes from labor Retirement Time τ : the immortal agent can choose when to retire Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 10 / 36
The Model Definition 1 (General Utility Function) A function u : ( 0 , ∞ ) → R is a utility function if it is strictly increasing, strictly concave, continuously differentiable and satisfies u ′ ( 0 +) � lim c ↓ 0 u ′ ( c ) = ∞ , u ′ ( ∞ ) � lim c ↑∞ u ′ ( c ) = 0 . Labor Income ǫ : the agent receives the income continuously Disutility l : disutility comes from labor Retirement Time τ : the immortal agent can choose when to retire Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 10 / 36
The Model Main Problem Main Problem Expected Utility Maximization Problem The immortal investor wants to maximize her/his expected utility: �� ∞ � e − β t � � J ( x ; c , π, τ ) � E u ( c t ) − l 1 { 0 ≤ t <τ } dt 0 i.e., V ( x ) = sup J ( x ; c , π, τ ) ( c ,π,τ ) ∈A ( x ) subject to the budget constraint and the borrowing constraint, where A ( x ) is the set of an admissible triple ( c , π, τ ) and β > 0 is the subjective discount rate. Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 11 / 36
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