On Hyperbolic Growth of Long-term Bayesian Optimal Power-utility - PowerPoint PPT Presentation

On Hyperbolic Growth of Long-term Bayesian Optimal Power-utility Hideaki Miyata † and Jun Sekine ‡ † Kyoto University and ‡ Osaka University Workshop on Stochastic Processes and Their Applications, March 8–9, 2012, NCTS, Hsinchu, Taiwan 1 / 30

Key Finding Bayesian CRRA-utility maximization: § 1. Introduction ⊲ Key Finding π E u ( γ ) ( X x,π U ( T,γ ) ( x ) := sup ⊲ Hyperbolic Growth T ) , ⊲ Lifetime Port. ⊲ Hyperbolic Disc. u ( γ ) ( x ) := x 1 − γ ⊲ Plan 1 − γ , γ > 1 . § 2. Setup § 3. Preliminary: Non-Bayesian Result § 4. Bayesian ■ 1 -riskless asset ( r : risk-free interest rate) and n -risky assets. CRRA-optimal Utility: Karatzas and Zhao ■ λ : market price of risk, hidden and unobservable r.v. § 5. Long-time Asymptotics § 6. Bayesian Lifetime ■ available information is generated by risky assets prices. Optimal Consumption § 7. Remarks 2 / 30

Key Finding Bayesian CRRA-utility maximization: § 1. Introduction ⊲ Key Finding π E u ( γ ) ( X x,π U ( T,γ ) ( x ) := sup ⊲ Hyperbolic Growth T ) , ⊲ Lifetime Port. ⊲ Hyperbolic Disc. u ( γ ) ( x ) := x 1 − γ ⊲ Plan 1 − γ , γ > 1 . § 2. Setup § 3. Preliminary: Non-Bayesian Result § 4. Bayesian ■ 1 -riskless asset ( r : risk-free interest rate) and n -risky assets. CRRA-optimal Utility: Karatzas and Zhao ■ λ : market price of risk, hidden and unobservable r.v. § 5. Long-time Asymptotics § 6. Bayesian Lifetime ■ available information is generated by risky assets prices. Optimal Consumption § 7. Remarks Are interested in ∂ T log U ( T,γ ) ( x ) as T ≫ 1 , �� T � U ( T,γ ) ( x ) = u ( γ ) ( x ) exp ∂ t log U ( t,γ ) ( x ) dt . 0 2 / 30

Hyperbolic Growth ∂ T log U ( T,γ ) ( x ) = (1 − γ ) r − n 1 § 1. Introduction T + ǫ ( T ) as T → ∞ . ⊲ Key Finding 2 ⊲ Hyperbolic Growth ⊲ Lifetime Port. ⊲ Hyperbolic Disc. ⊲ Plan ■ ǫ ( T ) : “smaller” than 1 /T , (depends on the law of λ ). § 2. Setup § 3. Preliminary: Non-Bayesian Result § 4. Bayesian CRRA-optimal Utility: Karatzas and Zhao § 5. Long-time Asymptotics § 6. Bayesian Lifetime Optimal Consumption § 7. Remarks 3 / 30

Hyperbolic Growth ∂ T log U ( T,γ ) ( x ) = (1 − γ ) r − n 1 § 1. Introduction T + ǫ ( T ) as T → ∞ . ⊲ Key Finding 2 ⊲ Hyperbolic Growth ⊲ Lifetime Port. ⊲ Hyperbolic Disc. ⊲ Plan ■ ǫ ( T ) : “smaller” than 1 /T , (depends on the law of λ ). § 2. Setup § 3. Preliminary: Non-Bayesian Result Sharp contrast to “non-Bayesian” case with exponential growth § 4. Bayesian CRRA-optimal Utility: Karatzas and Zhao � r + 1 � ∂ T log U ( T,γ ) ( x ) = (1 − γ ) 2 γ | λ | 2 . § 5. Long-time Asymptotics § 6. Bayesian Lifetime Optimal Consumption § 7. Remarks 3 / 30

Lifetime Portfolio Selection with Hyperbolic Discount § 1. Introduction � ∞ � t � � ⊲ Key Finding U ( γ ) ( x ) := sup exp − ρ ( u ) du u ( γ ) ( c t ) dt, E ⊲ Hyperbolic Growth ⊲ Lifetime Port. ( π,c ) 0 0 ⊲ Hyperbolic Disc. β ⊲ Plan ρ ( t ) := δ + 1 + αt, § 2. Setup § 3. Preliminary: Non-Bayesian Result ■ agent’s wealth ( X x,π,c ) t ≥ 0 . § 4. Bayesian t CRRA-optimal Utility: Karatzas and Zhao ■ Merton (1969, 1971): β = 0 . § 5. Long-time Asymptotics § 6. Bayesian Lifetime ■ Zervos (2008), Bjork and Murgoci (2010), Eckland, Mbodji and Optimal Consumption Pirvu (2011): finite horizon with a priori hyperbolic discounting. § 7. Remarks 4 / 30

“Endogeneous” Hyperbolic Discounting For Bayesian agents, § 1. Introduction ⊲ Key Finding ■ characterize the critical discount rate: ⊲ Hyperbolic Growth ⊲ Lifetime Port. ⊲ Hyperbolic Disc. β ⊲ Plan ρ ( t ) := δ + 1 + αt, § 2. Setup § 3. Preliminary: to ensure the solvability, i.e., | U ( γ ) ( x ) | < ∞ iff one of the Non-Bayesian Result § 4. Bayesian following conditions is satisfied, CRRA-optimal Utility: Karatzas and Zhao § 5. Long-time (a) δ > δ , or (b) δ = δ and β > β . Asymptotics § 6. Bayesian Lifetime Optimal Consumption § 7. Remarks 5 / 30

Plan 2. Setup: Bayesian CRRA-utility maximization § 1. Introduction ⊲ Key Finding ⊲ Hyperbolic Growth 3. Preliminary: Non-Bayesian Result ⊲ Lifetime Port. ⊲ Hyperbolic Disc. ⊲ Plan 4. Bayesian CRRA-optimal Utility § 2. Setup ◆ Karatzas and Zhao (2001). § 3. Preliminary: Non-Bayesian Result § 4. Bayesian CRRA-optimal Utility: 5. Long-time Asymptotics Karatzas and Zhao § 5. Long-time ◆ hyperbolic growth Asymptotics § 6. Bayesian Lifetime ◆ cost of uncertainty Optimal Consumption § 7. Remarks 6. Lifetime Optimal Consumption ◆ “endogeneous” hyperbolic discount rate (Miyata, 2012) 6 / 30

Market Model(1) ■ S 0 t := e rt : riskless, § 1. Introduction § 2. Setup ■ S := ( S 1 , . . . , S n ) ⊤ , S i := ( S i ⊲ Market Model(1) t ) t ≥ 0 : n -risky, ⊲ Market Model(2) ⊲ Bayes. Util. Max. � � ⊲ Related works r 1 dt + σ ( t, S t ) d ˜ S 0 ∈ R n dS t = diag ( S t ) W t , ++ § 3. Preliminary: Non-Bayesian Result § 4. Bayesian on (Ω , F , ˜ P , ( F t ) t ≥ 0 ) , F t := σ ( ˜ W u ; u ∈ [0 , t ]) , CRRA-optimal Utility: Karatzas and Zhao 0 < ∃ c 1 I ≤ σσ ⊤ ( t, y ) ≤ ∃ c 2 I . § 5. Long-time Asymptotics ■ (Ω , F , ˜ P ) : “reference” prob. space, § 6. Bayesian Lifetime Optimal Consumption W i := ( ˜ ◆ n -dim. BM. ˜ W := ( ˜ W 1 , . . . , ˜ W n ) ⊤ , ˜ W i t ) t ≥ 0 , § 7. Remarks W . ν ( dx ) := ˜ ⊥ ˜ ◆ n -dim. r.v. λ ⊥ P ( λ ∈ dx ) . ■ F t = σ ( S u ; u ∈ [0 , t ]) : “public” information. 7 / 30

Market Model(2) ■ ( G t ) t ≥ 0 , G t := F t ∨ σ ( λ ) , § 1. Introduction § 2. Setup ⊲ Market Model(1) ■ real-world prob. measure: P on (Ω , ∨ t ≥ 0 G t ) by ⊲ Market Model(2) ⊲ Bayes. Util. Max. W 0 ) − | λ | 2 � � ⊲ Related works d P | G t = Z t d ˜ λ ⊤ ( ˜ W t − ˜ P | G t , Z t := exp 2 t . § 3. Preliminary: Non-Bayesian Result § 4. Bayesian CRRA-optimal Utility: W := ( W t ) t ≥ 0 , W t := ˜ W t − λt : ( P , G t ) -BM. Karatzas and Zhao § 5. Long-time Asymptotics ■ P -dynamics: § 6. Bayesian Lifetime Optimal Consumption dS t = diag ( S t ) { µ ( t, S t ) dt + σ ( t, S t ) dW t } , § 7. Remarks µ ( t, y ) := r 1 + σ ( t, y ) λ λ := σ − 1 ( µ − r 1 ) : unobservable market price of risk. ⊥ λ under P , P ( λ ∈ dx ) = ˜ ■ W ⊥ P ( λ ∈ dx ) = ν ( dx ) . 8 / 30

Bayesian CRRA-Utility Maximization X x,π U ( T,γ ) ( x ) := sup � � § 1. Introduction E u ( γ ) T π ∈ A T § 2. Setup ⊲ Market Model(1) with CRRA-utility, ⊲ Market Model(2) ⊲ Bayes. Util. Max. ⊲ Related works x 1 − γ  § 3. Preliminary: if γ > 0 , � = 1 ,  Non-Bayesian Result u ( γ ) ( x ) := 1 − γ § 4. Bayesian log x if γ = 1 ,  CRRA-optimal Utility: Karatzas and Zhao § 5. Long-time where Asymptotics � n § 6. Bayesian Lifetime � n � � dS 0 dS i Optimal Consumption dX x,π = X x,π � � X x,π t t π i π i + 1 − , = x, § 7. Remarks t t 0 t t S 0 S i t t i =1 i =1 and � � n -dim. F t -prog. m’ble, � � A T := ( f t ) t ∈ [0 ,T ] . � T � 0 | f t | 2 dt < ∞ a.s. � 9 / 30

Related works Similar market models and problems with unobservable random § 1. Introduction market price of risks : § 2. Setup ⊲ Market Model(1) ⊲ Market Model(2) ■ Brennan and Xia (2001), Cvitani´ c et. al. (2006), Karatzas ⊲ Bayes. Util. Max. (1997), Karatzas and Zhao (2001), Lakner (1995), Pham and ⊲ Related works § 3. Preliminary: Quenez (2001), Rieder and B¨ auerle (2005), Xia (2001), and Non-Bayesian Result Zohar (2001), for example. § 4. Bayesian CRRA-optimal Utility: Karatzas and Zhao § 5. Long-time Asymptotics § 6. Bayesian Lifetime Optimal Consumption § 7. Remarks 10 / 30

Constant λ : Merton (1969, 1971) π ( γ ) := (ˆ π ( γ ) § 1. Introduction (A) The optimal investment strategy ˆ ) t ∈ [0 ,T ] ∈ A T : t § 2. Setup § 3. Preliminary: := 1 γ ( σσ ⊤ ) − 1 ( µ − r 1 )( t, S t ) = 1 π ( γ ) γ ( σ ⊤ ) − 1 ( t, S t ) λ, Non-Bayesian Result ˆ t ⊲ Constant λ ⊲ LT Opt. (1) ⊲ LT Opt. (2) : T -independent ! § 4. Bayesian CRRA-optimal Utility: Karatzas and Zhao X ( γ ) (B) The optimal wealth process ( ˆ ) t ∈ [0 ,T ] : t § 5. Long-time Asymptotics � 1 � 1 � � § 6. Bayesian Lifetime π ( γ ) γ λ ⊤ ( W t + λt ) + X ( γ ) ˆ := X x, ˆ 2 γ 2 | λ | 2 = x exp r − t . Optimal Consumption t t § 7. Remarks (C) The optimal utility: � ˆ � � 2 γ | λ | 2 � � r + 1 T X ( γ ) U ( T,γ ) ( x ) = E u ( γ ) � = u ( γ ) x e . T 11 / 30

Long-term Optimalities (1) ■ When γ = 1 , for ∀ π , § 1. Introduction § 2. Setup 1 ≤ 1 X (1) T log X x,π T log ˆ = − Γ ′ (1) a.s.. lim § 3. Preliminary: T T Non-Bayesian Result T →∞ ⊲ Constant λ ⊲ LT Opt. (1) ■ When 0 < γ < 1 , for ∀ π , ⊲ LT Opt. (2) 1 1 § 4. Bayesian X ( γ ) T log E u ( γ ) ( X x,π T log E u ( γ ) ( ˆ lim T ) ≤ lim T ) = Γ( γ ) . CRRA-optimal Utility: Karatzas and Zhao T →∞ T →∞ § 5. Long-time ■ When γ > 1 , for ∀ π , Asymptotics § 6. Bayesian Lifetime 1 1 Optimal Consumption X ( γ ) T log E u ( γ ) ( ˆ T log E u ( γ ) ( X x,π lim T ) ≥ lim T ) = Γ( γ ) . § 7. Remarks T →∞ T →∞ where � r + 1 � − Γ ′ (1) := r + 1 2 γ | λ | 2 2 | λ | 2 . Γ( γ ) := (1 − γ ) , 12 / 30