scholarly journals Portfolio Selection with Liability and Affine Interest Rate in the HARA Utility Framework

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Hao Chang ◽  
Kai Chang ◽  
Ji-mei Lu
Keyword(s):  
Interest Rate ◽  
Absolute Risk ◽  
Optimal Investment ◽  
Investment Strategy ◽  
Dynamic Programming Principle ◽  
Legendre Transform ◽  
Management Problem ◽  
Power Utility ◽  
Special Cases ◽  
Hara Utility

This paper studied an asset and liability management problem with stochastic interest rate, where interest rate is assumed to be governed by an affine interest rate model, while liability process is driven by the drifted Brownian motion. The investors wish to look for an optimal investment strategy to maximize the expected utility of the terminal surplus under hyperbolic absolute risk aversion (HARA) utility function, which consists of power utility, exponential utility, and logarithm utility as special cases. By applying dynamic programming principle and Legendre transform, the explicit solutions for HARA utility are achieved successfully and some special cases are also discussed. Finally, a numerical example is provided to illustrate our results.

scholarly journals Legendre Transform-Dual Solution for a Class of Investment and Consumption Problems with HARA Utility

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Hao Chang ◽  
Xi-min Rong
Keyword(s):  
Absolute Risk ◽  
Optimal Investment ◽  
Dynamic Programming Principle ◽  
Legendre Transform ◽  
Power Utility ◽  
Transform Method ◽  
Intermediate Consumption ◽  
Special Cases ◽  
Hara Utility ◽  
Optimal Investment And Consumption

This paper provides a Legendre transform method to deal with a class of investment and consumption problems, whose objective function is to maximize the expected discount utility of intermediate consumption and terminal wealth in the finite horizon. Assume that risk preference of the investor is described by hyperbolic absolute risk aversion (HARA) utility function, which includes power utility, exponential utility, and logarithm utility as special cases. The optimal investment and consumption strategy for HARA utility is explicitly obtained by applying dynamic programming principle and Legendre transform technique. Some special cases are also discussed.

scholarly journals Optimal Consumption and Portfolio Decision with Heston’s SV Model Under HARA Utility Criterion

2017 ◽  
Vol 5 (1) ◽  
pp. 21-33 ◽  
Author(s):  
Chunfeng Wang ◽  
Hao Chang ◽  
Zhenming Fang
Keyword(s):  
Absolute Risk ◽  
Closed Form Solution ◽  
Investment Strategy ◽  
Dynamic Programming Principle ◽  
Legendre Transform ◽  
Optimal Consumption ◽  
Stochastic Dynamic ◽  
Power Utility ◽  
Special Cases ◽  
Hara Utility

Abstract This paper studies the optimal consumption-investment strategy with Heston’s stochastic volatility (SV) model under hyperbolic absolute risk aversion (HARA) utility criterion. The financial market is composed of a risk-less asset and a risky asset, whose price process is supposed to be driven by Heston’s SV model. The risky preference of the individual is assumed to satisfy HARA utility, which recovers power utility, exponential utility and logarithm utility as special cases. HARA utility is of general framework in the utility theory and is seldom studied in the existing literatures. Legendre transform-dual technique along with stochastic dynamic programming principle is presented to deal with our problem and the closed-form solution to the optimal consumption-investment strategy is successfully obtained. Finally, some special cases are derived in detail.

Dynamic asset allocation for a bank under CRRA and HARA framework

2015 ◽  
Vol 02 (03) ◽  
pp. 1550031 ◽  
Author(s):  
Ryle S. Perera
Keyword(s):  
Interest Rate ◽  
Risk Aversion ◽  
Portfolio Choice ◽  
Asset Allocation ◽  
Absolute Risk ◽  
Optimal Investment ◽  
Stock Index ◽  
Investment Horizon ◽  
Power Utility ◽  
Dynamic Portfolio Optimization

This paper analyzes an optimal investment and management strategy for a bank under constant relative risk aversion (CRRA) and hyperbolic absolute risk aversion (HARA) utility functions. We assume that the bank can invest in treasuries, stock index fund and loans, in an environment subject to stochastic interest rate and inflation uncertainty. The interest rate and the expected rate of inflation follow a correlated Ornstein–Uhlenbeck processes and the risk premia are constants. Then we consider the portfolio choice under a power utility that the bank's shareholders can maximize expected utility of wealth at a given investment horizon. Closed form solutions are obtained in a dynamic portfolio optimization model. The results indicate that the optimal proportion invested in treasuries increases while the optimal proportion invested in the loans progressively decreases with respect to time.

scholarly journals Optimal Consumption and Portfolio Decision with Convertible Bond in Affine Interest Rate and Heston’s SV Framework

2016 ◽  
Vol 2016 ◽  
pp. 1-12
Author(s):  
Hao Chang ◽  
Xue-Yan Li
Keyword(s):  
Interest Rate ◽  
Stochastic Volatility ◽  
Separation Of Variables ◽  
Optimal Investment ◽  
Convertible Bond ◽  
Dynamic Programming Principle ◽  
Stochastic Dynamic ◽  
Special Cases ◽  
The Impact ◽  
Consumption Strategies

We are concerned with an optimal investment-consumption problem with stochastic affine interest rate and stochastic volatility, in which interest rate dynamics are described by the affine interest rate model including the Cox-Ingersoll-Ross model and the Vasicek model as special cases, while stock price is driven by Heston’s stochastic volatility (SV) model. Assume that the financial market consists of a risk-free asset, a zero-coupon bond (or a convertible bond), and a risky asset. By using stochastic dynamic programming principle and the technique of separation of variables, we get the HJB equation of the corresponding value function and the explicit expressions of the optimal investment-consumption strategies under power utility and logarithmic utility. Finally, we analyze the impact of market parameters on the optimal investment-consumption strategies by giving a numerical example.

scholarly journals Optimal Asset Allocation for CRRA and CARA Insurers under the Vasicek Interest Rate Model

2022 ◽  
Vol 2022 ◽  
pp. 1-14
Author(s):  
Hanlei Hu ◽  
Shaoyong Lai ◽  
Hongjing Chen
Keyword(s):  
Interest Rate ◽  
Risk Aversion ◽  
Asset Allocation ◽  
Investment Strategy ◽  
Interest Rate Risk ◽  
Dynamic Programming Principle ◽  
Legendre Transform ◽  
Rate Model ◽  
Rate Fluctuation ◽  
Interest Rate Model

This paper considers the reinsurance-investment problem with interest rate risks under constant relative risk aversion and constant absolute risk aversion preferences, respectively. Stochastic control theory and dynamic programming principle are applied to investigate the optimal proportional reinsurance-investment strategy for an insurer under the Vasicek stochastic interest rate model. Solving the corresponding Hamilton-Jacobi-Bellman equation via the Legendre transform approach, the optimal premium allocation strategies maximizing the expected utilities of terminal wealth are derived. In addition, several sensitivity analyses and numerical illustrations are given to analyze the impacts of different risk preferences and interest rate fluctuation on the optimal strategies. We find that the asset allocation and reinsurance ratio of the insurer are correlated with risk preference coefficient and interest rate fluctuation, and the insurance company may adjust the reinsurance-investment strategy to deal with interest rate risk.

scholarly journals Optimal Investment-Consumption Strategy under Inflation in a Markovian Regime-Switching Market

2016 ◽  
Vol 2016 ◽  
pp. 1-17 ◽  
Author(s):  
Huiling Wu
Keyword(s):  
Regime Switching ◽  
Stock Price ◽  
Optimal Investment ◽  
Investment Strategy ◽  
Power Utility ◽  
Optimal Investment Strategy ◽  
Admissible Strategies ◽  
Definition Of ◽  
Markovian Regime Switching ◽  
Optimal Investment And Consumption

This paper studies an investment-consumption problem under inflation. The consumption price level, the prices of the available assets, and the coefficient of the power utility are assumed to be sensitive to the states of underlying economy modulated by a continuous-time Markovian chain. The definition of admissible strategies and the verification theory corresponding to this stochastic control problem are presented. The analytical expression of the optimal investment strategy is derived. The existence, boundedness, and feasibility of the optimal consumption are proven. Finally, we analyze in detail by mathematical and numerical analysis how the risk aversion, the correlation coefficient between the inflation and the stock price, the inflation parameters, and the coefficient of utility affect the optimal investment and consumption strategy.

Do Time Preferences Matter in Intertemporal Consumption and Portfolio Decisions?

2019 ◽  
Vol 19 (2) ◽  
Author(s):  
Shou Chen ◽  
Shengpeng Xiang ◽  
Hongbo He
Keyword(s):  
Absolute Risk ◽  
Hyperbolic Discounting ◽  
Time Preferences ◽  
Hjb Equations ◽  
Portfolio Decisions ◽  
Special Cases ◽  
Intertemporal Consumption ◽  
Hamilton Jacobi Bellman ◽  
Hara Utility ◽  
Exponential Discounting

Abstract We study the intertemporal consumption and portfolio rules in the model with the general hyperbolic absolute risk aversion (HARA) utility. The equivalent approximation approach is employed to obtain the Hamilton-Jacobi-Bellman (HJB) equations, and a remarkable property is shown: portfolio rules are independent of the discount function. Moreover, both the consumption and portfolio rates are non-increasing functions of wealth. Particularly illustrative cases examined in detail are the models with the most adopted discount functions, including exponential discounting and hyperbolic discounting. Explicit solutions for intertemporal decisions are found for these special cases, revealing that individual’s time preferences affect the consumption rules only. Moreover, the time-consistent consumption rate under hyperbolic discounting is larger than its counterpart under exponential discounting.

scholarly journals Robust optimal asset-liability management with penalization on ambiguity

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yu Yuan ◽  
Hui Mi
Keyword(s):  
Reference Model ◽  
Optimal Investment ◽  
Investment Strategy ◽  
Programming Approach ◽  
Stochastic Dynamic ◽  
Economic Implication ◽  
Investment Strategies ◽  
Dynamic Programming Approach ◽  
Optimal Investment Strategy ◽  
Special Cases

<p style='text-indent:20px;'>In this paper, we study the robust optimal asset- problems for an ambiguity-averse investor, who does not have perfect information in the drift terms of the risky asset and liability processes. Two different kinds of objectives are considered: <inline-formula><tex-math id="M1">\begin{document}$ (i) $\end{document}</tex-math></inline-formula> Maximizing the minimal expected utility of the terminal wealth; <inline-formula><tex-math id="M2">\begin{document}$ (ii) $\end{document}</tex-math></inline-formula> Minimizing the maximal cumulative deviation. The ambiguity in both problems is described by a set of equivalent measures to the reference model. By the stochastic dynamic programming approach and Hamilton-Jacobi-Bellman (HJB) equation, we derive closed-form expressions for the value function and corresponding robust optimal investment strategy in each problem. Furthermore, some special cases are provided to investigate the effect of model uncertainty on the optimal investment strategy. Finally, the economic implication and parameter sensitivity are analyzed by some numerical examples. We also compare the robust optimal investment strategies in two different problems.</p>

scholarly journals Dynamic Mean-Variance Model with Borrowing Constraint under the Constant Elasticity of Variance Process

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Hao Chang ◽  
Xi-min Rong
Keyword(s):  
Stock Price ◽  
Investment Strategy ◽  
Hjb Equation ◽  
Dynamic Programming Principle ◽  
Legendre Transform ◽  
Order Partial Differential Equation ◽  
Constant Elasticity ◽  
Constant Elasticity Of Variance ◽  
Time Dynamic ◽  
Mean Variance

This paper studies a continuous-time dynamic mean-variance portfolio selection problem with the constraint of a higher borrowing rate, in which stock price is governed by a constant elasticity of variance (CEV) process. Firstly, we apply Lagrange duality theorem to change an original mean-variance problem into an equivalent optimization one. Secondly, we use dynamic programming principle to get the Hamilton-Jacobi-Bellman (HJB) equation for the value function, which is a more sophisticated nonlinear second-order partial differential equation. Furthermore, we use Legendre transform and dual theory to transform the HJB equation into its dual one. Finally, the closed-form solutions to the optimal investment strategy and efficient frontier are derived by applying variable change technique.

scholarly journals An Investment and Consumption Problem with CIR Interest Rate and Stochastic Volatility

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Hao Chang ◽  
Xi-min Rong
Keyword(s):  
Interest Rate ◽  
Stochastic Volatility ◽  
Optimal Investment ◽  
Stochastic Volatility Model ◽  
Power Utility ◽  
Variable Approach ◽  
Volatility Model ◽  
Stochastic Interest ◽  
Hamilton Jacobi Bellman ◽  
Optimal Investment And Consumption

We are concerned with an investment and consumption problem with stochastic interest rate and stochastic volatility, in which interest rate dynamic is described by the Cox-Ingersoll-Ross (CIR) model and the volatility of the stock is driven by Heston’s stochastic volatility model. We apply stochastic optimal control theory to obtain the Hamilton-Jacobi-Bellman (HJB) equation for the value function and choose power utility and logarithm utility for our analysis. By using separate variable approach and variable change technique, we obtain the closed-form expressions of the optimal investment and consumption strategy. A numerical example is given to illustrate our results and to analyze the effect of market parameters on the optimal investment and consumption strategies.

Sign in / Sign up

Export Citation Format

Share Document