CRRA Utility Function Research Articles

Non-constant discounting and consumption, portfolio and life insurance rules

Consumption, portfolio and life insurance rules are studied for an investor with an arbitrary but known distribution of lifetime with time-inconsistent preferences. Solutions are found for naive and sophisticated agents for the family of CARA and CRRA utility functions. Effects of non-constant discounting are illustrated numerically.

Optimal Investment Strategies for DC Pension with Stochastic Salary under the Affine Interest Rate Model

We study the optimal investment strategies of DC pension, with the stochastic interest rate (including the CIR model and the Vasicek model) and stochastic salary. In our model, the plan member is allowed to invest in a risk-free asset, a zero-coupon bond, and a single risky asset. By applying the Hamilton-Jacobi-Bellman equation, Legendre transform, and dual theory, we find the explicit solutions for the CRRA and CARA utility functions, respectively.

Optimal consumption and investment for markets with random coefficients

We consider an optimal investment and consumption problem for a Black–Scholes financial market with stochastic coefficients driven by a diffusion process. We assume that an agent makes consumption and investment decisions based on CRRA utility functions. The dynamic programming approach leads to an investigation of the Hamilton–Jacobi–Bellman (HJB) equation which is a highly nonlinear partial differential equation (PDE) of the second order. By using the Feynman–Kac representation, we prove uniqueness and smoothness of the solution. Moreover, we study the optimal convergence rate of iterative numerical schemes for both the value function and the optimal portfolio. We show that in this case, the optimal convergence rate is super-geometric, i.e., more rapid than any geometric one. We apply our results to a stochastic volatility financial market.

On efficiency of mean–variance based portfolio selection in defined contribution pension schemes

We consider the portfolio selection problem in the accumulation phase of a defined contribution (DC) pension scheme. We solve the mean–variance portfolio selection problem using the embedding technique pioneered by Zhou and Li [Appl. Math. Optimiz., 2000, 42, 19–33] and show that it is equivalent to a target-based optimization problem, consisting of the minimization of a quadratic loss function. We support the use of the target-based approach in DC pension funds for three reasons. Firstly, it transforms the difficult problem of selecting the individual's risk-aversion coefficient into the easier task of choosing an appropriate target. Secondly, it is intuitive, flexible and adaptable to the member's needs and preferences. Thirdly, it produces final portfolios that are efficient in the mean–variance setting. We address the issue of the comparison between an efficient portfolio and a portfolio that is optimal according to the more general criterion of maximization of the expected utility (EU). The two natural notions of Variance Inefficiency and Mean Inefficiency are introduced, which measure the distance of an optimal inefficient portfolio from an efficient one, focusing on their variance and on their expected value, respectively. We illustrate the general procedure for finding the mean–variance inefficiency for the HARA class of utility functions, and then focus on the popular classes of CARA and CRRA utility functions. In these cases we prove the intuitive but non-trivial result that the mean–variance inefficiency decreases with the risk aversion of the individual and increases with the time horizon and the Sharpe ratio of the risky asset. As a by-product we prove that the CARA and CRRA optimal portfolios are not mean–variance efficient. Numerical investigations stress the impact of the time horizon on extent of the mean–variance inefficiency of CARA and CRRA utility functions. While at the instantaneous level EU optimality and efficiency coincide (see Merton [J. Econ. Theory, 1971, 3, 373–413]), we find that, for short durations, they do not differ significantly. However, for longer durations, which are typical for pension funds, the extent of the inefficiency turns out to be remarkable and should be taken into account by pension fund investment managers seeking appropriate rules for portfolio selection. Indeed, we see this result as a fourth reason for supporting the use of the target-based approach in DC pension schemes.

Optimal asset allocation for DC pension plans under inflation

In this paper, the stochastic dynamic programming approach is used to investigate the optimal asset allocation for a defined-contribution pension plan with downside protection under stochastic inflation. The plan participant invests the fund wealth and the stochastic interim contribution flows into the financial market. The nominal interest rate model is described by the Cox–Ingersoll–Ross (Cox et al., 1985) dynamics. To cope with the inflation risk, the inflation indexed bond is included in the asset menu. The retired individuals receive an annuity that is indexed by inflation and a downside protection on the amount of this annuity is considered. The closed-form solution is derived under the CRRA utility function. Finally, a numerical application is presented to characterize the dynamic behavior of the optimal investment strategy.

THE EQUITY PREMIUM IN CONSUMPTION AND PRODUCTION MODELS

In this paper we use a simple model with a single Cobb–Douglas firm and a consumer with a CRRA utility function to show the difference between the equity premia in the production-based Brock model and the consumption-based Lucas model. With this simple example we show that the equity premium in the production-based model exceeds that of the consumption-based model with probability 1.

Heterogeneous Discounting in Consumption-Investment Problems: Time-Consistent Solutions

In this paper we analyze a stochastic continuous time model in finite horizon in which the agent discounts the instantaneous utility function and the final function at constant but different instantaneous discount rates of time preference. Within this context we can model problems in which, when the time $t$ approaches to the final time, the valuation of the final function increases compared with previous valuations in a way that cannot be explained by using a unique constant or a variable discount rate. We derive a dynamic programming equation whose solutions are time-consistent Markov equilibria. For this class of time preferences, we study the classical consumption and portfolio rules model (Merton, 1971) for CRRA and CARA utility functions for time-consistent agents, and we compare the different equilibria with the time-inconsistent solutions. The introduction of stochastic terminal time is also discussed.

On Efficiency of Mean-Variance Based Portfolio Selection in DC Pension Schemes

We consider the portfolio selection problem in the accumulation phase of a defined contribution (DC) pension scheme. We solve the mean-variance portfolio selection problem using the embedding technique pioneered by Zhou and Li (2000) and show that it is equivalent to a target-based optimization problem, consisting in the minimization of a quadratic loss function. We support the use of the target-based approach in DC pension funds for three reasons. Firstly, it transforms the difficult problem of selecting the individual's risk aversion coefficient into the easiest task of choosing an appropriate target. Secondly, it is intuitive, flexible and adaptable to the member's needs and preferences. Thirdly, it produces final portfolios that are efficient in the mean-variance setting.We address the issue of comparison between an efficient portfolio and a portfolio that is optimal according to the more general criterion of maximization of expected utility (EU). The two natural notions of Variance Inefficiency and Mean Inefficiency are introduced, which measure the distance of an optimal inefficient portfolio from an efficient one, focusing on their variance and on their expected value, respectively. As a particular case, we investigate the quite popular classes of CARA and CRRA utility functions. In these cases, we prove the intuitive but not trivial results that the mean-variance inefficiency decreases with the risk aversion of the individual and increases with the time horizon and the Sharpe ratio of the risky asset.Numerical investigations stress the impact of the time horizon on the extent of mean-variance inefficiency of CARA and CRRA utility functions. While at instantaneous level EU-optimality and efficiency coincide (see Merton, 1971), we find that for short durations they do not differ significantly. However, for longer durations -- that are typical in pension funds -- the extent of inefficiency turns out to be remarkable and should be taken into account by pension fund investment managers seeking appropriate rules for portfolio selection. Indeed, this result is a further element that supports the use of the target-based approach in DC pension schemes.

It’s Not My Money: An Experiment on Risk Aversion and the House-Money Effect

The house-money effect – people’s tendency to be more daring with easily-gotten money – is a behavioral pattern that poses questions about the external validity of experiments in economics: to what extent do people behave in experiments like they would have in a real-life situation, given that they play with easily-gotten house money? We ran an economic experiment with 66 students to measure the house-money effect on their risk preferences. They received an amount of money with which they made risky decisions involving losses and gains; a treatment group got the money 21 days in advance and a control group got it the day of the experiment. We find that, when facing possible losses, people in the treatment group showed a lower tolerance to risk than people in the control group. If the players are assumed to have a CRRA utility function and to behave according to expected-utility theory, the risk-attitude adjustment corresponds to an average increase of 1 in their risk aversion coefficient. While the exact pattern of this house-money adjustment differs by gender, it is not possible to determine the sign of this gender effect unambiguously. In any case, it is advisable to include credible controls for the house-money effect in experimental work in economics.

An extended CEV model and the Legendre transform–dual–asymptotic solutions for annuity contracts

This paper develops an extended constant elasticity of variance (E-CEV) model to overcome the shortcomings of the general CEV model. Under the E-CEV model, we study the optimal investment strategy before and after retirement in a defined contribution pension plan where benefits are paid by annuity. By applying the Legendre transform, dual theory and an asymptotic expansion approach, we respectively derive two asymptotic strategies for a CRRA and CARA utility functions in two different periods. Furthermore, we find that each asymptotic strategy can be decomposed into an optimal zero-order strategy and a perturbation strategy. The optimal zero-order strategy denotes an investment strategy where the current volatility is just equal to the mean level of the volatility, whereas the perturbation strategy provides an approximation solution to hedge the slow varying nature of the current volatility deviating from mean level. Finally, we find that the optimal zero-order strategy under given conditions will reduce to the results of Devolder et al. (2003), Xiao et al. (2007) and Gao (2009), respectively.

Optimal Portfolio for CRRA Utility Functions when Risky Assets are Exponential Additive Processes

In this paper, we analyse a market where the risky assets follow exponential additive processes, which can be viewed as time‐inhomogeneous generalizations of geometric Levy processes. In this market we show that, when an investor wants to maximize a CRRA utility function of his/her terminal wealth, his/her optimal strategy consists in keeping proportions of wealth in the risky assets which depend only on time but not on the current wealth level or on the prices of the risky assets. In the time‐homogeneous case, the optimal strategy is to keep constant proportions of wealth, a result already found by Kallsen which extends the classical Merton’s result to this market. While the one‐dimensional case has been extensively treated and the multidimensional case has been treated only in the time‐homogeneous case Callegaro and Vargiolu (2009), Kallsen (2000), and Korn et al. (2003) to the authors’ knowledge this is the first time that such results are obtained for exponential additive processes in the multidimensional case. We use these results to show that the optimal solution in the presence of jumps has the form of the analogous one without jumps but with the asset yields vector reduced by suitable quantities: in the one‐dimensional case, we extend a result by Benth et al. (2001). We conclude with four examples.

Overpriced Puts Puzzle in KOSPI 200 Options Market

The purpose of this paper is to examine the argument that the put options traded in the exchanges are too high, compared to the asset prices based on the classical CAPM model, and thus the short position of the put option would make a significant profit from trading. In order to explore the earlier report, this paper, using the KOSPI 200 index options market price, estimates the historical rate of return on several option trading strategies such as naked option, protective put, covered call, straddle, and strangle. Secondly this paper compares the historical rates of return on the option trading strategies and Sharpe ratios with those generated by Monte-Carlo simulation and examines whether the historical option returns are inconsistent with Black-Scholes model, Jump-diffusion model, Stochastic Volatility model, or Stochastic Volatility with Jump model. Thirdly, this paper computes the optimal asset allocation ratio among the risk-free asset, risky assets, and option trading strategies in the viewpoint of rational investors who maximize the CRRA utility function. The results show that the historical returns on short position of ATM and OTM puts are too high to explain based on the classical CAPM, and the optimal allocation ratios among put, risky asset, and the risk-free asset are different from those derived using Monte-Carlo simulation.

Optimal portfolios for DC pension plans under a CEV model

This paper studies the portfolio optimization problem for an investor who seeks to maximize the expected utility of the terminal wealth in a DC pension plan. We focus on a constant elasticity of variance (CEV) model to describe the stock price dynamics, which is an extension of geometric Brownian motion. By applying stochastic optimal control, power transform and variable change technique, we derive the explicit solutions for the CRRA and CARA utility functions, respectively. Each solution consists of a moving Merton strategy and a correction factor. The moving Merton strategy is similar to the result of Devolder et al. [Devolder, P., Bosch, P.M., Dominguez F.I., 2003. Stochastic optimal control of annunity contracts. Insurance: Math. Econom. 33, 227–238], whereas it has an updated instantaneous volatility at the current time. The correction factor denotes a supplement term to hedge the volatility risk. In order to have a better understanding of the impact of the correction factor on the optimal strategy, we analyze the property of the correction factor. Finally, we present a numerical simulation to illustrate the properties and sensitivities of the correction factor and the optimal strategy.

General equilibrium asset pricing under regime switching

We have investigated the asset pricing problem in a general equi- librium in an economy with two states. Based on the assumption of a CRRA utility function, we have derived a partial dierential equation satisfied by the representative agent's cost function. In the case when the representative agent doesn't have intermediate consumption, we have found an explicit so- lution of the cost function. A closed-form expression for the riskless interest rate has been derived. We have also provided a partial dierential equation satisfied by any contingent claim. Based on the stochastic discount factor computed, we have suggested an explanation for the equity premium puzzle.

On the Welfare Improving Trade in Marginal Tax Rates

In this paper we extend the dimensionality of the standard income taxation problem. Specifically, we allow agents to purchase at a price, from the government, discounts on their marginal tax rates. Consequently agents have two decision variables: the level of income they choose to earn and the magnitude of discount they want to purchase at a given price. By rationally choosing to purchase discounts agents face lower effective marginal tax rates, which enhances efficiency. At the same time agents pay a price for the discounts, which can compensate for a shortfall in revenue resulting from lower e?ective marginal tax rates. We show, in the case of the CRRA utility function, that for any tax function it is always possible to construct the corresponding discount price function that leads to a strict Pareto improvement through trade in discounts and that ensures that revenue collected is unaffected when trade in discounts is permitted. In addition, we show that a fat tax or a piece-wise linear tax function are never optimal in the Mirrlees's sense. Similarly, we argue that trade in discounts can lead to a Pareto improvement over the allocation induced by any tax function, including Mirrlees's solutions, that has an existing inverse and is differentiable. We also show that the standard asymptotic result of zero marginal tax rate at the top of income distribution can be extended to an open set. It is shown in the dynamic context that no tax function can be optimal unless it is conditioned on the entire income history.

Non-hyperbolic time inconsistency

The commonly used hyperbolic and quasi-hyperbolic discount functions have been developed to accommodate decreasing impatience, which is the prevailing empirical finding in intertemporal choice, in particular for aggregate behavior. However, these discount functions do not have the flexibility to accommodate increasing impatience or strongly decreasing impatience. This lack of flexibility is particularly disconcerting for fitting data at the individual level, where various patterns of increasing impatience and strongly decreasing impatience will occur for a significant fraction of subjects. This paper presents discount functions with constant absolute (CADI) or constant relative (CRDI) decreasing impatience that can accommodate any degree of decreasing or increasing impatience. In particular, they are sufficiently flexible for analyses at the individual level. The CADI and CRDI discount functions are the analogs of the well-known CARA and CRRA utility functions for decision under risk.

Does Pricing Profits Lead to a Higher Equity Premium

In this paper we use a simple model with a single Cobb‐Douglas firm and a consumer with a CRRA utility function to compute the dierence in the equity premia on returns based on profits and on returns based on consumption. With this simple example we show that the equity premium based on profits exceeds the premium based on consumption with probability 1.

Intertemporal Asset Allocation with Habit Formation in Preferences: An Approximate Analytical Solution

In this paper we derive an approximate analytical solution to the optimal consumption and portfolio choice problem of an infinitely-lived investor with power utility defined over the difference between consumption and an external habit. The investor is assumed to have access to two tradable assets: a risk free asset with constant return and a risky asset with a time-varying premium. We extend the approach proposed by Campbell and Viceira (1999), which builds on log-linearizations of the Euler equation, intertemporal budget constraint, and portfolio return, to also contain the log-linearized surplus consumption ratio. The 'difference habit model' implies that the relative risk aversion is time-varying which is in line with recent evidence from the asset pricing literature. We show that accounting for habit affects both the myopic and intertemporal hedge component of optimal asset demand, and introduces an additional component that works as a hedge against changes in the investor's habit level. In an empirical application, we calibrate the model to U.S. data and show that habit formation has significant effects on both the optimal consumption and portfolio choice compared to a standard CRRA utility function.

Utilitarianism and Fairness in Portfolio Positioning

The paper introduces the theory of optimal positioning of financial products. It is illustrated in the context of long-term intertemporal portfolio allocation and can be applied for example to asset allocation funds. We embed this problem in location theory: the portfolio is optimized within the investors' risk aversion dimension. For the CRRA utility functions, we compute explicitly the distance functions. For the first (utilitarian criterion), the average utility of the investors is maximized. For the second one (fairness criterion), the choice of portfolio is optimized so that the average monetary loss due to the lack of customization is minimized. Given the distribution of investors' risk aversion, we provide a solution method and an algorithm to optimally position standardized portfolio along one of these two criteria.

Utilitarianism and fairness in portfolio positioning

The paper introduces the theory of optimal positioning of financial products. It is illustrated in the context of long-term intertemporal portfolio allocation and can be applied for example to asset allocation funds. We embed this problem in location theory: the portfolio is optimized within the investors’risk aversion dimension. For the CRRA utility functions, we compute explicitly the distance functions. For the first (utilitarian criterion), the average utility of the investors is maximized. For the second one (fairness criterion), the choice of portfolio is optimized so that the average monetary loss due to the lack of customization is minimized. Given the distribution of investors’ risk aversion, we provide a solution method and an algorithm to optimally position standardized portfolio along one of these two criteria.