CRRA Utility Research Articles

Optimal Bankruptcy with a Continuous Debt Repayment

We investigate the optimal consumption and investment problem when a working debtor has an option to file for bankruptcy. By applying the duality approach, the closed-form solutions are obtained for the case of CRRA utility function. The optimal bankruptcy time is determined by the first hitting time when the financial wealth hits the wealth threshold derived from the optimal stopping time problem. Moreover, the numerical results show that the investment increases as the wealth approaches the threshold and the value gain from the bankruptcy option is vanished as wealth increases.

Prudence and the Convexity of Compensation Contracts

In a standard principal-agent model, we derive a new condition that relates the convexity of the optimal contract to the agent's risk preferences. This condition shows that the optimal contract is more convex than the likelihood ratio of the performance measure if and only if the coefficient of absolute prudence is larger than twice the coefficient of absolute risk aversion. With CRRA utility, this condition is satisfied if and only if relative risk aversion is less than one. We provide an economic intuition for the underlying tradeoff.

OPTIMAL ENDOGENOUS GROWTH WITH NATURAL RESOURCES: THEORY AND EVIDENCE

This paper considers an optimal endogenous growth model where the production function is assumed to exhibit increasing returns to scale and two types of resource (renewable and nonrenewable) are imperfect substitutes. Natural resources, labor, and physical capital are used in the final goods sector and in the accumulation of knowledge. Based on results in the calculus of variations, a direct proof of the existence of an optimal solution is provided. Analytical solutions for the planner case, balanced growth paths, and steady states are found for a specific CRRA utility and Cobb–Douglas production function. It is possible to have long-run growth where both energy resources are used simultaneously along the equilibrium path. As the law of motion of the technological change is not concave, reflecting the increasing returns to scale, so that the Arrow–Mangasarian sufficiency conditions do not apply, we provide a sufficient condition directly. Transitional dynamics to the steady state from the theoretical model are used to derive three convergence equations of output intensity growth rate, exhaustible resource growth rate, and renewable resource growth rate, which are tested based on OECD data on production and energy consumption.

The robust Merton problem of an ambiguity averse investor

We derive a closed form portfolio optimization rule for an investor who is diffident about mean return and volatility estimates, and has a CRRA utility. Confidence is here represented using ellipsoidal uncertainty sets for the drift, given a (compact valued) volatility realization. This specification affords a simple and concise analysis, as the agent becomes observationally equivalent to one with constant, worst case parameters. The result is based on a max–min Hamilton–Jacobi–Bellman–Isaacs PDE, which extends the classical Merton problem and reverts to it for an ambiguity-neutral investor.

The Liquidity Risk Premium Demanded by Large Investors: Dynamic Portfolio Choice with Stochastic Illiquidity

textabstractRecent empirical work documents large liquidity risk premiums in stock markets. We calculate the liquidity risk premiums demanded by large investors by solving a dynamic portfolio choice problem with stochastic price impact of trading, CRRA utility and a time-varying investment opportunity set. We find that, even with high trading-cost rates and substantial trading motives, the theoretically demanded liquidity risk premium is negligible, less than 3 basis points per year. Assuming forced selling during market downturn enlarges the liquidity risk premium to maximally 20 basis points per year, which is well below existing empirical estimates of the liquidity risk premium.

Optimal CAR simulation

The present paper is conceived to expose a new model of bank capital adequacy. Our analysis highly relies on the dynamics of the capital adequacy ratio (CAR) as computed from balance sheet items, namely, loans, securities and bank regulatory capital, in a stochastic dynamic setting. In addition, an attempt is made to demonstrate how the CAR can be optimized in terms of asset allocation and the rate at which the bank remains solvent. Consequently, a dynamic programming principle has been applied to solve the Hamilton–Jacobi–Bellman (HJB) equation explicitly in the case of CRRA utility function. The parameters’ estimation is based on the maximum likelihood method, using Tunisian data relevant to the period 2004–2012. As regard computations, they are carried out in MATLAB. The simulation results have shown the model relevance as a decision tool for bank risk management. Moreover, the relationship persisting between the macroeconomic activity and the bank capital adequacy has also been discussed.

Non-Keynesian Savings of Russians

The Russian recession of 2014-2015 began with a run on the ruble and a rise in the rate of inflation, the precise opposite of a Western-type deflationary slump combined with money hoarding. Does this mean that Russians need different micro-model to describe savings and consumption behavior? This study shows that the workhorse log-linearized rational SDF formula with the CRRA utility function still provides a good explanation for the behavior of Russian consumers. It explains dollarization, domestic equity market avoidance, preference for real estate, and, most importantly, a wary attitude towards the ruble. Expectations derived from past and interactive preferences lock the Russian economy in a state of steadfast distrust in the ruble as prone to inflation. At present, one should not expect a Keynesian-type deflationary cycle in Russia. The next recession is likely to be inflationary, requiring monetary tightening. This reasoning is generalized for other emerging countries. A free-floating currency and inflation targeting do not ensure an easy path for countries with recent experiences of high inflation.

The Non-Existence of Representative Agents

We characterize environments in which there exists a representative agent: an agent who inherits the structure of preferences of the population that she represents. The existence of such a representative agent imposes very strong restrictions on individual utility functions -- requiring them to be linear in the allocation and additively separable in any parameter that characterizes agents (e.g., a risk aversion parameter, a discount factor, etc.). In particular, commonly used classes of utility functions (exponentially discounted utility functions, CRRA or CARA utility functions, logarithmic functions, etc.) do not admit a representative agent.

Debt, Deficits, and Finite Horizons: The Continuous Time Stochastic Case

We present a continuous time stochastic perpetual youth model in the spirit of Blanchard (1985) and Farmer et. al (2011). Agents are assumed to have CRRA utility and face a constant probability of death rate. Investment assets are a risk-less government bond as well as one risky asset, modeled as geometric Brownian motion, representing the stock market. Using Martingale techniques, we determine aggregate variables and derive their dynamics.

Stochastic maximum principle with Lagrange multipliers and optimal consumption with Lévy wage

We show how a stochastic version of the Lagrange multiplier method can be combined with the stochastic maximum principle for jump diffusions to solve certain constrained stochastic optimal control problems. Two different terminal constraints are considered; one constraint holds in expectation and the other almost surely. As an application of this method, we study the effects of inflation- and wage risk on optimal consumption. To do this, we consider the optimal consumption problem for a budget constrained agent with a L\'evy income process and stochastic inflation. The agent must choose a consumption path such that his wealth process satisfies the terminal constraint. We find expressions for the optimal consumption of the agent in the case of CRRA utility, and give an economic interpretation of the adjoint processes.

Optimal dynamic asset allocation of pension fund in mortality and salary risks framework

In this paper, we consider the optimal dynamic asset allocation of pension fund with mortality risk and salary risk. The managers of the pension fund try to find the optimal investment policy (optimal asset allocation) to maximize the expected utility of terminal wealth. The market is a combination of financial market and insurance market. The financial market consists of three assets: cashes with stochastic interest rate, stocks and rolling bonds, while the insurance market consists of mortality risk and salary risk. These two non-hedging risks cause incompleteness of the market. By martingale method and dynamic programming principle we first derive the approximate optimal investment policy to overcome the difficulty, then investigate the efficiency of the approximation. Finally, we solve an optimal assets liabilities management(ALM) problem with mortality risk and salary risk under CRRA utility, and reveal the influence of these two risks on the optimal investment policy by numerical illustration.

Worst-Case Approach to Strategic Optimal Portfolio Selection Under Transaction Costs and Trading Limits

We study a worst-case scenario approach to the problem of strategic portfolio selection in presence of transaction costs and trading limits under uncertain stochastic process of market parameters. Unlike classic stochastic programming, the approach is model-free, solution of the arising Bellman-Isaacs equation can be easily found numerically under some general assumptions. All results hold for a general class of utility functions and several risky assets. For a special case of proportional transaction costs and CRRA utility, we present a numerical scheme which allows to reduce the dimension of the Bellman-Isaacs equation by a number of risky assets.

Optimal positioning in financial derivatives under mixture distributions

In this paper, we study and extend the optimal portfolio positioning problem introduced by Brennan and Solanki (1981) and by Leland (1980). For that purpose, we introduce mixtures of probability distributions to model the log returns of financial assets. In this framework, we provide and analyze the general solution for log return with mixture distributions, in particular for the mixture Gaussian case. Our solution is characterized for arbitrary utility functions and for any risk neutral probability. Moreover, we illustrate the solution for a CRRA utility and for the minimal risk-neutral probability. Lastly, we compare our solution with the optimal portfolio within ambiguity. Our results have significant implications to improve the management of structured financial portfolios.

Optimal control of markovian switching systems with applications to portfolio decisions under inflation

This article is concerned with a class of control systems with Markovian switching, in which an Itô formula for Markov-modulated processes is derived. Moreover, an optimal control law satisfying the generalized Hamilton-Jacobi-Bellman (HJB) equation with Markovian switching is characterized. Then, through the generalized HJB equation, we study an optimal consumption and portfolio problem with the financial markets of Markovian switching and inflation. Thus, we deduce the optimal policies and show that a modified Mutual Fund Theorem consisting of three funds holds. Finally, for the CRRA utility function, we explicitly give the optimal consumption and portfolio policies. Numerical examples are included to illustrate the obtained results.

Optimal savings management for individuals with defined contribution pension plans

Abstract The paper provides some guidelines to individuals with defined contribution (DC) pension plans on how to manage pension savings both before and after retirement. We argue that decisions regarding investment, annuity payments, and the size of death sum should not only depend on the individual’s age (or time left to retirement), nor should they solely depend on the risk preferences, but should also capture: (1) economical characteristics—such as current value on the pension savings account, expected pension contributions (mandatory and voluntary), and expected income after retirement (e.g. retirement state pension), and (2) personal characteristics—such as risk aversion, lifetime expectancy, preferable payout profile, bequest motive, and preferences on portfolio composition. Specifically, the decisions are optimal under the expected CRRA utility function and are subject to the constraints characterizing the individual. The problem is solved via a model that combines two optimization approaches: stochastic optimal control and multi-stage stochastic programming. The first method is common in financial and actuarial literature, but produces theoretical results. However, the latter, which is characteristic for operations research, has practical applications. We present the operations research methods which have potential to stimulate new thinking and add to actuarial practice.

Fund Ratings: The Method Reconsidered

This paper compares the performance of a quadratic utility function and discusses how to change its characteristic parameter, ARA, so that rating is consistent with return and risk measurements. In particular, this parameter is modified in such a way that a positive return Fund has always a rating higher than one with a negative yield. This modification confirms the possibility of building a new ranking procedure which is more coherent with the actual behaviour of investors.The paper moreover demonstrates that the CRRA utility function has not such behaviour, it is possible therefore a ranking in which a Fund with negative return has a rank greater than a Fund with positive return. This seems counterintuitive with respect to the expectations of investors. The CRRA utility function also shows a very slight link with the standard deviation and, consequently, the induced ranking depends mainly on the returns.The CRRA utility function here considered is that one used by Morningstar.

Portfolio Optimization with Ambiguous Correlation and Stochastic Volatilities

In a continuous-time economy, we investigate the asset allocation problem among a risk-free asset and two risky assets with an ambiguous correlation between the two risky assets. The portfolio selection that is robust to the uncertain correlation is formulated as the utility maximization problem over the worst-case scenario with respect to the possible choice of correlation. Thus, it becomes a maximin problem. We solve the problem under the Black-Scholes model for risky assets with an ambiguous correlation using the theory of G-Brownian motions. We then extend the problem to stochastic volatility models for risky assets with an ambiguous correlation between risky asset returns. An asymptotic closed-form solution is derived for a general class of utility functions, including CRRA and CARA utilities, when stochastic volatilities are fast mean-reverting. We propose a practical trading strategy that combines information from the option implied volatility surfaces of risky assets through the ambiguous correlation.

Changes in probability distributions and the form of compensation contracts

We consider a standard principal-agent setting where the first-order approach to the effort choice problem applies. We decompose the effect of a change in the probability distribution of performances on the form of the optimal contract into three additive components. We also consider the specific cases of a linear likelihood ratio, a rightward or leftward translation of the likelihood ratio, and CRRA utility. The results shed light on the differences between the optimal contracts derived in the literature, notably depending on whether a linear likelihood ratio or a lognormally distributed performance measure is assumed.

Learning to be Risk Averse?

The purpose of this research is to search for the best (highest performing) risk profile of agents who successively choose among risky prospects. An agent's risk profile is his attitude to perceived risk, which can vary from risk preferring to risk neutral (an expected-value decision maker) to risk averse. We use the Genetic Algorithm to search in the complex stochastic space of repeated lotteries. We find that agents with a CARA utility function learn to possess risk-neutral risk profiles. Since CARA utility functions are wealth-independent, this is not surprising. When agents have wealth-dependent, CRRA utility functions, however, they also learn to possess risk profiles that are about risk neutral (from slightly risk-averse to even slightly risk-preferring), which is surprising.

Risk Preferences, Probability Distributions, and the Form of Compensation Contracts

We consider a standard principal-agent setting where the first-order approach to the effort choice problem applies. We decompose the effect of a change in the probability distribution of performances on the form of the optimal contract into three additive components. We also consider the specific cases of a linear likelihood ratio, a rightward or leftward translation of the likelihood ratio, and CRRA utility. The results shed light on the differences between the optimal contracts derived in the literature, notably depending on whether a linear likelihood ratio or a lognormally distributed performance measure is assumed.