Utility Function Of Wealth Research Articles

Multi-period power utility optimization under stock return predictability

In this paper, we derive an analytical solution to the dynamic optimal portfolio choice problem in the case of an investor equipped with a power utility function of wealth. The results are established by solving the Bellman backward recursion under the assumption that the vector of asset returns follows a vector-autoregressive process with predictable variables. In an empirical study, the performance of the derived solution is compared with the one obtained by applying the numerical method. The comparison is performed in terms of the final wealth and its expected utility. It is documented that the application of the analytical solution to the multi-period portfolio choice problem leads to higher values of both the final wealth and the expected utility.

A Study on the Mechanism and Pricing of Drainage Rights Trading Based on the Bilateral Call Auction Model and Wealth Utility Function

In the context of the high incidence of flooding disputes among neighboring subjects in the basin, exploring the drainage rights trading mechanism and clarifying the pricing method of drainage rights are new ideas to manage flooding disasters. In this study, the trading mechanism of drainage rights is constructed using a bilateral call auction model, in which the price constraint, the control total quantity constraint, the clearing rule, and the matching rule are explained. In addition, we adopt the wealth utility concept to construct a function, apply the social welfare function to associate the wealth utility functions of both sides of the transaction, and draw on the environmental Gini coefficient concept to construct a pricing model for drainage rights transactions based on fairness and efficiency. Finally, we conduct an arithmetic simulation of the drainage rights trading mechanism and transaction pricing for China’s Jiangsu section of the Huaihe River Basin. The study’s results show that the trading mechanism model can give the optimal trading scheme when multiple subjects are trading drainage rights. This fills the gap in existing studies and can lead to efficient trading of drainage rights. At the same time, the pricing model can give the optimal transaction price that considers efficiency and fairness, providing a more practical pricing theory for drainage rights trading.

Range Utility Theory

We introduce range utility theory for decisions under risk. Two functions are implicated in the representation of preferences: a traditional utility function for wealth---or changes in wealth---and a range distortion function. The latter introduces a local deformation of the utility function on the range under consideration, possibly magnifying the sensitivity to intermediate outcomes. In a sense we make precise, range utility theory is an alternative way to extend the binary rank dependent model to more than two outcomes. It can account for the fourfold pattern, as well as for the preference reversal phenomenon. When the range is fixed, range utility is linear in probabilities. We illustrate this advantage in game theory applications.

Towards optimal portfolio strategy to control maximum drawdown

PurposeThe purpose of this paper is to present a discrete quantitative trading strategy to directly control a portfolio's maximum percentage of drawdown losses while trying to maximize the portfolio's long‐term growth rate.Design/methodology/approachThe loss control target is defined through a Rolling Economic Drawdown (REDD) with a constant look‐back time window. The authors specify risk aversion in the power‐law portfolio wealth utility function as the complement of maximum percentage loss limit and assume long‐term stable Sharpe ratios for asset class indexes while updating volatility estimation in dynamic asset allocation implementation.FindingsOver a test period of the past 20 years (1992‐2011), a risk‐based out‐of‐sample dynamic asset allocation among three broad based indexes (equity, fixed income and commodities) and a risk free asset, is robust against variations in capital market expectation inputs, and out‐performs the in‐the‐sample calibrated model and traditional asset allocation significantly.Research limitations/implicationsThe current proposal can lead to a new mathematical framework for portfolio selection. Besides investors' liquidity and behavioural constraints, macroeconomic and market cycle, and the potential of central bank interventions following a market crash, could be additionally considered for a more rigorous dynamic asset allocation model.Practical implicationsBesides the benefit of a clear mandate to construct suitable client portfolios, the portfolio approach can be applied to design invest‐able securities, such as principal‐guaranteed investment products, target risk asset allocation ETFs, and target‐date mutual funds with a glide path, etc. The formulation can also be implemented as a managed futures hedge fund portfolio.Originality/valueThe paper introduces the Rolling Economic Drawdown (REDD) concept and specifies risk aversion as the floor of maximum percentage loss tolerance. Dynamic asset allocation is implemented through updating estimation of asset class volatilities.

Optimal Portfolio Strategy to Control Maximum Drawdown - The Case of Risk Based Dynamic Asset Allocation

Draw-down losses from a previously reached maximum portfolio wealth level, is an important risk measure for investment management. In this study, we present a discrete trading strategy to directly control a portfolio’s maximum percentage of drawdown within a target level while maximizing the portfolio’s long term growth rate. Modifying the continuously rebalancing models proposed by Grossman and Zhou (1993) and Cvitanic and Karatzas (1995), we define the loss control target as a Rolling Economic Drawdown (REDD) with a constant look-back window progressing in time. Additionally, investor’s risk aversion in the power law wealth utility function is specified as complement of the maximum loss limit to construct a portfolio’s risk-return efficient frontier. We test the dynamic strategy using data of three broad asset class indexes: S&P 500 Total Return Index (SPTR), Barclays Capital 20 Year US Treasury Bond Index (TLT) and Dow-Jones UBS Commodity Total Return Index (DJUBS), with 3-month U.S. Treasury Bill as the risk-free asset. Over a test period of the past 20 years (1992-2011), a simplified risk-based out-of-sample dynamic asset allocation portfolio is robust against variations in capital market expectation inputs, and out-performs the in-the-sample calibrated model portfolio and traditional asset allocation portfolios significantly.

The Dividend Puzzle Unpuzzled

A claim of solving the dividend puzzle, as formulated by F. Black, is made. Various stock pricing formulas are established, extending substantially the classical Miller-Modigliani theory of valuation of shares in a way to contradict their main conclusion that dividends are irrelevant. Indeed, the main stock pricing formula obtained implies that if the company's costs depend on the available cash, then dividends affect the stock pricing function in a substantial manner. On the other hand, if the company's costs are independent of the available cash, if the dividend tax rate is constant, and under a further technical condition, the established formula is consistent with the Miller-Modigliani's conclusion of dividend irrelevancy. The presented theory is in the context of Miller-Modigliani's stream of dividends approach to valuation of shares, facilitated by the latest developments in the Black-Scholes type pricing theory for incomplete multivariable markets. Various cash flows are modeled, some using Ito SDEs, so that the value of the company is not assumed to be tradable, yielding market incompleteness. In spite of market incompleteness, under some simplifying assumptions, referred to as the Basic Equity Model, the established stock pricing formulas yield unique prices, i.e., prices independent of the investor's risk aversion. Some more general models when this property of prices does not hold are considered as well. The intrinsic non-uniqueness of fair stock prices in such cases is settled by choosing the value of the investor's HARA risk aversion parameter, while if the CARA wealth utility function is used, fair prices remain unique.

Asset Allocation and Long-Term Returns: An Empirical Approach

We present a discussion of the standard approaches to asset allocation, generally falling into two camps: Mean Variance Optimization and the maximization of the final value of a wealth utility function. After describing shortcomings in both of these standard approaches, we describe a heuristic, empirical approach that uses concepts of Shortfall Risk as an objective and actual data as a direct model of the stochastic market evolution. This empirical approach leads to fundamentally different conclusions than the standard approaches, and by definition fits the past data much better. Among other observations, we find that (1) for holding periods of about 15 years and greater, bonds are riskier than stocks in the practical sense that bonds are almost always expected to under perform stocks over 15-year or higher time horizons, (2) while short-term stock returns show fat tails, long-term stock returns show skinny tails, and (3) while stocks show mean reversion, bonds exhibit mean aversion.

On the Two-Box Paradox

On a game show, you are presented with two identical boxes. Both boxes contain positive monetary prizes, one twice the other. You are allowed to pick one box and observe the prize x > 0, after which you can choose to trade boxes. In terms of simple expected value, it is always better to trade since (2x) + () 5 > x. That is the paradox. Simple thought experiments suggest that a sufficiently large observed prize would cause a player not to trade, despite the mathematical computation of expected value. In individual cases, this creates some threshold, which depends on the observed prize, for ceasing to trade. A player may have in mind prior probabilities about what prizes the game show would offer, so that an observed prize of $10,000, for instance, would not yield equal judgmental odds of $20,000 or $5,000 in the unobserved box. The judgmental probability approach to the two-box problem seeks to develop optimal threshold strategies in terms of prior distributions on the set of possible prizes. Recent articles in this MAGAZINE have focused on the judgmental probability approach, although they have also discussed the second line of attack on this problem, expected utility [2, 3]. In expected utility theory, it is assumed than an individual has an underlying utility function for wealth. This utility function is increasing because it is presumed that an individual will always prefer more wealth to less wealth. In addition, the utility function is concave because it is presumed that an individual will have nonincreasing marginal utility for wealth. The utility function u is thus an increasing, concave function from the positive half line into the real line. The scaling on this function is unimportant because a positive linear transformation a + bu, with b > 0, is equivalent for individual REM 2. When n > 2 is even, integers a, b, and c satisfy a2 + b2 = cn if and 302 MATHEMATICS MAGAZINE

Risk Aversion and Expected-utility Theory: A Calibration Theorem

Within the expected-utility framework, the only explanation for risk aversion is that the utility function for wealth is concave: A person has lower marginal utility for additional wealth when she is wealthy than when she is poor. This paper provides a theorem showing that expected-utility theory is an utterly implausible explanation for appreciable risk aversion over modest stakes: Within expected-utility theory, for any concave utility function, even very little risk aversion over modest stakes implies an absurd degree of risk aversion over large stakes. Illustrative calibrations are provided.

Incomplete markets, transaction costs and liquidity effects

An agent's optimization problem of the expected terminal wealth utility in a trinomial tree economy is solved. At each transaction date, the agent can trade in a riskless asset, a primitive asset subject to constant proportional transaction costs, and a contingent claim characterized by some parameter kappa whose bid and ask price is defined by allowing for different equivalent martingale measures. In addition to the classical portfolio choice problem, the characteristic of the contingent claim κ is determined endogenously in the optimization problem. Under suitable conditions, it is proved that the optimal demand of the agent in the primitive risky asset is zero independently of the choice of the terminal wealth utility function: the agent prefers not to trade in the asset subject to transaction costs, which prevents the market from being complete, rather than trading in both assets. Next, the optimal choice of the contingent claim is characterized and the results are applied to European call and put options with fixed maturity and varying exercise price κ.

A Contextual Uncertainty Condition for Behavior Under Risk

Suppose that your choice between uncertain financial prospects is made more difficult by two independent contextual uncertainties concerning the size of your existing wealth. One contextual uncertainty has a greater spread than the other. If you could resolve one of these contextual uncertainties before making your choice, would you rather it be the larger or the smaller? In this paper we explore the intuitive notion that it ought to be more advantageous to resolve larger rather than smaller contextual uncertainties. The utility function for wealth u(w) = aw − be−cw (a, b, c ≥ 0) is the only utility function that satisfies this condition and that is also increasing and risk averse at all wealth levels.

Attitude towards Risk, Prospect Variability, and the Value of Imperfect Information

Choice in decision making under uncertainty often includes the option of choosing to utilize an information system, a set of potential messages that may improve current decisions and resulting outcomes. In decision theory the demand price for an information system is defined as the maximum non-stochastic cost, payable from initial wealth prior to the receipt of any message, that makes the decision maker (DM) indifferent between purchasing the system or not. The determinants of the value of information are important to both the buyers and the sellers in the marketplace for information products, and economists study the subject with both theoretical and practical motivations. Theoretical analyses seek to discover the characteristics of the economic and/or statistical environment that allow for definitive qualitative statements about the value of information. The most famous result, Blackwell's Theorem [1; 2], states the necessary and sufficient conditions under which any potential user values one information system more than another. Unfortunately, most results are of the negative variety: there is no general monotonic relationship between the information value and the degree of aversion towards risk [5], the amount of statistical information transmitted [8], the level of initial wealth [6], or the Rothschild-Stiglitz variability of the prior distribution on the state [4]. In practical situations, the buyer or the producer of an information system may desire to utilize quantitative estimates of the value of information. Complete quantitative analysis requires knowledge of the statistical characteristics of the information, identification of the payoff function for the decision problem at hand, and assessment of the information user's utility function for wealth. This last factor, the utility function, is specific to each individual DM and is unlikely to be known by the producer of the information. Unless it can be assumed that all potential buyers are neutral towards risk and hence have an easily assessed linear utility function, the seller of the information product (e.g., a forecasting service with many clients) faces the problem of estimating demand without knowledge of the demanders' utility functions. Policy makers, seeking to assess the value of publicly collected and disseminated information, face a similar problem [12]. Blair and Romano [3], henceforth B-R, recently have introduced a new problem-specific ap-

Optimal Consumption and Investment Policies Allowing Consumption Constraints and Bankruptcy

An agent can distribute his wealth between two investments, one with a fixed rate of return r and the other with a random rate of return (modeled as a diffusion) with mean r. The agent seeks to maximize total discounted utility from consumption over an infinite horizon. Consumption may be constrained from below. Various models for bankruptcy, including welfare, are considered. The agent has a strictly concave utility function for consumption; however, it is shown that the utility function for wealth may have convex portions, thus the agent may be risk seeking. The paper gives a complete treatment of the existence and nonexistence of optimal policies. New theorems for the optimal control of degenerate diffusions are given, as well as explicit formulas for the value function.

On the Theory of Risk Aversion

the existing theory by establishing the economic significance of the partial relative risk aversion function. Let u(t) be a utility function for wealth. The functions A(t) = -u(t)/u'(t) and R(t) -tu(t)/u'(t) are the Arrow-Pratt absolute and relative risk aversion functions. The importance of A(t) arises when considering an individual's aversion to risk as wealth is varied but the risk remains unchanged, while R(t) becomes relevant when wealth and the risk are changed in the same proportion. We shall demonstrate that the partial relative risk aversion function P(t; w) = -tu(t + w)/u'(t + w) is important when the risk is varied but wealth w remains fixed. In addition, we indicate the economic relationships between the functions A, R, and P; present some results about the behavior of P; and relate its behavior to that of A and R. The analysis in this paper is based on Pratt's risk premium which we feel is the only function which actually measures risk aversion for arbitrary risks.2 Our analysis differs from that of both Arrow and Pratt in that we do not use infinitesimal risks. Arrow and Pratt interpret A and R as local measures of absolute and relative risk aversion. The results of this paper show that the functions A, R, and P have a significance beyond their interpretation as local measures of risk aversion in that they determine the behavior of the risk premium in different comparative static contexts. In Section 2 basic concepts are briefly discussed. Section 3 contains our main results. In it the economic significance of A, R, and the new function P is established through their relationship with the risk premium, and we show how the behavior of these functions is relevant for the theory of risk aversion. Section 4 contains a comparison of our results with those of Arrow and Pratt, and indicates the usefulness of A, P, and R. for comparative static analysis of expected utility maximization models.

The Disutility of Uncertain Losses

ABSTRACT The ancient hypothesis that equal quantities of money gained or lost may have different utilities depending upon the cirumstances of the recipient has never been fully accepted into the main body of economic theory. To eliminate confusion, it is essential to abandon the attempt to quantify a utility function of wealth or income, and to develop instead a utility function of uncertain gains and losses, basing it upon relative values placed upon alternative contingencies by the same individuals. The basic facts of life insurance and of health insurance show that the majority of persons are prepared to pay approximately twice the actuarial value to avoid major contingencies. These results allow definition of a marginal utility coefficient for this range of losses. Small losses and moderate gains apparently have utility coefficients approximating unity. A working hypothesis based upon the above is consistent with evidence as to risk aversion from the securities markets. The most important characterist...