Portfolio Optimization Under Convex Incentive Schemes

The paper studies connections between risk aversion and martingale measures in a discrete‐time incomplete financial market. An investor is considered whose attitude toward risk is specified in terms of the index b of constant proportional risk aversion. Then dynamic portfolios are admissible if the terminal wealth is positive. It is assumed that the return (risk) processes are bounded. Sufficient (and nearly necessary) conditions are given for the existence of an optimal dynamic portfolio which chooses portfolios from the interior of the set of admissible portfolios. This property leads to an equivalent martingale measure defined through the optimal dynamic portfolio and the index 0 < b≤ 1. Moreover, the option pricing formula of Davis is given by this martingale measure. In the case of b= 1; that is, in the case of the log‐utility, the optimal dynamic portfolio defines the numéraire portfolio.

The paper studies connections between arbitrage and utility maximization in a discrete-time financial market. The market is incomplete. Thus one has several choices of equivalent martingale measures to price contingent claims. Davis determines a unique price for a contingent claim which is based on an optimal dynamic portfolio by use of a `marginal rate of substitution' argument. Here conditions will be given such that this price is determined by a martingale measure and thus by a consistent price system. The underlying utility function U is defined on the positive half-line. Then dynamic portfolios are admissible if the terminal wealth is positive. In case of the logarithmic utility function, the optimal dynamic portfolio is the numeraire portfolio.

We illustrate the classical Markowitz portfolio optimization theory in detail, and we give all essential proofs. Especially we show how to explicitly calculate the efficient border and the market portfolio in the case where short selling is possible without restrictions. In the case without short selling, we will work with Monte Carlo. We also introduce the single-index model as an alternative to estimate parameters needed for portfolio optimization. Then we formulate the optimal investment and consumption problem. We give a heuristic and intuitive introduction to the basic techniques of stochastic optimal control, especially to the Hamilton-Jacobi-Bellman equations and to their application. With the help of these concepts, we solve the optimal investment and consumption problem and discuss the results.

We study the stochastic control problem of maximizing expected utility from terminal wealth and/or consumption, when the portfolio is constrained to take values in a given closed, convex subset of $\\mathscr{R}^d$. The setting is that of a continuous-time, Ito process model for the underlying asset prices. General existence results are established for optimal portfolio/consumption strategies, by suitably embedding the constrained problem in an appropriate family of unconstrained ones, and finding a member of this family for which the corresponding optimal policy obeys the constraints. Equivalent conditions for optimality are obtained, and explicit solutions leading to feedback formulae are derived for special utility functions and for deterministic coefficients. Results on incomplete markets, on short-selling constraints and on different interest rates for borrowing and lending are covered as special cases. The mathematical tools are those of continuous-time martingales, convex analysis and duality theory.

Rank Matters - The Impact of Social Competition on Portfoilio Choice

Although dichotomous choice (DC) contingent valuation (CV) has been recommended by the US NOAA 'blue-ribbon' panel for large-scale contingent valuation studies, useful information can still be obtained from smaller, open-ended (OE) studies, often undertaken as a precursor to a DC survey. The CV study considered here was carried out in Greece and looked at willingness-to-pay (WTP) for protecting the Mediterranean monk seal (Monachus-monachus) in the Aegean area. This is the most endangered seal in the world, and the application of the CV methodology was the first such application in Greece. The OE data consist of two responses: first, a binary response detailing whether or not respondents were in principle prepared to pay for the protection of this seal; secondly, those respondents who answered 'yes' to the first question were then asked to state their maximum WTP for such protection. A multivariate binomial – log-normal mixture model is used to develop a bid function including explanatory variables such as income, sex, age and education. Such a modelling approach provides an alternative to more commonplace tobit estimation. However, the model is extended to include further information which was collected on: (a) an increased WTP amount given in response to information that the initial WTP amount may not be enough to prevent the extinction of the seal; (b) respondents were asked to divide their final WTP amount between use, option and existence values, the latter requiring a multivariate model with four binary and four continuous responses per individual in the same model. The discussion focuses on the methodological issues raised with some comment on the substantive interpretation of results.

Portfolio optimization of electricity markets participation using forecasting error in risk formulation

The aim of this paper is to find optimal portfolio strategies for an n-stock extension of the stochastic volatility model proposed in Ref. [2]. It is a modification of [19], and characterizes the dependence by the use of a factor structure. The idea of a factor structure is that the diffusion components of the stocks contain one Brownian motion that is unique for each stock, and a few Brownian motions that all stocks share. Our model can obtain strong correlations between the returns for different stocks without affecting their marginal distributions. This was not possible in Ref. [19]. Further, the number of model parameters does not grow too fast as the number of stocks n grows. We use dynamic programming to solve Merton’s optimization problem for power utility, with utility drawn from terminal wealth. Optimal strategies for n stocks are obtained. We also discuss how to estimate the model from data. This is illustrated with an example.

In this article, we are interested in an investment problem with stochastic volatilities and Portfolio constraints on proportions. We model the risky assets by jump diffusion processes and we consider a power utility function. The objective is to maximize the expected utility from terminal wealth. The value function is known to be a viscosity solution of an integro-differential Hamilton-Jacobi-Bellman (HJB) equation. Thanks to an exponential transformation, we reduce the nonlinearity of the HJB equation to a semi-linear equation. We prove existence of a smooth solution to the latter equation and we state a verification theorem which relates this solution to the value function. We derive the backward stochastic differential equation (BSDE) associated with the semi-linear equation. We propose a numerical scheme for the resolution of the BSDE based on iterative regressions on function bases and Monte-Carlo method. We illustrate an example which shows the importance of this reduction for the optimal portfolio numerical study. Therefore, we compute the optimal investment strategy by solving the associated BSDE.

We consider an investor facing a classical portfolio problem of optimal investment in a log-Brownian stock and a fixed-interest bond, but constrained to choose portfolio and consumption strategies that reduce a dynamic shortfall risk measure. For continuous- and discrete-time financial markets we investigate the loss in expected utility of intermediate consumption and terminal wealth caused by imposing a dynamic risk constraint. We derive the dynamic programming equations for the resulting stochastic optimal control problems and solve them numerically. Our numerical results indicate that the loss of portfolio performance is not too large while the risk is notably reduced. We then investigate time discretization effects and find that the loss of portfolio performance resulting from imposing a risk constraint is typically bigger than the loss resulting from infrequent trading.

We consider an investor maximizing his expected utility from terminal wealth with portfolio decisions based on the available information flow. This investor faces the opportunity to acquire some additional initial information ${\cal G}$ . His subjective fair value of this information is defined as the amount of money that he can pay for ${\cal G}$ such that this cost is balanced out by the informational advantage in terms of maximal expected utility. We study this value for common utility functions in the setting of a complete market modeled by general semimartingales. The main tools are a martingale preserving change of measure and martingale representation results for initially enlarged filtrations.

Portfolio Optimization in a Defaultable LLvy Driven Market Model

Optimal portfolio and insurance strategy with biometric risks, habit formation and smooth ambiguity

In spite of theoretical benefits, Markowitz mean-variance (MV) optimized portfolios often fail to meet practical investment goals of marketability, usability, and performance, prompting many investors to seek simpler alternatives. Financial experts Richard and Robert Michaud demonstrate that the limitations of MV optimization are not the result of conceptual flaws in Markowitz theory but unrealistic representation of investment information. What is missing is a realistic treatment of estimation error in the optimization and rebalancing process. The text provides a non-technical review of classical Markowitz optimization and traditional objections. The authors demonstrate that in practice the single most important limitation of MV optimization is oversensitivity to estimation error. Portfolio optimization requires a modern statistical perspective. Efficient Asset Management, Second Edition uses Monte Carlo resampling to address information uncertainty and define Resampled Efficiency(TM) (RE) technology. RE optimized portfolios represent a new definition of portfolio optimality that is more investment intuitive, robust, and provably investment effective. RE rebalancing provides the first rigorous portfolio trading, monitoring, and asset importance rules, avoiding widespread ad hoc methods in current practice. The Second Edition resolves several open issues and misunderstandings that have emerged since the original edition. The new edition includes new proofs of effectiveness, substantial revisions of statistical estimation, extensive discussion of long-short optimization, and new tools for dealing with estimation error in applications and enhancing computational efficiency. RE optimization is shown to be a Bayesian-based generalization and enhancement of Markowitz’s solution. RE technology corrects many current practices that may adversely impact the investment value of trillions of dollars under current asset management. RE optimization technology may also be useful in other financial optimizations and more generally in multivariate estimation contexts of information uncertainty with Bayesian linear constraints. Michaud and Michaud’s new book includes numerous additional proposals to enhance investment value including Stein and Bayesian methods for improved input estimation, the use of portfolio priors, and an economic perspective for asset-liability optimization. Applications include investment policy, asset allocation, and equity portfolio optimization. A final chapter includes practical advice for avoiding simple portfolio design errors. A simple global asset allocation problem illustrates portfolio optimization techniques. The presentation is intuitive, rigorous and informed with institutional management experience to appeal to investment management executives, consultants, fund trustees, brokers, academics, and anyone seeking to stay abreast of the future of investment technology. With its important implications for investment practice, Efficient Asset Management’s highly intuitive yet rigorous approach to defining optimal portfolios will appeal to investment management executives, consultants, brokers, and anyone seeking to stay abreast of current investment technology. Through practical examples and illustrations, Michaud and Michaud update the practice of optimization for modern investment management.

Optimal decision on dynamic insurance price and investment portfolio of an insurer