A Homogeneous Distribution Problem with Applications to Finance
TL;DR: In this paper, the authors consider the problem of determining the cumulative distribution function and/or moments of the optimal solution value of a nonlinear program dependent upon a single random variable and provide an algorithm to solve the distribution problem in the linear case based on extensions of the methods of parametric linear programming.
Abstract: We consider the problem of determining the cumulative distribution function and/or moments of the optimal solution value of a nonlinear program dependent upon a single random variable. This problem is difficult computationally because one must in effect determine the optimal solution to an infinite number of nonlinear programs. Bereanu [Bereanu, B., G. Peeters. 1970. A ‘Wait-and-See’ problem in stochastic linear programming. An experimental computer code. Cashiers Centre Etudes Rech. Oper. 12 (3) 133–148.] has provided an algorithm to solve the distribution problem in the linear case based on extensions of the methods of parametric linear programming. (See also [Bereanu, B. 1967. On stochastic linear programming, distribution problems: stochastic technology matrix. Z. f. Wahrscheinlichkeitstheorie u. oerw. Gerbieter 8 148–152; Bereanu, B. 1971. The distribution problem in stochastic linear programming: the Cartesian integration method. Center of Mathematical Statistics of the Academy of RSR, Bucharest, 71...
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TL;DR: How still sharper bounds may be generated based on the simple idea of sequentially applying the classic bounds to smaller and smaller subintervals of the range of the random variable is indicated.
Abstract: This paper is concerned with the determination of tight lower and upper bounds on the expectation of a convex function of a random variable. The classic bounds are those of Jensen and Edmundson-Madansky and were recently generalized by Ben-Tal and Hochman. This paper indicates how still sharper bounds may be generated based on the simple idea of sequentially applying the classic bounds to smaller and smaller subintervals of the range of the random variable. The bounds are applicable in the multivariate case if the random variables are independent. In the dependent case bounds based on the Edmundson-Madansky inequality are not available; however, bounds may be developed using the conditional form of Jensen's inequality. We give some examples to illustrate the geometrical interpretation and the calculations involved in the numerical determination of the new bounds. Special attention is given to the problem of maximizing a nonlinear program that has a stochastic objective function.
107 citations
TL;DR: In this paper, it is shown that knowledge of means or of means, variances, co-variances and n-moments are sufficient for the calculation of optimal decision rules.
Abstract: The cost of obtaining good information regarding the various probability distributions needed for the solution of most stochastic decision problems is considerable. It is important to consider questions such as: (1) what minimal amounts of information are sufficient to determine optimal decision rules; (2) what is the value of obtaining knowledge of the actual realization of the random vectors; and (3) what is the value of obtaining some partial information regarding the actual realization of the random vectors. This paper is primarily concerned with questions two and three when the decision maker has an a priori knowledge of the joint distribution function of the random variables. Some remarks are made regarding results along the lines of question one. Mention is made of assumptions sufficient so that knowledge of means, or of means, variances, co-variances and n-moments are sufficient for the calculation of optimal decision rules. The analysis of the second question leads to the development of bounds on...
44 citations
TL;DR: In this paper, a general approach to the development of deterministic equivalents of constraints to be satisfied within certain probability limits is presented, and a deterministic transformation of a stochastic programming problem with random variables in the objective function is presented.
17 citations
TL;DR: This article reviewed contributions to portfolio theory and practice by William T. Ziemba and his colleagues and reviewed the effects of parameter errors on optimal portfolio choice along with exit strategies from bubble like financial markets.
Abstract: This paper reviews contributions to portfolio theory and practice by William T. Ziemba and his colleagues. The paper covers static and dynamic portfolio and capital growth theory along with real applications to asset and asset-liability management and various types of trading and prediction and risk control models for a variety of asset classes. There is reference to and synopsis of many journal articles, academic and practical research and books. The effects of parameter errors on optimal portfolio choice is reviewed along with exit strategies from bubble like financial markets. Mispriced options, risk and pure arbitrage, political effects and stock market anomalies are used along with optimization in various applications.
6 citations
TL;DR: A survey of applied methods for solving the linear programming decision problem with uncertain parameters is presented, seeking particularly to provide an updated bibliography for the post 1975 period.
6 citations
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TL;DR: In this paper, the optimal consumption-investment problem for an investor whose utility for consumption over time is a discounted sum of single-period utilities, with the latter being constant over time and exhibiting constant relative risk aversion (power-law functions or logarithmic functions), is discussed.
Abstract: Publisher Summary This chapter reviews the optimal consumption-investment problem for an investor whose utility for consumption over time is a discounted sum of single-period utilities, with the latter being constant over time and exhibiting constant relative risk aversion (power-law functions or logarithmic functions). It presents a generalization of Phelps' model to include portfolio choice and consumption. The explicit form of the optimal solution is derived for the special case of utility functions having constant relative risk aversion. The optimal portfolio decision is independent of time, wealth, and the consumption decision at each stage. Most analyses of portfolio selection, whether they are of the Markowitz–Tobin mean-variance or of more general type, maximize over one period. The chapter only discusses special and easy cases that suffice to illustrate the general principles involved and presents the lifetime model that reveals that investing for many periods does not itself introduce extra tolerance for riskiness at early or any stages of life.
2,369 citations
"A Homogeneous Distribution Problem ..." refers background in this paper
...In the classic dynamic capital accumulation problem (see Phelps [11], Samuelson [13], and Hakansson [8]), the investor wishes to allocate his funds between various risky investments in each period to maximize the expected utility of lifetime consumption....
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TL;DR: Parkin this article discusses the application of portfolio selection to the behavior of a depository financial intermediary and provides a set of propositions and theorems with a wide variety of applications including intuitive insights into the gains from diversification, empirically useful algorithms for common stock selection and theoretical advances for understanding the demand for money and the general equilibrium of financial markets.
Abstract: It has long been recognized that the useful application of the portfolio selection theory of Markowitz [10] and Tobin [19], [20] is limited by the restrictive assumptions placed on the utility function, distribution of returns, wealth holdings, and dynamic behaviour of the individual economic unit. Recent research by numerous individuals has indicated, however, that many of the implications of the Markowitz-Tobin theory can be derived from a weaker set of assumptions. The net result is a remarkably strong set of propositions and theorems with a wide variety of applications including intuitive insights into the gains from diversification, empirically useful algorithms for common stock selection, and theoretical advances for understanding the demand for money and the general equilibrium of financial markets. In this paper we discuss the application of these principles of portfolio selection to the behaviour of a depository financial intermediary. This notion is not, of course, entirely novel. For one thing, the extensions of the Markowitz-Tobin theory to include short sales (see Lintner [9]) and mutual fund behaviour (see Merton [12] and the cited studies) indicate that at a sufficiently high level of abstraction the available theory may be directly applicable to financial intermediaries. For another, empirical studies are now available that attempt to explain financial intermediary behaviour on the basis of the principles of portfolio theory (see Parkin [13]). In the case of the extended Markowitz-Tobin model, however, it is to be stressed that it may be applied to depository financial intermediaries only by abstracting from many of the institutional and market factors that make these intermediaries unique, and thus there does remain the need to develop a theory that explicitly incorporates these factors. With respect to the empirical studies, the primary objective has been the estimation of demand functions for financial assets, and thus there has been little attempt to derive the theoretical propositions that become available when portfolio theory is applied to financial intermediaries.
211 citations
"A Homogeneous Distribution Problem ..." refers methods in this paper
...Theorem 1 can be used to prove a separation theorem due to Hart and Jaffee [9]....
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TL;DR: In this article, the optimal lifetime consumption strategy of an individual whose wealth holding possibilities expose him to the risk of loss is analyzed using a stochastic, discrete-time dynamic programming model.
Abstract: This chapter discusses the optimal lifetime consumption strategy of an individual whose wealth holding possibilities expose him to the risk of loss. The vehicle of analysis is a stochastic, discrete-time dynamic programming model that postulates an expected lifetime utility function to be maximized. All wealth consists of a single asset called capital. The chapter discusses the utility function and a stochastic capital growth process. It also discusses the structure of the optimal consumption policy, that is, the way in which consumption depends upon the individual's age and capital. Optimal consumption is an increasing function of both age and capital. Consumption cannot be expressed as a function of aggregate expected income because expected wage income and expected capital income have different variances.
173 citations
TL;DR: Mossin and Leland as discussed by the authors showed that the logarithmic function and the power function induce completely myopic utility functions, and that when the interest rate in each period is zero, all terminal wealth functions such that the risk tolerance index is linear in x, then the last class of wealth functions induces partially myopic utilities.
Abstract: Publisher Summary In a recent paper, Mossin attempts to isolate the class of utility functions of terminal wealth, f(x), which, in the sequential portfolio problem, induces myopic utility functions of intermediate wealth positions. Induced utility functions of shortrun wealth are myopic whenever they are independent of yields beyond the current period, that is, they are positive linear transformations of f(x). Mossin concludes that the logarithmic function and the power functions induce completely myopic utility functions, that when the interest rate in each period is zero, all terminal wealth functions such that the risk tolerance index—f’(x)/f”(x) is linear in x induce completely myopic utility functions of short-run wealth, that when interest rates are not zero, the last class of terminal wealth functions induces partially myopic utility functions, and that all of the preceding is true whether the yields in the various periods are serially correlated or not. With the exception of the last assertion, the same conclusions are reached by Leland. The second and third statements are true only in a highly restricted sense even when yields are serially independent and that when investment yields in the various periods are statistically dependent, only the logarithmic function induces utility functions of short-run wealth that are myopic. On optimal myopic portfolio policies, with and without serial correlation of yields in view of the difficulty of estimating future yields and their apparent serial correlation, the myopic property of the logarithmic utility functions is highly significant. However, this function also has other attractive properties.
167 citations
01 Jan 1975
TL;DR: Mossin and Leland as discussed by the authors showed that the logarithmic function and the power function induce completely myopic utility functions, and that when the interest rate in each period is zero, all terminal wealth functions such that the risk tolerance index is linear in x, then the last class of wealth functions induces partially myopic utilities.
Abstract: Publisher Summary In a recent paper, Mossin attempts to isolate the class of utility functions of terminal wealth, f(x), which, in the sequential portfolio problem, induces myopic utility functions of intermediate wealth positions. Induced utility functions of shortrun wealth are myopic whenever they are independent of yields beyond the current period, that is, they are positive linear transformations of f(x). Mossin concludes that the logarithmic function and the power functions induce completely myopic utility functions, that when the interest rate in each period is zero, all terminal wealth functions such that the risk tolerance index—f’(x)/f”(x) is linear in x induce completely myopic utility functions of short-run wealth, that when interest rates are not zero, the last class of terminal wealth functions induces partially myopic utility functions, and that all of the preceding is true whether the yields in the various periods are serially correlated or not. With the exception of the last assertion, the same conclusions are reached by Leland. The second and third statements are true only in a highly restricted sense even when yields are serially independent and that when investment yields in the various periods are statistically dependent, only the logarithmic function induces utility functions of short-run wealth that are myopic. On optimal myopic portfolio policies, with and without serial correlation of yields in view of the difficulty of estimating future yields and their apparent serial correlation, the myopic property of the logarithmic utility functions is highly significant. However, this function also has other attractive properties.
160 citations