Two methods are frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean--risk approaches. The former is based on an axiomatic model of risk-averse preferences but does not provide a convenient computational recipe. The latter quantifies the problem in a lucid form of two criteria with possible tradeo # analysis, but cannot model all risk-averse preferences. In particular, if variance is used as a measure of risk, the resulting mean--variance (Markowitz) model is, in general, not consistent with stochastic dominance rules. This paper shows that the standard semideviation (square root of the semivariance) as the risk measure makes the mean--risk model consistent with the second degree stochastic dominance, provided that the trade-o# coe#cient is bounded by a certain constant. Similar results are obtained for the absolute semideviation, and for the absolute and standard deviations in the case of symmetric or bounded distributions. In the analysis we...

Most individuals must decide how much of their marketable wealth should be annuitized at retirement. The natural alternative to annuitization is investing the wealth and withdrawing the exact same consumption stream as the annuity would have provided. Of course, this strategy risks under-funding retirement in the event of below average investment returns with above average longevity. This paper develops the framework for a third alternative. We propose a model in which retirees defer annuitization, via a "do-it-yourself " scheme, until it is no longer possible to beat the mortality-adjusted rate of return from a life annuity. We make use of a unique Canadian database to calibrate the insurance loads and interest rate parameters. We conclude that in the current environment, a sixty five year old female (male) has a ninety percent (eighty-five percent) chance of beating the rate of return from a life annuity, until age eighty.

In recent years, descriptive models of risky choice have incorporated features that reflect the importance of particular outcome values in choice. Cumulative prospect theory (CPT) does this by inserting a reference point in the utility function. SP/A (security-potential/aspiration) theory uses aspiration level as a second criterion in the choice process. Experiment 1 compares the ability of the CPT and SP/A models to account for the same within-subjects data set and finds in favor of SP/A. Experiment 2 replicates the main finding of Experiment 1 in a between-subjects design. The final discussion brackets the SP/A result by showing the impact on fit of both decreasing and increasing the number of free

review. Views or opinions expressed herein do not necessarily represent those of the Institute, its National Member Organizations, or other organizations supporting the work. –ii– We analyse relations between two methods frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean–risk approaches. The concept of α-consistency of these approaches is defined as the consistency within a bounded range of mean–risk trade-offs. We show that mean–risk models using central semideviations as risk measures are α-consistent with stochastic dominance relations of the corresponding degree if the trade-off coefficient for the semideviation is bounded by one.

Mathematics and statistics have transformed the day-to-day trading in the world's financial markets. This has lead to new ways to reduce (or "hedge") risks which provides an important service to society, but also a temptation to very big gambles, with a potential for extreme losses. This paper discusses some of the ways statistics and mathematics can be used to understand and protect against very large, "catastrophic" financial risks. We argue that means don't mean anything for catastrophic risk, that separate large financial risks often are better handled by separate companies, and that the mathematical aspects of risk can't be summarized into one number. We also believe that there is a large potential for improved risk management in financial institutions, where extreme value theory, a speciality of the present authors, may be a useful tool. Improvements, however will not come for free but require long and hard work, where mathematics is only one part of the total effort. ...

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Abstract. Mean-variance portfolio analysis provided the first quantitative treatment of the tradeoff between profit and risk. We describe in detail the interplay between objective and constraints in a number of single-period variants, including semivariance models. Particular emphasis is laid on avoiding the penalization of overperformance. The results are then used as building blocks in the development and theoretical analysis of multiperiod models based on scenario trees. A key property is the possibility of removing surplus money in future decisions, yielding approximate downside risk minimization.

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Introduction There has been a controversy in this journal about using downside risk measures in portfolio analysis. The downside risk measures supposedly are a major improvement over traditional portfolio theory. That is where the battle lines clashed when Rom and Ferguson (1993, 1994b) and Kaplan and Siegel (1994a, 1994b) engaged in a "tempest in a teapot". I should confess that I am strong supporter of downside risk measures and have used them in my teaching, research and software for the past two decades. Therefore, you should keep that bias in mind as you read this article. One of the best means to understand a concept is to study the history of its development. Understanding the issues facing researchers during the development of a concept results in better knowledge of the concept. The purpose of this paper is to provide an understanding of the measurement of downside risk. First, it helps to define terms. Portfolio theory is the application of decision-making tools unde

We investigate some portfolio problems that consist of maximizing expected terminal wealth under the constraint of an upper bound for the risk, where we measure risk by the variance, but also by the Capital-at-Risk (CaR). The solution of the mean-variance problem has the same structure for any price process which follows an exponential Levy process. The CaR involves a quantile of the corresponding wealth process of the portfolio. We derive a weak limit law for its approximation by a simpler Levy process, often the sum of a drift term, a Brownian motion and a compound Poisson process. Certain relations between a Levy process and its stochastic exponential are investigated.