A tool for evaluating reinsurance alternatives daniel d. Modern portfolio theory mpt, which originated with harry markowitzs seminal paper portfolio selection in 1952, has stood the test of time and continues to be the intellectual foundation for realworld portfolio management. A through guide covering modern portfolio theory as well as the recent developments surrounding it. This book offers an uptodate, comprehensive coverage of stochastic dominance and its related concepts in a unified framework. The secondorder stochastic dominance condition can be equivalently written by means of the integration operator j f x. This facilitates the derivation of the first order asymptotic properties of tests for stochastic spanning given some stochastic dominance relation. Exercises on first order and second order stochastic dominance. Theoretically, given a choice over two risky assets with equivalent expected re. Final revision march 1996 summary this paper introduces a new distribution free test for firstorder stochastic dominance.
Pdf this is a chapter of a book manuscript entitled topics in microeconomics. Introduction to probability theory for graduate economics. The cumulative distribution and stochastic dominance. Pdf higherdegree stochastic dominance optimality and. We characterize a range of stochastic dominance sd relations by means of finite systems of convex inequalities. Econometric analysis of stochastic dominance by yoonjae whang. The second order stochastic dominance criterion ssd adds the assumption of global risk aversion. Empirical tests for stochastic dominance optimality. A comparison of five models that predict violations of first order stochastic dominance in risky. Other readers will always be interested in your opinion of the books youve read. Investment decision making under uncertainty pdf, epub, docx and torrent then this site is not for you. Conversely, both distributions climb to 1 at w b 1, but f. To allow for a compact presentation, we focus on the most common criterion of second order stochastic dominance ssd. Jul 07, 2008 universal domination and stochastic domination.
Hence the property in your question defines a stronger relation than first order stochastic dominance also known as the usual stochastic order, and it is called the reversed hazard rate order. We derive properties of the cdf of random variables defined as saddletype points of real valued continuous stochastic processes. Spanning tests for markowitz stochastic dominance sciencedirect. Chapter 4 stochastic dominance mit opencourseware free. Stochastic dominance notes agec 662 a fundamental concern, when looking at risky situations is choosing among risky. Econ4510 finance theory diderik lund, 26 january 2012.
Let option l i realize a monetary outcome of at least x with probability pi x. Eumaximizer curves equal expected monetary or u 2 u 1 c 2 c 1 u 3 u 2 c 3 c 2 this corresponds to risk aversion. Munich personal repec archive on the third order stochastic dominance for riskaverse and riskseeking investors. It is intended as a means to promote research to interested readers. Risk aversion and stochastic dominance bruner 1 introduction this paper presents the results of an experiment intended to determine the frequency that risk averse individuals make choices that satisfy second order stochastic dominance ssd. For sd optimality of degree 1 to 4 and sd efficiency of degree 2 to 5. Thus a,b is the maximal support of all the probability distributions we will consider. Then x second order stochastically dominates y if z a l prx tdt z a l pry tdt for all a. Traditionally, stochastic dominance is tested pairwise. Count, the annals of mathematical statistics, 1968.
If youre looking for a free download links of stochastic dominance. For instance, bayes first order stochastic dominance is the same as berry and fristedts 1 concept of strongly to the right, which is useful in deriving comparative. Therefore, from now on we restrict ourselves to the case sx2. We conduct dominance test at the first and second orders. Eco 317 fall 2009 economics of uncertainty sample midterm examination answer key question 1 20 points a 7 points independence axiom machina p. First order stochastic dominance let us begin with the definition of preference given in equation 1 and the most general constraint on a utility function given in equation 2. On the third order stochastic dominance for riskaverse and. Dual stochastic dominance and related meanrisk models. Similarly, firstorder stochastic dominance is called.
The basic properties of first order stochastic dominance were initially. Two good references containing the above at least as far as i can remember are the books m\uller and stoyan 2002, comparison methods for. Stochastic dominance is a partial order between random variables. Between first and secondorder stochastic dominance request. Then x secondorder stochastically dominates y if z a l prx tdt z a l pry tdt for all a. We consider the problem of constructing meanrisk models which are consistent with the second degree stochastic dominance relation.
The motivation for such a continuum is that while decision makers have a preference for more is better, they are mostly risk averse. This notion is a direct extension of stochastic dominance to the case where full diversi. Exercises on stochastic dominance exercises on firstorder. From this graph we can see that firstorder stochastic dominance is equivalent. Equivalently, l 1 is fosd if there is an option l 2 such that. If ux is strictly increasing and piecewise differentiable, and cumulative f. This allows us to show that several models using quantiles and tail characteristics of the distribution are in harmony with the. Econ4510 finance theory diderik lund, 26 january 2012 secondorder stochastic dominance, ssd possible to show that \x a. It is however possible to generalize our analysis to first order or third order sd by changing the relevant order of the lower partial moments lpms, as in bawa et al in addition, we focus on the joint crosssectional. Returnrisk comparisons sometimes, it is very easy to predict a choice of an individual between two lotteries g.
Measuring risk first order stochastic dominance second order stochastic dominance mean and variance. Secondorder stochastic dominance, rewardrisk portfolio. First order stochastic dominance let g and f on 0, 1 be two continuous distribution functions, such that f. Portfolio optimization with stochastic dominance constraints. The concept arises in decision theory and decision analysis in situations where one gamble a probability distribution over possible outcomes, also known as prospects can be ranked as superior to another gamble for a broad class of decisionmakers. Assume a 2 goods2 agents economy and wellbehaved utility functions. Fishburn and vickson 1978 show that tsd and dara stochastic dominance are equivalent concepts when the means of the random alternatives are equal to one another. A method for ordering probability distributions, stochastic dominance has grown in importance recently as a way to measure comparisons in welfare economics, inequality studies, health economics, insurance wages, and trade patterns. It also takes advantage of statas ability to handle complex sampling design in the estimation of the standard errors.
Stochastic dominance two criteria for making decisions. Stochastic dominance and option pricing in discrete and. We develop a continuum of stochastic dominance rules, covering preferences from first to second order stochastic dominance. We define the concept of markowitz stochastic dominance spanning, and develop an analytical representation of the spanning property. A 1% increase in both the risk free rate and the expected rate of return on the market, so that the cml shifts in a parallel fashion 2. We first relax the ssd problem as a stochastic semiinfinite programming sip problem. Unfortunately, as is also usual, the strength of this assumption comes at the price of realism. In statistics, the monotone likelihood ratio property is a property of the ratio of two probability density functions pdfs. Estimation simultaneously under a broad class of loss functions hwang, jiunn tzon, the annals of statistics, 1985.
In practice a may be zero if there is limited liability, but. Violations of firstorder stochastic dominance as salience. But one of them or both can nevertheless be useful for narrowing down choices by excluding dominated alternatives. Sample midterm examination answer key question 1 20 points. Between first and secondorder stochastic dominance by alfred. Only recently kuosmanen 2004 and post 2003 have introduced the notion of stochastic dominance e. In that setting both authors derive statistics to test for stochastic. The rigorous way to determine whether there is a stochastic dominance relationship between two proposed gambles is to examine their probability distributions. If a riskfree opportunity is available with outcome x0, and if it is meaningful to talk. Put in another way, does first order stochastic dominance survive halftruncations. Third order stochastic dominance, ascending stochastic dominance.
On the third order stochastic dominance for riskaverse. By exploiting duality relations of convex analysis we develop the quantile model of stochastic dominance for general distributions. Stochastic dominance and conditional expectationan. Exercises on stochastic dominance exercises on first. Stochastic dominance and option pricing in discrete and continuous time. First degree stochastic dominance how is first degree. Stochastic dominance of the first and second order has a clear economic interpretation, i.
In this paper, we propose an incremental bundle method with inexact oracle for solving the portfolio optimization with stochastic second order dominance ssd constraints. Expected utility, risk aversion and stochastic dominance. Bayes first order stochastic dominance is not a new concept. He also demonstrates that third order stochastic dominance implies dominance under meanlower partial variance rule. Since the slope of the cumulative distribution function is the probability density function, the density f 1w must be less than the density f 2w for small w. The motivation for such a continuum is that while decision makers have preference for more is better, they are mostly risk averse but cannot assert that they would dislike any risk.
Below is a link to a program i wrote to conduct tests of stochastic dominance using stata. Between first and secondorder stochastic dominance. An increase in the expected rate of return on the market. An incremental bundle method for portfolio selection.
The secondorder stochastic dominance corresponds to a. For the particular case of sip problem, we exploit the improvement function and the idea of incremental technique for dealing with the. Sample midterm examination answer key question 1 20. The random payoff of lottery b, denoted by b x, is uniformly distributed on the interval 0,6. Empirical tests for stochastic dominance optimality thierry post september 15, 2015 abstract if a given risky prospect is compared with multiple choice alternatives, then a joint test for optimality is more appropriate than a series of pairwise stochastic dominance tests. The condition for fosd is that the cdf of the outcomes of gamble a be entirely below. Calculating discrete pdf from discrete cdf in python. Let us do this by defining an a and b terms which fit the integration by parts structure. The motivation for such a continuum is that while decision makers have a preference for more is better, they are mostly risk averse but cannot assert that they would dislike any risk. Simple portfolio problem, one risky, one risk free asset.
Stochastic dominance general structure here we suppose that the consequences are wealth amounts denoted by w, which can take on any value between a and b. Jan 14, 2015 the rigorous way to determine whether there is a stochastic dominance relationship between two proposed gambles is to examine their probability distributions. The random payoff of lottery a, denoted by a x, is uniformly distributed on the interval 3,5. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. First order stochastic dominance a random variable x a rst order stochastically dominates another random variable x.
Reference levels and loss aversion experimental evidence shows that people view consumption and wealth relative to a reference level and. The stochastic dominance ordering over probability distributions is one of the most. The firstorder stochastic dominance ordering of the singhmaddala. Expected utility, risk aversion and stochastic dominance 1. We develop a continuum of stochastic dominance rules, covering preferences from first to secondorder stochastic dominance. Stochastic dominance and conditional expectation 35 this generalizes the insight from the theory of. The first order stochastic dominance criterion fsd places on the form of the utility function no restriction beyond the usual requirement that it is nondecreasing, i. How to test firstorder and secondorder stochastic dominance. L2theory in this chapter we will study stochastic processes characterized by their rst two moments or second order processes. Request pdf between first and secondorder stochastic dominance we. Comparing probability distributions 1 first order stochastic. Tests for firstorder stochastic dominance teresa ledwina1 and grzegorz wylupek 2 1 institute of mathematics, polish academy of sciences, poland 2 institute of mathematics, university of wroc law, poland in a recent article ledwina and wy lupek, 2012a, we proposed and studied the two new tests.
This and other related concepts have been used in the literature. If the distribution of x is f and the distribution of y is g, then x. A comparison of five models that predict violations of. Random variable a has firstorder stochastic dominance over random variable b if for any outcome x, a gives at least as. In contrast to the meanvariance model, rewardrisk portfolio selection arises from an axiomatic definition of reward and risk measures based on a few basic principles, including consistency with second order stochastic dominance. The idea of interpolating integerdegree stochastic dominance rules has. A method for ordering probability distributions, stochastic dominance has grown in importance recently as a way to measure comparisons in welfare economics, inequality. Working paper series 201593dsc revised version of 201555dsc a working paper is the authors intellectual property.