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Spectral Risk Measure
A Spectral risk measure is a risk measure given as a weighted average of outcomes where bad outcomes are, typically, included with larger weights. A spectral risk measure is a function of portfolio (finance), portfolio returns and outputs the amount of the numeraire (typically a currency) to be kept in reserve. A spectral risk measure is always a coherent risk measure, but the converse does not always hold. An advantage of spectral measures is the way in which they can be related to risk aversion, and particularly to a utility function, through the weights given to the possible portfolio returns. Definition Consider a portfolio (finance), portfolio X (denoting the portfolio payoff). Then a spectral risk measure M_: \mathcal \to \mathbb where \phi is non-negative, non-increasing, Continuous function#Directional and semi-continuity, right-continuous, integrable function defined on [0,1] such that \int_0^1 \phi(p)dp = 1 is defined by :M_(X) = -\int_0^1 \phi(p) F_X^(p) dp where F_X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Risk Measure
In financial mathematics, a risk measure is used to determine the amount of an asset or set of assets (traditionally currency) to be kept in reserve. The purpose of this reserve is to make the risks taken by financial institutions, such as banks and insurance companies, acceptable to the regulator. In recent years attention has turned towards convex and coherent risk measurement. Mathematically A risk measure is defined as a mapping from a set of random variables to the real numbers. This set of random variables represents portfolio returns. The common notation for a risk measure associated with a random variable X is \rho(X). A risk measure \rho: \mathcal \to \mathbb \cup \ should have certain properties: ; Normalized : \rho(0) = 0 ; Translative : \mathrm\; a \in \mathbb \; \mathrm \; Z \in \mathcal ,\;\mathrm\; \rho(Z + a) = \rho(Z) - a ; Monotone : \mathrm\; Z_1,Z_2 \in \mathcal \;\mathrm\; Z_1 \leq Z_2 ,\; \mathrm \; \rho(Z_2) \leq \rho(Z_1) Set-valued In a situation w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Statistics
In statistics, the ''k''th order statistic of a statistical sample is equal to its ''k''th-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference. Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample median and other sample quantiles. When using probability theory to analyze order statistics of random samples from a continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution. Notation and examples For example, suppose that four numbers are observed or recorded, resulting in a sample of size 4. If the sample values are :6, 9, 3, 8, the order statistics would be denoted :x_=3,\ \ x_=6,\ \ x_=8,\ \ x_=9,\, where the subscript enclosed in parentheses indicates the th order statistic of the sam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as or \mathbb. History The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes ''in a fair way'' between two players, who have to end th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Shortfall
Expected shortfall (ES) is a risk measure—a concept used in the field of financial risk measurement to evaluate the market risk or credit risk of a portfolio. The "expected shortfall at q% level" is the expected return on the portfolio in the worst q\% of cases. ES is an alternative to value at risk that is more sensitive to the shape of the tail of the loss distribution. Expected shortfall is also called conditional value at risk (CVaR), average value at risk (AVaR), expected tail loss (ETL), and superquantile. ES estimates the risk of an investment in a conservative way, focusing on the less profitable outcomes. For high values of q it ignores the most profitable but unlikely possibilities, while for small values of q it focuses on the worst losses. On the other hand, unlike the discounted maximum loss, even for lower values of q the expected shortfall does not consider only the single most catastrophic outcome. A value of q often used in practice is 5%. Expected shortfall is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comonotonicity
In probability theory, comonotonicity mainly refers to the perfect positive dependence between the components of a random vector, essentially saying that they can be represented as increasing functions of a single random variable. In two dimensions it is also possible to consider perfect negative dependence, which is called countermonotonicity. Comonotonicity is also related to the comonotonic additivity of the Choquet integral. The concept of comonotonicity has applications in financial risk management and actuarial science, see e.g. and . In particular, the sum of the components is the riskiest if the joint probability distribution of the random vector is comonotonic. Furthermore, the -quantile of the sum equals of the sum of the -quantiles of its components, hence comonotonic random variables are quantile-additive. In practical risk management terms it means that there is minimal (or eventually no) variance reduction from diversification. For extensions of comonotonicity, s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent Risk Measure
In the fields of actuarial science and financial economics there are a number of ways that risk can be defined; to clarify the concept theoreticians have described a number of properties that a risk measure might or might not have. A coherent risk measure is a function that satisfies properties of monotonicity, sub-additivity, homogeneity, and translational invariance. Properties Consider a random outcome X viewed as an element of a linear space \mathcal of measurable functions, defined on an appropriate probability space. A functional \varrho : \mathcal → \R \cup \ is said to be coherent risk measure for \mathcal if it satisfies the following properties: Normalized : \varrho(0) = 0 That is, the risk when holding no assets is zero. Monotonicity : \mathrm\; Z_1,Z_2 \in \mathcal \;\mathrm\; Z_1 \leq Z_2 \; \mathrm ,\; \mathrm \; \varrho(Z_1) \geq \varrho(Z_2) That is, if portfolio Z_2 always has better values than portfolio Z_1 under almost all scenarios then the risk of Z_2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elsevier
Elsevier () is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as ''The Lancet'', ''Cell'', the ScienceDirect collection of electronic journals, '' Trends'', the '' Current Opinion'' series, the online citation database Scopus, the SciVal tool for measuring research performance, the ClinicalKey search engine for clinicians, and the ClinicalPath evidence-based cancer care service. Elsevier's products and services also include digital tools for data management, instruction, research analytics and assessment. Elsevier is part of the RELX Group (known until 2015 as Reed Elsevier), a publicly traded company. According to RELX reports, in 2021 Elsevier published more than 600,000 articles annually in over 2,700 journals; as of 2018 its archives contained over 17 million documents and 40,000 e-books, with over one billion annual downloads. Researchers have criticized Elsevier for its high profit marg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Banking And Finance
The ''Journal of Banking and Finance'' is a peer-reviewed academic journal covering research on financial institutions, capital markets, and topics in investments and corporate finance. In 1989 the journal absorbed ''Studies in Banking & Finance''. A 2011 study ranked it among six elite finance journals. It publishes theoretical and empirical research papers spanning all the major research fields in finance and banking. Geert Bekaert is currently the managing editor. See also * ''Journal of Money, Credit and Banking The ''Journal of Money, Credit and Banking'' is a peer-reviewed academic journal covering monetary and financial issues in macroeconomics. It is published by Wiley-Blackwell on behalf of the Ohio State University Department of Economics. The edi ...'' References External links * Elsevier academic journals English-language journals Finance journals Publications established in 1977 Monthly journals {{finance-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cumulative Distribution Function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by an ''upwards continuous'' ''monotonic increasing'' cumulative distribution function F : \mathbb R \rightarrow ,1/math> satisfying \lim_F(x)=0 and \lim_F(x)=1. In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables. Definition The cumulative distribution function of a real-valued random variable X is the function given by where the right-hand side represents the probability that the random variable X takes on a value less tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weighted Average
The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics. If all the weights are equal, then the weighted mean is the same as the arithmetic mean. While weighted means generally behave in a similar fashion to arithmetic means, they do have a few counterintuitive properties, as captured for instance in Simpson's paradox. Examples Basic example Given two school with 20 students, one with 30 test grades in each class as follows: :Morning class = :Afternoon class = The mean for the morning class is 80 and the mean of the afternoon class is 90. The unweighted mean of the two means is 85. However, this does not account for the difference in number of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Utility Function
As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosophers such as Jeremy Bentham and John Stuart Mill. The term has been adapted and reapplied within neoclassical economics, which dominates modern economic theory, as a utility function that represents a single consumer's preference ordering over a choice set but is not comparable across consumers. This concept of utility is personal and based on choice rather than on pleasure received, and so is specified more rigorously than the original concept but makes it less useful (and controversial) for ethical decisions. Utility function Consider a set of alternatives among which a person can make a preference ordering. The utility obtained from these alternatives is an unknown function of the utilities obtained from each alternative, not the sum of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
