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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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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 X (denoting the portfolio payoff). Then a spectral risk measure M_: \mathcal \to \mathbb where \phi is non-negative, non-increasing, right-continuous, integrable function defined on ,1/math> 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 is the cumulative distribution function for ''X''. If there are S equiprobable outcomes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Risk Metric
In the context of risk measurement, a risk metric is the concept quantified by a risk measure. When choosing a risk metric, an agent is picking an aspect of perceived risk to investigate, such as volatility or probability of default. Risk measure and risk metric In a general sense, a measure is a procedure for quantifying something. A metric is that which is being quantified. In other words, the method or formula to calculate a risk metric is called a risk measure. For example, in finance, the volatility of a stock might be calculated in any one of the three following ways: * Calculate the sample standard deviation of the stock's returns over the past 30 trading days. * Calculate the sample standard deviation of the stock's returns over the past 100 trading days. * Calculate the implied volatility of the stock from some specified call option on the stock. These are three distinct risk measures. Each could be used to measure the single risk metric volatility. Examples * De ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tail Conditional Expectation
Tail value at risk (TVaR), also known as tail conditional expectation (TCE) or conditional tail expectation (CTE), is a risk measure associated with the more general value at risk. It quantifies the expected value of the loss given that an event outside a given probability level has occurred. Background There are a number of related, but subtly different, formulations for TVaR in the literature. A common case in literature is to define TVaR and average value at risk as the same measure. Under some formulations, it is only equivalent to expected shortfall when the underlying distribution function is continuous at \operatorname_(X), the value at risk of level \alpha. Under some other settings, TVaR is the conditional expectation of loss above a given value, whereas the expected shortfall is the product of this value with the probability of it occurring. The former definition may not be a coherent risk measure in general, however it is coherent if the underlying distribution is c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
RiskMetrics
The RiskMetrics variance model (also known as exponential smoother) was first established in 1989, when Sir Dennis Weatherstone, the new chairman of J.P. Morgan, asked for a daily report measuring and explaining the risks of his firm. Nearly four years later in 1992, J.P. Morgan launched the RiskMetrics methodology to the marketplace, making the substantive research and analysis that satisfied Sir Dennis Weatherstone's request freely available to all market participants. In 1998, as client demand for the group's risk management expertise exceeded the firm's internal risk management resources, the Corporate Risk Management Department was spun off from J.P. Morgan as RiskMetrics Group with 23 founding employees. The RiskMetrics technical document was revised in 1996. In 2001, it was revised again in ''Return to RiskMetrics''. In 2006, a new method for modeling risk factor returns was introduced (RM2006). On 25 January 2008, RiskMetrics Group listed on the New York Stock Exchange ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Managerial Risk Accounting
Managerial Risk Accounting is concerned with the generation, dissemination and use of risk related accounting information to managers within organisations to enable them to judge and shape the risk situation of the organisation according to the objectives of the organisation. Subject As a part of the management accounting system and function, managerial risk accounting has the following two main purposes: * decision-facilitating or decisions-making * decision-influencing or stewardship These purposes are achieved by providing respectively relevant information to improve the ability and willingness of the employees to achieve the organisations’s goals and objectives. For the purpose of decision facilitation, decision makers should be provided with an accounting representation of the state-act-outcome set of the decision. Especially, it is necessary to provide statements concerning the likelihood or probability of states and outcomes. For the purpose of stewardship, it is neces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamic Risk Measure
In financial mathematics, a conditional risk measure is a random variable of the financial risk (particularly the downside risk) as if measured at some point in the future. A risk measure can be thought of as a conditional risk measure on the trivial sigma algebra. A dynamic risk measure is a risk measure that deals with the question of how evaluations of risk at different times are related. It can be interpreted as a sequence of conditional risk measures. A different approach to dynamic risk measurement has been suggested by Novak. Conditional risk measure Consider a portfolio's returns at some terminal time T as a random variable that is uniformly bounded, i.e., X \in L^\left(\mathcal_T\right) denotes the payoff of a portfolio. A mapping \rho_t: L^\left(\mathcal_T\right) \rightarrow L^_t = L^\left(\mathcal_t\right) is a conditional risk measure if it has the following properties for random portfolio returns X,Y \in L^\left(\mathcal_T\right): ; Conditional cash invariance ... [...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_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acceptance Set
In financial mathematics, acceptance set is a set of acceptable future net worth which is acceptable to the regulator. It is related to risk measures. Mathematical Definition Given a probability space (\Omega,\mathcal,\mathbb), and letting L^p = L^p(\Omega,\mathcal,\mathbb) be the Lp space in the scalar case and L_d^p = L_d^p(\Omega,\mathcal,\mathbb) in d-dimensions, then we can define acceptance sets as below. Scalar Case An acceptance set is a set A satisfying: # A \supseteq L^p_+ # A \cap L^p_ = \emptyset such that L^p_ = \ # A \cap L^p_- = \ # Additionally if A is convex then it is a convex acceptance set ## And if A is a positively homogeneous cone then it is a coherent acceptance set Set-valued Case An acceptance set (in a space with d assets) is a set A \subseteq L^p_d satisfying: # u \in K_M \Rightarrow u1 \in A with 1 denoting the random variable that is constantly 1 \mathbb-a.s. # u \in -\mathrmK_M \Rightarrow u1 \not\in A # A is directionally closed in M with A + u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function is a one-to-one (injective) and onto (surjective) mapping of a set ''X'' to a set ''Y''. The term ''one-to-one correspondence'' must not be confused with ''one-to-one function'' (an injective function; see figures). A bijection from the set ''X'' to the set ''Y'' has an inverse function from ''Y'' to ''X''. If ''X'' and ''Y'' are finite sets, then the existence of a bijection means they have the same number of elements. For infinite sets, the picture is more complicated, leading to the concept of cardinal number—a way to distinguish the various sizes of infinite se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by \sigma^2, s^2, \operatorname(X), V(X), or \mathbb(X). An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by \sigma^2, s^2, \operatorname(X), V(X), or \mathbb(X). An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
