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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 downside risk, risks taken by financial institutions, such as banks and insurance companies, acceptable to the regulator (economics), regulator. In recent years attention has turned to coherent risk measure, 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 \ ... [...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 * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tail Conditional Expectation
In financial mathematics, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Risk Management
Risk management is the identification, evaluation, and prioritization of risks, followed by the minimization, monitoring, and control of the impact or probability of those risks occurring. Risks can come from various sources (i.e, Threat (security), threats) including uncertainty in Market environment, international markets, political instability, dangers of project failures (at any phase in design, development, production, or sustaining of life-cycles), legal liabilities, credit risk, accidents, Natural disaster, natural causes and disasters, deliberate attack from an adversary, or events of uncertain or unpredictable root cause analysis, root-cause. Retail traders also apply risk management by using fixed percentage position sizing and risk-to-reward frameworks to avoid large drawdowns and support consistent decision-making under pressure. There are two types of events viz. Risks and Opportunities. Negative events can be classified as risks while positive events are classifi ... [...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 invarian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distortion Risk Measure
In financial mathematics and economics, a distortion risk measure is a type of risk measure which is related to the cumulative distribution function of the return of a financial portfolio. Mathematical definition The function \rho_g: L^p \to \mathbb associated with the distortion function g: ,1\to ,1/math> is a ''distortion risk measure'' if for any random variable of gains X \in L^p (where L^p is the Lp space) then : \rho_g(X) = -\int_0^1 F_^(p) d\tilde(p) = \int_^0 \tilde(F_(x))dx - \int_0^ g(1 - F_(x)) dx where F_ is the cumulative distribution function for -X and \tilde is the dual distortion function \tilde(u) = 1 - g(1-u). If X \leq 0 almost surely then \rho_g is given by the Choquet integral, i.e. \rho_g(X) = -\int_0^ g(1 - F_(x)) dx. Equivalently, \rho_g(X) = \mathbb^ X/math> such that \mathbb is the probability measure generated by g, i.e. for any A \in \mathcal the sigma-algebra then \mathbb(A) = g(\mathbb(A)). Properties In addition to the properties of genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Value-at-risk
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 ... [...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 ... [...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, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set. A function is bijective if it is invertible; that is, a function f:X\to Y is bijective if and only if there is a function g:Y\to X, the ''inverse'' of , such that each of the two ways for composing the two functions produces an identity function: g(f(x)) = x for each x in X and f(g(y)) = y for each y in Y. For example, the ''multiplication by two'' defines a bijection from the integers to the even numbers, which has the ''division by two'' as its inverse function. A function is bijective if and only if it is both injective (or ''one-to-one'')—meaning that each element in the codomain is mappe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variance
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. 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. It is 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 deviation; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard devi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
