|
Deviation Risk Measure
In financial mathematics, a deviation risk measure is a function to quantify financial risk (and not necessarily downside risk) in a different method than a general risk measure. Deviation risk measures generalize the concept of standard deviation. Mathematical definition A function D: \mathcal^2 \to [0,+\infty], where \mathcal^2 is the L2 space of random variables (random Portfolio (finance), portfolio returns), is a deviation risk measure if # Shift-invariant: D(X + r) = D(X) for any r \in \mathbb # Normalization: D(0) = 0 # Positively homogeneous: D(\lambda X) = \lambda D(X) for any X \in \mathcal^2 and \lambda > 0 # Sublinearity: D(X + Y) \leq D(X) + D(Y) for any X, Y \in \mathcal^2 # Positivity: D(X) > 0 for all nonconstant ''X'', and D(X) = 0 for any constant ''X''. Relation to risk measure There is a bijection, one-to-one relationship between a deviation risk measure ''D'' and an expectation-bounded risk measure ''R'' where for any X \in \mathcal^2 * D(X) = R(X - \mathbb[X]) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Financial Mathematics
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling in the Finance#Quantitative_finance, financial field. In general, there exist two separate branches of finance that require advanced quantitative techniques: Derivative (finance), derivatives pricing on the one hand, and risk management, risk and Investment management#Investment managers and portfolio structures, portfolio management on the other. Mathematical finance overlaps heavily with the fields of computational finance and financial engineering. The latter focuses on applications and modeling, often with the help of stochastic asset models, while the former focuses, in addition to analysis, on building tools of implementation for the models. Also related is quantitative investing, which relies on statistical and numerical models (and lately machine learning) as opposed to traditional fundamental analysis when investment ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Financial Risk
Financial risk is any of various types of risk associated with financing, including financial transactions that include company loans in risk of default. Often it is understood to include only downside risk, meaning the potential for financial loss and uncertainty about its extent. Modern portfolio theory initiated by Harry Markowitz in 1952 under his thesis titled "Portfolio Selection" is the discipline and study which pertains to managing market and financial risk. In modern portfolio theory, the variance (or standard deviation In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...) of a portfolio is used as the definition of risk. Types According to Bender and Panz (2021), financial risks can be sorted into five different categories. In their study, they apply an algorith ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Downside Risk
Downside risk is the financial risk associated with losses. That is, it is the risk of the actual return being below the expected return, or the uncertainty about the magnitude of that difference. Risk measures typically quantify the downside risk, whereas the standard deviation (an example of a deviation risk measure) measures both the upside and downside risk. Specifically, downside risk can be measured either with downside beta or by measuring lower semi-deviation. The statistic ''below-target semi-deviation'' or simply ''target semi-deviation'' (TSV) has become the industry standard. History Downside risk was first modeled by Roy (1952), who assumed that an investor's goal was to minimize his/her risk. This mean-semivariance, or downside risk, model is also known as “safety-first” technique, and only looks at the lower standard deviations of expected returns which are the potential losses. This is about the same time Harry Markowitz was developing mean-variance theory. Ev ... [...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 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] |
Standard Deviation
In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range. The standard deviation is commonly used in the determination of what constitutes an outlier and what does not. Standard deviation may be abbreviated SD or std dev, and is most commonly represented in mathematical texts and equations by the lowercase Greek alphabet, Greek letter Sigma, σ (sigma), for the population standard deviation, or the Latin script, Latin letter ''s'', for the sample standard deviation. The standard deviation of a random variable, Sample (statistics), sample, statistical population, data set, or probability distribution is the square root of its variance. (For a finite population, v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L2 Space
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line (-\infty, +\infty) is defined as follows. One may also speak of quadratic integrability over bounded intervals such as ,b/math> for a \leq b. An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable. For this to be true, the integrals of the positive and negative portions of the real part must both be finite, as well as those for the imaginary part. The vector space of (equivalence classes of) square integrable functions (with respect to Lebesgue measure) forms the L^p space with p = 2. Among the L^p spaces, the class of square integrable functions is unique in being compatible with an inner product, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Variables
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function in which * the domain is the set of possible outcomes in a sample space (e.g. the set \ which are the possible upper sides of a flipped coin heads H or tails T as the result from tossing a coin); and * the range is a measurable space (e.g. corresponding to the domain above, the range might be the set \ if say heads H mapped to -1 and T mapped to 1). Typically, the range of a random variable is a subset of the real numbers. Informally, randomness typically represents some fundamental element of chance, such as in the roll of a die; it may also represent uncertainty, such as measurement error. However, the interpretation of probability is philosophic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Portfolio (finance)
In finance, a portfolio is a collection of investments. Definition The term "portfolio" refers to any combination of financial assets such as stocks, bonds and cash. Portfolios may be held by individual investors or managed by financial professionals, hedge funds, banks and other financial institutions. It is a generally accepted principle that a portfolio is designed according to the investor's risk tolerance, time frame and investment objectives. The monetary value of each asset may influence the risk/reward ratio of the portfolio. When determining asset allocation, the aim is to maximise the expected return and minimise the risk. This is an example of a multi-objective optimization problem: many efficient solutions are available and the preferred solution must be selected by considering a tradeoff between risk and return. In particular, a portfolio A is dominated by another portfolio A' if A' has a greater expected gain and a lesser risk than A. If no portfolio dominates A ... [...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] |
Essential Infimum
In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for ''all'' elements in a set, but rather ''almost everywhere'', that is, except on a set of measure zero. While the exact definition is not immediately straightforward, intuitively the essential supremum of a function is the smallest value that is greater than or equal to the function values everywhere while ignoring what the function does at a set of points of measure zero. For example, if one takes the function f(x) that is equal to zero everywhere except at x = 0 where f(0) = 1, then the supremum of the function equals one. However, its essential supremum is zero since (under the Lebesgue measure) one can ignore what the function does at the single point where f is peculiar. The essential infimum is defined in a similar way. Definition As is ... [...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] |
Standard Deviation
In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range. The standard deviation is commonly used in the determination of what constitutes an outlier and what does not. Standard deviation may be abbreviated SD or std dev, and is most commonly represented in mathematical texts and equations by the lowercase Greek alphabet, Greek letter Sigma, σ (sigma), for the population standard deviation, or the Latin script, Latin letter ''s'', for the sample standard deviation. The standard deviation of a random variable, Sample (statistics), sample, statistical population, data set, or probability distribution is the square root of its variance. (For a finite population, v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
