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This groundbreaking publication extends conventional ways of chance size and portfolio optimization by way of combining distributional versions with danger or functionality measures into one framework. all through those pages, the specialist authors clarify the basics of chance metrics, define new ways to portfolio optimization, and speak about numerous crucial danger measures. utilizing quite a few examples, they illustrate more than a few purposes to optimum portfolio selection and chance idea, in addition to purposes to the realm of computational finance which may be worthy to monetary engineers.
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Additional resources for Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures
Un In this way, using the relationship between the copula and the distribution function, the density of the copula can be expressed by means of the density of the random variable. This is done by applying the chain rule of differentiation, c(FY1 (y1 ), . . , FYn (yn )) = fY (y1 , . . , yn ) . fY1 (y1 ) . . 4) In this formula, the numerator contains the density of the random variable Y and on the denominator we find the density of the Y but under the assumption that components of Y are independent random variables.
FXn (xn ). In the special case of n = 2, we can say that two random variables are said to be independently distributed, if knowing the value of one random variable does not provide any information about the other random variable. 1) equals P(X ≤ −2%|Y ≤ −10%) = P(X ≤ −2%)P(Y ≤ −10%) P(Y ≤ −10%) = P(X ≤ −2%). Indeed, under the assumption of independence, the event Y ≤ −10% has no influence on the probability of the other event. 5 Covariance and Correlation There are two strongly related measures among many that are commonly used to measure how two random variables tend to move together, the covariance and the correlation.
For example, suppose that a portfolio consists of a position in two assets, asset 1 and asset 2. Then there will be a probability distribution for (1) asset 1, (2) asset 2, and (3) asset 1 and asset 2. The first two distributions are referred to as the marginal probability distributions or marginal distributions. The distribution for asset 1 and asset 2 is called the joint probability distribution. Like in the univariate case, there is a mathematical connection between the probability distribution P, the cumulative distribution function F, and the density function f of a multivariate random variable (also called a random vector) X = (X1 , .