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Mutual Fund Separation Theorem
In portfolio theory, a mutual fund separation theorem, mutual fund theorem, or separation theorem is a theorem stating that, under certain conditions, any investor's optimal portfolio can be constructed by holding each of certain mutual funds in appropriate ratios, where the number of mutual funds is smaller than the number of individual assets in the portfolio. Here a mutual fund refers to any specified benchmark portfolio of the available assets. There are two advantages of having a mutual fund theorem. First, if the relevant conditions are met, it may be easier (or lower in transactions costs) for an investor to purchase a smaller number of mutual funds than to purchase a larger number of assets individually. Second, from a theoretical and empirical standpoint, if it can be assumed that the relevant conditions are indeed satisfied, then implications for the functioning of asset markets can be derived and tested. Portfolio separation in mean-variance analysis Portfolios can b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modern Portfolio Theory
Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of diversification in investing, the idea that owning different kinds of financial assets is less risky than owning only one type. Its key insight is that an asset's risk and return should not be assessed by itself, but by how it contributes to a portfolio's overall risk and return. It uses the variance of asset prices as a proxy for risk. Economist Harry Markowitz introduced MPT in a 1952 essay, for which he was later awarded a Nobel Memorial Prize in Economic Sciences; see Markowitz model. Mathematical model Risk and expected return MPT assumes that investors are risk averse, meaning that given two portfolios that offer the same expected return, investors will prefer the less risky one. Thus, an investor will take on increased risk only if compen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance Matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of a given random vector. Any covariance matrix is symmetric and positive semi-definite and its main diagonal contains variances (i.e., the covariance of each element with itself). Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions. As an example, the variation in a collection of random points in two-dimensional space cannot be characterized fully by a single number, nor would the variances in the x and y directions contain all of the necessary information; a 2 \times 2 matrix would be necessary to fully characterize the two-dimensional variation. The covariance matrix of a random vector \mathbf is typically denoted by \operatorname_ or \Sigma. Definition Throughout this article, boldfaced unsub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Portfolio Theories
Portfolio may refer to: Objects * Portfolio (briefcase), a type of briefcase Collections * Portfolio (finance), a collection of assets held by an institution or a private individual * Artist's portfolio, a sample of an artist's work or a case used to display artwork, photographs etc. * Career portfolio, an organized presentation of an individual's education, work samples, and skills * Electronic portfolio, a collection of electronic documents * IT portfolio, in IT portfolio management, the portfolio of large classes of items of enterprise Information Technology * Patent portfolio, a collection of patents owned by a single entity * Project portfolio, in project portfolio management, the portfolio of projects in an organization * Ministry (government department), the post and responsibilities of a head of a government department Computing * Atari Portfolio, a palmtop computer * Extensis Portfolio, a digital asset manager Media * '' The Portfolio'', a British fine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Stiglitz
Joseph Eugene Stiglitz (; born February 9, 1943) is an American New Keynesian economist, a public policy analyst, and a full professor at Columbia University. He is a recipient of the Nobel Memorial Prize in Economic Sciences (2001) and the John Bates Clark Medal (1979). He is a former senior vice president and chief economist of the World Bank. He is also a former member and chairman of the (US president's) Council of Economic Advisers. He is known for his support of Georgist public finance theory and for his critical view of the management of globalization, of ''laissez-faire'' economists (whom he calls " free-market fundamentalists"), and of international institutions such as the International Monetary Fund and the World Bank. In 2000, Stiglitz founded the Initiative for Policy Dialogue (IPD), a think tank on international development based at Columbia University. He has been a member of the Columbia faculty since 2001, and received the university's highest academic rank ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Cass
David Cass (January 19, 1937 – April 15, 2008) was a professor of economics at the University of Pennsylvania, mostly known for his contributions to general equilibrium theory. His most famous work was on the Ramsey–Cass–Koopmans model of economic growth. Biography David Cass was born in 1937 in Honolulu, Hawaii. He earned an A.B. in economics from the University of Oregon in 1958 and started to study law at the Harvard Law School as he thought of becoming a lawyer according to family tradition. As he hated studying law he left the program after one year and served in the army from 1959 to 1960. He then entered the economics Ph.D. program at Stanford University. Here he met Karl Shell, although the two began to work together only after both graduated. Cass's doctoral advisor was Hirofumi Uzawa, who also introduced him to Tjalling Koopmans, who at that time was a professor at Yale University. In 1965, Cass graduated with a Ph.D. in economics and statistics with a dissertati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Utility
In economics and finance, exponential utility is a specific form of the utility function, used in some contexts because of its convenience when risk (sometimes referred to as uncertainty) is present, in which case expected utility is maximized. Formally, exponential utility is given by: :u(c) = \begin (1-e^)/a & a \neq 0 \\ c & a = 0 \\ \end c is a variable that the economic decision-maker prefers more of, such as consumption, and a is a constant that represents the degree of risk preference (a>0 for risk aversion, a=0 for risk-neutrality, or a of final wealth ''W'' subject to :W = x'r + (W_0 - x'k) \cdot r_f where the prime sign indicates a vector transpose and where W_0 is initial wealth, ''x'' is a column vector of quantities placed in the ''n'' risky assets, ''r'' is a random vector of stochastic returns on the ''n'' assets, ''k'' is a vector of ones (so W_0 - x'k is the quantity placed in the risk-free asset), and ''r''''f'' is the known scalar return on the risk-free ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Function
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of is , or . The logarithm of to ''base'' is denoted as , or without parentheses, , or even without the explicit base, , when no confusion is possible, or when the base does not matter such as in big O notation. The logarithm base is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number as its base; its use is widespread in mathematics and physics, because of its very simple derivative. The binary logarithm uses base and is frequently used in computer science. Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators, scientists, engineers, surveyors and others to perform high- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Utility Function
In economics, the isoelastic function for utility, also known as the isoelastic utility function, or power utility function, is used to express utility in terms of consumption or some other economic variable that a decision-maker is concerned with. The isoelastic utility function is a special case of hyperbolic absolute risk aversion and at the same time is the only class of utility functions with constant relative risk aversion, which is why it is also called the CRRA utility function. It is : u(c) = \begin \frac & \eta \ge 0, \eta \neq 1 \\ \ln(c) & \eta = 1 \end where c is consumption, u(c) the associated utility, and \eta is a constant that is positive for risk averse agents. Since additive constant terms in objective functions do not affect optimal decisions, the term –1 in the numerator can be, and usually is, omitted (except when establishing the limiting case of \ln(c) as below). When the context involves risk, the utility function is viewed as a von Neumann–Morge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Absolute Risk Aversion
In finance, economics, and decision theory, hyperbolic absolute risk aversion (HARA) (Chapter I of his Ph.D. dissertation; Chapter 5 in his ''Continuous-Time Finance'').Ljungqvist & Sargent, Recursive Macroeconomic Theory, MIT Press, Second Edition refers to a type of risk aversion that is particularly convenient to model mathematically and to obtain empirical predictions from. It refers specifically to a property of von Neumann–Morgenstern utility functions, which are typically functions of final wealth (or some related variable), and which describe a decision-maker's degree of satisfaction with the outcome for wealth. The final outcome for wealth is affected both by random variables and by decisions. Decision-makers are assumed to make their decisions (such as, for example, portfolio allocations) so as to maximize the expected value of the utility function. Notable special cases of HARA utility functions include the quadratic utility function, the exponential utility funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Risk-free Interest Rate
The risk-free rate of return, usually shortened to the risk-free rate, is the rate of return of a hypothetical investment with scheduled payments over a fixed period of time that is assumed to meet all payment obligations. Since the risk-free rate can be obtained with no risk, any other investment having some risk will have to have a higher rate of return in order to induce any investors to hold it. In practice, to infer the risk-free interest rate in a particular currency, market participants often choose the yield to maturity on a risk-free bond issued by a government of the same currency whose risks of default are so low as to be negligible. For example, the rate of return on T-bills is sometimes seen as the risk-free rate of return in US dollars. Theoretical measurement As stated by Malcolm Kemp in chapter five of his book ''Market Consistency: Model Calibration in Imperfect Markets'', the risk-free rate means different things to different people and there is no consensus on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Condition
In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about the concavity of a function. The usefulness of derivatives to find extrema is proved mathematically by Fermat's theorem of stationary points. First-derivative test The first-derivative test examines a function's monotonic properties (where the function is increasing or decreasing), focusing on a particular point in its domain. If the function "switches" from increasing to decreasing at the point, then the function will achieve a highest value at that point. Similarly, if the function "switches" from decreasing to increasing at the point, then it will achieve a least value at that point. If the function fails to "switch" and remains increasing or remains decreasing, then no highest or least value is achieved. One can examine a func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Calculus
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities. This greatly simplifies operations such as finding the maximum or minimum of a multivariate function and solving systems of differential equations. The notation used here is commonly used in statistics and engineering, while the tensor index notation is preferred in physics. Two competing notational conventions split the field of matrix calculus into two separate groups. The two groups can be distinguished by whether they write the derivative of a scalar with respect to a vector as a column vector or a row vector. Both of these conventions are possible even when the common assumption is made that vectors shoul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
