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Hansen–Jagannathan Bound
Hansen–Jagannathan bound is a theorem in financial economics that says that the ratio of the standard deviation of a stochastic discount factor to its mean exceeds the Sharpe ratio attained by any portfolio. This result applies, among others, the Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is .... The Hansen-Jagannathan (H-J) bound is a type of mean-variance frontier. The main contribution is that it allows us to say something about moments of the stochastic discount factor, which is unobservable, in terms of moments of returns, which can be (in principle) observed. Specifically, given the observed Sharpe ratio (say, around 0.4), the bound tells us that the SDF must be at least just as volatile. References * * External links Hansen and Jagannathan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem
In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ''corollary'' for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formal system ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Financial Economics
Financial economics is the branch of economics characterized by a "concentration on monetary activities", in which "money of one type or another is likely to appear on ''both sides'' of a trade".William F. Sharpe"Financial Economics", in Its concern is thus the interrelation of financial variables, such as share prices, interest rates and exchange rates, as opposed to those concerning the real economy. It has two main areas of focus:Merton H. Miller, (1999). The History of Finance: An Eyewitness Account, ''Journal of Portfolio Management''. Summer 1999. asset pricing and corporate finance; the first being the perspective of providers of Financial capital, capital, i.e. investors, and the second of users of capital. It thus provides the theoretical underpinning for much of finance. The subject is concerned with "the allocation and deployment of economic resources, both spatially and across time, in an uncertain environment".See Fama and Miller (1972), ''The Theory of Finance'', ... [...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] |
Stochastic Discount Factor
The concept of the stochastic discount factor (SDF) is used in financial economics and mathematical finance. The name derives from the price of an asset being computable by "discounting" the future cash flow \tilde_i by the stochastic factor \tilde, and then taking the expectation. This definition is of fundamental importance in asset pricing. If there are ''n'' assets with initial prices p_1, \ldots, p_n at the beginning of a period and payoffs \tilde_1, \ldots, \tilde_n at the end of the period (all ''x''s are random (stochastic) variables), then SDF is any random variable \tilde satisfying :E(\tilde\tilde_i) = p_i, \text i=1,\ldots,n. The stochastic discount factor is sometimes referred to as the pricing kernel as, if the expectation E(\tilde\,\tilde_i) is written as an integral, then \tilde can be interpreted as the kernel function in an integral transform. Other names sometimes used for the SDF are the "marginal rate of substitution" (the ratio of utility of states, when ut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sharpe Ratio
In finance, the Sharpe ratio (also known as the Sharpe index, the Sharpe measure, and the reward-to-variability ratio) measures the performance of an investment such as a security or portfolio compared to a risk-free asset, after adjusting for its risk. It is defined as the difference between the returns of the investment and the risk-free return, divided by the standard deviation of the investment returns. It represents the additional amount of return that an investor receives per unit of increase in risk. It was named after William F. Sharpe, who developed it in 1966. Definition Since its revision by the original author, William Sharpe, in 1994, the '' ex-ante'' Sharpe ratio is defined as: : S_a = \frac = \frac, where R_a is the asset return, R_b is the risk-free return (such as a U.S. Treasury security). E _a-R_b/math> is the expected value of the excess of the asset return over the benchmark return, and is the standard deviation of the asset excess return. The t-sta ... [...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] |
Cauchy–Schwarz Inequality
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is considered one of the most important and widely used inequalities in mathematics. Inner products of vectors can describe finite sums (via finite-dimensional vector spaces), infinite series (via vectors in sequence spaces), and integrals (via vectors in Hilbert spaces). The inequality for sums was published by . The corresponding inequality for integrals was published by and . Schwarz gave the modern proof of the integral version. Statement of the inequality The Cauchy–Schwarz inequality states that for all vectors \mathbf and \mathbf of an inner product space where \langle \cdot, \cdot \rangle is the inner product. Examples of inner products include the real and complex dot product; see the examples in inner product. Every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Financial Economics
Financial economics is the branch of economics characterized by a "concentration on monetary activities", in which "money of one type or another is likely to appear on ''both sides'' of a trade".William F. Sharpe"Financial Economics", in Its concern is thus the interrelation of financial variables, such as share prices, interest rates and exchange rates, as opposed to those concerning the real economy. It has two main areas of focus:Merton H. Miller, (1999). The History of Finance: An Eyewitness Account, ''Journal of Portfolio Management''. Summer 1999. asset pricing and corporate finance; the first being the perspective of providers of Financial capital, capital, i.e. investors, and the second of users of capital. It thus provides the theoretical underpinning for much of finance. The subject is concerned with "the allocation and deployment of economic resources, both spatially and across time, in an uncertain environment".See Fama and Miller (1972), ''The Theory of Finance'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Financial Ratios
A financial ratio or accounting ratio states the relative magnitude of two selected numerical values taken from an enterprise's financial statements. Often used in accounting, there are many standard ratios used to try to evaluate the overall financial condition of a corporation or other organization. Financial ratios may be used by managers within a firm, by current and potential shareholders (owners) of a firm, and by a firm's creditors. Financial analysts use financial ratios to compare the strengths and weaknesses in various companies. If shares in a company are publicly listed, the market price of the shares is used in certain financial ratios. Ratios can be expressed as a decimal value, such as 0.10, or given as an equivalent percentage value, such as 10%. Some ratios are usually quoted as percentages, especially ratios that are usually or always less than 1, such as earnings yield, while others are usually quoted as decimal numbers, especially ratios that are usually mor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
