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is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives <span>a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate)

Black–Scholes model - Wikipedia Black–Scholes model From Wikipedia, the free encyclopedia (Redirected from Black–Scholes) Jump to: navigation, search The Black–Scholes /ˌblæk ˈʃoʊlz/ [1] or Black–Scholes–Merton model is a mathematical model of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead replacing the security's expected return with the risk-neutral rate). The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. [2]

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In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to th

h For the martingale betting strategy, see martingale (betting system). [imagelink] Stopped Brownian motion is an example of a martingale. It can model an even coin-toss betting game with the possibility of bankruptcy. <span>In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values. Contents [hide] 1 History 2 Definitions 2.1 Martingale sequences with respect to another sequence 2.2 General definition 3 Examples of martingales 4 Submartingales, super

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Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.

analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that

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span> Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the <span>linear functions defined on these spaces and respecting these structures in a suitable sense. <span><body><html>

analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that

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A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

company, see Vector Space Systems. [imagelink] Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w. <span>A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of v

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In mathematics, a Fourier series ( English: / ˈ f ʊər i ˌ eɪ / ) [1] is a way to represent a function as the sum of simple sine waves.

are wave [imagelink] Fourier transforms Continuous Fourier transform Fourier series Discrete-time Fourier transform Discrete Fourier transform Discrete Fourier transform over a ring Fourier analysis Related transforms <span>In mathematics, a Fourier series ( English: /ˈfʊəriˌeɪ/) [1] is a way to represent a function as the sum of simple sine waves. More formally, it decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or, equivalently, complex exponentials). The discrete-time Fourier transform is a periodic function, often defined in terms of a Fourier series. The Z-transform, another example of application, reduces to a Fourier series for

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It is also not happenstance that the downfall of the disputational culture roughly coincided with the introduction of new printing techniques in Europe by Johannes Gutenberg, around 1440.

ich is thoroughly disputational, with Meditations on First Philosophy (1641) by Descartes, a book argued through long paragraphs driven by the first-person singular. The nature of intellectual enquiry shifted with the downfall of disputation. <span>It is also not happenstance that the downfall of the disputational culture roughly coincided with the introduction of new printing techniques in Europe by Johannes Gutenberg, around 1440. Before that, books were a rare commodity, and education was conducted almost exclusively by means of oral contact between masters and pupils in the form of expository lectures in which

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Instead, early modern authors emphasise the role of novelty and individual discovery

tually unthinkable before the wide availability of printed books) was well-established. Moreover, as indicated by the passage from Descartes above, the very term ‘logic’ came to be used for something other than what the scholastics had meant. <span>Instead, early modern authors emphasise the role of novelty and individual discovery, as exemplified by the influential textbook Port-Royal Logic (1662), essentially, the logical version of Cartesianism, based on Descartes’s conception of mental operations and the prima

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the Wiener process or Brownian motion process, [a] used by Louis Bachelier to study price changes on the Paris Bourse

arkets have motivated the extensive use of stochastic processes in finance. [16] [17] [18] Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include <span>the Wiener process or Brownian motion process, [a] used by Louis Bachelier to study price changes on the Paris Bourse, [21] and the Poisson process, used by A. K. Erlang to study the number of phone calls occurring in a certain period of time. [22] These two stochastic processes are considered the mo

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If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead.

hangeably, often with no specific mathematical space for the set that indexes the random variables. [27] [29] But often these two terms are used when the random variables are indexed by the integers or an interval of the real line. [5] [29] <span>If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead. [5] [30] The values of a stochastic process are not always numbers and can be vectors or other mathematical objects. [5] [28] Based on their properties, stochastic processes can be d

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One of the simplest stochastic processes is the Bernoulli process, [60] which is a sequence of independent and identically distributed (iid) Bernoulli variables.

} -dimensional vector process or n {\displaystyle n} -vector process. [51] [52] Examples of stochastic processes[edit source] Bernoulli process[edit source] Main article: Bernoulli process <span>One of the simplest stochastic processes is the Bernoulli process, [60] which is a sequence of independent and identically distributed (iid) random variables, where each random variable takes either the value one or zero, say one with probability p {\displaystyle p} and zero with probability 1 − p {\displaystyle 1-p} . This process can be likened to somebody flipping a coin, where the probability of obtaining a head is p {\displaystyle p} and its value is on

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Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple random walk.

stant μ {\displaystyle \mu } , which is a real number, then the resulting stochastic process is said to have drift μ {\displaystyle \mu } . [84] [85] [86] <span>Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered a continuous version of the simple random walk. [49] [85] The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled, [87] [88] which is the subject of Donsker's theorem or inva

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The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process.

constant, then the process is called a homogeneous Poisson process. [99] [101] The homogeneous Poisson process (in continuous time) is a member of important classes of stochastic processes such as Markov processes and Lévy processes. [49] <span>The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. [102] [103] If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t {\displaystyle t} ,

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If the parameter constant of the Poisson process is replaced with some non-negative integrable function of , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant.

sses. [49] The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. [102] [103] <span>If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t {\displaystyle t} , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. [104] Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randoml

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body> If the parameter constant of the Poisson process is replaced with some non-negative integrable function of , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. <body><html>

sses. [49] The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. [102] [103] <span>If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t {\displaystyle t} , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. [104] Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randoml

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In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1 AP is a diagonal matrix.

Diagonalizable matrix From Wikipedia, the free encyclopedia Jump to: navigation, search This article is about matrix diagonalisation in linear algebra. For other uses, see Diagonalisation. <span>In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1 AP is a diagonal matrix. If V is a finite-dimensional vector space, then a linear map T : V → V is called diagonalizable if there exists an ordered basis of V with respect to which T is represented by a diagona

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In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms.

Module-like[show] Module Group with operators Vector space Linear algebra Algebra-like[show] Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra v t e <span>In mathematics, and more specifically in abstract algebra, an algebraic structure is a set (called carrier set or underlying set) with one or more operations defined on it that satisfies a list of axioms. [1] Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets,

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In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × V → K , where K is the field of scalars.

Bilinear form - Wikipedia Bilinear form From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map V × V → K, where K is the field of scalars. In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v) The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When K is the field of complex numbers C, one

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The forward-backward algorithm In the first pass, the forward–backward algorithm computes a set of forward probabilities which provide, for all , the probability of ending up in any particular state given the first observations in the sequence, i.e. . In the second pass, the algorithm computes a set of backward probabilities which provide the probability of observing the remaining observations given any starting point , i.e. . Thes

cific instance of this class. Contents [hide] 1 Overview 2 Forward probabilities 3 Backward probabilities 4 Example 5 Performance 6 Pseudocode 7 Python example 8 See also 9 References 10 External links Overview[edit source] <span>In the first pass, the forward–backward algorithm computes a set of forward probabilities which provide, for all k ∈ { 1 , … , t } {\displaystyle k\in \{1,\dots ,t\}} , the probability of ending up in any particular state given the first k {\displaystyle k} observations in the sequence, i.e. P ( X k | o 1 : k ) {\displaystyle P(X_{k}\ |\ o_{1:k})} . In the second pass, the algorithm computes a set of backward probabilities which provide the probability of observing the remaining observations given any starting point k {\displaystyle k} , i.e. P ( o k + 1 : t | X k ) {\displaystyle P(o_{k+1:t}\ |\ X_{k})} . These two sets of probability distributions can then be combined to obtain the distribution over states at any specific point in time given the entire observation sequence: P ( X k | o 1 : t ) = P ( X k | o 1 : k , o k + 1 : t ) ∝ P ( o k + 1 : t | X k ) P ( X k | o 1 : k ) {\displaystyle P(X_{k}\ |\ o_{1:t})=P(X_{k}\ |\ o_{1:k},o_{k+1:t})\propto P(o_{k+1:t}\ |\ X_{k})P(X_{k}|o_{1:k})} The last step follows from an application of the Bayes' rule and the conditional independence of o k + 1 : t {\displaystyle o_{k+1:t}} and o 1 : k {\displaystyle o_{1:k}} given X k {\displaystyle X_{k}} . As outlined above, the algorithm involves three steps: computing forward probabilities computing backward probabilities computing smoothed values. The forward and backward steps m

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rward probabilities which provide, for all , the probability of ending up in any particular state given the first observations in the sequence, i.e. . In the second pass, the algorithm computes a set of backward probabilities which provide <span>the probability of observing the remaining observations given any starting point , i.e. . These two sets of probability distributions can then be combined to obtain the distribution over states at any specific point in time given the entire observation sequence:

cific instance of this class. Contents [hide] 1 Overview 2 Forward probabilities 3 Backward probabilities 4 Example 5 Performance 6 Pseudocode 7 Python example 8 See also 9 References 10 External links Overview[edit source] <span>In the first pass, the forward–backward algorithm computes a set of forward probabilities which provide, for all k ∈ { 1 , … , t } {\displaystyle k\in \{1,\dots ,t\}} , the probability of ending up in any particular state given the first k {\displaystyle k} observations in the sequence, i.e. P ( X k | o 1 : k ) {\displaystyle P(X_{k}\ |\ o_{1:k})} . In the second pass, the algorithm computes a set of backward probabilities which provide the probability of observing the remaining observations given any starting point k {\displaystyle k} , i.e. P ( o k + 1 : t | X k ) {\displaystyle P(o_{k+1:t}\ |\ X_{k})} . These two sets of probability distributions can then be combined to obtain the distribution over states at any specific point in time given the entire observation sequence: P ( X k | o 1 : t ) = P ( X k | o 1 : k , o k + 1 : t ) ∝ P ( o k + 1 : t | X k ) P ( X k | o 1 : k ) {\displaystyle P(X_{k}\ |\ o_{1:t})=P(X_{k}\ |\ o_{1:k},o_{k+1:t})\propto P(o_{k+1:t}\ |\ X_{k})P(X_{k}|o_{1:k})} The last step follows from an application of the Bayes' rule and the conditional independence of o k + 1 : t {\displaystyle o_{k+1:t}} and o 1 : k {\displaystyle o_{1:k}} given X k {\displaystyle X_{k}} . As outlined above, the algorithm involves three steps: computing forward probabilities computing backward probabilities computing smoothed values. The forward and backward steps m

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Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive:

[imagelink] Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0. <span>In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space R n . For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word – specifically, 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. [1] Indeed, their existence is a non-trivial consequence of the axiom of choice. Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others. The ma

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A crucial development in the history of money was the promissory note . The process began when individuals began leaving their excess gold with goldsmiths, who would look after it for them. In turn the goldsmiths would give the depositors a receipt, stating

3; acts as a medium of exchange; provides individuals with a way of storing wealth; and provides society with a convenient measure of value and unit of account. <span>2.1.2. Paper Money and the Money Creation Process Although precious metals like gold and silver fulfilled the required functions of money relatively well for many years, and although carrying gold coins around was easier than carrying around one’s physical produce, it was not necessarily a safe way to conduct business. A crucial development in the history of money was the promissory note . The process began when individuals began leaving their excess gold with goldsmiths, who would look after it for them. In turn the goldsmiths would give the depositors a receipt, stating how much gold they had deposited. Eventually these receipts were traded directly for goods and services, rather than there being a physical transfer of gold from the goods buyer to the goods seller. Of course, both the buyer and seller had to trust the goldsmith because the goldsmith had all the gold and the goldsmith’s customers had only pieces of paper. These depository receipts represented a promise to pay a certain amount of gold on demand. This paper money therefore became a proxy for the precious metals on which they were based, that is, they were directly related to a physical commodity. Many of these early goldsmiths evolved into banks, taking in excess wealth and in turn issuing promissory notes that could be used in commerce. In taking in other people’s gold and issuing depository receipts and later promissory notes, it became clear to the goldsmiths and early banks that not all the gold that they held in their vaults would be withdrawn at any one time. Individuals were willing to buy and sell goods and services with the promissory notes, but the majority of the gold that backed the notes just sat in the vaults—although its ownership would change with the flow of commerce over time. A certain proportion of the gold that was not being withdrawn and used directly for commerce could therefore be lent to others at a rate of interest. By doing this, the early banks created money. The process of money creation is a crucial concept for understanding the role that money plays in an economy. Its potency depends on the amount of money that banks keep in reserve to meet the withdrawals of its customers. This practice of lending customers’ money to others on the assumption that not all customers will want all of their money back at any one time is known as fractional reserve banking . We can illustrate how it works through a simple example. Suppose that the bankers in an economy come to the view that they need to retain only 10 percent of any money deposited with them. This is known as the reserve requirement .2 Now consider what happens when a customer deposits €100 in the First Bank of Nations. This deposit changes the balance sheet of First Bank of Nations, as shown in Exhibit 2, and it represents a liability to the bank because it is effectively loaned to the bank by the customer. By lending 90 percent of this deposit to another customer the bank has two types of assets: (1) the bank’s reserves of €10, and (2) the loan equivalent to €90. Notice that the balance sheet still balances; €100 worth of assets and €100 worth of liabilities are on the balance sheet. Now suppose that the recipient of the loan of €90 uses this money to purchase some goods of this value and the seller of the goods deposits this €90 in another bank, the Second Bank of Nations. The Second Bank of Nations goes through the same process; it retains €9 in reserve and loans 90 percent of the deposit (€81) to another customer. This customer in turn spends €81 on some goods or services. The recipient of this money deposits it at the Third Bank of Nations, and so on. This example shows how money is created when a bank makes a loan. Exhibit 2. Money Creation via Fractional Reserve Banking First Bank of Nations Assets Liabilities Reserves €10 Deposits €100 Loans €90 Second Bank of Nations Assets Liabilities Reserves €9 Deposits €90 Loans €81 Third Bank of Nations Assets Liabilities Reserves €8.1 Deposits €81 Loans €72.9 This process continues until there is no more money left to be deposited and loaned out. The total amount of money ‘created’ from this one deposit of €100 can be calculated as: Equation (1) New deposit/Reserve requirement = €100/0.10 = €1,000 It is the sum of all the deposits now in the banking system. You should also note that the original deposit of €100, via the practice of reserve banking, was the catalyst for €1,000 worth of economic transactions. That is not to say that economic growth would be zero without this process, but instead that it can be an important component in economic activity. The amount of money that the banking system creates through the practice of fractional reserve banking is a function of 1 divided by the reserve requirement, a quantity known as the money multiplier .3 In the case just examined, the money multiplier is 1/0.10 = 10. Equation 1 implies that the smaller the reserve requirement, the greater the money multiplier effect. In our simplistic example, we assumed that the banks themselves set their own reserve requirements. However, in some economies, the central bank sets the reserve requirement, which is a potential means of affecting money growth. In any case, a prudent bank would be wise to have sufficient reserves such that the withdrawal demands of their depositors can be met in stressful economic and credit market conditions. Later, when we discuss central banks and central bank policy, we will see how central banks can use the mechanism just described to affect the money supply. Specifically, the central bank could, by purchasing €100 in government securities credited to the bank account of the seller, seek to initiate an increase in the money supply. The central bank may also lend reserves directly to banks, creating excess reserves (relative to any imposed or self-imposed reserve requirement) that can support new loans and money expansion. 2.1.3. Definitions of Money The process of money creation raises a fundamental issue: What is money? In an economy with money but without promisso

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The money multiplier is the amount by which a change in the monetary base is multiplied to calculate the final change in the money supply. Money Multiplier = 1/b, where b is the required reserve ratio. In

liquid of all assets, due to its function as the medium of exchange. However, many methods of holding money do not yield an interest return and the purchase power of money will decline during a time of inflation. <span>The Money Creation Process Reserves are the cash in a bank's vault and deposits at Federal Reserve Banks. Under the fractional reserve banking system, a bank is obligated to hold a minimum amount of reserves to back up its deposits. Reserves held for that purpose, which are expressed as a percentage of a bank's demand deposits, are called required reserves. Therefore, the required reserve ratio is the percentage of a bank's deposits that are required to be held as reserves. Banks create deposits when they make loans; the new deposits created are new money. Example Suppose the required reserve ratio in the U.S. is 20%, and then suppose that you deposit $1,000 cash with Citibank. Citibank keeps $200 of the $1,000 in reserves. The remaining $800 of excess reserves can be loaned out to, say, John. After the loan is made, the money supply increases by $800 (your $1,000 + John's $800). After getting the loan, John deposits the $800 with Bank of America (BOA). BOA keeps $160 of the $800 in reserves and can now loan out $640 to another person. Thus, BOA creates $640 of money supply. The process goes on and on. With each deposit and loan, more money is created. However, the money creation process does not create an infinite amount of money. The money multiplier is the amount by which a change in the monetary base is multiplied to calculate the final change in the money supply. Money Multiplier = 1/b, where b is the required reserve ratio. In our example, b is 0.2, so money multiplier = 1/0.2 = 5. Definitions of Money There are different definitions of money. The two most widely used measures of money in the U.S. are: The M1 Money Su

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Violations of moral principles can harm the community in a variety of ways.

Ethical conduct is behavior that follows moral principles. Ethical actions are those actions that are perceived as beneficial and conform to the ethical expectations of society. <span>Ethics encompass a set of moral principles ( code of ethics ) and standards of conduct that provide guidance for our behavior. Violations can harm the community in a variety of ways. <span><body><html>

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Demand and supply analysis is the study of how buyers and sellers interact to determine transaction prices and quantities.

croeconomics has its roots in microeconomics , which deals with markets and decision making of individual economic units, including consumers and businesses. Microeconomics is a logical starting point for the study of economics. <span>This reading focuses on a fundamental subject in microeconomics: demand and supply analysis. Demand and supply analysis is the study of how buyers and sellers interact to determine transaction prices and quantities. As we will see, prices simultaneously reflect both the value to the buyer of the next (or marginal) unit and the cost to the seller of that unit. In private enterprise market economies, which are the chief concern of investment analysts, demand and supply analysis encompasses the most basic set of microeconomic tools. Traditionally, microeconomics classifies private economic units into two groups: consumers (or households) and firms. These two groups give rise, respectively, to the theor

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Examples of long-lived financial assets include investments in equity or debt securities issued by other companies.

Long-lived assets , also referred to as non-current assets or long-term assets, are assets that are expected to provide economic benefits over a future period of time, typically greater than one year.1 Long-lived assets may be tangible, intangible, or financial assets. Examples of long-lived tangible assets, typically referred to as property, plant, and equipment and sometimes as fixed assets, include land, buildings, furniture and fixtures, machinery and equipment, and vehicles; examples of long-lived intangible assets (assets lacking physical substance) include patents and trademarks; and examples of long-lived financial assets include investments in equity or debt securities issued by other companies. The scope of this reading is limited to long-lived tangible and intangible assets (hereafter, referred to for simplicity as long-lived assets). The first issue in accounting for a long-lived asset is determining its cost at acquisition. The second issue is how to allocate the cost to expense over time. The costs of most long-lived assets are capitalised and then allocated as expenses in the profit or loss (income) statement over the period of time during which they are expected to provide economic benefits. The two main types of long-lived assets with costs that are typically not allocated over time are land, which is not depreciated, and those intangible assets with indefinite useful lives. Additional issues that arise are the treatment of subsequent costs incurred related to the asset, the use of the cost model versus the revaluation model, unexpected declines in the value of the asset, classification of the asset with respect to intent (for example, held for use or held for sale), and the derecognition of the asset. This reading is organised as follows. Section 2 describes and illustrates accounting for the acquisition of long-lived assets, with particular attention to the impact of ca

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Measured by daily turnover, the foreign exchange (FX) market—the market in which currencies are traded against each other—is by far the world’s largest market.

Measured by daily turnover, the foreign exchange (FX) market—the market in which currencies are traded against each other—is by far the world’s largest market. Current estimates put daily turnover at approximately USD4 trillion for 2010. This is about 10 to 15 times larger than daily turnover in global fixed-income markets and about 50 times larger than global turnover in equities. Moreover, volumes in FX turnover continue to grow: Some predict that daily FX turnover will reach USD10 trillion by 2020 as market participation spreads and deepens. The FX market is also a truly global market that operates 24 hours a day, each business day. It involves market participants from every time zone connected through electronic communications networks that link players as large as multibillion-dollar investment funds and as small as individuals trading for their own account—all brought together in real time. International trade would be impossible without the trade in currencies that facilitates it, and so too would cross-border capital flows that connect all financial markets globally through the FX market. These factors make foreign exchange a key market for investors and market participants to understand. The world economy is increasingly transnational in nature, with both production processes and trade flows often determined more by global factors than by domestic considerations. Likewise, investment portfolio performance increasingly reflects global determinants because pricing in financial markets responds to the array of investment opportunities available worldwide, not just locally. All of these factors funnel through, and are reflected in, the foreign exchange market. As investors shed their “home bias” and invest in foreign markets, the exchange rate—the price at which foreign-currency-denominated investments are valued in terms of the domestic currency—becomes an increasingly important determinant of portfolio performance. Even investors adhering to a purely “domestic” portfolio mandate are increasingly affected by what happens in the foreign exchange market. Given the globalization of the world economy, most large companies depend heavily on their foreign operations (for example, by some estimates about 40 percent of S&P 500 Index earnings are from outside the United States). Almost all companies are exposed to some degree of foreign competition, and the pricing for domestic assets—equities, bonds, real estate, and others—will also depend on demand from foreign investors. All of these various influences on investment performance reflect developments in the foreign exchange market. This reading introduces the foreign exchange market, providing the basic concepts and terminology necessary to understand exchange rates as well as some of the basics of ex

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3.1.4 Board of Directors A company’s board of directors is elected by shareholders to protect shareholders’ interests, provide strategic direction, and monitor company and management performance.

mally compensated through salary, bonuses, equity based remuneration (or compensation). As a result, managers may be motivated to maximize the value of their total remuneration while also protecting their employment positions. <span>3.1.4 Board of Directors A company’s board of directors is elected by shareholders to protect shareholders’ interests, provide strategic direction, and monitor company and management performance. 3.1.5 Customers Customers expect a company’s products or services to satisfy their needs and provide appropriate benefits given the price paid, as well as to meet a

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Corporate governance provides a framework that defines the rights, roles and responsibilities of various groups within an organization.

Corporate governance can be defined as: “the system of internal controls and procedures by which individual companies are managed. It provides a framework that defines the rights, roles and responsibilities of various groups . . . within an organization. At its core, corporate governance is the arrangement of checks, balances, and incentives a company needs in order to minimize and manage the conflicting interests between insiders and external shareowners.”

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At its core, corporate governance is the arrangement of checks, balances, and incentives a company needs in order to minimize and manage the conflicting interests between insiders and external shareowners.

Corporate governance can be defined as: “the system of internal controls and procedures by which individual companies are managed. It provides a framework that defines the rights, roles and responsibilities of various groups . . . within an organization. At its core, corporate governance is the arrangement of checks, balances, and incentives a company needs in order to minimize and manage the conflicting interests between insiders and external shareowners.”

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: navigation, search [imagelink] A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications to science and engineering. <span>Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions. [1] [2] These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of a

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It is not likely you will become proficient in High Probability Selling merely by reading about it. However, there are people who do. They read it frequently. Learning High Probability Selling usually requires many hours of rigorous interactive training and a desire to enjoy your work and be very good at what you do.

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span> It is not likely you will become proficient in High Probability Selling merely by reading about it. However, there are people who do. They read it frequently. Learning High Probability Selling usually requires many hours of rigorous interactive training and a desire to enjoy your work and be very good at what you do. <span><body><html>

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become proficient in High Probability Selling merely by reading about it. However, there are people who do. They read it frequently. Learning High Probability Selling usually requires many hours of rigorous interactive training and <span>a desire to enjoy your work and be very good at what you do. <span><body><html>

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What is it that the armed forces and salespeople have in common that requires them to be the largest users of motivational training? How many carpenters, mechanics, CPA's, claims adjusters or veterinarians need to attend motivatio

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formation between a coffee mug and a donut (torus) illustrating that they are homeomorphic. But there need not be a continuous deformation for two spaces to be homeomorphic — only a continuous mapping with a continuous inverse function. <span>In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος (homoios) = similar or same (French pareil) and μορφή (morphē) = shape, form, introduced to mathematics by Henri Poincaré in 189

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High Probability Selling trains salespeople how to discover whether there is a mutually acceptable basis for doing business – without using manipulative techniques.

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tive · Surjective · Bijective Constructions Restriction · Composition · λ · Inverse Generalizations Partial · Multivalued · Implicit v t e In mathematics, a function space is a set of functions between two fixed sets. <span>Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space have a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function space. Contents [hide] 1 In linear algebra 2 Examples 3 Functional analysis 4 Norm 5 Bibliography 6 See also 7 Footnotes In linear algebra[edit source] See also: Vector spac

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edia, the free encyclopedia Jump to: navigation, search [imagelink] The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis. <span>Real analysis (traditionally, the theory of functions of a real variable) is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, smoothness and related properties of real-valued functions. Contents [hide] 1 Scope 1.1 Construction of the real numbers 1.2 Order properties of the real numbers 1.3 Sequences 1.4 Limits and convergence 1.5 Continuity 1.5.1 Uniform

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Banach space - Wikipedia Banach space From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, more specifically in functional analysis, a Banach space (pronounced [ˈbanax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. [1]

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ce Cox's theorem never uses any fancy set theory to derive the basic rules of probability, this problem is avoided altogether. If I remember correctly, however, this benefit has some unfortunate drawbacks, and is kind of a double-edged sword--<span>I vaguely recall reading that probability as derived from Cox's theorem only possesses finite additivity, but not countable additivity, whereas the measure-theoretic derivation possesses both. So to conclude, the Kolmogorov axioms and Cox's theorem represent completely distinct ways of building up probability theory. Yes, measure theory can be used as a tool given the Baye

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es of functions and sequences.However, it is also possible to study these objects in finite dimensions, and this way one can use geometry to understand the difference between Hilbert spaces and (non Hilbert) Banach spaces. I would argue that <span>the difference between Hilbert spaces and (non Hilbert) Banach spaces is fundamentally geometric. Mathematicians have abstracted this geometry to a considerable degree, but the basic intuition can be reduced to pictures.Ok, so my approach to this question will be to consider X:=ℝ2X

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2 {\displaystyle BD^{2}+AC^{2}=2a^{2}+2b^{2}} Q.E.D. The parallelogram law in inner product spaces[edit source] [imagelink] Vectors involved in the parallelogram law. <span>In a normed space, the statement of the parallelogram law is an equation relating norms: 2 ‖ x ‖ 2 + 2 ‖ y ‖

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In a normed space, the statement of the parallelogram law is an equation relating norms:

2 {\displaystyle BD^{2}+AC^{2}=2a^{2}+2b^{2}} Q.E.D. The parallelogram law in inner product spaces[edit source] [imagelink] Vectors involved in the parallelogram law. <span>In a normed space, the statement of the parallelogram law is an equation relating norms: 2 ‖ x ‖ 2 + 2 ‖ y ‖

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a Banach space is a complete normed vector space. Normed: with a metric that allows the computation of vector length and distance between vectors Complete: a Cauchy sequence of vectors always converges to a well defined limit tha

Banach space - Wikipedia Banach space From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, more specifically in functional analysis, a Banach space (pronounced [ˈbanax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. [1]

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a Banach space is a complete normed vector space.

Banach space - Wikipedia Banach space From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, more specifically in functional analysis, a Banach space (pronounced [ˈbanax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. [1]

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a Banach space is a complete normed vector space. Normed: with a metric that allows the computation of vector length and distance between vectors Complete: a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

Banach space - Wikipedia Banach space From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, more specifically in functional analysis, a Banach space (pronounced [ˈbanax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. [1]

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A normed space has a metric that allows the computation of vector length and distance between vectors

Banach space - Wikipedia Banach space From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, more specifically in functional analysis, a Banach space (pronounced [ˈbanax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. [1]

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A normed space has a metric that allows the computation of vector length and distance between vectors

Banach space - Wikipedia Banach space From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, more specifically in functional analysis, a Banach space (pronounced [ˈbanax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. [1]

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A normed space has a metric that allows the computation of vector length and distance between vectors

Banach space - Wikipedia Banach space From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, more specifically in functional analysis, a Banach space (pronounced [ˈbanax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. [1]

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The scope of working capital management includes transactions, relations, analyses, and focus: Transactions include payments for trade, financing, and investment. Relations with financial institutions and trading partners must be maintained t

and collecting on this credit, managing inventory, and managing payables. Effective working capital management also requires reliable cash forecasts, as well as current and accurate information on transactions and bank balances. <span>Both internal and external factors influence working capital needs; we summarize them in Exhibit 1. Exhibit 1. Internal and External Factors That Affect Working Capital Needs Internal Factors External Factors Company size and growth rates Organizational structure Sophistication of working capital management Borrowing and investing positions/activities/capacities Banking services Interest rates New technologies and new products The economy Competitors The scope of working capital management includes transactions, relations, analyses, and focus: Transactions include payments for trade, financing, and investment. Relations with financial institutions and trading partners must be maintained to ensure that the transactions work effectively. Analyses of working capital management activities are required so that appropriate strategies can be formulated and implemented. Focus requires that organizations of all sizes today must have a global viewpoint with strong emphasis on liquidity. In this reading, we examine the different types of working capital and the management issues associated with each. We also look at methods of evaluating the effectiveness of working capital management. <span><body><html>

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The scope of working capital management includes transactions, relations, analyses, and focus: Transactions include payments for trade, financing, and investment. Relations with financial institutions and trading partners must be maintained to ensure that the transactions work effectively. Analyses of working capit

and collecting on this credit, managing inventory, and managing payables. Effective working capital management also requires reliable cash forecasts, as well as current and accurate information on transactions and bank balances. <span>Both internal and external factors influence working capital needs; we summarize them in Exhibit 1. Exhibit 1. Internal and External Factors That Affect Working Capital Needs Internal Factors External Factors Company size and growth rates Organizational structure Sophistication of working capital management Borrowing and investing positions/activities/capacities Banking services Interest rates New technologies and new products The economy Competitors The scope of working capital management includes transactions, relations, analyses, and focus: Transactions include payments for trade, financing, and investment. Relations with financial institutions and trading partners must be maintained to ensure that the transactions work effectively. Analyses of working capital management activities are required so that appropriate strategies can be formulated and implemented. Focus requires that organizations of all sizes today must have a global viewpoint with strong emphasis on liquidity. In this reading, we examine the different types of working capital and the management issues associated with each. We also look at methods of evaluating the effectiveness of working capital management. <span><body><html>

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ead> The scope of working capital management includes transactions, relations, analyses, and focus: Transactions include payments for trade, financing, and investment. Relations with financial institutions and trading partners must be maintained to ensure that the transactions work effectively. Analyses of working capital management activities are required so that appropriate strategies can be formulated and implemented. Focus requires that orga

and collecting on this credit, managing inventory, and managing payables. Effective working capital management also requires reliable cash forecasts, as well as current and accurate information on transactions and bank balances. <span>Both internal and external factors influence working capital needs; we summarize them in Exhibit 1. Exhibit 1. Internal and External Factors That Affect Working Capital Needs Internal Factors External Factors Company size and growth rates Organizational structure Sophistication of working capital management Borrowing and investing positions/activities/capacities Banking services Interest rates New technologies and new products The economy Competitors The scope of working capital management includes transactions, relations, analyses, and focus: Transactions include payments for trade, financing, and investment. Relations with financial institutions and trading partners must be maintained to ensure that the transactions work effectively. Analyses of working capital management activities are required so that appropriate strategies can be formulated and implemented. Focus requires that organizations of all sizes today must have a global viewpoint with strong emphasis on liquidity. In this reading, we examine the different types of working capital and the management issues associated with each. We also look at methods of evaluating the effectiveness of working capital management. <span><body><html>

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Transactions include payments for trade, financing, and investment. Relations with financial institutions and trading partners must be maintained to ensure that the transactions work effectively. <span>Analyses of working capital management activities are required so that appropriate strategies can be formulated and implemented. Focus requires that organizations of all sizes today must have a global viewpoint with strong emphasis on liquidity. <span><body><html>

and collecting on this credit, managing inventory, and managing payables. Effective working capital management also requires reliable cash forecasts, as well as current and accurate information on transactions and bank balances. <span>Both internal and external factors influence working capital needs; we summarize them in Exhibit 1. Exhibit 1. Internal and External Factors That Affect Working Capital Needs Internal Factors External Factors Company size and growth rates Organizational structure Sophistication of working capital management Borrowing and investing positions/activities/capacities Banking services Interest rates New technologies and new products The economy Competitors The scope of working capital management includes transactions, relations, analyses, and focus: Transactions include payments for trade, financing, and investment. Relations with financial institutions and trading partners must be maintained to ensure that the transactions work effectively. Analyses of working capital management activities are required so that appropriate strategies can be formulated and implemented. Focus requires that organizations of all sizes today must have a global viewpoint with strong emphasis on liquidity. In this reading, we examine the different types of working capital and the management issues associated with each. We also look at methods of evaluating the effectiveness of working capital management. <span><body><html>

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ing partners must be maintained to ensure that the transactions work effectively. Analyses of working capital management activities are required so that appropriate strategies can be formulated and implemented. <span>Focus requires that organizations of all sizes today must have a global viewpoint with strong emphasis on liquidity. <span><body><html>

and collecting on this credit, managing inventory, and managing payables. Effective working capital management also requires reliable cash forecasts, as well as current and accurate information on transactions and bank balances. <span>Both internal and external factors influence working capital needs; we summarize them in Exhibit 1. Exhibit 1. Internal and External Factors That Affect Working Capital Needs Internal Factors External Factors Company size and growth rates Organizational structure Sophistication of working capital management Borrowing and investing positions/activities/capacities Banking services Interest rates New technologies and new products The economy Competitors The scope of working capital management includes transactions, relations, analyses, and focus: Transactions include payments for trade, financing, and investment. Relations with financial institutions and trading partners must be maintained to ensure that the transactions work effectively. Analyses of working capital management activities are required so that appropriate strategies can be formulated and implemented. Focus requires that organizations of all sizes today must have a global viewpoint with strong emphasis on liquidity. In this reading, we examine the different types of working capital and the management issues associated with each. We also look at methods of evaluating the effectiveness of working capital management. <span><body><html>

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es and amounts of information that must be provided to users of financial statements, including investors and creditors, so that they may make informed decisions. This reading focuses on the framework within which these standards are created. <span>An understanding of the underlying framework of financial reporting standards, which is broader than knowledge of specific accounting rules, will allow an analyst to assess the valuation implications of financial statement elements and transactions—including transactions, such as those that represent new developments, which are not specifically addressed by the standards. <span><body><html>

Financial reporting standards provide principles for preparing financial reports and determine the types and amounts of information that must be provided to users of financial statements, including investors and creditors, so that they may make informed decisions. This reading focuses on the framework within which these standards are created. An understanding of the underlying framework of financial reporting standards, which is broader than knowledge of specific accounting rules, will allow an analyst to assess the valuation implications of financial statement elements and transactions—including transactions, such as those that represent new developments, which are not specifically addressed by the standards. Section 2 of this reading discusses the objective of financial statements and the importance of financial reporting standards in security analysis and valuation. Section 3