on 27-Jan-2018 (Sat)

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Él seguía deprimido, aunque dijo que había dormido bien.

In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points.

Definition. Let (X, d) be a metric space. Then a map T : X → X is called a contraction mapping on X if there exists q ∈ [0, 1) such that for all x, y in X.

Banach Fixed Point Theorem. Let (X, d) be a non-empty complete metric space with a contraction mapping T : X → X. Then T admits a unique fixed-point x* in X (i.e. T(x*) = x*). Furthermore, x* can be found as follows: start with an arbitrary element x₀ in X and define a sequence {x_n} by x_n = T(x_n−1), then x_n → x* .

A graphical model or probabilistic graphical model (PGM) or structured probabilistic model is a probabilistic model for which a graph expresses the conditional dependence structure between random variables.

In a Bayesian network, the network structure of the model is a directed acyclic graph, the model represents a factorization of the joint probability of all random variables. More precisely, if the events are then the joint probability satisfies

where is the set of parents of node . In other words, the joint distribution factors into a product of conditional distributions.

Formally, Bayesian networks are DAGs whose:

1. nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses.
2. Edges represent conditional dependencies;
3. nodes that are not connected (there is no path from one of the variables to the other in the Bayesian network) represent variables that are conditionally independent of each other.
4. Each node is associated with a probability function that takes, as input, a particular set of values for the node's parent variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node.

A Markov random field, also known as a Markov network, is a model over an undirected graph.

d-separation is a criterion for deciding, from a given a causal graph, whether a set X of variables is independent of another set Y, given a third set Z.

The idea (of d-separation) is to associate "dependence" with "connectedness" (i.e., the existence of a connecting path) and "independence" with "unconnected-ness" or "separation".
The only twist on this simple idea is to define what we mean by "connecting path", given that

1. we are dealing with a system of directed arrows
2. in the graph some vertices correspond to measured variables, whose values are known precisely.

To account for the orientations of the arrows we use the terms "d-separated" and "d-connected" (d connotes "directional").

The forward–backward algorithm is an inference algorithm for hidden Markov models which computes the posterior marginals of all hidden state variables given a sequence of observations

The foreward-backward algorithm makes use of the principle of dynamic programming to compute efficiently the values that are required to obtain the posterior marginal distributions in two passes. The first pass goes forward in time while the second goes backward in time; hence the name forward–backward algorithm.

In a general sense, economics is the study of production, distribution, and consumption and can be divided into two broad areas of study: macroeconomics and microeconomics. Macroeconomics deals with aggregate economic quantities, such as national output and national income. Macroeconomics 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.

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 theory of the consumer and theory of the firm as two branches of study. The theory of the consumer deals with consumption (the demand for goods and services) by utility-maximizing individuals (i.e., individuals who make decisions that maximize the satisfaction received from present and future consumption). The theory of the firm deals with the supply of goods and services by profit-maximizing firms. The theory of the consumer and the theory of the firm are important because they help us understand the foundations of demand and supply. Subsequent readings will focus on the theory of the consumer and the theory of the firm.

Investment analysts, particularly equity and credit analysts, must regularly analyze products and services, their costs, prices, possible substitutes, and complements, to reach conclusions about a company’s profitability and business risk (risk relating to operating profits). Furthermore, unless the analyst has a sound understanding of the demand and supply model of markets, he or she cannot hope to forecast how external events—such as a shift in consumer tastes or changes in taxes and subsidies or other intervention in markets—will influence a firm’s revenue, earnings, and cash flows.

Having grasped the tools and concepts presented in this reading, the reader should also be able to understand many important economic relations and facts and be able to answer questions, such as:

• Why do consumers usually buy more when the price falls? Is it irrational to violate this “law of demand”?

• What are appropriate measures of how sensitive the quantity demanded or supplied is to changes in price, income, and prices of

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Government securities are coupon bearing instruments which are issued by RBI on behalf of Government of India. Government securities have maturity dates ranging from less than 1 year to a max of 30 year.

Debentures are bonds issued by a company. It has fixed rate of interest usually payable half-yearly, on specific dates and the principal amount repayable on a particular date on redemption of debenture. It is an unsecured debt

The Mumbai Interbank Offered Rate (MIBOR) is calculated everyday by the National Stock Exchange of India (NSEIL) as a weighted average of lending rates of a group of banks, on funds lent to first-class borrowers. 26. It is the interest rate at which banks can borrow funds, in marketable size, from other banks in the Indian interbank market. 27. The MIBOR was launched on June 15, 1998 by the Committee for the Development of the Debt Market, as an overnight rate.

LIBOR or ICE LIBOR (previously BBA LIBOR) is a benchmark rate that some of the world’s leading banks charge each other for short-term loans.
22. It stands for Intercontinental Exchange London Interbank Offered Rate and serves as the first step to calculating interest rates on various loans throughout the world.

23. LIBOR is administered by the ICE Benchmark Administration (IBA), and is based on five currencies: U.S. dollar (USD), Euro (EUR), pound sterling (GBP), Japanese yen (JPY) and Swiss franc (CHF).
24. It serves seven different maturities: overnight, one week, and 1, 2, 3, 6 and 12 months. There are a total of 35 different LIBOR rates each business day. The most commonly quoted rate is the three- month U.S. dollar rate.

Features of Government Securities 1. Issued at face value. 2. No default risk as the securities carry sovereign guarantee. 3. Ample liquidity as the investor can sell the security in the secondary market. 4. Interest payment on a half yearly basis on face value. 5. No tax deducted at source. 6. Can be held in Demat form

The Reserve Bank of India was established on April 1, 1935 in accordance with the provisions of the Reserve Bank of India Act, 1934.

The NSEIL launched the 14-day MIBOR on November 10, 1998, and the one month and three month MIBORs on December 1, 1998. 29. Since the launch, MIBOR rates have been used as benchmark rates for the majority of money market deals made in India.

FII can invest up to 49 % in Stock Exchange in India.

Corporatization is the process of converting the organizational structure of the Stock Exchange from a non-corporate to a corporate structure. 4. Demutualization refers to the transition process of a Stock Exchange from a mutually owned association to a shareholders-owned company.

Commercial papers are borrowing of a company from the market. These money market instruments are issued for 90 days.

Any company making a public issue or a listed company making a RI of a value of more than Rs 50 lacs is required to file a draft offer document with SEBI for its observations. This observation period is only 3 months.

DIP stands for Disclosure and Investor Protection guidelines

Offer document means prospectus in case of public issue. 20. Offer document means an offer for sale and letter of offer in case of a RI. 21. Offer documents are filed with Registrar of Companies and Stock Exchanges. 22. A draft offer document means the offer document in a draft stage. 23. The draft offer documents are filed with SEBI. 24. The period of filing draft offer document is at least 21 days prior to that of offer document.

RHP (Red Herring Prospectus) is a prospectus which doesn’t have details of either price of number of shares being offered or the amount of issue. But the number of shares and the upper and lower price bands are disclosed. 26. In case of FPO, the RHP can be filed with Registrar of Companies without the price band. The price band is notified one day prior to the opening of the issue by way of an advertisement.

By now it should be clear that economists are model builders. In the previous reading, we examined one of their most fundamental models, the model of demand and supply. And as we have seen, models begin with simplifying assumptions and then find the implications that can then be compared to real-world observations as a test of the model’s usefulness. In the model of demand and supply, we assumed the existence of a demand curve and a supply curve, as well as their respective negative and positive slopes. That simple model yielded some very powerful implications about how markets work, but we can delve even more deeply to explore the underpinnings of demand and supply. In this reading, we examine the theory of the consumer as a way of understanding where consumer demand curves originate. In a subsequent reading, the origins of the supply curve are sought in presenting the theory of the firm.

This reading is organized as follows:

Section 2 describes consumer choice theory in more detail.

Section 3 introduces utility theory, a building block of consumer choice theory that provides a quantitative model for a consumer’s preferences and tastes.

Section 4 surveys budget constraints and opportunity sets. Section 5 covers the determination of the consumer’s bundle of goods and how that may change in response to changes in income and prices.

Section 6 examines substitution and income effects for different types of goods.

In studying decision making by consumers and businesses, microeconomics gives rise to the theory of the consumer and theory of the firm as two branches of study.

The theory of the consumer is the study of consumption—the demand for goods and services—by utility-maximizing individuals. The theory of the firm, the subject of this reading, is the study of the supply of goods and services by profit-maximizing firms. Conceptually, profit is the difference between revenue and costs. Revenue is a function of selling price and quantity sold, which are determined by the demand and supply behavior in the markets into which the firm sells/provides its goods or services. Costs are a function of the demand and supply interactions in resource markets, such as markets for labor and for physical inputs. The main focus of this reading is the cost side of the profit equation for companies competing in market economies under perfect competition. A subsequent reading will examine the different types of markets into which a firm may sell its output.

The study of the profit-maximizing firm in a single time period is the essential starting point for the analysis of the economics of corporate decision making. Furthermore, with the attention given to earnings by market participants, the insights gained by this study should be practically relevant. Among the questions this reading will address are the following:

• How should profit be defined from the perspective of suppliers of capital to the firm?

• What is meant by factors of production?

• How are total, average, and marginal costs distinguished, and how is each related to the firm’s profit?

• What roles do marginal quantities (selling prices and costs) play in optimization?

This reading is organized as follows:

Section 2 discusses the types of profit measures, including what they have in common, how they differ, and their uses and definitions.

Section 3 covers the revenue and cost inputs of the profit equation and the related topics of breakeven analysis, shutdown point of operation, market entry and exit, cost structures, and scale effects. In addition, the economic outcomes related to a firm’s optimal supply behavior over the short run and long run are presented in this section.