The MDA of the Frechet Distribution
The Frechet distribution is written as Oa x exp -x a . Let's start by observing that the tail of the Frechet distribution decays as an inverse power law. In fact, we can write 1 - Oa x 1 - exp -x a x a for x gt . It can be demonstrated that a distribution function F belongs to the MDA of a Frechet distribution Oa x , a gt 0 if and only if there is a slowly varying function L such that F x x aL x . In this case, the constants assume the values We can rewrite this condition more compactly as...
Point Process of Exceedances or Peaks over Threshold
We have now reviewed the behavior of sums, maxima, and upper order statistics of continuous random variables. Yet another approach to EVT is based on point processes herein we will use point processes only to define the point process of exceedances. Point processes can be defined in many different ways. To illustrate the mathematics of point processes, let's first introduce the homogeneous Poisson process. A homogeneous Poisson process is defined as a process N t that starts at zero, i.e., N 0...
The Multifactor Capm Merton
The CAPM assumes that the only risk that an investor is concerned with is uncertainty about the future price of a security. Investors, however, are usually concerned with other risks that will affect their ability to consume goods and services in the future. Three examples would be the risks associated with future labor income, the future relative prices of consumer goods, and future investment opportunities. Recognizing these other risks that investors face, in 1976 Robert Merton extended the...
Price Diffusion Bachelier
In 1900, the Sorbonne University student Louis Bachelier presented a doctoral dissertation, Th orie de la Sp culation, that was to anticipate much of today's work in finance theory. Bachelier's advisor was the great French mathematician Henri Poincar . There were three notable aspects in Bachelier's thesis He argued that in a purely speculative market stock prices should be random. He developed the mathematics of Brownian motion. He computed the prices of several options. To appreciate the...
The Lag Operator L
Let's now simplify the notation by introducing the lag operator L. The lag operator L is an operator that acts on an infinite series and produces another infinite series shifted one place to the left. In other words, the lag operator replaces every element of a series with the one delayed by one time lag The n-th power of the lag operator shifts a series by n places Negative powers of the lag operator yield the forward operator F, which shifts places to the right. The lag operator can be...
The Conditional Capm
As we have seen, the CAPM is embodied in a static linear regression of asset returns over the market portfolio whose explanatory power has been questioned by, among others, Fama and French.14 Ravi Jagan-nathan and Zheniu Wang15 suggested a solution They made the CAPM regression coefficients conditional on some global information set, thereby generalizing the model. Called the Conditional CAPM or C CAPM , this model represents each expected return given the information set at time t by the...
Markov Coefficients and ARMA Models
For the theoretical analysis of ARMA processes, it is useful to state what conditions on the Markov coefficients ensure that the process admits an ARMA representation. Consider a process xt, stationary or not, which admits a moving-average representation either as The process admits an ARMA representation if and only if there is an integer q and a set of p matrices Ai, i 0, , p such that the Markov coefficients Hj satisfy the following linear difference equation starting from q Therefore, any...
The Class of FatTailed Distributions
Many important classes of fat-tailed distributions have been defined each is characterized by special statistical properties that are important in given application domains. We will introduce a number of such classes in order of inclusion, starting from the class with the broadest membership the class , which is defined as follows. Suppose that F is a 1 M.C. Bryson, Heavy-Tailed Distributions, in N.L. Kotz and S. Read eds. , Encyclopedia of Statistical Sciences, Vol. 3 New York John Wiley amp...
Eigenvalues And Eigenvectors
Consider a square matrix A of order n and the set of all n-dimensional vectors. The matrix A is a linear operator on the space of vectors. This means that A operates on each vector producing another vector and that the following property holds Consider now the set of vectors x such that the following property holds Any vector such that the above property holds is called an eigenvector of the matrix A and the corresponding value of X is called an eigenvalue. To determine the eigenvectors of a...
Financial Modeling and Investment Management
To Dominique, Leila, Guillaume, and Richard To my beautiful wife Donna and my children, Francesco, Patricia, and Karly Copyright 2004 by John Wiley amp Sons, Inc. All rights reserved. Published by John Wiley amp Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under...
Capital Asset Pricing Model Sharpe Lintner And Mossin
Absence of arbitrage is a powerful economic principle for establishing relative pricing. In itself, however, it is not a market equilibrium model. William Sharpe in 1964 ,16 John Lintner in 1965 ,17 and Jan Mossin in 1966 ,18 developed a theoretical equilibrium model of market prices called the Capital Asset Pricing Model CAPM . As anticipated 60 years earlier by Walras and Pareto, Sharpe, Lintner, and Mossin developed the consequences of Markowitz's portfolio selection into a full-fledged...
ddy [ft et M y dt
where each equation corresponds to a specific white noise path. However this definition would be meaningless in the domain of ordinary functions. In other words, it would generally not be possible to find a family of functions y -,w that satisfy the above equations for each white-noise path and that form a reasonable stochastic process. The key problem is that it is not possible to define a white noise process as a zero-mean stationary stochastic process with independent increments and...
The Principles Of Investment Markowitz
Just how an investor should allocate his resources has long been debated. Classical wisdom suggested that investments should be allocated to those assets yielding the highest returns, without the consideration of correlations. Before the modern formulation of efficient markets, speculators widely acted on the belief that positions should be taken only if they had a competitive advantage in terms of information. A large amount of resources were therefore spent on analyzing financial information....
x P af a
In its original formulation, the CAPM was derived as a general equilibrium theory the actual asset price process is the fixed point where the collective action of all agents trying to maximize their utility does not produce any change in the price process, thus the situation of equilibrium. CAPM assumes the joint normality of returns and the independence of returns from one period to another the single factor evolves as an arithmetic random walk. This version of the CAPM is conceptually...
Hankel Matrix
For the theoretical analysis of the autoregressive integrated moving averages ARMA processes described in Chapter 11, it is important to understand a special type of matrix, a Hankel matrix. A Hankel matrix is a matrix where for each antidiagonal the element is the same. For example, consider the following square Hankel matrix 17 16 15 24 16 15 24 33 15 24 33 72 24 33 72 41 Each antidiagonal has the same value. Now consider the elements of the antidiagonal running from the second row, first...
Stock Market Indicators
Stock market indicators have come to perform a variety of functions, from serving as benchmarks for evaluating the performance of professional money managers to answering the question, How did the market do today Thus, stock market indicators indexes or averages have become a part of everyday life. Even though many of the stock market indicators are used interchangeably, it is important to realize that each indicator applies to, and measures, a different facet of the stock market. The most...
Laplace Transform
Given a real-valued function f, its one-sided Laplace transform is an operator that maps f to the function L s L f x defined by the improper integral L s L f x J e sxf x dx 0 The Laplace transform of a real-valued function is thus a real-valued function. The one-sided transform is the most common type of Laplace transform used in physics and engineering. However in probability theory Laplace transforms are applied to density functions. As these functions are defined on the entire real axis, the...
Pension Funds
A pension plan is a fund that is established for the payment of retirement benefits. The entities that establish pension plans called plan sponsors are private business entities acting for their employees, state and local entities on behalf of their employees, unions on behalf of their members, and individuals for themselves. In the United States, corporate pension plans are governed by the Employee Retirement Income Security Act of 1974 ERISA . Pension funds are exempt from taxation. There are...
Nature of Liabilities
The nature of an institutional investor's liabilities will dictate the general investment strategy to pursue. Depository institutions, for example, seek to generate income by the spread between the return that they earn on their assets and the cost of their funds. Life insurance companies are in the spread business. Pension funds are not in the spread business, in that they themselves do not raise funds in the market. Certain types of pension funds seek to cover the cost of pension obligations...
Role of Brokers and Dealers in Real Markets
Common occurrences in real markets keep them from being theoretically perfect. Because of these occurrences, brokers and dealers are necessary to the smooth functioning of a secondary market. One way in which a real market might not meet all the exacting standards of a theoretically perfect market is that many investors may not be present at all times in the marketplace. Further, a typical investor may not be skilled in the art of the deal or completely informed about every facet of trading in...
Hafize Gaye Erkan
We are grateful to Professor Ren-Raw Chen of Rutgers University for coau-thoring Chapter 22 Credit Risk Modeling and Credit Default Swaps . The application of mean-variance analysis to asset allocation in Chapter 16 is from the coauthored work of Frank Fabozzi with Harry Markowitz and Francis Gupta. The discussion of tracking error and risk decomposition in Chapter 18 draws from the coauthored work of Frank Fabozzi with Frank Jones and Raman Vardharaj. In writing a book that covers a wide range...