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Svetlozar T. Rachev, PhD, Doctor of Science, is Chair-Professor at the University of Karlsruhe in the School of Economics and Business Engineering; Professor Emeritus at the University of California, Santa Barbara; and Chief-Scientist of FinAnalytica Inc.

Stoyan V. Stoyanov, PhD, is the Chief Financial Researcher at FinAnalytica Inc.

Frank J. Fabozzi, PhD, CFA, is Professor in the Practice of Finance and Becton Fellow at Yale University's School of Management and the Editor of the Journal of Portfolio Management.

De la contraportada

S ince the 1990s, significant progress has been made in developing the concept of a risk measure from both a theoretical and a practical viewpoint. This notion has evolved into a materially different form from the original idea behind traditional mean-variance analysis. As a consequence, the distinction between risk and uncertainty, which translates into a distinction between a risk measure and a dispersion measure, offers a new way of looking at the problem of optimal portfolio selection.

In Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization, the authors assert that the ideas behind the concept of probability metrics can be borrowed and applied in the field of asset management in order to construct an ideal risk measure which would be "ideal" for a given optimal portfolio selection problem. They provide a basic introduction to the theory of probability metrics and the problem of optimal portfolio selection considered in the general context of risk and reward measures.

Generally, the theory of probability metrics studies the problem of measuring distances between random quantities. There are no limitations in the theory of probability metrics concerning the nature of the random quantities, which makes its methods fundamental and appealing. Actually, it is more appropriate to refer to the random quantities as random elements: they can be random variables, random vectors, random functions, or random elements of general spaces. In the context of financial applications, we can study the distance between two random stocks prices, or between vectors of financial variables building portfolios, or between entire yield curves that are much more complicated objects. The methods of the theory remain the same, no matter the nature of the random elements.

Using numerous illustrative examples, this book shows how probability metrics can be applied to a range of areas in finance, including: stochastic dominance orders, the construction of risk and dispersion measures, problems involving average value-at-risk and spectral risk measures in particular, reward-risk analysis, generalizing mean-variance analysis, benchmark tracking, and the construction of performance measures. For each chapter where more technical knowledge is necessary, an appendix is included.

De la solapa interior

S ince the 1990s, significant progress has been made in developing the concept of a risk measure from both a theoretical and a practical viewpoint. This notion has evolved into a materially different form from the original idea behind traditional mean-variance analysis. As a consequence, the distinction between risk and uncertainty, which translates into a distinction between a risk measure and a dispersion measure, offers a new way of looking at the problem of optimal portfolio selection.

In Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization, the authors assert that the ideas behind the concept of probability metrics can be borrowed and applied in the field of asset management in order to construct an ideal risk measure which would be "ideal" for a given optimal portfolio selection problem. They provide a basic introduction to the theory of probability metrics and the problem of optimal portfolio selection considered in the general context of risk and reward measures.

Generally, the theory of probability metrics studies the problem of measuring distances between random quantities. There are no limitations in the theory of probability metrics concerning the nature of the random quantities, which makes its methods fundamental and appealing. Actually, it is more appropriate to refer to the random quantities as random elements: they can be random variables, random vectors, random functions, or random elements of general spaces. In the context of financial applications, we can study the distance between two random stocks prices, or between vectors of financial variables building portfolios, or between entire yield curves that are much more complicated objects. The methods of the theory remain the same, no matter the nature of the random elements.

Using numerous illustrative examples, this book shows how probability metrics can be applied to a range of areas in finance, including: stochastic dominance orders, the construction of risk and dispersion measures, problems involving average value-at-risk and spectral risk measures in particular, reward-risk analysis, generalizing mean-variance analysis, benchmark tracking, and the construction of performance measures. For each chapter where more technical knowledge is necessary, an appendix is included.

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Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization

John Wiley & Sons

Chapter One

1.1 INTRODUCTION

Will Microsoft's stock return over the next year exceed 10%? Will the one-month London Interbank Offered Rate (LIBOR) three months from now exceed 4%? Will Ford Motor Company default on its debt obligations sometime over the next five years? Microsoft's stock return over the next year, one-month LIBOR three months from now, and the default of Ford Motor Company on its debt obligations are each variables that exhibit randomness. Hence these variables are referred to as random variables. In this chapter, we see how probability distributions are used to describe the potential outcomes of a random variable, the general properties of probability distributions, and the different types of probability distributions. Random variables can be classified as either discrete or continuous. We begin with discrete probability distributions and then proceed to continuous probability distributions.

1.2 BASIC CONCEPTS

An outcome for a random variable is the mutually exclusive potential result that can occur. The accepted notation for an outcome is the Greek letter [omega]. A sample space is a set of all possible outcomes. The sample space is denoted by [OMEGA]. The fact that a given outcome [[omega].sub.i] belongs to the sample space is expressed by [[omega].sub.i] [member of] [OMEGA]. An event is a subset of the sample space and can be represented as a collection of some of the outcomes. For example, consider Microsoft's stock return over the next year. The sample space contains outcomes ranging from 100% (all the funds invested in Microsoft's stock will be lost) to an extremely high positive return. The sample space can be partitioned into two subsets: outcomes where the return is less than or equal to 10% and a subset where the return exceeds 10%. Consequently, a return greater than 10% is an event since it is a subset of the sample space. Similarly, a one-month LIBOR three months from now that exceeds 4% is an event. The collection of all events is usually denoted by [**]. In the theory of probability, we consider the sample space [OMEGA] together with the set of events [**], usually written as ([OMEGA], [**]), because the notion of probability is associated with an event.

1.3 DISCRETE PROBABILITY DISTRIBUTIONS

As the name indicates, a discrete random variable limits the outcomes where the variable can only take on discrete values. For example, consider the default of a corporation on its debt obligations over the next five years. This random variable has only two possible outcomes: default or nondefault. Hence, it is a discrete random variable. Consider an option contract where for an upfront payment (i.e., the option price) of $50,000, the buyer of the contract receives the payment given in Table 1.1 from the seller of the option depending on the return on the S&P 500 index. In this case, the random variable is a discrete random variable but on the limited number of outcomes.

The probabilistic treatment of discrete random variables is comparatively easy: Once a probability is assigned to all different outcomes, the probability of an arbitrary event can be calculated by simply adding the single probabilities. Imagine that in the above example on the S&P 500 every different payment occurs with the same probability of 25%. Then the probability of losing money by having invested $50,000 to purchase the option is 75%, which is the sum of the probabilities of getting either $0, $10,000, or $20,000 back. In the following sections we provide a short introduction to the most important discrete probability distributions: Bernoulli distribution, binomial distribution, and Poisson distribution. A detailed description together with an introduction to several other discrete probability distributions can be found, for example, in the textbook by Johnson et al. (1993).

1.3.1 Bernoulli Distribution

We will start the exposition with the Bernoulli distribution. A random variable X is Bernoulli-distributed with parameter p if it has only two possible outcomes, usually encoded as 1 (which might represent success or default) or 0 (which might represent failure or survival).

One classical example for a Bernoulli-distributed random variable occurring in the field of finance is the default event of a company. We observe a company ITLITL in a specified time interval I, January 1, 2007, until December 31, 2007. We define

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The parameter p in this case would be the annualized probability of default of company ITLITL.

1.3.2 Binomial Distribution

In practical applications, we usually do not consider a single company but a whole basket, [ITLITL.sub.1], ..., [C.sub.n], of companies. Assuming that all these n companies have the same annualized probability of default p, this leads to a natural generalization of the Bernoulli distribution called binomial distribution. A binomial distributed random variable Y with parameters n and p is obtained as the sum of n independent and identically Bernoulli-distributed random variables [X.sub.1], ..., [X.sub.n]. In our example, Y represents the total number of defaults occurring in the year 2007 observed for companies [ITLITL.sub.1], ..., C.sub.n. Given the two parameters, the probability of observing k, 0 [less than or equal to] k [less than or equal to] n defaults can be explicitly calculated as follows: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

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Recall that the factorial of a positive integer n is denoted by n! and is equal to n(n - 1)(n - 2) x ... x 2 x 1.

Bernoulli distribution and binomial distribution are revisited in Chapter 4 in connection with a fundamental result in the theory of probability called the Central Limit Theorem. Shiryaev (1996) provides a formal discussion of this important result.

1.3.3 Poisson Distribution

The last discrete distribution that we consider is the Poisson distribution. The Poisson distribution depends on only one parameter, [lambda], and can be interpreted as an approximation to the binomial distribution when the parameter p is a small number. A Poisson-distributed random variable is usually used to describe the random number of events occurring over a certain time interval. We used this previously in terms of the number of defaults. One main difference compared to the binomial distribution is that the number of events that might occur is unbounded, at least theoretically. The parameter [lambda] indicates the rate of occurrence of the random events, that is, it tells us how many events occur on average per unit of time.

The probability distribution of a Poisson-distributed random variable N is described by the following equation:

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1.4 CONTINUOUS PROBABILITY DISTRIBUTIONS

If the random variable can take on any possible value within the range of outcomes, then the probability distribution is said to be a continuous random variable. When a random variable is either the price of or the return on a financial asset or an interest rate, the random variable is assumed to be continuous. This means that it is possible to obtain, for example, a price of 95.43231 or 109.34872 and any value in between. In practice, we know that financial assets are not quoted in such a way. Nevertheless, there is no loss in describing the random variable as continuous and in many times treating the return as a continuous random variable means substantial gain in mathematical tractability and convenience. For a continuous random variable, the calculation of probabilities is substantially different from the discrete case. The reason is that if we want to derive the probability that the realization of the random variable lays within some range (i.e., over a subset or subinterval of the sample space), then we cannot proceed in a similar way as in the discrete case: The number of values in an interval is so large, that we cannot just add the probabilities of the single outcomes. The new concept needed is explained in the next section.

1.4.1 Probability Distribution Function, Probability Density Function, and Cumulative Distribution Function

A probability distribution function P assigns a probability P(A) for every event A, that is, of realizing a value for the random value in any specified subset A of the sample space. For example, a probability distribution function can assign a probability of realizing a monthly return that is negative or the probability of realizing a monthly return that is greater than 0.5% or the probability of realizing a monthly return that is between 0.4% and 1.0%.

To compute the probability, a mathematical function is needed to represent the probability distribution function. There are several possibilities of representing a probability distribution by means of a mathematical function. In the case of a continuous probability distribution, the most popular way is to provide the so-called probability density function or simply density function.

In general, we denote the density function for the random variable X as [f.sub.X](x). Note that the letter x is used for the function argument and the index denotes that the density function corresponds to the random variable X. The letter x is the convention adopted to denote a particular value for the random variable. The density function of a probability distribution is always nonnegative and as its name indicates: Large values for [F.sub.X](x) of the density function at some point x imply a relatively high probability of realizing a value in the neighborhood of x, whereas [F.sub.X](x) = 0 for all x in some interval (a, b) implies that the probability for observing a realization in (a, b) is zero.

Figure 1.1 aids in understanding a continuous probability distribution. The shaded area is the probability of realizing a return less than b and greater than a. As probabilities are represented by areas under the density function, it follows that the probability for every single outcome of a continuous random variable always equals zero. While the shaded area in Figure 1.1 represents the probability associated with realizing a return within the specified range, how does one compute the probability? This is where the tools of calculus are applied. Calculus involves differentiation and integration of a mathematical function. The latter tool is called integral calculus and involves computing the area under a curve. Thus the probability that a realization from a random variable is between two real numbers a and b is calculated according to the formula,

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The mathematical function that provides the cumulative probability of a probability distribution, that is, the function that assigns to every real value x the probability of getting an outcome less than or equal to x, is called the cumulative distribution function or cumulative probability function or simply distribution function and is denoted mathematically by [f.sub.X] (x). A cumulative distribution function is always nonnegative, nondecreasing, and as it represents probabilities it takes only values between zero and one. An example of a distribution function is given in Figure 1.2.

The mathematical connection between a probability density function f, a probability distribution P, and a cumulative distribution function F of some random variable X is given by the following formula:

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Conversely, the density equals the first derivative of the distribution function,

[f.sub.X] (x) = d[F.sub.x](x)/dx.

The cumulative distribution function is another way to uniquely characterize an arbitrary probability distribution on the set of real numbers. In terms of the distribution function, the probability that the random variable is between two real numbers a and b is given by

P(a < X [less than or equal to] b) = [F.sub.X](b) - [F.sub.X](a).

Not all distribution functions are continuous and differentiable, such as the example plotted in Figure 1.2. Sometimes, a distribution function may have a jump for some value of the argument, or it can be composed of only jumps and flat sections. Such are the distribution functions of a discrete random variable for example. Figure 1.3 illustrates a more general case in which [F.sub.X](x) is differentiable except for the point x = a where there is a jump. It is often said that the distribution function has a point mass at x = a because the value a happens with nonzero probability in contrast to the other outcomes, x [not equal to] a. In fact, the probability that a occurs is equal to the size of the jump of the distribution function. We consider distribution functions with jumps in Chapter 7 in the discussion about the calculation of the average value-at-risk risk measure.

1.4.2 The Normal Distribution

The class of normal distributions, or Gaussian distributions, is certainly one of the most important probability distributions in statistics and due to some of its appealing properties also the class which is used in most applications in finance. Here we introduce some of its basic properties.

The random variable X is said to be normally distributed with parameters [mu] and [sigma], abbreviated by X [member of] N ([mu], [[sigma].sup.2]), if the density of the random variable is given by the formula,

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The parameter [mu] is called a location parameter because the middle of the distribution equals [mu] and [sigma] is called a shape parameter or a scale parameter. If [mu] = 0 and [sigma] = 1, then X is said to have a standard normal distribution.

An important property of the normal distribution is the location-scale invariance of the normal distribution. What does this mean? Imagine you have random variable X, which is normally distributed with the parameters [mu] and [sigma]. Now we consider the random variable Y, which is obtained as Y = aX + b. In general, the distribution of Y might substantially differ from the distribution of X but in the case where X is normally distributed, the random variable Y is again normally distributed with parameters and [**] = a[mu] + b and [**] = a]sigma]. Thus we do not leave the class of normal distributions if we multiply the random variable by a factor or shift the random variable. This fact can be used if we change the scale where a random variable is measured: Imagine that X measures the temperature at the top of the Empire State Building on January 1, 2008, at 6 A.M. in degrees Celsius. Then Y = 9/5 X + 32 will give the temperature in degrees Fahrenheit, and if X is normally distributed, then Y will be too.

Another interesting and important property of normal distributions is their summation stability. If you take the sum of several independent random variables that are all normally distributed with location parameters [[mu].sub.i] and scale parameters [[sigma].sub.i], then the sum again will be normally distributed. The two parameters of the resulting distribution are obtained as

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(Continues...)

Excerpted from Advanced Stochastic Models, Risk Assessment, and Portfolio Optimizationby Svetlozar T. Rachev Stoyan V. Stoyanov Frank J. Fabozzi Copyright © 2008 by Svetlozar T. Rachev. Excerpted by permission.
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