Über die Autorin bzw. den Autor

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.

Von der hinteren Coverseite

Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization

The finance industry is seeing increased interest in new risk measures and techniques for portfolio optimization when parameters of the model are uncertain.

This groundbreaking book extends traditional approaches of risk measurement and portfolio optimization by combining distributional models with risk or performance measures into one framework. Throughout these pages, the expert authors explain the fundamentals of probability metrics, outline new approaches to portfolio optimization, and discuss a variety of essential risk measures. Using numerous examples, they illustrate a range of applications to optimal portfolio choice and risk theory, as well as applications to the area of computational finance that may be useful to financial engineers. They also clearly show how stochastic models, risk assessment, and optimization are essential to mastering risk, uncertainty, and performance measurement.

Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization provides quantitative portfolio managers (including hedge fund managers), financial engineers, consultants, and academic researchers with answers to the key question of which risk measure is best for any given problem.

Aus dem Klappentext

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.

Auszug. © Genehmigter Nachdruck. Alle Rechte vorbehalten.

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

1.4 CONTINUOUS PROBABILITY DISTRIBUTIONS

If the random...