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This book is intended for the Risk Management and Insurance course where Risk Management is emphasized. When we think of large risks, we often think in terms of natural hazards such as hurricanes, earthquakes or tornados. Perhaps man-made disasters come to mind such as the terrorist attacks in the U.S. on September 11, 2001. Typically we have overlooked financial crises, such as the credit crisis of 2008. However, these types of man-made disasters have the potential to devastate the global marketplace. Losses in multiple trillions of dollars and in much human suffering and insecurity are already being totaled, and the global financial markets are collapsing as never before seen. We can attribute the 2008 collapse to financially risky behavior of a magnitude never before experienced. The 2008 U.S. credit markets were a financial house of cards. A basic lack of risk management (and regulators' inattention or inability to control these overt failures) lay at the heart of the global credit crisis. This crisis started with lack of improperly underwritten mortgages and excessive debt. Companies depend on loans and lines of credit to conduct their routine business. If such credit lines dry up, production slows down and brings the global economy to the brink of deep recession—or even depression. The snowballing effect of this failure to manage the risk associated with providing mortgage loans to unqualified home buyers have been profound, indeed. When the mortgages failed because of greater risk- taking on the Street, the entire house of cards collapsed. Probably no other risk-related event has had, and will continue to have, as profound an impact world wide as this risk management failure. How was risk in this situation so badly managed? What could firms and individuals have done to protect themselves? How can government measure such risks (beforehand) to regulate and control them? These and other questions come to mind when we contemplate the consequences of this risk management fiasco. Standard risk management practice would have identified sub-prime mortgages and their bundling into mortgage-backed-securities as high risk. People would have avoided these investments or would have put enough money into reserve to be able to withstand defaults. This did not happen. Accordingly, this book may represent one of the most critical topics of study that the student of the 21st century could ever undertake. Risk management will be a major focal point of business and societal decision—making in the 21st century. A separate focused field of study, it draws on core knowledge bases from law, engineering, finance, economics, medicine, psychology, accounting, mathematics, statistics and other fields to create a holistic decision-making framework that is sustainable and value- enhancing. This is the subject of this book.
This is a book about how to prove theorems. Until this point in your education, you may have regarded mathematics primarily as a computational discipline. You have learned to solve equations, compute derivatives and integrals, multiply matrices and find determinants; and you have seen how these things can answer practical questions about the real world. In this setting, your primary goal in using mathematics has been to compute answers. But there is another approach to mathematics that is more theoretical than computational. In this approach, the primary goal is to understand mathematical structures, to prove mathematical statements, and even to invent or discover new mathematical theorems and theories. The mathematical techniques and procedures that you have learned and used up until now have their origins in this theoretical side of mathematics. For example, in computing the area under a curve, you use the fundamental theorem of calculus. It is because this theorem is true that your answer is correct. However, in your calculus class you were probably far more concerned with how that theorem could be applied than in understanding why it is true. But how do we know it is true? How can we convince ourselves or others of its validity? Questions of this nature belong to the theoretical realm of mathematics. This book is an introduction to that realm. This book will initiate you into an esoteric world. You will learn and apply the methods of thought that mathematicians use to verify theorems,explore mathematical truth and create new mathematical theories. This will prepare you for advanced mathematics courses, for you will be better able to understand proofs, write your own proofs and think critically and inquisitively about mathematics. This text has been used in classes at:Virginia Commonwealth University, Lebanon Valley College, University of California - San Diego, Colorado State University, Westminster College, South Dakota State University, PTEK College - Brunei, Christian Brothers High School, University of Texas Pan American, Schola Europaea, James Madison University, Heriot-Watt University, Prince of Songkla University, Queen Mary University of London, University of Nevada - Reno, University of Georgia - Athens, Saint Peter's University, California State University,Bogaziçi University, Pennsylvania State University, University of Notre Dame