Financial Modeling and Valuation Study Guide of Final
Financial Modeling and Valuation Study Guide of Final BU.230.620.W4.SP16
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Date Created: 05/03/16
FINAL 1. CHAPTER 24 In this chapter we discuss techniques for computing random numbers. We use random numbers extensively in Chapters 25–30 to simulate stock prices, investment strategies, and option strategies. In this chapter we show how to produce both uniformly distributed and normally distributed random numbers. A random-number generator on a computer is a function that produces a seemingly unrelated set of numbers. The question of what is a random number is a philosophical one. In this chapter we will ignore philosophy and con- centrate on some simple random-number generators—primarily the Excel random-number generator Rand( ) and the VBA random-number 2 generator Rnd. We will show how to use these generators to produce uniform random numbers and subsequently random numbers that are normally distributed. At the end of the chapter we use the Cholesky decomposition to produce corre- lated random numbers. To imagine a set of uniformly distributed random numbers think of an urn filled with 1,000 little balls, numbered 000, 001, 002, ... , 999. Suppose we perform the following experiment: Having shaken the urn to mix up the balls, we draw one ball out of the urn and record the ball’s number. Next we put the ball back into the urn, shake the urn thoroughly so that the balls are mixed up again, and then draw out a new ball. The series of numbers produced by repeat- ing this procedure many times should be uniformly distributed between 000 and 999. A random-number generator on a computer is a function that imitates this procedure. The random-number generators considered in this chapter are sometimes termed pseudo-random-number generators, since they are actually deterministic functions whose values are indistinguishable from random numbers. All pseudo-random-number generators have cycles (i.e., they eventu- ally start to repeat themselves). The trick is to find a random- number generator with a long cycle. The Excel Rand( ) function has very long cycles and is a respectable random-number generator. If you’ve never used a random-number generator, open an Excel spreadsheet and type =Rand( ) in any cell. You will see a 15-digit number between 0.000000000000000 and 0.999999999999999. Every time you recalculate the spreadsheet (e.g., by pressing the F9 key), the number changes. We leave the technical details of how Rand( ) works for the exercises to this chapter, where we show you how to design your own random-number generator. Suffice it to say, however, that the series of numbers produced by the function should be (to use Lehmer’s terminology from footnote 1) “unpredictable to the uninitiated.” In this chapter we shall deal with several kinds of random-number genera- tors: We first examine the uniform random-number generators which come with Excel and VBA. Subsequently we generate normally distributed random numbers. Finally we generate correlated random numbers using the Cholesky decomposition. 2. CHAPTER 25 “Monte Carlo” (MC) methods refer to a variety of random simulations used to determine the values of parameters. In this introductory chapter to MC methods, we use MC to determine the value of . In subsequent chapters we use MC to gain insight into investment and option strategies. The Monte Carlo method has its source in physics, where it is often used to determine 1 model values for which there is no analytical solution. One use of Monte Carlo in finance is similar: Monte Carlo methods use simulation to price assets whose prices are not readily determined by analytical means. In short: If there isn’t a formula for computing the value of an asset, maybe we can determine its value with a simulation. 3. CHAPTER 32 A matrix with only one row is also called a row vector; a matrix with only one column is also called a column vector. A matrix with an equal number of rows and columns is called a square matrix. A single letter is often used to denote a matrix or a vector. In this case we often write, for example, B [b ], ijere b stanij for the entry in row i and column j of the matrix. For a vector we might write A [a] or C [i]. Thus fir the examples given above: a =4, b =10, c =13, d =0 3 22 1 41 The matrix B above is symmetric, meaning that b b . (The varianceij ji covariance matrices used in the portfolio discussion of Chapters 8–13 are symmetric.) 4. CHAPTER 8 In this chapter we review the basic mechanics of portfolio calculations. We start with a simple example of two assets, showing how to derive the return distributions from historical price data. We then discuss the general case of N assets; for this case it becomes convenient to use matrix notation and exploit Excel’s matrix handling capabilities. It is useful before going on to review some basic notation: Each asset i (assets may be stocks, bonds, real estate, or whatever, although our numerical examples will be largely confined to stocks) is characterized by several statis- tics: E(r), thi expected return on asset i; Var(r), the variance i of asset i’s return; and Cov(r, r), thi cjvariance of asset i’s and asset j’s returns. Occasionally we will use to denote tie expected return on asset i. In addition, it will often be convenient to write Cov(r, r) as and Var(i) js ij i 2 iiinstead of ⌠ , asiusual). Since the covariance of an asset’s returns with itself, iiv(r, r)i isiin fact the variance of the asset’s returns, this notation is not only economical but also logical. 5. CHAPTER 10 In order to calculate efficient portfolios, we must be able to compute the variance-covariance matrix from return data for stocks. In this chapter we discuss this computation, showing how to do the calculations in Excel. The most obvious calculation is the sample variance-covariance matrix: This is the matrix computed directly from the historic returns. We illustrate several methods for calculating the sample variance-covariance matrix, including a direct calculation in the spreadsheet using the excess return matrix and an implementation of this method with VBA. While the sample variance-covariance matrix may appear to be an obvious choice, a large literature recognizes that it may not be the best estimate of variances and covariances. Disappointment with the sample variance- covariance matrix stems both from its often unrealistic parameters and from its inability to predict. These issues are discussed briefly in sections 10.5 and 10.6. As an alternative to the sample matrix, sections 10.7–10.10 discuss so-called “shrinkage” 1ethods for improving the estimate of the variance- covariance matrix. Before starting this chapter, you may want to peruse Chapter 34 which discusses array functions. These are Excel functions whose arguments are vectors and matrices; their implementation is slightly different from standard Excel functions. This chapter makes heavy use of the array functions Trans- pose( ) and MMult( ) as well as some other “home- grown” array functions. 6. CHAPTER 9 This chapter covers the theory and calculations necessary for both versions of the classical capital asset pricing model (CAPM)—both that which is based on a risk-free asset (also known as the Sharp-Lintner-Mossin model) and Black’s (1972) zero-beta CAPM (which does not require the assumption of a risk-free asset). You will find that using a spreadsheet enables you to do the necessary calculations easily. The structure of the chapter is as follows: We begin with some preliminary definitions and notation. We then state the major results (proofs are given in the appendix to the chapter). In succeeding sections we implement these results, showing you: • How to calculate efficient portfolios. • How to calculate the efficient frontier. This chapter includes more theoretical material than most chapters in this book: Section 9.2 contains the propositions on portfolios which underlie the calculations of both efficient portfolios and the security market line (SML) in Chapter 11. If you find the theoretical material in section 9.2 difficult, skip it at first and try to follow the illustrative calculations in section 9.3. This chapter assumes that the variance-covariance matrix is given; we delay a dis- cussion of various methods of computing the variance-covariance matrix until Chapter 10. 7. CHAPTER 28 Value at risk (VaR) measures the worst expected loss under normal market conditions over a specific time interval at a given confidence level. As one of our references states: “VaR answers the question: how much can I lose with x% probability over a pre-set horizon?” (J. P. Morgan, RiskMetrics— Technical Document ). Another way of expressing this is that VaR is the lowest quantile of the potential losses that can occur within a given portfolio during a specified time period. The basic time period T and the confidence level (the quantile) q are the two major parameters that should be chosen in a way appropriate to the overall goal of risk measurement. The time horizon can differ from a few hours for an active trading desk to a year for a pension fund. When the primary goal is to satisfy external regulatory requirements, such as bank capital require- ments, the quantile is typically very small (e.g., 1% of worst outcomes). However, for an internal risk management model used by a company to control the risk exposure, the typical number is around 5% (visit the Internet sites in Selected References for more details). A general introduction to VaR can be found in Linsmeier and Pearson (1996) and in Jorion (1997). In the jargon of VaR, suppose that a portfolio manager has a daily VaR equal to $1 million at 1%. This statement means that there is only one chance in 100 that a daily loss bigger than $1 million occurs under normal market conditions.
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