Financial Econometrics

Transcription

1 Financial Econometrics Carlos Martins-Filho Department of Economics IFPRI University of Colorado 2033 K Street NW Boulder, CO , USA & Washington, DC , USA carlos.martins@colorado.edu c.martins-filho@cgiar.org Voice: Voice: August, 2017

2 Chapter 1 Returns 1.1 Introduction Let P t denote the price of a financial asset at time period t where t T = {0, ±1, ±2, }. If the asset is bought at time period t h, for h = 1, 2,, and sold at time period t, we define the net return over these h periods by R t,h = P t P t h P t h for P t h 0. (1.1) If h = 1, we drop the h subscript and denote by R t the net return over one time period. The time that elapses between t and t + 1 can be any measure of time, i.e., minutes, hours, days, etc. Observe that the set T is countable, and in this case we say that prices and returns are indexed by a discrete, as opposed to a continuous, measure of time. Also, since P t (0, ) we have R t,h ( 1, ). as There are two additional related concepts. Logarithm returns or log-returns 1 are defined and gross returns are defined as r t,h = log(1 + R t,h ) = log P t log P t h, for P t h 0, (1.2) ρ t,h = P t P t h for P t h 0. (1.3) As in the case of net returns, when h = 1 we write r t,1 r t and ρ t,1 ρ t. Note that r t,h (, ) and ρ t,h (0, ). 1 In these notes, log always refers to the natural logarithmic (base e). Whenever a different base b is used we will write log b. 1

3 If net returns are sufficiently small they can be well approximated by log-returns. To see this, let 0 < x < 1, (0, x) be an (open) interval and g(y) = log(1 + y). Note that g has a derivative at each point y (0, x), given by g (y) = 1, and is continuous at 0 and x. 1+y Hence, by the Mean Value Theorem, there is a point x (0, x) such that g(x) g(0) = 1 x. 1+ x Furthermore, since x 0 and g(0) = 0 we write g(x) x = x g(x) and lim x 0 x = 1. (1.4) Hence, for R t,h sufficiently close to 0, log(1 + R t,h )/R t,h = r t,h /R t,h will be close to log(1+x),x Net returns Figure 1.1: Graph of log returns and net returns Figure 1.1 shows the graphs of log-returns (blue line) and net returns (green line) against net returns. Note that the distance between the two graphs increases with net returns (see the MATLAB code returns.m to produce Figure 1.1). An advantage of working with log-returns is that the log-returns from holding an asset for h periods can be written as the sum of h one period log-returns. Indeed, r t,h = logp t logp t h = ( ) Pt P t 1 log Pt (h 1) P t 1 P t 2 P t h = r t,1 + r t 1,1 + + r t (h 1),1 = r t + r t r t (h 1) h = r t (j 1). j=1 This additivity of log-returns will prove convenient in subsequent developments. Since at time period t h it is not possible to know for sure what the price will be in (a future) time period t, there is uncertainty about R t,h, r t,h and ρ t,h. This uncertainty can be 2

4 0.15 Log returns on future contracts for wheat Log returns Time Index (days) Figure 1.2: Graph of log returns on future contracts for wheat from the Chicago Board of Trade particularly problematic when P t < P t h, that is R t,h, r t,h < 0 or ρ t,h < 1. This risk of losing money from buying an asset in period t h for P t h and selling it in period t for P t < P t h is a fundamental problem of investing. Because of this uncertainty, it is useful to think of R t,h, r t,h and ρ t,h as random variables. The insight that returns can be viewed as random variables is the foundation of our approach to the study of empirical finance (Bachelier (1900)). It allows for the construction of statistical models that represent the evolution of returns through time and for the testing and evaluation of various models developed in theoretical Finance. For example, take for simplicity the case where h = 1. At any time period t a collection (sample) of n observed log-returns on a financial asset can be represented by {r t (n 1),, r t 1, r t }. Without loss of generality, we set t = n and write {r 1,, r n 1, r n } = {r t } n t=1. Figure 1.2 shows a graph of one such sample associated with log returns on future contracts for wheat from the Chicago Board of Trade (see MATLAB code wheatret 1.m that produces Figure 1.2). Perhaps the simplest assumption that can be made about {r t } n t=1 is that it is a collection of independently and identically distributed random variables. If r t has a distribution function F, denoted by r t F, then using a sample of past returns, the distribution F can be estimated and important practical questions, such as: 3

5 What is E(r t )? What is the volatility, i.e., V (r t )? What is the probability that r t < a, i.e., P (r t < a) for a > 0? can be answered. Satisfactorily answering these questions depends, in turn, on constructing statistical models that reasonably represent the stochastic process that generates prices or returns. Construction of such models will lead, for example, to questions such as: Is it reasonable to assume that {r t } n t=1 is a collection of independently and identically distributed random variables? If V (r t ) exists, can it be suitably expressed as a parametric function of past returns? If r t and r t h are statistically correlated, is there a parametric model that suitably captures this correlation? It is apparent that the study of empirical finance depends on a solid understanding of probability and statistics. 4

6 Bibliography Bachelier, M. L., Théorie de la Spéculation. Annales Scientifique de Supérieure III-17, l École Normale 5