Financial Time Series Analysis (FTSA) Lecture 1: Introduction

Transcription

1 Financial Time Series Analysis (FTSA) Lecture 1: Introduction Brief History of Time Series Analysis Statistical analysis of time series data (Yule, 1927) v/s forecasting (even longer). Forecasting is often the goal of a time series analysis. In business and economics: o To study dynamic structures of a process o To investigate the dynamic relationship between variables o To perform seasonal adjustment of economic data (eg: GNP) o To improve regression analysis when errors are serially correlated o To produce point and interval forecasts for both level and volatility series Definition of a Time Series A time series is a chronological sequence of observations on a particular variable. Components of a Time Series Trend Cycle Seasonal Variations Irregular Fluctuations

2 Trend Trend refers to the upward or downward movement that characterizes a time series over a period of time. That is, a long-run growth or decline in the time series. Some factors that affect trend are: Technological change in the industry Changes in consumer tastes Market growth Inflation or deflation, etc. Cycle Cycle refers to recurring up and down movements around the trend levels. These fluctuations can have a duration of anywhere from 2 to 10 years or even longer when measured from peak to peak or trough to trough. Extremely long time series data are required to be able to detect cycles. Seasonal Variation Seasonal Variations are periodic patterns in a time series that complete themselves with the period of a calendar year. These generally repeat every year and are most often caused by weather and customs. Generally, monthly or quarterly data over a few years is needed to be able to detect seasonality in time series. Irregular Fluctuations Irregular fluctuations are erratic movements in a time series that follow no recognizable or regular pattern. Such movements represent what is left over in a time series after trend, cycle, and seasonal variations have been accounted for. Most are caused by events one cannot forecast such as: earthquakes, accidents, hurricanes, etc. Time Series Analysis Lecture 2 Edited: Sept 28, 2009 Page 2

3 Time Series Analysis Lecture 2 Edited: Sept 28, 2009 Page 3

4 Motivation for FTSA Financial time series analysis is defined as the, "theory and practice of asset valuation over time. What make FTSA different? Highly empirical Uncertainty -- concept of volatility Dependent more on statistical theory and methods for development of robust models. Objective of the Course Provide some basic knowledge of financial time series To introduce some statistical tools and econometric models useful for analyzing FTS To gain empirical experience in analyzing FTS Provide a flavor of current research in the field Course Outline review syllabus Time Series Analysis Lecture 2 Edited: Sept 28, 2009 Page 4

5 Lecture 1: Financial Time Series and Their Characteristics Asset Returns Simple definition: The profit realized on a base product. Why returns in FTSA? For most investors return of an asset is a complete and scale-free summary of the investment opportunity. Return series are easier to handle -- normalizing effect compared to the price series of the underlying investment. They have more attractive and widely studied statistical properties. Types of Returns in Financial Analysis One Period Simple Gross Return or Where: R t = return at time t P t = price of an asset at time t One Period Simple Net Return (or Simple Return) The k-period (or multi-period) simple gross return is: Time Series Analysis Lecture 2 Edited: Sept 28, 2009 Page 5

6 The k-period simple net return is: Note: actual time interval for the study is very important monthly, annual, daily, etc. When not specified, assume time interval to be one year. That is k=1. Annualized (Average) Return When the investment horizon is longer than a year (k > 1) it is customary to report the returns as annualized (average) returns as follows: Aside: Continuous Compounding & Present Value The concept is important in finance since most investors invest an initial amount (C) in the market for an extended period. When interest r is paid n times a year the principal on which the interest is computed changes and thus the net asset value (A) is different than that derived using a simple return computation. Thus, Or the present value is: A = C * exp(r * n) C = A * exp(-r * n) Time Series Analysis Lecture 2 Edited: Sept 28, 2009 Page 6

7 Effects of Compounding -- Table 1.1 Type # payments Interest Net Annual % $ Semi-Annual % $ Quarterly % $ Monthly % $ Weekly % $ Daily % $ Continuously infinity $ Continuously Compounded (or log) Return Where: p t = ln(p t ) Therefore the multi-period continuously compounded return can be stated as: Two advantages: Multiperiod return is simply the sum of continuously compounded single period returns Statistical properties of log returns are easier to study and well-developed in the literature. Time Series Analysis Lecture 2 Edited: Sept 28, 2009 Page 7

8 Portfolio Return Simple return of portfolio p at time t where each instrument i is invested with weight w, is: However, the continuously compounded return of portfolio p is stated as: Dividend Payments When the investment instrument has dividends they are simply added into the price components as follows: And, Excess Returns The difference between the assets return R and return R 0 on some reference asset at time t is stated as: Simple excess return: Log excess return: Time Series Analysis Lecture 2 Edited: Sept 28, 2009 Page 8

9 Aside: The excess return is considered to be a payoff for an arbitrage portfolio that goes long in an asset and short in the reference asset without an initial investment. Long Position: When an investor purchases shares in the market. Short Position: When an investor sells shares in the market that he/she does not own. Essentially, he/she has borrowed the shares from another investor that has purchased the shares. The borrowed shares must be paid back along with any accrued dividend. The hope of the short seller is that a decline in price in the market on the shares will allow him/her to profit. In Summary, and Time Series Analysis Lecture 2 Edited: Sept 28, 2009 Page 9

10 Distributional Properties of Returns Objective to understand the behavior of asset returns Across assets --- allows for diversification of portfolio Across time allows for long range planning Recall: A return of an asset is a random variable. Therefore, to compute the distribution properties one needs to only compute the moments of the random variable. For our discussion we will compute the first four moments of r it Moments General Definition of the l th Moment Where: E is the expectation f(x) is the probability density function of X First Moment Mean or Expectation of X Measures the central location of the distribution of X and is denoted as μ x. The sample mean is computed as: General Definition of the lth Central Moment Time Series Analysis Lecture 2 Edited: Sept 28, 2009 Page 10

11 Second Central Moment Variability of X This is more specifically called the variance of X and is denoted as variance is computed as:. The sample Third Central Moment Symmetry of X This is more specifically called the skewness of X or S(x). If X is normally distributed skewness is zero. Also, asymptotically. Test for symmetry Time Series Analysis Lecture 2 Edited: Sept 28, 2009 Page 11

12 Fourth Central Moment Tail Thickness of X This is more specifically called the kurtosis of X or K(x). If X is normally distributed kurtosis is three. Also, asymptotically. Test for tail thickness Reject H o of normal tails if K* > Z α/2 or p-value is less than α. Testing for Normality The JB Test A simple t-test may be performed on and to test on normality. Additionally, Jarque and Bera test may be performed to check for normality by combining the individual tests as follows: Reject H o of normality if p-value is less than the significance level (or α). Time Series Analysis Lecture 2 Edited: Sept 28, 2009 Page 12

13 Distribution of Returns An assets return is treated as a continuous random variable. Therefore, when N assets are studied over T time-periods we have a joint distribution of all asset returns. The CAPM (Capital Asset Pricing Model) assumes such a joint-distribution. Additionally, we are also interested in the distribution of a set of returns for N assets given the distribution of the i th asset or the conditional distribution. For this course, we will focus only on the joint distribution, more specifically the marginal distributions, as they will form the basis of the univariate time-series analysis. In this regard several distributions have been proposed as discussed next. Normal Distribution Assumption: Simple returns are iid and N(μ,σ 2 ). Unrealistic and false: Lower bound of a simple return is -1. However, a normal distribution is not bounded. Multi-period return is a product of one-period returns and thus is not normally distributed even if the one-period return is Empirically, it has been shown that returns have fat-tails, or excess kurtosis. Time Series Analysis Lecture 2 Edited: Sept 28, 2009 Page 13

14 Time Series Analysis Lecture 2 Edited: Sept 28, 2009 Page 14

15 Log-Normal Distribution Assumption: Log returns are iid and N(μ,σ 2 ). Realistic: There is no lower bound of a log return. Multi-period log returns is a sum of one-period log returns and thus is normally distributed. Unrealistic: Empirically, it has been shown that log returns have fat-tails, or excess kurtosis. Time Series Analysis Lecture 2 Edited: Sept 28, 2009 Page 15

16 Stable Distribution An extension of the normal distributions in that they are stable under addition. They solve the problem of capturing excess kurtosis but most stable distributions do not have a finite variance which is in conflict with most finance theories. An example of a stable distribution is a Cauchy distribution which is symmetric around the median but has an infinite variance. Scale Mixture of Normal Distributions New distributions being studied in literature. Basically they attempt to overcome the limitations of the normal distribution. Time Series Analysis Lecture 2 Edited: Sept 28, 2009 Page 16

17 Empirical Properties of Returns << WinORSe-ai bring in the returns and plot both simple returns and log returns >> Point: Both simple return and log-return will have a similar scatter plot. Thus for computation purposes use log-returns. Time Series Analysis Lecture 2 Edited: Sept 28, 2009 Page 17

18 Other Processes Considered Volatility useful in the study of option pricing and risk management. New studies focus on realized volatility, and conditional variance. Continuous Time Processes or high-frequency data analysis which is the focus of Chapter 5. Extreme Events focus of chapter 7 where the size, frequency, and impact of an extreme event will be studied. Time Series Analysis Lecture 2 Edited: Sept 28, 2009 Page 18