Financial Data Analysis, WS08/09. Roman Liesenfeld, University of Kiel 1

Transcription

1 Financial Data Analysis, WS08/09. Roman Liesenfeld, University of Kiel 1 Data sets used in the following sections can be downloaded from Exercise Sheet 1 1. Consider the monthly stock returns of Alcoa (aa), General Motors (gm), Walt Disney (dis), and Hershey Foods (hsy) from January 1962 to December 1999 for 456 observations and those of American Express (axp) and Mellon Financial Corporation (mel) from January 1973 to December 1999 for 324 observations. Again, you may obtain the data directly from CRSP or from the files on the Web. Tick symbols and years involved are used to create file names (e.g., m-mel7399.dat contains the monthly log returns, in percentage, of Mellon Financial Corporation stock from January 1973 to December 1999). (a) Compute the sample mean, variance, skewness, excess kurtosis, and minimum and maximum of the monthly log returns. (b) Transform the log returns into simple returns. Compute the sample mean, variance, skewness, excess kurtosis, and minimum and maximum of the simple returns. (c) Are the sample means of log returns statistically different from zero? Use the 5% significance level to draw your conclusion and discuss their practical implications. 2. Focus on the monthly stock returns of Alcoa from 1962 to (a) What is the average annual log return over the data span? (b) What is the annualized (average) simple return over the data span? (c) Consider an investment that invested one dollar on the Alcoa stock at the beginning of What was the value of the investment at the end of 1999? Assume that there were no transaction costs. 3. Obtain the histograms of daily simple and log returns of American Express stock from January 1990 to December Compare them with normal distributions that have the same mean and standard deviation.

2 Financial Data Analysis, WS08/09. Roman Liesenfeld, University of Kiel 2 Exercise Sheet 2 1. Suppose that the simple return of a monthly bond index follows the MA(1) model R t = a t + 0.2a t 1, where {a t } is a Gaussian white noise series with mean zero and standard deviation σ a = Assume that a 100 = Compute the 1-step and 2-step ahead forecasts of the return at the forecast origin t = 100. What are the standard deviations of the associated forecast errors? Also compute the lag-1 and lag-2 autocorrelations of the return series. 2. Suppose that the daily log return of a security follows the model r t = r t 2 + a t, where {a t } is a Gaussian white noise series with mean zero and variance What are the mean and variance of the return series r t? Compute the lag-1 and lag-2 autocorrelations of r t. Assume that r 100 = 0.01, and r 99 = Compute the 1- and 2-step ahead forecasts of the return series at the forecast origin t = 100. What are the associated standard deviations of the forecast errors? 3. Consider the monthly log returns of CRSP equal-weighted index from January 1962 to December 1999 for 456 observations. You may obtain the data from CRSP directly or from the file m-ew6299.dat on the Web. (a) Build an AR model for the series and check the fitted model. (b) Build an MA model for the series and check the fitted model. (c) Compute 1- and 2-step ahead forecasts of the AR and MA models built in the previous two questions. (d) Compare the fitted AR and MA models. 4. Column 3 of the file d-hwp3dx8099.dat contains the daily log of returns of the CRSP equal-weighted index from January 1980 to December (a) Build an AR model for the series and check the fitted model. (b) Build an MA model for the series and check the fitted model. (c) Use the fitted AR model to compute 1-step to 7-step ahead forecasts at the forecast origin h = 5049.

3 Financial Data Analysis, WS08/09. Roman Liesenfeld, University of Kiel 3 Exercise Sheet 3 1. The file m-bnd.dat contains simple returns of monthly indexes of U.S. government bonds with maturities in 30 years, 20 years, 10 years, 5 years, and 1 year (in column order). The data are obtained from CRSP, and the sample period is January 1942 to December Build an AR or MA model for simple return of bond index with maturity 5 years. Is the fitted model adequate? 2. Consider the sampling period from January 1990 to December Are the daily log returns of ALCOA stock predictable? You may test the hypothesis using (a) the first 5 lags of the autocorrelation function, and (b) the first 10 lags of autocorrelation function. Draw your conclusion by using the 5% significance level. The data are available from CRSP. 3. Consider the daily log returns of Hewlett-Packard stock, value-weighted index, equal-weighted index, and S&P 500 index from January 1980 to December 1999 for 5056 observations. The returns include all distributions and are in percentages. The data can be obtained from CRSP or from the file d-hwp3dx8099.dat, which has four columns with the same ordering as stated before. For each return series, test the hypothesis H 0 : ρ 1 =... = ρ 10 = 0 versus the alternative hypothesis H a : ρ i 0 for some i {1,..., 10}, where ρ i is the lag i autocorrelation. Draw your conclusion based on the 5% significance level. Compare the results between returns of individual stocks and market indexes. 4. (Martingale)Based on current and past observations (P t, P t 1,..., P 0 ) the stock price P t+1 shall be forecasted. Show that the forecast E(P t+1 P t, P t 1,..., P 0 ) exhibits the minimum mean squared error (MSE). Verify that a stock price process {P t } modelled according to has the martingale property. P t = P t 1 + ɛ t, ɛ t IID(0, σ 2 ) 5. (Uncorrelated and stochastic independent random variables)suppose the random variables X and Y have the following joint distribution: y x / /4 0 1/ /4 0 (a) Show that the two random variables are uncorrelated. (b) Calculate Cov(X 2, Y 2 ).

4 Financial Data Analysis, WS08/09. Roman Liesenfeld, University of Kiel 4 6. (Conditional Moments) Suppose for the sequence of random variables {P t } t=0 the following model is valid P t = µ + φp t 1 + ɛ t, φ 1, ɛ t IID(0, σ 2 ) Show that the conditional expectation E(P t P 0 ) and the conditional variance V ar(p t P 0 ) can be expressed by the following equations: t 1 E(P t P 0 ) = µ φ i + φ t P 0 i=0 t 1 V ar(p t P 0 ) = σ 2 φ 2i. 7. (Theoretical Autocorrelation) Calculate for the following AR(1)- model: i=0 r t = φr t 1 + ɛ t, φ < 1, ɛ t IID(0, σ 2 ) and for the following MA(1) model: r t = ɛ t + ψɛ t 1, ɛ t IID(0, σ 2 ) the corresponding autocorrelation functions for r t. 8. (Forecasting capabilities in the context of stock returns: empirical tests) The EViews file HETT.wf1 contains transaction changes of the Henkel stock. The time span ranges from 1st July 1999 to 31st August Only trades of the XETRA system between 9.30h and 16.00h were recorded. (a) Develop adequate descriptive measures to pinpoint the distributional and times series properties of transaction changes of the Henkel stock. (b) In order to assess the validity of the random walk model for the price process use the changes in transaction prices and perform the Ljung-Box and the Run test. Interpret your results. (c) Repeat your analysis done in a) and b) for the daily closing prices of the Daimler-Benz stock. The data refers to the time range from 2nd January 1990 to 31st Mai 1994 and records the closing prices of floor trading at Frankfurt stock exchange. The data is listed in the EViews file DAID.wf1.

5 Financial Data Analysis, WS08/09. Roman Liesenfeld, University of Kiel 5 Exercise Sheet 4 1. Derive multistep ahead forecasts for a GARCH(1,2) model at the forecast origin h. 2. Derive multistep ahead forecasts for a GARCH(2,1) model at the forecast origin h. 3. Suppose that r 1,..., r n are observations of a return series that follows the AR(1)- GARCH(1,1) model r t = µ + φ 1 r t 1 + a t, a t = σ t ɛ t, σ 2 t = α 0 + α 1 a 2 t 1 + β 1 σ 2 t 1, where ɛ t is a standard Gaussian white noise series. Derive the conditional log likelihood function of the data. 4. In the previous equation, assume that ɛ t follows a standardized Student-t distribution with υ degrees of freedom. Derive the conditional log likelihood function of the data. 5. The file m-mrk.dat contains monthly simple returns of Merck stock. There are three columns - namely, monthly simple returns, years, and months. Transform the simple returns to log returns. (a) Is there evidence of ARCH effects in the log returns? Use Ljung- Box statistics for the squared returns with 5 and 10 lags of auto- correlation and 5% significance level to answer the question. (b) Use the PACF of the squared log returns to identify an ARCH model for the data and fit the identified model. Write down the fitted model. 6. The file m-gmsp5099.dat contains the monthly log returns, in percentages, of General Motors stock and S&P 500 index from 1950 to The GM stock returns are in column 1. Build a GARCH model with Gaussian innovations for the log returns of GM stock. Check the model and write down the fitted model. 7. The file d-ibmln.dat contains the daily log returns, in percentages, of IBM stock from July 1962 to December 1997 with 8938 oberservations. The file has only one column. Fit a GARCH(1,1) model to the series. What is the fitted model?

6 Financial Data Analysis, WS08/09. Roman Liesenfeld, University of Kiel 6 Exercise Sheet 5 1. Consider the analysis presented in chapter 4 (Hausman, Lo and MacKinnlay (1992)). The EViews file ibmprobit.wf1 lists the following variables for transactions from 1st November 1990 to 31st January The data set contains only transactions made between 10.00h and 16.00h each day. The variables are volume (vol), bid price (bid), ask price (ask), spread (spread), transaction price (trans) transaction price change (transdiff) and the difference between two transactions. Furthermore, the variable class classifies the changes in transaction prices into 5 categories: class = 2 if transdiff < -1/8, 1 if transdiff = -1/8, 0 if transdiff = 0, 1 if transdiff = 1/8, 2 if transdiff > -1/8. The indicator variable ind is dened as 1 if trans < (bid+ask)/2, ind = 0 if trans = (bid+ask)/2, 1 if trans > (bid+ask)/2. Build and estimate a suitable ordered probit model and interpret your results.

7 Financial Data Analysis, WS08/09. Roman Liesenfeld, University of Kiel 7 Exercise Sheet 6 1. Explain the basic idea (Null hypothesis, construction of the test statistic, distribution) of the Jarque-Bera test on nonnormality. 2. Show that a random variable can exhibit excess kurtosis for the following case of a mixed normal distribution: r t = zn 1 + (1 z)n 2, such that N 1 N(0, b), N 2 N(0, 1 b), where b (0, 1), z is a random variable with z = 1 with probability a, z = 0 with probability 1 a, and (z, N 1, N 2 ) are jointly independent. 3. Let the stochastically independent random variables Z 1 to Z 4 be distributed as follows: Z 1 N(0, 1), Z 2 t(r 2 ), Z 3 X 2 (r 3 ) and Z 4 X 2 (r 4 ). Which distributions do the following statistics have: a)z1, 2 Z b) 1 Z, c)z 3 + Z 4, d) 4 /r 4, e) Z 2 Z3 /r 3 Z3 /r Estimate CAPMs for Deutsche Bank and Schering. Take the DAX as proxies for market portfolio. Furthermore, the 3-month Euribor rate may represent the riskfree interest rate. Data can be obtained in the file DAX97.xls. Interpret your results.