Chen-wei Chiu ECON 424 Eric Zivot July 17, Lab 4. Part I Descriptive Statistics. I. Univariate Graphical Analysis 1. Separate & Same Graph

Transcription

1 Chen-wei Chiu ECON 424 Eric Zivot July 17, 2014 Part I Descriptive Statistics I. Univariate Graphical Analysis 1. Separate & Same Graph Lab 4

2 Time Series Plot Bar Graph The plots show that the returns on the three companies rise and fall in a similar behavior; all rise around the same period, and all fall in the same period. However, both FMAGX and SBUX have higher volatility than VBLTX. All three have a huge fall in the returns toward the end of 2008 at the beginning of the financial crisis.

3 2. Plot of Cumulative Returns While VBLTX experienced a steady increase on the cumulative returns over the years, SBUX had a rapid growth from 2002 to 2007 and experienced a shock during the financial crisis in 2007 and However, SBUX had another rapid growth toward the end of On the other hand, returns on FMAGX had not grown much throughout the years, and it had a sharp fall in the August of Given the plot, SBUX has the best future values over the investment horizon while FMAGX has the worst. 3. VBLTX

4 FMAGX SBUX

5 Comparison of All Three The return series for SBUX and FMAGX do not look normally distributed. SBUX and FMAGX have negative skewedness. However, SBUX has a wider range of distribution on returns while VBLTX has the more narrow range of return distribution.

6 II. Univariate Numerical Summary Statistics 1. Calculated with R: > table.stats(lab4returns.z) VBLTX FMAGX SBUX Observations NAs Minimum Quartile Median Arithmetic Mean Geometric Mean Quartile Maximum SE Mean LCL Mean (0.95) UCL Mean (0.95) Variance Stdev Skewness Kurtosis From the descriptive statistics, SBUX has the highest arithmetic mean and VBLTX has the highest geometric mean. SBUX and FMAGX are more negatively skewed. The excess kurtosis means that all three distributions have fatter tails. Although SBUX seem to have a relatively higher return, it also bears the highest risk of all three because it has the highest variance and standard deviation. 2. Calculated with R: > # annualized cc mean > 12*apply(ret.mat, 2, mean) VBLTX FMAGX SBUX > # annualized simple mean > exp(12*apply(ret.mat, 2, mean)) - 1 VBLTX FMAGX SBUX The estimated annual simple returns are consistent with the annual continuously compounded returns, all three with a higher value.

7 3. Calculated with R: > # annualized sd values > sqrt(12)*apply(ret.mat, 2, sd) VBLTX FMAGX SBUX The estimated annualized standard deviations are all roughly 4 times higher than the standard deviations calculated in the descriptive statistics in Part 1. III. Bivariate Graphical Analysis From the plot, we can see that there s no correlation between VBLTX and FMAGX. There appears to have no correlation between VBLTX and SBUX as well. There is a weak, positive correlation between FMAGX and SBUX. IV. Bivariate Numerical Summary Statistics > var(ret.mat) VBLTX FMAGX SBUX VBLTX 6.748e e FMAGX 8.035e e SBUX e e > cor(ret.mat) VBLTX FMAGX SBUX VBLTX FMAGX SBUX

8 Using the var() and cor() function in R, we can find a very weak positive correlation VBLTX and FMAGX and a very weak negative correlation between VBLTX and SBUX. There is a positive correlation between FMAGX and SBUX, which is about V. Time Series Summary Statistics From the plots of autocorrelation functions graphed in R, the returns appear to be uncorrelated over time. Part II Constant Expected Return Model 1. Calculated with R: > cbind(muhat.vals,sigma2hat.vals,sigmahat.vals) muhat.vals sigma2hat.vals sigmahat.vals VBLTX FMAGX SBUX > cbind(covhat.vals,rhohat.vals) covhat.vals rhohat.vals VBLTX,FMAGX 8.035e VBLTX,SBUX e FMAGX,SBUX 3.035e Part I. The estimates (i.e. sample covariance and correlation) match our computation in

9 2. Calculated with R: > cbind(muhat.vals,se.muhat) muhat.vals se.muhat VBLTX FMAGX SBUX > cbind(sigma2hat.vals,se.sigma2hat) sigma2hat.vals se.sigma2hat VBLTX FMAGX SBUX > cbind(sigmahat.vals,se.sigmahat) sigmahat.vals se.sigmahat VBLTX FMAGX SBUX > cbind(rhohat.vals,se.rhohat) rhohat.vals se.rhohat VBLTX,FMAGX VBLTX,SBUX FMAGX,SBUX The precision of the variance and standard deviation are pretty high, as the standard errors are small for them. However, the precision of mean and correlation are low because the standard errors for them are substantially higher. 3. Calculated with R: 95% Confidence Interval > cbind(mu.lower,mu.upper) mu.lower mu.upper VBLTX FMAGX SBUX > cbind(sigma2.lower,sigma2.upper) sigma2.lower sigma2.upper VBLTX FMAGX SBUX > cbind(sigma.lower,sigma.upper) sigma.lower sigma.upper

10 VBLTX FMAGX SBUX > cbind(rho.lower,rho.upper) rho.lower rho.upper VBLTX,FMAGX VBLTX,SBUX FMAGX,SBUX % Confidence Interval > cbind(mu.lower,mu.upper) mu.lower mu.upper VBLTX FMAGX SBUX > cbind(sigma2.lower,sigma2.upper) sigma2.lower sigma2.upper VBLTX FMAGX SBUX > cbind(sigma.lower,sigma.upper) sigma.lower sigma.upper VBLTX FMAGX SBUX > cbind(rho.lower,rho.upper) rho.lower rho.upper VBLTX,FMAGX VBLTX,SBUX FMAGX,SBUX The CI for σ 2 and σ are fairly narrow, and that for opposite for μ and ρ are wider. 4. Calculated with R: > Value.at.Risk(ret.mat,p=0.05,w=100000) VBLTX FMAGX SBUX > Value.at.Risk(ret.mat,p=0.01,w=100000) VBLTX FMAGX SBUX VBLTX has the lowest value-at-risk.

11 Part III Ruppert Exercise 1. Plot with R: The time series look stationary. The fluctuations in the series seem to be of constant size, but it seems to be more volatile toward the end of 1997 and Plot with R: The series look stationary. The fluctuations in the series seem to be of constant size, roughly in between and Plots with R:

12 The marginal distribution of each series appear to be symmetric. However, the tails of these distributions do not appear to be normal and seem to be heavier than normal.