Chen-wei Chiu ECON 424 Eric Zivot July 17, 2014 Part I Descriptive Statistics I. Univariate Graphical Analysis 1. Separate & Same Graph Lab 4
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.
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 2008. However, SBUX had another rapid growth toward the end of 2008. On the other hand, returns on FMAGX had not grown much throughout the years, and it had a sharp fall in the August of 2008. Given the plot, SBUX has the best future values over the investment horizon while FMAGX has the worst. 3. VBLTX
FMAGX SBUX
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.
II. Univariate Numerical Summary Statistics 1. Calculated with R: > table.stats(lab4returns.z) VBLTX FMAGX SBUX Observations 143.0000 143.0000 143.0000 NAs 0.0000 0.0000 0.0000 Minimum -0.0894-0.2434-0.4803 Quartile 1-0.0098-0.0205-0.0487 Median 0.0090 0.0099 0.0181 Arithmetic Mean 0.0053 0.0019 0.0113 Geometric Mean 0.0050 0.0002 0.0035 Quartile 3 0.0198 0.0397 0.0868 Maximum 0.1077 0.1302 0.2775 SE Mean 0.0022 0.0047 0.0100 LCL Mean (0.95) 0.0010-0.0075-0.0084 UCL Mean (0.95) 0.0096 0.0112 0.0311 Variance 0.0007 0.0032 0.0143 Stdev 0.0260 0.0566 0.1194 Skewness -0.1439-1.0434-0.9083 Kurtosis 2.9528 2.7027 2.7061 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 0.06355 0.02266 0.13576 > # annualized simple mean > exp(12*apply(ret.mat, 2, mean)) - 1 VBLTX FMAGX SBUX 0.06561 0.02292 0.14541 The estimated annual simple returns are consistent with the annual continuously compounded returns, all three with a higher value.
3. Calculated with R: > # annualized sd values > sqrt(12)*apply(ret.mat, 2, sd) VBLTX FMAGX SBUX 0.08998 0.19617 0.41360 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-04 8.035e-05-0.0001764 FMAGX 8.035e-05 3.207e-03 0.0030347 SBUX -1.764e-04 3.035e-03 0.0142556 > cor(ret.mat) VBLTX FMAGX SBUX VBLTX 1.00000 0.05463-0.05689 FMAGX 0.05463 1.00000 0.44883 SBUX -0.05689 0.44883 1.00000
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 0.45. 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 0.005296 0.0006748 0.02598 FMAGX 0.001889 0.0032069 0.05663 SBUX 0.011313 0.0142556 0.11940 > cbind(covhat.vals,rhohat.vals) covhat.vals rhohat.vals VBLTX,FMAGX 8.035e-05 0.05463 VBLTX,SBUX -1.764e-04-0.05689 FMAGX,SBUX 3.035e-03 0.44883 Part I. The estimates (i.e. sample covariance and correlation) match our computation in
2. Calculated with R: > cbind(muhat.vals,se.muhat) muhat.vals se.muhat VBLTX 0.005296 0.002172 FMAGX 0.001889 0.004736 SBUX 0.011313 0.009984 > cbind(sigma2hat.vals,se.sigma2hat) sigma2hat.vals se.sigma2hat VBLTX 0.0006748 0.0000798 FMAGX 0.0032069 0.0003793 SBUX 0.0142556 0.0016859 > cbind(sigmahat.vals,se.sigmahat) sigmahat.vals se.sigmahat VBLTX 0.02598 0.001536 FMAGX 0.05663 0.003349 SBUX 0.11940 0.007060 > cbind(rhohat.vals,se.rhohat) rhohat.vals se.rhohat VBLTX,FMAGX 0.05463 0.08337 VBLTX,SBUX -0.05689 0.08335 FMAGX,SBUX 0.44883 0.06678 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 0.0009515 0.00964 FMAGX -0.0075827 0.01136 SBUX -0.0086557 0.03128 > cbind(sigma2.lower,sigma2.upper) sigma2.lower sigma2.upper VBLTX 0.0005152 0.0008344 FMAGX 0.0024484 0.0039654 SBUX 0.0108838 0.0176274 > cbind(sigma.lower,sigma.upper) sigma.lower sigma.upper
VBLTX 0.02290 0.02905 FMAGX 0.04993 0.06333 SBUX 0.10528 0.13352 > cbind(rho.lower,rho.upper) rho.lower rho.upper VBLTX,FMAGX -0.1121 0.2214 VBLTX,SBUX -0.2236 0.1098 FMAGX,SBUX 0.3153 0.5824 99% Confidence Interval > cbind(mu.lower,mu.upper) mu.lower mu.upper VBLTX -0.001221 0.01181 FMAGX -0.012318 0.01610 SBUX -0.018640 0.04127 > cbind(sigma2.lower,sigma2.upper) sigma2.lower sigma2.upper VBLTX 0.0004354 0.0009142 FMAGX 0.0020691 0.0043447 SBUX 0.0091979 0.0193133 > cbind(sigma.lower,sigma.upper) sigma.lower sigma.upper VBLTX 0.02137 0.03058 FMAGX 0.04658 0.06668 SBUX 0.09822 0.14058 > cbind(rho.lower,rho.upper) rho.lower rho.upper VBLTX,FMAGX -0.1955 0.3047 VBLTX,SBUX -0.3070 0.1932 FMAGX,SBUX 0.2485 0.6492 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 -3674-8722 -16896 > Value.at.Risk(ret.mat,p=0.01,w=100000) VBLTX FMAGX SBUX -5364-12177 -23390 VBLTX has the lowest value-at-risk.
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 1998. 2. Plot with R: The series look stationary. The fluctuations in the series seem to be of constant size, roughly in between -0.02 and 0.02. 3. Plots with R:
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.