# T df 4 normal 005 003 001 000 5 10 15 20 left tail

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t: df= 4 normal -0.05 -0.03 -0.01 0.00 0 5 10 15 20 left tail Density 0.00 0.01 0.02 0.03 0.04 0.05 0 5 10 15 20 right tail Density Figure 11: A plot of modeling log returns of the DAX index using: a kernel density estimate (KDE), a t -distribution with four degrees of freedom, and an normal distribution. is the Gaussian kernel. Problem 8 For the CAC index the QQ plots using t -distributions with different degrees of freedom are computed in the R code for this problem. From those plots it looks visually like six degrees of freedom fits the data best. Using this value we plot the parametric t -distribution with a KDE of the return data. These plots are given in Figure 12. Note that visually the curves agree quite well. Problem 9-10 The price series does not appear stationary. We see a steady climb of the assets value. The return series appears more stationary than the series of prices. Notice that the variance of the returns is non-constant. 35
-0.05 0.00 0.05 0 10 20 30 40 50 60 both tails N = 1859 Bandwidth = 0.001961 Density KDE t: df= 6 normal -0.05 -0.03 -0.01 0.00 0 5 10 15 20 left tail Density 0.00 0.01 0.02 0.03 0.04 0.05 0 5 10 15 20 right tail Density Figure 12: A plot of modeling log returns of the CAC index using: a kernel density estimate (KDE), a t -distribution with six degrees of freedom, and an normal distribution. Histogram of LogRet LogRet Density -0.04 0.00 0.02 0.04 0 10 20 30 40 50 60 -0.04 0.00 0.02 0.04 -3 -2 -1 0 1 2 3 Normal Q-Q Plot Sample Quantiles Theoretical Quantiles Figure 13: The plots for Problem 11. 36
Problem 11 These plots are given in Figure 13. The log returns do not appear normal (the Shapiro- Wilks) test will also agree with this statement). From the QQ plot we see that the data has heavier tails than a normal distribution. The log returns appear to be mostly symmetrically distributed with perhaps a bit more left skew. The tails appear to have about the same weighting i.e. none appears heavier than the other. Exercises See the R script chap 4.R where the exercises for this chapter are worked. Exercise 4.1 Part (a): We can use the summary command to compute some of these statistics > summary(ford.s[,2]) Min. 1st Qu. Median Mean 3rd Qu. Max. -0.1810000 -0.0099070 0.0000000 0.0007601 0.0112900 0.1040000 and the sd command to compute the standard deviation where we find the value of 0.01831557. Part (b): See Figure 14 where we plot a normal QQ plot for these returns. In that plot we see the classic mismatch that normal models have with return data, that is, the normal model does not fit the tails of the distribution very well. Part (c): The result from using the shapiro.test command is Shapiro-Wilk normality test data: ford.s[, 2] W = 0.9639, p-value < 2.2e-16 With a P -value this we can reject the null hypothesis of a normal distribution for the returns. Part (d): See Figure 15 where we produce QQ plot for a t -distribution of the Ford returns for various degrees of freedom. From that plot we see that the line that appears to match the data best is for six degrees of freedom.