## 6.5 Problems: The GWN Return Mode

Exercise 6.1 Consider the GWN model for cc returns:

$r_{t}=\mu+\epsilon_{t},\,\epsilon_{t}\sim\mathrm{GWN}(0,\sigma^{2}).$

It implies that the log price follows a random walk with drift:

$\ln P_{t}=\ln P_{t-1}+r_{t}=\ln P_{0}+\mu t+\sum_{s=1}^{t}\epsilon_{s}.$

Show that $$E[\ln P_{t}]$$ and $$\mathrm{var}[\ln P_{t}]$$ depend on $$t$$ so that $$\ln P_{t}$$ is non-stationary.

Exercise 6.2 Consider using Monte Carlo simulation to evaluate the GWN model for the monthly simple returns on Amazon stock and the S&P 500 index over the 5 year period January 2009, through January 2014.
1. Daily adjusted closing prices on Amazon stock and the S&P 500 index are in the IntroCompFinR "xts" objects amznDailyPrices and sp500DailyPrices, respectively. Using these prices, create monthly returns over the period January 2009, through January 2014.
2. Create a time plot showing the monthly returns on the two assets. Do the monthly returns from the two assets look like realizations from a covariance stationary stochastic process? Why or why not?
3. Compare and contrast the return characteristics of the two assets. In addition, comment on any common features, if any, of the two return series.
4. Using the IntroCompFinR function fourPanelPlot(), create graphical descriptive statistics for the monthly returns. Also, create the scatterplot between the two return series. Do the returns look normally distributed? Is there any evidence for linear time dependence? Do the return appear to contemporaneously correlated?
5. Compute sample descriptive statistics (mean, standard deviation, skewness, kurtosis, and correlation between Amazon and the S&P 500 index).
6. In the GWN model for simple returns prices do not follow a random walk but instead evolve over time according to $P_{t} = P_{t-1}(1 + R_{t})$

To be completed…