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I. Financial Price and Return
Consider a 60-month (5 year) investment in two assets: the Vanguard S&P 500 index (VFINX) and Amazon stock (AMZN). Suppose you buy one share of the S&P 500 fund and one share of Amazon stock at the end of January, 2011 for Pvfinx ,t−60 = 108, Pamzn ,t−60 = 170 , and then sell these shares at the end of January, 2016 for Pvfinx ,t = 179, Pamzn ,t = 587 . (Note: these are actual adjusted closing prices taken from Yahoo!). In this question, you will see how much money you could have made if you invested in these assets right after the financial crisis.
a. What are the simple 60-month (5-year) returns for the two investments?
b. What are the continuously compounded (cc) 60-month (5-year) returns for the two investments? Why are the cc returns smaller?
c. Suppose you invested $1,000 in each asset at the end of January, 2011. How much would each investment be worth at the end of January, 2016?
d. What are the compound annual returns on the two 5 year investments?
e. At the end of January, 2011, suppose you plan to invest in a portfolio of VFINX and AMZN over the next 60 months (5 years). Suppose you purchase 10 shares of the VFINX mutual fund (at $108/share) and 10 shares of AMZN stock (at $170/share). What is the value of your initial investment? What are the portfolio weights in the two assets as of the end of January, 2011?
f. Using the results from part a. compute the 5-year simple and cc portfolio returns. What is the value of your portfolio at the end of January, 2016?
g. Go to http://finance.yahoo.com and download monthly data on a stock of your choice (except Starbucks and Amazon) over the period January, 2015 to January, 2024. Read the data into Excel and make sure to reorder the data so that time runs forward. Delete all columns except those containing the dates and the adjusted closing prices. Save the file as a .csv (comma separated value) file and call it examprice.csv.
(i) Plot the monthly closing price data of your stock using the plot() function. Please add a legend.
(ii) Compute monthly simple and continuously compounded returns. Plot these returns separately first. Then also plot on the same graph.
(iii) Calculate the growth of $1 invested in your stock, and report the plot of future values.
II. Constant Expected Return and Single Index Model
Consider the constant expected return (CER) model
rit = μi + εit , εit ~ iid N(0,σi2)
cov(rit , rjt ) = σij , cor(rit, rjt ) = ρij
for the monthly simple returns on the Vanguard S&P500 index (VFINX) and Amazon stock (AMZN) presented in part I above. Below are simulated returns and some graphical descriptive statistics for VFINX and AMZN from the CER model calibrated using the sample estimates of the CER model parameters for the two assets (these are the sample statistics from the table of the previous question).
a. Which features of the actual returns shown in part I are captured by the simulated CER model returns and which features are not?
b. Does the CER model appear to be a good model for VFINX and AMZN returns? Why or why not?
c. For each asset, compute estimated standard errors for ˆμ, ˆσ and ˆρ. Using the table below, show the estimates in one row and the standard errors in another row. (i) Briefly comment on the “co-movement” between the two assets. (ii) What do you expect for the sign of beta of AMZN if you estimate the Single Index Model for AMZN?
d. For Amazon (AMZN) only, compute 95% confidence intervals for mu. From this result, do we have any expected positive return for AMZN (statistically)?