Problem 1 (70p): Risk modeling of a portfolio of stocks

Select a stock with ticker symbol that starts with your last name initial followed by first

name initial, for which there is available stock prices for the period 1/2/2009 to

11/18/2016. For example, if your name is Ann Brown, you can select BA, BAC, BAX

etc. In the exceptional case you cannot find such stocks, you can choose stocks that start

with your first name initial followed by last name initial, eg AB, ABC, ABX, ABT, or

with the first two letters identical to your first name or last name, eg AAP, BBY.

Download the historical daily prices (adj. close) for the selected stock and for Exxon

(ticker symbol XOM) for the period 1/2/2009 to 11/18/2016. You can obtain the data at

finance.yahoo.com

a. On 1/3/2011 (first trading day of 2011), you acquire 100 shares of XOM and 100

shares of the stock with your initials. Report the starting value (acquiring cost) of

your portfolio, and the value of your portfolio as of 11/18/2016. Assume that you

executed the trade few seconds before the market close, at the reported adjusted

close prices for that day. (For example, if the XOM price for 1/3/2011 is

$68.65/share, acquiring 100 shares costs $6865.)

b. Calculate the daily returns of your portfolio going back to January 2009, that is, as

if you owned the 100 shares of each stock starting in 1/2/2009. Report only the

average and standard deviation of these daily returns.

c. For the daily returns on your portfolio (going back to 2009), present a histogram

with a superimposed normal density with the same mean and standard deviation

(calculated previously). Produce also a qq-plot (versus a fitted normal) for the

portfolio returns. Are the returns on your portfolio normally distributed?

d. Compute the daily 5% Historical VaR and 5% Analytic VaR for your portfolio (in

percentages), for each day you hold the portfolio (that is, from 1/3/2011 to

11/18/2016), using all the available historical data up to that date (that is using the

data going back to 2009). Report only the HVaR and AVaR for 11/18/2016.

e. Backtest the Historical VaR and Analytic VaR model using Kupiecs test and

report the number of exceptions and your findings. Do you accept or reject the

model at 10%-significance level?

Problem 2 (10p): Stock Market Anomalies The Halloween Effect

Download the SP500 prices (adj. close) at monthly frequency for the period October 1979

to October 2016. You can obtain the data at finance.yahoo.com. The purpose of this

exercise is to confirm or disprove The Halloween effect. This effect refers to the popular

saying ``sell in May and go away, where it is believed that acquiring stocks (going long)

on the last trading day in October), and selling those positions at the end of April

produces superior returns when compared to a buy-and-hold strategy. Assume that Ann

invested $1000 in the SP500 at the end of October 1979. Ann was an ``active investor

and followed the ``sell in May and go away advice. At the end of each April she sold the

stock and placed her money in a checking account until the end of October. The checking

account pays 1% interest (for those 6 months). Then in the last trading day of October

(right before market closing time) she reinvested back all her money into the SP 500. Bob

invested $1000 in the SP500 at the end of October 1979. Bob was a ``passive investor

and let his money invested in the SP 500. How much money do Ann and Bob have at the

end of October 2016? Based on these calculations, is there a Halloween effect in the SP

500 index?

Problem 3 (20p): Simulating tosses of a fair coin

Consider a fair coin, with equal probability of a Head (H) or Tail (T). In a sequence of

coin tosses,

a. Which pattern is more probable to occur first, HH or HT?

b. What is the expected number of tosses until the pattern HH, respectively HT,

occurs?

c. What is the expected number of tosses needed until the pattern HH or TT occurs?

Answer the questions both analytically (providing exact results, with proofs), and via a

simulation, reporting the simulation results and checking that they are close to the

analytic results.

I was able to download the data for home depot and XOM but not quite sure what to do next