FNCE 5321: Finanical Risk Modeling II
Spring 2018
Group Project 1
Instructions: You can work in groups of up to 5 students. Please hand in one hard copy
of project report per group, and indicate clearly the members of your group on the rst page
of your submitted report.
Question 1
For Question 1, use the data in Q1Data.csv, which contains daily closing prices on a
stock market index for the period from July 2nd 1962 to December 30th 2016.
(A) Calcuate daily log returns Rt+1 ln(St+1) ln(St), where St+1 is the closing price on
day t+ 1, St is the closing price on day t, and ln( ) is the natural logarithm. Plot the
closing prices and returns over time.
(B) Calculate and report the mean and standard deviation of daily log returns.
(C) Calculate the rst through 100th lag autocorrelations of daily log returns. Plot the
autocorrelations against the lag order.
(D) Calculate the rst through 100th lag autocorrelations of squared daily log returns. Plot
the autocorrelations against the lag order. Do squared daily log returns have more or
less autocorrelations compared with daily log returns?
(E) Suppose you own a portfolio which is 100% invested in this market index. Estimate
the 1-day, 1% VaRs on each day assuming daily log returns are normally distributed
and updating volatility estimates with the RiskMetrics model, i.e.
2t+1 = 0:94 2t + 0:06R2t
where Rt is the daily log return at day t, and t+1 is the standard deviation of Rt+1.
Plot the estimated VaRs. (Hint: Use the standard deviation from Part (B) as 1 to
start the iteration.)
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(F) Estimate the 1-day, 1% VaRs on each day using Historical Simulation with a 250-day
moving estimation window. Plot the estimated VaRs.
(G) What are the major di erences between the two VaRs estimated in Part (E) and Part
(F)?
(H) Estimate the 1-day, 5% VaRs on each day using Historical Simulation with a 250-day
moving estimation window. Are the 5% VaRs smaller or larger than the 1% VaRs
estimated in (F)? Explain why with intuition.
(I) Formally test whether the daily closing prices and the daily log returns are stationary
using the augmented Dickey-Fuller tests.
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Question 2
(A) Suppoese Xt follows an AR(1) model: Xt = 0:9Xt 1 + t, where = 0:5. Simulate
an episode of Xt with 1 million observations (Hint: Use the R command arima.sim).
Calculate the rst through 10th lag autocorrelations of the simulated series. Compare
the estimated autocorrelations with the theoretical counterparts (Hint: the theoretical
autocorrelations for AR(1) is = 1, where is the lag.)
(B) Suppose Zt = Xt +Yt, where Xt is the same as in (A), and Yt follows an AR(1) model:
Yt = 0:5Yt 1 + t, where = 0:5. Is an AR(1) model appropriate to describe Zt?
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Question 3
For Question 3, use the data in Shillerdata.csv, which contains the monthly closing
price, dividends, earnings, and the PE ratio for S&P 500.
(A) Formally test whether the dividend series is stationary.
(B) Calculate the PD ratio on S&P 500 as follows
PD = PriceDividends
Both PD and PE are called valuation ratios. What is the unconditional correlation
between the two series? Formally test whether the PD ratio is stationary.
(C) Rede ne the PD ratio in logs
PD = log(Price) log(Dividends)
Formally test whether the PD ratio is stationary.