F537.s2024.HW06

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Finance 537 Financial Risk Management Spring 2024 Neil D. Pearson 1 Homework 6 Due Sunday, March 3, 2024 at 10:00 p.m. US central time Total 10 points Instructions . This is a group assignment. Groups may include up to 4 people. Please submit a Word or .pdf document with your solutions via the Canvas site prior to 10:00 p.m. on Sunday, March 3. You should also upload the computer codes and data files you use. Assignment. This assignment asks you use continuously compounded returns on the XLF to estimate several GARCH, NGARCH, and ARCH models and use them to forecast volatility. Column F of the .csv file “ETFreturns.csv contains simple returns on the XLF. You should convert these simple returns to continuously compounded returns, and then use the most recent 1,000 returns. These are the returns in rows 4,789 through 5,788. 1. GARCH(1,1) estimated 2 ways. (total 2 points) (a) (1 point) The GARCH(1,1) likelihood can be written as a function of the four parameters α , β , σ , and σ 1 . Use the 1,000 continuously compounded returns on the XLF to estimate the GARCH(1,1) parameters, treating all four as unknown parameters to be estimated. What are the estimate of the four parameters α , β , σ , and σ 1 ? (Note that the question asks for the estimates of σ , and σ 1 , not σ 2 and σ 1 2 .) Remark : You should do this using R and one of the general optimization functions such as optim() . To do this you will have to write your own R function to compute the negative of the log likelihood. The last page of this assignment displays an example of a function that computes the (negative of the) GARC(1,1) log likelihood. (b) (1 point) Using the same data, set both the long-run variance σ 2 and the initial variance σ 1 2 equal to the sample variance of the data used for estimation. If you do this, the GARCH(1,1) likelihood can be written as a function of the two unknown parameters α and β . Use maximum likelihood to estimate these two parameters. What are your estimates of the long-run standard deviation σ and the initial standard deviation σ 1 ? What are your estimates of the parameters α and β ? 2. Using the GARCH(1,1) model to forecast variance and standard deviation ( total 2 points ) This question asks you to use the estimates from Question 1(a) to compute forecast variance and standard deviation. (a) (1/2 point) Use the estimates from Question 1(a) to forecast the return variance and standard deviation for January 3, 2023, which is the first trading day of 2023. What are your estimates of the variance and standard deviation? (b) (1 point) Next, use the parameter estimates from Question 1(a) to forecast the variance of the 21-day return ∑ 𝑅𝑅 𝑡𝑡+𝑘𝑘 21 𝑘𝑘=1 . In this formula, assume that the current date t is just after the close of trading on December 30, 2022, and you have estimated the GARCH(1,1) model in Question 1(a). What is your forecast of the variance of the 21-day return ∑ 𝑅𝑅 𝑡𝑡+𝑘𝑘 21 𝑘𝑘=1 ?

Finance 537 Financial Risk Management Spring 2024 Neil D. Pearson 2 (c) (1/2 point) Express your forecast in terms of the annualized volatility. (Take the square root, and then annualize by multiplying by 21 / 252 . Or else annualize by multiplying by 252/21, and then take the square root.) 3. GARCH(1,1) using existing R functions (total 2 points) In this question use the same continuously compounded returns you used in Question 1. (a) (1 point) Use the R function garch() from the R package tseries to estimate the parameters of a GARCH(1,1) model. ( https://www.rdocumentation.org/packages/tseries/versions/0.10- 42/topics/garch ) Does the function return estimates of all four parameters α , β , σ , and σ 1 ? If not, which parameters does it estimate? What are the estimated parameters? Hint : You might want to use your estimates from Question 1(a) as your initial guesses. (But you are not required to do this.) (b) (1 point) Now use the R function garchFit() from the R package fGarch to estimate the parameters of a GARCH(1,1) model. (See https://cran.r- project.org/web/packages/fGarch/fGarch.pdf ) Does the function return estimates of all four parameters α , β , σ , and σ 1 ? If not, which parameters does it estimate? What are the estimated parameters? 4. N GARCH(1,1) (total 2 points) The NGARCH(1,1) likelihood can be written as a function of the five unknown parameters α , β , θ , σ , and σ 1 . (a) (1 point) Use the continuously compounded XLF returns to estimate the NGARCH(1,1) parameters, treating all five as unknown parameters to be estimated. What are the estimated parameters α , β , θ , σ , and σ 1 ? Remark : You should do this using R and one of the general optimization functions such as optim() . To do this you will have to write your own R function to compute the (negative of the) log likelihood. (b) (1 point) Using the results from Questions 1(a) and 4(a), carry out a likelihood ratio test of the hypothesis that θ = 0. 5. ARCH(1) (2 points) The ARCH(1) model is a special case of the GARCH(1,1) model in which β = 0. (a) (1 point) Please use the XLF returns data to estimate the parameters α , σ , and σ 1 of the ARCH(1) model. (b) (1 point) Compare the performance of the GARCH(1,1) model from Question 1(a) and the ARCH(1) model by carrying out a likelihood ratio test of the hypothesis that β = 0.

Finance 537 Financial Risk Management Spring 2024 Neil D. Pearson 3 Example of a function to compute the (negative of the) GARCH(1,1) log likelihood #Objective function (negative log likelihood) garch11a <- function(x) { sigmasqhat = rep(0.0,length(ret)) sigmasqhat[1] = x[4]^2 #(negative) log likelihood if (x[1]+x[2]>=1 || x[1]<0 || x[2]<0 || x[3]<0.001|| x[4]<0.001){ NeglogLH = 9999 } else { for (i in 1:(length(ret)-1)) { sigmasqhat[i+1] = (1-x[1]-x[2])*x[3]^2+x[1]*ret[i]^2+x[2]*sigmasqhat[i] } f <-(1/(sqrt(2*pi*sigmasqhat)))*exp(-0.5*ret^2/sigmasqhat) NeglogLH = -sum(log(f)) } return(NeglogLH) }

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