Hi Could you help me to solve question 5 and 6? Thank you so much!

RSM\t432:\tWinter\t2015 Risk\tManagement for\tFinancial\tManagers Assignment\t1 Assignment 1 This assignment must be submitted electronically by 9 pm on their due date. Due date: All sections: Friday February 13th, 2015 by 9pm Late submission will be penalized. The assignment questions are to be completed in a group. Do not share your work with students from the other groups. Only original work done by your group members are to be submitted. Plagiarism will not be tolerated in this class. You can complete all the problems entirely in the MS Excel spreadsheet. You can also complete it using word documents with Excel spreadsheet attachments. The assignment will be marked based on (1) How you arrive at the solution, (2) Is the solution correct and/or does it make sense? (3) The presentation of your results. Remember, you must present your work in a clear and concise manner. Show your work! Q1 (10 points) - Simple application of VaR A company estimates that its profit over the next five years is normally distributed with a mean of 1% of assets per year and the standard deviation of 2% of assets per year. (a) How much equity (as a percentage of assets) does the company currently need to have in order to be 99% sure that its equity will not fall below 3% of its assets at the end of two years. (b) If the company currently has equity that is 3% of its assets, what is the probability that all of its equity will be below 1% of its assets at the end of three years? Ignore taxes. Remember to show your work. Page 1 of 6 RSM\t432:\tWinter\t2015 Risk\tManagement for\tFinancial\tManagers Assignment\t1 Q2 (20 points) - VaR at different horizons A portfolio manager estimates that the volatility of her daily portfolio returns is 4.3% in a monthly term. She also expects this portfolio to bring a return of 5% per year. Assume that there are 252 trading days in a year. The current value of her portfolio is $10,000,000. (a) (5 points) Calculate a 5-day VaR ($) at the 99% confidence level (b) (5 points) What is the 95% confidence interval of the value of this portfolio after 21 days? (c) (5 points) Suppose the value of a portfolio dropped by 5% in 10 days. At what confidence level, would this be considered as breaching a VaR ($)? (d) (5 points) Up to what horizons (number of days) can she hold this portfolio with a 99% confidence that her total loss will not be more than 40%? Show your work. State your answer in number of days. Q3 (10 points) - Simple volatility question Suppose that in a GARCH(1,1) model, = 0.000003 (or 3*10-6) and persistence is 97%. a) (2 points) What is the long-run average daily variance? What is the long-run average monthly volatility? Assume that there are 21 trading days in a month. b) (4 points) If the current daily volatility is 0.5% per day, what is your estimate of the annualized volatility in 10, 30, and 50 days? Is your estimate increasing or decreasing in number of days? Briefly explain why it is increasing or decreasing. c) (4 points) Suppose that there is an event that increases the current daily volatility to 2% per day. Estimate the effect on the volatility in 10, 30, and 50 days. Again, is your estimate increasing or decreasing in number of days after the event? Discuss why. Page 2 of 6 RSM\t432:\tWinter\t2015 Risk\tManagement for\tFinancial\tManagers Assignment\t1 Q4 (30 points) - Simple VaR using real financial data You will need the data set provided to you on the course website. Use the data reported in the worksheet labeled \"DataProblem4\". The data set contains the daily prices of four securities from January 1997 to December 1998. The four securities are: S&P500 index, $/Yen exchange rate, Crude Oil price, and the TSE300 index. (a) (5 points) Compute annualized return and volatility for each of these four securities. Express your results in percentage terms. Don't forget that the data are in daily frequency and use logarithmic return when computing daily returns. That is, ln Compare the returns and volatilities between these four securities. (b) (25 points) Compute 1-day and 10-day VaR for these four securities at the 99% confidence level. Express VaR in percentage terms. (i) Construct a portfolio of these four securities. The weight that you will assign to each security is up to you. However, you must use all four securities and short selling is not allowed. Recall that your portfolio weights must sum to 1. That is, 1 . State clearly your choice of the portfolio weights. The portfolio weights that you choose will be used in this question as well as in questions 5 & 6. Compute and plot your daily portfolio returns. (ii) Assume that the four securities are normally and independently distributed; calculate the daily volatility of the portfolio that you constructed. Under this assumption, the variance of the portfolio is given by where , is the variance of the security i. Using the volatility that you calculated using Eq. 1, compute the 1-day VaR of your portfolio at the 99 % confidence level. Page 3 of 6 (Eq. 1) RSM\t432:\tWinter\t2015 Risk\tManagement for\tFinancial\tManagers Assignment\t1 (iii) Do not assume that these four securities are independently distributed. Consequently, Eq. 1 no longer applies. Calculate the volatility of your portfolio directly using your daily portfolio returns. After, calculate the 1-day and 10-day VaR of your portfolio at the 99% confidence level. (iv) Compare the 1-day VaR values that you calculated from (ii) and (iii). What do you find? What do you think about the assumption that these four securities are independently distributed? (v) Compute 1-day Expected Shortfall (ES) of your portfolio at the 99 % confidence level using your estimated 1-day 99 % VaR from (iii). Assume that your portfolio return is normally distributed. Q5 (20 points) - Back testing your VaR calculated from Question 4 We will test whether your 1-day VaR calculated from Question 4 is a good risk measure. Use the VaR calculated using the actual volatility of your portfolio returns (i.e., from (iii) in Q4). We will back test your 1-day VaR using the security prices from January 1999 to December 2001. The data is located in the worksheet labelled \"DataProblem5and6\". (a) (5 points) Assuming that your portfolio holding is the same as in Question 4, calculate daily returns of your portfolio from 1999 - 2001. Plot the results. (b) (7 points) Report the number of times that 1-day 99% VaR of your portfolio is violated during 1999 - 2001. Note that the violation occurs when the portfolio return falls greater than what your VaR measure suggests. Calculate the violation rate of your 1-day VaR measure. Show your work. If your VaR measure is good, what do you expect the violation rate to be? (c) (5 points) Using the statistical method introduced in class (Binomial distribution), test whether your 1-day 99% VaR measure is an acceptable model. Show your work. Page 4 of 6 RSM\t432:\tWinter\t2015 Risk\tManagement for\tFinancial\tManagers Assignment\t1 (d) (3 points) Based on the violations you observed in (b), compute 1-day Expected Shortfall (ES) of your portfolio at the 99 % confidence level. Do not assume that your portfolio return is normally distributed. Compare the results that you calculated from Question 4. b. (v). Q6 (20 Points) - Applying Risk Metric model to improve your VaR measure We will try to improve on your previous VaR measure by estimating the volatility of your portfolio daily using the Risk Metric model. Recall from your textbook, the Risk Metric model updates the next period variance according to 0.94 0.06 (a) (10 points) Apply the risk metric model to estimate the daily variance of your portfolio from January 1999 to December 2001. For the starting value of variance, on t=0, you can use the longrun variance estimated using your portfolio returns from 1997-1998. Using the new daily volatilities that you estimated, calculate the 1-day VaR of your portfolio return at the 99% confidence level. Plot the results of your 1-day VaR. Show your work. (b) (5 points) Report the number of times that your 1-day 99% VaR calculated using the Risk Metric model is violated during 1999-2001. Calculate the violation rate of your 1-day VaR measure. Show your work. (e) (5 points) Using the statistical method introduced in class (Binomial distribution), test whether your 1-day 99% VaR calculated using the Risk Metric model is an acceptable measure. Show your work. Page 5 of 6 RSM\t432:\tWinter\t2015 Risk\tManagement for\tFinancial\tManagers Assignment\t1 Q7 (15 Points) - S&P 500 index returns For this section, you will be using the S&P 500 index price data that is provided to you in the worksheet labeled \"Problem7Data\". The data contains S&P 500 index price from 1962 to 2013. (a) (5 points) Explain what is the S&P 500 index? What kinds of firms are included in this index and who decides the composition of the index? Be concise with your answer. (b) (10 points) Select two consecutive years of data from the S&P 500 price after 2000. For example, you may choose 2011-2012, or 2009-2010, etc. You can choose the years that interest you. (i) Plot the cumulative S&P 500 returns over the two years that you choose. (ii) Calculate daily variance of your selected S&P 500 returns data using the Risk Metric model. Plot the time-series of the daily volatilities that you calculated. Now, assume the following parameters of GARCH (1,1) model: = 0.12, = 0.86, = 6.0E-07. Calculate daily variances of your selected S&P 500 returns data using GARCH (1,1) with the given parameters. Plot the time-series of the daily volatilities you calculated. Page 6 of 6