Hi

my question is NO.3 in the attached file. I have calculated the limit. (a) is zero, and (b) is infinite. is that right? I just want to know how I can comment on the implications of my result for piratical applications in risk management.

and about question 4, I am quite confused about the question and the code? and how to write the code.

ACTSC 445/845 - Fall 2016 ASSIGNMENT 1 Due: Thursday, October 6, at 3:50pm. There is a (physical) drop box in MC across from MC 4066 to hand in hard copies of assignments. There is also a (virtual) dropbox on the course website where you can submit soft copies. Notes: You may work on this assignment individually or in groups of up to two people. If you work in a group, you should submit a single solution with each student's name and ID number. If you work in a group, each student in the group will receive the same mark. There will be no extra credit for students who work individually. 1. (30 points) Research and write a brief (6 2 pages) summary of one of the following failures of risk management: The bankruptcy of Lehman Brothers in September 2008. The bankruptcy of Orange County in 1994. The collapse of the hedge fund Long Term Capital Management in 1998. The Space Shuttle Columbia disaster in 2003. The nuclear accident on Three Mile Island in 1979. In your summary, be sure to include an identification and description of the types of risks that were taken, a description of how the risk management processes in place failed, and recommendations for improved risk management processes that could help prevent such disasters in the future. The grading scheme is as follows. 5 points for the description of the disaster, 10 points for the identification and description of the types of risks that were taken, 10 points for identifying risk management failures and making recommendations for improvements, and 5 points for style, organization, proper citation of references, etc. 1 2. (3 points) Suppose that X and Y are i.i.d. Bernoulli random variables with P(X = 0) = 0.96, P(X = 1) = 0.04. Calculate CTE0.95 (X), CTE0.95 (Y ), and CTE0.95 (X + Y ). 3. (12 points) Evaluate: R = lim 1 CTE (L) VaR (L) when: (a) L is normally distributed with mean and standard deviation . (b) The cumulative distribution function of L is F (x) = 1 (1 + x) , x > 0, > 1. Comment on the implications of your result for practical applications in risk management. 4. (20 points) The continuously compounded annual risk-free rate is r = 0.02. The current price of a stock is S0 = 100. You have just sold one million straddles (one put and one call option, with the same strike and maturity) with strike price K = 100 on the stock, and T = 1 year to maturity for 15.85 (million) and invested the proceeds in a risk-free zero coupon bond, also maturing in one year (recall that the payoff of the put option will be max(K ST , 0), where ST is the stock price at time T , and the payoff of a call will be max(ST K, 0)). Let = 0.08 and = 0.2. You are considering two possible models for the random behaviour of the stock of the form ST = S0 exp( + Z). In Model A, Z has a standard normal distribution, while in model B, Z has a Laplace distribution with probability density f (z) = 21 e|z| , cumulative distribution function: ( 1 z e, z (, 0] 2 F (z) = 1 z 1 2 e , z [0, ) and quantile function: ( ln(2), [0, 12 ] F 1 () = ln(2(1 )), [ 21 , 1] For both models A and B: 2 Using whatever software you like generate a pseudo-sample of 2000 values for the payoff at T from the portfolio. Use your pseudo-sample to estimate a 95% confidence interval for the 99% VaR for the loss at T . Use your pseudo-sample to estimate the 90% CTE for the loss at T. Estimate a 95% confidence interval for the 99% CTE. Briefly comment on your results. Submit a soft copy of your code and sample output to the dropbox on the course webpage. (Note: for both models A and B, the random variable Z can be simulated by first simulating a uniform random number U and then setting Z = FZ1 (U ).) 3