4. We collect the daily log-returns of the S&P500 index for the past one year {r}(let us consider that 1 year has 250 business days). Based on which, we obtain the esti- mates of the annualized mean and standard deviation as ê= -5.05% and ô = 21.67%, respectively. After standardization, we obtain standardized daily log-returns (sr) where sr₂ = (r₂ - At)/(√At) and At = 1/250. Fitting the standardized daily log- returns with Generalized Error Distribution (GED) with the shape parameter , we found that the MLE of is 1.477. Our aim is to obtain the 5-day Value-at-Risk (VaR) of S&P500 log-return at a 99% confidence level. Write pseudo-codes that calculate the desired VaR, where you should simulate 5 daily log-returns (sum of them will be 5-day log-return) in each iteration as the GED fitting is for daily data. Within the algorithm, you may need to determine the upper bound of ratio function in acceptance-rejection method (the alternative density is suggested to be double exponential density). Please report the upper bound value (find it numerically but it is not necessary to post the process) as well as the final VaR value, based on 1000 simulations, i.e. you should implement the pseudo-codes. [Optional exercise] you may repeat the exercise above but obtain 1-day Value-at-Risk instead. Based on this and the previous results, examine whether the holding period adjustment formula is applicable or not. [Optional Survey] How do you rate the difficulty of this assignment (1=easy, 5-hard)? Any feedback about lectures, assessment, and labs/tutorials overall?

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4. We collect the daily log-returns of the S&P500 index for the past one year {r}(let us consider that 1 year has 250 business days). Based on which, we obtain the esti- mates of the annualized mean and standard deviation as ê= -5.05% and ô = 21.67%, respectively. After standardization, we obtain standardized daily log