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1. (10 points) Solve the set of nonlinear equations

in .x; y; z/0.

3x exp.􏰀y/ C exp.y/ C z D 5 logxClogyClogz D 􏰀1

xyCz D 2:

2. (10+10+15+10 points) Consider the data set with daily stock returns that was created in week 1. The dataset to be used in this problem is posted together with this problem set on nestor (don’t use another version).

Important measures in risk management are the Value-at-Risk (VaR) and the expected shortfall. The Value-at-Risk is a level of losses that is not exceeded with large probability. If we denote losses by the random variable L, with distribution function FL .l I 􏰅 /, the Value-at-Risk is defined formally as

VaR ̨ DF􏰀1. ̨I􏰅/; L

with 􏰅 the parameters of the distribution of losses. Associated with the Value-at-Risk is the expected shortfall ES ̨ which are the expected losses given that the Value-at-Risk is exceeded. Formally,

ES ̨ D E.LjL > VaR ̨/:
If losses follow a normal distribution (L 􏰄 N.􏰆;􏰇2/, one can derive that

VaR ̨ D 􏰆C􏰇ˆ􏰀1. ̨/; (1) 􏰈 􏰀ˆ􏰀1. ̨/􏰁

ES ̨ D 􏰆C􏰇 1􏰀 ̨ ; (2) with ˆ.􏰁/ the distribution function of the standard normal distribution, and

􏰈.􏰁/ the density of the standard normal distribution.
(a) Verify the validity of equations (1) and (2) by means of a simple sim-

ulation experiment.

You have seen during the lectures that the tails of the normal distribution are too thin to model daily returns correctly. One alternative is to model the returns according to a normal mixture distribution. The density is given by

􏰅x 􏰀 􏰆1 􏰊 􏰅x 􏰀 􏰆2 􏰊 f .x/ D 􏰃􏰈 􏰇 C .1 􏰀 􏰃/􏰈 􏰇 ;

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with 􏰈.􏰁/ the standard normal density, and 0 < 􏰃 < 1.

1. (b)  Fit normal mixture distribution to the daily returns of the AEX by programming a loglikelihood function and optimizing that function. Estimate all five parameters. Report both the point estimates and their standard errors.
2. (c)  Estimatetheleveloflossesthatisnotexceedwith99%probabilityus- ing the normal mixture distribution estimated in the previous subques- tion, and provide a 95% confidence interval using asymptotic normal- ity of the maximum likelihood estimator, and the delta method. Enter your estimate for VaR0:99 in the Google form on nestor (you need to login with your University of Groningen account).
3. (d)  Estimate the expected shortfall corresponding to the 99% Value-at- Risk, using the results of the previous two subquestions.

Note: as the data are given as daily returns (and not in Euro’s), the answers to the last two subquestions are expected to be in terms of daily returns as well.

3. (10 points) Let f .x1; x2/ D .x1 C x2/2. Show that the numerical approxi- mation to the Hessian as discussed in week 3 is exact.

4. (10+15 points) In practice, sometimes data are presented in the form of a table with (relative) frequencies. For example a table with individual pay- ments on a general loss insurance is presented in table 1.

Assume that the individual data underlying the frequencies in table 1 follow a translated Gamma distribution (Y follows a translated Gamma distribution if we have Y D c C X, with X 􏰄 􏰉. ̨; ˇ/, and c an additional parameter to be estimated).

1. (a)  Use the information in the table to fit a translated Gamma distribution by optimizing a loglikelihood function.
2. (b)  Estimate the probability an arbitrary payment exceeds 150, and pro- vide a 95% confidence interval for that estimated probability. State precisely any assumptions you make.