Description
1. Mean excess function (1.5 points) For each threshold u ∈ {0.010, 0.012, 0.014, …, 0.05}
compute the mean excess loss e(u) given by the second displayed equation on slide 11 of the
lecture notes EVT and VaR.ppt. (You should compute 21 different mean excess losses.) Prepare
a graph that shows the mean excess loss e(u) as a function of the threshold u. (The threshold u
should be on the horizontal axis.)
2. Losses beyond the threshold (0.5 point) Examining the mean excess function, you decide
that the mean excess function is linear beyond u = 0.022. Thus, you decide that the GPD can be
used to approximate the conditional density of excess losses beyond the threshold u = 0.022.
How many losses are greater than or equal to 0.022?
3. Maximum likelihood estimation of the GPD parameters (5 points) Using the losses X
such that X ≥ 0.022, compute the excess losses X – u = X – 0.022. Then use the excess losses,
the density function of the GPD, and the method of maximum likelihood to estimate the
parameters ξ and β of the GPD. What are your estimates of the parameters ξ and β?
4. Conditional density function (1 point) Using the density function of the GPD and your
parameter estimates from Question 3, create a figure that shows the conditional density function
of the losses, condition on the loss being greater than or equal to 0.022. Your figure should show
the conditional density for losses between 0.022 and 0.10.
5. Probabilities of various losses (2 points) What is your estimate of the probability that the
loss X is greater than or equal to 0.022? (That is, what is your estimate of P(X ≥ 0.022)?) What
is your estimate of P(X ≥ 0.05)? What is your estimate of P(X ≥ 0.10)? Create a figure showing
P(X ≥ x), for x ∈ [0.022, 0.10].
