Christian-Albrechts-Universität zu Kiel
Mathematisches Seminar
Prof. Dr. Jan Kallsen
Mark Feodoria
WS 2013/14
Sheet 11
Risk Management
Exercises for participants of the programme Quantitative Finance
C-Exercise 19
(a) Write a scilab-function
tau = Kendall(x),
which estimates and returns Kendall’s tau ρτ (X1 , X2 ) for iid samples of a random vector
X = (X1 , X2 ).
(b) Write a scilab-function
rho = Spearman(x),
which estimates and returns Spearman’s rho ρS (X1 , X2 ) for iid samples of a random
vector X = (X1 , X2 ).
(c) Assume that the log returns of DAX and S&P 500 time series on webpage of this course
are iid samples from a random vector (X1 , X2 ). Estimate the correlation coefficients
ρ(X1 , X2 ), Kendall’s tau ρτ (X1 , X2 ) and Spearman’s rho ρS (X1 , X2 ). Plot the common
daily log returns.
(d) Estimate the mean µ and the covariance matrix Σ of (X1 , X2 ) with appropriate estib Simulate N = 5901 iid samples of a N(µ
b distribution. Plot these
b and Σ.
b , Σ)
mators µ
samples and estimate Kendall’s tau and Spearman’s rho.
Please turn over.
C-Exercise 20
(a) Write a scilab-function
[VaR, ES] = VaR_ES_historic_biv (x_data, l, alpha),
c α of the historical simulation method for
d α and ES
which computes the estimates VaR
given historical risk factor changes x_data = (x1 , . . . , xn ) ∈ Rn×2 , a two-dimensional
loss operator l : R2 → R and level α ∈ (0, 1).
(b) Consider a portfolio with initial value of 1000 e, that always invests 50% of the current
portfolio value in the DAX and 50% in the S&P 500. (We interprete the current
index levels as stock prices in e.) Compute for each trading day m = 254, . . . , 5814
the estimates for value at risk and expected shortfall at level α = 0.99. Apply the
function from (a) on the last n = 252 risk factor changes (xm , xm−1 , . . . , xm−n+1 ). Plot
the estimates. Compute the number of violations, i.e. the days when the actual loss lies
above the estimated VaR, and compare it with the theoretical number of violations.
Please save your solution of each C-Exercise in a file named Exercise_##.sce, where
## denotes the number of the exercise. Please include your name(s) as comment in the
beginning of the file.
Submit until: Friday, 07.02.2014, 08:30 (before the lecture)
Discussion:
in tutorials on Fri, 07.02.2014, 14:15-15:45 (LMS4-R.124)

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