ASSESSMENT OF MARKET RISK IN HOG PRODUCTION USING VALUE-AT-RISK AND EXTREME VALUE THEORY



k1

M* (kl ) = TT!n,,X^^ - lnX'-Λ+ 1У and                                       (20)

kI i=1
k1

ξ* (k1 ) = 1 У lnX* i -lnX* k T                                                       (21)

n x 1 / i                   n1 ,i nj,kj+ 1                                                                                                              ` '

k1 i=1

Then find kl 0 (n1 ), i.e. the value of k1, which minimizes the AMSE (19):

k*1,o (n1 ) = argmin Q (n1, k1 )                                                                           (22)

= (n1 )/n yields k**O (n2 ) .


(23)


A second step completely analogous to the first one but with a smaller sample size n2

Next calculate

lnn -lnkj o(n )

k (n)= (k*,o(n1l> I (lnk(n1))2      ɔ    lnn1

0 k(n2Ц(2lnn1 -lnk*o(n1 ))2 y

This allows to calculate the reciprocal tail index estimator:

^

^       1  ko

^ n I ko I = ɪ У lnXι,j

× × k   i=1

o0


ln X


*

n1,ko+1


(24)


This estimation procedure for k depends on two parameters, the number of bootstrap resamples, l, and the sample
size, n1. The number of resamples is in general determined by the available computational facilities. The application
presented in section 4 utilizes 10000 repetitions giving very stable results. Evaluation and optimization of n1,
necessitates a further step. Calculate the ratio

jQ(n μ (Q(nirt,o))2

R(n1)=Жх!

(25)


and determine n1 = argmin R (n1 ) numerically. If n * differs from the initial choice n 1, the previous steps should be
repeated. Remember that the quantile estimates derived from EVT are only valid for the tails of the profit-and-loss-
distribution. To allow inferences about quantiles in the interior of the distribution, Danielsson and de Vries (2000)
propose to link the tail estimator with the empirical distribution function at the threshold Xk+1. Thus the particular
advantages of the EVT and the HS are combined.



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