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

3.2

Estimation of the tail index

Several methods exist to estimate the tail index of a fat tail distribution from empirical data. The most popular is the
Hill estimator (Diebold et al. 1998). To implement this procedure the observed losses X are arranged in ascending

order: X, > X, > — > X, > — X . The tail index _a = 1/- then can be estimated as follows
12     k   n

( 1 /

^a(k )= 7∑ ^{lnX  ln}X

k

i=     г=1

1^-1

k+1

J

(15)

The function L(x) in (12) is usually approximated by a constant C. An estimator for C is (Embrechts et al. 1997, p.
334):

k ^k Y'

Ck     Xk+1

n

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This leads to the following estimator for the tail probabilities and the p-quantile:

-—-

F (_x}-V-( ( Xk+1 I^a X >X

F^(x) = ^p = ^-∣ — I ,^χ > Xk+1
n < x J

i.e.

(17)

1

.           ( k 1 â

x p = F -¹ (x )= Xk+11-I
< ⁿP J

(18)

It can be shown that the Hill Estimator is consistent and asymptotic normal distributed. (Diebold et al. 1998).

The implementation of the estimation procedure requires to determine the threshold value X_k , i.e. the sample size
k, on which the tail estimator is based. Unfortunately, the estimation results are strongly influenced by the choice of
k. Moreover, a trade-off exists: the more data are included in the estimation of the tail index a, the smaller the
variance becomes; however the bias increases at the same time, because the power function in (12) applies only to
the tail of the distribution. In order to solve this problem, Danielsson et al. (2001) develop a bootstrap method for the
determination of the sampling fraction k/n. The different steps of this iterative procedure are described below.

First resamples N = ʃX₁,...,X_n ) of predetermined size n ₁<n are drawn from the data set N_n = ʃX₁

^x, )

with replacement. For any k ₁ the asymptotic mean square error (AMSE) Q (n₁ ,k₁ ) is calculated:

Q⁽ⁿ1 ^,k1 ⁾= E ⁽M*1_I ^(k1 ⁾ - ²fe*1₁ ^(k1 ⁾⁾² У |Wn wi^th

(19)