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

plim means the limit of a probability for n → ∞ and H(x) denotes the GEV, which is defined as follows:

(-(1 + ₂x)-¹'^ξ)1 if ξ ≠ 0
exp (- e^x ) ∫ if ξ = ⁰

The GEV includes three extreme value distributions as special cases, the Frechet-distribution (ζ > 0), the Weibull-
distribution (ζ < 0), and the Gumbel-distribution (ζ = 0) . Depending on the parameter ζ a distribution F is
classified as fat tailed (ζ > 0), thin tailed (ζ = 0) and short tailed (ζ < 0). In the present context the focus is on
the first class of distributions, which includes for example the t-distribution and the Pareto-distribution, but not the
normal distribution. Embrechts et al. (1997, p. 131) prove that the sample maxima of a distribution exhibiting fat
tails converges towards the Frechet-distribution Φ ( x ) = exp (x^a ), if the following condition is satisfied:

1 - F(x) = x ¹ $ L ( x )

(12) requires that the tails of the distribution F behave like a power function. L(x) is a slowly varying function and
a = 1∣ζ is the tail index of the distribution. The smaller a is, the thicker are the tails. Moreover, (12) indicates that
inferences about extreme quantiles of a possibly unknown distribution of F can be made as soon as the tail index a
and the function L have been determined. Section 3.2 describes an estimation procedure for a.

The results of the EVT are also relevant for the aforementioned task of converting short-term VaRs into long-term
VaRs. Assume P ( X∣ > x) = Cx ^a applies to a single-period return X for large x, then due to the linear additivity
of the tail risks of fat tailed distributions for a h-period return can be deduced (Danielsson and de Vries 2000):

P(X₁ + X₂ + - + Xh > x) = hCx^-a                                               (13)

For a multi-period-VaR forecast of fat tailed return distribution follows under the iid assumption:

VaR (h ) = VaR (i)h^{1 a}

If the returns have finite variances then a > 2 and thus a smaller scaling factor applies than postulated by the
square-root-rule (Danielsson et al. 1998). Obviously the square-root-rule is not only questionable if the iid
assumption is violated, but also if the return distribution is leptocurtic.

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