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

assumed to be stochastically independent. Therefore Danielsson and de Vries (2000) recommend to derive
predictions about extreme events from unconditional distributions.

2.4 Long-Term-Value-at-Risk

Much of this paper is motivated by the supposition that the relevant VaR horizon in agricultural applications in
general will be longer than in a financial context, where one-day or few-day forecasts dominate. Moreover, the
desired horizon will often be larger than the frequency of the data. For example, a farmer having in mind the length
of a production cycle wishes to determine the VaR for three or six months on the basis of weekly price data.
Basically two methods exist to calculate long-term-VaRs: either one measures the value changes that occur during
the entire holding period, that means, the VaR is estimated on the basis of three or six month's returns. Alternatively,
a short-term VaR is extrapolated to the desired holding period (time scaling). The first procedure is applicable
independently of the return distribution. However, it has the serious drawback that the number of observations is
reduced strongly. If, for instance, weekly data are available over a period of 10 years a six-month-VaR is based on
20 observations only, since the measurement periods should be non-overlapping. The second method avoids this
problem. In practice the time-scaling is conducted by means of the square-root-rule.

VaR(h) = VaR(I) ∙ √h                                                                         (8)

VaR(1) and VaR(h) denote the one-period-VaR and the h-period-VaR, respectively. Diebold et al. (1997) point out
that the correctness of the square-root-rule relies on three conditions. First, the structure of the considered portfolio
may not change in the course of time. Secondly, the returns must be identically and independently distributed, and
thirdly, they must be normally distributed. Section 3.1 discusses the consequences of non-normality for time
aggregation. At this point we ask what would happen if the iid assumption is not fulfilled. Though a general answer
to this question is not available, Drost and Nijman (1993) provide a formula for the correct time aggregation of a
GARCH process. For the GARCH(1,1)-process described above the h -period-volatilities can be determined from
the one-period-volatilities as follows:

σ Λ (h) = ω(h)² + δ ( h)X^2 (h) + β ( h)σ t ( h)                                        (9)

with ω(h) = hω

δ(h) = (δ + β)^h - β(h)

and ∖β(h) < 1 as solution of the quadratic equation
β(h)  = a - (δ + β)^h - b

1 + β²(h) ~ a(1 + (δ + β)^2h) - 2b

The coefficients a und b are defined as:

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