The article presents a model that can be used for modelling movements in the prices of financial instruments in the market and which can be utilised in order to calculate VaR (Value at Risk). The model emphasises the autoregressive dependence of market returns on previous observations. Moreover, it also enables the generation of market returns which are characterised by heavy-tailed distributions, i.e. with deviant observations. However, apart from examining the dependence between returns themselves, it is also important to examine the variance of such returns and the dependence of this variance on the variance of returns (or also the squares of the returns themselves) from previous periods. A simple model with variance which is variable and dependent on market returns from previous periods has been developed on the basis of ARCH/GARCH models which enables the generation of time series that are directly dependent on historical information. It has also turned out that this model shows well the heavy-tail effect, and VaR figures approximate the values yielded with the help of stress testing procedure. This is the so-called MR-ARCH(p,q) model (Modified Randomised-ARCH). It is characterised by a feature that during certain periods, returns may deviate noticeably from the mean but due to certain technical tweaks the series does not approach infinity for large initial values (large absolute return values).

The variance in this model depends on the sum of products of random variables and squares of past process observations as well as on other, independent random variables. It was obtained on the basis of the variance of the ARCH model through the randomisation and modification of all model parameters — hence the name Modified Randomised ARCH. Due to the inclusion of random factors, whose distribution parameters may be modelled at will to yield sufficiently satisfactory results, the random effect is achieved together with simulation of returns while at the same time we can give them a desirable probability distribution and thus obtain precise results.

The MR-ARCH model has been analysed with regard to the foreign exchange market, and specifically with regard to the daily exchange rate of the pound sterling (GBP/PLN) during the period from 1 October 1996 to 30 September 2003 (1,764 observations). The autocorrelation graph of squared returns demonstrated short-range correlation between them, which allows us to assume that these returns may be modelled with the aid of heavy-tailed series of the ARCH/GARCH/MR-ARCH type. On the other hand, the autocorrelation graph of the returns themselves did not demonstrate any dependence between them. This excludes the application of series of the ARMA type, which are also heavy-tailed, but exhibit, however, a dependence between the returns themselves and no dependence between squared returns. This confirms the suitability of the MR-ARCH model application for VaR calculations. The calculations show that VaR values for different variants of the MR-ARCH model are usually larger (more “extreme”) than the VaR values yielded by classical methods. Moreover, the simulation-yielded returns for the MR-ARCH model deviate from the mean value to an extent similar to that of empirical returns, and exhibit heavy tails. This means that this model makes it possible to estimate potential VaR for shock price movements which confirms this model's suitability for analysing extreme market behaviours.