# . The Risk Factors Evolution Model

To describe this evolution, the following AR(1)-GARCH(1,1) model (Posedel 2005) is applied for each risk factor:

rt = д C mr,_i + £

£t = atS,  a,2 = ! C Pi£:_1 C «ia,

where rt—return at time t, д—basic value of return, £—model error, which is decomposed to St—stochastic component and at—conditional standard deviation at time t, ! —basic value of at.

The stochastic component of error St is often considered as a simple random variable with standard normal distribution. However from empirical data one can clearly see that this cannot be true, because it’s distribution usually has heavy tails.

In this paper we use Pareto distribution from extreme value theory to simulate this feature in the following way:

• The AR-GARCH model fitted onto historical returns gives historical values for it;

• Historical data on St allows to build its distribution;

• The modeled distribution of St used for the AR-GARCH forecast. Distribution St was constructed in the following way: The central part of the density curve obtained with univariate kernel density estimator in the form:

where Xi—sample, K—smoothing kernel (function which satisfies / K(x)dx = 1), h—bandwidth parameter. Here we used Gaussian kernel:

where P—scaling, 1/£—tail index. Fig. 1 Returns simulated by the AR(1)-GARCH(1,1) model (fitted onto the second half of 2008): black line—historical data, red line—simulated data (starts with 15th of May) Fig. 2 Returns simulated by the AR(1)-GARCH(1,1) model (fitted onto the first half of 2013): black line—historical data, red line—simulated data (starts with 15th of May) Fig. 3 Prices simulated by the AR(1)-GARCH(1,1) model (fitted onto the second half of 2008): black line—historical data, red line—simulated data (starts with 15th of May)

The proposed evolution model was applied to two historical periods:

1. Second half of 2008 (crisis conditions);

2. First half of 2013 (stable conditions).

Samples of returns and price dynamics forecast by this model are shown in Figs. 1, 2, 3, and 4. One can see that it catches the volatility clustering effect and correctly transfers initial the historical market conditions to the forecast period. Fig. 4 Prices simulated by the AR(1)-GARCH(1,1) model (fitted onto the first half of 2013): black line—historical data, red line—simulated data (starts with 15th of May)

Table 1 t-copula parameter estimation (second half of 2008), number of freedom degrees = 5

 HYDR GAZP GMKN LKOH ROSN SBER SBERP SNGS URKA VTBR HYDR 1.00 0.58 1.00 0.99 1.00 0.71 1.00 0.75 1.00 1.00 GAZP 0.58 1.00 0.55 0.54 0.55 0.87 0.55 0.81 0.55 0.55 GMKN 1.00 0.55 1.00 0.99 1.00 0.68 1.00 0.73 1.00 1.00 LKOH 0.99 0.54 0.99 1.00 0.99 0.67 1.00 0.72 0.99 0.99 ROSN 1.00 0.55 1.00 0.99 1.00 0.68 1.00 0.73 1.00 1.00 SBER 0.71 0.87 0.68 0.67 0.68 1.00 0.68 0.94 0.68 0.69 SBERP 1.00 0.55 1.00 1.00 1.00 0.68 1.00 0.72 1.00 1.00 SNGS 0.75 0.81 0.73 0.72 0.73 0.94 0.72 1.00 0.73 0.73 URKA 1.00 0.55 1.00 0.99 1.00 0.68 1.00 0.73 1.00 1.00 VTBR 1.00 0.55 1.00 0.99 1.00 0.69 1.00 0.73 1.00 1.00