The Risk Factor Interrelation Model

The dependence structure of risk factors was described by a t-copula (Genest et al. 2009). Copulas were used instead of the well-known Pearson’s linear correlation because the latter one has many drawbacks (Schimdt 2006) such as:

• It is impossible to capture the full dependency composition of risk factors;

• If the correlation is equal to zero it does not mean that the factors are independent;

• It does not work correctly for distributions with heavy tails because it supposes that risk factor variances are finite (which contradicts the empirical data).

The maximum likelihood method was used to estimate the parameters of the t – copula (Charpentier 2006). Historical data for stochastic component St of the AR(1)- GARCH(1,1) model error was used as a sample for this estimation. The results of the estimation are illustrated in Table 1, Table 2 and Fig. 5.

HYDR

GAZP

GMKN

LKOH

ROSN

SBER

SBERP

SNGS

URKA

VTBR

HYDR

1.00

0.61

1.00

0.94

0.79

1.00

1.00

0.72

1.00

1.00

GAZP

0.61

1.00

0.61

0.52

0.79

0.61

0.61

0.85

0.61

0.60

GMKN

1.00

0.61

1.00

0.94

0.79

1.00

1.00

0.72

1.00

1.00

LKOH

0.94

0.52

0.94

1.00

0.72

0.94

0.94

0.64

0.95

0.95

ROSN

0.79

0.79

0.79

0.72

1.00

0.79

0.79

0.93

0.79

0.79

SBER

1.00

0.61

1.00

0.94

0.79

1.00

1.00

0.72

1.00

1.00

SBERP

1.00

0.61

1.00

0.94

0.79

1.00

1.00

0.72

1.00

1.00

SNGS

0.72

0.85

0.72

0.64

0.93

0.72

0.72

1.00

0.72

0.72

URKA

1.00

0.61

1.00

0.95

0.79

1.00

1.00

0.72

1.00

1.00

VTBR

1.00

0.60

1.00

0.95

0.79

1.00

1.00

0.72

1.00

1.00

Table 2 t-copula parameter estimation (first half of 2013), number of freedom degrees = 5

image060Подпись: 0 0 0.2 0.4 0.6 0.8 1.0 LKOH Подпись: 0.0 0.2 0.4 0.6 0.8 1.0 LKOH image063image064Подпись:Подпись: LKOH 0.0 0.2 0.4 0.6 0.8 1.0 SBER Подпись: 0.0 0.2 0.4 0.6 0.8 1.0 SBER

Fig. 5 Simulated correlation between different stocks (as scatter plots) obtained on the basis of t-copula estimation and simulation: red color—second half of 2008, black points—first half of

The dependence structure simulation in Fig. 5 reproduces the well-known empirical fact that in a period of crisis and instability the correlation among separate stocks increase. But also the simulation shows a discorrelation in some cases. Perhaps this can be explained by links between the stocks in the portfolio under consideration.

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