Simulation of financial risk spillover effect based on ARMA-GARCH and fuzzy calculation model

Abstract

Convergence and spillover are the characteristics shown in the process of financial development. By verifying whether there is convergence and spillover in financial development within a certain region and between regions, the stage of financial development in the region can be more accurately judged. This paper combines the actual needs of financial analysis to construct a financial risk spillover effect model based on ARMA-GARCH and fuzzy calculation. The model uses ARMA-GARCH and fuzzy algorithm to verify the financial multiple risk factors. Moreover, in order to verify the effect of the model, this paper uses case data analysis to study the model effect and combines mathematical statistics to process the model data. The research results show that the model constructed in this paper has a certain effect, and the ARMA-GARCH model is suitable for analysis and research on financial risk spillover effects. At the same time, when the statistical distribution is used to fit its error distribution, the fitting and prediction effect of the model is better.

Keywords

Machine learning fuzzy calculation financial risk spillover effect

1 Introduction

In the context of regional development, agglomeration of industrial institutions has become a common phenomenon in modern economic operations, and agglomeration of financial institutions has emerged. The aggregation of financial institutions uses financing and capital operations as the main methods to optimize the allocation of resources within the region and accelerate the economic growth of the region and surrounding areas to a certain extent through spatial radiation spillover. At present, the three major financial institutions of my country’s banking, insurance, and securities are concentrated in the eastern coastal areas. Among them, the Yangtze River Delta, the Pearl River Delta and the Bohai Rim Economic Circle are the regions with the highest concentration of banking institutions and insurance institutions, and Beijing is the region with the highest concentration of securities institutions. Moreover, the eastern region has already formed a cluster of financial institutions in my country, while the western region generally has fewer financial institutions. Due to the large geographic distance between the east and the west, the concentrated radiation of financial institutions in the east is difficult to spread to the west. Moreover, although the western cities of Chengdu, Chongqing, and Xi’an have shown a trend of agglomeration of financial institutions, their spatial radiation radius is relatively small. Therefore, the western region urgently needs to seek new momentum for sustainable economic growth under the advantages of existing financial institutions, use space radiation to develop the regional economy, and reduce regional imbalances. In the study of financial development and economic growth, the research on the spatial spillover effects of the agglomeration of financial institutions and economic growth is a topic worthy of focus. How the agglomeration of financial institutions produces spatial spillover effects on economic growth is the focus of research. At present, my country is in an important period of economic transformation. At this time, the spatial spillover effect of the agglomeration of financial institutions plays a vital role in the transformation of the western economy from a speed-based growth to a high-quality growth, stimulating the weak links of China’s economy to achieve balanced and coordinated growth of the national economy, and completing the smooth economic transition [1].

With the development of economy and society, the economic behaviors of neighboring provinces depend on and influence each other spatially. At this time, more and more attention has been paid to the strategy of using regional spillover effects to drive the development of surrounding areas to make the entire economy develop in a balanced and stable direction. The mobility of financial factors between regions makes the interaction and mutual influence between regions play an important role that cannot be ignored in my country’s financial support for the construction of new urbanization. The development of new-type urbanization is not only affected by the financial support of the region, but also by the financial spillover effect of neighboring regions to support the new-type impact. Moreover, the spatial distribution of financial resources in various regions of my country is uneven, and the level of financial development varies greatly. Therefore, the impact of financial support in this region and neighboring regions on the new urbanization is different [2]. At present, the spatial distribution of financial resources in various regions of my country is uneven, and the level of financial development varies greatly. With the rapid economic growth, the inter-regional interaction and mutual influence play an important role that cannot be ignored in my country’s financial support for the construction of new urbanization. Therefore, it is important to include the dimension of space in the research scope of financial support for new urbanization. Based on this, this paper is dedicated to analyzing whether China’s new urbanization has spatial relevance, exploring whether financial support has spatial spillover effects on new urbanization, that is, analyzing whether new urbanization in a region is not only affected by financial support in the region, but also Affected by financial support from neighboring regions. Moreover, this paper studies the magnitude and direction of the spatial spillover effects of financial support in this region and neighboring regions on the new urbanization in the presence of spatial spillover effects. Through the exploration of the above problems, this paper puts forward practical countermeasures and suggestions to better play the role of financial support in the construction of new urbanization [3].

2 Related work

The development of foreign securities markets is relatively long, and the overall market system and investors are relatively mature. Therefore, foreign economists and econometric statisticians have studied the volatility of their securities markets for a long time [4]. The literature [5] observed that speculative price changes and changes in the rate of return have variable periods and stable periods, that is, price fluctuations are clustered, and the size of the variance changes with time. Subsequently, foreign scholars conducted a lot of research on the price fluctuation characteristics of financial investment and concluded that clustering and persistence are the most significant characteristics of financial market fluctuations. In order to better describe the clustering and persistence of financial market volatility to further study the internal characteristics of financial markets, the literature [6] proposed an autoregressive conditional heteroscedasticity model.

The ARCH model distinguishes the conditional variance from the variance and allows the conditional variance to change as a function of describing past errors, thus providing a new way to solve the heteroscedasticity problem of financial series. As a brand-new theory, the ARCH model has achieved extremely rapid development in the past two decades and has been widely used in the description of laws in financial theory and the forecast and decision-making of financial markets. In addition, economists and econometric economists try to continuously tap the potential of this model to continuously enhance our ability to explain and predict the market. The literature [7] uses the dual ARCH model to study the inflation rate in the United States. The literature [8] compares and fuses the ARcH model with the bilinear time series model. The literature [9] proposed a generalized autoregressive conditional heteroscedasticity model.

The literature [10] found that the distribution of financial asset returns has two major characteristics. 1. Biased, and its skewness is often greater than o, that is, the probability distribution is biased to the right and is asymmetric. 2. The obvious peak and fat tail, and its kurtosis is usually much greater than 3, indicating that in the financial market, the rate of return of financial assets fluctuates sharply. That is, in the actual financial market, due to the superposition of complex external factors and changes in investor psychological factors, the possibility of extreme events is greater than the probability of extreme events under the assumption of normal distribution. These also pointed out the shortcomings of the traditional assumption that the rate of return obeys a normal distribution. Although the standard GARCH model usually defaults to its error distribution as a normal distribution, which can better deal with heteroscedasticity problems, in many financial time series, this model does not reflect its bias [11]. From the perspective of finance, returns and risks should be equal. Changes in the risk of financial assets themselves over time will have a greater impact on the rate of return, and a higher conditional variance value will increase the conditional mean accordingly. Therefore, introducing the conditional variance as a variable into the conditional mean equation is more helpful to accurately describe the financial sequence. This is the generalized autoregressive conditional heteroscedasticity model in which the volatility term enters the mean equation proposed in the literature [12]. The literature [13] found that the volatility of stock prices has a leverage effect, that is, the expected price drop (bad news) and the expected price increase (good news) have asymmetric effects on volatility. Generally, investors are more sensitive to bad news due to investors’ risk aversion. Therefore, bad news is more likely to have a greater impact on the market than good news.

For another feature of financial data, that is, the obvious peak and fat tail, before the literature [14] proposed that financial data has a sharp peak and thick tail, scientists have never raised any major doubts about the normal distribution. The literature [15] proposed that previous studies have neglected the obvious peak and fat tail characteristics of financial asset distribution. The empirical research in the literature [16] shows that financial data has fat tail, and the residuals in the model follow the fat tail distribution, which is more in line with the actual situation. On the basis of observing a large number of financial assets including stock prices, exchange rates, yields and interest rates, the literature [17] summarizes five important characteristics of financial assets, and the most important of them are clustering and thick tail. At present, the characteristics of obvious peak and fat tail and heteroscedasticity seem to have been widely agreed. Therefore, the GARCH model with spikes and thick tails is still one of the main measurement tools for financial market research. Based on the partial sum of squares of the observations, the literature [18] used the classic R/S analysis method and used the maximum likelihood estimation to obtain the spectrum curve, thereby obtaining the semi-parametric estimation of the heteroscedasticity density of the long memory condition and its properties.

3 Basic content of VaR method

VaR is a method of applying mathematical statistics to measure financial risks. This method expresses risk through a density function or a cumulative function. Considering from the aspect of probability and statistics, the maximum loss that an asset will suffer within a certain period of time and at a certain probability level can be expressed as [19]: $Prod (Δ P > VaR) = 1 - c$ (1)

Among them, ΔP = ln (p_t) - ln (p_t-1) is the loss of the value of the asset during the holding period Δt, and VaR is the value at risk when the confidence level is c.

If an institution has a risk value of 9.8 million yuan with a one-day holding period at a 99% confidence level, the VaR concept can be understood as: within the next day, the organization will lose more than 9.8 million yuan with a 99% probability.

The accuracy of VaR measurement is affected by holding period and confidence level.

The holding period is the time range for calculating VaR, usually one day, one month, or one quarter. The longer the holding period, the greater the value of VaR. The selection of the holding period is determined by the risk manager according to his own needs. Generally, the transaction VaR is reported on a daily basis. Too long or too short holding period is not conducive to supervision. Generally, financial institutions have a holding period of one day.

The choice of confidence level reflects the importance of financial institutions on financial security. The greater the confidence level is selected, the more important it is to financial security. Financial institutions generally choose to ensure the safety and effectiveness of the financial system and the confidence level that the bank’s risk capital requirements are not too high [20].

4 Calculation theory of VaR method

At present, the commonly used VaR calculation methods include: historical simulation method, Monte Carlo simulation method, parameter method and extreme value theory method. Among them, the historical simulation method and the Monte Carlo simulation method are non-parametric methods. The characteristics of this method are that it does not need to make assumptions about the statistical distribution of the return rate series, and it can fit the financial series well. The parametric method needs to make assumptions about the distribution of the return sequence, and then fit the return sequence. The extreme value theory method only fits the tail data of the return sequence without making assumptions about the statistical distribution of the return sequence, so it can be described as a semi-parametric method.

According to the distribution of risk factors in the past period of time, a new return and return distribution is constructed, and then the VaR value is calculated according to the corresponding quantile. Historical data is used to simulate future market risk fluctuations, and parameters are not required to be estimated during the period. This is also the advantage of historical simulation.

If the financial institution contains m types of assets, the data of day N + 1 is selected from historical data, and there are [21]: ${PV}_{t}^{p} = \sum_{i = 1}^{m} ω_{i} V_{it}, t = - 1, - 2, \dots, - N$ (2)

Among them, V_it represents the income (i = 1, 2, ⋯ , m, t = -1, - 2, ⋯ , - N) of the i-th asset at time t, and ω_i is the investment weight of the i-th asset at t = 0. The historical return value ${{PV}_{t}^{p}}_{t = - 1, - 2, \dots, - N}$ is sorted from small to large, the empirical distribution function of the historical actual price of the market factor is obtained, and the VaR value under the corresponding quantile with different confidence levels is estimated.

Advantages of historical simulation method:

The calculation method is simple and easy to implement.

There is no need to make assumptions about the distribution of the return rate series, and there is no need to estimate the parameters.

Disadvantages of historical simulation method:

The accuracy of the calculation of this method largely depends on the length of the sample interval. If the sample interval is small, the estimation of VaR will be inaccurate.

The historical simulation method is a method to predict the fluctuation of future risk factors based on the fluctuation of historical data. If the fluctuation increases rapidly in a short period of time, the measurement result of VaR will be inaccurate.

The Monte Carlo simulation method is mainly to re-simulate the process of risk factor changes. It is assumed that the return rate sequence is subject to a certain distribution, a large number of simulations of future fluctuations are carried out, various possible changes are sorted in a certain order, and the empirical distribution of risk factor changes is obtained, so the VaR value corresponding to the quantile is obtained.

The first step is to build a model based on asset prices [22]: ${dS}_{t} = μ_{t} S_{t} dt + σ_{t} S_{t} d ω_{t}$ (3)

Among them, dS_t is the price change at time t, μ_t is the drift of the asset return at time t, and σ_t is the standard deviation of the asset return. At the same time, $d ω_{t} \overset{i . i . d}{\sim} N (0, dt)$ is Brownian motion. After simplification, the change process of (0, T) asset price is obtained: ${\begin{matrix} \begin{matrix} Δ S_{t} = S_{t - 1} (μ_{t} Δ t + σ_{ɛ_{t}} \sqrt{δ t}), \\ t = 1, 2, \dots, N, N Δ t = T \end{matrix} \\ S_{t + 1} = S_{t} + S_{t} (μ Δ t + σ Δ t \sqrt{Δ t}) \end{matrix}$ (4)

Performing N operations on the above formula can simulate the price at each point in time. When S_N = S_T, the operation is stopped.

In the second step, it is assumed that the selected data obey N (0, 1), the sequence ɛ₁, ɛ₂, ⋯ , ɛ_N is extracted from it and substituted into ${\begin{matrix} \begin{matrix} Δ S_{t} = S_{t - 1} (μ Δ t + σ_{ɛ_{t}} \sqrt{δ t}), \\ t = 1, 2, \dots, N, N Δ t = T \end{matrix} \\ S_{t + 1} = S_{t} + S_{t} (μ Δ t + σ Δ t \sqrt{Δ t}) \end{matrix}$ (5)

The predicted price sequence S₁, S₂, ⋯ , S_N can be obtained, and there is S_N = S_T.

The third step is to perform K operations on the previous step to obtain K estimated prices $S_{T}^{1}, S_{T}^{2}, \dots, S_{T}^{K}$ at time T and calculate the income distribution.

In the fourth step, at a given quantile α%, the corresponding VaR value is calculated.

Advantages of MonteCarlo simulation method [23]:

The calculation accuracy and reliability are higher.

There is no need to make assumptions about the distribution of the return rate series, and there is no need to estimate the parameters.

Disadvantages of MonteCarlo simulation method:

It has a large amount of calculation, inconvenient use, and high cost.

It is dependent on the selected random process and historical data.

The parameter method is also called the variance-covariance method. This method needs to make assumptions about the statistical distribution of the return sequence, and then obtain the parameter values of the return distribution through historical data, and then obtain the VaR value of the sequence. Among them, p represents the asset, σ is the standard deviation, and Z_α is the quantile value at the confidence level 1 - α. $VaR = z_{α} σ_{p} = z_{α} \sqrt{\sum_{i = 1}^{I} \sum_{j = 1}^{J} x_{i} x_{j} ρ_{i, j} σ_{i} σ_{j}}$ (6)

In the above formula, σ_p represents the standard deviation, ρ_i, ρ_j represents the standard deviation of i and j, and ρ_i,j represents the correlation coefficient of i and j. At the same time, x_i represents the sensitivity of the change in i to the entire portfolio, sometimes called Δ. If it is a normal distribution, x_i is the sum of i and Δ.

The research content of this paper only needs to fit the conditional variance, so this paper uses the ARMA-GARCH family model to fit the conditional variance [24].

5 ARMA-GARCH family model

If it is assumed that {Y_t, t = 1, 2, ⋯ , T } is a sample of distribution F (x) and its sample observation value is set to {y_t, t = 1, 2, ⋯ , T }, the sample observation value y_t satisfies the following expression: ${\begin{matrix} y_{t} = μ + \sum_{{i = 1}^{m} λ_{i} y_{t - 1} + \sum_{{j = 1}^{n} η_{j} ɛ_{t - j} + ɛ_{t}}} \\ ɛ_{t} = σ_{t} z_{t} \\ σ_{t}^{2} = ω + \sum_{{k = 1}^{p} α_{k} ɛ_{t - k}^{2} + \sum_{{l = 1}^{q} β_{l} σ_{t - l}^{2}}} \end{matrix}$ (7)

We call: Y_t obeys the ARMA (m, n) - GARCH (p, q) model.

If it is assumed that {Y_t, t = 1, 2, ⋯ , T } is a sample of distribution F (x) and its sample observation value is set to {y_t, t = 1, 2, ⋯ , T }, the sample observation value y_t satisfies the following expression: ${\begin{matrix} y_{t} = μ + \sum_{{i = 1}^{m} λ_{i} y_{t - i} + \sum_{{j = 1}^{n} η_{j} ɛ_{t - j} + ɛ_{t}}} \\ ɛ_{t} = σ_{t} z_{t} \\ In (σ_{t}^{2}) = ω + \sum_{{k = 1}^{p} α_{k} g (z_{t - k}) + \sum_{{l = 1}^{q} β_{l} In (σ_{t - l}^{2})}} \end{matrix}$ (8)

Among them, g (z_t) = θz_t + γ (|z_t| - E|z_t|). μ is the mean value of the return time series y_t, σ_t is the conditional standard deviation, and $σ_{t}^{2} = Var (ɛ_{t} | I_{t - 1})$ is the conditional heteroscedasticity of I_t-1 in period t - 1. At the same time, z_t obeys the generalized error distribution, that is, z_t ∼ GED (0, 1). We call: Y_t obeys the ARMA (m, n) - EGARCH (p, q) model.

If it is assumed that {Y_t, t = 1, 2, ⋯ , T } is a sample of distribution F (x) and its sample observation value is set to {y_t, t = 1, 2, ⋯ , T }, the sample observation value y_t satisfies the following expression: ${\begin{matrix} y_{t} = μ + \sum_{{i = 1}^{m} λ_{i} y_{t - i} + \sum_{{j = 1}^{n} η_{j} ɛ_{t - j} + ɛ_{t}}} \\ ɛ_{t} = σ_{t} z_{t} \\ σ_{t}^{2} = ω + \sum_{{k = 1}^{p} (α_{k} + ψ d_{t - k}) ɛ_{l = 1}^{q} + \sum_{{l = 1}^{q} β_{l} σ_{t - l}^{2}}} \end{matrix}$ (9)

Among them: d_t is an indicative variable of stock price information, if ɛ_t < 0, then d_t = 1, and if ɛ_t ⩾ 0, then d_t = 0.

We call Y_t obeys the ARMA (m, n) - TGARCH (p, q) model.

If {Y_t, t = 1, 2, ⋯ , T } is assumed to be a sample of distribution F (x) and its sample observation value is set to {y_t, t = 1, 2, ⋯ , T }, the sample observation value y_t satisfies the following expression [25]: ${\begin{matrix} y_{t} = μ + \sum_{{i = 1}^{m} λ_{i} y_{t - i} + \sum_{{j = 1}^{n} η_{j} ɛ_{t - j} + ɛ_{t} + c σ_{t}^{2}}} \\ ɛ_{t} = σ_{t} z_{t} \\ σ_{t}^{2} = ω + \sum_{{k = 1}^{p} α_{k} ɛ_{t - k}^{2} + \sum_{{l = 1}^{q} β_{l} σ_{t - l}^{2}}} \end{matrix}$ (10)

c is a parameter indicating the risk premium, and its positive or negative indicates whether the stock market return rate is positively correlated with the past volatility or negatively correlated. c > 0 is positively correlated, and c < 0 is negatively correlated.

We call: Y_t obeys the ARMA (m, n) - GARCH (p, q) - M model.

If it is assumed that {Y_t, t = 1, 2, ⋯ , T } is a sample of distribution F (x) and its sample observation value is set to {y_t, t = 1, 2, ⋯ , T }, the sample observation value y_t satisfies the following expression: ${\begin{matrix} y_{t} = μ + \sum_{{i = 1}^{m} λ_{i} y_{t - i} + \sum_{{j = 1}^{n} η_{j} ɛ_{t - j} + ɛ_{t} + c σ_{t}^{2}}} \\ ɛ_{t} = σ_{t} z_{t} \\ σ_{t}^{2} = ω + \sum_{{k = 1}^{p} (α_{k} + ψ d_{t - k}) ɛ_{t - k}^{2} + \sum_{{l = 1}^{q} β_{l} σ_{t - l}^{2}}} \end{matrix}$ (11)

Parameter c is the same as ARMA-GARCH-M, and other parameters are the same as ARMA-TGARCH.

We call: Y_t obeys the ARMA (m, n) - TGARCH (p, q) - M model.

If it is assumed that {Y_t, t = 1, 2, ⋯ , T } is a sample of distribution F (x) and its sample observation value is set to {y_t, t = 1, 2, ⋯ , T }, the sample observation value y_t satisfies the following expression: ${\begin{matrix} y_{t} = μ + \sum_{{i = 1}^{m} λ_{i} y_{t - i} + \sum_{{j = 1}^{n} η_{j} ɛ_{t - j} + ɛ_{t} + c σ_{t}^{2}}} \\ ɛ_{t} = σ_{t} z_{t} \\ In (σ_{t}^{2}) = ω + \sum_{{k = 1}^{p} α_{k} g (z_{t - k}) + \sum_{{l = 1}^{q} β_{l} In (σ_{t - l}^{2})}} \end{matrix}$ (12)

Parameter c is the same as ARMA-GARCH-M, and other parameters are the same as ARMA-EGARCH.

We call: Y_t obeys the ARMA (m, n) - EGARCH (p, q) - M model.

Advantages of ARMA-GARCH family model:

It can fit the volatility of the return sequence well.

It is easy to operate and feasible.

Disadvantages of ARMA-GARCH family models:

(1) Only the VaR of the normal quantile can be estimated, and the measurement of extreme risk is not very accurate.

Facing the instability of the current financial market, extreme value theory methods have high research value. This method does not need to make assumptions about the distribution, and can better describe the thick tail phenomenon of the sequence. It has the same function as the central limit theory in describing random variables and their distribution. The extreme value theory methods are divided into two categories: (1) Grouped maximum value method based on generalized extreme value distribution, (2) Peak limit method based on generalized pareto distribution. Moreover, two approaches are used for tail data modeling.

This method first divides the variables that meet independent and identical distribution into several continuous parts according to certain standards. Secondly, the maximum values are found from each part, and these maximum values are reconstituted into a new sequence. From a mathematical point of view, the sequence also satisfies the condition of independent and identical distribution. Finally, the recombined sequence is fitted into a generalized extreme value distribution, and from the assumption of the parameters, the overall distribution of the time series is obtained. However, the grouped maximum (BM) method has higher requirements for sample data, and is usually suitable for seasonal or periodic sample data. In addition, from the perspective of principle, the BM method sometimes cannot fully reflect all the information of the sample data. Therefore, this paper adopts the Peak Limit (POT) method of extreme value theory.

The POT model is widely used to measure extreme risks. The POT method can model all data exceeding the threshold from limited extreme data. The POT method is used to accurately measure the extreme risks of my country’s stock market. The POT model is also called the threshold model. Among them, the maximum value higher than the threshold in any continuous observation value is defined as the peak value. Over-threshold is the difference between the peak value and the threshold value. In the POT method, the generalized Pareto distribution is used to fit the sample distribution beyond the threshold.

We assume that {Z_t, t = 1, 2, ⋯ , m } is the return sequence and set X_t = - Z_t, t = 1, 2, ⋯ , m. Sample {X_t, t = 1, 2, ⋯ , m } obeys F (x) = P (X_t < x) distribution. The threshold is taken as u, y = x - u as the over-threshold part, and the conditional over-limit distribution function is defined as: $F_{u} (y) = P (X - u ⩽ y | X > u), X_{F} - u ⩾ y ⩾ 0$ (13) Among them, X_F< ∞ is the right end value of F distribution, then there is $\begin{matrix} F_{u} (y) = \frac{P (X - u ⩽ y, X > u)}{P (x > u)} = \\ \frac{P (X - u) - F (u)}{1 - F (u)} = \frac{F (x) - F (u)}{1 - F (u)} \end{matrix}$ (14)

If the threshold u is sufficiently large, the generalized Pareto distribution (GPD) approaches F_u (y), that is $F_{u} (y) \approx G_{ξ, σ} (y) = {\begin{matrix} 1 - {(1 + \frac{ξ}{σ} y)}^{- \frac{1}{ξ}} & ξ \neq 0 \\ 1 - e^{- \frac{y}{σ}} & ξ = 0 \end{matrix}$ (15)

Or $F_{u} (y) \approx G_{ξ, σ} (y) = {\begin{matrix} 1 - {(1 + \frac{ξ}{σ} (x - u))}^{- \frac{1}{ξ}} & ξ \neq 0 \\ 1 - e^{- \frac{(x - u)}{σ}} & ξ = 0 \end{matrix}$ (16)

The Mean excess function method is a common method for selecting the threshold. It is calculated based on the average excess function e (u) of the generalized Pareto distribution G (x, ξ, σ), $\begin{matrix} e (u) = E (X - u | X > u) = \frac{σ}{1 - ξ} = \\ \frac{σ + ξ u}{1 - ξ}, u \in D (ξ, σ), ξ < 1 \end{matrix}$ (17)

From the sample sequence X₁, ⋯ , X_n, the average excess function of the sequence is $e (u) = \frac{1}{N_{u}} \sum_{i}^{N_{u}} (X_{i} - u), u > 0$ (18)

Here, N_u represents the number of excess.

Three, parameter maximum likelihood estimation

In order to calculate the parameter ξ, σ, we adopt the method of maximum likelihood estimation.

G_ξ,σ (y) is set as the generalized Pareto distribution function, and its density function is $g_{ξ, σ} = \frac{1}{σ} {(1 + \frac{ξ}{σ} y_{i})}^{- \frac{1}{ξ} - 1} (ξ \neq 0)$ . A set of samples of G_ξ,σ (y) is set as y₁, y₂, ⋯ , y_k, then the log likelihood function is: $I (ξ, σ) = - kIn (σ) - (\frac{1}{ξ} + 1) \sum_{i = 1}^{k} (1 + \frac{ξ}{σ} y_{i}), (ξ \neq 0)$ (19)

Respectively seeking partial derivatives of ξ, σ, we get: ${\begin{matrix} \frac{\partial l (ξ, σ)}{\partial ξ} = \frac{1}{ξ^{2}} k - \frac{k}{σ} \bar{y} \\ \frac{\partial l (ξ, σ)}{\partial σ} = - \frac{k}{σ} (1 + \frac{\bar{y}}{σ} + \frac{ξ \bar{y}}{σ}) \end{matrix}$ (20)

We set ${\begin{matrix} \frac{1}{ξ^{2}} k - \frac{k}{σ} \bar{y} = 0 \\ - \frac{k}{σ} (1 + \frac{\bar{y}}{σ} + \frac{ξ \bar{y}}{σ}) = 0 \end{matrix}$ (21)

ξ, σ can be solved.

The prerequisite for the application of extreme value theory is that the amount of excess threshold is independent of the time when the threshold occurs. However, the amount of excess threshold obtained by using the POT model directly on the sequence may not meet this condition. Therefore, it is necessary to consider both the ARMA-GARCH family model and extreme value theory method to construct ARMA-GARCH family-EVT model.

6 Estimation of parameters μ_t+1 and σ_t+1 of ARMA-GARCH family-M model

We assume that ${y_{t}}_{t = T - n + 1}^{T}$ represents the logarithmic return sequence of financial assets, and assume that the time series meets the ARMA-GARCH family-M model, that is, the conditional mean meets the autoregressive moving average model, and the conditional variance meets the GARCH family-M model, expressed by the formula as follows:

(1) ARMA-GARCH-M

If it is assumed that {Y_t, t = 1, 2, ⋯ , T } is a sample of distribution F (x), and its sample observation value is set to {y_t, t = 1, 2, ⋯ , T }, the sample observation value y_t satisfies the following expression: ${\begin{matrix} y_{t} = μ + \sum_{{i = 1}^{m} λ_{i} y_{t - i} + \sum_{{j = 1}^{n} η_{j} ɛ_{t - j} + ɛ_{t} + c σ_{t}^{2}}} \\ ɛ_{t} = σ_{t} z_{t} \\ σ_{t}^{2} = ω + \sum_{{k = 1}^{p} α_{k} α_{t - k}^{2} + \sum_{{l = 1}^{q} β_{l} σ_{t - l}^{2}}} \end{matrix}$ (22)

{z_t} is a white noise process and satisfies the F distribution, and the parameter is ω > 0, α ⩾ 0, β ⩾ 0. The conditional mean is μ = E (r_t|F_t-1), the conditional variance is σ_t = VaR (r_t|F_t-1), and F_t-1 is the historical information of the return sequence up to day t - 1.

We call: Y_t obeys the ARMA (m, n) - GARCH (p, q) - M model.

(2) ARMA-EGARCH-M model

If it is assumed that {Y_t, t = 1, 2, ⋯ , T } is a sample of distribution F (x) and its sample observation value is set to {y_t, t = 1, 2, ⋯ , T }, the sample observation value y_t satisfies the following expression: ${\begin{matrix} y_{t} = μ + \sum_{{i = 1}^{m} λ_{i} y_{t - i} + \sum_{{j = 1}^{n} η_{j} ɛ_{t - j} + ɛ_{t} + c σ_{t}^{2}}} \\ ɛ_{t} = σ_{t} z_{t} \\ In (σ_{t}^{2}) = ω + \sum_{{k = 1}^{p} α_{k} g (z_{t - k}) + \sum_{{l = 1}^{q} β_{l} In (σ_{t - l}^{2})}} \end{matrix}$ (23)

Among them, there is g (z_t) = θz_t + γ (|z_t| - E|z_t|). μ is the mean value of the return time series y_t, σ_t is the conditional standard deviation, and $σ_{t}^{2} = Var (ɛ_{t} | I_{t - 1})$ is the conditional heteroscedasticity of I_t-1 in period t - 1. At the same time, z_t obeys the generalized error distribution, that is, z_t ∼ GED (0, 1).

We call: Y_t obeys the ARMA (m, n) - EGARCH (p, q) - M model.

3) ARMA-TGARCH-M model

If it is assumed that {Y_t, t = 1, 2, ⋯ , T } is a sample of distribution F (x), and its sample observation value is set to {y_t, t = 1, 2, ⋯ , T }, the sample observation value y_t satisfies the following expression: ${\begin{matrix} y_{t} = μ + \sum_{{i = 1}^{m} λ_{i} y_{t - i} + \sum_{{j = 1}^{n} η_{j} ɛ_{t - j} + ɛ_{t} + c σ_{t}^{2}}} \\ ɛ_{t} = σ_{t} z_{t} \\ σ_{t}^{2} = ω + \sum_{{k = 1}^{p} (α_{k} + ψ d_{t - k}) ɛ_{t - k}^{2} + \sum_{{l = 1}^{q} β_{l} σ_{t - l}^{2}}} \end{matrix}$ (24)

Among them: d_t is an indicative variable of stock price information. If ɛ_t < 0, then there is d_t = 1, and if ɛ_t ⩾ 0, then there is d_t = 0.

We call: Y_t obeys the ARMA (m, n) - TGARCH (p, q) - M model.

Because the ARMA-GARCH family-M model has normal residuals, the parameter estimates can be obtained by maximizing the likelihood function, which is still reasonable and feasible when the distribution of the fitted model is unknown.

We can estimate the conditional mean estimate sequence ${{\hat{μ}}_{T - N + 1}, {\hat{μ}}_{T - 1}, {\hat{μ}}_{T}}$ and the standard deviation estimate sequence ${{\hat{σ}}_{T - N + 1}, {\hat{σ}}_{T - 1}, {\hat{σ}}_{T}}$ of the asset return sequence. From this, we can get the residual sequence formula: $\begin{matrix} {z_{T - N + 1}, z_{T - N + 2}, \dots, z_{T}} = \\ {\frac{y_{T - N + 1} - {\hat{μ}}_{T - N + 1}}{{\hat{σ}}_{T - N + 1}}, \frac{y_{T - N + 2} - {\hat{μ}}_{T - N + 2}}{{\hat{σ}}_{T - N + 2}}, \dots, \frac{y_{T} - {\hat{μ}}_{T}}{{\hat{σ}}_{T}}} \end{matrix}$ (25)

Finally, according to the conditional mean and standard deviation estimated value sequence on day t, we obtain the piece mean estimated value and standard deviation estimated value sequence on day t + 1, as follows: ${\begin{matrix} {\hat{μ}}_{T + 1} = {\hat{ψ}}_{0} + \hat{ψ} y_{T} \\ {\hat{σ}}_{T + 1}^{2} = \hat{ω} + \hat{α} {(y_{T} - {\hat{μ}}_{T})}^{2} + \hat{γ} {(y_{T} - {\hat{μ}}_{T})}^{2} ∥ y_{T} - γ > 0 + \hat{β} σ_{T}^{2} \end{matrix}$ (26)

7 Model data analysis

This paper selects A region for research. Table 1 and Fig. 1 show the statistical table and diagram of financial correlation rates of the national and A region from 2014 to 2019.

Table 1
Statistical Table of financial correlation rate

Nationwide Area A

2014 2.78 2.32

2015 2.69 2.39

2016 3.01 2.59

2017 3.13 2.79

2018 3.24 2.91

2019 3.27 2.99

Average 3.02 2.67

	Nationwide	Area A
2014	2.78	2.32
2015	2.69	2.39
2016	3.01	2.59
2017	3.13	2.79
2018	3.24	2.91
2019	3.27	2.99
Average	3.02	2.67

Fig. 1

Statistical diagram of financial correlation rate.

It can be seen from Table 1 and Fig. 1 that compared with the national average, the financial correlation rate of A region is always low, but it has continued to increase rapidly since 2014. The reason is that with the further advancement of the Western Development, financial institutions in A region gradually increase their credit funds. In addition, in response to the global financial crisis that occurred in 2019, it has further increased its credit efforts to promote sustained and rapid economic growth.

The statistical table and diagram of the output value of the financial industry are shown in Table 2 and Fig. 2, respectively.

Table 2

Statistical table of the output value of the financial industry

	Financial industry output output value	The financial industry accounts for the tertiary industry	Financial industry output value as a proportion of GDP
2014	62.22	6.14	2.30
2015	73.21	6.27	2.30
2016	89.15	6.48	2.59
2017	109.43	6.94	2.64
2018	130.57	7.08	2.68
2019	154.99	7.41	2.75

Fig. 2

Statistical diagram of the output value of the financial industry.

From the perspective of the output value of the financial industry, the financial industry accounts for a relatively small proportion of the tertiary industry and has a low contribution to the province’s GDP. It can be seen from Table 2 that in 2019, the output value of the financial industry in Region A reached 15.346 billion yuan, accounting for 2.72% of GDP. However, during the same period, the proportion in developed coastal areas has reached 10% to 20%, and the national average has reached about 5%. In addition, the output value of the financial industry in A region accounts for a relatively low proportion of the tertiary industry. By the end of 2019, this proportion is 7.34%, which was lower than the national average of 9.24%, and far lower than developed regions such as Beijing, Shanghai and Guangdong. This all reflects that the overall level of development of the financial industry in A region is relatively low, and it has not yet become an advantageous industry in the province. However, from a month-on-month perspective, the output value of the financial industry in A region has shown a gradual increase since 2014, and the output value of the financial industry as a percentage of GDP and the tertiary industry also hit a new high in the past six years in 2019. The growth rate of financial industry output value in A region has accelerated in recent years, especially in 2018 and 2019, the growth rate reached 19.32% and 18.70% respectively, indicating that the development level of the financial industry in A region is gradually improving.

The statistical table and chart of the number of branches in the banking, insurance and securities industries are shown in Table 3 and Fig. 3, respectively.

Table 3

Statistical table of the number of branches in the banking, insurance and securities industries

	banking	Securities Industry	Insurance
2014	4482	40	795
2015	4183	42	989
2016	4350	48	1206
2017	4244	53	1236
2018	4488	59	1324
2019	4436	62	1621

Fig. 3

Statistical table of the number of branches in the banking, insurance and securities industries.

It can be seen from Table 3 and Fig. 3 that, compared with developed provinces and cities, financial institutions in area A are dominated by banks. Among them, banking financial institutions are mainly large commercial banks, with joint-stock commercial banks accounting for a relatively small proportion, and no foreign financial institutions have entered. Moreover, the number of provincial financial institutions is relatively small, with only two city commercial banks, one securities company, and one futures company.

The financing structure table is shown in Table 4 and Fig. 4.

Table 4

Financing structure table

Financing amount (100 million yuan)		loan	Proportion(%) (100 million yuan)
2014	335.72	88.68	2014	335.72
2015	521.56	90.40	2015	521.56
2016	1242.30	75.25	2016	1242.30
2017	911.73	87.67	2017	911.73
2018	1510.76	78.28	2018	1510.76
2019	1831.33	81.20	2019	1831.33

Fig. 4

Financing structure diagram.

From the perspective of financing channels, indirect financing is still the mainstay. As shown in Table 4 and Fig. 4, in 2019, the total social financing of A area was 181.32 billion yuan. Among them, loans accounted for 80.4% of the total financing, and bond financing and stock financing accounted for 14.7% and 4.9% of the total financing respectively. Compared with 2014, the proportion of loans in the total financing of the whole society has declined, but this proportion has always been above 75%. It can be seen from this that indirect financing is still the main means of financing for enterprises in the province, which is closely related to the fact that the financial industry in A region is mainly composed of the banking industry.

In order to better understand the financial development of each region in the A region, this paper divides the A region into three regions: A-1, A-2, and A-3. The five-year changes in the distribution of RMB deposits in financial institutions in the three regions are shown in Fig. 5 and Table 5.

Fig. 5

Statistical diagram of changes in the distribution of RMB deposits in financial institutions in three regions over the past five years.

Table 5

Five-year changes in the distribution of RMB deposits in financial institutions in the three regions

	1997	2002	2007	2012	2017	2019
A-1	Various deposits	437273.44	1472011.37	3102774.54	5999880.76	13432410.16	A-1
	Proportion	24.54%	24.53%	22.99%	21.81%	22.70%
A-2	Various deposits	460885.22	1387760.2	3105741.92	6228670	16634456.59	A-2
	Proportion	25.87%	23.13%	23.01%	22.63%	20.73%
A-3	Various deposits	901639.12	3201052.59	7422907.13	15562758.72	35841284.2	A-3
	Proportion	50.60%	53.35%	54.99%	56.56%	57.56%

The five-year changes in the per capita deposit balance of financial institutions in the three regions are shown in Table 6 and Fig. 6.

Table 6

Five-year changes in the per capita deposit balance of financial institutions in the three regions

	A-1	A-2	A-3
1997	1058.72	324.53	2275.47
2002	3156.69	872.46	7556.20
2007	5625.55	1624.62	14838.24
2012	11871.70	3081.63	31205.71
2017	29393.18	6713.12	83925.94
2019	32829.32	8174.79	10764.44

Fig. 6

Statistical diagram of the five-year changes in the per capita deposit balance of financial institutions in the three regions.

The five-year growth rate comparison of per capita deposit balance of financial institutions in A region is shown in Fig. 7 and Table 7.

Fig. 7

Comparison diagram of 5-year growth rate of per capita deposit balance of financial institutions.

Table 7

Comparison table of 5-year growth rate of per capita deposit balance of financial institutions

	A-1	A-2	A-3
2002	24.42	21.87	27.13
2007	12.25	13.24	14.45
2012	16.11	13.66	16.03
2017	17.88	16.85	21.88
2019	17.62	16.67	20.29

It can be seen from the above analysis that the overall level of financial development in A region is relatively low, the composition of financial industry institutions is relatively single, and the financing channels rely more on bank credit, and direct financing is not developed. From the perspective of the regional distribution of financial resources, the A-3 area concentrates more than half of the province’s total deposits and loans, which is significantly higher than the A-1 and A-2 areas. From the perspective of change trends, the growth rate of deposits and loans per capita and financial correlation rate of A-1, A-2 and A-3 from 1997 to 2019 shows a trend of first decline and then rise, and the growth rate of A-3 is higher than that of A-2 and A-1.

8 Conclusion

How to reasonably use the spillover effects of financial agglomeration in agglomeration areas to actively promote the economic development of surrounding areas, how to reasonably play the role of other influencing factors, and how to effectively estimate the spillover effects of financial risks are issues that need to be resolved in the current regional financial development.

This paper selects A region for research. According to the perspective of modern financial development theory, existing studies generally use the financial correlation rate (FIR) to measure the level of regional financial development, that is, the ratio of the total regional financial resources to GDP. The higher the financial correlation rate, the higher the level of regional financial development. However, the financial development of A region is lagging behind, the financial structure is mainly composed of the banking industry, and the RMB deposits and loans of financial institutions account for a large proportion of total financial assets. Therefore, the ratio of the balance of deposits and loans of financial institutions to GDP is generally used as a substitute indicator of the financial correlation rate to measure the level of financial development in A region. The results of verification and analysis through statistical methods show that the results obtained in this paper are consistent with the actual situation, and the model constructed in this paper has certain effects.

Footnotes

Acknowledgments

This paper was supported by (1) The Natural Science Foundation of Anhui Higher Education Institutions of China under grant no. KJ2020A0024; (2) The Natural Science Foundation of Anhui Higher Education Institutions of China under grant no. KJ2018A035;(3) the Nature Science Foundation of Anhui Province under grant No.1908085MF186.

References

Fichtner

, Heemskerk

E.M.

and Garcia-Bernardo

, Hidden power of the Big Three? Passive index funds, re-concentration of corporate ownership, and new financial risk, Business and Politics 19(2) (2017), 298–326.

Cueva

, Roberts

R.E.

, Spencer

, et al., Cortisol and testosterone increase financial risk taking and may destabilize markets, Scientific Reports 5(1) (2015), 1–16.

Lee

C.J.

and Andrade

E.B.

, Fear, excitement, and financial risk-taking, Cognition and Emotion 29(1) (2015), 178–187.

Bartram

S.M.

, Brown

G.W.

and Waller

, How important is financial risk? Journal of Financial and Quantitative Analysis 50(4) (2015), 801–824.

Cooper

W.W.

, Kingyens

A.T.

and Paradi

J.C.

, Two-stage financial risk tolerance assessment using data envelopment analysis, European Journal of Operational Research 233(1) (2014), 273–280.

Montford

and Goldsmith

R.E.

, How gender and financial self-efficacy influence investment risk taking, International Journal of Consumer Studies 40(1) (2016), 101–106.

Wanjohi

S.M.

, Wanjohi

J.G.

and Ndambiri

J.M.

, The effect of financial risk management on the financial performance of commercial banks in Kenya, International Journal of Finance and Banking Research 3(5) (2017), 70–81.

Magendans

, Gutteling

J.M.

and Zebel

, Psychological determinants of financial buffer saving: the influence of financial risk tolerance and regulatory focus, Journal of Risk Research 20(8) (2017), 1076–1093.

Cole

E.S.

, Walker

, Mora

, et al., Identifying hospitals that may be at most financial risk from Medicaid disproportionate-share hospital payment cuts, Health Affairs 33(11) (2014), 2025–2033.

10.

Zalik

, Resource sterilization: reserve replacement, financial risk, and environmental review in Canada’s tar sands, Environment and Planning A 47(12) (2015), 2446–2464.

11.

Lucarelli

, Uberti

and Brighetti

, Misclassifications in financial risk tolerance, Journal of Risk Research 18(4) (2015), 467–482.

12.

Chan

E.Y.

, Physically-attractive males increase men’s financial risk-taking, Evolution and Human Behavior 36(5) (2015), 407–413.

13.

West

and Worthington

A.C.

, Macroeconomic conditions and Australian financial risk attitudes, 2001–2010, Journal of Family and Economic Issues 35(2) (2014), 263–277.

14.

Tanimura

, Jaramillo

, Weil

, et al., Financial burden for tuberculosis patients in low-and middle-income countries: a systematic review, European Respiratory Journal 43(6) (2014), 1763–1775.

15.

Scharding

T.K.

, Imprudence and immorality: A Kantian approach to the ethics of financial risk, Business Ethics Quarterly 25(2) (2015), 243–265.

16.

and Furman

, A form of multivariate Pareto distribution with applications to financial risk measurement, A, STIN Bulletin: The Journal of the IAA 47(1) (2017), 331–357.

17.

Wang

G.W.Y.

, Woo

S.H.

and Mileski

, The relative efficiency and financial risk assessment of shipping companies, Maritime Policy & Management 41(7) (2014), 651–666.

18.

Luck

L.J.

, Justice and systemic financial risk, Journal of Applied Philosophy 34(3) (2017), 331–352.

19.

Allen

and Petsoulas

, Pricing in the English NHS quasi market: a national study of the allocation of financial risk through contracts, Public Money & Management 36(5) (2016), 341–348.

20.

Thomä

and Chenet

, Transition risks and market failure: a theoretical discourse on why financial models and economic agents may misprice risk related to the transition to a low-carbon economy, Journal of Sustainable Finance & Investment 7(1) (2017), 82–98.

21.

Sosnovska

and Zhytar

, Financial architecture as the base of the financial safety of the enterprise, Baltic Journal of Economic Studies 4(4) (2018), 334–340.

22.

Acemoglu

, Ozdaglar

and Tahbaz-Salehi

, Systemic risk and stability in financial networks, American Economic Review 105(2) (2015), 564–608.

23.

Castellano

, Cerqueti

and Rotundo

, Exploring the financial risk of a temperature index: A fractional integrated approach, Annals of Operations Research 284(1) (2020), 225–242.

24.

Cohn

, Engelmann

, Fehr

, et al., Evidence for countercyclical risk aversion: An experiment with financial professionals, American Economic Review 105(2) (2015), 860–885.

25.

Gallardo Millán

L.A.

, Ochoa

E.A.

and Estructura De Capital

, Riesgo Financiero: Evidencia Empírica En Pymes Hoteleras (Capital Structure and Financial Risk: Empirical Evidence in SMEs), Revista Global de Negocios 3(4) (2015), 1–10.