Mathematical modelling of enterprise financial risk assessment based on risk conduction model

Some scholars believe that the market volatility is positively correlated with the value at risk, while the stock price in the financial market has a significant volatility clustering effect. Then the value at risk closely related to volatility should also have similar properties. For this reason, the author proposes a CAViaR model with autocorrelation characteristics based on the idea of quantile regression to directly measure the value at risk of financial markets [2]. However, the CAViaR model is mainly suitable for analysing the dynamic risk characteristics of a single sector. Still, it cannot capture the complex relationship of risk contagion between different sectors (or different markets). Some scholars have extended the traditional CAVi-aR model to the MVMQ-CAViaR model to overcome the shortcomings of the above models. This model extends the idea of single-equation quantile regression to the structured equation of vector autoregression. At the same time, the model reveals that the value at risk of a single sector (or a single market) is affected by its market and risk spillovers from other markets. The specific expression is as follows: (1) {q1t(θ)=c1+a11|Y1t−1|+a12|Y2t−1|+b11q1t−1(θ)+b12q2t−1(θ)q2t(θ)=c2+a21|Y1t−1|+a22|Y2t−1|+b21q1t−1(θ)+b22q2t−1(θ) \left\{\begin{array}{l} q_{1 t}(\theta)=c_{1}+a_{11}\left|Y_{1 t-1}\right|+a_{12}\left|Y_{2 t-1}\right|+b_{11} q_{1 t-1}(\theta)+b_{12} q_{2 t-1}(\theta) \\ q_{2 t}(\theta)=c_{2}+a_{21}\left|Y_{1 t-1}\right|+a_{22}\left|Y_{2 t-1}\right|+b_{21} q_{1 t-1}(\theta)+b_{22} q_{2 t-1}(\theta) \end{array}\right.

q_it(θ) represents the conditional quantile of the market rate of return Y_it−1 under the θ probability. VaR represents the value-at-risk of the market rate of return corresponding to the probability of θ. |Y_it−1| represents the absolute value of the return rate of the market index i. This parameter represents a market shock item and implies that a positive shock and a negative shock lagging have the same effect on the current VAR. q_it−1 represents the lagged conditional quantile, which can well describe the autocorrelation of the financial market tail distribution. Taking the banking sector and the securities sector as examples, suppose q_1t represents the Var of the bank and q_2t represents the Var of the securities. Eq. (1) shows that the bank’s risk value q_1t is affected by its market and the extreme risk q_2t−1 of the securities sector and the market shock item |Y_2t−1|.

Model (1) assumes that the market shock item does not have a ‘leverage effect’. The positive shocks and adverse shocks of this market and other markets have equivalent effects on the value at risk. In real life, the investor’s utility function often puts a greater weight on the negative utility of losses while giving a smaller weight to the positive utility of returns. Therefore, this paper expands Eq. (1) to the asymmetric MVMQ-CAViaR model. The model is expressed as follows: (2) {q1t(θ)=c1+a11(Y1t−1)++a12(Y1t−1)−+d11(Y2t−1)++d12(Y2t−1)−+b11q1t−1(θ)+b12q2t−1(θ)q2t(θ)=c2+a21(Y1t−1)++a22(Y1t−1)−+d21(Y2t−1)++d22(Y2t−1)−+b21q1t−1(θ)+b22q2t−1(θ) \left\{\begin{array}{l} q_{1 t}(\theta)=c_{1}+a_{11}\left(Y_{1 t-1}\right)^{+}+a_{12}\left(Y_{1 t-1}\right)^{-}+d_{11}\left(\begin{array}{l} Y \\ 2 t-1 \end{array}\right)++d_{12}\left(\begin{array}{l} Y \\ 2 t-1 \end{array}\right)-+b_{11} q_{1 t-1}(\theta)+b_{12} q_{2 t-1}(\theta) \\ q_{2 t}(\theta)=c_{2}+a_{21}\left(Y_{1 t-1}\right)^{+}+a_{22}\left(Y_{1 t-1}\right)^{-}+d_{21}\left(\begin{array}{l} Y \\ 2 t-1 \end{array}\right)++d_{22}\left(\begin{array}{l} Y \\ 2 t-1 \end{array}\right)-+b_{21} q_{1 t-1}(\theta)+b_{22} q_{2 t-1}(\theta) \end{array}\right.

(Y_1t−1)⁺ and (Y_1t−1)⁻, respectively, represent the positive and negative part of the slow rate of return. When the two plates negatively impact, it is likely to be a joint impact [3]. This kind of shock will undermine public investment confidence and investors’ panic, amplifying the original market risk. To further study the asymmetric effects on Var when different financial sectors are simultaneously subjected to adverse shocks, we further extend model (2) into a joint asymmetric MVMQ-CAViaR model. The specific model is as follows: (3) {q1t(θ)=c1+a11(Y1t−1)++a12(Y1t−1)−+d11(Y2t−1)++d12(Y2t−1)−+e1(Y1t−1)−(Y2t−1)−+b11q1t−1(θ)+b12q2t−1(θ)q2t(θ)=c2+a21(Y1t−1)++a22(Y1t−1)−+d21(Y2t−1)++d22(Y2t−1)−+e2(Y1t−1)−(Y2t−1)−+b21q1t−1(θ)+b22q2t−1(θ) \left\{\begin{array}{l} q_{1 t}(\theta)=c_{1}+a_{11}\left(Y_{1 t-1}\right)^{+}+a_{12}\left(Y_{1 t-1}\right)^{-}+d_{11}\left(Y_{2 t-1}\right)^{+}+d_{12}\left(Y_{2 t-1}\right)^{-}+e_{1}\left(Y_{1 t-1}\right)^{-} \\ \left(Y_{2 t-1}\right)^{-}+b_{11} q_{1 t-1}(\theta)+b_{12} q_{2 t-1}(\theta) \\ q_{2 t}(\theta)=c_{2}+a_{21}\left(Y_{1 t-1}\right)^{+}+a_{22}\left(Y_{1 t-1}\right)^{-}+d_{21}\left(Y_{2 t-1}\right)^{+}+d_{22}\left(Y_{2 t-1}\right)^{-}+e_{2}\left(Y_{1 t-1}\right)^{-} \\ \left(Y_{2 t-1}\right)^{-}+b_{21} q_{1 t-1}(\theta)+b_{22} q_{2 t-1}(\theta) \end{array}\right.

From Eqs (1)–(3), we can see that risks between different financial industry sectors will be transmitted to each other. The market shock item (namely Y_it) of a single market will directly affect the value at risk of itself and other markets. The current market shock will change investors’ expectations of the risk value of financial assets in the next period and directly lead to changes in size VaR_t. In addition, since funds in the market can flow freely in different industries, investors will change their investment portfolios in different sectors according to changes in risk conditions [4]. Therefore, the linkage effect of this sector will indirectly cause changes in other markets VaR_t. To investigate the interaction between variables, the more straightforward method is impulse response analysis. This article needs to examine the dynamic influence process of market shocks on the tail of the rate of return. Therefore, we use QIRF. The specific calculation steps of this method are as follows:

First, suppose that the formation process of the return rate data of two different sector indexes satisfies [Y1tY2t]=[at0βtγt][ε1tε2t]\left[\begin{array}{l} Y_{1 t} \\ Y_{2 t} \end{array}\right]=\left[\begin{array}{ll} a_{t} & 0 \\ \beta_{t} & \gamma_{t} \end{array}\right]\left[\begin{array}{l} \varepsilon_{1 t} \\ \varepsilon_{2 t} \end{array}\right], and the intensity of the market shock depends on the Cholesky decomposition matrix [at0βtγt]\left[\begin{array}{ll} a_{t} & 0 \\ \beta_{t} & \gamma_{t} \end{array}\right].

Second, at time t, a one-time positive or negative impact of 1 unit of new interest ε_1t will make the actual rate of return Y_it become Y~it.Y~it=Yit+ΔYit\widetilde{Y}_{i t} . \widetilde{Y}_{i t}=Y_{i t}+\Delta Y_{i t} in the current period. However, the rate of return at other moments remains unchanged [5]. Similarly, when analysing the dynamic impact of the combined negative shock, we simultaneously apply 1 unit of a negative shock to ε_1t and ε_2t.

Finally, the ΔY_it value is obtained according to the impact of different plates. At the same time, we use the coefficient values estimated by the models (1)–(3) to further analyse the dynamic influence process of the VaR on the value at risk of different sectors.

The MVMQ-CAViaR model belongs to the category of multiple quantile regression. We can use the Least Absolute Deviation Method (LAD) estimation, and the objective function it needs to optimise is: (4) mina1T∑t=1T∑i=1n[ρ(θi)(Yit−qit(θi,a))] \min _{a} \frac{1}{T} \sum_{t=1}^{T} \sum_{i=1}^{n}\left[\rho\left(\theta_{i}\right)\left(Y_{i t}-q_{i t}\left(\theta_{i}, a\right)\right)\right]

ρ(θ_i) = θ_i − I(Y_it < q_it(θ_i, a)). I() is an indicative function, where n represents the number of financial markets, and T is the total number of samples. Eq. (4) shows that the absolute deviation of the structured equation is minimised overall. Compared with the parameter estimation method of multivariate joint distribution, this method has the following advantages. First of all, we don’t need to make any prior assumptions about the joint distribution of income data, which effectively avoids misconfiguration of the distribution. Second, quantile regression is to regress the data under a specific quantile, so it has relative robustness to the outliers in the data. Finally, it can directly measure the size of the tail risk in the dependent market, without the need to separately estimate the conditional mean and conditional volatility equations to solve indirectly [6]. This greatly simplifies the number of model estimation coefficients.

This article takes the conditional quantile of θ =5% for different sector indexes of the financial industry. We take the quantile corresponding to the first 100 observations to initialise q_i1 and use the simplex algorithm and the quasi-Newton algorithm to optimise the model. This paper adopts a two-step estimation method to improve the estimation efficiency of the structured model [7]. The first step is to estimate the univariate quantile SAV model and use the estimated result as the initial trial estimated coefficient of the second step optimisation. In the second step, we optimise the multivariate quantile model (1)–(3) as a whole to minimise the objective function (4).

The in-sample performance of the model cannot simply be extended outside the sample. To further demonstrate the effect of this model on predicting VAR, we must also perform an out-of-sample robustness test. Some scholars have assumed that the model can effectively predict risk, proving that the sequence of ‘hit events’ obeys the Bernoulli distribution. Scholars construct unconditional test statistics of likelihood ratio as the following: (5) LR=−2ln⁡[(1−p)N−n×pn]+2ln⁡[(1−n/N)N−n×(n/N)nx2 L R=-2 \ln \left[(1-p)^{N-n} \times p^{n}\right]+2 \ln \left[(1-n / N)^{N-n} \times(n / N)^{n} x^{2}\right.

where p is the significance level, N is the total number of predicted samples and n is the number of hits in the prediction sample. When the LR statistic is greater than the critical value of the chi-square distribution under a given confidence level, the original model is rejected. On the contrary, when the statistic is less than the critical value, the model is accepted.

Eq. (5) shows that the Kupiec likelihood ratio test quantity is an index that characterises the degree to which the actual number of hits is close to the theoretical number of hits. In addition to testing the failure rate, it should also be tested whether there is a correlation between hit events. If there is a significant correlation between the observations that fail to predict the VaR, then a loss that continuously exceeds the VaR may occur. This will bring huge losses to investors [8]. The hit sequence of an accurate and reliable risk measurement model should be unbiased and non-autocorrelation. Therefore, some scholars proposed a dynamic quantile test. They used the bullish risk as an example to define a new hit sequence: (6) HITθ,t=I(yt<−VaRt)−θ $HIT_{\theta ,t} = I\left( {y_t &lt; - VaR_t } \right) - \theta$

θ is the given quantile. When y_t < −VaR_t is HIT_{θ, t} = 1 − θ ; when y_t < −VaR_t is HIT_{θ, t} = −θ. If the model parameters are estimated correctly, then E(HIT_{θ, t}|Ω_t−1) = 0. This shows that HIT_{θ, t} should not be relevant to any lagging HIT_{θ, t−k} and predicted VaR_t−k. We construct the following regression equation: (7) HITθ,t=β0+β1HITθ,t−1+β2HITθ,t−2+⋯+βpHITθ,t−p+βp+1VaRt+ut H I T_{\theta, t}=\beta_{0}+\beta_{1} H I T_{\theta, t-1}+\beta_{2} H I T_{\theta, t-2}+\cdots+\beta_{p} H I T_{\theta, t-p}+\beta_{p+1} V a R_{t}+u_{t}

We express the above model as a matrix form, HIT_{θ, t} = Xβ + u_t, where X is the T × K matrix-vector, and we take p = 5, k = 7. In the case of the null hypothesis β =0, the DQ test statistic we constructed is: (8) DQ=βols′X′Xβolsθ(1−θ)→x2(k) D Q=\frac{\beta_{o l s}{ }^{\prime} X^{\prime} X \beta_{o l s}}{\theta(1-\theta)} \rightarrow x^{2}(k)

From the results in Table 1, we can see that in the sample interval selected in this article, the average yields of banks and securities are both positive. Among them, the average yield of securities reached 0.033, while the insurance industry was hostile. Thus, if the stock price index changes reflect the overall expectations of the industry’s operating performance, from this perspective, the securities industry is more prosperous than the insurance industry. At the same time, the standard deviation of the securities sector is relatively the largest, and the high risk also brings a higher risk premium [9]. This is in line with the ‘small-cap stock effect’ in finance.

Furthermore, the three plate indices all have typical ‘negative bias’ and ‘sharp peaks and thick tails.’ Finally, according to the P-value of the J-B statistic and the ADF statistic, we further found that all the index returns are non-normal and stationary time series. Thus, the trends of banking, securities and insurance remained the same.

Table 2 shows the mutual risk transmission results of banks and securities, securities and insurance, and banks and insurance at the 5% quantile level. All the coefficients b₁₁ and b₂₂ reject the null hypothesis at the 1% significance level, and the coefficient values are all greater than 0.8. This shows that the risk levels of different financial sectors have a high degree of serial correlation. Other specific results are as follows:

First, the coefficient a₁₁ in the multiple regression model of banks and securities reaches a significance level of 10%. This shows that the bank’s previous yield has a significant negative impact on Var, and the coefficient a₂₂ is also significantly different from zero. This shows that the market impact of the securities sector will also increase the risk value of the market. In addition, we also found that the coefficients a₂₁ and a₂₁ have reached the 10% significance level. This shows that the banking sector’s extreme risks and market shocks will be transmitted to the securities sector, and the direction of influence will be harmful [10]. However, the securities sector does not have a significant risk transmission effect on the banking sector.

Second, for the regression results of banking and insurance, we can find that the coefficient a₂₂ is not statistically significant. This shows that the insurance sector risk is not affected by previous market shocks. Coefficient a₂₁ significantly indicates that extreme risks of banks will be transmitted to the insurance sector, while risks in the insurance industry have not been transmitted to banks.

Third, from the empirical results between securities and insurance, it can be seen that the extreme risks of insurance have a significant one-way spillover effect on securities, and securities do not have the function of actively transmitting risks.

Table 2 does not distinguish between the different effects of the rise and fall of the sector index on Var. To further explore the leverage effect of VaR in the financial industry, Table 3 gives the empirical results of the asymmetric MVMQ-CAViaR model. It can be seen from this that the estimation results of the three multi-quantile regression models are consistent with Table 2, and the main difference is reflected in the asymmetric coefficient. From the significance of the coefficients a₁₁, a₁₂ and a₂₁, a₂₂, it is easy to know that the risk of the banking sector is significantly affected by its own negative market shock. In contrast, the impact of the positive market shock is not wholly significant. The securities sector was significantly affected by both positive and negative market shocks [11]. In addition, the insurance sector is also occasionally affected by adverse market shocks. And the absolute values of all parameters are not correspondingly equal. This means that the market shock of the Chinese financial industry has apparent asymmetric effects on the Var of different sectors, and the impact of the decline of the sector index is more significant than the rise of the sector index. From the estimation results of the coefficients a₂₁, a₂₂ and a₁₁, a₁₂, it can be seen that the negative market impact of banks will spread to the securities sector and the insurance sector and significantly increase the risk value of these two sectors. However, the positive market shock coefficients are not significant. Negative insurance information will also spread to the securities sector, and there is no risk spillover effect on other markets.

Figures 1 and 2 show the quantile impulse response results of the banking and securities sectors, respectively. Whether it is a standard deviation information shock from the banking sector or the securities sector, we find that a negative information shock has a more significant impact on the original market than a positive information shock. The positive impact of the banking sector has an initial positive effect on securities and then quickly decays to negative [12]. Comparing Figures 1 and 2, we can find that when there is a joint negative information shock, the intensity of this shock is significantly greater than the negative shock of a single market. The risk expansion value of the more volatile securities sector is significantly greater than that of the bank.

Fig. 1

Fig. 2

To summarise, this part explains the significant asymmetry of risk transmission between different sectors of the Chinese financial industry according to the two methods of coefficient value and quantile impulse response of the regression results. It is further discovered that Chinese banks have significant risk contagion effects on other sectors, while securities are at a disadvantageous position to receive risks from other sectors passively.

Table 5 shows the backtest results of different sectors under the 5% quantile. At the 5% significance level, the Kupiec likelihood ratio results of the three models all passed the robustness test. Judging from the accuracy of prediction, the asymmetric MVMQ−CAViaR model is significantly better than the original MVMQ−CAViaR model. The joint asymmetric MVMQ −CAViaR model is slightly better than the asymmetric MVMQ-CAViaR model. In addition, it can be seen from the DQ test that the traditional model rejected the null hypothesis three times, while the asymmetric MVMQ−CAViaR model rejected the null hypothesis only once. It can be seen from this that the joint asymmetric MVMQ −CAViaR model has wholly passed the DQ test in the statistical sense. In short, the two new models proposed in this paper can significantly improve the accuracy of risk prediction in different sectors of finance, and the joint asymmetric F model has relatively more competitive advantages. This shows that the leverage decomposition of market shock items has a theoretical basis and practical significance.