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A Study on the Application of Quantile Regression Equation in Forecasting Financial Value at Risk in Financial Markets

Publié en ligne: 15 Jul 2022
Volume & Edition: AHEAD OF PRINT
Pages: -
Reçu: 28 Apr 2022
Accepté: 17 Jun 2022
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Magazine
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01 Jan 2016
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Introduction

Financial market is a relatively complex system, which contains a large number of uncertainties, such as national macroeconomic policies, interest rate changes and exchange rate fluctuations. These uncertainties can affect the development of financial markets and may even lead to financial crises. Therefore, when studying financial market risks, it is necessary to take into account the macroeconomic environment and the internal management of micro enterprises[1]. After the gradual establishment and improvement of China's market economy system, the financial market has also ushered in a period of rapid development. At this stage, people pay more and more attention to the financial situation of enterprises, but because China's financial industry is still in the process of starting and exploring development, many enterprises also face more difficulties and challenges, how to identify and assess the financial situation of enterprises and take effective measures to solve these problems has become an important issue in the development of the financial market.

Risks and benefits in the financial market go hand in hand, and under certain conditions, there is a positive and negative correlation. In theory, these are economic factors such as interest rates and exchange rates, which can affect the financial situation of an enterprise. However, with the continuous improvement of China's market economy system and the increase of capital circulation brings the trend of financial investment expansion, which makes the financial market risk more prominent[2, 3]. The financial situation is the most important factor affecting the development of enterprises, so it is important to take these aspects into account when conducting research on them. It has also led to an increase in interest rate volatility, resulting in sharp changes in asset prices, which can cause problems such as lower yields and higher losses[4]. Therefore, it is important to study how to identify and measure risk, which requires in-depth analysis and investigation.

Financial market risk forecasting is the analysis of economic development conditions and trends in a certain period of time in the future, and then it will be based on this information to develop appropriate countermeasures and take effective measures to minimize potential losses[5]. In practice, many investors tend to focus on the current profit or income level and ignore the profitability of the enterprise, which makes many enterprises are facing huge financial risks. In the financial market, stocks, bonds and other investment varieties are high yield and stable growth, investors are often only concerned about whether their profits or income is substantial, but ignore the profitability of enterprises. This leads to a weak control and management of risk when investors choose to invest. Therefore, how to use financial market data to predict stock price movement trends and identify potential losses is a topic that financial managers need to study. By introducing a large amount of data to build a model to predict the various scenarios and influencing factors that a financial portfolio may encounter. The deviation and the size of the error between the sample stock price and the expected return can be calculated using regression equations[6]. Then, according to the obtained error value and the degree of deviation, a financial management method that is suitable for the development of the company itself and targeted can be selected, so as to reduce the risk loss and increase the level of return[7].

Financial value forecasting algorithm based on quantile regression function
Quantile regression function

Regression analysis is the process of describing, estimating and explaining the relationship between variables based on known data using statistical methods. The quantile regression model is shown in Figure 1. In financial market risk research, people usually build models to predict or assess how things may change in the future. The empirical studies are mainly based on the influence of factors and the quantitative approach to infer the characteristics that are related to them with certain regularity, which is what the mathematical tools used are regression equations[8]. Regression equations also use statistical methods to analyze and explain the correlations that exist between variables.

Figure 1

Quantile regression model

The quantile regression function has the following description:

Let the distribution function of the random variable Z be as shown in Equation (1): F(z)=P(Zz) F\left( z \right) = P\left( {Z \le z} \right)

Then the τth quantile of Z is shown in Equation (2), and then the median is not F−1 (1/2) F1(τ)=inf{ z:F(z)τ } {F^{ - 1}}\left( \tau \right) = \inf \left\{ {z:F\left( z \right) \ge \tau } \right\}

For a group of random samples of Z {z1, z2...., Zn}, the sample mean is the optimal solution to equation (3). The sample median is the sum of the absolute values of the minimized residuals, as shown in Equation (4): mean=mini=1n(ziξ)2 {\rm{mean}} = \min \sum\limits_{i = 1}^n {{{\left( {{{\rm{z}}_{\rm{i}}} - \xi } \right)}^2}} F1(1/2)=argminξRi=1| ziξ | {F^{ - 1}}\left( {1/2} \right) = \arg \mathop {\min}\limits_{\xi \in R} \sum\limits_{i = 1} {\left| {{z_i} - \xi } \right|}

And for the other τ quantile one can derive Equation (5): minβRP[ i{ i:ziξ }τ| ziξ |+i{ i:ziξ }(1τ)| ziξ | ] {\min _{\beta \in {R^P}}}\left[ {\sum\limits_{i \in \left\{ {i:{z_i} \ge \xi } \right\}} {\tau \left| {{z_i} - \xi } \right| + } \sum\limits_{i \in \left\{ {i:{z_i} \ge \xi } \right\}} {\left( {1 - \tau } \right)\left| {{z_i} - \xi } \right|} } \right]

The equivalence is shown in Equation (6). minξRi=1Pτ(ziξ) \mathop {\min}\limits_{\xi \in R} \sum\limits_{i = 1} {{P_\tau }\left( {{z_i} - \xi } \right)}

Value at Risk (VaR) approach

Financial market risk refers to the loss of business due to unpredictable factors, resulting in lower expected returns or loss of principal for investors. In practical analysis, we usually use the “VaR” model to describe the various types of potential losses that affect economic activities. This method can be used to estimate and judge future uncertainty events, it can also be used to calculate the magnitude of changes in the value of the corresponding assets under various factors and the direction of their changes, it can also predict a series of adverse consequences of the occurrence of risk and take appropriate preventive measures to reduce the degree of economic losses or reduce the rate of economic losses, which can also provide investors with more basis for investment decisions This can also provide investors with more basis for investment decisions[9]. The maximum possible loss VaR at confidence level is shown in Equation (7), and the definition of VaR is shown in Equation (8). prob(Pt+1Pt<VaR)=α prob\left( {{P_{t + 1}} - {P_t} < - VaR} \right) = \alpha VaR=inf{ x|prob(ΔP<x)>α } VaR = - \inf \left\{ {x\left| {prob\left( {\Delta P < x} \right) > \alpha } \right.} \right\}

In general, the VaR is calculated using the rate of return on financial assets, as shown in Equation (9). prob(ri,j<VaR)=α prob\left( {{r_{i,j}} < VaR} \right) = \alpha

If the financial asset return ri, j is a random variable of continuous type and the probability density function is f(.), then the VaR of the financial asset return is the lower α quantile of the distribution f(.) of the lower alpha quantile of the distribution as shown in equation (10): VaRf(t)dt=α \int_{ - \infty }^{ - VaR} {f\left( t \right)dt = \alpha }

For a discrete distribution of financial asset returns, VaR is expressed as shown in Equation (11). ri,j<VaRprob(ri,j)=α \sum\limits_{{r_{i,j}} < - VaR} {prob\left( {{r_{i,j}}} \right) = \alpha }

The relative VaR of the portfolio over the future holding period at confidence level 1-α is defined as shown in Equation (12). VaR=E(P)P*=P0(r*μ) VaR = E\left( P \right) - P* = - {P_0}\left( {{r^*} - \mu } \right)

If the absolute loss is expressed in terms of the expected average payoff u = 0, it is called the absolute VaR, which is defined in Equation (13) as follows. VaR=P0P*=P0r* VaR = {P_0} - P* = - {P_0}{r^*}

Let f(P) be the probability density function of the asset portfolio value, at the confidence level of 1-α, to find the minimum value P*, so that the probability of a possible return higher than P* is 1-α, it can be expressed as shown in Equation (14): 1α=prob(PP*)=P*f(P)dP 1 - \alpha = prob\left( {P \ge {P^*}} \right) = \int_{{P^*}}^\infty {f\left( P \right)dP} or the probability that the possible return is lower than P is α, as shown in equation (15). α=prob(PP*)=P*f(P)dP \alpha = prob\left( {P \ge {P^*}} \right) = \int_\infty ^{{P^*}} {f\left( P \right)dP}

Financial market risk financial value forecasting model-VAR

VAR model is an important financial risk evaluation index, and the method is widely used in financial market research for financial analysis and forecasting, etc. The graph of influencing factors of exchange rates studied by VAR model is shown in Figure 2. From the definition of VaR under general distribution conditions, it can be seen that the estimation of the portfolio benefit rate of the investment is the key to calculate VaR. Calculating a more accurate value of VaR not only allows predicting various risks that may occur in financial markets, but also provides investors with an effective basis for decision making. Conversely, it can affect the management effectiveness, it will make the expected return on investment deviate from the actual. This requires investors to effectively assess financial risk to provide a basis for business decisions. Most of the VaR methods are based on the estimation of the benefit rate in the form of a specific distribution. The values calculated by the model are compared with the actual returns, and the results are analyzed and the corresponding preventive measures are proposed based on the data and conclusions obtained. The current VaR calculation method is divided into parametric and non-parametric methods.

Figure 2

VAR model research method

Parametric method refers to the use of statistical methods to establish the corresponding model according to the characteristics of the sample data, and by testing whether its parameters conform to the law of normal distribution. The parametric method needs to make reasonable assumptions about its parameters when calculating VaR, and combine these theories with reality to get a correct and effective prediction model, which makes the VaR results can also have large deviations. The models commonly used in the parametric method are RiskMetrics model, analytical method (variance-covariance method), for RiskMetrics mixed normal model and ARCH model.

The RiskMetatial model is a multiple linear regression analysis method proposed by an American economist in 1965. The model uses a quadratic polynomial, i.e., there is a relationship and homogeneity between the dependent and independent variables. The core idea is to estimate the variance-covariance matrix of the distribution of the portfolio benefit rate, and the RiskMetrics model takes an exponentially weighted average method to calculate VaR, the structure of the RiskMetatial model is shown in Figure 3, and its expressions are as (16) follows: σ12=(1λ)i=1nλt1Ri12 \sigma _1^2 = \left( {1 - \lambda } \right)\sum\limits_{i = 1}^n {{\lambda ^{t - 1}}} R_{i - 1}^2

Figure 3

Structure of the RiskMetatial model

The analytical method is the most typical representative of the method of calculating VaR, which is a frequently used in financial analysis. It enables a quantitative description of a particular risk and divides the degree of loss faced and the influencing factors into several components based on that characteristic, then classifies the risk level according to the degree of loss of each component, and finally classifies them into different types so as to rationalize the analysis and finally draw conclusions.

The RiskMetatics mixed-normal model was proposed by Kohnen, a well-known economist, in 1991. The method uses a linear regression equation, which means that problems with multiple variables in a continuous time series can be solved using their corresponding equations. The algorithm is able to interrelate multiple factors, it enables high accuracy and stability of the object of study, and it simplifies the processing and computational analysis of the data, thus improving the prediction accuracy as well as reducing the error rate, which enables a high accuracy of the model's parameter estimation. The RiskMetrics mixed-normal model is shown in Figure 4.

Figure 4

RiskMetrics mixed-normal plot

The ARCH model refers to taking the historical price data of a variable as the basis, and then using specific parameters for regression analysis, and then establishing mathematical functions and corresponding econometric equations[10, 11]. This method can provide reasonable explanations for a large number of complex problems. Engle proposed the ARCH model, followed by Bollerslev's GARCH model. The ARCH model is shown in Figure 5. Its structure is as Equation (17): rt=μ+εtσ12=α0+αtεt12+β1σt12++βqσtq2 \matrix{ {{r_t} = \mu + {\varepsilon _t}} \hfill \cr {\sigma _1^2 = {\alpha _0} + {\alpha _t}\varepsilon _{t - 1}^2 + \cdots {\beta _1}\sigma _{t - 1}^2 + \cdots + {\beta _q}\sigma _{t - q}^2} \hfill \cr }

Figure 5

ARCH model

Non-parametric methods is used to predict the overall risk profile by analyzing sample data and comparing it with actual economic indicators to draw conclusions. Commonly used methods are historical simulation and Monte Carlo's simulation method. Historical simulation is based on a large amount of actual data and models, and it obtains the corresponding regression equation by statistical analysis of the sample. The method treats the object of study as a uniform standard variable. Based on the obtained information and conclusions, it predicts the direction of development of future economic trends, the pattern of changes, and predicts possible future economic events or financial situations. Monte Carlo simulation method is a probabilistic-based, which uses the principle of randomization on a computer to build a mathematical model to predict various risk events that may occur in the financial markets at a certain moment in the future. It can be calculated and analyzed according to the actual economic phenomena and statistical laws, and predict the possible future risk events, so as to reduce losses and increase the rate of return[12]. The Monte Carlo simulation method is shown in Figure 6.

Figure 6

Monte Carlo simulation method

Empirical study

This will use the established non-parametric VaR model to analyze the risk profile of our financial markets and to forecast them. It uses the data of 2013 and uses regression analysis to build a risky asset price model for the financial market. Firstly, this paper introduces the problems of interest rate, credit and capital management faced by commercial banks in the financial market in China in recent years, followed by the construction of the corresponding mathematical method based on the regression equation and the sample data. The sample data of 2013 has about 751, and transforming the daily benefit price P, into the daily benefit rate can yield 750. In order to better measure the value at risk, regression model will be adopted to analyze the financial market and combine the research results with the actual situation. In this paper, we use SPSS22.0 statistical software and multiple linear transformation methods such as principal component factor extraction and variance decomposition to establish the data set and sample selection, which can be divided into 125 days, 250 days and 500 days as shown in Table 1.

Summary of data

data statistics the number of days in the sample period Days outside the sample period
capacity 125 days 250 days 500 days 250 days
mean −0.191 −0.046 −0.067 0.031
standard deviation 1.290 3.120 2.210 1.301
Skewness coefficient −0.501 −0.699 −0.329 −0.542
Kurtosis coefficient 3.987 4.702 4.731 4.562
J-BP value 0.003 0.000 0.000 0.000

According to this table, the index benefit of Shanghai securities is higher in terms of standard deviation in each period. This means that the portfolio is vulnerable to financial crises when there is a high degree of uncertainty in the current period. The series of benefit ratios during the period shows a left-skewed pattern, and it shows a stable upward trend. This indicates that there is a large uncertainty in the financial market during this period. In addition, the results of the Jarqe-Bera normal test show that the parameters of the rate of return series have a relationship with the profitability of the firm. That is, investors consider the profitability factor in their portfolios, but tend to ignore other influencing factors. Therefore, investors should take into account the factors affecting profitability when investing in order to reduce risk.

The six models above were adopted to estimate VaR and the results were calculated in the form of box line graphs, and the regression models were built using Excel tables, and the results were analyzed and conclusions were drawn. The results of the study are: with the rapid development of China's financial markets, increasing size, diversification of transaction structures and increase in the number of outputs, which leads to a decrease in VaR values; interest rate risk and exchange rate fluctuations are the most important factors affecting changes in stock market returns. The empirical tests led to the following conclusions: in our financial markets, risk factors influence stock market returns, but at the same time the magnitude of interest rate fluctuations is positively correlated with stock price changes, while on the other hand with the decline in stock market prices, diversification of trading structures and increase in the number of assets, this leads to a greater degree of impact on stock index futures price changes.

Thereafter, the VaR values already calculated for the first six models were tested and we adopted the technique of the relative error of the square root of the mean and also the failure rate p = N/T to assess the accuracy and variability of VaR and obtained the factors that affect the VaR values. In the case of financial market risk management, we use SPSSM to perform the analysis. This method can reflect the various risks faced by companies more accurately. The relative errors of the square root of the mean of VaR for different confidence levels and view windows are summarized in Table 2 below.

Summary of the relative error of the square root of the mean of VaR

different confidence levels RMSRB value alpha = 99%
125 250 500 125 250 500
VaR1 0.3027 0.3085 0.3654 0.2319 0.1452 0.2430
VaR2 0.1901 0.1453 0.2109 0.1742 0.1321 0.1980
VaR3 0.4421 0.2452 0.2314 0.6034 0.3896 0.2789
VaR4 0.2103 0.1533 0.2753 0.2213 0.2078 0.2785
VaR5 0.1890 0.1899 0.2863 0.2090 0.1200 0.2540
VaR6 0.1675 0.1633 0.1632 0.2706 0.1989 0.2432

This table shows that the RMSRB calculated by each model differs when the confidence level reaches 99%, and the differences in risk estimation by these six models are also significant. Therefore, we use regression models to study the risk of financial markets. The factors that affect the magnitude of volatility of our stock market returns are obtained by establishing VARP, and the calculated risk values are also close to the average risk values. When the confidence level reaches 95%, model 3 is at the confidence level, then the stability of model 3 is better. So, we use the difference between VAR and actual values to build the regression equation.

The failure test method is to test how much the actual loss probability is for exceeding the VaR. It takes the value of the actual loss probability as the dependent variable, and then determines what risks exist in the financial market based on that prediction, so that it can be controlled or intervened. For the first two confidence level scenarios, the results of the failure method to detect VaR are given as shown in Table 3.

Results of the failure method to detect VaR

different confidence levels RMSRB value alpha = 99%
125 250 500 125 250 500
VaR1 0.034 0.023 0.031 0.062 0.056 0.051
VaR2 0.021 0.015 0.021 0.045 0.047 0.039
VaR3 0.013 0.018 0.017 0.051 0.054 0.050
VaR4 0.011 0.007 0.018 0.049 0.043 0.048
VaR5 0.017 0.009 0.012 0.067 0.061 0.056
VaR6 0.005 0.016 0.009 0.043 0.049 0.054
Conclusion

In this paper, by using regression analysis to study the data related to the financial market in 2013, and using SPS software to analyze the linear correlation that exists between the independent variables and compare them with the actual data, the regression equation has a strong applicability in the prediction of financial market risk. Finally, the following conclusions are drawn: In our financial market, stock market, bonds and funds are the main factors affecting stock returns, while stock prices have a large degree of influence on interest rate changes. Therefore, investors need to analyze these data and choose the appropriate investment strategy to obtain higher returns. Regression models can be used to predict stock price volatility as well as investment risk. By building a VAR model to investigate whether there is a linear relationship between cash flow and asset price changes in sample firms, we can determine whether stock return volatility and asset price changes have an impact on the changes in firm cash flow. Multiple stepwise regression was used to study the impact of correlations of different industries in the financial market on stock market returns and stock price changes in 2013, and it was concluded that asset price changes of firms had an impact on earnings per share and this was positively correlated with stock price volatility.

Figure 1

Quantile regression model
Quantile regression model

Figure 2

VAR model research method
VAR model research method

Figure 3

Structure of the RiskMetatial model
Structure of the RiskMetatial model

Figure 4

RiskMetrics mixed-normal plot
RiskMetrics mixed-normal plot

Figure 5

ARCH model
ARCH model

Figure 6

Monte Carlo simulation method
Monte Carlo simulation method

Summary of data

data statistics the number of days in the sample period Days outside the sample period
capacity 125 days 250 days 500 days 250 days
mean −0.191 −0.046 −0.067 0.031
standard deviation 1.290 3.120 2.210 1.301
Skewness coefficient −0.501 −0.699 −0.329 −0.542
Kurtosis coefficient 3.987 4.702 4.731 4.562
J-BP value 0.003 0.000 0.000 0.000

Results of the failure method to detect VaR

different confidence levels RMSRB value alpha = 99%
125 250 500 125 250 500
VaR1 0.034 0.023 0.031 0.062 0.056 0.051
VaR2 0.021 0.015 0.021 0.045 0.047 0.039
VaR3 0.013 0.018 0.017 0.051 0.054 0.050
VaR4 0.011 0.007 0.018 0.049 0.043 0.048
VaR5 0.017 0.009 0.012 0.067 0.061 0.056
VaR6 0.005 0.016 0.009 0.043 0.049 0.054

Summary of the relative error of the square root of the mean of VaR

different confidence levels RMSRB value alpha = 99%
125 250 500 125 250 500
VaR1 0.3027 0.3085 0.3654 0.2319 0.1452 0.2430
VaR2 0.1901 0.1453 0.2109 0.1742 0.1321 0.1980
VaR3 0.4421 0.2452 0.2314 0.6034 0.3896 0.2789
VaR4 0.2103 0.1533 0.2753 0.2213 0.2078 0.2785
VaR5 0.1890 0.1899 0.2863 0.2090 0.1200 0.2540
VaR6 0.1675 0.1633 0.1632 0.2706 0.1989 0.2432

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