Volatility, Global Proxy Index, V-A-R: Empirical Study on Pakistan And China Stock Exchanges

This study postulates that propose global proxy index is a significant conduit to evaluate the shocks in volatile stock markets i.e. PSX and SSE, alike. The two separate models i.e. Log-GARCH (1, 1) and ARMA-GARCH (1, 1) have been used along with the value at risk (V-a-R) @ 5% criteria for choosing best-fitted model. The study results showed Log-GARCH (1, 1) model proves to the best. This study results are not driven by political-level risks and thus independent study can be conducted to evaluate the detrimental consequences on investment opportunities under volatile environments.


INTRODUCTION
Generally, risk is defined as a probability of loss due to unexpected changes in financial market and has been widely interpreted with in specific context. Piroozfar [1] describe differrent types of financial risk in his study as Equity risk, Exchange rate risk, Interest rate risk, Commodity risk, and Liquidity risk. Korner, K.F., Kneafsey, K.P., & Claessens [2] and Mark R. Manfredo and Raymond M. Leuthold [3] used models to meausre the volatility in comodity market and found that the commodity risk stems from the change in future income level and market volatility. DZ BANK [4] define equity risk under its annual report as a risk of loss due to negative change in value of stocks. Risk measurement considered as an essential part of financial manager's job in every era. Different tools have been used to analyze and measure the risk at different levels. These risk measurement tools enable the financial managers to identify expected risk, generate best possible mechanism, implementing and tracking risk. Due to globalization and more advancement if finance sector, stock markets all around the world are more aligned and effected with the movement in country specific and other economic level risks.
In 1960's Fama gives efficient market hypothesis (EMH) that describes the informational efficiency of financial market. Efficient market presents the true value of securities. Efficient capital market refers to rapidly adjustment of prices in response to new information. Fama's efficient market theory describes that it might be a situation that investor is assured that price of security is fully consistent with all the information in market. Fama and Schwert [5] further categorize efficient market hypothesis in three different levels starting form weak-form efficient market to semi-strong-form and semi-strong form to strong-form efficient market level of hypothesis. This study posits that the investment opportunities are closely connected to the stock exchanges across the globe so they possess the greater sensitivity towards the efficiency level of these markets.
Changes in returns are due to variations that exist in the markets and the common measure to capture these movements is the standard deviation. These variations are connected to the sources of risk out of which the most important source is the volatility. Sudden changes in seurities prices due to any reason in stock market is reffered as stock market volatility [6]. Values of highly volatile instruments suffer with higher probability of increase or decrease in end values. Unexpected changes, either due to negative or positive in prices of instrument, will a bit of concern for the stakeholders expectations.
The traditional methods which are typically used to estimate and measure the volatility are standard deviation or variance that are unconditional and cannot capture the characteristics of financial time-series data [7]. For that one of the accepted model for the estimation and measurement of volatility is Generalized Autoregressive Conditional Heteroskedasticity (GARCH) [7]. It has been extracted from the initial system, to gauge volatility for financial series, as proposed by Engle [8] known as Autoregressive Conditional Heteroskedasticity (ARCH) system. Under this the said series of data posit the variations tend to be greater at some points compared to others which is caused by heteroskedasticity phenomenon. Instead of treating the risk as a linear trend, ARCH/GARCH treats it as a variance in the model, by relaxing the normality condition (Engle, 2001). Another study showed that the scholars used standard error of empirical quantile estimate by using monte carlo simulation [9].
This GARCH (1, 1) model as a non-parametric model consider negative and positive error terms to have symmetric effects on volatility, i.e. that negative and positive shocks have the same effect on volatility [10]. A general understanding is that negative returns tend to be followed by periods of greater volatility than positive returns of equal size. Similarly, the bad news as compared to good news, tend to increase more volatility [11]. An explanation of the asymmetric response of return volatility to the sign of the shock is that positive and negative shocks lead to different values of a firm financial leverage, which in turn will result in different volatilities [10]. In order to capture the asymmetry in return volatility, the non-parametric model approach is an appropriate tool to investigate the resulted effect of volatility phenomenon.

RESEARCH METHODOLOGY
This study firstly use normal log-GARCH(1, 1) and ARMA-GARCH(1, 1) models i.e moving average model, for the computation of value at risk(V-a-R) at 5% level. This methodolgy includes the adjustments of previous historical data on markets under consideration to reveal the significance among market variables and existing volaitlity of the market as proposed by (White, 1998). Also, the author used filter historical simulation in in his paper for the computation of V-a-R to measure the volatility in order to meet up the current challenges, which is align to the past work done by Christoffersen [12].
Let assume that rt; t=1,...., T that represent a continuous change in stock prices returns for a specific holding time t. if the vt is the value/price of stock then rt=ln(vt)-ln(vt), where ln is the natural logarithm. So we can write the model as: rt+1= c+1rt + 2rt-1+…+krt+1-k + Ǯ1X1,t+1 + Ǯ2X2,t+1+…..+ ǮsXs,t+1 + σ t+1Ϛ t+1 ; r 2 t+1 = Ϣ+αResid 2 t + βσ 2 t ; t=1,2,3,4….T where residt =( rt -c -Σirt-1 -ΣǮjxj,t); Ϛt present the white noise with mean and variance of zero and internal factors that influence the rt are the α+β < 1, X1….. Xs. . Another model has been used in this study is known as moving average model i.e. ARMA-GARCH(1, 1) model and can be expressed as under: C is constant, ℘ is parameters used in AR and ℴ are the parameters used in MA while ℇt refer as the white noise error term. Jorion [6] mentioned different types of methods for risk managers to estimate the possible financial risk out of which V-a-R is known as an advanced model to estimate the volatility factoe. V-a-R method explained the max possible risk at any certain day at time period t. we generally quote the value or percentage with confidence level of 10%, 05% and 01%. In case of extremely positive or negative returns V-a-R computed on normal distribution will gave vague values. The 100α% one day ahead V-a-R (λα,t) is defined as; Assumptions on which V-a-R Model is calculated do not change over the specific holding time period. These assumptions are only applicable for the short holding time period. Historical simulation is commonly used non-parametric tool. Filtered Historical Simulation is considering the more appropriate and accurate predictor of risk then the other Risk assessment models.Calculation of V-a-R is quite simple using filtered historical simulation through given formula;

V-a-R= CI × Zα√ℎt
Where CI is the value of cash invested in capital market, Zα is the value normal distribution and ht is the conditional variance of r returns series. Inorder to determine which model is best fit out of some the 'akaike information criteria' (AIC) and 'schwarz information critria' (SIC) is the best choice [13]. Here the V-a-R @ 95% level of confidence has been incorporated on two models i.e. log-GARCH(1, 1) and ARMA-GARCH(1, 1) respectively. Therefore, dispersion resulted from the V-a-R estimation @ 5% significant level based upon two models, such as; ΩA= ∑ (rt -V-a-RA t,α ) 2 (1) ΩB= ∑ (rt -V-a-RB t,α ) 2 (2) Where as, ΩA, represents a log normal model; ΩB, represents a ARMA model; rt, represents the log normal returns for respective series; α, represents confidence interval level of 5%.

RESULTS AND FINDINGS 3.1. Quantitative Model Analysis
Pakistan-China Stock Markets. The standard procedure recommended using the non-parametric models like (ARCH)/GARCH (1, 1) is to run the test of ARCH LM test statistics. This test postulates that if the p-value is insignificant after running the regression model then there is no heteroskedasticity condition existed in the proposed model and hence no non-parametric model should be used and vice versa. Estimated results given in Tables 1-4 have shown that the existence of ARCH effect in both the regression equations with respect to Pakistan and China perspective. Thus, the results have confirmed the existence of volatility in both the countries' stock exchanges and hence cannot be tested by using parametric models. Table 1 shows the model log-GARCH (1, 1) after keeping returns of Shanghai Stock exchange as an exogenous variable in the quantitative model equation. By looking at the p values of the global proxy index such as gold (R_GOLD), returns of oil (R_OIL) and returns of YUAN (R_YUAN), there found to be an insignificant impact over all. However, the variable i.e. R_SEE representing the Shanghai composite index returns have shown significant impact on KSE-100 index. The variance equation has shown that ARCH and GARCH terms are significant too, with the p-value below 1%.

Log-GARCH (1, 1) -Model 1. The below
The below Table 2 shows the model log-GARCH (1, 1) after keeping returns of Karachi Stock exchange as an exogenous variable in the quantitative model equation. By looking at the p values of the global proxy index such as gold (R_GOLD), returns of oil (R_OIL) and returns of YUAN (R_YUAN), there found to be a significant impact over all. However, the variable i.e. R_PSE representing the Karachi -100 index returns have shown an insignificant impact on Shanghai composite index. The variance equation has shown that ARCH and GARCH terms are significant too, with the p-value below 1% as shown in the table below.  Similarly, the ARMA-GARCH (1, 1) model applied in context of Shanghai stock market showed that the significant p values at 1% achieved by 1 autoregressive and 1 moving average term in a mean equation. Also, in the variance equation, both ARCH and GARCH terms are statistically significant as evident by the p values in Table 4. Interesting findings has reported by the variable R_PSE i.e. an exogenous variable that has shown insignificant impact on the volatility of Shanghai Stock exchange. However, the ARCH and GARCH terms are statistically significance showing that the model fully explains the magnitude of volatility and shocks observed in the Shanghai stock exchange.  Similarly, the log-GARCH (1, 1) model has been applied to examine the volatility existed in the R_Oil variable in the global proxy index with respective stock exchanges in Table 6. Only one of the variables in the mean equation of the model i.e. R_GOLD has shown the significant impact on the volatility of oil price indicator. In the variance equation the respective ARCH and GARCH terms are significant at 1% level showing the credibility of the model. Also, exogenous variables such as R_PSE and R_SSE i.e. the returns of both the exchanges have shown the significant impact too for explaining the possible variations in oil indicator. The volatility of a final indicator in the global proxy index i.e. Gold price has been examined through the log-GARCH (1, 1) against the variations exists in both the stock exchanges. The Table  7 has reported the two separate equations i.e. mean and variance. The former one showed that only the oil price indicator in global proxy index has shown the significant association in explaining the volatility existed in R_Gold at 1% confidence of interval. Whereas, in the variance equations except the R_PSE, all the variables have shown significant impact in explaining the volatility in R_Gold.  Table 8 showed the results in favor of using this model with respect to the variance equation. All the ARCH and GARCH terms along with the exogenous variables i.e. R_PSE and R_SSE are statistically significant at the 1% level. This showed that the existing volatility in R_Yuan currency has been affected by both of the stock exchanges. Also, ARMA-GARCH (1, 1) model has been applied to examine the volatility existed in the R_Oil variable in the global proxy index with respective stock exchanges in Table 9. Interestingly, all the ARMA terms in the mean and variance equation alike are statistically significant at 1% level. The moving average terms also have a significant impact on the R_Oil, as a global proxy index. Moreover, the exogenous variables like R_PSE and R_SSE have fully explained the volatility on Oil prices with a significant impact.   There is an indication of market crash in Pakistan stock exchange too, during the year 2008 showed by the little horizontal line. When applied the model 2 equation for assessing the maximum loss incurred in composite period analysis, it is evident that this model has shown the close association with the respective volatility and shocks in PSX.

Risk assessment tool -Value at risk (V-a-R) Composite period analysis. Given graph below describes the relationship between the V-a-R and
On the basis of choosing a best model out of two, it is observed that the resulted dispersion out of Pakistan stock exchange price indices is less in the model 2 nd , compared to the 1 st one. The details are based upon the whole data of approximately 15 years and the respective models' values are given as under; ΩA,PSE = 2.8081 ΩB,PSE = 2.7874 The evidence is given in the graph represent by Figure 1, which states the situation for composite years analysis. Clearly the Model 2 nd line is above the Model 1 st line and the dispersion of returns of price indices of Pakistan stoack exchange move with tendem to respective two models. The graph shows the relation between the bandwidth of V-a-R and the Pakistan stock variations at the 5% confidence level. The analysis has addressed that there is a similarity exists in both the trends generated by both the models under study.
The evidence is given from the graph represents the Figure 3, which shows the model 1 st line is above the model 2 nd line on majority of the cases throughout the years.
Global proxy index analysis. Given graph below describes the relationship between the V-a-R and the returns of Yuan when incorporated the model 1 equation in assessing the magnitude of risk as per Figure 3.The bandwidth comprises of two independent series 1 and 2 i.e. V-a-R = −1. 65√ℎt and V-a-R = 1. 65√ℎt respectively. The arrangement from the model 1 shows that the maximum loss with confidence level of 5% over the composite period with respect to Yuan currency as a one of the global proxy indexes. Results evident from the year 2006 till 2009 showed the more volatility as compared to initial stage of the data. The maximum loss occurs at year 2005 is approximately -0.02. It followed by the distress situation in the year 2015, which showed a panic situation in international markets with respect to currency in China market. The model 2 specifications have been arranged in order to apply V-a-R with the confidence level of 05% to evaluate the volatility in Yuan. The similarity existed with the model 1 and the bandwidth generated from the V-a-R statistics has incorporated composite period variation too.
Similarly, the V-a-R @ 05% confidence level to assess the loss or deviations has been applied on the Oil prices, as an indicator of global proxy index. The arrangement from the model 1 has shown hint of volatility is highest throughout the composite period. There is also evidence that the V-a-R bandwidth approximately incorporated all the variations existed in this volatile industry. The maximum loss is upto -0.17 during the year 2001, which is less than the existed volatility during the crisis of 2007-08. The model 2 arrangement has been shown in Figure 4, which shows that the association of V-a-R bandwidth and that of the variance of Oil prices is significantly close.
The last indicator in the global proxy index is the Gold prices and its significant has been evaluated against the respected model's arrangements with V-a-R bandwidth. The bandwidth margin comprises of values of two independent series i.e. −1. 65√ℎ and 1. 65√ℎ, respectively. The model 1 arrangement has been based upon the log-GARCH (1, 1) specifications in relations to assess the volatility in gold prices against the respective indicators of global proxy index and   the stock exchanges. The maximum loss reported is during the year 2013 which is approximately -0.097. There found to be the close association among the volatility and the bandwidth proposed through V-a-R @ 5%. Similarly, the model 2 specifications based on the arrangement which incorporates the volatility exist in the gold prices as last indicator of global proxy index has been presented in the Figure 4. There also found to be the close association among the respective variations in the returns of the gold price and proposed bandwidth of V-a-R @ 5% level of loss assessment. The model best criteria have been applied in relation to the global proxy index on the basis of V-a-R @ 5% by Model 1 st and Model 2 nd . The resulted dispersion from model 1 st is less compared to the model 2 nd on the basis of composite year's analysis in case of all three indicators i.e. Yuan, Oil and Gold prices returns. The graph represents by the Figure 5 displays the results in confirmation to the Model 1 st as a best fitted model.

CONCLUSION
Value at risk approach handles market volatility at best that relates to its usage under certain conditions. The Risk has been most fundamental notion for experts and analysts in assessing their investment opportunities with specific to stock markets volatility. This study has incorporated filtered historical simulation for stock exchanges in order to access volatility condition based on V-a-R criteria. The findings reveals that Log-GARCH (1,1) model best explain the existing volatility in Shanghai stock exchange and dictates that the majority of the global proxy indicators shows their significant association in defining the volatility in said stock exchange. Due to weak statistical impact of these indicators on Pakistan stock exchange ARMA-GARCH (1, 1) is the best model to explain volatility in the said Stock Exchange. V-a-R has confirmed this @ 5% confidence interval that the model first as shown less dispersion against the existing returns of price Indexes from respective stock exchanges. Since, the filtered historical simulation is the best method to assess betterly the risk from the historical data and incorporate the volatility, it is suggested that for future prospect the use of other statistical models should be used with the aim of assessing the loss as an outcome of risk through the other applications such as modified V-a-R and Sharpe ratios etc. Also, the study results are not driven by political-level risks and thus independent study can be conducted to evaluate the detrimental consequences of it on investment efficiency in such volatile stock exchanges.