Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Comparing Forecast Performance of Stock Market and Macroeconomic Volatilities: An US Approach Kaya Tokmakçıoğlu* and Oktay Taş† Up to the present few researches focused on analyzing macroeconomic volatility of national economies. The aim of the paper is to compare the forecast performance of stock market and macroeconomic volatility of US economy between 2007-2010. Accordingly, two different types of financial time series are generated, namely weekly stock returns and quarterly return on investment. Firstly, the appropriate model is determined via time series analysis. Secondly, the relevant ARCH-type model is implemented. Finally, conditional variance forecast performance of models is presented with respect to confidence interval. Furthermore, coefficient of correlation between squared residuals and coefficient of conditional variance is given. Field of Research: Finance 1. Introduction: Financial agents (analysts, experts, spectators etc.) put great emphasis on volatility for the last two decades. For those who are interested in financial environment, volatility simply signifies risk where high volatility is regarded as a symptom of market disruption. In case of market volatility, it can be assumed that the capital market is not functioning well and financial securities are not being priced fairly. Therefore, analyzing, forecasting volatility and managing problems arisen from it are of great importance. The standard deviation of returns is one of the most important methods of measuring volatility of asset returns. In the past, it is assumed that volatility is constant whereas it is calculated as the standard deviation of returns for a given period (based on the closing prices), which is called historical volatility. Today, for most of the researchers it is obvious that the majority of asset returns displays volatility clustering which leads to high volatility when it has recently been high and vice versa. There are three different ways, which reveal these results. Realized volatility, which is obtained from direct measures, option priced implied volatility and GARCH and stochastic volatility, which are estimated via parametric time series models, help to analyze and forecast the volatility of asset returns. Among these, the most appropriate model for analyzing time-varying conditional volatility is the GARCH (General Autoregressive Conditional Heteroscedasticity) model whereby the time varying variance of returns is modeled as a function of lagged squared residuals and lagged conditional variance. In comparison to other models, GARCH outperforms them with its efficient capturing of the dynamics of volatilities and its simplicity of estimation. * Res. Asst. Kaya Tokmakçıoğlu, Faculty of Management, Department of Management Engineering, Istanbul Technical University, Turkey, Email: [email protected] † Assoc. Prof. Oktay Taş, Faculty of Management, Department of Management Engineering, Istanbul Technical University, Turkey, Email: [email protected] There are two different types of volatility used in the study, namely stock market and macroeconomic. The paper is organized as follows: Section 2 sums up the literature. Section 3 outlines the relevant methodology and data used in the study and Section 4 presents the findings. In order to compare the forecast performance of ARCH-type models in volatility modeling firstly the appropriate model is determined via time series analysis that can be found in the fourth section. Secondly, the relevant ARCHtype model is implemented. Finally, conditional variance forecast performance of models is presented with respect to confidence interval in the same section. Section 5 concludes. 2. Literature Review: Financial econometrics and economics have been dealing with modeling asset return volatility. For the simplicity of calculation, it has to be assumed that returns are zeromean variable that are distributed normally. A common model for returns is defined by t ~ i.i.d .(0,1) rt t t (1) where t is the error term with zero-mean, white noise and t is the time-varying volatility. This model focuses on the dynamics in the variance whereas it mostly neglects the possible dynamics in the mean. It can be asserted that returns conditional on t are normal if error terms are assumed to be white noise. Moreover, the unconditional distribution of t becomes leptokurtic whereby the normality condition is mostly valid for the conditional distribution. As mentioned above, conditional heteroscedasticity is largely modeled by ARCH-type models, which are firstly introduced by Engle (1982). In addition to this, Bollerslev (1986) developed the most popular GARCH(1,1) model which is widely used in the financial analysis. t ~ i.i.d .(0,1) r t21 rt t t 2 t 2 t 1 (2) Equation 2 presents the mean and variance equation separately. It is assumed that t is normal. Moreover, the variance depends on the previous information whereby the returns are normally distributed conditionally on the variance. The stationarity condition is ensured by the value of the parameters 1 . Furthermore, Nelson (1991) developed an asymmetric GARCH model, which is called the Exponential GARCH (EGARCH). With the help of this model, asymmetric impact of returns on conditional variance is captured. For an EGARCH(1,1) the Equation 2 is modified to ln t2 ln t21 rt 1 t 1 rt 1 t 1 (3) In Equation 3 the quantification of the asymmetry is guaranteed whereby the positive conditional variance is ensured by the logarithmic formulation. In general, ARCH-type models are estimated very easily by maximum likelihood method because of its normality assumption. In case of the absence of the normality assumption, a Studentt distribution is applied, which captures the fat tails of the actual distribution of the returns quite easily (Baillie and Bollerslev, 1989). GARCH models also simplify the forecast process with respect to other models by using one step ahead forecast which is fully determined. In literature there are some papers, which deal and refer to Clark's estimation of international macroeconomic market value. Coccia (2007), Davey (2007), Simpson (2008), Mullins et al. (2008), Ortiz et al. (2006), Yapraklı and Güngör (2007) mention on sovereignty, political risk etc. all of which use the estimation of balance sheet approach as a tool of calculating market value of national economies. Karmann and Maltritz (2002), in their discussion paper on sovereign risk, underlined the importance of calculation of market value of a national economy by using the cash flows of future net exports discounted at the economy’s rate of return. They also stated that the economy's annual rates of return are calculated as (ΔV +NX)/V where V is the market value of a national economy and NX are cash flows of future net exports. Miller (2006) put an emphasis on the model adopted by Clark (2002). His method assumes all asset returns are denominated in a single currency and therefore makes it possible to disentangle currency from systematic risk. Since a single currency is used for the world market portfolio the econometric analysis is simplified. Oshiro and Saruwatari (2005) laid stress on the estimation of the probability of default. They employed an accumulated fixed capital formulation as the underlying asset in the option-pricing model, which was introduced by Clark (1991). Furthermore, they emphasized the concept of macroeconomic balance sheet proposed by Clark (1991) and calculated the equity values of Thailand and Argentina. Finally, Asteriou (2008) used the Clark (2002) and Clark and Kassimatis (2004) methodology to calculate the market value of three Asian countries (Indonesia, Malaysia and Philippines) for each year over the period 1990–2004 and to estimate the macroeconomic financial risk premium from 1990 to 2004. He emphasized the correspondence of earnings on the country’s entire stock of assets to the earnings before interest and taxes in corporate finance that was also underlined by Clark and Marois (1996). 2. Methodology and Data: The theoretical background of the methodology used in the analysis of macroeconomic volatility can be found in Clark (1991), Clark & Marois (1996), Clark (2002) and Clark & Kassimatis (2004). The value of an asset or a project can be defined as the net present value of future cash flows that are discounted at the relevant discount rate, which depends on the risk of relevant the asset or project. For this reason, in order to calculate the macroeconomic volatility the discounted cash flow model is used whereby the relationship between macroeconomic balance sheet and cash flows are established. The general form of discounted cash flow models is as follows: Vt (bt at ) (bt 1 at 1 ) R 1 ... (bn an ) R ( nt ) (4) where “V” represents the net present value, “b” represents the cash income, “a” represents expenditure and “R” represents the discount factor with R=1+r where r is the relevant discount rate. For the calculation of macroeconomic volatility the equation of balance of payments should be stated: BPt X t M t FS t Ft (5) In Equation 5 the difference between the foreign exchange value of exports and imports of all goods and services plus (minus) the net inflow (outflow) of foreign capital indicates the change of official foreign reserves. It has to be noted that official and private unrequited transfers are included in X and M. From this point of view, the market value of a national economy can be calculated. With the assumption that all transactions occur at the beginning of each period, Equation 4 presents Vt as the market value of a national economy at the beginning of period t. In the same equation, “b”s are equal to the foreign exchange value of internal sales of domestically produced final goods and services. Total consumption less imports of consumption goods (CT-McT) gives the domestically produced final goods and services, which can be seen in Equation 6: bT = XT +(CT - M cT ) (6) Similarly, “a”s are equal to the foreign exchange value imports plus the foreign exchange value of internal expenditure for domestically produced final goods and services. Its equation can be written as follows: aT M T (CT M cT ) (7) It has been known that the foreign exchange value of internal expenditure on domestically produced final goods and services is exactly equal to the foreign exchange value of internal sales of domestically produced final goods and services. Therefore, (bT-aT) will always be equal to (XT-MT). With this background of knowledge, the foreign exchange value of the economy at the time T can be calculated with the help of the cash flows of income and expenditure. Hereupon, the foreign exchange value of the economy at time T is: VT E (bT aT ) (bT 1 aT 1 ) R 1 ... (bn an ) R ( nT ) (8) where VT stands for the capital value of the economy at the beginning of period T, R=1+r, and E is the expectations operator. Writing the formula for VT+1 gives: VT+1 = E éë(bT+1 - aT+1 )+ (bT+2 - aT+2 )R-1 +... + (bn - an )R-(n-(T+1)) ùû (9) By using Equation 6, 7, 8 and 9 and rearranging the resulting formula, the national accounting equation for period T can be found and is as follows: r VT aT bT aT bT VT 1 VT (10) where aT represents cost, bT represents income, r VT aT bT stands for profits before interest and dividends paid abroad and VT 1 VT is the net investment. Moreover, rearranging the equation by means of Equation 8, 9 and 10, it gives Equation 11: r VT M T X T CT X T M T CT VT 1 VT (11) In Equation 11, the right-hand side is a derivative presentation of net domestic product. The economy’s earnings before interest and dividends paid abroad plus consumption is given on the left-hand side of the equation. For net national product is well known to economists, the approach of national accounts is favorable. The availability of the macroeconomic data makes it also preferable. Moreover, net national product is given in a more convenient format on the left-hand side of the equation. In addition to these, the rate of growth of the country’s capacity to generate net foreign exchange value is a more relevant measure of economic performance from the perspective of the international investor. Generally, to estimate the growth rate return on investment (ROI) and rate of reinvestment out of profits are used. Return on investment can be defined as the ratio of money gained or lost on an investment relative to the amount of money invested. Following this approach, the country’s ROI can be estimated from current profits and the balance sheet and is given in Equation 12: ROI X T M T VT 1 VT T V t 0 t 1 Vt (12) With respect to this theoretical background, the international macroeconomic market value of a national economy and its return on investment can be calculated step by step as follows: First of all, the net fixed capital formation is gathered for US national economy. Statistically, gross fixed capital formation quantifies the value of acquisitions of new or existing fixed assets by the business sector, governments and "pure" households (excluding their unincorporated enterprises) subtracted by the disposals of fixed assets. Thus, net fixed capital formation can be found by subtracting the depreciation of existing assets from gross capital formation. With the help of balance of payments of national accounts, net capital value for each time period is calculated by adding the net investment (gross investment minus depreciation) to the value of the US economy for the last time period. At this point, net investment indicates an activity of expending which increases the availability of means of production or fixed capital goods. It can be calculated by subtracting replacement investment from total spending on new fixed investment. Thus, the economy’s market value in dollars is calculated at the end of each time period. Furthermore, using the Equation 12, the return series for US national economy is generated. For the analysis of stock market volatility, the data consists of weekly stock index closing prices of United States of America. The index is collected for the period from July 13, 2001 to January 1, 2010 due to availability and the weekly index price is taken a natural logarithm (ln) in front. It is expressed in US dollars and the source of the data is Datastream. For the macroeconomic volatility, the data consists of quarterly ROI (return on investment) of the national economy of US between 19802010. The calculation of ROI is given above and all the necessary data for the analysis of macroeconomic volatility is also gathered from Datastream. 4. Findings: According to the study, there are two different types of volatility, namely stock market and macroeconomic. In order to save space, weekly stock index and quarterly ROI data are not presented, however they can be provided upon request. For comparing the forecast performance of ARCH-type models in volatility modeling firstly the appropriate model is determined via time series analysis. The appropriate conditional variance equation of stock market and macroeconomic returns can be seen in Equation 13 and 14, respectively. ln t2 0.32 0.97 ln t21 0.09 ln t2 8.32 0.38 ln t21 1.31 rt 1 0.13 rt 1 0.35 t 1 t 1 rt 1 (13) rt 1 (14) t 1 t 1 Secondly, the relevant ARCH-type model is implemented. Finally, conditional variance forecast performance of models is presented with respect to confidence interval. Particularly, one-step ahead forecast is carried on using static forecast. Due to data availability, last three years of the analysis is forecasted and number of outliers is counted in order to contrast the weekly and quarterly volatility results. Furthermore, coefficient of correlation between squared residuals and coefficient of conditional variance is given. The results can be found in the following figures and tables: Figure 1: Forecast performance of the volatility model for US stock market return Figure 2: Forecast performance of the volatility model for US return on investment (ROI) Table 1: Coefficient of correlation between squared residuals and coefficient of conditional variance for US stock market and macroeconomic return (ROI) US (stock market) 0.451143 US (macroeconomic) -0.339238 According to the results, it can be asserted that forecast performance of the appropriate model for stock market volatility is much better than that of for macroeconomic volatility. Between years 2007 and 2010, which is the actual forecast period, the coefficient of conditional variance (GARCH term) of stock market returns is in most of the observations within the confidence interval. On the contrary, there is a downward trend in the comovement of GARCH terms and the squared residuals of macroeconomic returns (ROI). Moreover, the coefficient of correlation between squared residuals and coefficient of conditional variance is also negative for return on investment of US economy. Therefore, it can be affirmed that the forecast performance of stock market volatility is superior to that of macroeconomic volatility. 5. Summary and Conclusions: For traders, risk managers and investors, as well as researchers, who seek to understand market dynamics, precise volatility forecasts are important. Estimates of future volatility are not only needed to understand market structure but also to comprehend a country’s economic performance. This paper has compared two basic approaches to forecast volatility in the US stock market and national economy. The first approach analyses stock market volatility and the second one deal with macroeconomic volatility. The results suggest that forecast performance of the appropriate model for stock market volatility is much better than that of for macroeconomic volatility. For further research, a relevant model for forecasting macroeconomic volatility could be suggested. Macroeconomic volatility is an important tool for capturing the dynamics between macroeconomic indicators. Therefore, analyzing and measuring the macroeconomic volatility is of vital importance for evaluating the fragility of world market economy. References: Asteriou, D. 2008, “Country financial and political risk: the case of Indonesia, Malaysia and Philippines”, International Journal of Monetary Economics and Finance, Vol. 1, No. 2, pp. 162-176. Baillie, R.T. & Bollerslev, T. 1989, “The message in daily exchange rates: a conditional variance tale”, Journal of Business and Economic Statistics, Vol. 7, pp. 297-305. Bollerslev, T. 1986, “Generalized autoregressive conditional heteroscedasticity”, Econometrics, Vol. 31, pp. 307-327. Clark, E. 1991, Cross Border Investment Risk, London: Euromoney Publications. Clark, E. & Marois, B. 1996, Managing Risk in International Business, London: International Thomson Business Press. Clark, E. 2002, “Measuring Country Risk as Implied Volatility”, Wilmott Magazine, September, pp. 64-67. Clark, E. & Kassimatis, K. 2004, “Country financial risk and stock market performance: the case of Latin America”, Journal of Economics and Business, Vol. 56, No. 1, pp. 21-41. Coccia, M. 2007, “A New Taxonomy of Country Performance and Risk Based on Economic and Technological Indicators”, Journal of Applied Economics, Vol. 10, No. 1, pp. 29-42. Davey, P.P. 2007, “Analysing the changing relationship between the Brazilian stock market and global economic indicators”, Unpublished dissertation thesis in MA Finance and Investment, University of Nottingham. Engle, R.F. 1982, “Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. inflation”, Econometrica, Vol. 50, pp. 987-1008. Karmann, A. & Maltritz, D. 2002, “Sovereign Risk in a Structural Approach”, Dresden Discussion Paper Series in Economics, 7/03 Miller, S.M. 2006, “Global Equity Systematic Risk Hedging And The Asian Crisis”, 11th Finsia-Melbourne Centre for Financial Studies Banking and Finance Conference Paper, 25-26 September, 2006. Mullins, M., Murphy, F. & Garvey, J.F. 2008, “Do Credit Derivatives Dampen Political Risk: The Case of Brazil Post 1998”, Journal of Globalization, Competitiveness and Governability, Vol. 2, No. 2, pp. 46–59. Nelson, D.B. 1991, “Conditional heteroscedasticity in asset returns: a new approach”, Econometrica, Vol. 59, pp. 347-370. Ortiz, E., Cabello, A. & de Jesus, R. 2006, “Long-Run Inflation and Exchange Rates Hedge of Stocks in Brazil and Mexico”, Global Economy Journal, Vol. 6, No. 3, pp. 1-29. Oshiro, N. & Saruwatari, Y. 2005, “Quantification of sovereign risk: Using the information in equity market prices”, Emerging Markets Review, Vol. 6, No. 4, pp. 346-362. Simpson, J.L. 2008, “Cointegration and Exogeneity in Eurobanking and Latin American Banking: Does Systemic Risk Linger?”, The Financial Review, Vol. 43, No. 3, pp. 439-460. Yapraklı, S. & Güngör, B. 2007, “The Impact of Country Risk on Stock Prices: An Investigation of ISE 100” (in Turkish), Journal of Faculty of Social Sciences of Ankara University, Vol. 62, No. 2, pp. 199-218.