Download Comparing Forecast Performance of Stock Market and Macroeconomic Volatilities: An US Approach:

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   t21
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  t21  
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 ( nt )
(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 ( nT )

(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  t21  0.09
ln  t2  8.32  0.38 ln  t21  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.
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