Risk-return relationship in US equity market: Evidence from Markov regime-switching model
Peiming Wanga+, Ihsan Badshahb
aFaculty of Business and Law, Auckland University of Technology, Private Bag 92006, Auckland 1142, New Zealand
(Email:[email protected])
bFaculty of Business and Law, Auckland University of Technology, Private Bag 92006, Auckland 1142, New Zealand
(Email:[email protected])
Abstract
We propose a Markov regime-switching model to examine the market risk-return relation in the US equity market under different market conditions with monthly data of S&P500 stock market and VIX indices. Our findings show a positive relation between market volatility and market returns and a negative relation between change in market volatility and market returns, which is consistent with the ICAPM. Our results also provide evidence of the presence of asymmetric volatility in the US stock market as well as leverage effect during periods of market turbulence. Our results contain implications for investment strategies under different market conditions.
JEL Codes: C22, G12
Keywords: Asymmetric volatility; Leverage effect; Markov regime-switching; VIX
+ Corresponding Author. E-mail: [email protected]. Tel: +64-9 921 9999 extn: 5393. Fax: +64 9 921 9940.
1. Introduction
Merton (1973) introduces an intertemporal capital asset pricing model (ICAPM) in which an asset’s expected return depends on its covariance with the market portfolio and with state variables that proxy for changes in investment opportunity set. One of the fundamental implications of Merton’s model is that market volatility and returns are positively related.
According to Merton’s model, investors require the market risk premium as compensation not only for the level of market risk but also for the risk of changes in the future investment opportunity set. Campbell (1993, 1996) extends Merton’s model by proposing a variant version of ICAPM in which the risk of changes in the future investment opportunity set is approximated by the risk of changes in the future volatility. A large number of studies in finance literature test the significance of risk-return relation in an aggregate stock market. However, the existence of a positive risk-return tradeoff for market indices has not been universally found in the existing literature. For example, a positive risk-return relation in an equity market is supported by studies of Campbell and Hentschel (1992), Scruggs (1998), Mayfield (2004), Ghysels et al.
(2005), Guo and Whitelaw (2006), Bae et al. (2007) and Kanas (2013), while a risk-return relation in an equity market is found by studies of French et al. (1987), Campbell (1987), Glosten et al. (1993), Whitelaw (1994), and Brandt and Kong (2004).
Several studies in the literature attempt to resolve the puzzling risk-return relation with various alternative models. Scruggs (1998) proposes a conditional two-factor model of the market risk premium in which the market risk premium is a function of conditional market variance and conditional market covariance with a variable that describes the state of investment opportunities in the economy, and uses it to explain for the failure of providing empirical evidence for a positive risk-return relationship based on a conditional single-factor model with a simple linear relation between the market risk premium and conditional market
covariance. Like many other studies in the literature1, Mayfield (2004) proposes a two-state Markov regime-switching model for estimating the market risk premium that accounts for shifts in investment opportunities by explicitly modeling the underlying process governing the level of market volatility. According to Mayfield’s model, market volatility is governed by a two-state Markov chain with two constant levels of the market volatility corresponding to the two states of a stock market and the change in the expected future market variance represents the change in the future investment opportunity set; and the market risk premium is related to both the expected market variance and the change in the expected future market variance. His results confirm a positive risk-return relation in the US stock market. Similarly, Bae et al.
(2007) follow Mayfield’s specification of the expected market variance and the change in the expected future market variance for the market risk premium, but employ a two-state Markov regime-switching GARCH model to examine risk-return relation while controlling for leverage. Their findings also support a positive risk-return relation in the US stock market.
In this paper, we propose a two-state Markov regime-switching model to investigate the risk-return relation in the US equity market under different market conditions with monthly data of the S&P 500 stock and VIX indices. The proposed model extends Mayfield’s framework in several ways. Specifically, we use the VIX index as a measure of the expected market volatility in the specification of the market risk premium as proposed by Mayfield (2004); we include lagged returns in the conditional volatility to examine leverage effect under different market conditions; and we allow the likelihood of being a particular market condition to be associated with the state of economy. The empirical evidence from the proposed two- regime Markov model supports a positive relationship between the market volatility and returns, while there is no evidence for a positive risk-return relationship based on the single-
1 For instance, Turner et al. (1989), and Kim et al. (2004) use a two-state Markov regime-switching model to estimate the market risk premium.
regime model. Furthermore, our findings show that during periods of market tranquility (regime 1), the positive relationship between the market volatility and market returns is significant, but the leverage effect is insignificant; and during periods of market turbulence (regime 2), there is insignificant evidence that the market volatility is correlated with the market returns, but the leverage effect is significant. On the other hand, the change in the future volatility is negatively associated with stock returns regardless whether the stock market is in the tranquil or turbulent state, providing evidence of asymmetric volatility in the US stock market. 2 This result is also consistent with the intuition provided by Campbell (1993, 1996).
Finally we provide evidence that the proposed two-regime Markov model is more appropriate than the single-regime model for characterizing the risk-return relation in the US stock market.
Hence our results contain implications for investment strategies under different market conditions.
The VIX index is a popular measure of the implied volatility of S&P 500 index options and represents the measure of the market's expectation of stock market volatility over the next 30-day period. The use of the VIX index as a proxy for expected market volatility is more favourable than alternative measures of the expected market volatility based on realized returns. The VIX index incorporates most information of volatility available in market; and it is a consensus on the expectation of the future volatility among all market participants. Hence the VIX index inherently is a forward-looking measure of market volatility, while alternative measures of volatility based on realized returns are backward-looking by their construction.
Many studies show that the VIX index is a powerful predictor of the future volatility (e.g., Christensen and Prabhala, 1998; Fleming, 1998; Poon and Granger, 2003).
2 Asymmetric volatility in stock markets is a phenomenon in which the market volatility is usually higher after the stock market falls than after it rises, so stock returns are negatively correlated with the volatility of subsequent returns.
This paper is organized as follows. Section 2 describes the model and data. Section 3 presents the empirical results. Section 4 concludes the paper.
2. Methodology and data
2.1 The model
Mayfield (2004) proposes a two-state Markov regime-switching model for market risk premium based on the relationship between volatility and expected return, in which the state of an equity market is assumed to follow a two-state first-order Markov chain with the two states (or regimes) representing the two different levels of the variance of returns on the equity market and constant transition probabilities. Furthermore, Mayfield assumes that investors know the current regime with certainty but face the possibility of a shifts in the regime at each point in time. Hence, under Mayfield’s specification, there is no uncertainty over the magnitude of the future change in volatility, and uncertainty exists only over the time at which the level of volatility will change. Mayfield shows that the market risk premium for a period is related to both the expected variance of returns and the expected change in the variance of returns over that period, which can be expressed by
2 2 2
1 2 1
( t) ( t ) ( t ) ( t ) ,
E R E E E (1) where E R( t) is the expected excess return on an equity market over an risk-free rate of return for period t, and E(t2) is the expected variance of returns for period t at the beginning of that period.
In our empirical study, we apply Mayfield’s framework to analyze the risk-return relationship between the excess US equity market returns over the Treasury bill rate and its volatility with monthly data by relaxing some of his assumptions. Specifically, let St be a
discrete indictor to represent the state of the US equity market for time period t. We assume that St follows a two-state first-order Markov chain with the transition probabilities given by
1 ,1 ,1 1
1 1
( | ) ( ) logit( ) ( 1, 2)
( | ) 1 ( | ) ( )
t t
t t ii S S t
t t t t
P S i S i p t USCCI i
P S j S i P S i S i j i
(2)
where USCCIt1 is the change in the U.S. consumer confidence index (USCCI) over time period (t-1) and is associated with the transition probabilities through the logistic function with parameters i,0 and i,1 for state i (i = 1,2).3 Unlike Mayfield’s assumption of constant transition probabilities, we allow the time-varying transition probabilities that may depend on the change in the USCCI that acts as the proxy for economic and financial market conditions.
In addition, unlike Mayfield’s assumption that investors know the current state for sure, we assume that investors only know the structure of the Markov process generating the states, but are unsure of the prevailing state in the past, present and future. 4 Hence investors must base investment decisions for period t on the inference about the state using the information which is available at the beginning of period t.
Conditional onSt, the excess return on the US equity market index (over the risk- free rate) for period t, denoted as Rt, is determined by
, t
t t S t
R (3)
where the conditional mean of market returns, ,
t St
, is defined as
3 As evidenced in the literature (e.g., Schwert, 1989, Filardo, 1994 and Bae et al. 2007), the likelihood of regime
shifts may be affected by change in variables capturing information on the state of an economy. The USCCI measures the degree of optimism on the state of the US economy that consumers are expressing through their activities of savings and spending. The change in this variable is useful to time business cycle in the US.
4 Our assumption is more realistic and in line with many studies on risk premium (e.g., Turner et al. 1989, and
Liu et al. 2012).
2 2 2
, ,1 1 ,2( 1),
t t t
t S S VIXt S VIXt VIXt
(4)
and ~ (0, 2, )
t N t St
with the conditional volatility of stock market returns linked to the lagged return through the exponential function with parameters ,1
St
and ,2
St
for a state:
,t exp( t,1 t,2 1).
t S S S Rt
(5)
Note that VIXt2 is the observed level of the implied variance of returns on the US equity market index for next month at time t, which is regarded as the expected variance of returns for period t+1 by all market participants at time t. Thus,(VIXt2VIXt21) measures the change in the expected variance of returns on the US equity market from the beginning of period t to the end of period t.
Unlike Mayfield’s assumption of constant level of state-dependent market volatility, we allow the time-varying level of state-dependent market volatility that is associated with lagged return as specified by Equation (5). This controls for possible leverage effect between return and volatility. Particularly, ifi,2 0, it indicates the leverage effect under regime i (i = 1, 2).5
Under our setting, the market risk premium for period t is the unconditional expectation of excess return determined by
,1 ,2
2 2 2
1,1 2,1 1 1,2 2,2 1
( ) P r( , 1) P r(t , 2)
(P r(t , 1) P r(t , 2) ) (P r(t , 1) P r(t , 2) )( ),
t t t
t t t
E R t
V IX V IX V IX
m m
b b - b b -
= +
= + + + - (6)
5 As in the literature (e.g., Bollerslev et al. 2006), the leverage effect can be measured by a negative correlation between volatility and past returns.
whereP r( , )t i = P r(St = i),(i =1,2), is the probability that the equity market is in state i for period t. Equation (6) has the same form as in Equation (1) in the sense that the expected variance is measured by the squared VIX and the two coefficients in Equation (1) may be time- varying. Our specification of the risk premium also is in line with Merton’s (1973) ICAPM in which changes in investment opportunities are related to not only the level but also the change in the level of market volatility.
We obtain the maximum likelihood estimates of the model parameters via the EM algorithm using the method described in Hamilton (1994, Chapter 4).
2.2 Data
Our sample runs from January 1990 to February 2013with monthly data on the S&P 500 stock market index, VIX volatility index, 90-day Treasury-bill rate, and U.S. consumer confidence index obtained from the DataStream. Figure 1 plots the data of the S&P 500 stock market (solid line) and VIX volatility (dotted line) indices, and Table 1 reports summary statistics of the variables used in the study. The monthly excess stock return over the Treasury- bill rate ranges from -18.38% to 14.60% with a sample mean of 0.28% and standard deviation of 4.54%. The excess returns are negatively skewed with heavier tails than the normal distribution as indicated by the skewness and kurtosis statistics. The results of the augmented Dickey-Fuller test (ADF) suggest that the data series for each variable is a stationary time series.
Figure 1: S&P500 stock index, VIX, periods of high and low volatility.
Table 1: Summary statistics for the variables in the study
Statistic Rt ΔUSCCIt‐1
Mean 0.283 4.801 -0.016 -0.174
Standard deviation 4.537 4.635 3.228 6.393
Maximum 14.598 46.936 18.121 21.700
Minimum -18.377 1.063 -30.936 -23.000
Skewness -0.707 4.347 -2.207 -0.142
Kurtosis 5.079 31.859 38.104 4.330
P-value of ADF test 0.001 0.001 0.001 0.001 Number of observation 276 276 276 276
3. Empirical results
3.1 The risk-return relationship based on single regime model
We first examine the risk-return relationship based on a single regime model (St º 1) by fitting the data to the normal distribution with mean and standard deviation determined by Equations (4) and (5) respectively. The last column in Table 2 reports the maximum likelihood estimates, log-likelihood, AIC and BIC values for the single-regime model. As the coefficient of V IXt2-1 is insignificant, it provides evidence that the stock market return is not related to the
0 10 20 30 40 50 60 70
0 200 400 600 800 1000 1200 1400 1600 1800
1990 1991 1991 1992 1993 1994 1995 1996 1996 1997 1998 1999 2000 2001 2001 2002 2003 2004 2005 2006 2006 2007 2008 2009 2010 2011 2011 2012 VIX
S&P500 Futures Index
State 2 S&P 500 Index VIX
level of market volatility. This contradicts to the risk-return relationship based on the ICAPM.
On the other hand, since 1,2 and 1,2 are both negative and significant, this provides evidence that the stock market return is negatively related to the change in the future market volatility, and there is leverage effect in the US stock market.
Table 2: Parameter estimates (standard errors), and log-likelihood value for the two-regime Markov model
Component Variable
Two-regime Markov model Single-regime model Regime 1
(low volatility)
Regime 2 (high volatility)
Regime 1
Estimate Estimate Estimate Expected return
St
*0.178 (0.071)
-0.055 (0.046)
-0.011 (0.032)
**-1.580 (0.170)
**-1.094 (0.105)
**-1.162 (0.080) Volatility St Constant **0.557 (0.070) **1.348 (0.064) **1.145 (0.042)
Rt-1 0.028
(0.029)
**-0.031 (0.010)
**-0.038 (0.007) Transition
probability p tii( )
Constant **4.252
(1.440)
**4.721 (1.300) ΔUSCCIt-1 0.295
(0.222)
*-0.301 (0.120)
Log-likelihood -680.4 -706.4
AIC 1384.8 1420.8
BIC 1428.6 1435.3
Note that ** and * stand for significance at level of 1% and 5% respectively.
3.2 The risk-return relationship based on two-regime Markov model
The maximum likelihood estimates, log-likelihood, AIC and BIC values for the two- regime Markov model are also reported in Table 2. The coefficient of VIXt21 is positive and significant at level of 5% for regime 1, but insignificant for regime 2. This indicates that when the stock market is in regime 1, the market return is positively related to the market volatility.
Furthermore, the coefficient of (VIXt2VIXt21) is negative and significant at level of 1% for both regimes, providing evidence of negative correlation between the stock market return and
the change in the future market volatility regardless whether the stock market is in regime 1 or 2. Therefore, in terms of the unconditional expected return as in Equation (6), the market risk premium is positively associated with the squared VIX and negatively associated with the change in the squared VIX.
As for the conditional volatility equation, the coefficient of the lagged return is negative and significant at level of 1% for regime 2, but insignificant for regime 1. This means that there is evidence for the presence of leverage effect only when the stock market is in regime 2.
For the transition probabilities of remaining at the same regime over two consecutive periods, the change in the USCCI is negative and significant at level of 5% for regime 2, and insignificant for regime 1. This shows that the transition probabilities are time-varying.
Particularly if the stock market is at regime 2 for a period, an increase in the USCCI means that the stock market will be more likely to switch from regime 2 to regime 1 for the next period.
Moreover, either of the two regimes is highly persistent as the average of the transition probabilities is 0.946 for p t11( ) and 0.964 for p t22( ) respectively.
We also plot the estimated conditional volatilities for the two regimes in Figure 2 where the dotted (solid) line is for regime 1 (2). Since the conditional volatility for regime 2 is larger, we characterize the two regimes as low- and high-volatility regimes respectively. The shaded (unshaded) areas of the graph represent the periods during which the stock market is at the high (low)-volatility regime because the estimated smoothing probabilities of being at regime 2 (1) exceeds 0.5.
Figure 2: Plot of the estimated conditional volatilities for the two regimes.
Note: The shaded (unshaded) areas indicate the periods during which the stock market is in regime 2 (1) as the estimated smoothing probability of being regime 2 (1) exceeds 0.5.
Finally, for comparison between the two models, both AIC and BIC lead to the selection of the two-regime Markov model because the values of AIC and BIC for the two-regime Markov model are smaller than those for the single-regime model. This provides evidence that the two-regime Markov model is more appropriate to characterize the risk-return relationship in the US stock market
3.3 Discussion
According to Equation (6), our findings show that the market return is positively correlated with the level of the market volatility, while it is negatively correlated with the change in the market volatility. This empirical evidence of a positive risk-return relation in the stock market is supported by the CAMP and ICAPM, and is consistent with several studies in
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
1990 1990 1991 1992 1993 1993 1994 1995 1996 1996 1997 1998 1999 1999 2000 2001 2002 2002 2003 2004 2005 2005 2006 2007 2008 2008 2009 2010 2011 2011 2012
State 2 sigma2 sigma1
the literature (e.g., Ghysels, Santa-Clara, and Valkanov (2005) and Guo and Whitelaw (2006)).
The evidence of a negative association between the change in market volatility and return is also consistent with Campbell's (1993, 1996) version of the ICAPM that risk-averse investors want to hedge against changes in market volatility because volatility positively affects future expected market returns, as in Merton (1973).
Our findings also show that during the periods of low volatility, the stock market returns are positively correlated with the market volatility, and there is insignificant evidence of leverage effect in the US stock market. On the other hand, during the periods of high volatility, the correlation between the stock market returns and volatility is insignificant, while leverage effect is significant. This suggests that volatility is less persistent during the periods of high volatility. Furthermore, the change in the future volatility is negatively associated with stock returns regardless whether the stock market is in the tranquil or turbulent state, providing evidence of asymmetric volatility in the US stock market.Finally, our findings imply the need for different investment strategies during different periods of market tranquility and turbulence because of the different relationships between risk premium and volatility.
4. Conclusions
We have proposed a two-state Markov regime-switching model to analyze the risk- return relationship in the US stock market under different market conditions with monthly data of the S&P 500 stock and VIX indices. The proposed model enables us to examine the risk- return relation when the dynamics of the market condition is governed by a first-order Markov process with time-varying transition probabilities. We provide empirical evidence that stock market returns are positively associated with market volatility and negatively associated with change in market volatility. Our empirical findings are broadly consistent with the implications from the ICAPM that investors care about risks both from the market return and from changes in forecasts of future market returns. Our empirical evidence demonstrate that there is the
asymmetric behavior of volatility in the US stock market and the leverage effect is significant only during the periods of high volatility. Our results contain investment implications for investors under different market conditions.
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