According to Merton (1973) inter-temporal capital asset pricing model (ICAPM), the conditional expected excess return on the stock market should be a linear function of the stock market's conditional variance as well as a hedging component which captures the investor's motive to hedge for future investment opportunities: where Et[Rt+1] and Vart[Rt+1] are the expectation and the variance of the market excess return which are conditional on the information available at the beginning of the return period denoted as time t, while ÃÆ’MF,t is the conditional covariance between the expected excess return and variation in the investment opportunity set. J (W,F,t) is the indirect utility function in wealth W and any variables F, describing the evolution of the investment opportunity set over time. The term [-JwwW/Jw] is associated with the coefficient of relative risk aversion which is usually assumed to be positive, whereas the other term is the adjustment to the conditional risk premium arising from innovations to the state variables that describe variation in the investment opportunity set. Since Jw is positive, the sign of the adjustment component is determined by (1) By ignoring the hedging component under certain circumstances, the original ICPAM extends to a more general model which can be approximately expressed as where ÃŽÂ³ describes the coefficient of relative risk aversion of the representative investor and ÃŽÂ¼ should be equal to zero. To put it another way, the investor who bears more risk would require higher risk premium for compensation. This risk-return trade-off is known as the "first fundamental law of finance" in financial economics (Ghysels, Santa-Clara and Valkanov, 2005). However, in spite of the great contribution of Merton's research, it is exploratory which does not include hypotheses test. This positive risk-return trade-off has been proved to be hard to find in the data by previous literature. The main difficulty in testing the ICAPM relation is the way to filter the unobservable market conditional variance from the raw data of past returns. Thus, the conflicting findings of past empirical literature result mostly from the different approaches to model the conditional variance. Generally, it can be classificated into three types of models namely ARIMA, GARCH type models, Markov switching models and a pretty new approach-MIDAS. The remainder part of this section reviews the representative work with each type of models more specifically. French, Schwert and Stambaugh (1987) find evidence that the expected market risk premium is positively related to the predictable stock market volatility in a generalized autoregressive conditional heteroskedasticity (GARCH) approach. Also, they observe a strong negative relation between unexpected stock market returns and unexpected change in the stock market volatility which they interpret as the indirect proof of a positive ex ante risk-return relation by using a univariate autoregressive integrated-moving average (ARIMA) model. To start with, they decompose estimates of monthly volatility into predictable and unpredictable components by ARIMA model. The variance of monthly return is computed as the sum of the squared daily returns added by twice the sum of the products of adjacent returns, owing to the autocorrelation of daily returns especially at lag one caused by non-synchronous trading of securities (Fisher 1966, Scholes and Williams 1977):
, where Nt is the number of days and rit is daily return in month t. Compared with the rolling 12-month standard deviation (Officer 1973 and Merton 1980), the accuracy of this standard deviation estimate is increased not only for any particular interval but also for any month by sampling more frequently and using merely within-month returns. Besides, the adjacent rolling 12-month estimators share 11 returns while non-overlapping data of returns is adopted in their estimates above. By plotting the monthly standard deviation estimates, they suggest the first differences of lnÃÆ’mt follow a third-order moving average process:
. The conditional variance forecast is constructed as
, where Thus, they regress excess returns on the predictable components of the stock market variance and contemporaneous unexpected changes in market variance as follows:
, where Consequently, the estimates of both regressions above do not provide any evidence of a positive ex ante risk-return relation but rather a reliable evidence of negative relation between excess holding period returns and the unpredictable component of volatility. If the positive relation between expected excess return and predictable volatility dose exist, the upward unpredictable change in volatility is likely to increase future expected excess returns and decrease current stock prices. The magnitude of this negative relation is too large to be attributed solely to the leverage effects (Black 1976 and Christie 1982), so French et al. regard the negative relation as indirect proof of ICAPM. In their second approach, the GARCH-in-mean model (Engel, Lilien and Robins 1987) is employed to estimate the ex ante relation between risk premiums and volatility. The conditional variance and risk-return relation are estimated as: and respectively, where As a result, a strong positive relation between expected risk premiums and volatility is perceived which is consistent with the interpretation of ARIMA results. Overall, though their work does not include other variables which also have impact on expected risk premiums along with other measures of time-varying risk, they does great contribution to the empirical analysis of ICAPM. However, from Nelson (1991)'s perspective, there are several important limitations regarding the application of the GARCH models in asset pricing. The GARCH models rule out the results that there is a negative relation between stock market returns and the changes in volatility of returns by assumption. But empirically, investigation in this field since Black (1976) dose report a negative correlation that volatility is prone to go up in response to lower return than expected and vice versa. Another weakness arises from the non-negative constraints imposed on the parameters which the estimated coefficients often violate and may in turn inappropriately restrict the dynamics of the conditional variance process. What is worse, it is difficult to examine the persistence of shocks to conditional variance due to the lack of agreement for measuring persistence. Aimed at these objections that ordinary GARCH model cannot meet, Nelson (1991) put forward an alternative model called Exponential GARCH or EGARCH. By using the EGARCH model, they find evidence of a negative but insignificant relation between the expected risk premium and conditional variance. This result stands in contradiction with the positive relation found by other researchers such as French, Schwert and Stambaugh (1987) who adopt the standard GARCH-M model. In addition, the estimated AR root they obtained indicates a large extent of persistence of shocks in daily data. To sum up, having said the EGARCH model overcomes some of the limitations of ordinary GARCH models, nevertheless, with respect to the simplicity and flexibility, ARIMA model used by French et al. (1987) may be preferable especially when presenting the conditional variance. Similarly, Glosten, Jagannathan and Runkle (1993) argue that the standard GARCH-M is misspecified. In logic, they suggest although the rational risk-averse investors may require a relatively higher risk premium in general within a given time period when the security is more risky, it is not necessarily the case across time. For one thing, investors may be more able to live with particular types of risk during time periods which are relatively more risky. For another, investors may prefer saving relatively more if with the expectation of more risky future. According to Glosten et al. (1993), if all the productive assets available for transferring wealth to the future carry risk and no risk-free investment opportunities are available, then the price of the risky asset may be raised significantly, thereby reducing the risk premium. Based on the conflicting predictions about the risk-return relationship above, they consider a more general specification of the GARCH-M model that is known as the Threshold-GARCH or TARCH model. The most advanced feature about the TRACH model is to allow for such asymmetry that positive and negative unpredicted returns have different influences on the conditional variance. They also introduce some dummy variables in the modified GARCH-M model to characterize seasonal patterns in volatility, following the procedures proposed by Glosten, Jagannathan and Runkle (1988). The nominal interest rate is incorporated into the prediction of the conditional variances. As a consequence, a weak but significantly negative relation between conditional expected monthly return and conditional variance of monthly return is observed. Besides, they combine the EGARCH-M model suggested by Nelson (1991) with the modifications mentioned above for robustness. The result remains unchanged with the EGARCH-M model integrating seasonal effects or nominal interest rate or both. By the revised GARCH-M model they also find that the time series properties of monthly excess returns are substantially different from the reported properties of daily excess returns. First, persistence of conditional variance in excess returns is quite low in monthly data. That is inconsistent with Nelson (1991)'s finding of high persistence in daily data. In addition, positive and negative unanticipated returns have significantly different effects on future conditional variance. That is, a positive unexpected return is supposed to have a negative impact on future conditional variance whereas a negative unanticipated return is expected to have a positive impact on the conditional variance. In contrast, Engle and Ng (1993) report that positive and negative unexpected returns both lead to the increase in future conditional variance in spite of different effects.