On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks

Abstract

We find support for a negative relation between conditional expected monthly return and conditional variance of monthly return, using a GARCH-M model modified by allowing (1) seasonal patterns in volatility, (2) positive and negative innovations to returns having different impacts on conditional volatility, and (3) nominal interest rates to predict conditional variance. Using the modified GARCH-M model, we also show that monthly conditional volatility may not be as persistent as was thought. Positive unanticipated returns appear to result in a downward revision of the conditional volatility whereas negative unanticipated returns result in an upward revision of conditional volatility. There is general agreement that investors, within a given time period, require a larger expected return from a security that is riskier. However, there is no such agreement about the relation between risk and return across time. Whether or not investors require a larger risk premium on average for investing in a security during times when the security is more risky remains an open question. At first blush, it may appear that rational risk-averse Most of the support for a zero or positive relation has come from studies that use the standard GARCH-M model of stochastic volatility.2 Other studies, using alternative techniques, have documented a negative relation between expected return and conditional variance. In order to resolve this conflict we examine the possibility that the standard GARCH-M model may not be rich enough to capture the time series properties of the monthly excess return on stocks. We consider a more general specification of the GARCH-M model. In particular, (1) we incorporate dummy variables in the GARCH-M model to capture seasonal effects using the procedure first suggested by Glosten, Jagannathan, and Runkle ( THE TRADEOFF BETWEEN RISK and return has long been an important topic in II. Estimating the Model A. Econometric IssuesThe parameter p in the model given by (2) cannot be estimated without specifying how variances change over time, since Var(xt + 1? Ft) is not directly observed by the econometrician. To appreciate the difficulties involved, project both sides of (2) on Gt, the econometrician's information set, which is a strict subset of the agents' information set Ft. The term on the left in equation (4) Since the estimated slope coefficient cl is a consistent estimate of /3 bl, and d1 provides a consistent estimate of b1, the ratio of any two corresponding elements of cl and d1 provides a consistent estimate of /8. If zt-, is not a scalar, then we may impose the constraint that the slope coefficients in (5) and the slope coefficients in (6) differ only by the scale factor, /3. Such a restriction also provides a natural test for the validity of the model specification. We call this approach Campbell's Instrumental Variable Model. Another approach, the GARCH-M model, assumes that Var(vtllGt_-) is identically zero, and that zt-1 consists of innovations and variances that, while unobservable, can be...

Pricing of risk and volatility dynamics on an emerging stock market: evidence from both aggregate and disaggregate data

“…Although negative risk premium contradicts the fundamental portfolio theory (i.e. Markowitz, 1952), it still has been determined in the empirical financial press (see, for example, Balios, 2008;Elyasiani & Mansur, 1998;Fraser & Power, 1997;Glosten et al, 1993;Lebaron, 1989;Lettau & Ludvigson, 2010;Mandimika & Chinzara, 2012). It is supported by the four reasons given in the financial literature.…”

“…US dollar), then there is a high probability that a positive risk premium can become evident. The third and fourth reasons rest on the argument of Elyasiani and Mansur (1998) and Glosten et al (1993) who documented that the negative risk premium might be due to the fact that the riskier period coincides with the period when investors are relatively better in bearing risk or that if the investors are interested in saving more during a riskier period while holding all risky assets, the competition may increase the asset prices and, hence, decrease the risk premium.…”

“…However, (1) several studies in the financial press (e.g. see Balios, 2008;Campbell & Hentschel, 1991;Dimitriou & Simos, 2011;Fraser & Power, 1997;French, Schwert, & Stambaugh, 1987;Glosten, Jagannathan, & Runkle, 1993;Hansson & Hordahl, 1998;Jiranyakul, 2011;Koutmos, Negakis, & Theodossiou, 1993;Lanne & Saikkonen, 2004;Lebaron, 1989;Lettau & Ludvigson, 2010;Li, Yang, & Hsiao, 2005;and Mandimika & Chinzara, 2012), ascribed mixed evidence regarding the existence of risk premium in the stock markets of Australia, the USA, Europe, Asia and Africa by applying the GARCH-M model. For instance, Li et al (2005) concluded negative risk premium for six out of 12 markets, and so does by Mandimika and Chinzara (2012) while examining the South African stock market.…”

An international integration history of the Zagreb Stock Exchange

“…Here we find the dynamic conditional correlation model of Engle and Sheppard (2001) Our methodological procedure consists of two steps. In the first part we find the best-fitted GARCH model among several possibilities: Bollerslev's (1986) GARCH model, the exponential GARCH model of Nelson (1991), and the GJR-GARCH model of Glosten et al (1993). In the second part we feed the residuals from the first step into the DCC model and the ADCC model to choose the best fitted model between the two.…”

Estimating and Forecasting Volatility of Financial Markets Using Asymmetric GARCH Models: An Application on Turkish Financial Markets

“…Asymmetric GARCH models can be listed as Exponential GARCH (EGARCH) model by Nelson (1991), GJR model by Glosten et al (1993), Threshold GARCH (TGARCH) model by Zakoian (1994), Asymmetric Power GARCH (APGARCH or PGARCH) model by Ding et al (1993), Quadratic GARCH (QGARCH) model, Conditional AutoRegresive Range (CARR), Dynamic Asymmetric (DAGARCH) by Caporin and McAleer (2006), Integrated GARCH (IGARCH), Component GARCH (CGARCH), Fractional Integrated GARCH (FIGARCH), Volatility Switching ARCH (VS-ARCH) so on. Nelson (1991) introduced one of the well-known asymmetric GARCH model as EGARCH by working up Exponential ARCH.…”