As previous authors have done, we consider a market with a risky security S and a risk-free security with annual return r and suppose that the log return
for period [0, T] (denoted as [X.sub.T]) has a Gram-Charlier distribution under the risk-neutral (or "pricing") measure, which we denote as Q.
Empirical results from this study show the presence of the leverage effect in the log returns
of the JSE index.
First, the log return
on the market (LogRm) is the log gross returns on the S&P 500 Index (For robustness, we also use the value-weighted and equal-weighted Center for Research in Security Prices [CRSP] index).
t-test: [H.sub.0] states that the average daily log return
of the investigated day is equal to the average daily log return
of other weekdays.
For the WTI series, six input combinations based on previous log return
of daily oil prices are evaluated to estimate current prices value.
Table 6 presents the expected log return
under risk-based solvency requirements using [epsilon] = 0.025.
In the analysis of the performance of mutual funds in Slovenia from 2005 to 2009, we used the monthly log returns
This table shows the net return rather than the log return
in order to include firms with payoffs of zero.
The monthly log returns
of the two series are presented in Figure 1(A).
The expected log return
on equity, like the expected log return
on any other asset, is just a constant plus relative risk aversion times expected consumption growth.
In purely statistical terms, the standard CAPM beta--as well as our more complicated expression for beta--is the covariance of firm i's excess log return
with the U.S.
We indicate the expectation and standard deviation of the return, and the skewness and kurtosis of the log return
(to better illustrate the comparison with the Gaussian distribution).