Calendar Based Risk and the Random Walk Hypothesis Revisited

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Independent variance contributions of long and short horizon returns for calendar and non-calendar periods are examined for the stochastic processes of time-varying returns. The random walk hypothesis is tested to determine if individual firm's total and idiosyncratic risk have significant calendar based components. We find the second moment of individual firm's time-varying returns by estimating the non-parametric power spectral density. Finite length return signals are analyzed using a stochastic additive noise model and small sample digital signal processing methods for the first time to obtain high resolution of calendar risk contributions. We compute idiosyncratic risk by treating long and short horizon components of market risk as a known risk source and filter it from individual security return process total risk. The empirical test rejects the white noise hypothesis and finds the time-varying return signals of individual firm's have significant short and long horizon periodic memory. No significant calendar based risk is found for the market index returns. All individual security samples have significant one-year calendar based risk. In contrast to previous studies large firms are found to be strongly affected by four-year calendar based risk (two-year mean reversion) while all other firms are not. We find a mid-cap effect with significant two-year and six-month calendar based risk. Small firms have no significant four-year or two-year horizon risk but are dominated by January-like risk with significant power at yearly, six-month, quarterly, and monthly calendar periods. Rejection of the random walk hypothesis for total and idiosyncratic risk components does not imply stock market inefficiency since these calendar risk anomalies appear to be coincident to calendar return anomalies reported in previous empirical studies. The results of the test differ from previous mean reversion and long memory studies that have been unable to identify multiple periodic effects. The calendar based risk model adds a periodic calendar time horizon dimension to the theory of risk. We conclude that calendar and non-calendar based risk should be considered in valuation and timing decision.