Implications of the integral approach and earnings management for alternate annual reporting periods

Abstract

We compare the last 12 months’ earnings ending in quarter four (i.e., fiscal year earnings), three, two and one. Lipe and Bernard (2000) offer two competing explanations for higher volatility in fourth quarter earnings relative to other quarters. First, under the integral approach, any estimation errors in the earlier quarters are corrected through fourth quarter earnings, which could make them more volatile. Second, earnings management concentrated in the fourth quarter renders fourth quarter earnings more volatile. While both explanations have similar implications for the properties of quarterly earnings, their implications differ for the properties of annual earnings ending in each quarter. Our result comparing earnings variability is more consistent with earnings management than the integral approach. We examine the relative earnings attributes and find that fiscal year earnings attributes rank lower. Finally, we re-investigate the accrual anomaly and find that the accrual anomaly is more pronounced for fiscal year earnings.

Appendix: annual earnings computed over different intervals

In this appendix we consider a setting without earnings management and provide sufficient conditions under which, consistent with our null hypothesis, the fiscal year-end would not affect the variance or skewness of annual earnings.

For exposition, we consider a firm whose fiscal year coincides with the calendar year and denote its quarterly earnings for year y and quarter q by x _y,q . Further assume that quarterly earnings are mean reverting and follow a seasonal first order autoregressive process, such that

$$ x_{y,q} = m_{q} + \rho \left( {x_{y - 1,q} - m_{q} } \right) + \widetilde{e}_{y,q} , $$

where m _q is the level of quarter q earnings; ρ is the autocorrelation parameter that captures the speed of adjustment to the mean reversion; $\widetilde{e}_{y,q}$ is the error term that is independently distributed with mean zero and quarter-specific variance, σ ² _q . Annual earnings can be calculated as

$$ X_{y} = x_{y,1} + x_{y,2} + x_{y,3} + x_{y,4} $$

Under the above assumptions,

$$ X_{y} = \mu + \rho \left( {X_{y - 1} - \mu } \right) + \widetilde{\varepsilon }_{y} , $$

where $\mu = \sum\nolimits_{q = 1}^{4} {m_{q} }$ is the long-run level of annual earnings, and $\widetilde{\varepsilon }_{y} = \sum\nolimits_{q = 1}^{4} {\widetilde{e}_{y,q} }$ is the random component in annual earnings that is distributed with mean zero and variance, $\sigma^{2} = \sum\nolimits_{q = 1}^{4} {\sigma_{q}^{2} }$. In this setting, fiscal year annual earnings do not exhibit seasonality because the aggregation is over the business cycle.

In this paper we use quarterly earnings and aggregate to annual earnings in three additional ways that differ from the fiscal year. In particular, we consider the following three alternative annual time-series:

$$ \begin{array}{*{20}c} {X_{y}^{[1]} = x_{y,2} + x_{y,3} + x_{y,4} + x_{y + 1,1} } \hfill \\ {X_{y}^{[2]} = x_{y,3} + x_{y,4} + x_{y + 1,1} + x_{y + 1,2} } \hfill \\ {X_{y}^{[3]} = x_{y,4} + x_{y + 1,1} + x_{y + 1,2} + x_{y + 1,3} } \hfill \\ \end{array} $$

It can easily be verified that

$$ X_{y}^{[n]} = \mu + \rho \left( {X_{y - 1}^{[n]} - \mu } \right) + \widetilde{\varepsilon }_{y} $$

for n = 1, 2, 3. Therefore the assumptions provided above suffice to ensure that the choice of annual period for calculating annual earnings does not matter. This holds true even though quarterly data exhibit seasonality (as suggested by Oyer 1998), captured by quarterly variation in the level of earnings, m _q , and even though the variance of earnings, $\sigma_{q}^{2}$, may change with the calendar quarter, q.

Note, however, that when working with these annualized earnings observations an overlapping observations problem arises. The problem of overlapping observations causes the estimator of the variance of earnings for different annualized data on the same firm to be dependent. This means that standard statistical significance levels on the tests for difference in variances do not apply because they presume independent samples. We circumvent this overlapping observations problem by using ranks for any given firm.