While the use of matching principle in accounting has long been established, there has been very little research on the scientific and theoretical rationale of its use. Recent research in accounting (Gibbins and Willett 1997, Su 2005c) has shown that matching principle or accrual accounting can induce a smoother income effect (measured by the volatility of accounting earnings) as compared to cash flow accounting. This result was demonstrated theoretically in Su (2005c) which presents approximation formulae to calculate the Expected Sample Variance (ESV) to measure the volatility of matched and unmatched earnings. The approximation formulae were used as a conservative test to examine the benefit of matched earnings over unmatched earnings in terms of long-term profitability estimation. However, the approximation formulae developed there tend to overestimate, and this paper discusses and corrects the overestimation biases by deriving the exact formulae for the ESV of matched and unmatched earnings.

Su (2005c) used the principles of Statistical Activity Cost Theory (SACT) (Willett 1987, 1988, 1991) to model accounting earnings as random variables under two scenarios. First, expenses and revenues are matched at the point of sale (the matched earnings scenario) and second, the unmatched scenario, where revenue and expenses are recognized in the period in which they are incurred. Su (2005c) used an approximate solution to the Expected Sample Variance (ESV) of matched and unmatched accounting earnings. As explained in that paper, ESV can be thought of as a measurement risk or error associated with the estimation of long-term profitability. The principal interest in Su (2005c) was not to find the exact formulae for the ESV of matched and unmatched earnings but to compare them. It will be shown in Section 2 that, while the approximation formulae provide a conservative test of the variance reduction effect of matching, they do have overestimation biases. These biases are often marginal, as indicated by the graphs in Appendix 4.2, which replicate the analysis done in Su (2005c). These graphs show there are differences between the approximation formulae used in Su (2005c) and the exact formulae developed in this paper.

The derivation of the exact formulae is important. While the formulae developed in Su (2005c) can be applied reasonably successfully to minimize the volatility of earnings using overhead cost allocations as was done in Su (2005b), it is sometimes necessary to find exact formulae for the ESV of unmatched and matched earnings. In the case of depreciation for example, where there are frequent unmatched and matched earnings calculations with costs being apportioned over time and matched with eventual gain or loss from sale, it is necessary to use the exact formulae calculation to avoid accumulating errors (Su 2005a).

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