Depreciation in a model of probabilistic investment

A pervasive theme in both accounting and statistics is aggregation. However, in contrast to statistics, a customary standard for determining the best aggregation rule in accounting is unavailable or, at least, not explicitly defined. Also, most accounting procedures follow a well-specified recursive algorithm of updating a summarized history number (a beginning balance sheet number) by the current period's activities (changes). In this paper, we present a setting in which the best accounting aggregation rule arises naturally, resembles observed depreciation schedules, and proceeds recursively in a manner analogous to the above outlined stock-flow updating process. Our main results are (1) in every period, the performance of the BLU estimate based on active investments can be replicated by the period's depreciation amount and (2) in every period, the performance of the BLU estimate based on the entire history of investments can be replicated by a recursive procedure that updates the BLU estimate of the previous period with the current perioda's investment realization. Depreciation successfully satisfies multiple objectives - it serves as a periodic allocation of realized investment amounts and as a statistic for the unknown investment population mean. Depreciation schedules commonly used in practice, straight-line, accelerated and declining balance, are shown to be best in particular settings.