The going concern assumption is that a business is planned to continue existing for some span of time into the future, usually assumed to be at least 1 year. In order to be a going concern, a business must either be profitable or at least be running a loss small enough that current assets will suffice to keep the business solvent during that period. Most accounting systems require that all businesses must be going concerns to file normal accounting statements, and otherwise must file for bankruptcy.
The historical cost method means that whenever a business purchases an asset, they report the value of that asset in future accounting reports as the actual amount paid when the asset was bought. The alternative is current cost accounting, on which assets are "marked to market" and accounted at their current market value.
Since most assets depreciate, the historical cost can overestimate what those assets would be worth if they were sold today, though most accounting methods do include depreciation for this reason. The problem is that depreciation for accounting purposes may not be the actual rate of depreciation of that asset in the market.
Thus, a company that appears to be solvent under historical cost accounting could actually be insolvent if all of its assets were sold today. Historical cost accounting could make a business appear to be a going concern even though it should really be bankrupt.
The opposite is also possible, if the asset has appreciated; a business that appears insolvent at historical cost accounting could in reality be solvent. This is less of a problem, however, as the business could simply sell the asset at the higher market price and receive the capital gain in cash, thus proving their solvency.
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