To overcome the shortcoming of using the balance at the beginning of the year and the balance at the end of the year, some accountants use 13 balances: the balance at the beginning of the year plus the balances at the end of each month. For example, the 2012 average would be calculated using the ending balance on December 31, 2011 + the ending balance on January 31, 2012 + the ending balance on February 28, 2012 and so on. Obviously, this is better than using only the two December 31 balances. Of course you need to have access to the monthly balances.
Still better is to compute the average for the year by using the balances on each of the 365 days in the year—thereby considering fluctuations that occur within a month.
If you need a monthly average for accounts receivable, the logic is the same: 30 or 31 points of data is far better than merely using the beginning of the month balance and the end of the month balance.
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