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This is a great question and one that I am very familiar with.
In general an operating budget is based on accrual accounting and, therefore, doesn't necessarily coincide with a cash budget.
In general, the cash budget uses the operating budget for operating expenses. However, the cash collected differs from the revenue stream in timing only (assuming the company sells on credit). The payment stream also differs from the operating expense budget in timing for those vendors that get paid on credit terms, and also based on the payroll timing.
Here is what I mean by timing:
Say you sell $100,000 of an item to a customer on January 1st. If the terms for that customer are net 30 (payment due within 30 days), then you shouldn't expect to see that money until around Feb 1. Take this approach an apply it to the revenue budget and you will see that the cash budget is offset form the revenue budget by a month.
The same concept applies to expenses that are paid on credit.
Additionally, to create the cash budget, you will need to know other cash expenditures that aren't part of the operating budget. These include capital expenditures, debt payments (principal, not interest because interest should be in the operating budget), and dividends. Remember you must know the amounts and timing of these payments to create a good cash budget.
So, to summarize, the relationship between an operating budget and a cash budget is this:
A cash budget, while largely derived from the operating budget accounts for the timing differences between revenues and expenses and when the cash flows in and out respectively for these items. It also accounts for non-operating cash flows out and in like principal payments on debt, capital expenditures, dividends.