When I budget for me, I usually count savings part of the expenses.
Is it the correct way to do it? Because of this, when I look at my budget at the end of the month, it feels like my expenses are a bit high.
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I have found it useful to have four (yes, four) categories in a budget, rather than the traditional two (income and expenses). This may not be the "correct" way to do it, but it makes sense to me.
First, income. This one is a no-brainer and is needed in any budget. In this scheme income also includes money taken out of deferred-spending savings (see below). I don't consider money taken out of savings "income", however.
Second, expenses. This is the money you actually spent in the given period (month, with most budgets, but if you are paid on a different schedule it may be bi-weekly, quarterly, or whatever else makes sense in your particular situation). Plain and simple whatever goes into "expenses" is money that is no longer available to you in any way shape or form. I'd put credit card spending into this category as well (treat it the same as debit card spending) but as long as you aren't carrying a balance, that's simply a matter of taste. I'd also put loan payments into this category since "getting the money back" then means taking out a new loan.
Third, deferred spending. This category is for setting money aside for big-ticket items that you know are coming and which you either aren't sure exactly when they will turn into real expenses (money spent), or know when they will and they are simply too large to handle comfortably with the day-to-day cash flow. An example of the former might be a new car fund, and the latter might be something like my buying dog food about once every four months; the cost for such a batch is too large to handle comfortably within a single month's budget without making large adjustments elsewhere, so I set money aside each month and then use it every four months (or thereabouts). In this categorization, setting the money aside classifies as deferred spending, and taking it back out classifies as income. If done right, this category will trend toward zero over time, but may hold a substantial balance at times and will rarely or never actually be at zero. When you are just starting out, it may very well be negative unless you compensate by taking money from elsewhere. Note that some may refer to this category as targeted savings, which to me certainly is an overlapping term but does not carry quite the same meaning.
Fourth, savings. This is long term savings and investments that are not earmarked for a particular purpose or the purpose for which it is earmarked is very far into the future. How you define "particular purpose" and "very far into the future" are really a matter of definition, here, and you will have to come up with distinctions that work for you. The point is to separate saving for the future (savings) from saving to cover upcoming expenses that you have already decided on or committed to (deferred spending).
The deferred spending and savings categories can be further subdivided in much the same way as income and expenses, so you can keep track of the money going into your "new car" fund and your "travelling around the world" fund separately. The actual place where the money ends up is on your balance sheet, not the budget. (If you really want to keep track of your money, you will need both.)
In the end, the budget's bottom line becomes (income + money out of deferred spending) - (expenses + money into deferred spending + savings). Just like a regular income/expenses budget, this will be zero if your budget is balanced.
It's a matter of semantics.
If a budget is done to include every last item, by definition there would be no extra. I save about 20% of my income, but if that's in my budget, the budget is a 100% view of my gross income, a pie chart (or table) of all cash flow.
Some people prefer to think of their budget as the spending with what remains going into saving. In the end, that produces the same result. An accounting of all of your money, although this approach strikes me as having the potential of the savings aspect appearing an afterthought.
Most financial advisors would council a client to save "off the top" and not with what remains after everything else.
From looking at some of the other answers I see the issue is one of accounting. If it helps the matter, the piece you are missing in the discussion is the rest of the balance sheet, the assets accruing. If you gross $2000, account for $2000, but $400 went to savings, your balance sheet has improved $400.
By the way, paying a debt's principal will also improve the balance sheet. Your net worth grows for saving a dollar or paying off a dollar's worth of debt.
Savings isn't really an expense but I understand what your trying to say, you are allocating a specific amount to save. Which is definitely a good idea. If you're going over the budget then you have to decide what to cut down on. You can sacrifice some luxuries or if you feel comfortable enough you can cut back on how much you save. Clearly the more conservative choice would be to cut back on luxuries.