I call our mid-term savings an 'emergency fund' on this board, just to keep the lingo similar. However, we actually have three levels of savings (short-term, mid-term, and long-term) and that is how we think of them. Plus, we keep a cushion in our checking account to smooth out the highs and lows in any given month. I think it is important to figure out what all this means to you and your family versus following a standard formula.
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To compare our finances to common terms I see here:
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~ Our baby savings fund (typically seen around here as $1K) is the cushion in our checking account. However, I keep it anywhere between $500 - 1000.
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~ Instead of separate "sinking funds", we budget an entire year (called our "spending plan"). We average all expenses over all 12 months for the "budget" aspect, which I call the "monthly averages". The actual spending plan shows the proper amounts in the correct month (when paid...cash based accounting methods). For example, one car registration is due in January each year and the other is due in February. The estimated figure is in the month it is due and $0 are in the remaining months. The very last column in our spending plan has the total of the two car registrations divided by 12. I see our actual inflows and outflows each month and I also know the average of what it should be in advance. It is easy to see when a category is underfunded or overfunded on paper.
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~ We separate out healthcare costs in an HSA because we were eligible for 12 months, which spanned two calendar years and we were able to max it out per law (prorated) for both years. We didn't need to spend it all, so we are still allowed to keep it open and use it for qualified expenses. We also separate out vacation/travel expenses and home improvement projects. We have one account (short-term savings) that we contribute to monthly and one year we use all the money we've put in that account for home improvement projects and the next year we use it all for vacations/travel. We figure if we are doing a lot of home improvement projects, then we don't have extra time for traveling....and if we are spending money on vacations, then we aren't able to save it for home improvement projects. Win-win for our family! 
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~ Our mid-term savings is comprised of multiple accounts, including insured money market accounts and investment money market accounts and stocks and mutual funds (various levels of liquidity and risk -- my risk tolerance is fairly high, so YMMV). Our purpose for this money is multi-fold. First, it is intended to cover our needs in the event of an emergency income situation (job loss, for example). Next, it is meant to reduce our dependence on credit. We use debt on purpose, but with wide open eyes. We paid most of each car (different years) in cash, but we did purposely take out auto loans as well. In May 2009, we had a major flood and we took on two zero-interest consumer loans to pay for big ticket repairs/items while the insurance check sat in insured accounts earning short-term bonus interest. We paid them off before any interest had accrued and came out ahead. (FWIW, our credit scores are each over 800.) Another purpose for this money could be college tuition for our child. We'd rather have the money technically be in our name versus our child's name for potential financial aid purposes. (Although, we actually have a different plan for college, so this is more of a back-up.) Lastly, the ultimate use of these accounts will be (unofficial) supplemental retirement. As newlyweds, we aimed for one month of living expenses. As new parents, we aimed for three months of living expenses. Over the last few years, we've slowly increased it to 6-9-12 months (one year of essential living expenses is our year-end 2010 goal). We have minimum balances we require of ourselves as we slowly add to these accounts -- once we pass certain thresholds, we don't allow the aggregate amount to ever drop below those thresholds.
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~ Our long-term savings is formal retirement (IRA/Roth). We had 401ks at earlier points in our careers and we rolled them over as things changed (firms no longer offered 401k programs or changing jobs and new company didn't have 401k and old company was no longer in biz, etc). DH's company doesn't offer a 401k program, so all of our retirement funding is limited to the government's cap of $5K per IRA per year. That sounds like a lot when you are in your 20s and early 30s. About mid-30s, it seems do-able. We're in our early 40s now and it seems too little. Hence the above bullet point of supplemental retirement. We live in a high COL, so YMMV.
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We've only been able to accomplish any of this by continually reducing expenses and living within our means as a family (especially purchasing a home we could afford from the get-go). DH brought a lot of debt to our relationship and I had plenty of debt while putting myself through college (years before we met). It may sound all rosy now (and it certainly is easier), but it has been challenging along the way.
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Anyway, I typed all this out in order to encourage the OP (and others) to critically assess your own finances and what money means to you and your family. All the "popular terms" are fine, but do they make sense for YOU? As unpopular as debt is right now (and I caution people to use it wisely, not go crazy), I strongly feel credit/debt is an important finance tool and it has it's place right alongside savings. I advocate building savings AS you reduce debt, concurrently (at the same time, in different percentages), not successively (pay off all debt, THEN build savings). Too many of life's uncertainties can and do pop up when you least expect them and having cash on hand simply puts you in a better place to handle them.
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Therefore, great job on saving up $10K!
 What does that money mean to you and your spouse? (No need to answer, just food for thought...)
Edited by sunnysandiegan - 12/18/10 at 10:13pm