Funky Math

My sister and I have been independently growing more cognizant of money. One interesting finance site is Mint. It looks very convenient, but I feel more than a little squeamish about putting my financial information in a third party website.

Mint also has a blog that’s chock-full of useful tips. In one post, they give a quick overview of the practical financial aspects of car-buying. One thing that caused me to raise an eyebrow, however, was their stance on down payments. In essence, they noted that paying a higher monthly interest rate and while keeping a “down payment” in a high-yield savings account can be better than using the down payment to decrease the size of your car loan. That’s pretty unintuitive! They’re saying that taking an extra $4,000 worth of car-payment-loan at 7.5% and putting $4000 in a savings account at 4.75% is BETTER the alternative, which is using that four thousand in the lower-rate account to “pay off” the higher-interest-rate loan. That doesn’t make sense at first glance; usually, you’re best off when you pay off loans in decreasing order of interest rates.

I checked their math; sure enough, you’re better off taking the larger loan. Strange! I realized a few days later, while their comparison is useful and reflects what goes on in “the real world,” it’s not entirely apples to apples. The reason lies not only in how much money is loaned, paid, and invested, but also when. Also, as an aside: In calculating interest on money you invest, Mint assumed no compounding. I redid the numbers to be a little more realistic, compounding once a month.

As a quick overview: In both cases the car is $20,000, and you have a 48-month repayment plan at 7.5%.

In case 1, you pay $4,000 down and pay $387 a month. $4,000 + 48 * $387 = $22,576.

In case 2, you pay nothing down, and your monthly payment is $484, and the total amount you pay at the end is $23,232, $656 more than case 1. At this point, it looks like the down payment is a good idea: you pay less money! But if you take the down payment and put it in a CD at 4.75% APY, you’ll earn $835, and will come out $179 richer than case 1.

So what gives?

Well, the thing is that case 2 assumes you have more money to start with. If you simply took a down payment, the most money you’ve ever invested in the car at any given point in time is $22,576, on the day you send in your last car payment. In contrast, look what happens in case 2. Your car payments total $23,232, PLUS you have to pay an ADDITIONAL $4,000 to earn interest in the CD. That means that right after you send in the last car payment, and before you close out the CD, you are $27,232 poorer than the day you signed the loan. After you close the CD, of course, your total cost comes out to a final value of $22,472.

So, then, what’s the apples-to-apples comparison? You need to give yourself 27,232 – 22,576 = $4,656 more to invest over the course of the 48 months. That way, when you’re done with the payments in case 1, you’ll have set aside exactly as much money in total as you did in case 2. Coincidentally enough (just kidding!), $4,656/48 comes out to $97 per month, which is the exact difference between the monthly payments in the two scenarios!

This makes sense: now, in both cases, we have an initial outlay of $4,000 that we use for something-or-other, and monthly payments of $484. For our “new” case 1, we use $4,000 for a down payment, and invest the extra $97 per month this frees up. Assuming we our only use of the extra money is to make deposits into our CD or savings account, we will earn $598 interest on our principal of $4,656. This is excellent: we earn less interest, but we also pay much less interest, and our total cost is a mere $21,978. Instead of being $179 richer than we were originally, we’re now $598 richer. Yay! For completeness, I’d also note that this is essentially the same ($8 less) than if we could avoid paying for the higher-interest-rate car loan by both making a down payment AND giving the higher monthly payment over a smaller number of months.

What can we conclude? The best strategy is to make as large a down payment as possible at the beginning, and then, each month, put as much leftover money as we can into a high-yield savings account. Shocking, isn’t it?

As an aside, the economic cost of the car itself (ignoring practical things like title fees and insurance, as well as the monkey wrench known as inflation) is not $20,000, nor is it $21,978. The future value of the money used to pay for the car is $30,466. That’s how much you could have had if you didn’t buy the car in the first place. This means that the opportunity cost of the car is more than 150% of the sticker price!

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