Optimal down payment amount
I need to buy a car; this is a necessity, my current one is on its last legs.
I have enough cash on hand to comfortably pay for any car that I would consider purchasing.
However, there is value in taking out loans, both to have more cash liquid, and hedge inflation.
I would like to keep cash on hand, but also, of course, minimize interest.
Is there a quick formula I can plug numbers into to get the optimal down payment amount, balancing both the time value of money, and interest?
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The optimal down payment is 0% IF your interest rate is also 0%. As the interest rate increases, so does the likelihood of the better option being to pay for the car outright. Note that this is probably a binary choice. In other words, depending on the rate you will pay, you should either put 0% down, or 100% down. The interesting question is what formula should you use to determine which way to go?
Obviously if you can invest at a higher return than the rate you pay on the car, you would still want to put 0% down. The same goes for inflation, and you can add these two numbers together. For example, if you estimate 2% inflation plus 1% guaranteed investment, then as long as the rate on your car is less than 3%, you would want to minimize the amount you put down. The key here is you must actually invest it.
Other possible reasons to minimize the down payment would be if you have other loans with higher rates- then obviously use that money to pay down those loans before the car loan.
All that being said, some dealers will give you cash back if you pay for the car outright. If you have this option, do the math and see where it lands. Most likely taking the cash back is going to be more attractive so you don't even have to hedge inflation at all.
Tip: Make sure to negotiate the price of the car before you tell them how you are going to pay for it. (And during this process you can hint that you'll pay cash for it.)
The optimal down payment is 100%. The only way you would do anything else when you have the cash to buy it outright is to invest the remaining money to get a better return. When you compare investments, you need to take risk into account as well. When you make loan payments, you are getting a risk free return. You can't find a risk-free investment that pays as much as your car loan will be. If you think you can "game the system" by taking a 0% loan, then you will end up paying more for the car, since the financing is baked into the sales [price in those cases (there is no such thing as free money). If you pay cash, you have much more bargaining power.
Buy the car outright (negotiating as hard as you can), start saving what you would have been making as a car payment as an emergency fund, and you'll be ahead of the game.
For the inflation hedge - you need to find investments that act as an inflation hedge - taking a loan does not "hedge" against inflation since you'll still be paying interest regardless of the inflation rate. The fact that you'll be paying slightly less interest (in "real" terms) does not make it a hedge.
To answer the actual question, if your "reinvestment rate" (the return you can get from investing the "borrowed" cash) is less than the interest rate, then the more you put down, the greater your present value (PV). If your reinvestment rate is less than the interest rate, then the less you put down the better (not including risk). When you incorporate risk, though, the additional return is probably not worth the risk.
So there is no "optimal" down payment in between those mathematically - it will depend on how much liquid cash you need (knowing that every dollar that you borrow is costing you interest).
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