Cap'n Arbyte's



Other sites

Auto Incentives

I'm in the market for a new car. I've been saying that for a year already and haven't bought one yet, but let's ignore that and play along for a while.

As a Buick Regal driver ("you drive an old people car!?" — average age 68!) I'm looking at the new LaCrosse, which replaces the Regal. GM recently started advertising the vehicle on television, which miffed me a bit because (1) I gave them my contact information on their website months ago, inviting them to market directly to me, but they never did and (2) the Chevrolet commercials from last January were much better.

Having never purchased a new car before, I need to figure out all these crazy incentives, and since I'm going to do the math anyway I might as well post it as a public service.

The LaCrosse incentives (for the Western U.S.) are… confusing. The generic GM incentives page breaks it down a bit better. Between the two of them, I hesitantly conclude that the real offers are between:

  • 0% financing (36mo) and $1000 "GMAC Bonus"
  • $2000 with an unknown breakdown between "Bonus Cash" and "GMAC Bonus"

I think the "GMAC Bonus" applies only when financing through GMAC. However, I'm not interested in financing a vehicle unless I could do it at 0%; otherwise I'd be inclined to buy it outright and wouldn't get that bonus. Assuming the second offer breaks down as $1000 "Bonus Cash" and $1000 "GMAC Bonus", my options are between 0% financing and $1000 off, or buying outright and $1000 off. This one's easy — I'll take the 0%, thanks, and let my savings earn interest.

Confident that it cannot possibly be this straightforward, I want to solve for the general case, so I can go in armed with a calculator and get the best deal no matter what the details are. I'm also curious to know exactly how much benefit there is to 0% financing given my meager understanding of the incentives. Gather around, children, this is why we study mathematics in school. Let's define some variables:

  • p = price of vehicle before incentives ($)
  • d = discount if buying outright ($)
  • s = interest rate earned on savings (%)
  • r = interest rate if financing (%)
  • t = term of financing (years)
  • o = down payment if financing ($)
  • x = discount if financing ($) applied to principal
  • y = loan fees ($) rolled into loan

Assume my savings is equal to p. If I buy outright, I pay (p-d) and earn interest on the remaining d of my savings. If I finance, I need to keep simultaneous track of the balances of both the loan and my savings. Every month each is adjusted for both the car payment and the interest on the respective accounts. The loan is for (p-o-x+y) over t years at r% while my savings (p-o) earns s% and is drawn from over t years to make the loan payments. The option I should choose is the one that maximizes my savings balance at the end of the term.

The payment amount for a loan is (P(1+r)nr)/((1+r)n-1). In my terms, and for monthly payments, ((p-o-x+y)(1+r/12)t*12r/12)/((1+r/12)t*12-1).

The amount of my savings balance if I finance is (p-o)(1+s/12)t*12 - payment(((1+s/12)t*12-1)/(s/12)). If I buy outright, it's d(1+s/12)t*12.

That's the general solution. Now I'll plug in a few values I know.

  • p = $32820 (Sticker price with gobs of options; a worst-case figure)
  • s = 5% (I'd actually use my savings to pay principal on my 5% mortgage)
  • r = 0%
  • t = 3 years
  • d, o, x, y = unknown

At a 0% interest rate we immediately hit a divide-by-zero problem in the general equation. No matter, 0% is the straightforward case where we don't need fancy math. If I finance, the monthly payment would be (32820-o-x+y)/36 and my final savings balance would be (32820-o)1.1615-(payment*38.7534).

If I buy outright, my final savings balance would be d*1.1615.

If I make reasonable assumptions of d=$1000, o=$5000, x=$1000, y=$268.20 (1% of the loan amount), I'm looking at monthly payments of $752.45 and a financing final savings balance of $3152.16. Buying outright, my balance would be $1161.47. Over three years, I'm $1990.69 better off — that's over $50/mo!

(However, my actual benefit would be lower. I couldn't actually put all the savings into my mortgage because my current discretionary income wouldn't cover the car payments. I'd have to keep some savings in the bank to draw from, earning … let me see … 0.9%. But those details are beyond the scope of my purpose here.)

There you have it, kids. Through the Wonders of Math™, I have quantified the benefit of 0% financing over buying outright, and have the tools to quickly make comparisons of this nature under different conditions — viz., for different vehicles, incentives, interest rates, and loan durations.

Math. It's like money in the bank.

Somebody cut me a check for public service.

UPDATE 2004-12-13 07:33:30 UTC: Added the link about the average age of a Buick owner being 68. Am I not in the Buick demographic, or am I forty years ahead of my time? :)

Tiny Island