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I would like to buy a car, and found out about the 20/4/10 rule today. I definitely have more than 20% downpayment saved up, due to that I am not concerned with the 20 part. I am more concerned about the 4/10 part.

Given the rate(r), time(t), and payment frequency(f) I want to find how much money I will end up spending at the end of the 4 years AND how much of that money will be interest paid.

So far I know that the best interest rate I can get is 4.7% (annual whatever that means) for 4 years. And I know that I can make monthly payments of 303$ per month. So given this info, how much money would I have paid by the end of the 4 years, and how much of it would be lost paying off interest. Would that number change if I change from 303$ monthly payment to 140$ bi weekly or 70$ weekly?

I actually do not want anyone to solve my problem for the numbers I provided, I am more interested in what math is involved and what formula is used to calculate this information. Unfortunately due to my poor mastery of English language I couldn't google proper terms needed to find an answer. All the answers I find seem to know the amount loaned, but I want to know how much I will be able to take out as a loan given only rate(r), time(t), and payment frequency(f).

  • Perhaps I'm misunderstanding what you actually want to know. As the question is phrased now, what you're asking seems trivially simple. The total amount paid is $303 * 48 months, the amount spent on interest is that minus the stated price of the car. – jamesqf Dec 7 '18 at 4:03
  • @jamesqf OP wants to know what the purchase price of the car would be if he had a 48 month loan at $303/month. – Ben Miller Dec 7 '18 at 5:19
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    Why are you using so little of a down payment to buy a car? It should be 100% of the "out the door" price. If you don't have that much cash, buy a less expensive car. – Pete B. Dec 7 '18 at 11:39
  • What do you mean by the 20/4/10 rule? – yoozer8 Dec 7 '18 at 12:55
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    People who advocate that kind of thing do not understand risk management. When the economy goes south people lose jobs and their investments lose value. How are you going to make your car payment then? Better to pay cash. – Pete B. Dec 7 '18 at 15:34
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The 20/4/10 Rule, for anyone who has never heard of it before, says that for a car to remain affordable, the car buyer should have at least a 20% down payment, should take out a loan for a maximum of 4 years, and the loan payment and insurance premiums should be less than 10% of the buyer's income. It's not a bad rule of thumb (although I prefer the 100/0/0 Rule).

To correctly apply the "4/10" part of the rule, you need to know your monthly income. 10% of that is the maximum amount that your loan payment + monthly insurance premium can be. Your question seems to be focused on the loan payment portion of that payment, so that is what the rest of my answer will discuss. But keep in mind that you will also have insurance payments to deal with.

The formula for the principal amount of a loan is:

$$ P = A \frac{(1+r)^n-1}{r(1+r)^n} $$

where:

  • P is the principal amount of the loan
  • A is the payment per time period
  • r is the interest rate per time period (for monthly, it is annual rate divided by 12)
  • n is the number of time periods of the loan (for monthly, it is years * 12)

For your example, where you get a 4 year loan (48 months) at 4.7% rate (0.003917 monthly) and your monthly payment is $303, the formula above tells us that a loan of $13235.11 would result in those terms.

The amount of total interest and the total loan cost are very easy to calculate if you know the monthly payment. The total loan cost is simply the monthly payment times the number of months. The amount of interest is simply the total loan cost minus the principal of the loan (purchase price of the car after your downpayment). In your example, the total loan cost would be $14544, and if we subtract the $13235.11 principal of the loan, the interest you would have paid would be $1308.89.

If you are keeping the number of months and the purchase price of the car the same, switching to a biweekly or weekly payment would have a negligible effect on the total interest paid. Sometimes you will hear about switching to a biweekly payment on a mortgage being a way to save money on interest, but that is because with that scheme the borrower ends up making an additional payment every year, shortening the length of the loan.

  • Insurance premiums should be less than 0% of the buyer's income? – Hart CO Dec 7 '18 at 5:04
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    @HartCO That would be nice, wouldn't it? :) My 100/0/0 Rule (which I just made up, so it can be whatever I want), does not take insurance premium amounts into account. – Ben Miller Dec 7 '18 at 5:11
  • I wonder who comes up with these rules, I'd never heard of this 20/4/10 one before. I like the idea of not financing a car purchase, but last time I bought it was 0.9% interest on a used thing and their incentives to push financing meant they had more wiggle room on price if financing than if paying cash, very odd. – Hart CO Dec 7 '18 at 5:19
  • For x/y/0 to work with any finite income, the car must cost nothing. :) – Lawrence Dec 7 '18 at 11:19
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    @Lawrence The last number in my rule says that your loan payments needs to be no more than z% of your monthly income. If you have no loan payment, you are at 0%. – Ben Miller Dec 7 '18 at 12:07
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So far I know that the best interest rate I can get is 4.7% (annual whatever that means) for 4 years. And I know that I can make monthly payments of 303$ per month. So given this info, how much money would I have paid by the end of the 4 years, and how much of it would be lost paying off interest

The total Payment made is simple.
Total Payment = 303[Monthly Payments]*12[Months]*4[Years]= 14,544

The Loan Amount you will get can be easily determined by the PV Function in Excel.

  • PV = (Rate,Nper,Pmt,Fv,Type)
  • Rate = Rate for the Payment period. If 4.7% is yearly, then 4.7/12 Monthly; like wise if you are doing it fortnightly, it would be 4.7/24. or 4.7/52 for weekly.
  • Nper = Number of Payments. 12*4 for monthly. 24*4 for fortnightly and 52*4 for weekly.
  • Pmt = is the EMI you are paying, 303 for monthly, etc. Use a -ve sign to indicate you are payment so that PV is positive; you are getting money.
  • Fv and Type; Leave Blank.

Plugging in for Monthly;
PV = (4.7/12,12*4,303) = 13,235.11 - This is the Loan you are getting.

The total Interest paid is Total Payment - PV
Interest = 14,544 - 13,235 = 1,309

Like wise you can check for other methods, fortnightly or weekly.

Please note, this assumes that your Car Loan works truly on Monthly reducing and allows weekly / fortnightly payment schedules.

Generally Mortgages allow these. In Auto/Car Loan, generally its only Monthly. Some Financial Institutions would give you the flexibility of making weekly/Bi-Weekly payments but they will not get applied to the Loan immediately. Thus making this similar to Monthly payments.

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With

s = principal
r = periodic rate
d = periodic payment
n = number of periods

the basic loan formula is

s = (d - d (1 + r)^-n)/r

With an effective annual rate of 4.7% the periodic rates are

monthly,  rm = (1 + 0.047)^(1/12) - 1 = 0.00383474
biweekly, rb = (1 + 0.047)^(2/52.1775) - 1 = 0.00176204
weekly,   rw = (1 + 0.047)^(1/52.1775) - 1 = 0.000880632

assuming 52.1775 weeks per year for accuracy.

For various payment amounts the corresponding principals for 4 year loans are

monthly:  s = (303 - 303 (1 + rm)^-(4*12))/rm        = 13260.80
biweekly: s = (140 - 140 (1 + rb)^-(4*52.1775/2))/rb = 13334.48
weekly:   s = ( 70 -  70 (1 + rw)^-(4*52.1775))/rw   = 13340.35

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