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Is there a rough, quick and dirty, formula that I can do in my head for calculating the total interest on a loan? Something like:

TotalInterest = Principal * (Yearly Interest% * Term In Years / 2);

Extra points for giving details for margins of error.

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Your calculation is actually about as good as you're going to get using only simple operations (add, subtract, multiply, divide). My calculations indicate that a value of 9/16ths would be somewhat more accurate than the 1/2 you use, but that may or may not be worth the additional complexity. –  stannius Jan 30 '13 at 1:01

3 Answers 3

up vote 4 down vote accepted

What you show is good for short term loans, it underestimates by less than 7% since the amortization is not linear. $12K loan, 5 years, 6% (a car loan, say). Your equation shows $1800 interest while the true number is $1919. Not bad for a calculation done on your fingers or napkin.

Edit to answer comment below - The error increases as the rate rises and the term lengthens. e.g. at zero interest, a 30 year loan is paid linearly, but as the rate rises, the amortization chart bows north. At 1%, a 30 year loan of $100K has $15,790 total interest, vs your equation $15,000 (darn close!) but at 4%, it's $71,869 vs $60,000 (now a near-20% difference!)

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So then for a longer term loan will the error increase? –  Matt Parkins Jan 24 '13 at 14:30
    
Yes, edit in above. By the way, you can create a spreadsheet to compare your equation to actual, then eyeball the correction factor and get pretty close to actual right in your head. –  JoeTaxpayer Jan 24 '13 at 15:52
    
This of course assumes fixed rate and fixed payment. There are all kinds of different loan types... –  littleadv Jan 24 '13 at 18:56
    
@littleadv - I thought to mention that, then thought it too obvious, but I'll just agree with you, here. An ARM would certainly mess things up. Matt's equation is great for car loans, the shorter the better, and fixed rate. –  JoeTaxpayer Jan 24 '13 at 19:09
    
ARM compound balloon loan. Now go estimate:) –  littleadv Jan 24 '13 at 19:11

The Rule of 78 would suggest that using

TotalInterest = Principal * (Yearly Interest% * (Term In Years + 1)/ 2)

instead of

TotalInterest = Principal * (Yearly Interest% * Term In Years / 2);

will give an upper bound on the total interest since the Rule or 78 computation definitely favors the lender if the loan is paid off early. In this instance, the average of the simple interest calculation ($1800) that you provide and the Rule of 78 calculation ($2160) is $1980 which is a closer approximation, though still an upper bound, to the exact figure obtained by more detailed calculation.

For what it is worth, Wikipedia says that US law forbids the use of the Rule of 78 for mortgage refinancing and consumer loans exceeding 61 months, but there is no reason why one cannot use it for back of the envelope calculations.

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How is the Rule of 78's relevant here? The rule of 78's is applied to the total interest after it is calculated, so you have to know the total interest as in input to it. You could use it to estimate the interest paid to date on a loan, once you know the total interest. –  stannius Jan 30 '13 at 0:10
    
With principal amount $P, annual interest rate I%, and variable monthly payment equal to $P/12 principal repayment plus (I/12)% interest on balance owing, the loan is paid off at the end of the year, and the total interest paid is exactly ($P)*(I%)*(6/12). This is an underestimate of the actual interest paid when the loan is paid off via equal monthly payments. The Rule of 78 estimates the total interest paid via equal monthly payments as ($P)*(I%)*((78/12)/12) = ($P)(I%)*(6.5/12) which is an overestimate. Note that 6.5/12 = 0.54166... not too different from your 9/16 = 0.5625. –  Dilip Sarwate Jan 30 '13 at 2:53

The true calculation is:

iA/[1 - (1+i)^-N]*N-A

Where

  • N = term (number of payments, multiplied e.g. by 12 in the case of a monthly payment)
  • i = interest rate per term (divided by 12 for monthly payment)
  • A = starting principal

Unfortunately with the Nth root in there, small changes in the inputs can cause large swings in the output (especially when a 30 year mortgage has an N of 360). I haven't been able yet to figure out a simple way to work around that.

Perhaps we can work from the Rule of 72 to get a rough estimate (which as I commented above, works out to pretty much what you've already figured). If this were an investment, you'd divide 72 by the interest rate times 100 to get the number of years until it doubled, with compound interest. Then, we can take that number, and divide the term of the loan in years by it, to make a guess at how many times it would double in the given time period. Then, multiply it by the starting balance. That would give:

(Y/72/(i*100))*A

Simplify the fractions and you get

Y * I * A * 100/72

If you calculate this out, you'll quickly see that it significantly overestimates the interest (about double). This makes sense, since you pay the principle down over time, lowering your interest. The calculation above would (roughly) hold if you never paid any payments and just let the balance grow (ignoring penalties and forclosure/reposession :) However, I did the calculation in a spreadsheet against a grid of values, varying both term and interest rate, and found that it tracked pretty well against the true value as calculated above. You can correct for that using an additional constant. Varying term from 1 to 30 years and rate from 4% to 15%, I found that the constant varied from 37 to 52, with an average of about 41 (also, the higher value was for things like 12% interest for 30 years). 41/72 = 0.57, which is fairly close to the 0.5 you use in your own estimating formula, but IMHO you could make it more accurate by using a constant of 0.6.

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Thanks for this - everyone's head is different, but I'm going to struggle to do this one in my head. Thanks for answering though! –  Matt Parkins Jan 29 '13 at 10:33
    
Wish I could +1 this more than once. –  Matt Parkins Jan 30 '13 at 11:18

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