Given a loan amount, time period, and annual interest rate, what is the formula for calculating the monthly payments?
For example, if I borrow a 100 grand (that’s a 100K for you kids) on a 15 year loan at 6% interest, what will my monthly payments be?
When I bought my house a few years ago and was shopping for loans, I asked a few banks’ loan officers this question. The answer was, “I don’t know. We look it up in a table or a computer program does it.” Unbelievable!
Now I’m asking this question because when I try to calculate it myself with a calculator, my answer doesn’t agree with my lender. However, the answer I get from a MSMoney program does agree with the lender, so I figure I must be the one making a mistake.
Also, if it’s a mortgage loan, you will need to add in the amount the lender holds out in your escrow account for things like property tax, insurance, etc. The total payment is what is called PITI (Principal, Interest, Taxes, Insurance).
Great. This matches my loan-calculating program output.
Of further interest is the questionable practice of using 0.06/12 as a “monthly” interest. This is strictly incorrect.
For instance, if I borrow $100,000 at 6% annual interest, make no payments at all, then after 1 year I should owe a total of
P x (1 + R )[sup]t[/sup]
= $100,000 x ( 1.06 )[sup]1[/sup]
= $106,000
However, if I use the “monthly” rate, we get
P x (1 + R/12)[sup]N[/sup]
= $100,000 x (1.005)[sup]12[/sup]
= $106,167.78
Clearly, it is to the advantage of the lender to use the “monthly” rate! But how do they get away with calling the above calculation “6% annual rate”? Is it just considered “common knowledge” that the above monthly calculation is what is really meant by 6% annual rate?
Now I know why my calculations came out wrong. I was basing my derivation of the monthly payment formula on the strict use of annual rate, not the “monthly” rate actually used.
Lord Vor -
There is no compounding of interest on a loan. You don’t pay interest on interest. This is different from savings account where the bank does pay you interest on your earned interest.
Francis E Dec, Esq -
I’m not quite sure what you calculated, but I don’ t think that is your total payout over the year.
If you paid the loan off in one year, you would have
M = 100,000 * (0.005/(1-(1+0.005)^-12) = 8606.64
After one year, you would have paid:
8606.64*12 = 103279.72
So the interest you would have paid would be 3,279.72.
Compare this with the 6,000 you’d pay if you made one payment at the end of the year.
Lord Vor does make an important point, the theory is the same. Most mortgage loans are based on a interest on “diminishing balance”. In other words you do not pay interest on the principal amount you have already paid back. As different from “simple interest” where the principal amount is constant throughout the period of the loan.
If you think this is confusing, it was put to good use in my home country (India) a few years ago where banks started quoting “flat rates” for loans that were about half that of the prime lending rate. The flat rate was the interest calculated on a the principal amount remaining constant throughout.
Francis, there are some very good formulas available in MS Excel where you can reference the formula to cells and change the various parameters to check for the best option. There are other formulas that separate the principal portion and the interest portion of each repayment, this helps calculate the equity you hold in the house at any point in time. I could send you the spreadsheet if you really want it.
At the risk of taking this into the intracies of different loan types:
“Simple interest” implies the interest you owe is determined by multiplying your outstanding principal times your rate times the number of periods (at that rate).
Of course most mortgages are based on “diminishing balance”. However, they are also mostly “simple interest” (as opposed to “amortized interest”). Your payment remains constant, not your principal. With each payment more and more goes to principal and less and less goes to interest.
When LordVor talks about compounding interest and e, he is remembering what happens with savings accounts. I state again: You are not charged interest on interest which is where e would be used. The lending institution can charge fees for late interest and payments, but they cannot charge you interest on the interest you owe.
I have no idea how you can calculate loan terms for a closed end loan where the principal does not diminish. Your principal must diminish in order to finally owe nothing. Can you provide more information on these loans? What does your amortization schedule look like on one of these?
I thought of a better way to clarify the (1+R/12)[sup]N[/sup] versus (1+R)[sup]N/12[/sup] question.
Suppose that the time frame for the loan is infinity. That should correspond to monthly payments exactly equal to the interest and such that you never make a dent in the principle.
So for M = P x (J / 1 - (1 + J)-[sup]N[/sup]) with N = infinity, we get simply
M = P x J
since J = R/12, in one year we would have paid
12M = P x R
which is just the desired answer, and annual interest of P x R.
I believe that the reason using J=R/12 works is precisely because, as cmosdes pointed out, you are not paying interest on interest.