How to calculate the number of months until a loan is paid off (given principal, APR and payment amount)?
The formula for determining the number of payments (months) you'll need to make on your loan is:  where i=monthly interest rate (annual rate / 12), A=loan amount (principal), and P=monthly payment. To determine the total interest that you will pay, you can use the following formula:  where P=monthly payment, N=number of payments (from above formula), and A=loan amount (principal). A quick example: using the numbers in the screenshot above ($10,000 loan, $500 monthly payment, 10% APR), the number of payments ends up to be 21.97 (which means that payment number 22 is slightly less than the rest).  In the second formula, you take that number times your $500 payment and determine that you have paid $10,984.81 over the course of the entire loan period.  Subtracting the principal, you have paid $984.81 in total interest. On your spreadsheet, the function you are looking for is NPER: NPER(rate, payment_amount, present_value, [future_value, end_or_beginning]) rate - The interest rate. (This should be the monthly rate, or the annual rate divided by 12.) payment_amount - The amount of each payment made. (For a loan payment, this should be a negative number.) present_value - The current value of the annuity. (The initial principal of the loan) future_value - [ OPTIONAL ] - The future value remaining after the final payment has been made.  (This should be 0, the default if omitted.) end_or_beginning - [ OPTIONAL - 0 by default ] - Whether payments are due at the end (0) or beginning (1) of each period. Here is the derivation of the formula, with The loan is equal to the sum of the repayments discounted to present value.
I don't understand the calculations in the comments by the OP. He says My monthly savings after mandatory expense is around USD 2000. This includes rent, expenses, emergency fund savings, and the monthly required payment of my auto loan.  (emphasis added) He has $2000 USD left over after monthly expenses (which includes rent, food, utilities etc, contribution towards emergency funds, and the required monthly payment on the auto loan). He claims that by applying the $2000 USD per month towards reducing the debt, it would take him 30-36 months to be debt-free.  But is it not the case that applying the $2000 to the student loan of $18K+ (while continuing to make the auto loan payments) will pay the student loan off in less than 10 months? If no payments are made on that $18K+ student loan, the accrued interest of about $2K in 10 months (this is (18.25*13.7%*)(10/12) for a total of $20K+). In actuality, with the loan being paid down, the interest will be much less. Once the student loan is paid off, the extra $2000 can go towards what is left of the $10K auto loan each month and  pay it off in another 4 or 5 months or so. So we are talking of 15 months max instead of 30-36 months. Of course, as Carlos Briebiescas points out, the car is more valuable as an asset than can be sold in case of job loss  creating  a need for cash etc, and so paying it off first might be better, but that is a different calculation. The second choice is a normal payment, just made early. This guards you against forgetting to make the payment later and incurring late payment fees -- which in this kind of loan are added to the balance and themselves accrue compounded interest. The first option is an extra payment, applied entirely to the principal. That lets you avoid years of accrued interest on that portion of the loan, and reduces the loan's actual cost. I think the extra payment is a better investment. The equation is the same one used for mortgage amortization.  You first want to calculate the PV (present value) for a stream of $50K payments over 20 years at a10% rate.  Then that value is the FV (future value) that you want to save for, and you are looking to solve the payment stream needed to create that future value. Good luck achieving the 10% return, and in knowing your mortality down to the exact year. Unless this is a homework assignment, which need not reflect real life.  Edit - as indicated above, the first step is to get that value in 20 years:  The image is the user-friendly entry screen for the PV calculation. It walks you though the need to enter rate as per period, therefore I enter .1/12 as the rate. The payment you desire is $50K/yr, and since it's a payment, it's a negative number. The equation in excel that results is: =PV(0.1/12,240,-50000/12,0) and the sum calculated is $431,769 Next you wish to know the payments to make to arrive at this number:  In this case, you start at zero PV with a known FV calculated above, and known rate. This solves for the payment needed to get this number, $568.59 The excel equation is: =PMT(0.1/12,240,0,431769) Most people have access to excel or a public domain spreadsheet application (e.g. Openoffice). If you are often needing to perform such calculations, a business finance calculator is recommended. TI used to make a model BA-35 finance calculator, no longer in production, still on eBay, used.  One more update- these equations whether in excel or a calculator are geared toward per period interest, i.e. when you state 10%, they assume a monthly 10/12%. With that said, you required a 20 year deposit period and 20 year withdrawal period. We know you wish to take out $4166.67 per month. The equation to calculate deposit required becomes - 4166.67/(1.00833333)^240= 568.59  HA! Exact same answer, far less work. To be clear, this works only because you required 240 deposits to produce 240 withdrawals in the future. This calculation arrives at the correct answer.  However, it uses the formula for an annuity due.  This means the payments are made at the beginning of the month and the last month of the 10 year period has interest accrued. See the section, Calculating the Future Value of an Annuity Due. The rate is given as an effective rate. with  In Excel, =FV((1+0.12)^(1/12)-1,120,3500,0,1) "I got $3394.83 The first problem with this is that it is backwards.  The NPV (Net Present Value) of three future payments of $997 has to be less than the nominal value.  The nominal value is simple:  $2991.   First step, convert the 8% annual return from the stock market to a monthly return.  Everyone else assumed that the 8% is a monthly return, but that is clearly absurd.  The correct way to do this would be to solve for m in  But we often approximate this by dividing 8% by 12, which would be .67%.  Either way, you divide each payment by the number of months of compounding.   Sum those up using m equal to about .64% (I left the calculated value in memory and used that rather than the rounded value) and you get about $2952.92 which is smaller than $2991.   Obviously $2952.92 is much larger than $2495 and you should not do this.  If the three payments were $842.39 instead, then it would about break even.   Note that this neglects risk.  In a three month period, the stock market is as likely to fall short of an annualized 8% return as to beat it.  This would make more sense if your alternative was to pay off some of your mortgage immediately and take the payments or yp pay a lump sum now and increase future mortgage payments.  Then your return would be safer.   Someone noted in a comment that we would normally base the NPV on the interest rate of the payments.  That's for calculating the NPV to the one making the loan.  Here, we want to calculate the NPV for the borrower.  So the question is what the borrower would do with the money if making payments and not the lump sum.   The question assumes that the borrower would invest in the stock market, which is a risky option and not normally advisable.  I suggest a mortgage based alternative.  If the borrower is going to stuff the money under the mattress until needed, then the answer is simple.  The nominal value of $2991 is also the NPV, as mattresses don't pay interest.  Similarly, many banks don't pay interest on checking these days.  So for someone facing a real decision like this, I'd almost always recommend paying the lump sum and getting it over with.  Even if the payments are ""same as cash"" with no premium charged." Add a few more cells to your header that list the interest paid in in the next 3 to 4 years on your current mortgage. Use the cumulative interest function from your spreadsheet program. In the main body of your spreadsheet, add columns that summarize the total cost over 3-4 years for each loan. Add columns that list the interest cost, total cost of interest + refi cost, and the difference between that approach and the interest costs from your current loan. Add 6 columns total: a set for 3 years and a set for 4.  Something like this: Repeat that 3 year block off to the right and plug in the 4 year numbers.  You requested that we factor in a 3.5% penalty against the money that goes to the discount fee. You could do that by adding a column that calculates this, like Joe described in his answer. Add that 3.5% accrual into the total calculation above, which in turn will knock down the amount of savings for each refi loan. PS: How are you going to earn 3.5% over 48 months? A lease payment is composed of an interest portion (borrowed money) and depreciation amount (purchase - residual). The Monthly payment is then Monthly Interest Cost + Monthly Depreciation Cost The Money Factor is used to estimate the amount of interest due in a single month of a lease so you can figure out the monthly payment. If you are borrowing $100,000 then over the entire loan of repayment from a balance of $100,000 to a balance of $0, the average amount you owed was $50,000 (1/2 of principal).   You are repaying this loan monthly (1/12 of a year) and percents are expressed as decimals (1/100). 6 * 1/2 (for principal) * 1/12 (for monthly) * 1/100 (to convert percentage from 6% to .06)  = 6 * 1/2400. 2400 is the product of 3 consecutive conversion (1/2 * 1/12 * 1/100) to convert from an interest rate to a money factor. 6/2400 = Money factor of 0.0025 which can be multiplied against the total amount being borrowed to know what the monthly interest would roughly equal. Some Money Factor info: https://www.alphaleasing.com/resources/articles/MoneyFactor.asp You're looking for the amortization calculation. This calculation is essentially solving for the series of cash flows that will yield a zero balance after the specified term, at the given rate for a loan amount. Once you have solved for the payment amount, you can add additional principal amount to each payment period and see the interest payments and time to zero balance drop. "The other answers have touched on amortization, early payment, computation of interest, etc, which are all very important, but I think there's another way to understand the importance of knowing the P/I breakdown. The question mentions the loan payment as ""cash outflow"". That is true, but from an accounting perspective (disclaimer: I am not an accountant, but I know enough of the basics to be dangerous), the outflow needs to be directed to different accounts. The loan principal appears as a liability on your personal balance sheet, which you could use, for example, in determining net worth. The principal amount in your payment should be applied to reduce the liability account. The interest payment goes into the expense account.  Another way to look at it is that the principal, while it does reduce your cash account, can be thought of as an internal transfer to the liability account, thus reducing the size of the liability. The interest payment cannot. Aside: From this perspective, the value of the home is an asset, and the difference between the asset account and the loan liability account is the equity in the house (as pointed out in different language by the accepted answer). Of course, precisely determining the value of an illiquid asset like a house at any given moment pretty much requires you to actually sell it, so those accounts are hard to maintain in real-time (the liability of the loan is much easier to track)." "The monthly bill should reduce as required by Loan A no longer requiring payment. This will occur only when Loan A is fully payed off, not before. If you're going to do this, make sure you tell them that any extra money is principal reduction, and not ""prepayment""  Lets say you do pay off loan A, and you continue to pay $11 a month.  If you specify ""principle reduction"" for the $1 extra, they must reduce the loan balance by $1.  If you do not specify, or you specify ""prepayment"", they ""may"" apply $0.20 to principal reduction and $0.80 to interest." Using the following loan equations where  and With the balance b[n] in period n given by Applying the OP's figures Check & demonstration Switching to $96 payment every 10 days, with 365.2422 days per year  Paying $96 every 10 days saves $326.85 and pays the loan down 2.68 months quicker. Edited to incorporate the comments elsewhere of @Atkins Assuming, (apparently incorrectly) that duration is time to maturity: First, note that the question does not mention the coupon rate, the size of the regular payments that the bond holder will get each year.  So let's calculate that. Consider the cash flow described.  You pay out 1015 at the start of Year #1, to buy the bond.  At the end of Years #1 to #5, you receive a coupon payment of X.  Also at the end of Year #5, you receive the face value of the bond, 1000.  And you are told that the pay out equals the money received, using a time value of money of 4.69% So, if we use the date of maturity of the bond as our valuation date, we have the equation: Maturity + Future Value of coupons = Future value of Bond Purchase price 1000 + X *( (1 + .0469)^5-1)/0.0469 = 1015 * 1.0469^5 Solving this for X, we obtain 50.33;  the coupon rate is 5.033%.  You will receive 50.33 at the end of each of the five years.   Now, we can take this fixed schedule of payments, and apply the new yield rate to the same formula above;  only now, the unknown is the price paid for the bond, Y. 1000 + 50.33 * ((1 + 0.0487)^5 - 1) / .0487 = Y * 1.0487^5 Solving this equation for Y, we obtain: Y = 1007.08 First of all any loan will have the last payment be slightly different than the rest. Even if the interest rate is zero it is hard to have a perfect monthly payment amount. By perfect I mean the amount of the loan divided by the number of months would be $ddd.cc0000...  When looking at the amortization table the last payment will either be slightly higher or lower to adjust for the rounding that was done. My suspicion is that the rounding would have resulted in a left over 12 cents. It is also possible that you are paying it off slightly early and the computer is simply taking the leftover amount an spreading it over the remaining period of the loan. You could be paying it early because for the main part of the loan was paid to a company that rounds all payments to the higher dollar, or has a minimum monthly payment amount. I looked at the company you mentioned in the question, and searched the site for an amortization tool. I found it, and used the pre-filled in example. https://www.edfinancial.com/amortizationschedule?pmts=120&intr=6.8&prin=25000 If that last payment is not adjusted the borrower will still owe $0.12. The fact it equals your situation is a coincidence. But is does show what can happen. Easier to copy paste than type this out. Credit: www.financeformulas.net Note that the present value would be the initial loan amount, which is likely the sale price you noted minus a down payment.   The loan payment formula is used to calculate the payments on a loan.   The formula used to calculate loan payments is exactly the same as the   formula used to calculate payments on an ordinary annuity. A loan, by   definition, is an annuity, in that it consists of a series of future   periodic payments. The PV, or present value, portion of the loan payment formula uses the   original loan amount. The original loan amount is essentially the   present value of the future payments on the loan, much like the   present value of an annuity. It is important to keep the rate per period and number of periods   consistent with one another in the formula. If the loan payments are   made monthly, then the rate per period needs to be adjusted to the   monthly rate and the number of periods would be the number of months   on the loan. If payments are quarterly, the terms of the loan payment   formula would be adjusted accordingly. I like to let loan calculators do the heavy lifting for me. This particular calculator lets you choose a weekly pay back scheme. http://www.calculator.net/loan-calculator.html "A major thing to consider when deciding whether to invest or pay off debt is cash flow. Specifically, how each choice affects your cash flow, and how your cash flow is affected by various events. Simply enough, your cash flow is the amount of money that passes through your finances during a given period (often a month or a year). Some of this is necessary payments, like staying current on loans, rent, etc., while other parts are not necessary, such as eating out. For example, you currently have $5,500 debt at 3% and another $2,500 at 5%. This means that every month, your cashflow effect of these loans is ($5,500 * 3% / 12) + ($2,500 * 5% / 12) = $24 interest (before any applicable tax effects), plus any required payments toward the principal which you don't state. To have the $8,000 paid off in 30 years, you'd be paying another $33 toward the principal, for a total of about $60 per month before tax effects in your case. If you take the full $7,000 you have available and use it to pay off the debt starting with the higher-interest loan, then your situation changes such that you now: Assuming that the repayment timeline remains the same, the cashflow effect of the above becomes $1,000 * 3% / 12 = $2.50/month interest plus $2.78/month toward the principal, again before tax effects. In one fell swoop, you just reduced your monthly payment from $60 to $5.25. Per year, this means $720 to $63, so on the $7,000 ""invested"" in repayment you get $657 in return every year for a 9.4% annual return on investment. It will take you about 11 years to use only this money to save another $7,000, as opposed to the 30 years original repayment schedule. If the extra payment goes toward knocking time off the existing repayment schedule but keeping the amount paid toward the principal per month the same, you are now paying $33 toward the principal plus $2.50 interest against the $1,000 loan, which means by paying $35.50/month you will be debt free in 30 months: two and a half years, instead of 30 years, an effective 92% reduction in repayment time. You immediately have another about $25/month in your budget, and in two and a half years you will have $60 per month that you wouldn't have if you stuck with the original repayment schedule. If instead the total amount paid remains the same, you are then paying about $57.50/month toward the principal and will be debt free in less than a year and a half. Not too shabby, if you ask me. Also, don't forget that this is a known, guaranteed return in that you know what you would be paying in interest if you didn't do this, and you know what you will be paying in interest if you do this. Even if the interest rate is variable, you can calculate this to a reasonable degree of certainty. The difference between those two is your return on investment. Compare this to the fact that while an investment in the S&P might have similar returns over long periods of time, the stock market is much more volatile in the shorter term (as the past two decades have so eloquently demonstrated). It doesn't do you much good if an investment returns 10% per year over 30 years, if when you need the money it's down 30% because you bought at a local peak and have held the investment for only a year. Also consider if you go back to school, are you going to feel better about a $5.25/month payment or a $60/month payment? (Even if the payments on old debt are deferred while you are studying, you will still have to pay the money, and it will likely be accruing interest in the meantime.) Now, I really don't advocate emptying your savings account entirely the way I did in the example above. Stuff happens all the time, and some stuff that happens costs money. Instead, you should be keeping some of that money easily available in a liquid, non-volatile form (which basically means a savings account without withdrawal penalties or a money market fund, not the stock market). How much depends on your necessary expenses; a buffer of three months' worth of expenses is an often recommended starting point for an emergency fund. The above should however help you evaluate how much to keep, how much to invest and how much to use to pay off loans early, respectively."