Calculate APR for under 1 year loan
Is the pay cycle every 2 weeks? So 30% each two week period is 1.3^26 = 917.33 or an APR of 91633%.  Loansharks charge less, I believe standard vig was 2%/week for good customers. Only 180% per year.
Part 2 = 12%  -5000/0.12 (1 - 1/(1.12^2 ) = -8450.26 Then subtract year 0s 5000 payment if you havent = -13450.26 and then compare to option 2 -2000/0.12 (1 - 1/(1.12^10) = -11300.46 and subtract 20000 -31300.46  As you can see both values are substantially smaller compared to a 4% discount rate. "Assuming the numbers you gave are forecasted 2013 annual income, you should really use an average and give the lender 1 number, as long as you can provide documentation to back it up. Lenders aren't as sophisticated as considering your monthly income fluctuations into their underwriting algorithm.  If you're not tied down to your existing lender, I highly recommend you to shop around. There isn't an ""universal lending requirement"". You'll be surprised at how flexible they are. Not as a recommendation to get around the rules, but just finding a lender that'll work with your situation. Try personal finance forums such as FatWallet or Slickdeal to find low-cost lenders: http://goo.gl/vIojT" You need the Present Value, not Future Value formula for this. The loan amount or 1000 is paid/received now (not in the future). The formula is $ PMT = PV (r/n)(1+r/n)^{nt} / [(1+r/n)^{nt} - 1] $  See for example http://www.calculatorsoup.com/calculators/financial/loan-calculator.php With PV = 1000, r=0.07, n=12, t=3 we get PMT = 30.877 per month 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 question states :- Our insurance company is offering a 30% discount on an $8200/year   commercial policy, if we install sprinklers. The insurance is paid in   two installments. ... This appears to mean six-monthly payments, so I'll make some comparison calculations using six-monthly loan repayments to keep things simple. Without the loan or sprinklers the insurance costs $4100 every six months. Using this loan payment formula, the calculation below shows, with the 30% discounted insurance, sprinkler maintenance and loan repayment, you would be paying $4655.28 every six months.  The discount required to break even is 43.5%.  I.e. rearranging the equation :-  Alternatively, with the discount of 30% you would break even if the six-monthly repayment amount was $1030.  Solving the payment equation for s gives an equation for the loan :-  So with the 30% discount you would break even if the loan required was $25989. Checking by back-calculating the periodic payment amount, a :-  Likewise we can keep the loan at $40000 and solve for t to find the break-even loan term :-  (Note, in this formula Log denotes the natural logarithm.)  Now we can set some values :-  So with break-even payments the $40000 loan is paid off in just under 65.5 years.  I.e. checking :-  This just beats the $4100 cost of proceeding without the sprinklers. Notes If your loan repayment was monthly it would reduce the cost of the loan slightly.  The periodic interest rate is calculated from the APR according to the method used in the EU and in some cases in US.  The calculations above were run using Mathematica. The 1.140924% is calculated by taking 13.69%/12 = 1.140924%. Dividing this number by 100 gives you the answer 1.140924 / 100 = .01140924. When dealing with decimals it's important to remember the relationship between a decimal and a percent. 1% = .01 To return .01 to a percent you must multiple that number by 100.  So .01 x 100 = 1% In order to get a decimal from a percent, which is what is used in calculations, you must divide by 100. So, here if we are trying to calculate how much interest you are paying each month we can do this: 9800 * .1369 = $1341.62 (interest you will pay that year IF the principal balance never changed) 1341.62 / 12 = ~111.81 Now, month two 9578.34 * .1369 = 1311.274746 1311.274746 / 12 = 109.28 In order to get your monthly payments (which won't change) for the life of the loan, you can use this formula: Monthly payment = r(PV) / (1-(1+r)^-n) Where:  r= Interest Rate (remember if calculating monthly to do .1369/12) PV= Present Value of loan n=time of loan ( in your case 36 since we are talking monthly and 12*3 = 36) from here we get: [(.1369/12)*9800]/(1-(1+.1369/12)^-36) = $333.467 when rounding is $333.47 As far as actual applied interest rate, I'm not even sure what that number is, but I would like to know once you figure out, since the interest rate you're being charged is most definitely 13.69%. Question is, what is this number 0.01140924 13.69/12=0.01140924 In addition, how does one come out with the EIR as 13.69% pa? When calculating payments, PV = 9800, N=36 (months), PMT=333.47, results in a rate of 1.140924% per period, and rate of 13.69%/yr. No idea how they claim 7.5% In Excel, type =RATE(36,333.47,-9800,0,0) And you will get 1.141% as the result.  36 = #payments, 333.47 = payment per period, -9800 is the principal (negative, remember this) And the zeros are to say the payments are month end, second zero is the guess.  Edit - I saw the loan is from a Singapore bank. It appears they have different rules on the rates they quote. As quid's answer showed the math, here's the bank's offer page -  The EIR is the rate that we, not just US, but most board members, are used to. I thought I'd offer an example using a 30 year mortgage.   Yo can see above, a 6% fixed rate somehow morphs into a 3.86% AR. No offense to the Singapore bankers, but I see little value in this number. What surprises me most, is that I've not seen this before.  What's baffling is when I change a 15yr term the AP drops to less than half. It's still a 6% loan and there's nothing about it that's 2 percent-ish, in my opinion.  Now we know. "I am trying to set up a formula that will find interest-rate behind first pencil sheets at car dealerships. I am not a car finance expert so I need someone who is intimate with how these loans really work.   The points of data I get at first pencil are: 1) Amount Financed 2) Period 3) Monthly Payment  The data I need to extrapolate: 1) Rate in percentile so that I can compare to my bank's offer.  I have tried this and many other stock ""find rate"" formulas with no accurate results: R=(A/P^(1/n)-1)n" "To calculate the balance (not just principal) remaining, type into your favorite spreadsheet program: It is important that the periods for ""Periods"" and ""Rate"" match up.  If you use your annual rate with quarterly periods, you will get a horribly wrong answer. So, if you invest $1000 today, expect 6% interest per year (0.5% interest per month), withdraw $10 at the end of each month, and want to know what your investment balance will be 2 years (24 months) from now, you would type: And you would get a result of $872.84. Or, to compute it manually, use the formula found here by poster uart:   This is often taught in high-school here as a application of geomentric series. The derivation goes like this.  Using the notation :  r = 1 + interest_rate_per_term_as_decimal p = present value a = payment per term eot1 denotes the FV at end of term 1 etc.  eot1: rp + a   eot2: r(rp + a) + a = r^2p + ra + a   eot3: r(r^2p + ra + a) + a = r^3p + r^2a + ra + a ...   eotn: r^np + (r^(n-1) + r^(n-2) + ... 1)a = p r^n + a (r^n - 1)/(r-1)  That is, FV = p r^n + a (r^n - 1)/(r-1).   This is precisely what exel [sic] computes for the case of payments made at the end of each term (payment type = 0). It's easy enough to repeat the calculations as above for the case of payments made at the beginning of each term. This won't work for changing interest rates or changing withdrawal amounts.  For something like that, it would be better for you (if you don't want online calculators) to set up a table in a spreadsheet so you can adjust different periods manually." I have answered your question in detail here https://stackoverflow.com/questions/12396422/apr-calculation-formula The annuity formula in FDIC document is at first finding PVIFAD present value annuity due factor and multiplying it with annuity payment and then dividing it by an interest factor of (1+i) to reduce the annuity to an ordinary annuity with end of period payments They could have simply used PVIFA and multiplying it with annuity payment to find the present value of an ordinary annuity In any case, you should not follow the directions in FDIC document to find interest rate at which the present value of annuity equals the loan amount. The method they are employing is commonly used by Finance Professors to teach their students how to find internal rate of return. The method is prone to lengthy trial and error attempts without having any way of knowing what rate to use as an initial guess to kick off the interest rate calculations So this is what I would suggest if you are not short on time and would like to get yourself familiar with numerical methods or iterative techniques to find internal rate of return There are way too many methods at disposal when it comes to finding interest rates some of which include All of the above methods use a seed value as a guess rate to start the iterative calculations and if results from successive calculations tend to converge within a certain absolute Error bound, we assume that one of the rates have been found as there may be as many rates as the order of the polynomial in this case 36 There are however some other methods that help find all rates by making use of Eigenvalues, but for this you would need a lengthy discourse of Linear Algebra One of the methods that I have come across which was published in the US in 1969 (the year I was born :) ) is called the Jenkins Traub method named after the two individuals who worked jointly on finding a solution to all roots of a polynomial discarding any previous work on the same subject I been trying to go over the Jenkins Traub algorithm but am having difficulty understanding the complex nature of the calculations required to find all roots of the polynomial In summary you would be better of reading up on this site about the Newton Raphson method to find IRR The reason it's broken out is very specific: this is showing you how much interest accrued during the month.  It is the only place that's shown, typically. Each month's (minimum) payment is the sum of [the interest accrued during that month] and [some principal], say M=I+P, and B is your total loan balance.  That I is fixed at the amount of interest that accrued that month - you always must pay off the accrued interest.  It changes each month as some of the principal is reduced; if you have a 3% daily interest rate, you owe (0.03*B*31) approximately (plus a bit as the interest on the interest accrues) each month (or *30 or *28).  Since B is going down constantly as principal is paid off, I is also going down. The P is most commonly calculated based on an amortization table, such that you have a fixed payment amount each month and pay the loan off after a certain period of time.  That's why P changes each month - because it's easier for people to have a constant monthly payment M, than to have a fixed P and variable I for a variable M. As such, it's important to show you the I amount, both so you can verify that the loan is being correctly charged/paid, and for your tax purposes. Unless you are getting the loan from a loan shark, it is the most common case that each payment is applied to the interest accrued to date and the rest is applied towards reducing the principal.  So, assuming that fortnightly means 26 equally-spaced payments during the year, the interest accrued at the end of the first fortnight is $660,000 x (0.0575/26) = $1459.62 and so the principal is reduced by $2299.61 - $1459.62 = $839.99 For the next payment, the principal still owing at the beginning of that fortnight will be $660,000-$839.99 = $659,160.01 and the interest accrued will be $659,160.01 x (0.0575/26) = $1457.76 and so slightly more of the principal will be reduced than the $839.99 of the previous payment.  Lather, rinse, repeat until the loan is paid off which should occur at the end of 17.5 years (or after 455 biweekly payments). If the loan rate changes during this time (since you say that this is a variable-rate loan), the numbers quoted above will change too. And no, it is not the case that just %5.75 of the $2300 is interest, and the rest comes off the principle (sic)? Interest is computed on the principal amount still owed ($660,000 for starters and then decreasing fortnightly). not the loan payment amount. Edit  After playing around with a spreadsheet a bit, I found that if  payments are made every two weeks (14 days apart) rather than 26 equally spaced payments in one year as I used above, interest accrues at the rate of 5.75 x (14/365)% for the 14 days rather than at the rate of (5.75/26)% for the time between payments as I used above each 14 days, $2299.56 is paid as the biweekly mortgage payment instead of the $2299.61 stated by the OP, then 455 payments (slightly less than 17.5 calendar years when leap years are taken into account) will pay off the loan.  In fact, that 455-th payment should be reduced by 65 cents.  In view of rounding of fractional cents and the like, I doubt that it would be possible to have the last equal payment reduce the balance to exactly 0. At time = 0, no interest has accrued. That's normal. And the first payment is due after a month, when there's a month's interest and a bit of principal due.  Note - I missed weekly payments. You'd have to account for this manually, add a month's interest, then calculate based on weekly payments. In my opinion, the simplest way to run these numbers is to first assume you are borrowing the full amount, including the points, if any.  They run a spreadsheet, and while using the new rate, apply your full current payment each month.  Then compare balances at month 48. You'll find it easy to calculate the breakeven. In the case of the negative points, it's immediate. For higher points, the B/E is later but then you are further ahead each month. Illustrating with a shorter example:  Suppose I deposit 1,000 USD. Every year I deposit another 100 USD. I want to know how much money will be on that savings account in 4 years.  The long-hand calculation is Expressed with a summation  And using the formula derived from the summation (as shown by DJohnM) So for 20 years Note in year 20 (or year 4 in the shorter example) the final $100 deposit does not have any time to accrue interest before the valuation of the account.