Having trouble with APR calculation
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 Check your calculation of A**. I was able to duplicate their calculations using excel. Make you sure have accounted for all the terms, it can be easy to be one off. They are making a guess at the interest rate which will be wrong, then they are adjusting it to see how wrong it is, then making another adjustment. They will repeat until they see no movement in the guesses.
The formula you need is:    M = (r * PV) / (1 - ((1+r)^(-n)))    M = monthly payment ($350)  r = interest per period (7.56% / 12) = 0.63%  n = number of periods (36 months)  PV = present value, or here, your max loan amount given M    Therefore:  $350 = (0.63% * PV) / (1 - ((1+0.63%)^(-36)))   The denominator on the right ends up equal to ~ 0.2025 when you do the math in your calculator. Carry that over to the by multiplying both sides of the equation by 0.2025    This results in $70.82 = 0.63% * PV    Divide $70.82 by 0.63% to get PV = $11,242 (roughly).    Hope this helps explain it algebraically!! Interest is calculated daily. Doing the math: Between 6-17 and 7-25 are 38 days, 200.29 / 38 = 5.27 interest per day. Between 7-25 and 8-17 are 23 days. 120.02 / 23 = 5.22 interest per day. The minimal difference is because the principal has already gone down a little bit. So you should expect ~5.20 x number of days for the next interest number coming up; slowly decreasing as the remaining principal debt decreases. Note that this is equivalent of an annual interest rate of over 20 %, which is beyond acceptable. In the current economy, this is ridiculously high. I recommend trying to get a refinancing with another provider; you should be able to get it for a third of that. This is a clear example of annuities, where you are trying to find Net Present Value (NPV). To find a quick solution you can use the excel function (=NPV). Solving it without excel is slightly more complicated. You have to use the annuities formula:  (Payment/rate) * (1 - 1/(1+r)^n )  This formula can be written in many different ways, this is one of the simplest.  So how do we translate that to your problem? For the first option: -5000/0.04 * (1 - 1/(1.04^2 ) = -9430.47 If you haven't made this years 5000 payment you have to subtract this years 5000, so option 1 would equal -14430,47.  Option 2 is similar -2000/0.04 (1 - 1/(1.04^10 ) = -16221.79 WAIT! You also have to subtract the initial cost so option 2 is -36221.79  So in essence, option 1 is better (but you will have to buy a machine sooner or later).  Part 2 (discount rate = 12%) is for you to understand the importance of high rates in long periods of time (like option 2's 10 years)    EDIT: For clearer formulas. Also made a mistake calculating option 2 In this case, it looks like the interest is simply the nominal daily interest rate times number of days in the period. From that you can use a spreadsheet to calculate the total payment by trial and error. With the different number of days in each period, any formula would be very complicated. In the more usual case where the interest charge for each period is the same, the formula is: m=P*r^n*(r-1)/(r^n-1) where * is multiplication ^ is exponentiation / is division (Sorry, don't know if there's a way to show formulas cleanly on here) P=original principle r=growth factor per payment period, i.e. interest rate + 100% divided by 100, e.g. 1% -> 1.01 n=number of payments Note the growth factor above is per period, so if you have monthly payments, it's the rate per month. The last payment may be different because of rounding errors, unequal number of days per period, or other technicalities. Using that formula here won't give the right answer because of the unequal periods, but it should be close. Let's see: r=0.7% times an average of 28.8 days per period gives 20.16% + 1 = 1.2016. n=5 P=500 m=500*1.2016^5*(1.2016-1)/(1.2016^5-1) =167.78 Further off than I expected, but ballpark. The calculation can be made on the basis that the loan is equal to the sum of the repayments discounted to present value.  (For more information see Calculating the Present Value of an Ordinary Annuity.) With Deriving the loan formula from the simple discount summation.  As you can see, this is the same as the loan formula given here. In the UK and Europe APR is usually quoted as the effective interest rate while in the US it is quoted as a nominal rate.  (Also, in the US the effective APR is usually called the annual percentage yield, APY, not APR.) Using the effective interest rate finds the expected answer. The total repayment is £30.78 * n = £1108.08 Using a nominal interest rate does not give the expected answer. 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. ADP does not know your full tax situation and while the standard exemption system (actually designed by the IRS not ADP) works fairly well for most people it is an approximation.  This system is designed so most people will end up with a small refund while some people will end up owing small amounts.  So, while it is possible that ADP has messed up the calculations it is unlikely this is the cause.  The most likely cause is that approximation ends ups making you pay less tax during the year than you actually owe. A few people like your friend may end up owing large amounts due to various circumstances.  It is always your responsibility to make sure you pay enough tax throughout the year.  While this technically means that you need to do your taxes every quarter during the year to make sure you pay the correct tax during the year, for most people this ends up being unnecessary as the approximation works fine.  It is possible the exemption system failed your friend, but much more commonly people owe penalties because they put the wrong number of exemptions or had other side income. On a related note, most people in finance would argue that your situation where you owe some money at tax time, but not so much that you have to pay a penalty, is actually the best way to go.  Getting a tax refund actually means you paid more tax than you needed to.  This is similar to giving an interest-free loan to the government. "There is no formula that can be applied to most variations of the problem you pose.  The reason is that there is no simple, fixed relationship between the two time periods involved: the time interval for successive payments, and the time period for successive interest compounding. Suppose you have daily compounding and you want to make weekly payments (A case that can be handled).  Say the quoted rate is  4.2% per year, compounded daily Then the rate per day is 4.2/365, or 0.0115068 % So, in one week, a debt would grow through seven compoundings.  A debt of $1 would grow to 1 * (1+.000225068)^7, or 1.000805754 So, the equivalent interest rate for weekly compounding is 0.0805754% Now you have weekly compounding, and weekly payments, so the standard annuity formulas apply. The problem lies in that number ""7"", the number of days in a week.  But if you were trying to handle daily / monthly, or weekly / quarterly, what value would you use? In such cases, the most practical method is to convert any compounding rate to a daily compounding rate, and use a spreadsheet to handle the irregularly spaced payments." The equation for the payment is  This board does not support Latex (the number formatting code) so the above is an image, the code is M is the payment calculated, n is the number of months or periods to pay off, and i is the rate per period. You can see that with i appearing 3 times in this equation, it's not possible to isolate to the form i=....  so a calculator will 'guess,' and use, say, 10%. It then raises or lowers the rate until the result is within the calculator's tolerance. I've observed that unlike other calculations, when you hit the button to calculate, a noticeable time lag occurs. I hope I haven't read too much into your question, it seemed to me this was what you asked. TWRR = (2012Q4 x 2013Q1 x 2013Q2) ^ (1/3) = ?? (1.1 * .809 * 1.29) ^ (1/3) = 1.047 or 4.7% return. No imaginary numbers needed.  But. Your second line there is wrong $15,750 - $15,000 - $4,000 ? The $15K already contains the $4k, why did you subtract it again?  This a homework problem? "Wow, this turns out to be a much more difficult problem than I thought from first looking at it. Let's recast some of the variables to simplify the equations a bit. Let rb be the growth rate of money in your bank for one period. By ""growth rate"" I mean the amount you will have after one period. So if the interest rate is 3% per year paid monthly, then the interest for one month is 3/12 of 1% = .25%, so after one month you have 1.0025 times as much money as you started with. Similarly, let si be the growth rate of the investment. Then after you make a deposit the amount you have in the bank is pb = s. After another deposit you've collected interest on the first, so you have pb = s * rb + s. That is, the first deposit with one period's growth plus the second deposit. One more deposit and you have pb = ((s * rb) + s) * rb + s = s + s * rb + s * rb^2. Etc. So after n deposits you have pb = s + s * rb + s * rb^2 + s * rb^3 + ... + s * rb^(n-1). This simplifies to pb = s * (rb^n - 1)/(rb - 1). Similarly for the amount you would get by depositing to the investment, let's call that pi, except you must also subtract the amount of the broker fee, b. So you want to make deposits when pb>pi, or s*(ri^n-1)/(ri-1) - b > s*(rb^n-1)/(rb-1) Then just solve for n and you're done! Except ... maybe someone who's better at algebra than me could solve that for n, but I don't see how to do it. Further complicating this is that banks normally pay interest monthly, while stocks go up or down every day. If a calculation said to withdraw after 3.9 months, it might really be better to wait for 4.0 months to collect one additional month's interest. But let's see if we can approximate. If the growth rates and the number of periods are relatively small, the compounding of growth should also be relatively small. So an approximate solution would be when the difference between the interest rates, times the amount of each deposit, summed over the number of deposits, is greater than the fee. That is, say the investment pays 10% per month more than your bank account (wildly optimistic but just for example), the broker fee is $10, and the amount of each deposit is $200. Then if you delay making the investment by one month you're losing  10% of $200 = $20. This is more than the broker fee, so you should invest immediately. Okay, suppose more realistically that the investment pays 1% more per month than the bank account. Then the first month you're losing 1% of $200 = $2. The second month you have $400 in the bank, so you're losing $4, total loss for two months = $6. The third month you have $600 in the bank so you lose an additional $6, total loss = $12. Etc. So you should transfer the money to the investment about the third month. Compounding would mean that losses on transferring to the investment are a little higher than this, so you'd want to bias to transferring a little earlier. Or, you could set up a spreadsheet to do the compounding calculations month by month, and then just look down the column for when the investment total minus the bank total is greater than the broker fee. Sorry I'm not giving you a definitive answer, but maybe this helps." Thanks it is problem for class and I don't want to give the problem I just want to understand how to actually do it and the book hasn't been much help. Only additional information I can provide is   In Inventory – 15 days Accounts Receivable outstanding – 35 days Vendor credit – 40 days Operating Cycle – 50 days If the APR is an effective rate. If the APR is a nominal rate compounded monthly, first convert it to an effective rate. If your APR is quoted as nominal rate compounded monthly, the APR is 108.6 %. Here is the calculation, (done in Mathematica ). The sum of the discounted future payments (p) are set equal to the present value (pv) of the loan, and solved for the periodic interest rate (r).  Details of the effective interest rate calculation can be found here. http://en.wikipedia.org/wiki/Effective_interest_rate#Calculation "Taking the last case first, this works out exactly. (Note the Bank of England interest rate has nothing to do with the calculation.) The standard loan formula for an ordinary annuity can be used (as described by BobbyScon), but the periodic interest rate has to be calculated from an effective APR, not a nominal rate.  For details, see APR in the EU and UK, where the definition is only valid for effective APR, as shown below. 2003 BMW 325i £7477 TYPICAL APR 12.9% 60 monthly payments £167.05 How does this work? See the section Calculating the Present Value of an Ordinary Annuity.  The payment formula is derived from the sum of the payments, each discounted to present value. I.e. The example relates to the EU APR definition like so.  Next, the second case doesn't make much sense (unless there is a downpayment). 2004 HONDA CIVIC 1.6 i-VTEC SE 5 door Hatchback £6,999 £113.15 per month ""At APR 9.9% [as quoted in advert], 58 monthly payments"" 58 monthly payments at 9.9% only amount to £5248.75 which is £1750.25 less than the price of the car. Finally, the first case is approximate. 2005 TOYOTA COROLLA 1.4 VVTi 5 door hatchback £7195 From £38 per week ""16.1% APR typical, a 60 month payment, 260 weekly payments"" A weekly payment of £38 would imply an APR of 14.3%."