Calculate Finance Rate, Interest Amount when we have below line Fees
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.
These are the steps I'd follow: $200 today times (1.04)^10 = Cost in year 10.  The 6 deposits of $20 will be one time value calculation with a resulting year 7 final value. You then must apply 10% for 3 years (1.1)^3 to get the 10th year result.  You now have the shortfall. Divide that by the same (1.1)^3 to shift the present value to start of year 7. (this step might confuse you?) You are left with a problem needing 3 same deposits, a known rate, and desired FV. Solve from there.  (Also, welcome from quant.SE. This site doesn't support LATEX, so I edited the image above.) The number you are trying to calculate is called the Internal Rate of Return (IRR).  Google Spreadsheets (and excel) both have an XIRR function that can do this for you fairly simply. Setup a spreadsheet with 1 column for dates, 1 column for investment. Mark your investments as negative numbers (payment to invest).  All investments will be negative. Mark your last row with today's date and today's valuation (positive). All withdrawals will be positive, so you are pretending to withdrawal your entire account for the purpose of calculation. Do not record dividends or other interim returns unless you are actually withdrawing money. The XIRR function will calculate your internal rate of return with irregularly timed investments. Links: Article explaining XIRR function (sample spreadsheet in google docs to modify) You can get really nerdy with the EV calc, but I would just add that it's important to deduct any non-operating, non-consolidated assets in addition to the minority interest adjustment - e.g. unconsolidated subsidiaries, excess real estate, excess working capital, etc. The Finance functions in spreadsheet software will calculate this for you.  The basic functions are for Rate, Payment, PV (present value), FV (Future value), and NPER, the number of periods.  The single calculation faces a couple issues, dealing with inflation, and with a changing deposit. If you plan to save for 30 years, and today are saving $500/mo, for example, in ten years I hope the deposits have risen as well. I suggest you use a spreadsheet, a full sheet, to let you adjust for this.  Last, there's a strange effect that happens. Precision without accuracy. See the results for 30-40 years of compounding today's deposit given a return of 6%, 7%, up to 10% or so. Your forecast will be as weak as the variable with the greatest range. And there's more than one, return, inflation, percent you'll increase deposits, all unknown, and really unknowable.  The best advice I can offer is to save till it hurts, plan for the return to be at the lower end of the range, and every so often, re-evaluate where you stand. Better to turn 40, and see you are on track to retire early, than to plan on too high a return, and at 60 realize you missed it, badly.  As far as the spreadsheet goes, this is for the Google Sheets -  Type this into a cell =nper(0.01,-100,0,1000,0) It represents 1% interest per month, a payment (deposit) of $100, a starting value of $0, a goal of $1000, and interest added at month end. For whatever reason, a starting balance must be entered as a negative number, for example - =nper(0.01,-100,-500,1000,0) Will return 4.675, the number of months to get you from $500 to $1000 with a $100/mo deposit and 1%/mo return. Someone smarter than I (Chris Degnen comes to mind) can explain why the starting balance needs to be entered this way. But it does show the correct result. As confirmed by my TI BA-35 financial calculator, which doesn't need $500 to be negative. Line one shows your 1M, a return with a given rate, and year end withdrawal starting at 25,000.  So Line 2 starts with that balance, applies the rate again, and shows the higher withdrawal, by 3%/yr.  In Column one, I show the cumulative effect of the 3% inflation, and the last number in this column is the final balance (903K) but divided by the cumulative inflation.  To summarize - if you simply get the return of inflation, and start by spending just that amount, you'll find that after 20 years, you have half your real value.  The 1.029 is a trial and error method, as I don't know how a finance calculator would handle such a payment flow. I can load the sheet somewhere if you'd like.  Note: This is not exactly what the OP was looking for. If the concept is useful, I'll let it stand. If not, downvotes are welcome and I'll delete. This depends on the loan calculation methodology. If it is on reducing balance then yes. Else not much difference 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%. The question is valid, you just need to work backwards. After how much money-time will the lower expense offset the one time fee?  Lower expenses will win given the right sum of money and right duration for the investment. For this, the internal rate of return is preferred. In short, all cash flows need to be discounted to the present and set equal to 0 so that an implied rate of return can be calculated. You could try to work this out by hand, but it's practically hopeless because of solving for roots of the implied rate of return which are most likely complex. It's better to use a spreadsheet with this capability such as OpenOffice's Calc. The average return on equity is 9%, so anything higher than that is a rational choice. Example Using this simple tool, the formula variables can easily be input. For instance, the first year has a presumed cash inflow of $2,460 because the insurance has a 30% discount from $8,200 that is assumed to be otherwise paid, a cash inflow of $40,000 to finance the sprinklers, a cash outflow of $40,000 to fund the sprinklers, a $400 outflow for inspection, and an outflow in the amount of the first year's interest on the loan. This should be repeated for each year.  They can be input undiscounted, as they are, for each year, and the calculator will do the rest. The solution to this problem is somewhat like grading on a curve. Use the consumption ratio multiplied by the attendance (which is also a ratio, out of 100 days) to calculate how much each person owes. This will leave you short. Then add together all of the shares in a category, determine the % increase required to get to the actual cost of that category, and increase all the shares by that %. 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 1: Low fees means: a Total Expense Ratio of less than 0,5%. One detail you may also want to pay attention to whether the fund reinvests returns (Thesaurierender Fonds) which is basically good for investing, but if it's also a foreign-based fund then taxes get complicated, see http://www.finanztip.de/indexfonds-etf/thesaurierende-fonds/ 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 Here's the purely mathematical answer for which fees hurt more. You say taking the money out has an immediate cost of $60,000. We need to calculate the present value of the future fees and compare it against that number. Let's assume that the investment will grow at the same rate either with or without the broker. That's actually a bit generous to the broker, since they're probably investing it in funds that in turn charge unjustifiable fees. We can calculate the present cost of the fees by calculating the difference between: As it turns out, this number doesn't depend on how much we should expect to get as investment returns. Doing the math, the fees cost: 220000 - 220000 * (1-0.015)^40 = $99809 That is, the cost of the fees is comparable to paying nearly $100,000 right now. Nearly half the investment! If there are no other options, I strongly recommend taking the one-time hit and investing elsewhere, preferably in low-cost index funds. Details of the derivation. For simplicity, assume that both fees and growth compound continuously. (The growth does compound continuously. We don't know about the fees, but in any case the distinction isn't very significant.) Fees occur at a (continuous) rate of rf = ln((1-0.015)^4) (which is negative), and growth occurs at rate rg. The OPs current principal is P, and the present value of the fees over time is F. We therefore have the equation P e^((rg+rf)t) = (P-F) e^(rg t) Solving for F, we notice that the e^rg*t components cancel, and we obtain F = P - P e^(rf t) = P - P e^(ln((1-0.015)^4) t) = P - P (1-0.015)^(4t) I was typing up a long response and lost it to a backspace.. so, I apologize but I don't intend on rewriting it all.  You'll have to use a method called bootstrapping to get the forward rates.  Essentially you're looking at the spot rate today, and the forward rates, then filling in what must be the rate to make them equal out in the end.  Sorry I'm not more help!