How does compounding of annual interest work?
The general concept is that your money will grow at an accelerating rate because you start getting interest paid on your returns in addition to the original investment.  As a simple example, assume you invest $100 and get 10% interest per year paid annually.   -At the end of the first year you have your $100 + $10 interest for a total of $110. -So you start the second year with $110 and so 10% would be $11 for a total of $121. -The third year you start with $121 so 10% would be $12.10 for a total of $133.10 See how the amount it goes up each year increases? If we were talking a higher initial amount or a larger number of years that can really add up. That is essence is compound interest. Most of the complicated looking formulas you see out there for compound interest are just shortcuts so you don't have to iteratively go through the above exercise a bunch of times to find out how much you would have after some number of years. This formula tells you how much you would have(A) after a certain number of years(t)  at a given interest rate(r) assuming they pay interest n times per year, for example you would use 12 for n if it paid interest monthly instead of yearly. P represents the  amount you started out with.   If you keep investing monthly (as shown in your example) instead of just depositing it and letting it sit, you have to use a more complicated formula. Finance people refer to this as calculating the future value of an annuity.  That formula looks like this: A = PMT [((1 + r)N - 1) / r] x (1+r)  A : Is the amount you would have at the end of the time period.   N : The number of compounding periods (months if you get interest calculated monthly)   PMT : The total amount you are putting in each period (N)    r: Just like before, the interest rate you are getting paid. Be sure to adjust this to a monthly number if N represents months (divide APR by 12)* *Most interest rates are quoted as APR, which is the annualized interest rate not counting compounding. Don't confuse this with APY, which has compounding built into it and is not appropriate for use in this formula. Inserting your example: r (monthly interest rate) = 15% APR / 12 = .0125 n = 30 years * 12 months/year = 360 months   A = $150 x [((1 + .0125)360 - 1) / .0125] x (1+.0125) A = $1,051,473.09 (rounded) Here's how I have worked it out.  Different answer to the one expected.  Pretty sure it's right though.
Firstly, it should be noted that the period of the investment is  all of the years 2007 to 2017 inclusive. This totals (temporarily removing one shoe and sock to extend counting range) 11 years!  Not the 9 of the OP and the accepted answer... Secondly, there are various definitions of average rate of return.  One would be : What constant annual rate of return, compounded annually, will yield   the same result as the given investment? Unfortunately, this results in an equation that cannot be solved by ordinary algebraic methods.  If r is the desired annual rate, then the equation is: (1+r)^11 * 10000 = 14567 Using logarithms: log(1 + r) = log(1.4567)/11 r = 3.4789% Could someone please explain to me how interest rates work? I like to think of interest rates as the price of money. It is specified as a percentage paid per unit of time (for example, 3%/year). To figure out how much interest money you get (or have to pay) for a given amount and time, multiply the amount with the interest rate and then divide by the time divided by the interest rate's specified time. That sounds awfully complicated, so let's look at a simple example instead. You deposit $1,000 at a fixed interest rate of 2% per year, for two and a half years, where the interest is paid at the end of the term. This means that you earn $1,000 * 2% = $20 per year in interest. Multiply this by [2.5 years] / [year] = 2.5, and you will have received $20 * 2.5 = $50 in interest over 2.5 years. If the interest is paid yearly, this gets slightly more complicated, but the principle is the same. Now imagine that you deposit $5,000 at a fixed 3% per year, for half a year. Again, the interest is paid at the end of the term. You now earn $5,000 * 3% [per year] * [[0.5 years] / [year]] = $75 in interest over six months. Variable interest rates makes this a little more complicated, but it is exactly the same thing in principle: calculate the interest paid for each period (taking any compounding into account), then add up all periods to get the total amount of interest paid over time. It also works the same way if you take out a loan rather than depositing money. Tax effects (capitals gains taxes or interest expense deductions) may make the actual amount paid or received different, but that does not change the fundamental aspect of how to calculate interest. Do CD's make more money with higher interest rates, or is it the other way around? Usually fixed interest rate instruments such as certificates of deposit, or loans with fixed rates, pay a higher interest rate for longer terms. This is because it is harder to judge credit risk in a longer term, so whoever gives the loan usually wants a premium for the additional risk. So a 6-month CD will normally pay a smaller percentage interest per year than a five-year CD. Note that this is not always the case; the technical term for when this does not hold is inverted yield curve. Interest rates are almost always formally specified in terms of percent per year, which makes it easy to compare rates. If you buy a $100 6-month CD paying 1% (I told you these were only examples :)) and then reinvest the money at the end of the term in another 6-month CD also paying 1%, the total amount paid will be ($100 * 1 + (1% * 6/12)) = $100.50 for the first term, then ($100.50 * 1 + (1% * 6/12)) = $101.0025 at the end of the second term. As you can see, the compounding of the interest makes this return slightly more than a single $100 12-month CD ($100 * 1 + 1% = $101), but unless you are dealing with large amounts of money, the difference is small enough to be negligible. If you were to put $100 in a 2% one-year CD, you'd get back $102 at the end of the year. Put the same amount in a 5% one-year CD, and you get back $105. So yes, higher interest rates means more interest money paid, for loans as well as deposits. Keep in mind that loans and deposits really are essentially the same thing, and interest calculations work the same way for both. The interest rate of a normal certificate of deposit does not change if the variable interest rates change, but rather is locked in when the money is deposited (or the CD is bought, whichever way you prefer to look at it). I've found the systems that seem to work. Firstly, you need to find how much money is required to pay for the withdrawals after retirement, while still accruing interest.  I couldn't seem to do this with an equation, but this bit of javascript worked: yearsToLast: Number of years of yearly withdrawals yearlyWithdrawal: Amount to withdraw each year interest: Decimal form of yearly compounding interest Now that we have how much is required at the beginning of the retirement, to figure out how much to add yearly to hit this mark, you'd use: amount: Previously found required amount to reach interest: Decimal form of yearly compounding interest yearsSaving: Number of years saving till amount needs to be hit I hope this helps some other poor soul, because I could find squat on how to do this. Max There is no interest outstanding, per se.  There is only principal outstanding.  Initially, principal outstanding is simply your initial loan amount. The first two sections discuss the math needed - just some arithmetic. The interest that you owe is typically calculated on a monthly basis.  The interested owed formula is simply (p*I)/12, where p is the principal outstanding, I is your annual interest, and you're dividing by 12 to turn annual to monthly. With a monthly payment, take out interest owed.  What you have left gets applied into lowering your principal outstanding.  If your actual monthly payment is less than the interest owed, then you have negative amortization where your principal outstanding goes up instead of down. Regardless of how the monthly payment comes about (eg prepay, underpay, no payment), you just apply these two calculations above and you're set.  The sections below will discuss these cases in differing payments in detail. For a standard 30 year fixed rate loan, the monthly payment is calculated to pay-off the entire loan in 30 years.  If you pay exactly this amount every month, your loan will be paid off, including the principal, in 30 years.  The breakdown of the initial payment will be almost all interest, as you have noticed.  Of course, there is a little bit of principal in that payment or your principal outstanding would not decrease and you would never pay off the loan. If you pay any amount less than the monthly payment, you extend the duration of your loan to longer than 30 years.  How much less than the monthly payment will determine how much longer you extend your loan.  If it's a little less, you may extend your loan to 40 years.  It's possible to extend the loan to any duration you like by paying less.  Mathematically, this makes sense, but legally, the loan department will say you're in breach of your contract. Let's pay a little less and see what happens.  If you pay exactly the interest owed = (p*I)/12, you would have an infinite duration loan where your principal outstanding would always be the same as your initial principal or the initial amount of your loan.   If you pay less than the interest owed, you will actually owe more every month.  In other words, your principal outstanding will increase every month!!!  This is called negative amortization.  Of course, this includes the case where you make zero payment.  You will owe more money every month. Of course, for most loans, you cannot pay less than the required monthly payments.  If you do, you are in default of the loan terms. If you pay more than the required monthly payment, you shorten the duration of your loan.  Your principal outstanding will be less by the amount that you overpaid the required monthly payment by.  For example, if your required monthly payment is $200 and you paid $300, $100 will go into reducing your principal outstanding (in addition to the bit in the $200 used to pay down your principal outstanding).  Of course, if you hit the lottery and overpay by the entire principal outstanding amount, then you will have paid off the entire loan in one shot! When you get to non-standard contracts, a loan can be structured to have any kind of required monthly payments.  They don't have to be fixed.   For example, there are Balloon Loans where you have small monthly payments in the beginning and large monthly payments in the last year.   Is the math any different?  Not really - you still apply the one important formula, interest owed = (p*I)/12, on a monthly basis.  Then you break down the amount you paid for the month into the interest owed you just calculated and principal.  You apply that principal amount to lowering your principal outstanding for the next month. Supposing that what you have posted is accurate, the most likely scenario is that you have a structured 5 year car loan where your monthly payments are smaller than the required fixed monthly payment for a 5 year loan, so even after 2 years, you owe as much or more than you did in the beginning!  That means you have some large balloon payments towards the end of your loan.  All of this is just part of the contract and has nothing to do with your prepay. Maybe I'm incorrect in my thinking, but I have a question about   prepaying a loan. When you take out a mortgage on a home or a car   loan, it is my understanding that for the first years of payment you   are paying mostly interest. Correct. So, let's take a mortgage loan that allows prepayment without penalty.   If I have a 30 year mortgage and I have paid it for 15 years, by the   16th year almost all the interest on the 30 year loan has been paid to   the bank and I'm only paying primarily principle for the remainder of   the loan. Incorrect.  It seems counter-intuitive, but even in year 16, about 53% of your monthly payment still goes to interest!!!  It is hard to see this unless you try to do the calculations yourself in a spreadsheet. If suddenly I come into a large sum of money and decide I want to pay   off the mortgage in the 16th year, but the bank has already received   all the interest computed for 30 years, shouldn't the bank recompute   the interest for 16 years and then recalculate what's actually owed in   effect on a 16 year loan not a 30 year loan? It is my understanding   that the bank doesn't do this. What they do is just tell you the   balance owed under the 30 year agreement and that's your payoff   amount. Your last sentence is correct.  The payoff amount is simply the principal outstanding plus any interest from (p*I)/12 that you owe.  In your example of trying to payoff the rest of your 30 year loan in year 16, you will owe around 68% of your original loan amount. That seems unfair. Shouldn't the loan be recalculated as a 16 year   loan, which it actually has become? In fact, you do have the equivalent of a 15 year loan (30-15=15) at about 68% of your initial loan amount.  If you refinanced, that's exactly what you would see.  In other words, for a 30y loan at 5% for $10,000, you have monthly payments of $53.68, which is exactly the same as a 15y loan at 5% for $6,788.39 (your principal outstanding after 15 years of payments), which would also have monthly payments of $53.68. A few years ago I had a 5 year car   loan. I wanted to prepay it after 2 years and I asked this question to   the lender. I expected a reduction in the interest attached to the car   loan since it didn't go the full 5 years. They basically told me I was   crazy and the balance owed was the full amount of the 5 year car loan.   I didn't prepay it because of this. That is the wrong reason for not prepaying.  I suspect you have misunderstood the terms of the loan - look at the Variable Monthly Payments section above for a discussion.   The best thing to do with all loans is to read the terms carefully and do the calculations yourself in a spreadsheet.  If you are able to get the cashflows spelled out in the contract, then you have understood the loan. "APY stands for Annual Percentage Yield, a calculation done by the financial institution to make simple comparisons of account value after 1 year between competing accounts. The APY includes the effects of compound interest regardless of the rate of interest, so the simple answer is: no, your return is only your principle multiplied by the APY after a year. Credit Unions are more member participatory than a bank, so the name ""share"" implies that you own a share of the credit union and it's future. It's possible that the CU could elect to pay a dividend on top of your interest rate. Since you have the option to credit the dividend to the share certificate account or another place, it seems that interest would be paid on the dividends left in the SC on the compounding schedule at the contracted rate.  You would have to look at the terms of the account to verify precisely when the dividend is payed, whether it's a value above and beyond the interest, and how it is compounded into our account." At what point does my investment benefit from compounded interest? Monthly? Quarter? Yearly? Does it even benefit? I think you are mixing things. There is no concept of interest or compounding in Mutual Funds. When you buy a mutual fund, it either appreciates in value or depreciates in value; both can happen depending on the time period you compare. Now, let's assume at the end of the year I have a 5% return. My $10,000 is now $10,500. The way you need to look at this is Given you started with $10,000 and its now $10,500 the return is 5%. Now if you want to calculate simple return or compounded return, you would have to calculate accordingly. You may potentially want to find a compounded return for ease of comparison with say a Bank FD interest rate or some other reason.  So if $10,000 become $10,500 after one year and $11,000 after 2 year. The absolute return is 10%, the simple yearly return is 5%. Or the Simple rate of return for first year is 5% and for second year is 4.9%. Or the Average Year on Year return is 4.775%. Perhaps there is no single formula that accounts for all the time intervals, but there is a method to get formulas for each compound interest period. You deposit money monthly but there is interest applied weekly. Let's assume the month has 4 weeks.  So you added x in the end of the first month, when the new month starts, you have x money in your account. After one week, you have x + bx money. After the second week, you have x + b(x + bx) and so on. Always taking the previous ammount of money and multiplying it by the interest (b) you have. This gives you for the end of the second month:  This looks complicated, but it's easy for computers. Call it f(0), that is: It is a function that gives you the ammount of money you would obtain by the end of the second month. Do you see that the future money inputs are given with relation to the previous ones? Then we can do the following, for n>1 (notice the x is the end of the formula, it's the deposit of money in the end of the month, I'm assuming it'll pass through the compound interest only in the first week of the next month):  And then write:  There is something in mathematics called recurrence relation in which we can use these two formulas to produce a simplified one for arbitrary b and n. Doing it by hand would be a bit complicated, but fortunately CASes are able to do it easily. I used Wolfram Mathematica commands: And it gave me the following formula:  All the work you actually have to do is to figure out what will be f(0) and then write the f(n) for n>0 in terms of f(n-1). Notice that I used the command FullSimplify in my code, Mathematica comes with algorithms for simplyfing formulas so if it didn't find something simpler, you probably won't find it by yourself! If the code looks ugly, it's because of Mathematica clipboard formatting, in the software, it looks like this:  Notice that I wrote the entire formula for f(0), but as it's also a recurrence relation, it can be written as:  That is: f(0)=g(4). This should give you much simpler formulas to apply in this method. "Sure, stocks don't pay interest. I just looked up the word ""compound"" in a couple of dictionaries and the relevant definition in all of them just mentioned interest and not growth in the value of stock. So it may be technically inaccurate to talk about ""compound growth"" of a stock. I'll yield to someone more knowledgeable about the technical language of finance to answer that part. But regardless of whether the word strictly applies, the concept certainly does. Suppose you put $1000 into a mutual fund and the fund grows by 10%. You now have $1100. The next year the fund grows by 15%. So you gain 15% of what? Of your original $1000? No, of your present balance, $1100. The effect is the same as compound interest. There is the fundamental difference that interest is normally a fixed rate: you get such-and-such percent a year as spelled out in a contract. But change in the value of a stock depends on many factors, none of them guaranteed." I don't understand the logic of converting a cost of funds of 4% to a monthly % and then subtracting that number from an annual one (the 1.5%).  Unfortunately without seeing the case I really can't help you...there was likely much you have left out from above. Let P denote the amount of the investment, R the rate of return and I the rate of inflation.  For simplicity, assume that the payment p is made annually right after the return has been earned.  Thus, at the end if the year, the investment P has increased to P*(1+R) and p is returned as the annuity payment. If I = 0, the entire return can be paid out as the payment, and thus p = P*R. That is, at the end of the year, when the dust settles after the return P*R has been collected and paid out as the annuity payment, P is again available at the beginning of the next year to earn return at rate R. We have P*(1+R) - p = P If I > 0, then at the end of the year, after the dust settles, we cannot afford to have only P available as the investment for next year. Next year's payment must be p*(1+I) and so we need a larger investment since the rate of return is fixed.  How much larger?  Well, if the investment at the beginning of next year is P*(1+I), it will earn exactly enough additional money to pay out the increased payment for next year, and have enough left over to help towards future increases in payments. (Note  that we are assuming that R > I. If R < I, a perpetuity cannot be created.)  Thus, suppose that we choose p such that P*(1+R) - p = P*(1+I)   Multiplying this equation by (1+I), we have [P(1+I)]*(1+R) - [p*(1+I)] = P*(1+I)^2 In words, at the start of next year, the investment is P*(1+I) and the return less the increased payout of p*(1+I) leaves an investment of P*(1+I)^2 for the following year.  Each year, the payment and the amount to be invested for the following year increase by a factor of (1+I). Solving P*(1+R) - p = P*(1+I) for p, we get  p = P*(R-I)  as the initial perpetuity payment and the payment increases by a factor (1+I) each year.  The initial investment is P and it also increases by a factor of (1+I) each year. In later years, the investment is P*(1+I)^n at the start of the year, the payment is p*(1+I)^n and the amount invested for the next year is P*(1+I)^{n+1}. This is the same result as obtained by the OP but written in terms that I can understand, that is, without the financial jargon about discount rates, gradients, PV, FV and the like. The really simple answer is that compound interest is compound not linear.  Money invested for longer earns more interest, and the sooner you start investing, the longer it has to earn interest.  These ideas come out of pension investment where 65 is the usual retirement age and what you invest in the 1st ten years of your pension (or any other compound interest fund) accounts for over 50% of what you will get out. 25 to 65 is forty years and $100 invested at 7% for 40 years is $1400.  $100 invested every year for 40 years the pot would be worth just under $20,000. At 30 years, it would be worth under $10,000, and at 20 years it would be worth only $4099.  If you double your investment amount every 10 years you would have invested $15700, and the pot would be worth $45,457. Do exactly the same but starting at 35 instead of 25 and your pot would only be worth $14,200. 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 "This answer is better served as a comment but I don't have enough rep.  It is not guaranteed that they 'do not accrue interest while you are a full time student'. Some student loans can capitalize the interest - before pursuing leveraged investing, be sure that your student loan is not capitalizing. https://www.salliemae.com/student-loans/manage-your-private-student-loan/understand-student-loan-payments/learn-about-interest-and-capitalization/ Capitalized interest Capitalized interest is a second reason your loan may end up costing   more than the amount you originally borrowed. Interest starts to accrue (grow) from the day your loan is disbursed   (sent to you or your school). At certain points in time—when your   separation or grace period ends, or at the end of forbearance or   deferment—your Unpaid Interest may capitalize. That means it is added   to your loan’s Current Principal. From that point, your interest will   now be calculated on this new amount. That’s capitalized interest."" https://www.navient.com/loan-customers/interest-and-taxes/how-student-loan-interest-works/ Capitalized Interest If you accrue interest while you are in school – as with Direct   Unsubsidized, FFELP Unsubsidized, Direct and FFELP PLUS Loans, and   Private Loans – you will have capitalized interest if it is unpaid.   Unpaid accrued interest is added to the principal amount of your loan   after you leave school and finish any applicable grace period. Simply put, there will be interest to be paid on both the principal of   the loan and on the interest that has already accumulated. To minimize the effects of the capitalized interest on the amount you   will pay overall, you can pay the interest during college instead of   waiting until after graduation. That way, you start with the original   principal balance (minus any fees) when you begin repayment." "Typically mutual funds will report an annualized return. It's probably an average of 8% per year from the date of inception of the fund. That at least gives some basis of comparison if you're looking at funds of different ages (they will also often report annualized 1-, 3-, 5-, and 10- year returns, which are probably better basis of comparison since they will have experience the same market booms and busts...).  So yes, generally that 8% gets compounded yearly, on average. At that rate, you'd get your investment doubled in roughly 9 years... on average... Of course, ""past performance can't guarantee future results"" and all that, and variation is often significant with returns that high. Might be 15% one year, -2% the next, etc., hence my emphasis on specifying ""on average"". EDIT: Based on the Fund given in the comments: So in your fund, the times less than a year (1 Mo, 3 Mo, 6 Mo, 1 Yr) is the actual relative change that of fund in that time period. Anything greater is averaged using CAGR approach. For example. The most recent 3 year period (probably ending end of last month) had a 6.19% averaged return. 2014, 2015, and 2016 had individual returns of 8.05%, 2.47%, and 9.27%. Thus that total return over that three year period was 1.0805*1.0247*1.0927=1.21 = 21% return over three years. This is the same total growth that would be achieved if each year saw consistent 6.5% growth (1.065^3 = 1.21).  Not exactly the 6.19%, but remember we're looking at a slightly different time window. But it's pretty close and hopefully helps clarify how the calculation is done." I've been hearing storied from Germans that this is happening in Germany, too, but at the bank level. All anecdotal, people I've met telling me their personal stories, but they follow the same pattern. Go to the bank, try to take out a few grand for a vacation or large purchase, bank tells them they can't have that much and that they just have to do with less, even if the account balance covers the withdrawal.