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.
So my Question is this, in reality is investment in equities like the   stock market even remotely resemble the type of growth one would   expect if investing the same money in an account with compounding   interest? Generally no as there is a great deal of volatility when it comes to investing in stocks that isn't well represented by simply taking the compounded annual growth rate and assuming things always went up and never went down. This is adding in the swings that the market will take that at times may be a bit of a rude surprise to some people. Are all these prognosticators vastly underestimating how much savers   need to be socking away by overstating what is realistic in terms of   growth in investment markets? Possibly but not probably. Until we know definitively what the returns are from various asset classes, I'm not sure I'd want to claim that people need to save a ton more. I'll agree that the model misses how wide the swings are, not necessarily that the averages are too low or overstated. Compound interest means that the interest in each time period is calculated taking into account previously earned interest and not only the initial sum. Thus, if you had $1000 and invested it so that you'd earn 5% each year, than if you would withdraw the earnings each year you in 30 years you would earn 0.05*30*1000 = $1500, so summarily you'd have $2500, or 150% profit. However, if you left all the money to earn interest - including the interest money - then at the end of 30 years you'd have $4321 - or 330% profit.  This is why compound interest is so important - the interest on the earned interest makes money grow significantly faster. On the other hand, the same happens if you owe money - the interest on the money owed is added to the initial sum and so the whole sum owed grows quicker.  Compound interest is also important when calculating interest by time periods. For example, if you are told the loan accumulates 1% interest monthly, you may think it's 12% yearly. However, it is not so, since monthly interest is compounded - i.e., in February the addition not only February's 1% but also 1% on 1% from January, etc. - the real interest is 12.68% yearly. Thus, it is always useful to know how interest is compounded - both for loans and investments - daily, monthly, yearly, etc. I would like to know how they calculated such monthly payment The formula is:  Your values would come out to be: r = (1+3.06/(100*365))^31-1=0.002602 (converting your annual percentage to a monthly rate equivalent of daily compounded interest) PV = 12865.57 n = 48 Inserting your values into the formula: P = [r*(PV)]/[1-(1+r)^(-n)] P = [0.002602*(12865.57)]/[1-(1.002602)^(-48)] P = 285.47 "When we talk about compounding, we usually think about interest payments. If you have a deposit in a savings account that is earning compound interest, then each time an interest payment is made to your account, your deposit gets larger, and the amount of your next interest payment is larger than the last. There are compound interest formulas that you can use to calculate your future earnings using the interest rate and the compounding interval.  However, your mutual fund is not earning interest, so you have to think of it differently. When you own a stock (and your mutual fund is simply a collection of stocks), the value of the stock (hopefully) grows. Let's say, for example, that you have $1000 invested, and the value goes up 10% the first year. The total value of your investment has increased by $100, and your total investment is worth $1100. If it grows by another 10% the following year, your investment is then $1210, having gained $110.  In this way, your investment grows in a similar way to compound interest. As your investment pays off, it causes the value of the investment to grow, allowing for even higher earnings in the future. So in that sense, it is compounding. However, because it is not earning a fixed, predictable amount of interest as a savings account would, you can't use the compound interest formula to calculate precisely how much you will have in the future, as there is no fixed compounding interval.  If you want to use the formula to estimate how much you might have in the future, you have to make an assumption on the growth of your investment, and that growth assumption will have a time period associated with it. For example, you might assume a growth rate of 10% per year. Or you might assume a growth rate of 1% per month. This is what you could use in a compound interest formula for your mutual fund investment.  By reinvesting your dividends and capital gains (and not taking them out in cash), you are maximizing your ""compounding"" by allowing those earnings to cause your investment to grow." You want the net expense of the surcharge minus the rewards to be no more than the interest that you would pay otherwise.   Where t is the compounding period for the rate D expressed as a fraction of the overall period for D.  So if D is an annual rate (not the APR, the simple rate), it would be expressed as something like 1/365 if compounded daily.  That is the number of years in the compounding period.  If a monthly rate or weekly compounding, that would change.   And p is the number of such time periods in the grace period.  So if the grace period were one month, this might be 30.   Other variables are as used in the question, all expressed as percentages (which is why I'm dividing by 100).  The D rate should be the simple rate, like 6% not the APR of 6.24% or whatever.   Note that I'm saying <=.  When equal, there is no financial advantage or disadvantage.  You could choose either method for the same cost.  Now, one method may be more annoying to implement, in which case you might add a fee for it on one side or the other of the equation.  Or simply change the less than or equal to be just less than.   I may be missing something that you should consider but I don't know.  The problem is generic enough that pertinent details might be hidden.  But hopefully this at least gives you a framework under which to consider it. I read Q#4 as  Will $250 in one account earn more interest than $250 in five accounts? in which case Excel says no, assuming a constant interest rate for all accounts. I dunno if the same holds true for banks. 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). Okay, I think I managed to find the precise answer to this problem! It involves solving a non-linear exponential equation, but I also found a good approximate solution using the truncated Taylor series. See below for a spreadsheet you can use. Let's start by defining the growth factors per period, for money in the bank and money invested:   Now, let S be the amount ready to be invested after n+1 periods; so the first of that money has earned interest for n periods. That is,  The key step to solve the problem was to fix the total number of periods considered. So let's introduce a new variable: t = the total number of time periods elapsed So if money is ready to invest every n+1 periods, there will be t/(n+1) separate investments, and the future value of the investments will be:  This formula is exact in the case of integer t and n, and a good approximation when t and n are not integers. Substituting S, we get the version of the formula which explicitly depends on n:  Fortunately, only a couple of terms in FV depend on n, so we can find the derivative after some effort:  Equating the derivative to zero, we can remove the denominator, and assuming t is greater than zero, we can divide by the constant ( 1-G t ):  To simplify the equation, we can define some extra constants:    Then, we can define a function f(n) and write the equation as:  Note that α, β, γ, G, and R are all constant. From here there are two options: Use Newton's method or another numerical method for finding the positive root of f(n). This can be done in a number of software packages like MATLAB, Octave, etc, or by using a graphics calculator. Solve approximately using a truncated Taylor series polynomial. I will use this method here. The Taylor series of f(n), centred around n=0, is:  Truncating the series to the first three terms, we get a quadratic polynomial (with constant coefficients):  Using R, G, α, β and γ defined above, let c0, c1 and c2 be the coefficients of the truncated Taylor series for f(n):    Then,  n should be rounded to the nearest whole number. To be certain, check the values above and below n using the formula for FV. Using the example from the question: For example, I might put aside $100 every week to invest into a stock   with an expected growth of 9% p.a., but brokerage fees are $10/trade.   For how many weeks should I accumulate the $100 before investing, if I   can put it in my high-interest bank account at 4% p.a. until then? Using Newton's method to find roots of f(n) above, we get n = 14.004. Using the closed-form approximate solution, we get n = 14.082. Checking this against the FV with t = 1680 (evenly divisible by each n + 1 tested): Therefore, you should wait for n = 14 periods, keeping that money in the bank, investing it together with the money in the next period (so you will make an investment every 14 + 1 = 15 weeks.) Here's one way to implement the above solution with a spreadsheet. StackExchange doesn't allow tables in their syntax at this time, so I'll show a screenshot of the formulae and columns you can copy and paste: Formulae:  Copy and paste column A: Copy and paste column B: Results:  Remember, n is the number of periods to accumulate money in the bank. So you will want to invest every n+1 weeks; in this case, every 15 weeks.