Retirement formula for annual compound interest with changing principal
The equation is the same one used for mortgage amortization.  You first want to calculate the PV (present value) for a stream of $50K payments over 20 years at a10% rate.  Then that value is the FV (future value) that you want to save for, and you are looking to solve the payment stream needed to create that future value. Good luck achieving the 10% return, and in knowing your mortality down to the exact year. Unless this is a homework assignment, which need not reflect real life.  Edit - as indicated above, the first step is to get that value in 20 years:  The image is the user-friendly entry screen for the PV calculation. It walks you though the need to enter rate as per period, therefore I enter .1/12 as the rate. The payment you desire is $50K/yr, and since it's a payment, it's a negative number. The equation in excel that results is: =PV(0.1/12,240,-50000/12,0) and the sum calculated is $431,769 Next you wish to know the payments to make to arrive at this number:  In this case, you start at zero PV with a known FV calculated above, and known rate. This solves for the payment needed to get this number, $568.59 The excel equation is: =PMT(0.1/12,240,0,431769) Most people have access to excel or a public domain spreadsheet application (e.g. Openoffice). If you are often needing to perform such calculations, a business finance calculator is recommended. TI used to make a model BA-35 finance calculator, no longer in production, still on eBay, used.  One more update- these equations whether in excel or a calculator are geared toward per period interest, i.e. when you state 10%, they assume a monthly 10/12%. With that said, you required a 20 year deposit period and 20 year withdrawal period. We know you wish to take out $4166.67 per month. The equation to calculate deposit required becomes - 4166.67/(1.00833333)^240= 568.59  HA! Exact same answer, far less work. To be clear, this works only because you required 240 deposits to produce 240 withdrawals in the future. 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
It would help if we had numbers to walk you through the analysis. Current balance, rate, remaining term, and the new mortgage details.  To echo and elaborate on part of Ben's response, the most important thing is to not confuse cash flow with savings. If you have 15 years to go, and refinance to 30 years, at the rate rate, your payment drops by 1/3. Yet your rate is identical in this example.  The correct method is to take the new rate, plug it into a mortgage calculator or spreadsheet using the remaining months on the current mortgage, and see the change in payment. This savings is what you should divide into closing costs to calculate the breakeven. It's up to you whether to adjust your payments to keep the term the same after you close.  With respect to keshlam, rules of thumb often fail. There are mortgages that build the closing costs into the rate. Not the amount loaned, the rate. This means that as rates dropped, moving from 5.25% to 5% made sense even though with closing costs there were 4.5% mortgages out there. Because rates were still falling, and I finally moved to a 3.5% loan. At the time I was serial refinancing, the bank said I could return to them after a year if rates were still lower. In my opinion, we are at a bottom, and the biggest question you need to answer is whether you'll remain in the house past your own breakeven time. Last - with personal finance focusing on personal, the analysis shouldn't ignore the rest of your balance sheet. Say you are paying $1500/mo with 15 years to go. Your budget is tight enough that you've chosen not to deposit to your 401(k). (assuming you are in the US or country with pretax retirement account options) In this case, holding rates constant, a shift to 30 years frees up about $500/mo. In a matched 401(k), your $6000/yr is doubled to $12K/year.  Of course, if the money would just go in the market unmatched, members here would correctly admonish me for suggesting a dangerous game, in effect borrowing via mortgage to invest in the market. The matched funds, however are tough to argue against. What is the formula for calculating the total cost of a loan with extra payments towards the principal? The formula you require is the standard one for calculating the time to repay.  With larger repayments the time to completely repay the loan is reduced. Where The total cost of the loan is then n * d. Explanation & Calculation The formula for a loan is derived from the sum of the cash flows discounted to present value being equal to the principal.  For further info see the section here titled: Calculating the Present Value of an Ordinary Annuity The summation can be reduced to a closed form by induction:  Rearranging for d and n With the OP's figures The original monthly repayment is $1,006.96 Adding $200 each month ... With the higher repayment the loan is repaid in 257.36 months instead of 360.  (Of course a bank would simply take a reduced payment in month 258, but the amounts work out the same.) The saving is $51,882.37 Addendum If the repayments increase was made part-way through the term of the loan the summation and formula would be  Where Then For example, if for the first ten years the payments are $1,006.96 and for the remaining time the repayments are $1,206.96 The loan is completely repaid in 302.528 months. The saving is Plotting over a range of m months: Yes your basic math is correct. If your tax bracket never changes, then either type of retirement account will end up in the same place. Assuming that there are no income restrictions that will limit your ability to contribute to the type of account you want. Now your job is to guess what your tax bracket will be each and every year for the next 3 or 4 decades. Events that will influence your bracket: getting married; having children; buying a house; selling a house; paying for college; the cost of medical care; moving to a state with a different state tax structure. Of course that assumes that you don't get a big bonus one year or that congress changes the tax brackets. That is why many people have both types of retirement accounts: Roth and non-Roth. "Well, the first one is based on the ""Pert"" formula for continuously-compounded present value, while the second one is the periodically-compounded variant.  Typically, the continuously-compounded models represent the ideal; as the compounding period of time-valued money shrinks towards zero, and the discount rate (or interest rate if positive) stays constant over the time period examined, the periodic equation's results approach that of the continuously-compounded equation. Those two assumptions (a constant rate and continuous balance adjustment from interest) that allow simplification to the continuous form are usually incorrect in real-world finance; virtually all financial institutions accrue interest monthly, for a variety of reasons including simpler bookkeeping and less money paid or owed in interest. They also, unless prohibited by contract, accrue this interest based on a rate that can change daily or even more granularly based on what financial markets are doing. Most often, the calculation is periodic based on the ""average daily balance"" and an agreed rate that, if variable, is based on the ""average daily rate"" over the previous observed period. So, you should use the first form for fast calculation of a rough value based on estimated variables. You should use the second form when you have accurate periodic information on the variables involved. Stated alternately, use the first form to predict the future, use the second form in retrospect to the past." "Hey guys I have a quick question about a financial accounting problem although I think it's not really an ""accounting"" problem but just a bond problem.  Here it goes    GSB Corporation issued semiannual coupon bonds with a face value of $110,000 several years ago. The annual coupon rate is 8%, with two coupons due each year, six months apart. The historical market interest rate was 10% compounded semiannually when GSB Corporation issued the bonds, equal to an effective interest rate of 10.25% [= (1.05 × 1.05) – 1]. GSB Corporation accounts for these bonds using amortized cost measurement based on the historical market interest rate. The current market interest rate at the beginning of the current year on these bonds was 6% compounded semiannually, for an effective interest rate of 6.09% [= (1.03 × 1.03) – 1]. The market interest rate remained at this level throughout the current year. The bonds had a book value of $100,000 at the beginning of the current year. When the firm made the payment at the end of the first six months of the current year, the accountant debited a liability for the exact amount of cash paid. Compute the amount of interest expense on these bonds for the last six months of the life of the bonds, assuming all bonds remain outstanding until the retirement date.  My question is why would they give me the effective interest rate for both the historical and current rate?  The problem states that the firm accounts for the bond using historical interest which is 10% semiannual and the coupon payments are 4400 twice per year.  I was just wondering if I should just do the (Beginning Balance (which is 100000 in this case) x 1.05)-4400=Ending Balance so on and so forth until I get to the 110000 maturity value. I got an answer of 5474.97 and was wondering if that's the correct approach or not." 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 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. First, assuming you are making payments for a savings deposit.  The present value of the deposit is the sum of the all the payments discounted to present value.  In this case they would be discounted by the rate of inflation: £100 deposited next year is worth less than £100 today because it will be eroded by inflation.  With a higher rate of inflation the payments are discounted more heavily, so the present value decreases.  A deposit, or annuity due (see Calculating the Present Value of an Annuity Due), can be expressed mathematically like so:-  ∴ by induction  So for example, the following annuity has a present value of £1,136.76 The total amount that will be paid for the annuity is 12 x £100 = £1,200.  With a higher rate of inflation, say 2% per month, and with the same 12 x £100 payments, the present value of the annuity decreases. In Excel (£1,078.68) A similar case is that for a loan, or ordinary annuity (see Calculating the Present Value of an Ordinary Annuity), except the discounting factor is the loan interest rate rather than inflation and repayments are made at the end of each period rather than at the start. The present value of a loan is the value of the all the future repayments discounted to present value.  With a higher interest rate the payments are discounted more heavily, so the present value decreases.  A loan can be expressed mathematically like so:-  ∴ by induction  So for example, the following values fulfill a loan worth £1,125.51 The total amount that will be paid for the loan is 12 x £100 = £1,200.  With a higher rate of interest, say 2% per month, and the same 12 x £100 repayments, the present value of the loan that can be obtained decreases. In Excel (£1,057.53) Retirement calculation, in general, should be based on the amount of money needed per year/month and the expected life expectancy. Life expectancy, if calculated to 90 years (let's say) indicates that post retirement age (60 yrs.) your accumulated/invested money should generate adequate income to cover your expenses till 90 years. The problem in general is not how long you shall live but what would be your expected spending from retirement to end of life expectancy. The idea is at the minimum your investments should generate income that is inflation adjusted. One way to do this is to consider your monthly expense now i.e. the expense that is absolute minimum for carrying on (food, electricity, water, medicines, household consumables, car petrol, insurance, servicing, entertainment, newspaper etc.) this does not contain the amortizable liabilities (home loan, child's education, other debts). It is better to take this amount per family rather than per person and yearly rather than monthly (as we tend to miss a lot of yearly expenses). This amount that you need today will increase at a Compounded Annual Growth Rate (CAGR) of the average inflation. For example, if today you spend 100 per year in 7 years you will need to spend appx. 200 at 10% inflation. Now, your investments will not increase post your retirement, so your current investment needs to do two things (1) give you your yearly requirement (2) grow by a fixed amount so that next year it can give you CAGR adjusted returns. In general, this kind of investment grows by high net amounts initially and slowly the growth decrease. The above can be calculated by Net Present value (NPV) formulae (http://en.wikipedia.org/wiki/Net_present_value). The key is to remember that the money that is invested when you retire should be able to give you inflation adjusted returns to cover your yearly expenses. How much money you need depends on your life style/expectation and how much return is received depends on the instruments that you invest on. As for your question above on the difference between the age of you and your spouse, it better to go with the consolidated family requirement and get an idea of how much investment is necessary and provision the same as soon as possible from your as well as your spouse's income. Hope this helps.- thanks "The formula you are looking for is pretty complicated. It's given here: http://itl.nist.gov/div898/handbook/eda/section3/eda3661.htm You might prefer to let somebody else do the grunt work for you. This page will calculate the probability for you: http://stattrek.com/online-calculator/normal.aspx. In your case, you'd enter mean=.114, standard deviation=.132, and ""standard score""= ... oh, you didn't say what you're paying on your debt. Let's say it's 6%, i.e. .06. Note that this page will give you the probability that the actual number will be less than or equal to the ""standard score"". Enter all that and click the magic button and the probability that the investment will produce less than 6% is ... .34124, or 34%. The handy rule of thumb is that the probability is about 68% that the actual number will be within 1 standard deviation of the mean, 95% that it will be within 2 standard deviations, and 99.7% that it will be within 3. Which isn't exactly what you want because you don't want ""within"" but ""less than"". But you could get that by just adding half the difference from 100% for each of the above, i.e. instead of 68-95-99.7 it would be 84-98-99.9. Oh, I missed that in a follow-up comment you say you are paying 4% on a mortgage which you are adjusting to 3% because of tax implications. Probability based on mean and SD you gave of getting less than 3% is 26%. I didn't read the article you cite. I assume the standard deviation given is for the rate of return for one year. If you stretch that over many years, the SD goes down, as many factors tend to even out. So while the probability that money in a given, say, mutual fund will grow by less than 3% in one year is fairly high -- the 25 - 35% we're talking here sounds plausible to me -- the probability that it will grow by an average of less than 3% over a period of 10 or 15 or 20 years is much less. Further Thought There is, of course, no provably-true formula for what makes a reasonable risk. Suppose I offered you an investment that had a 99% chance of showing a $5,000 profit and a 1% chance of a $495,000 loss. Would you take it? I wouldn't. Even though the chance of a loss is small, if it happened, I'd lose everything I have. Is it worth that risk for the modest potential profit? I'd say no. Of course to someone who has a billion dollars, this might be a very reasonable risk. If it fails, oh well, that could really cut in to what he can spend on lunch tomorrow." Since the compounding period and payment period differs (Compounded Daily vs Paid Monthly), you need to find the effective interest rate for one payment period (month).   This means that each month you pay 0.33387092772% of the outstanding principal as interest.  Then use this formula to find the number of months:  Where PV = 21750, Pmt = 220, i = 0.0033387092772  That gives 120 Months.  Depending on the day count convention, (30/360 or 30.416/365 or Actual/Actual), the answer may differ slightly. Using Financial Calculator gives extremely similar answer.  The total cash paid in the entire course of the loan is 120 x $220 = $26,400 "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." With 10% return over three years, depositing $900 each month, in three years $34,039.30.  Re. downvote. I guess this is too brief and without explanation, but I was rushing.  If you want further explanation of how this is calculated check the link already posted by JoeTaxpayer, and have a look at the formula for continuously compounded return.  Also, try out the numbers in the simplified example below yourself.  E.g.  Addendum mhoran_psprep has pointed out that I didn't read the OP's post closely enough.  With rolling investments the total return will be:  Where n is the month number i.e. 36, 37, etc. 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. If you need to shop coins, you could do your personal improvement, however, this may be tricky because of even easy documentation errors fee extra. 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