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
As mentioned, the main advantage of a 15-year loan compared to a 30-year loan is that the 15-year loan should come at a discounted rate.  All things equal, the main advantage of the 30-year loan is that the payment is lower. A completely different argument from what you are hearing is that if you can get a low interest rate, you should get the longest loan possible.  It seem unlikely that interest rates are going to get much lower than they are and it's far more likely that they will get higher.  In 15 years, if interest rates are back up around 6% or more (where they were when I bought my first home) and you are 15 years into a 30 year mortgage, you'll being enjoying an interest rate that no one can get. You need to keep in mind that as the loan is paid off, you will earn exactly 0% on the principal you've paid.  If for some reason the value of the home drops, you lose that portion of the principal.  The only way you can get access to that capital is to sell the house.  You (generally) can't sell part of the house to send a kid to college.  You can take out another mortgage but it is going to be at the current going rate which is likely higher than current rates. Another thing to consider that over the course of 30 years, inflation is going to make a fixed payment cheaper over time.  Let's say you make $60K and you have a monthly payment of $1000 or 20% of your annual income.  In 15 years at a 1% annualized wage growth rate, it will be 17% of your income.  If you get a few raises or inflation jumps up, it will be a lot more than that.  For example, at a 2% annualized growth rate, it's only 15% of your income after 15 years. In places where long-term fixed rates are not available, shorter mortgages are common because of the risk of higher rates later.  It's also more common to pay them off early for the same reason.   Taking on a higher payment to pay off the loan early only really only helps you if you can get through the entire payment and 15 years is still a long way off.  Then if you lose your job then, you only have to worry about taxes and upkeep but that means you can still lose the home.  If you instead take the extra money and keep a rainy day fund, you'll have access to that money if you hit a rough patch.  If you put all of your extra cash in the house, you'll be forced to sell if you need that capital and it may not be at the best time.  You might not even be able to pay off the loan at the current market value. My father took out a 30 year loan and followed the advice of an older coworker to 'buy as much house as possible because inflation will pay for it'.  By the end of the loan, he was paying something like $250 a month and the house was worth upwards of $200K.  That is, his mortgage payment was less than the payment on a cheap car.  It was an insignificant cost compared to his income and he had been able to invest enough to retire in comfort.  Of course when he bought it, inflation was above 10% so it's bit different today but the same concepts still apply, just different numbers.  I personally would not take anything less than a 30 year loan at current rates unless I planned to retire in 15 years. 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. 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. Sending your money off to do the heavy lifting is a way of saying that compound returns can do the bulk of your retirement investing work.  Check out the image below, I swiped this from a quick google search so I cannot claim graphic credit. But as you can see the earning potential of your money as you approach retirement is many times higher than your annual contributions. With the aim of having your money earn interest/returns to pay your annual living expenses, replacing your previous annual income.  https://i.stack.imgur.com/fpZPN.jpg I tried to post the image but do not have enough rep. "The best way to do this is to use IRR.  It's a complicated calculation, but will take into account multiple in/out cash flows over time along with ""idle periods"" where your money may not have been doing anything.  Excel can calculate it for you using the XIRR function" Set your xirr formula to a very tall column, leaving lots of empty rows for future additions. In column C, instead of hardcoding the value, use a formula that tests if it's the current bottom entry, like this: =IF(ISBLANK(A7),-C6, C6) If the next row has no date entered (yet), then this is the latest value, and make it negative. Now, to digress a bit, there are several ways to measure returns. I feel XIRR is good for individual positions, like holding a stock, maybe buying more via DRIP, etc. For the whole portfolio it stinks. XIRR is greatly affected by timing of cash flows. Steady deposits and no withdrawals dramatically skew the return lower. And the opposite is true for steady withdrawals. I prefer to use TWRR (aka TWIRR). Time Weighted Rate of Return. The word 'time' is confusing, because it's the opposite. TWRR is agnostic to timing of cashflows. I have a sample Excel spreadsheet that you're welcome to steal from: http://moosiefinance.com/static/models/spreadsheets.html (it's the top entry in the list). Some people prefer XIRR. TWRR allows an apples-to-apples comparison with indexes and funds. Imagine twin brothers. They both invest in the exact same ideas, but the amount of cash deployed into these ideas is different, solely because one brother gets his salary bonus annually, in January, and the other brother gets no bonus, but has a higher bi-weekly salary to compensate. With TWRR, their percent returns will be identical. With XIRR they will be very different. TWRR separates out investing acumen from the happenstance timing of when you get your money to deposit, and when you retire, when you choose to take withdrawals. Something to think about, if you like. You might find this website interesting, too: http://www.dailyvest.com/ ~~Dividends~~  edit:  Sorry; misunderstood your question.  Subsidiary losses. If NI is -$1000 and you own 80%, then your adjustment for year 20XX is $-200. If the accumulated minority interest is &lt;$200, the end balance of non-controlling interest at 20XX+1 would be negative.   I can also imagine a scenario due to negative value of the sub's net identifiable assets using partial goodwill method. It's easiest to get your payment from the PMT function in Excel or Google Sheets.  So a $100,000 30 year mortgage at 3% looks like this: The basic calculation is pretty simple.  You take the annual interest rate, say 3%, divided by 12, times the existing principal balance: The idea is that borrowers would like to have a predictable payment.  The earlier payments are proportionally more interest than principal than later payments are but that's because there is much more principal outstanding on month 1 than on month 200.