Calculating Future Value: Initial deposit and recurring deposits of a fixed but different Value
But how do I bring the initial deposit into the equation? Basically, you can't.  Unless you combine two different formulas from Math of Finance into a single expression. The single initial deposit of $1000 will compound for 20 years at 5% compounded annually. The final amount for this part of the deposit will be: V1 = 1000 x (1.05)^20 In addition the series of 20 payments will be an ordinary annuity with a regular payment of $100, with the value on the occasion of the 20th payment given by:  So the final total amount in the account at the end of 20 years will be the sum of these two values... If I is the initial deposit, P the periodic deposit, r the rent per period, n the number of periods, and F the final value, than we can combine two formulas into one to get the following answer: F = I*(1+r)n + P*[(1+r)n-1]/r In this case, you get V = 1000*(1.05)20 + 100*[(1.05)20-1]/0.05 = 5959.89 USD. Note that the actual final value may be lower because of rounding errors. Illustrating with a shorter example:  Suppose I deposit 1,000 USD. Every year I deposit another 100 USD. I want to know how much money will be on that savings account in 4 years.  The long-hand calculation is Expressed with a summation  And using the formula derived from the summation (as shown by DJohnM) So for 20 years Note in year 20 (or year 4 in the shorter example) the final $100 deposit does not have any time to accrue interest before the valuation of the account.
"There are some assumptions which can be made in terms of the flexibility you have - I will start with the least flexible assumption and then move to more flexible assumptions. If you must put down a number 1, your go-to for this(""Change the start period to 1""), is pretty good, and it's used frequently for other divide-by-zero calculations like kda in a video game. The problem I have with '1' is that it doesn't allow you to handle various scales.  Some problems are dealt with in thousands, some in fractions, and some in hundreds of millions. Therefore, you should change the start period to the smallest significantly measurable number you could reasonably have. Here, that would take your example 0 and 896 and give you an increase of 89,500%. It's not a great result, but it's the best you can hope for if you have to put down a number,  and it allows you to keep some of the ""meaning in the change."" If you absolutely must put something This is the assumption that most answers have taken - you can put down a symbol, a number with a notation, empty space, etc, but there is going to be a label somewhere called 'Growth' that will exist. I generally agree with what I've seen, particularly the answers from Benjamin Cuninghma and Nath. For the sake of preservation - those answers can be summarized as putting 'N/A' or '-', possibly with a footnote and asterisk. If you can avoid the measurement entirely The root of your question is ""What do my manager and investors expect to see?"" I think it's valuable to dig even further to ""What do my manager and investors really want to know?"". They want to know the state of their investment. Growth is often a good measurement of that state, but in cases where you are starting from zero or negative, it just doesn't tell you the right information. In these situations, you should avoid % growth, and instead talk in absolute terms which mention the time frame or starting state. For example:" So would it be more sensible to park the money in 3 separate fixed deposits created each month or to invest this money as usual in debt and mutual funds and instead just liquidate the recurring deposit and use it to pay my taxes in July? Most taxes would already have been paid by March 31. So what you are talking here is less than Rs 10,000/- of taxes. The optimization you are looking at is very small and you are better of choosing one that is more convenient. 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. EDIT: After reading one of the comments on the original question, I realized that there is a much more intuitive way to think about this. If you look at it as a standard PV calculation and hold each of the cashflows constant. Really what's happening is that because of inflation the discount rate isn't the full value of the interest rate. Really the discount rate is only the portion of the interest rate above the inflation rate. Hence in the standard perpetuity PV equation PV = A / r r becomes the interest rate less the inflation rate which gives you PV = A / (i - g). That seems like a much better way to get to the answer than all the machinations I was originally trying. Original Answer: I think I finally figured this out. The general term for this type of system in which the payments increase over time is a gradient series annuity. In this specific example since the payment is increasing by a percentage each period (not a constant rate) this would be considered a geometric gradient series. According to this link the formula for the present value of a geometric gradient series of payments is: Where P is the present value of this series of cashflows.   A_1 is the initial payment for period 1 (i.e. the amount you want to withdraw adjusted for inflation).   g is the gradient or growth rate of the periodic payment (in this case this is the inflation rate)   i is the interest rate   n is the number of payments This is almost exactly what I was looking for in my original question. The only problem is this is for a fixed amount of time (i.e. n periods). In order to figure out the formula for a perpetuity we need to find the limit of the right side of this equation as the number of periods (n) approaches infinity. Luckily in this equation n is already well isolated to a single term: (1 + g)^n/(1 + i)^-n}. And since we know that the interest rate, i, has to be greater than the inflation rate, g, the limit of that factor is 0. So after replacing that term with 0 our equation simplifies to the following: Note: I don't do this stuff for a living and honestly don't have a fantastic finance IQ. It's been a while since I've done any calculus or even this much algebra so I may have made an error in the math. 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. "Option pricing models used by exchanges to calculate settlement prices (premiums) use a volatility measure usually describes as the current actual volatility.  This is a historic volatility measure based on standard deviation across a given time period - usually 30 to 90 days.   During a trading session, an investor can use the readily available information for a given option to infer the ""implied volatility"".  Presumably you know the option pricing model (Black-Scholes).  It is easy to calculate the other variables used in the pricing model - the time value, the strike price, the spot price, the ""risk free"" interest rate, and anything else I may have forgotten right now.  Plug all of these into the model and solve for volatility.  This give the ""implied volatility"", so named because it has been inferred from the current price (bid or offer). Of course, there is no guarantee that the calculated (implied) volatility will match the volatility used by the exchange in their calculation of fair price at settlement on the day (or on the previous day's settlement).  Comparing the implied volatility from the previous day's settlement price to the implied volatility of the current price (bid or offer) may give you some measure of the fairness of the quoted price (if there is no perceived change in future volatility).  What such a comparison will do is to give you a measure of the degree to which the current market's perception of future volatility has changed over the course of the trading day.   So, specific to your question, you do not want to use an annualised measure.  The best you can do is compare the implied volatility in the current price to the implied volatility of the previous day's settlement price while at the same time making a subjective judgement about how you see volatility changing in the future and how this has been reflected in the current price." i will not calculate it for you but just calculate the discounted cash flow (by dividing with 1.1 / 1.1^2 / 1.1^3   ...)of each single exercise as stated and deduct the 12.000 of the above sum.  in the end compare which has the highest npv "You can use the 6900 to make an investment.  Or to buy something.  That's why people keep reminding you that you could make interest.  Because most people think of either 7200 now or paying the same 7200 over time.  So you could just be storing the 7200 under the mattress until you pay it out.  Obviously in that case, inflation doesn't matter (""is not applicable"").  You've given up the use of the 7200 from the beginning.   Think of it instead as 7200 now and twenty-three payments of 300 each.  So 14,100 total.  Then you can spend 6900 on something else at the beginning or spend 300 a month on other things.  The difference between spending 6900 now and 300 in each of twenty-three months would be measured in inflation.  Of course, this requires you to have both 7200 now and an income stream producing at least 300 a month.   Another way of doing things is to take 6900 and invest it.  Each month you remove 300 and use it to make a payment.  We're now back to just one 7200, plus the interest over time.  I would argue that this is still an inflation advantage.  It's just that instead of spending the money, you invested it.  And that of course is your prerogative.   The point being that you would not have that opportunity if you paid up front.   Now to calculate the difference in monthly payments of 300 euro using the above webpage what should I do?   As already said, you would have to calculate twenty-three values and then sum them with the 300 you pay up front in the monthly.   There are other ways to calculate it, if you are not using that particular tool.  For example, there are formulas to calculate the net present value of an annuity.  E.g. see Investopedia.   Where: P = the present value of an annuity stream PMT = the dollar amount of each annuity payment r = the inflation rate n = the number of periods in which payments will be made Investopedia talks about interest rates, but you can put inflation there for this purpose.  In that case, r might better be called the ""discount rate"".   PMT is 300. r is whatever estimate you are using for inflation.  E.g. .003 per period. n is 23.   Note that the monthly inflation rate is smaller than the annual rate.  So .003 is about 3.66% annually.  3.04% annually is more like .0025 a month.   I found calculators for this with search terms ""present value of annuity calculator"".  Some of the calculators will take the annual rate (3.66%) and number of periods per year as input.  Or the calculator may take a monthly rate as a percentage (.3%) rather than as a decimal (.003).  So be careful of the inputs.   This gives me a net present value (NPV) of 6,657.69 for the 23 payments of 300, assuming .3%.  Or 6957.69 is the NPV of the monthly payments and 7200 is the NPV of the 7200 up front.   Obviously if you can pay less than 6957.69 up front rather than 7200, then it makes more sense to pay up front.   Even without a discount though, it still may make more sense to pay up front.   How much those intangibles are worth is up to you." what do you mean exactly?  Do you have a future target price and projected future dividend payments and you want the present value (time discounted price) of those? Edit: The DCF formula is difficult to use for stocks because the future price is unknown.  It is more applicable to fixed-income instruments like coupon bonds.  You could use it but you need to predict / speculate a future price for the stock. You are better off using the standard stock analysis stuff: Learn Stock Basics - How To Read A Stock Table/Quote The P/E ratio and the Dividend yield are the two most important.  The good P/E ratio for a mature company would be around 20.  For smaller and growing companies, a higher P/E ratio is acceptable.  The dividend yield is important because it tells you how much your shares grow even if the stock price stays unchanged for the year. HTH "When using Time Value of Money equations, you need to know when the flow starts. A mortgage for example, has a first payment at the end of the first time period, usually 1 month.  For savings, one can start the account with a deposit of course, or start by saying ""I will deposit $XXX at the end of each month.  The answer really depends on the exact details of the situation. In your example, I'm inclined to suggest first flow is 1 year out." Fairly straightforward to match the result from the calculator soup link.  There is a formula to calculate n from the future value s (using natural logs) In Excel This was derived as shown  To calculate n from the inflation-adjusted future value si requires using a solver since an algebraic formula cannot be formulated.  As demonstrated  Calculations done using Mathematica 7. "I think we'd need to look at actual numbers to see where you're running into trouble. I'm also a little confused by your use of the term ""unexpected expenses"". You seem to be using that to describe expenses that are quite regular, that occur every X months, and so are totally expected. But assuming this is just some clumsy wording ... Here's the thing: Start out by taking the amount of each expense, divided by the number of months between occurrences. This is the monthly cost of each expense. Add all these up. This is the amount that you should be setting aside every month for these expenses, once you get a ""base amount"" set up. So to take a simple example: Say you have to pay property taxes of $1200 twice a year. So that's $1200 every 6 months = $200 per month. Also say you have to pay a water bill once every 3 months that's typically $90. So $90 divided by 3 = $30. Assuming that was it, in the long term you'd need to put aside $230 per month to stay even. I say ""in the long term"" because when you're just starting, you need to put aside an amount sufficient that your balance won't fall below zero. The easiest way to do this is to just set up a chart where you start from zero and add (in this example) $230 each month, and then subtract the amount of the bills when they will hit. Do this for some reasonable time in the future, say one year. Find the biggest negative balance. If you can add this amount to get started, you'll be safe. If not, add this amount divided by the number of months from now until it occurs and make that a temporary addition to your deposits. Check if you now are safely always positive. If not, repeat the process for the next biggest negative. For example, let's say the property tax bills are April and October and the water bills are February, May, August, and November. Then your chart would look like this: The biggest negative is -370 in April. So you have to add $370 in the first 4 months, or $92.50 per month. Let's say $93. That would give: Now you stay at least barely above water for the whole year. You could extend the chart our further, but odds are the exact numbers will change next year and you'll have to recalculate anyway. The more irregular the expenses, the more you  will build up just before the big expense hits. But that's the whole point of saving for these, right? If a $1200 bill is coming next week and you don't have close to $1200 saved up in the account, where is the money coming from? If you have enough spare cash that you can just take the $1200 out of what you would have spent on lunch tomorrow, then you don't need this sort of account." The value does change from 12.61% to 13.48%. The difference between re-investing cashflows at 14% vs 12% is not big enough to change the rounded value. Edit: The initial cashflow is discounted at t0, meaning it's already equal to its present value and the finance rate doesn't have an effect. It does impact future outgoing cashflows, as you've noted. The short answer is that you would want to use the net inflow or net outflow, aka profit or loss.   In my experience, you've got a couple different uses for IRR and that may be driving the confusion.  Pretty much the same formula, but just coming at it from different angles.   Thinking about a stock or mutual fund investment, you could project a scenario with an up-front investment (net outflow) in the first period and then positive returns (dividends, then final sale proceeds, each a net inflow) in subsequent periods.  This is a model that more closely follows some of the logic you laid out. Thinking about a business project or investment, you tend to see more complicated and less smooth cashflows.  For example, you may have a large up-front capital expenditure in the first period, then have net profit (revenue less ongoing maintenance expense), then another large capital outlay, and so on.   In both cases you would want to base your analysis on the net inflow or net outflow in each period.  It just depends on the complexity of the cashflows trend as to whether you see a straightforward example (initial payment, then ongoing net inflows), or a less straightforward example with both inflows and outflows.   One other thing to note - you would only want to include those costs that are applicable to the project.  So you would not want to include the cost of overhead that would exist even if you did not undertake the project. If you mean the internal rate of return, then the quarterly rate of return which would make the net present value of these cash flows to be zero is 8.0535% (found by goal seek in Excel), or an equivalent compound annual rate of 36.3186% p.a. The net present value of the cash flows is: 10,000 + 4,000/(1+r) - 2,000/(1+r)^2 - 15,125/(1+r)^3, where r is the quarterly rate. If instead you mean Modified Dietz return, then the net gain over the period is: End value - start value - net flow = 15,125 - 10,000 - (4,000 - 2,000) = 3,125 The weighted average capital invested over the period is: 1 x 10,000 + 2/3 x 4,000 - 1/3 x 2,000 = 12,000 so the Modified Dietz return is 3,125 / 12,000 = 26.0417%, or 1.260417^(1/3)-1 = 8.0201% per quarter, or an equivalent compound annual rate of 1.260417^(4/3)-1 = 36.1504%. You are using an inappropriate formula, because we know for a fact that the flows take place at the beginning/end of the period. Instead, you should be combining the returns for the quarters (which have in fact been provided in the question). To calculate this, first calculate the growth factor over each quarter, then  link them geometrically to get the overall growth factor. Subtracting 1 gives you the overall return for the 3-quarter period. Then convert the result to a quarterly rate of return. Growth factor in 2012 Q4 is 11,000/10,000 = 1.1 Growth factor in 2013 Q1 is 15,750/15,000 = 1.05 Growth factor in 2013 Q2 is 15,125/13,750 = 1.1 Overall growth factor is 1.1 x 1.05 x 1.1 = 1.2705 Return for the whole period is 27.05% Quarterly rate of return is 1.2705^(1/3)-1 = 8.3074% Equivalent annual rate of return is 1.2705^(4/3)-1 = 37.6046% ========= I'd recommend you to refer to Wikipedia.