Calculating theoretical Present Value
If you are using an Excel, the Function PV should be able to easily calculate this. Excel Formulae PV = (Rate,Nper,Pmt,Fv,Type)   Where Rate: Rate of return. In this case you can use Inflation or assumed rate that would cost you. Say 3-5%. Note the Rate has to be for Nper. i.e. in Nper if you are counting yearly payments, then rate is yearly, if you are counting as monthly, then the rate should be monthly.   NPer: Number of periods. If yearly in your case it would be 20. If Monthly 20*12, if Quarterly 20*4 etc.   Pmt: Expected Payments for Nper. If you are saying 20 million over 20 years, it would be 1 million per year. Fv and Type can be blank So assuming a rate of 3%, and yearly payments of 1 million over 20 years. PV = $14,877,474.86 [It would show negative, just ignore the sign] The example from the following website: Investopedia - Calculating The Present And Future Value Of Annuities  specifically the section 'Calculating the Present Value of an Annuity Due' shows how the calculation is made.  Using their figures, if five payments of $1000 are made over five years and depreciation (inflation) is 5%, the present value is $4545.95  There is also a formula for this summation, (ref. finance formulas)
"Re. question 2 If I buy 20 shares every year, how do I get proper IRR? ... (I would   have multiple purchase dates) Use the money-weighted return calculation: http://en.wikipedia.org/wiki/Rate_of_return#Internal_rate_of_return  where t is the fraction of the time period and Ct is the cash flow at that time period. For the treatment of dividends, if they are reinvested then there should not be an external cash flow for the dividend.  They are included in the final value and the return is termed ""total return"".  If the dividends are taken in cash, the return based on the final value is ""net return"". The money-weighted return for question 2, with reinvested dividends, can be found by solving for r, the rate for the whole 431 day period, in the NPV summation. Now annualising And in Excel" "I got $3394.83 The first problem with this is that it is backwards.  The NPV (Net Present Value) of three future payments of $997 has to be less than the nominal value.  The nominal value is simple:  $2991.   First step, convert the 8% annual return from the stock market to a monthly return.  Everyone else assumed that the 8% is a monthly return, but that is clearly absurd.  The correct way to do this would be to solve for m in  But we often approximate this by dividing 8% by 12, which would be .67%.  Either way, you divide each payment by the number of months of compounding.   Sum those up using m equal to about .64% (I left the calculated value in memory and used that rather than the rounded value) and you get about $2952.92 which is smaller than $2991.   Obviously $2952.92 is much larger than $2495 and you should not do this.  If the three payments were $842.39 instead, then it would about break even.   Note that this neglects risk.  In a three month period, the stock market is as likely to fall short of an annualized 8% return as to beat it.  This would make more sense if your alternative was to pay off some of your mortgage immediately and take the payments or yp pay a lump sum now and increase future mortgage payments.  Then your return would be safer.   Someone noted in a comment that we would normally base the NPV on the interest rate of the payments.  That's for calculating the NPV to the one making the loan.  Here, we want to calculate the NPV for the borrower.  So the question is what the borrower would do with the money if making payments and not the lump sum.   The question assumes that the borrower would invest in the stock market, which is a risky option and not normally advisable.  I suggest a mortgage based alternative.  If the borrower is going to stuff the money under the mattress until needed, then the answer is simple.  The nominal value of $2991 is also the NPV, as mattresses don't pay interest.  Similarly, many banks don't pay interest on checking these days.  So for someone facing a real decision like this, I'd almost always recommend paying the lump sum and getting it over with.  Even if the payments are ""same as cash"" with no premium charged." The formula for determining the number of payments (months) you'll need to make on your loan is:  where i=monthly interest rate (annual rate / 12), A=loan amount (principal), and P=monthly payment. To determine the total interest that you will pay, you can use the following formula:  where P=monthly payment, N=number of payments (from above formula), and A=loan amount (principal). A quick example: using the numbers in the screenshot above ($10,000 loan, $500 monthly payment, 10% APR), the number of payments ends up to be 21.97 (which means that payment number 22 is slightly less than the rest).  In the second formula, you take that number times your $500 payment and determine that you have paid $10,984.81 over the course of the entire loan period.  Subtracting the principal, you have paid $984.81 in total interest. On your spreadsheet, the function you are looking for is NPER: NPER(rate, payment_amount, present_value, [future_value, end_or_beginning]) rate - The interest rate. (This should be the monthly rate, or the annual rate divided by 12.) payment_amount - The amount of each payment made. (For a loan payment, this should be a negative number.) present_value - The current value of the annuity. (The initial principal of the loan) future_value - [ OPTIONAL ] - The future value remaining after the final payment has been made.  (This should be 0, the default if omitted.) end_or_beginning - [ OPTIONAL - 0 by default ] - Whether payments are due at the end (0) or beginning (1) of each period. The IRR is the Discount Rate r* that makes Net Present Value NPV(r*)==0. What this boils down to is two ways of making the same kind of profitability calculation.  You can choose a project with NPV(10%)>0, or you can choose based on IRR>10%, and the idea is you get to the same set of projects.  That's if everything is well behaved mathematically. But that's not the end of this story of finance, math, and alphabet soup. For investments that have multiple positive and negative cash flows, finding that r* becomes solving for the roots of a polynomial in r*, so that there can be multiple roots.  Usually people use the lowest positive root but really it only makes sense for projects where NPV(r)>0 for r<r* and NPV(r)<0 for r>r*.   To try to help with your understanding, you can evaluate a real estate project with r=10%, find the sum future discounted cash flows, which is the NPV, and do the project if NPV>0.   Or, you can take the future cash flows of a project, find the NPV as a function of the rate r, and find r* where NPV(r*)==0. That r* is the IRR.  If IRR=r*>10% and the NPV function is well behaved as above, you can also do the project. When we don't have to worry about multiple roots, the preceding two paragraphs will select the same identical sets of projects as meeting the 10% return requirement. You can illustrate why expense ratio fees are in the numerator with an extreme example:  Let's say you have $100 in a mutual fund, their expense ratio is 50%, your nominal return is 900% and inflation is 900%.  Thus, without the expense, your investment would give you $100 in present value (because your return and inflation are identical), and $1000 in future value.  So with the expense ratio of 50% and no change in present value, you can reason that you would expect the expense ratio will eat half the present value.  If you apply your equation and include expenses in the numerator, you end up with:  ((100 - 100(.50))*(1+9))/(1+9) = $50 present value as you would expect  If you apply the manager's assumption that fees are applied external to inflation, then you end up with:  (100 * (1 + 9))/(1+9) - (100 * (1+9) * .50) = $-400 present value.  With this example you can see applying the fees externally acts as though they are charging you the fees on future returns today.  *Edit:  It's probably not worth fighting with someone senior to you over, as inflation rates are noisy estimates to begin with and the difference between these is typically not material to the decision being made; but pissing off someone senior by showing them their math is off will probably have a material impact on you. Ok so I used Excel solver for this but it's on the right track.  Latest price = $77.19  Latest div = $1.50  3-yr div growth = 28%  g = ???  rs = 14%  So we'll grow out the dividend 3 years @ 28%, and then capitalize them into perpetuity using a cap rate of [rs - g], and take the NPV using the rs of 14%. We can set it up and then solve g assuming an NPV of the current share price of $77.19.  So it should be:  NPV = $77.19 =   [$1.50 / (1+0.14)^0 ] +  [$1.50 x (1+0.28)^1 / (1+0.14)^1 ] +  ... +  [$1.50 x (1+0.28)^3 / (1+0.14)^3 ] +  [$1.50 x (1+0.28)^3 x (1+g) / (0.14-g) / (1+0.14)^4 ]  Which gives an implied g of a little under 9%. Let me know if this makes sense, and definitely check the work... "A real simple definition or analogy of present Value would be the ""Principal"" or ""Loan Amount"" being lent and the future value as being returning the Principal along with cost of borrowing The (1+i)^n is the interest you earn on present value The (i+i)^-n is the interest you pay on future value The first one is the FVIF or future value of a $1 The second one is the PVIF or present value of a $1 Both these interest factors assume interest is paid annually, if the interest payment is made more often within the payment year then interest factors look this way m is the frequency of interest payment, the higher this frequency the more of interest you pay or earn and you pay or earn the most interest when compounding occurs on each small fraction of time This entails here e is the Euler's e  Thus the interest factors turn to this The preceeding examples only considered a single repayment at future date. Now if you were obliged to make periodic loan repayments say in amount of $1 for n number of periods. Then the present value of all such periodic payment is the ""Principal"" or amount you borrowed. This is the sum of discounted periodic payments as  if we replace 1/1+i with x then this turns out to be geometric series of the form This simplifies to  replacing (1/1+i) for x we get which is the present value of periodic payment in amount of $1 The future value of periodic payments in amount of $1 can be arrived at multiplying the PVIFA by (1+i)^n giving Once again the interest is compounded per annum and for intra-year compounding you would have to at first find the annual effective yield AEY to use as the effect rate is the PVIFA and FVIFA calculation for continuous compounding All the calculation discussed thus far did not take inflation into consideration, if we were to adjust the amounts for a growth of g% then the present value of a $1 would be as follow Once again you would have to use AEY if compounding frequency of interest is intra-year Now assume that each loan repayment increases or decreases by an extra amount Q per period. To find the present value of series of payments P that increase or decrease per period by an amount Q we would do the following calculations Here and All of these calculations have been available in tadXL add-in for finance and incrementally being offered as JavaScript financial functions library tadJS. Please note that the tad series of the financial functions library for various environments such as Excel, JavaScript, PHP, Ruby, Microsoft.net and others are property of the author writing this post. All of these libraries except one for Excel are available for FREE for public use. And the future value of such payments with increments may be found by multiplying the PV by (1+i)^n as follows Here" 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. Thank you very much for this thoughtful response.  In my opinion the judges care more about the why behind your valuation rather than a how.  Anyone can use a formula, but it takes so much more to understand why to use the formula.    Personally, the 'why' is going to be the toughest part for me understand and wrap my head around.  Once again thank you for the advice and the tip. This is a clear example of annuities, where you are trying to find Net Present Value (NPV). To find a quick solution you can use the excel function (=NPV). Solving it without excel is slightly more complicated. You have to use the annuities formula:  (Payment/rate) * (1 - 1/(1+r)^n )  This formula can be written in many different ways, this is one of the simplest.  So how do we translate that to your problem? For the first option: -5000/0.04 * (1 - 1/(1.04^2 ) = -9430.47 If you haven't made this years 5000 payment you have to subtract this years 5000, so option 1 would equal -14430,47.  Option 2 is similar -2000/0.04 (1 - 1/(1.04^10 ) = -16221.79 WAIT! You also have to subtract the initial cost so option 2 is -36221.79  So in essence, option 1 is better (but you will have to buy a machine sooner or later).  Part 2 (discount rate = 12%) is for you to understand the importance of high rates in long periods of time (like option 2's 10 years)    EDIT: For clearer formulas. Also made a mistake calculating option 2 The MWRR that you showed in your post is calculated incorrectly. The formula that you use... ($15,750 - $15,000 - $4,000) / ($15,000 + 0.5 x $4,000) Translates into a form of the DIETZ formula of (EMV-BMV-C)/(BMV + .5 x C) The BMV is the STARTING balance. And as a matter of fact, the starting balance was NOT 15,000. It was IN FACT 11,000. See, the starting value for a month MUST BE the ending value of the prior month. So the BMV of 11,000 would give you the correct answer. Because if you added 4,000 at the start of the month (on day 1), it would have to have been ADDED to the 11,000 of the PRIOR month's ENDING value. Make sense? That would also mean that the addition of 4000 to the 11000 would imply that you started day 1 with 11,000. Make sense? Summary: When doing the calculations, you may use the ending value on the last day of the month to get your EMV. BUT YOU MAY NOT take the ending value on day 1 to get the BMV. That simply can not make sense since you already added a bunch of money during the day. Think about it. Davie The Finance functions in spreadsheet software will calculate this for you.  The basic functions are for Rate, Payment, PV (present value), FV (Future value), and NPER, the number of periods.  The single calculation faces a couple issues, dealing with inflation, and with a changing deposit. If you plan to save for 30 years, and today are saving $500/mo, for example, in ten years I hope the deposits have risen as well. I suggest you use a spreadsheet, a full sheet, to let you adjust for this.  Last, there's a strange effect that happens. Precision without accuracy. See the results for 30-40 years of compounding today's deposit given a return of 6%, 7%, up to 10% or so. Your forecast will be as weak as the variable with the greatest range. And there's more than one, return, inflation, percent you'll increase deposits, all unknown, and really unknowable.  The best advice I can offer is to save till it hurts, plan for the return to be at the lower end of the range, and every so often, re-evaluate where you stand. Better to turn 40, and see you are on track to retire early, than to plan on too high a return, and at 60 realize you missed it, badly.  As far as the spreadsheet goes, this is for the Google Sheets -  Type this into a cell =nper(0.01,-100,0,1000,0) It represents 1% interest per month, a payment (deposit) of $100, a starting value of $0, a goal of $1000, and interest added at month end. For whatever reason, a starting balance must be entered as a negative number, for example - =nper(0.01,-100,-500,1000,0) Will return 4.675, the number of months to get you from $500 to $1000 with a $100/mo deposit and 1%/mo return. Someone smarter than I (Chris Degnen comes to mind) can explain why the starting balance needs to be entered this way. But it does show the correct result. As confirmed by my TI BA-35 financial calculator, which doesn't need $500 to be negative. Annualize quarterly returns:      AR = (1+QR)^4  Where *QR* is a decimal return, e.g. 0.05.  Standard deviations are similar:      Annual SD = SD * sqrt(T)  If you have quarterly deviations, *T=4*, if you have daily, *T=252*, etc.  As an aside, for work with money riding on it, it is *not* okay to aggregate standard deviations if there's autocorrelation amongst observations at a smaller time scale. Volatility is often quoted this way and that's fine, but it is dangerous to do any sort of risk management with this and you'll require more due diligence. It's a good enough approximation for napkin math, though. "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:" "First, don't use Yahoo's mangling of the XBRL data to do financial analysis.  Get it from the horse's mouth: http://www.sec.gov/edgar/searchedgar/companysearch.html Search for Facebook, select the latest 10-Q, and look at the income statement on pg. 6 (helpfully linked in the table of contents).  This is what humans do. When you do this, you see that Yahoo omitted FB's (admittedly trivial) interest expense.  I've seen much worse errors. If you're trying to scrape Yahoo... well do what you must.  You'll do better getting the XBRL data straight from EDGAR and mangling it yourself, but there's a learning curve, and if you're trying to compare lots of companies there's a problem of mapping everybody to a common chart of accounts. Second, assuming you're not using FCF as a valuation metric (which has got some problems)... you don't want to exclude interest expense from the calculation of free cash flow.  This becomes significant for heavily indebted firms. You might as well just start from net income and adjust from there... which, as it happens, is exactly the approach taken by the normal ""indirect"" form of the statement of cash flows.  That's what this statement is for.  Essentially you want to take cash flow from operations and subtract capital expenditures (from the cash flow from investments section).  It's not an encouraging sign that Yahoo's lines on the cash flow statement don't sum to the totals. As far as definitions go... working capital is not assets - liabilities, it is current assets - current liabilities.  Furthermore, you want to calculate changes in working capital, i.e. the difference in net current assets from the previous quarter.  What you're doing here is subtracting the company's accumulated equity capital from a single quarter's operating results, which is why you're getting an insane result that in no way resembles what appears in the statement of cash flows.  Also you seem to be using the numbers for the wrong quarter - 2014q4 instead of 2015q3. I can't figure out where you're getting your depreciation number from, but the statement of cash flows shows they booked $486M in depreciation for 2015q3; your number is high. FB doesn't have negative FCF."