Talk:Net present value

Simple Formula

The formula below is most easier that table

$NPV={\frac {1-(1+i)^{-n}}{i}}-S$

and also used by many ERP systems now a days as a base formula to determine NPV why people make any article their personal matter i added this formula for public's best interest but it is removed many time if anyone think that this formula is improper then discuss here

Needs Example

The article needs an example of NPV of a bond paying 5% yearly coupon for 10 years and a face value of 10,000.00. as well as a zero coupon bond maturing in 10 years and a face value of 10,000.00.

The article needs a discussion of how to compare the two bonds above to

• a risk free rate NPV such as 10 year US treasuray
• inflation assumptions for 10 years (using for example, the 10 year US TIPS inflation adjusted security).

you can view detailed cases on bonds and how to compare them at the following link

http://www.oopine.com/view_topic.php?ti=121

however by the time i will update this article also

Question:

Two alternative scenarios for a project involve expensing the up-front costs when incurred or capitalizing them and taking capital cost allowance over future periods. Cash flow values are identical with the exception of income tax payments. When up-front costs are expensed, income taxes are lowered in year 0, and when up-front costs are capitalized, income taxes are lowered in future periods as capital cost allowances are claimed.

NPV of the alternative where up-front costs are expensed is higher than the alternative where costs are capitalized.

Does it make sense to favor one alternative over the other for this reason alone? Would/could companies alter their capital accounting practises based on this reason?

72.2.16.10 (talk) 15:54, 28 October 2009 (UTC)

Formula

The formula was deleted by a graffiti edit on 02:07, 22 June 2009. I restored the formula from the previous edit.

Is it me or is the formula not appearing anymore in the article ? —Preceding unsigned comment added by 80.13.188.216 (talk) 09:45, 7 September 2009 (UTC)

Corrected the formula. The formula does not match the example and formulas I found in text books. Instead of ${\mbox{NPV}}=\sum _{t=1}^{N}{\frac {C_{t}}{(1+i)^{t}}}$ it should be ${\mbox{NPV}}=\sum _{t=0}^{N}{\frac {C_{t}}{(1+i)^{t}}}$ If not, T0 in the example would be wrong. —The preceding unsigned comment was added by 134.130.104.136 (talk) 13:49, 22 October 2006 (UTC).

$t_{0}$ is the period you already have results for, i.e. the period just past. Do you really want to discount last period's cash flows by the discount rate? You should only discount future cash flows. Am I missing something? --JHP 09:04, 16 March 2007 (UTC)

$t_{0}$ is the current period. Often this is the period in which the initial investment is made. The discount rate at time zero is 1 (i.e., no discount). The summation in the formula can, therefore, include t=0. An alternative -- as shown now -- is to start the summation at t=1 and add a separate C0 term.

• --Grieger (talk) 13:35, 17 December 2007 (UTC) The formula could be written either like

${\mbox{NPV}}=\sum _{t=1}^{n}{\frac {C_{t}}{(1+r)^{t}}}+{C_{0}}$ or like ${\mbox{NPV}}=\sum _{t=0}^{n}{\frac {C_{t}}{(1+r)^{t}}}$

The subtraction of $C_{0}$ makes no sense, once $C_{0}<0$ if it is an outlay. Per definition, one should sum the present value of the cash flow with the outlay or inlay that already exists.

—Daniel.gruno (talk) 21:53, 10 February 2008 (UTC)
In principle the formula should be close to what you state, ${\mbox{NPV}}=\sum _{t=0}^{N}{\frac {C_{t}}{(1+r)^{t}}}$, with the slight alteration, that N is prefered to n (we want a full population of cash flows, not just a random sample). In principle! (explanation below)

Mr. unsigned from 134.130.104.136: Assuming that you meant to include $C_{0}+$ in your formula, plenty of text books list the formula you first submitted (ie Essentials of Corporate Financial Management by Glen Arnold, Pearson Ed. 2007), and the sole reason they cling on to this oh so tedious version of the formula (which clearly is a waste of 2 cm of space which could've been used for other doodles) is to emphasize (for learning purposes) that $C_{0}$ refers to the initial investments put into the project (or whatever you are calculating). Though I too prefer the shortened version of the formula, I think it remains important for Wikipedia as a source of information for not only us hardcore financial idiots, but also people in the midst of a learning process, for whom the added emphasis of $C_{0}$ would prove useful.
Furthermore - and yes, this is Nitpicking Extreme - the variable r should actually be a k when dealing with (opportunity) cost of capital, an i when dealing with discounting and r when dealing with the internal rate of return...but I simply cannot be bothered to even try fixing that up. —Preceding unsigned comment added by Daniel.gruno (talk • contribs) 01:46, 11 February 2008 (UTC)

Without Title

added "and no cash inflow for the 12 months of Year 0." It makes the example more complete, and highlights that cashflow in applies to Year 0 just as it does for years 1 to 6. The first monkey 12 months of a projects life is Year 0 and many projects do generate cash within the first 12 months of life.

Removed "and no cash inflow for the 12 months of t=0" as previous poster has misunderstood the time periods. This project does generate cashflow in during the first 12 months, and that is what is shown at the end of that first year as the entry for t=1.

Questions

The calculations are using 10% per MONTH discount rate, not 10% per year. The whole example should be made a ten year project, which is much more realistic. Doing NPV on a ten month horizon is very rare, since the cost of money/inflation in less than a year is so small (except in very high inflationary economies). If the example is to be month-based, the interest rate is 0.10/12, or 0.0083, for a 10% annual discount rate. Krementz (talk) —Preceding undated comment added 16:11, 16 April 2012 (UTC).

Why is it $5000 / 1.10^1 and not $5000 * 0.9? If I'm adjusting for inflation and the inflation rate is 10% the money would be worth $5000*0.9 after one year.--Jerryseinfeld 18:32, 18 July 2005 (UTC)

This is because you want to know how much money you need today to buy later a product that will be priced 10% more. If the price today is $4545,45, after the period of time which inflation is 10%, the product will cost $5000 ($4545,45 * 1.1). It's just the inverse operation.

From memory, NPV is a method of comparing the financial return of a project to the return on the same cash in a term deposit. Liberator 09:38, July 19, 2005 (UTC)

I have looked at several people's attempts to try to explain what NPV is, and nobody gets it right. The best explaination I have found is in Brierly and Myers textbook, the title I forget.

I think it would be better to start with a simple explaination along the lines of -

You have £100 which you place in a savings account for one year. This account has, for the sake of argument, a 10% interest rate. Thus after one year you get £110. Therefore, £110 one year in the future is worth the same as £100 now, at a 10% discount rate.

Actually, this is a description of present value. NPV would be £110-£100=£10.

This article ought to refer to PV, (with a link) and then point out the difference between PV and NPV.

On the Danish Wikipedia we're having a discussion whether the term also refers to a serie of previous payments - and not only future payments? Personally I haven't meet the expression in such connection. Have anybody else perhaps? --83.91.231.222 16:01, 31 January 2006 (UTC)

The math would work, but it would be simplier just to use future value calculations here, e.g. $100/ (1.10)^-3 = $100 * 1.10^3. NPV is in essence a forward looking tool so the past is irrelevant (sunk costs). Somebody's thinking though. Smallbones 18:01, 5 March 2006 (UTC)

Restructuring, debate

This article is a bit of a mess. can someone go ahead and clean up the bit with the excel sheets? I know it represents the information, but I'm sure a better screenshot/diagram could be used. EmileVictor 00:06, 3 April 2007 (UTC)

I'm looking at the proposal to merge this article with Discounted cash flow from March 2007 with some concern. For those of you who are financial wizzards just remember that there is an engineering aspect to this subject where ordinary people have to understand what you're talking about. KISS is important to the rest of us. Lin 01:39, 21 June 2007 (UTC)

Agreed. I don't work in Finance, but have worked on Financial systems and projects, and most of this is over my head. The introduction should really explain it in plain English as opposed to the current financial jargon. There is a (slightly) clearer version at http://www.investopedia.com/terms/n/npv.asp#axzz2HJHRYdu4 ("The difference between the present value of cash inflows and the present value of cash outflows. NPV is used in capital budgeting to analyze the profitability of an investment or project. NPV analysis is sensitive to the reliability of future cash inflows that an investment or project will yield"). However, even that could be simplified into something like:

"Is a snapshot of the cash flowing into an organisation, minus that flowing out, converted into today's values by taking into account the effects of inflation. It is used to analyse, or estimate, the profit generated by an investment or project. It can be unreliable if the cash-flow and inflation estimates are inaccurate."

I won't replace the existing version as I am not 100% sure if my understanding is correct. Can someone with a Financial background confirm my lay-woman's understanding? ComplyAnt (talk) —Preceding undated comment added 14:11, 8 January 2013 (UTC)

Is there something wrong with this calculation?

A while ago I had posted a link to the finance wonk's version of net present value calculation at http://finance wonk.blogspot.com/2006/05/present-value-analysis-fundamentals.html

(Remove the space in the first word in the address to use the link, the blacklist kills it here too!)

Recently it was removed and I was notified the site had been blacklisted?

Did I do something wrong? Or is something wrong with the analysis on the destination site?

Looking at the finance wonk page (I haven't reread it in detail) it still looks like a neat runthrough and I like the plots, they do a lot more for me than the equations on the current wikipedia page. Especially the third and fourth ones.

I also like the discussion of how to do NPV in excel, although I see someone has found a possibly better site for that and added it to external links (cool!).

It still looks like the plots contribute a lot, I would add them into the Wikipedia write up except that it would violate copyright since that site obviously owns them (or so I assume).

I read Wikipedia's policy on external links and the link would seem to qualify on the basis of the cool plots alone, and financewonk doesn't ACTUALLY show up on the blacklist (http://meta.wikimedia.org/wiki/Spam_blacklist) so I can't even tell if there's an overall site problem or what.

Mostly I'm intrigued by the process and hoping the person who did the blacklisting/removal could comment. Is it because the blogspot page has google ads on it? There is hardly any of it and it's kind of out of the way, but I could see if that were a policy.

Full disclosure -- I do know the guy who writes that page. I consider a (small) handful of his work link-worthy, most of it being rather more specific and of limited application.

The controversy of the formula

Have a look at this page

http://www.investopedia.com/terms/n/npv.asp —The preceding unsigned comment was added by 72.52.66.10 (talk) 13:49, 8 January 2007 (UTC).

Fixed. Although, I don't know why you subtract last period's cash flows, unless it's because you are spending it to buy capital equipment. That would make sense for capital equipment purchases, but not for financial asset purchases. Perhaps this needs to be explained in the article. --JHP 09:10, 16 March 2007 (UTC)

—Daniel.gruno (talk) 22:03, 10 February 2008 (UTC)
They don't subtract last period's cash flow, they subtract the initial cash flow but place it at the end of the formula.
Nonetheless, the cash flow should not be subtracted, it should be added since any investment is a negative cash flow, and thus adding it will...subtract it! And I'd probably put $C_{0}$ to the left of the whole sum thing in the formula to show which flows comes first - but I guess it's a chicken-or-the-egg kind of thing.

Interest Rate v Discount Rate or Discount Factor

In the body of the page we talk a lot about interest rate applied to the cash flows. I am not comfortable with that. It implies that the discount factor is somehow directly linked to interest rates. I prefer to use discount factor in the body. I believe that depending on the context of the analysis, the discount factor may or may not be related to some interest rate. Comments?Kenckar 15:31, 26 March 2007 (UTC)

Agree, very good point. Be bold and edit! It would be good to sign your comments.--Gregalton 18:23, 27 February 2007 (UTC)

Disagree. Interest rate is directly linked to discount rate. An example will be fixed term deposit. The interest rate to use in calculating future value should be the discount rate in calculating present value. Gp10kenkwok 10:04, 16 April 2007 (UTC)

They are similar but distinct concepts. Interest rate is a convenient shortcut but not the same. For your fixed term deposit it may or may not be appropriate. To put a different way in relation to your example: interest rate may be the contractual amount you will get paid, but this is not necessarily the rate you should discount at; you may decide that the interest being paid does not sufficiently compensate you for risk, for example. This is particularly true for NPV/Cash flow discounting, where the discount rate for investment projects would be substantially more than any relevant interest rate.

That said, it should be easy enough to write this in such a way as to be clear that interest rate is often used as the discount rate.--Gregalton 10:13, 16 April 2007 (UTC)

I do not agree as value is able to move backward and forward. This can been seen with Present Value and Future Value relationship. Interest rate is the rate to be used to move the present value forward, and discount rate is the rate to be used to move the future value backward. This two rate has to be the same or else the relationship will not hold. Gp10kenkwok 08:10, 17 April 2007 (UTC)

Ah, I see the objection. Although I don't think it makes much difference in any case (call it something else "going forward" if you like, such as accumulation rate), the article is about net present value - which refers to discounting future cash flows to the present.--Gregalton 09:12, 17 April 2007 (UTC)

I see the point as well. Ideally, we could create a new term, e.g. present value factor or something, but most references speak of discount rate. It is often thought of as a weighted combination of cost of equity and cost of debt, although some people promote the use of other rates, such as the corporate investment rate. In any case, it might or might not be the same as some interest rate. Kenckar 04:09, 30 April 2007 (UTC)

From a pure linguistic point of view, I would definitely prefer discount rate to any other term. You could just as easily nitpick about how opportunity cost of investment should be opportunity cost of capital and vice versa, but in reality it all comes down to the fact, that the term discount - as far as I can gather - is mistaken for something that merely "cheapens" a quality of merchandise, cash etc, when it in fact means that you simply dis-count a portion of the capital due to any of the various fun terms we have.Daniel.gruno (talk) 22:16, 10 February 2008 (UTC)

Discount Rate and Timing

This might be beyond the scope of this article. But if T1 is one year later than T0, and the discount rate I'm using is 10% per year, then if I decided to use six-month intervals (i.e. T1 is six months later than T0), would I then cut the interest rate in half? Or is the math more complex than that? Or should I suck it up and divide into years?

Either way, I think that there should be a mention of how the time and the rate are related. DRogers 16:58, 9 August 2007 (UTC)

See effective interest rate and nominal interest rate. The short form is, the math is a bit more complex than that. Not much, though. If your example is 10% annually and compounding twice a year (the intervals), the effective rate will be slightly higher. If I've understood your question correctly.--Gregalton 20:22, 9 August 2007 (UTC)

Yes, thank you. That makes more sense. DRogers 21:47, 9 August 2007 (UTC)

This is actually a quite valid point to make and should perhaps be emphasized in the article, since most (primarilly online) means of calculating NPV, IRR etc. disregard the fact that compound intervals are not static Daniel.gruno (talk) 22:10, 10 February 2008 (UTC)

Dead link

The first external link: http://www.odellion.com/pages/online%20community/NPV/financialmodels_npv_definition.htm seems to go to an expired domain? —Preceding unsigned comment added by 90.231.187.94 (talk) 19:34, 2 March 2008 (UTC)

Continuous functions?

I'm confused: wouldn't it make more sense to express these equations as continous functions. Granted financial transactions are discrete, but in as much as interest compounds continuously, wouldn't it make more sense to express

${\mbox{NPV}}=\sum _{t=0}^{N}{\frac {C_{t}}{(1+r)^{t}}}$

as

${\mbox{NPV}}=\int _{0}^{t_{\mathrm {final} }}{\frac {C(t)}{(1+r)^{t}}}\,dt$

or am I missing something? Also, it looks like this can be expressed as an inner product:

Let $g(t):=(1+r)^{-t}$ then

${\mbox{NPV}}=\int _{0}^{t_{\mathrm {final} }}C(t)g(t)\,dt=\langle C,g\rangle$

do finance people think of it that way? —Ben FrantzDale (talk) 00:19, 24 April 2008 (UTC)

From a mathematical point of view, sure, you could but it like you write. That's just not how economists usually tackle this issue. The solutions you present, albeit not totally bogus, are complicated beyond what is necessary, and believe it or not, I don't think all economists would understand the mechanics of NPV if you tried to present the formula as you did. The equation would be, and is, perfectly suited for statistical and physical/chemical/etc calculations, but having an equation that considers floating point values and negative/positive numbers in a world that only deals with positive integers is like hunting rabbits with a tank.

Furthermore, we shirt-and-tie people prefer to use Annuity and Perpetuity calculations to find the value of unending cash flows (typically, we'll just mash together a string of successive years and make it look like one single cash flow instead), and as such, there's no real need to start using more complex math equations, as it'd just be more confusing than helpful. And, well, lastly, writing C(t) in a formula would, even though I know it's just a conceptual idea, implicate that all cash flows can be extracted from the same function, which is usually not the case. Typically, you would have a variety of different scenarios requiring different calculations to get the final result of the cash flow (these could be initial costs, operating profits/expenditures, missed opportunities, scrap income or cost and so on). Of course, you could state that C(t) is an abstract function, but then again, that would just make it even more useless to include a function that isn't really needed.Daniel V. Gruno (talk) 00:49, 15 May 2008 (UTC)

That's about how I figured it. The mathematician in me is tempted to argue that this "tank" I'm huntin' waskaly wabbits wit' is a simpler, more beautiful weapon, but point taken. :-) —Ben FrantzDale (talk) 00:28, 16 May 2008 (UTC)

The continous function shall not use interest i but a modified continous interest ln(1 + i) --Gunnar (talk) 01:52, 5 December 2012 (UTC)

References

Added a reference to Expected commercial value as Cooper suggests it as a possible replacement for NPV in his "Winning at New Products" book

52 cents

In the sentence, "The sum of all these present values is the net present value, which equals $8,881.52." Maybe drop the cents in the sentence to be consistent with the PVs in the box above the sentence? As it is, it does not add up. SWAK ! --Seahappy (talk) 22:04, 13 March 2009 (UTC)

Graph update?

Shouldn't the graph for discounted and undiscounted cash flows (CumCF.jpg) be updated so it doesn't say "undiscounted cum" and "discounted cum" in the key at the side? Just an observation. Daneel (talk) 23:33, 21 June 2009 (UTC)

Removed some misleading content

"If NPV is less than 0, which is to say, negative, the project should not be immediately rejected. Sometimes companies have to execute an NPV-negative project if not executing it creates even more value destruction." That would only happen if you had missed out some important issues in your NPV calculations. Could be rephrased as something like: "Not including all the relevant issues in the NPV calculations".

"Another issue with relying on NPV is that it does not provide an overall picture of the gain or loss of executing a certain project. To see a percentage gain relative to the investments for the project, usually, Internal rate of return is used complimented to the NPV method." Sigh. Oh dear. This must have been written by someone who does not really understand DCF. This is precisely why people should use NPV and not IRR to evaluate projects. For example you could have two projects that you have to choose between - you cannot do both. Project A has a 200% IRR, but only has a NPV of £10. Project B has a 5% IRR, but has a NPV of £50. Clearly, Project B is the one you should choose.

"Another issue with relying on NPV is that it does not provide an overall picture of the gain or loss of executing a certain project." Sorry to say but that is complete nonsense - completely untrue - the opposite of the truth in fact. It is true to say this about IRR, but not NPV. Which is why you should use NPV.

Brealey and Myers Principles Of Corporate Finance has an excellent chapter on Why You Should Use NPV and I suggest the original editor reasds that. 84.13.28.161 (talk) 16:17, 4 January 2010 (UTC)

Removed the same misleading content again after someone put it back

Please leave it alone. 84.13.28.161 (talk) 16:21, 4 January 2010 (UTC)

Example needs an overhaul

I'm not a finance guy (never took a business or accounting class even), but it seems like the Example section desperately needs an overhaul. As noted above the inputs are for a 12 month project life with a 10% monthly discount rate. That can't be right. Also, the cash flows in the example don't seem to relate to the numbers in the 2 charts except for the t=0. Anybody with Finance 101 under their belt able to straighten this out? — Preceding unsigned comment added by 68.0.45.66 (talk) 00:24, 23 March 2013 (UTC)

Example (not mine originally) was changed to be an annual model. Prior calculations were verified correct.Rick (talk) 14:47, 15 July 2013 (UTC)

Bank Loan Interest

As a example if we obtain 5 million bank loan to invest a 10 million project; when we are calculating NPV do we need to take the instalment repayment and interest payment as an outflow from the project? — Preceding unsigned comment added by Lasith.gunasekara (talk • contribs) 09:42, 15 July 2013 (UTC)

Here is the "MBA" (but not the PHD in finance) answer. Don't get various financing options confused with capital project analysis. IE - do not consider "installment repayment and interest payment as an outflow". Here's why - in your discount rate used in NPV calculation you are setting that high enough to cover the risk you are assuming on the project, including financing. Implicit in that rate is the time value of money (the risk free rate) plus a reasonable risk premium. You then LATER shop for financing - some combo of your money (or equity) plus borrowing (debt).

You are very astute however in pointing out any debt holder must make all payments WHEN DUE, according to the exact debt agreement. If you have installment debt, and your capital projects runs cash flow negative for the first several years, you have to still meet those installments during those years. You may have to borrow more just to cover those payments. More at Accounting liquidity vs. Solvency and Insolvency and Miller, M (1977). NPV makes a heroic assumption - you can meet debt payments as they come due, which is of course not necessarily true.

You can use Internal rate of return on the project (ignoring all financing cash flows) vs. your financing cost (APR) for the project. (Read however problems on IRR so you know the limitations.) In general if IRR > APR by a good margin, commensurate with risk taken, its a favorable project. However if there are personal guarantees, or a low APR is obtained via other assets and obligations pledged, and the project is indeed quite risky, you need to have much more IRR to make it a good deal. Read the fine print in debt agreement, bond covenant, etc. very carefully. See above Accounting liquidity too. Rick (talk) 15:42, 15 July 2013 (UTC)

Too complicated

This article - or at least the introduction - seems too complicated to me. I'm a medical doctor with a PhD, and I need to understand this term for a business proposal. An encyclopedia article introduction should be comprehensible to an interested layman. This one isn't, I'm afraid.

Consider the first sentence: "In finance, the net present value (NPV) or net present worth (NPW) of a time series of cash flows, both incoming and outgoing, is defined as the sum of the present values (PVs) of the individual cash flows of the same entity."

There's no way that could make sense to anyone but an economist - but I don't think those ought to be the target of the introductory sentence.

Please can someone who understands the concept, rewrite the introduction so that it is comprehensible to people from outside the field?

I suggest focusing on the phrase "better to invest in the project than to do nothing", found buried deep in the example, to explain the concept in accessible terms.

Thanks. Dubbin^{u | t | c} 18:17, 20 November 2014 (UTC)

I agree completely. Time value of money does this right. --JMT32 (talk) 00:08, 6 May 2016 (UTC)

Example too complicated and unclear

There has already been a suggestion that the article is too technical for a layperson. I find the example unnecessarily complicated for someone unfamiliar with the concept, particularly the text describing the example which considers far too many variables to get the point across. Unless anyone objects, I will simplify the example to include an initial outgoing cash flow and identical incoming cash flows (with a nice number, say 10,000). Otherwise, I think the concept of discounting gets lost on a reader. I eat BC Fish (talk) 19:08, 29 December 2014 (UTC)

t=0 versus t=1

@Spalasamudram:, the discrete summation does indeed start from t=0 rather than t=1 (even though this paradoxically allows for N+1 cashflows in a summation over N years). This is to allow for any initial costs in the project. Example academic references:

• Financial Mathematics for Actuaries, Chapter 1: Interest Accumulation and Time Value of Money, page 52 (formula 1.36): http://www.mysmu.edu/faculty/yktse/FMA/S_FMA_1.pdf
• Subject CT1 Financial Mathematics Core Technical, Core Reading for the 2016 exams, Unit 9, page 6 (implicit in examples, noting the undiscounted terms): https://www.liverpool.ac.uk/media/livacuk/maths/docs/IandF_CT1_2016_FINAL_cr_v1.pdf
• MATH1510, Financial Mathematics I, page 38 (3.1.2): http://www1.maths.leeds.ac.uk/~jitse/math1510/notes-all.pdf

I'll revert your edit. -Stelio (talk) 10:48, 6 March 2018 (UTC)

Rewrite

I encourage other editors to help improve this article.--Jonathan G. G. Lewis 09:32, 26 March 2019 (UTC) — Preceding unsigned comment added by Jonazo (talk • contribs)

Deleted reference to Calculator Soup website

I have a concern about using the Calculator Soup web site as an inline citation. It's an online calculator web site. It gives the author's name but says nothing about his credentials. Before we cite it in this article, I'd like to get some comfort that Calculator Soup satisfies at least some of the Wikipedia guidelines for reliable secondary sources, such as "a reputation for fact-checking and accuracy" and "editorial oversight." I'm going to be bold and delete the citation. But I'm willing to consider other points of view. This is the same issue I raised about a similar site being cited in the Internal Rate or Return article. Cordially, ~~~ BuzzWeiser196 (talk) 11:50, 16 June 2023 (UTC)