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Reinvestment assumption of NPV
Is there an implied reinvestment assumption in the mathematics of NPV and the discounting process? In other words, when we discount future cash flows, do we implicitly assume those cash flows will be reinvested at that same discount rate for the life of the asset/project in question?
I have seen conflicting arguments in different financial textbooks.
I know this is often brought up when comparing IRR and NPV (and when looking at the YTM of bonds), but some financial textbooks argue that while this reinvestment assumption does exist in IRR and YTM, it does not apply to NPV. This seems odd since the methods are mathematically equivalent.
If we do assume reinvestment of all cash flows, what does that say about dividends and free cash flows? Are we assuming that those cash flows will never be used for anything else but reinvestment? If this is true, it sounds very strange.
Please help me understand this better.
Many thanks,
David
I have seen conflicting arguments in different financial textbooks.
I know this is often brought up when comparing IRR and NPV (and when looking at the YTM of bonds), but some financial textbooks argue that while this reinvestment assumption does exist in IRR and YTM, it does not apply to NPV. This seems odd since the methods are mathematically equivalent.
If we do assume reinvestment of all cash flows, what does that say about dividends and free cash flows? Are we assuming that those cash flows will never be used for anything else but reinvestment? If this is true, it sounds very strange.
Please help me understand this better.
Many thanks,
David
Tags: assumption, reinvestment
Views: 589
Replies to This Discussion
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Permalink Reply by Ignacio Vélez Pareja on
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Hello David
Yes, there is. NPV assumes (when using any discounting formula or Excel function) that the reinvestment rate is the same discount rate; it is the same with the Benefit/Cost ratio. The IRR, on the contrary, assumes that the reinvestment is at the same IRR. This is one of the assumptions that might induce to the classical problem of contradiction between IRR and NPV. The other assumption has to do with the differences in the initial invested amount.
The NPV reinvestment assumption makes sense when you think that the WACC or the cost of equity are considered as opportunity costs. The equity and debt holders will at least, reinvest their money at the same expected or desired rate for them.
I have a paper on making IRR and B/C match with NPV decisions: The Weighted Internal Rate of Return (WIRR) and Expanded Benefit-Cost Ratio (EB/CR). http://papers.ssrn.com/abstract=242867. Hopefully this will help you in the understanding of the problem.
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Permalink Reply by David Marsh on -
Thank you for the link to your paper. I will make sure to download and read it.
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Permalink Reply by Amol Sangeet on
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Hi David,
Yes, in NPV calculation there is an implied assumption that the cash flows will be reinvested at the discounting rate. This appears to be a more realistic and conservative assumption when compared with the IRR where the cash flows are assumed to be reinvested at the IRR itself.
If you want to specify a different reinvestment rate, you can use the MIRR.
If you are discounting Free Cash Flow to the Firm (FCFF) dividends are irrelevant; if you are discounting Free Cash Flow to Equity (FCFE), you discount the cash flow before dividend payout by the cost of equity and hence assume that the dividends, had they been reinvested, would have been reinvested at the cost of equity.
Both FCFF and FCFE are calculated after deducting capital expenditures, among other outflows. Hence, those are factored into the DCF analysis.
Best,
Amol Sangeet.
P.S. Textbooks being textbooks, confusion is bound to be there. When the noise becomes too loud, I listen to Aswath Damodaran. :)
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Permalink Reply by David Marsh on
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Great, thanks!
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Permalink Reply by Umed Saidov on
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David,
Your questions are very practical. I think the financial/mathematical side of it was pretty much covered by previous respondents... So allow me to focus on structural side of your question.
In reality money is fungible meaning it is hard to say which dollar was spent and where. Investment professionals usually find out post-factum (through audited financial statements) what happened to the amounts they have invested into a company. For debt holders teh deals can be structured in a number of ways to increase the probability of expected returns (loan covenants, etc.). Investment amounts can also be tied to specific projects (an area in finance called project finance) and can only used only for approved sets of activities (which in turn are tied to expected returns for investors).
For equity investors dividends are a part of their return (which they may assume to be reinvested at their preferred rate). The logic here is that if the company they invested in promises better returns they will opt-out of their dividends. Assuming the equity markets are efficient, their share prices will increase (based on expected growth) allowing these investors to sell a part of their shares and realize "homemade" dividends if needed.
hope this helps
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Permalink Reply by David Marsh on
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Thank you for your reply.
One question though: If we assume reinvestment of all future cash flows to justify today's asset prices (assuming we price assets using DCF), doesn't that mean that we also assume that no future cash flows will ever be taken out of the investment for discretionary purposes (e.g., buying clothes and food, going on vacations, etc.)?
It seems that by discounting future dividends or FCFs to price stocks, we assume that marginal investors will always reinvest their dividends or FCFs. However, they may be just as likely to use the dividends or FCFs to buy snack food.
Isn't constant reinvestment a heck of an assumption to make when pricing assets?
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Permalink Reply by Umed Saidov on
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Not really....I am not sure how other investors approach this, but I don't mix my investment CFs with those related to my personal needs (food, clothes, travel, etc.) Investments by their very nature are excess cash flows that (i) you don't need immediately and (ii) can deliver better returns/utility than immediate spending.
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Permalink Reply by HARIHARAN.G on
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Good one !!
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Permalink Reply by Robert Donohue on -
David,
The assertion that the IRR (or the NPV) depends upon the rate at which proceeds are reinvested is quite strange.
I believe it is one of the most frequently stated errors in discussions of the IRR.
If you calculate the IRR for the cashflows under Beg. Balance (see below), you will find that the IRR is 15%.
The IRR is both a discount rate and a yield.
Therefore we can invest the initial cashflow @15%, draw down the period payments and determine the ending balances (as below).
If this is carried through to the end, we find that the remaining balance is zero showing that cash remaining in the investment consistently earns the IRR.
Yet nowhere is there any need for the reinvestment of the cashflows flowing from the investment at any rate to attain this rate.
I believe one source of this error is the HP-12C handbook which, for years, has described a method for handling negative cashflows occurring in a series. HP directs one to discount all negative cashflows back to a PV using a "safe rate,", and to compound forward all positive cashflow using a "reinvestment rate." As a result one winds up with only a Present Value total and a Future Value total. Calculating the yield (MIRR) from these two values becomes very simple.
Unfortunately the handbook doesn't explain where the reinvestment rate comes from: cost of funds, hurdle rate, opportunity cost ??? But if we accept this method, it is quite evident that the Yield (IRR, MIRR) depends upon the rate at which the periodic cashflows are reinvested. If one selects the rate, then why not just use the NPV method?
(In passing, the calculation of the MIRR as directed in the HP handbook, and parroted in Excel, is the worst possible example of cash management.)
I corresponded with a N.Y. University professor of finance a couple years ago, offered my evidence but he remained adamant in his position. So I expect that all his students will be taught this error, as it is in so many B schools.
If you told me that this same academic was also a Keynesian, I would not be surprised. Look how long it took to convince people that the world is not flat.
Bob Donohue CCIM
Sorry, the table will not format on this system.
Beg. Balance Interest Rate Principal + Int. Withdrawals Remain. Balance
479.59 15.0% 551.53 -100 451.53
451.53 15.0% 519.26 -125 394.26
394.26 15.0% 453.40 -150 303.40
303.40 15.0% 348.91 -175 173.91
173.91 15.0% 200.00 -200 0
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Permalink Reply by sergei on -
To my mind, this question is not that simple to answer. We should keep the math tight with the logics of the model.
Let us consider the 2 different discounting problems in a risk-free setting: 1) discounting actual cash flows 2) discounting free cash flows.
When dealing with the actual cash flows, I think, there is no reinvestment assumption for the whole life of the project. We just assume that the cash flow is paid to an investor and that this meets her requirements. Where is reinvestment in this model? Consider, that the life of the project is 5 years and that the $1000 is to paid at the end of year 1. Let the risk free discount rate be 10%. We evaluate this payment as $909. I just cannot see any reinvestment in this situation.
When dealing with the potential (or free) cash flows, we assume that the cash flow received at, say, the end of year 2 will be paid to an investor sooner or later without any loss in its value. That is there is an assumption that the cash flow is paid at the moment when the potential cash flow is received. This assumption can also be interpreted in so way that the cash flow is held by a firm and the firm maintains the value of the cash flow until the moment of actual payment.
As for IRR, I recall that the IRR calculation is based on the assumption that the NPV is equal to 0. This intends to estimate the maximum discount rate tolerable. If a discount rate is greater the cash flow does not meet investor's requirements.
The next use of IRR is to measure the yield of the project. But again we consider either the actual cash payments or assumed payments of free cash flows. Where is a reinvestment assumption here?
I think, that the reinvestment assumption is a myth which is engendered by a fact that the mathematical equation under the above mentioned assumptions of actual or potential payments is identical to the equation of cash flows paid at the end of the project life and reinvested at the same discount rate in interim periods.
This is my view. Objections are welcome.
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Permalink Reply by The Hunter on
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The reinvestment assumption is implicit, not explicit, but it is very important.
Assume you have an investment which requires $100 today, and will pay you $161 in 5 years time. The IRR of this investment is 10% per year, and the NPV is zero if you have a cost of capital of 10%. In this case, there is no need to consider reinvestment because there is no periodic cash flow to reinvest.
Now imagine a similar situation which requires $100 today and will pay you $10 per year, and return your $100 at the end. IRR is still 10% and NPV is zero at 10% cost of capital. However if you received those $10 payments and chose not to reinvest them at all, your final cash on hand in year 5 would only be $150 and not $161.
NPV/IRR theory would say you are indifferent between these two cash flows at 10% discount rate. But the only way to arrive at the same year 5 value is to have reinvested those proceeds at 10% to arrive at $161. This is the implicit reinvestment assumption.
www.alpha-hunter.com
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Permalink Reply by David Marsh on
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That makes sense. So what does this imply for, say, the DDM model? Are we to assume that all proceeds (dividends + stock buybakcs) will always be reinvested?
While I'm now perfectly clear on the math behind discounting/compounding and the implied reinvestment assumption, I'm struggeling to understand how this applies to real investors. We assume that most people save/invest to increase their wealth for future consumption. So how can we then use a DCF model that assumes perpetual reinvestment?
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