Dear Graham N. Bornholt

In fact, I realize that both IRR and NPV are suffer with this assumption (the reinvestment of a project's interim cash flows at the same rate), that is one of the drawback of these methods in evalute the project.

But, in contrast if we let the reinvestment rate is fluctuated, so how does if fluctuate? We face to another problem: we have to assume the fluctuated reinvestment rate? Is it more difficultly or easier?

In my opinion, I accept this assumption and then I do the risk analysis such as scenario, the Monter Carlo simulation to evaluate the risk from this project, in which we have incoprate the fluctuation of reinvestment risk.

In hope these ideas are helpful for you

Cheers

Phuc Canh

If IRR is used to select one of many investment projects, the only relevant rule is to use the same method in determining IRR for all projects compared.

If it used with a benchmark value in mind, a certain IRR value that is determined empirically as a minimum threshold for feasibility, than the rule is to use the same method used for the generation of the empirical determined value.

However, when annual values of net cash-flows before discounting are calculated, some outflows such as installments, interests and profits can be included, and i suppose they should be included.

IRR and NPV are just ways to incorporate the change in the significance of the measurement unit, the currency. On dollar today has a different value than yesterday, and a different value than tomorrow.

To summarize, the reinvested interim cash-flows are included or not in the annual values of the cash-flow, before using discounting. The inclusion or non-inclusion depends on the way the investment is designed, but, if such outflows as installments and interests or profits are planned, they should be included.

Thanks Nguyen and Bradut. Just to clarify: My question is very focused. I am referring to the IRR calculation (not the rule). Some believe that the IRR calculation implicitly assumes that net cash inflows received before the end of the project are reinvested until the end of the project at the IRR rate. If you believe that the IRR calculation involves any implicit reinvestment assumption, can you justify your belief?

Unfortunately for the feasibility studies i made reinvestment was part of the plan, and part of the investors intentions. But studies i have done were confined to SME's. Probably for a listed company, it may be a different case.

Dear sir,

It is correct to say that IRR implicitly assumes that the cash flows are reinvested at IRR itself. My justification will be as follows.

One of the drawbacks associated with IRR is that it implicitly assumes that the cash flows are reinvested at the IRR itself. This is incorrect because in the actual scenario, the cash flows have to be reinvested at the prevailing discount rate. IRR is simply the rate at which the initial investment of a project will be equal to the present value of all the subsequent cash flows and hence it is merely a cutoff rate that can be used to assess whether a project is profitable or not.

Just assume that we are investing a sum of money in a bank and in return we will get a compounded interest. Compounded interest is the interest earned on interest. In this case, the bank uses the prevailing interest rate for compounding purpose and hence all the cash flows are reinvested at the prevailing interest rate. Discounting is the reverse of compounding and time value of money requires us to find the discounted value of future cash flows to compare with today's investment.

In the case of IRR, we are just finding the cutoff rate that equates the project’s discounted future cash flows to the initial outlay. Hence the cash flows would be discounted at the IRR itself. That implies that the future cash flows are reinvested at the IRR itself. NPV is superior to IRR because we are discounting future cash flows at the prevailing discount rate to assess the profitability of the project.

Best regards

As a matter of fact all DCF based capital budgeting rules and valuation models assumes reinvestment.  whether it is NPV, IRR, YTM on bonds....Therefore, Modified IRR is the answer. Even cost of capital may not be appropriate, use a realistic discount rate, 

Dear Sir,

First of all, I must tell you that, I too sometimes get intrigued by this question. What I feel is although there is no firm proof to justify the implicit reinvestment assumption at IRR from its calculations, the terminal values calculated for the project has often been used by researchers to prove their point.

Thanks Thushari, Shyam and Rohit.

To Thushari: Your argument in your last paragraph is that since the IRR discounts future cash flows at the IRR rate then this implies future cash flows are reinvested at the IRR rate. Let's see what this means for a specific example. Suppose a small investment of $50 returns $100 in one year and a further $1 in 20 years. The IRR is 100.000004% p.a. Your argument is that because $100 is discounted 1 year in the IRR calculation then this implies that the $100 will also be reinvested for 19 years (to the end of the project). What is your reason for thinking this? (Especially because, if correct, this would mean that the IRR calculation assumes the $50 will grow to more than $52,000,000 by the end of the project!)

To Shyam: My question is why you think that "as a matter of fact" the IRR assumes reinvestment of future cash inflows. [Aside: You say modified IRR is the answer but Modified IRR is the answer to the question: If the cash flows from this project are reinvested in other projects that earn y% per period until period z then what is the overall rate of return? As such, MIRR is a multi-project rate of return. In the above example, MIRR = 13.3% given a reinvestment rate of 10%. Clearly the hypothetical reinvestment projects drive the MIRR result.]

To Rohit: Yes, in the past a number of people have proposed justifications that involve terminal values. However, so far none of these arguments have proven the existence of an implicit IRR reinvestment assumption.

To Graham N. Bornholt,

I think you answered your own question. The example you gave in fact shows that IRR assumes a reinvestment at IRR. I think the real question is if this assumption is a reasonable assumption or not.

Specifically, if you invest $50, and receive $100 at year 1 and $1 at year 20, the IRR calculation shows IRR = 100.0000038 (and so on). If you were to reinvest the $100 at the end of year 1 at this rate, at the end of year 20 you would have a FV = $52,428,819. Now if you add the $1 from period 20, you have $52,428,820. That is the future value of your project in 20 years if we assume reinvestment at IRR. Now if you calculate the PV of this number using the IRR as the discount rate, you get PV = $50.00 exactly.

Therefore, mathematically, IRR in fact dues assume you are reinvesting at IRR. This is the answer to your specific question. Now if you are asking if this is reasonable, I would argue it is not and therefore another example of the shortcoming of IRR. MIRR would give you the flexibility to adjust the reinvestment rate and in fact you can set a different discount rate and reinvestment rate using MIRR.

Thanks Kevin for your answer.

Your second paragraph calculation shows that if the $100 is reinvested in another project for 19 years at the same IRR as the original project then the combined project (original +hypothetical) has the same IRR as the original IRR. This is what you would expect, much like the average speed calculation. Sure, if you extend a trip at the same speed as the first stage’s speed then the whole trip’s average speed is the same as the first stage’s speed. However, this does not mean that if I drive for 1 hour at an average speed of 60 m.p.h. then the speed calculation itself implicitly assumes that I will continue to drive at an average speed of 60 m.p.h. for the next 19 hours.

I do agree with the following statement S: “Investing the proceeds from the original project in a second project with the same IRR would lead to a combined project with the same IRR.”

However, while we can both agree on the validity of S, you have not explained why the validity of statement S means that the IRR calculation assumes reinvestment of interim cash flows at the IRR.

Since your problem is for 20 years. If you reinvest in another project at the same IRR, your second project also will end at the same time. That is the assumption, that you reinvest at IRR. That could be another project or reinvesting into the same project. Remember IRR is the rate at which NPV of the entire project equal 0. Which is what your example showed.

This is why IRR has a draw back because you cannot always assume you can reinvest at IRR in real life.

This is simply a mathematical truth about the IRR calculation. NPV N the other hand assumes you reinvest at some discount rate (for example WACC). This assumption is also not always correct bit of is more realistic.

Looking at your speed calculation example, you said if you average 60 mph for the first hour it does not mean you will average 60 mph in the next 19 hours. However if you had an extra bit of information like you know that the entire trip will be 1200 miles and it takes you 20 hours total, then you do in fact know that you will average 60mph for the entire trip.

The fact that you have an initial investment "price" in your example is that extra bit of information.

Thanks Kevin. I think I see your argument more clearly now.

Let’s call my original project A. That is, project A requires an initial investment of $50 and returns $100 cash in 1 year and $1 cash at the end of year 20. Project B requires an initial investment of $50 and returns $100 cash in 1 year. These projects have almost identical NPVs and IRRs (100.000004% and 100%, respectively). Also, Project A’s Macaulay duration is 1.00000036 years and Project B’s Macaulay duration is 1 year. So for both projects, on a per dollar basis the average length of time invested is approximately 1 year.

You however see these projects as dramatically different projects. You want your rate of return to be determined from the terminal wealth at the time when a project ends by the formula:

$50(1+r)n = terminal wealth,    (1)

where n is the project length.

In the case of project A, no such terminal wealth is available so you need to assume a reinvestment project to get you to year 20. This means that, in general, the r you seek in (1) is a multi-project rate of return since it must involve a sequence of projects. Thus, you are not seeking a single-project rate of return. Note, this is your personal preference and is not an assumption of the IRR.

It is then an exercise in simple algebra to work out that if the reinvestment project involves reinvesting the original project’s interim cash flows at the IRR rate then the multi-project r in equation (1) will equal the original project’s IRR. Thus, your preference leads you to interpret this result as meaning that the IRR calculation assumes reinvestment at the IRR rate, when in truth it is your preference for using equation (1) that produces the misconception. IRR is a single-project rate of return that requires no reinvestment whatsoever. Project A is a 20-year project of 1-year duration but equation (1) assumes a project with n-year duration. There is no reason to think anyone would re-invest the $100 for 19 years, which means using equation (1) to work out Project A’s return is quite pointless.

Your comment on the speed example reinforces my point. Of course, if you knew the overall speed then you could work out what the second-stage speed must have been. Similarly, if you know what the terminal wealth will be from the combination of the original project and a reinvestment project, then we could work out the second stage. However, there is no way we can know from the original project’s cash flows alone what the terminal wealth would be from reinvestment. That is the point. We don’t even know if there will be any reinvestment, or for how long any reinvestment will occur.

You and I are almost in agreement now. However, I would argue you do know the terminal value. It is the future value. How do I know this, because it is FV=PV(1+i)^N and since you stated the PV as equal to $50 and by definition IRR is the discount rate that would cause NPV = 0, then I know the terminal value.

As for my preference of reinvesting the money, that is really the point. What exactly would a company do with the money? Reinvest it in another project and most like into a project with a different IRR. However if you assume a reinvestment at a different IRR, then you are simply using MIRR. So what you are asserting about IRR not explicitly assuming reinvesting at IRR is true for the modified internal rate of return.

Graham, I am actually enjoying this discussion quite a bit. Let me try one more thing. Assume we have two different projects A and B. For project A, you invest $100 and receive $100 at the end of year 1 and $100 at the end of year 2. For project B you invest $100 and at the end of year 2 you receive $261.80 (rounding off here).

If you calculate IRR, both come out to 61.803%. Project B has no reinvestment issues since you receive the money at the end of the project. Project A on the other hand has $100 you receive in the middle of the project. Now if we are looking at IRR only as our criterion, then both projects are equivalent.

The only mathematical way for both projects to be equivalent is to reinvest the $100 at the end of year 1 at IRR. The $100 will grow to $161.80. Adding that to the $100 at the end of the second year give you the same value as project B. Therefore, the fact that IRR would consider project A and B to be equivalent is tantamount to saying that project A is assumed to reinvest its intermediate cash flows at IRR.

This assumption does not prevent companies or individuals from deciding not to reinvest or deciding to reinvest at a different rate. However, this is the assumption made when calculating IRR.

Thanks Kevin for your last two replies. I will respond to each:

Response to the earlier reply:

In your first paragraph you say “I would argue you do know the terminal value. It is the future value. How do I know this, because it is FV=PV(1+i)^N and since you stated the PV as equal to $50 and by definition IRR is the discount rate that would cause NPV = 0, then I know the terminal value.” This is not a correct result of NPV = 0 in this case, since 50 = 100/(1+IRR) + 1/(1+IRR)^20 means that the $50 is the present value of $100 in 1 year and $1 in 20 years. Thus, the future values from the $50 are $100 in 1 year and $1 in $20 years, combined.

My point about your preference for calculating rate of return from a sequence involving hypothetical projects is the unrealistic nature of what you want to do. Suppose Apple undertakes a 10-year project. Do you imagine that they invest the year seven cash flow in a 3-year project, the year eight cash flow in a two-year project and the year nine cash flow in a 1-year project just so all these projects’ cash flows end at a common point in order for your calculation to make sense?

Response to later reply:

Yes, I am enjoying the discussion too. Regarding your two examples A and B. They may have the same IRR but A obviously has the shorter duration, different NPV, etc.

In general, I am only concerned here with IRR as a rate of return (rather than in a decision rule, for example). So I view these projects are far from equivalent. I know IRR has a number of problems, and I am not actually advocating IRR since it obviously has flaws in terms of interpretability, possible multiple values and other problems. What I am concerned about is that there is this misconception that IRR has an additional problem (reinvestment at the IRR) that is not justified, and that this is used to provide an incorrect motivation for people to use MIRR.

1. I have to reiterate that all DCF based models, rules and techniques (e.g. IRR, NPV, Profitability Index (PI,) Equivalent Annual Charge(EAC), Discounted Payback(DPP) , Yield to Maturity(YTM) and Duration on bonds imply reinvestment of intermediate cash flows at the discount rate used in the computation. 

2. IRR rule is at flaw only when it turns out be very high and when reinvestment rate could be changing during the life of the project. In these cases Modified IRR (MIRR) is more desirable (assuming that a realistic reinvestment is chosen)

3. In case of zero coupon bond or any one cash input-one cash output investments (e.g. real estate, art and antiques) where no intermediate cash flows are present, IRR is the best criteria with no flaws at all!!!!

My two-cents worth!

Dear All,

There are so many misconceptions about IRR that it's hard to know where to start.

I think this article intuitively explains much of the modern confusion about IRR:

http://www.propertymetrics.com/blog/2014/06/09/what-is-irr/

In case anybody wanted to discuss some of this in more depth, please feel free to ask.

This is not to say that IRR is perfect. IRR simply doesn't work when the cashflows are not one sided (i.e., there are inflows as well as outflows). In such situations MIRR would be much more appropriate.

Hope this provides some clarity into IRR.

http://www.propertymetrics.com/blog/2014/06/09/what-is-irr/

Thanks Dawid for your reply. The purpose of my question is very specific.

It has nothing to do with the pros and cons of the IRR, but rather the question aims to draw out the reasons why some users mistakenly believe that the IRR calculation implicitly assumes that the inflows from the project will be reinvested into other projects (that also earn the IRR rate) until the end of the actual project. So far, all the justifications put forward in answers have been shown to fail. Of course, some will continue to believe there is an implied reinvestment assumption even if they cannot articulate why they hold that belief.

Your propertymetrics blog fortunately does not make the mistake: the IRR calculation does not have an implicit reinvestment assumption.

Dear Graham.

I totally agree with you.

In my opinion the flawed assumption relies on the fact that compound interest calculated in time is simply unintuitive. My go-to quote on this point is from Richard Price (1777): "A penny, put out to 5 per cent. coumpound interest at our Saviour’s birth, would, by this time, have increased to more money than would be contained in 150 millions of globes, each equal to the earth in magnitude, and all solid gold."

One can easily see this when comparing cashflows of plain-vanilla bonds against PIK (payment-in-kind) bonds and zero-coupon bonds[, assuming the same nominal interest rate on each instrument].

But I think that, since we agree on this point, it may not be necessary to argue it between us any further.

Dear All,

Thank you for your past participation and answers to my original question. Some more comments about the reinvestment assumption can be found in my paper on a project's implied rate of return just published in the Early View section of the Abacus journal:

http://onlinelibrary.wiley.com/doi/10.1111/abac.12093/full

Cheers, Graham

I think I could help focus this debate.  If anybody believes that the IRR assumes that the net cash inflows each year are reinvested at the IRR, I ask the question, "Into what are these funds being invested into?"  In particular, are these being invested into an independent project that would be undertaken whether or not the project being analyzed is undertaken?  If not, these funds would be reinvested into dependent projects, projects that are only undertaken if the original project is undertaken.    Answering this question in a non-wishy-washy way could help lead to a resolution of the confusion between the two sides.  I will wait to get people to specify what type of project these funds would be reinvested into.

David, that fact that your question (for those who believe that IRR assumes reinvestment of interim positive cash flows) "Into what are these being invested into?" remains unanswered, suggests perhaps that such believers are being forced to reconsider the justification for their belief. Well done!

Graham and other Economists:

1. Normal Net cash flow (NCF) and some non-normal  NCF do not involve reinvestment at IRR or at hurdle rate;

2. However, some NNCF do involve reinvestment  and those NNCF leads to multiple IRR.

3. The new method based on capital amortization schedule (CAS) to estimate IRR and NPV (that perfectly match the IRR and NPV by DCF method) is more transparent clearly indicate the reinvestment or not. A modified CAS method eliminate the reinvestment and leads to a unique IRR (multiple IRR is resolved)>

The illustrations and proof are available in the following papers (modified versions):

a. A Resolution to the Problem of Multiple IRR: A Modified Capital Amortization Schedule (MCAS) Method for Non-Normal Cash flow (NNCF) to Obtain a Unique IRR (July 11, 2017). Available at this link: SSRN: https://papers.ssrn.com/sol3/papers.cfm?abstractid=3000729

b. "IRR Performs Better than NPV: A Critical Analysis of Cases of Multiple IRR and Mutually Exclusive and Independent Investment Projects" that question the conventional wisdom of using NPV for mutually exclusive projects and the problem of no or negative or multiple IRR. Please let me have any comments.https://ssrn.com/abstract=2913 905

Regards  Kannan

I think it is telling that the authors of one of the leading corporate finance textbooks felt compelled to dismiss the "reinvestment assumption". Here is the full quote:

"We will take a stand on one issue that frequently comes up in this context. The value of a project does not depend on what the firm does with the cash flows generated by that project. A firm might use a project’s cash flows to fund other projects, to pay dividends, or to buy an executive jet. It doesn’t matter: How the cash flows are spent in the future does not affect their value today. As a result, there is generally no need to consider reinvestment of interim cash flows."

(S. A. Ross, R. W. Westerfield, B. D. Jordan, Fundamentals of Corporate Finance, New York: McGraw-Hill 2013, p. 290.)

Dawid:  Agree, ".. there is generally no need to consider reinvestment of interim cash flows."  But the question is  Does the DCF method estimated IRR and NPV involves reinvestment at IRR or at hurdle rate, respectively, ? The fact of the matter  there are two issues: a. data problem and b. methodological problem. First normal NCF and some non-normal NCF(NNCF) data will not lead to reinvestment. However, some NNCF leads to reinvestment and reinvestment in turn leads to multiple IRR and multiple NPV too!.  Second, the DCF ,methods is not suitable to deal with NNCF data. 

The assertions that a. With all NCF and NNCF data, IRR and NPV involves reinvestment (false assertion) and therefore NPV is preferred because reinvestment is at hurdle rate and also assuming MIRR solves this problem is a fallacy (see my papers referred above) b. Some NNCF data leads to reinvestment that could be identified using a CAS method introduced in my paper to estimate IRR and NPV . Reinvestment happens when the CAS have one or more positive opening balance (interest income). c. The modified CAS (MCAS) method introduced in my paper eliminate the reinvestment income and resolve the problem of multiple IRR and also reveals with reinvestment the NPV is also wrong. 

Kannapiran: In your first response, you make a number of assertions about reinvestment but direct the reader elsewhere to your working papers for your justifications (apparently based on something else called the ‘capital amortization schedule’ (CAS)). One assertion you make is that “some NNCF do involve reinvestment”, however you provide no justification here for this. You cannot expect readers to spend time reading your working papers if you cannot at least show how the calculation of IRR involves reinvestment in a straightforward example (in an explanation that does not involve CAS, since there is no reason to view a project’s cash flows through this construct).

In your second response, which you direct to Dawid, you make more assertions without justification. Just one example of a project for which you are able to show that IRR assumes reinvestment is all that is needed to change your assertions into a justification. That is the challenge that my question offers those who believe IRR assumes reinvestment. So far, no one has been able to provide a convincing justification. Can you?

Thanks Graham and appreciate your comments. You know more than me that its hard to explain the numerical evidence of reinvestment with most NNCF investments here in this site. Thats the main reasons why I referred to my papers that present evidences than assertions.

The IRR and NPV estimated by the CAS method match 100% with the IRR and NPV estimated by DCF method. That being the case if CAS method reveals reinvestment with some NNCF and then that is equally applicable to DCF method with the same results. Now, what have been teaching and publishing in Texts for over a half century are based on assertion that there is reinvestment or no-reinvestment.

I have furnished worked out illustrations that there is reinvestment with some (Not all) NNCF. When there is reinvestment that leads to multiple IRR. I introduced a new method (modified CAS - MCAS) that handle well the reinvestment and resolve the multiple IRR problem and leads to a real and unique IRR. All these are explained with numerical evidences in my papers. None of my argument is based on assertion but based on numerical evidence.

Researchers, if serious, must read those papers and then can either question/reject or accept my findings. Thats is the prupose of my posting here to alert readers to those papers. I am sure in 5 or ten years, teaching and texts will move away from DCF to CAS method which is more transparent and resolve most of the problems that led to various criticisms of CBA and CIA.

Hope, my reply is OK with you. Cheers Kannan

Well Kannan I took up your challenge to read one of your papers. Since your paper b. is so provocatively titled ("IRR Performs Better than NPV: A Critical Analysis of Cases of Multiple IRR and Mutually Exclusive and Independent Investment Projects") this is the one I chose. My comments are as follows:

If I tell you that I have bought a painting that will give me a return of 30% p.a., how rich will it make me? Will I become a multi-millionaire? Of course, there is no way for you to know the answer because 2 aspects of the investment are missing: How much did I invest and what is the duration of the investment. Similarly, if I have to choose between buying a painting that earns 30% p.a. and buying a painting that earns 40% p.a., there is not enough information to determine which choice I should make. This is why rates of return methods such as maximising IRR can never be the appropriate criterion for selecting mutually exclusive investment projects. As many corporate finance texts discuss, differences in SAT (project scale and cash flow timing) make IRR unreliable as a decision-making criterion.

Given this background, the academic preference for NPV over IRR is unassailable and has been settled a long time ago. It remains to spot the flaw in your arguments. I see immediately that you do not distinguish between present values and discounted cash flows. IRR is the discount rate that would make NPV = 0, but this does not mean that when you discount cash flows with the project’s IRR that you are finding the cash flows’ present values. Let me explain. Suppose I am entitled to $1000 in 1 year. The PV of this $1000 is an amount now that I would accept to give up my right to the $1000 in 1 year. Say it is $930, implying an NPV discount rate of 7.5%. Now I might be a bit uncertain about the appropriate discount rate for some risky investment projects, should the NPV discount rate be 8%, 9%, 10%, maybe even 12%? (This is the reason that sensitivity analysis is recommended.) Suppose I choose 10% for the NPV discount rate for project B which has an IRR of 20.9%. If I then discount B’s cash flows with a different discount rate (say 33.7%), and sum them up then I know that the answer I get is definitely not project B’s NPV. Why should we care what that number is? Yet your method finds that number useful, and its sign the determining factor (your set 3).

In short, your paper does not address the key fact that maximising IRR can lead to inferior decisions because of differences in project scale or differences in cash flow timing. In your set 3, for example, your method chooses the small project A with the larger IRR even though the alternative project would have produced more wealth.

Thanks Graham. It’s interesting to read your comments and let me respond to each one of your comment:

1. If I tell you that I have bought a painting that will give me a return of 30% p.a., how rich will it make me? Will I become a multi-millionaire? Of course, there is no way for you to know the answer because 2 aspects of the investment are missing: How much did I invest and what is the duration of the investment. Similarly, if I have to choose between buying a painting that earns 30% p.a. and buying a painting that earns 40% p.a., there is not enough information to determine which choice I should make.

My Reply: In CBA or CIA, the unknow variable is the rate of return and the known variable is the capital investment. If we know the rate of return (IRR), we can simply arrive the how much is the value added or how rich the investment will make you. It’s that simple. With NPV you do not know the rate of return and no one can say with NPV how much the return on investment. NPV can at best indicate that the rate of return is above the hurdle rate! NPV is like saying that a student passed the exam but no one knows how much is his mark (in percentage term it’s like IRR). That’s why I discussed that NPV only provide an incomplete information and ONLY IRR could provide needed full information.

2. This is why rates of return methods such as maximizing IRR can never be the appropriate criterion for selecting mutually exclusive investment projects. As many corporate finance texts discuss, differences in SAT (project scale and cash flow timing) make IRR unreliable as a decision-making criterion.

My Reply: The weakness with NPV, as an incomplete indicator, is that it could not provide the maximum rate of return generated by mutually exclusive projects (A and B). If you estimate the IRR (as done in my paper), the full return on investment will be revealed that will rank the projects differently than by NPV. SAT equally affect NPV and IRR, and if you say IRR is only affected by SAT then the mathematical relationship: at IRR, NPV =0 is fundamentally questioned without evidence. Therefore, I commented that the published text must be rewritten.

3. Given this background, the academic preference for NPV over IRR is unassailable and has been settled a long time ago.

My Reply: In research, if one considers an assumption as unassailable then the world should have been flat forever! Academic may be settled for reasons best known to them but company executives and practitioners continue to prefer IRR.

4. I see immediately that you do not distinguish between present values and discounted cash flows. IRR is the discount rate that would make NPV = 0, but this does not mean that when you discount cash flows with the project’s IRR that you are finding the cash flows’ present values.

My Reply: The DCF method is fundamentally built on discounted cash flow and NPV is the discounted flow at hurdle rate but unutilized DCF whereas IRR is the rate that makes the DCF zero (NPV = 0) thereby revealing the maximum return or the full utilization of DCF. IRR can ONLY reveal the return on investment and NPV is a partial indicator at hurdle rate.

5. Let me explain. Suppose I am entitled to $1000 in 1 year. The PV of this $1000 is an amount now that I would accept to give up my right to the $1000 in 1 year. Say it is $930, implying an NPV discount rate of 7.5%. Now I might be a bit uncertain about the appropriate discount rate for some risky investment projects, should the NPV discount rate be 8%, 9%, 10%, maybe even 12%? (This is the reason that sensitivity analysis is recommended.) Suppose I choose 10% for the NPV discount rate for project B which has an IRR of 20.9%. If I then discount B’s cash flows with a different discount rate (say 33.7%), and sum them up then I know that the answer I get is definitely not project B’s NPV. Why should we care what that number is? Yet your method finds that number useful, and its sign the determining factor (your set 3).

My Reply: Whatever the discount rate one uses, IRR is fixed (multiple IRR is the problem caused by reinvestment of intermediate income, see my other paper on resolution of multiple IRR) whereas, NPV keep changing with discount rates. The question here is the given the NCF what is the return on investment (IRR)? Are we testing whether the NCF leads to some unutilized NCF (PV of unutilized NCF is the NPV)? NPV is exactly doing that it could not reveal the return on investment for a given NCF!

6. In short, your paper does not address the key fact that maximising IRR can lead to inferior decisions because of differences in project scale or differences in cash flow timing. In your set 3, for example, your method chooses the small project A with the larger IRR even though the alternative project would have produced more wealth.

My Reply: My paper is not arguing that maximizing IRR as the target but rather IRR is the highest rate of return a NCF can support nothing! I am sure you know DCF method is based on mathematical relationship. Whether is a small sample or large sample the result will be the same. For your information, I have conducted CBA for major projects funded by World Bank and ADB with 25 yrs NCF (also in QLD) but the conclusion is IRR is the best criteria and NPV is an incomplete indicator.

Kannan, there is some miscommunication going on. Perhaps you can call me if you want to discuss your points.

Graham,

Would you agree that YTM does not require reinvestment of coupons at the YTM rate throughout the term of the bond? I found the this article and thought it was similar to the discussion on this thread..... http://www.economics-finance.org/jefe/econ/ForbesHatemPaulpaper.pdf

I am reviewing for the CFA but found the following video that seems to assume reinvestment income is part of a YTM....https://www.youtube.com/watch?v=jq-yJoACYHA

Would you say the video is wrong?

Michael,

Yes I agree that YTM does not require reinvestment at the YTM over the life of the bond, and that journal article has some good points to make.

My comments start with a caveat: While the video is wrong, it is in complete accord with the CFA Body of Knowledge (and CFA texts) which are also wrong. Thus if you are studying for a CFA exam, you need to pretend it is correct for the purposes of the exam.

The reasons why YTM does not assume or require reinvestment of the bond’s cash inflows are as follows:

Let’s look at the video's specific example. A 20-year semiannual 7% coupon bond selling at par of $100. This bond pays $3.50 interest every six months for 19.5 years then a final payment of $103.50 in year 20. YTM is also 7% (it is annualized), and the discount rate =IRR = YTM/2 = 3.5%. Macaulay’s duration (the effective center of the inflows) for this bond is 11.1 years. In contrast, what the video calls the ‘Realised Yield’ is the return calculated as if the bond had a duration of 20 years. It doesn’t. One way to get to a 20-year duration in this case would be to use all those 39 lots of $3.50 to buy 39 other bonds or investments that must all mature at the same date as the original bond. That is the assumption of the Realised Yield calculation. In other words, the Realised Yield is the overall return from a combination of 40 bonds or investments not 1 bond. Realised Yield should be renamed ‘Hypothetical Realised Yield’ because, except for investors with fixed investment horizons, most investors do not typically invest their coupon income from a bond in further investments that all expire at the same time as the original bond.

So what does YTM mean? Well, for a semiannual bond, investing $100 at a compound rate of YTM/2 per six months would be just sufficient to reproduce all of the bonds cash flows. In other words, 3.5% is the bond’s equivalent compound rate of return. The whole of the $100 earns 3.5% per period, but of course not all of it is invested for the full 20 years (of the $100 we can regard $26.14 as invested for 20 years since $26.14 = $103.50/1.035^40 [implying that $26.14*(1.035)^40 = $103.50]). The concept of the equivalent compound rate of return is explained more fully in my recent Abacus paper, and in an earlier draft which is publicly available at SSRN: https://ssrn.com/abstract=3055473. An important point to note that in this bond case, the IRR is the equivalent compound rate of return but this is not true for all investment projects.

Just edited my previous answer to correct it.

Hi

I read all the above interesting comments and opinions and is left with uncertainties. I would appreciate if you could give your thoughts on my 2 cents.

Since there is no clear consensus on the reinvestment assumption. My approach of understanding the reinvestment is to categorize cash inflow in terms of nominal value and real value.

Here is an example of a project: Period 0 with an initial investment of -100$, Period 1's cash flow 200$, and Period 3's cash flow is 0$. The cost of capital is 10%.

IRR is 100% and MIRR is 48%.

So basically IRR assumes the Nominal value of 200$ to be 400$ (Real value being still 200$) at period 3.

MIRR on the other hand, assumes the Nominal value of 200$ to be 220$ (Real value still 200$) at period 3. Which is way realistic because if we do not assume the nominal value to increase, then the nominal value of 200$ will be still 200$ which makes the real value 180$ at period 3.

So there is no real reinvestment case, just giving time value to each cash flow.

I would like to hear your thoughts on this. (I'm not saying this is a bullet proof assumption just an idea)

Thank you

Thanks Ulziitushi, your comment raises interesting issues to address (I will not use your real versus nominal naming convention as this contrast is usually reserved for inflation adjusting). Some background first. Most capital budgeting in firms involves calculating both NPV and IRR, with NPV providing the net gain in wealth from accepting the project in present value terms and IRR providing an (often inadequate) measure of the project’s rate of return. In principle, there is no conflict between the measures themselves. Let’s look at your example, call it A = (-100, 200, 0) with NPV discount rate (k) of 10%. NPV = $81.82 and IRR = 100%. Now all that our choosing k = 10% means is that we are prepared to accept certain cash flows at different times as being equivalent in value. So, we are saying, for example, that $181.82 now has the same value as $200 in year 1. Also, that $200 in year 1 has the same value as $220 in year 2. Applying the latter case, this means that we would regard project A as having the same value as a hypothetical project B = (-100, 0, 220). That is all it means. Projects A and B differ in many important ways. Project A is real while project B is imaginary, A has a return of 100% and a duration of 1 year whereas B is imaginary with a duration of 2 years and an IRR of 48.3%. There is not even a reason to assume investors would prefer B to A even if it was to become available.

Now, we could imagine synthetically creating B by first investing in A and then reinvesting the $200 from A in another project C that earns 10% for 1 year. However, there are two problems. Firstly, there is no evidence that C will exist in 1 year, and secondly, that the firm would want to invest in it. In short, C is not a real project either, so the MIRR calculation is imaginary as it is the outcome of hypothetical projects. This is why MIRR has made few inroads into the practice of capital budgeting.

Coming to IRR, in general (and for many, unintuitively) it is defined as the discount rate that would make NPV = 0. Care must be taken in interpreting what this means. The NPV discount rate is still 10% and NPV = $81.82. All the IRR definition means is that to get NPV = 0 you would need the NPV discount rate to be 100% (but it is not). Thus, the IRR rate has no implications about equivalences in value, this is the role of the correct NPV discount rate. For example, using IRR =100% does not mean that you regard $200 in year 1 as having the same value as $400 in year 2. It clearly does not. We should still regard $200 in year 1 as having the same value as $220 in year 2.

An interesting effect of compensating for the reinvestment rate assumption by using MIRR rather than IRR is seen when your IRR is below your discount rate. Of course, assuming "orthodox" cash flows and a single solution for the IRR, that would yield a negative NPV, and you would not invest in the project.

Now, if the IRR is used to calculate CFROI (cash flow return on investment - HOLT methodology by Credit Suisse) for an enterprise (or a subsidiary), and it is lower than the weighted-average cost of capital, it doesn't necessarily mean that you will divest. On the other hand, if you use the MIRR formula and mechanically enter the WACC as your return on reinvestments, you will overstate long-term performance, as CFROI is calculated across multiple years, including estimated future or terminal cash flows.

If you believe in the reinvestment assumption, you could argue that the MIRR represents the minimum potential performance of the enterprise, provided all future investments yield the WACC as minimum rate of return. The problem with that line of reasoning is that MIRR/CFROI assumes that also historical cash flows were reinvested at the WACC or another rate. Who invested those cash flows if the returns do not show up in the company’s P&L? The answer will have to be the shareholders, but surely the enterprise cannot take credit for how shareholders reinvested their dividends.

When applied to enterprise level, the reinvestment rate assumption fallacy becomes more obvious.

Reinvestment is a reality with non- orthodox NCF THAT LEAD TO MULTIPLE IRR. MIRR IS A SPURIOUS RATE. EVEN WITH ORTHID OZ NCF THERE ARE INSTANCES OF REINVESTMENT THAT AFFECT NOV MORE THAN IRR..IF YOU NEED EVIDENCE PLEASE REVIEW MY PAPERS in SSRN. REGARDS KANNAN

Thanks Christian for your comments. From what I can see of the proprietary approach of Credit Suisse, they estimate the firm's future cash flows and when combined with today's market value this allows them to calculate an implied IRR which they call CFROI. Clearly this approach need not rely on reinvestment at any particular rate. We know in general that the IRR calculation does not imply reinvestment at any particular rate. How well the firm's cash flows are reinvested in the future will depend on future investment opportunities and on the ability of the firm to take advantage of those opportunities. In such circumstances, making any assumption that cash flows will be reinvested at any specific rate such as WACC or IRR or anything else will turn results into no more than a thought experiment. I think most analysts realise that MIRR is to be avoided for this reason.

My question invites people who believe that IRR implicitly assumes reinvestment at the IRR rate to justify their belief. So far, no one has been to able to provide a credible argument for the belief here in this forum. Hopefully, readers find the discussions are informative.

Kannan, you have repeated your claim from an earlier answer that IRR assumes reinvestment at the IRR in some cases. Once again, you have been unable to provide your reasoning here. This forum is for people to try to justify their claims.

To Graham and others,

1. In case of zero coupon bond or any one cash input-one cash output investments (e.g. real estate, art and antiques) where no intermediate cash flows are present, IRR is indeed the implicit reinvestment rate and I believe it is the best decision making rule, with no flaws at all!!!!

2. It is a mathematical truism that all DCF based models, rules and techniques (e.g. IRR, NPV, Profitability Index (PI,) Equivalent Annual Charge(EAC), Discounted Payback(DPP) , Yield to Maturity(YTM) and Duration on bonds, imply reinvestment of intermediate cash flows at the discount rate used in the computation. The big question is, whether intermediate cash flows are, can and will be reinvested at that rate? If not then all DCF models are flawed!

3. IRR based decisions are wrong only when it turns out be very high and when reinvestment rate could be changing during the life of the project. In these cases Modified IRR (MIRR) is more desirable (assuming that a realistic reinvestment is chosen).

Enough of this discussion as far as I am concerned.

Thanks Shyam. This question does not deal with IRR based decisions but with IRR itself. You claim that it is a mathematical truism that IRR (and other DCF techniques) "imply reinvestment of intermediate cash flows at the discount rate used in the computation." This is a very big claim that you provide no justification for. Surely a mathematical truism should be the easiest thing in the world to demonstrate! Having said that, you are entitled to your beliefs.

Shyam and Graham, Both os you are right as well as wrong. With NNCF investment there is reinvestment and with NCF no reinvestment with IRR but there is reinvestment in somce cases of NPV. Here is one of my recent paper posted in SSRN that expalin the logic with graphs.

Non-Monotonic NPV Function Leads to Spurious NPV and Multiple IRR Problems: A Critical analysis using a modified capital amortization method that Resolves These Problems Posted: Kannapiran Arjunan Date Written: August 3, 2018 Abstract: This analysis is conducted using some popular non-normal net cash flow (NNCF) investment data available in public domain and other hypothetical NNCF data. The methodology is mainly based on capital amortization schedule (CAS) and modified CAS (MCAS) methods along with a comparison of the results with the common DCF method. The findings are summarised here: a. The problem of multiple IRR is caused by reinvestment income and the resultant non-monotonic NPV function. The CAS methods clearly indicate whether there is any reinvestment. Non-monotonic NPV function of NNCF investment leads to multiple IRRs or spurious IRRs, NPVs and MIRRs. With non-monotonic NPV functions the DCF estimated criteria are all spurious. b. The MCAS method eliminates the reinvestment thereby leads to monotonic NPV function and resolves the problem of reinvestment, spurious NPVs, MIRRs, IRRs and or multiple IRR. c. Neither the NPV nor the MIRR could resolve the problem of multiple IRR. With normal NCFs and some of the NNCFs also, there are no reinvestment at IRR or at hurdle rate as wrongly asserted in many published works. d. It is normal for the estimated IRR to be either ‘nil or zero or negative’ when the sum of net benefits or NCF is zero or negative. Such IRRs are consistent with NCF or net benefit. IRR of ‘zero or negative or no’ is not a weakness or problem but it reveals the real or consistent return. e. MCAS is an appropriate method to estimate the rate of return (IRR and NPV) for both normal NCF and NNCF and resolves the multiple IRR problem and eliminates spurious NPVs and MIRRS. The estimated IRR and NPV by MCAS method are consistent with NCF. f. Ultimately, IRR and NPV, estimated by MCAS, are the best criteria available to investment, project and cost-benefit analysis. In summary, NPV and IRR estimated by MCAS method are equally appropriate and therefore one cannot be the best substitute for the other. The multilateral and bilateral organizations and corporate managements may wish to revisit their recommendation to use the NPV only and not the IRR while dealing with multiple IRRs associated with NNCF investments. Keywords: Capital Budgeting, Investment Analysis, Non-normal NCF investments, non-monotonic NPV functions, Multipe IRR and spurious NPVs and MIRR, New method Resolves These Problems JEL Classification: D, D61, G3, G31, O2, O22, O12 Suggested Citation: Suggested Citation Arjunan, Kannapiran, Non-Monotonic NPV Function Leads to Spurious NPV and Multiple IRR Problems: A Critical analysis using a modified capital amortization method that Resolves These Problems (August 3, 2018).

Available at SSRN: https://ssrn.com/abstract=3225559

Graham N. Bornholt,

Thank you for starting this discussion - it truly does hit on a contested point in finance today.

I think semantics play a role in the confusion. IRR creates an average return and average cost of capital that is than applied over a spread of cash inflows/outflows.

So I believe it follows to say that we are essentially assuming that these interim cash flows have a reinvestment rate = to IRR. And we can show this mathematically in a very simple fashion.

We can think of these incremental cash flows as simply adding to an investment pool on which a return is earned (only hypothetically as is implied by the calc). (It matters little whether a reinvestment actually occurs) it simply means capital base has increased and the IRR must be adjusted to reflect this. So people often than think that the calc has implied there must have been reinvestment in a project when in actuality this is not the case.

The key here is that reinvestment does not imply investment in a separate project - or a project at all - it simply takes into account an opportunity cost for the cash flow that on average will make it so all cash flows' NPV = 0 given the IRR.

And the only way this is mathematically possible is if IRR is that investment rate.

To conclude - we keep talking about investment rates when there is no assumption of reinvestment of interim cash flows. So I'd say there is just an overall misunderstanding on both sides.

Please refer to the following articles for analytical evidence on reinvestment and its impacts:

Non-Monotonic NPV Function Leads to Spurious NPV and Multiple IRR Problems: A Critical analysis using a modified capital amortization method that Resolves These Problems (August 3, 2018), Economic Papers , Volume 38, Issue1 March 2019 Pages 56-69. SSRN: https://ssrn.com/abstract=3225559

Arjunan, Kannapiran, Validity of NPV Rule and IRR Criterion for Capital Budgeting and CBA (December 17, 2019).Available at SSRN: https://ssrn.com/abstract=3505058

Cheers Kannan