Dear Karnen,

Please find enclosed my latest paper on AIRR, recently published in the Journal of Mathematical Economics.   Section 8, specifically devoted to practitioners, shows three different examples, including HomeNet’s example from Berk and DeMarzo’s textbook.

I welcome your comments and thank you very much for your attention

Best regards

Carlo Alberto Magni, Associate Professor
Department of Economics, University of Modena and Reggio Emilia
The Engineering Economist – Area Editor

I forward Prof. C.A. Magni’s above paper to Prof. Peter DeMarzo to seek his comments.

Personally, I see Magni’s paper on AIRR is quite comprehensive and convincing.

Hi Prof. Peter DeMarzo,I sent herewith one article written by Prof. C.A. Magni, in which he
shows AIRR implementation by using the HomeNet’s example taken from your book (see page 70 and 71).Though of course, we could just jump to NPV, yet, by putting something into “rate of return” (in %) it is much easier to get the point across to the other side of the table in many project analysis discussions.

It works for somebody without or with short finance course in the backdrop. Though I see a lot of Corporate Finance traditional textbooks explaining away on how to get the IRR, but it is not really touching the bone of this IRR. One article back in 1976 by C.B. Akerson, I guess, appropriately quite well in giving us a better idea about what this IRR is. From this paper, IRR seems this concept is built around the savings bank account analysis, in which the intermediate value is pretty clear to forecast.Akerson, C.B., The Internal Rate Of Return in Real Estate Investments,

A Research Monograph, Prepared for the American Society of Real Estate Counselors, 1976. Looking forward to hearing your opinion on this.



Prof. Peter DeMarzo:


Hi Karnen,

I have looked briefly at this but still remain unconvinced that it is very practical.  Like IRR, it does not improve upon NPV.  And worse, it tempts users to rank projects by their returns – don’t you agree?

Ignacio Velez-Pareja (IVP)

Listen, I don’t like rates of returns, in general. Imagine this: you surely calculate THE internal rate of return of  one project. However, if you do correctly the valuation, as you know, you have DIFFERENT discount rates for each period. N discount rates. What do you do with ONE IRR and N discount rates? Which discount rate is the one you choose to compare IRR with it?

On the last line from Ignacio Velez-Pareja:

What do you do with ONE IRR and N discount rates? Which discount rate is the one you choose to compare IRR with it?

I guess, the question is not quite apple-to-apple. IRR is, as we know, an “internal” rate of return, calculating by only need to know two things : the cash flows and the period (underlying assumption : the interval of the cash flow from one period to another period is the same), which means, under normal condition, we should have one or single rate. Since this is a single rate, then we need to compare it with another ‘single’ rate, which we could use CAGR, or many methods to come up with one single rate.


Dear Karnen

Remember that when correctly done, your cashflow valuation depends on two variables, among others: inflation rate and leverage. Also remember the circularity when using some methods of valuation. Hence, if discount rates change with time, (inflation and leverage) you will end up with N different discount rates. Which one will you choose for comparing the IRR with? The highest? The lowest? The simple average? Some kind of weighted average? Weighted on which basis?

Remind me what is CAGR, please.


CAGR = Compounded Annual Growth Rate. Let’s say we were earning a total three-year holding rate of  certain %, then CAGR is the annualized rate of return over that 3-year investment period.

Of course, a single rate might be a bit questioning if we are talking for more than 2-3 year investment horizon, yet I guess, the interest in certain countries could be quite stabile over a couple of years.

How about Ku (cost of unlevered equity)? Do you think business risk of the company will stay relatively the same? I guess, again, for certain industries, especially the mature ones, we could have used one Ku or one discount rate to compare to against IRR.



Dear Karnen


I am happy to read that authors start discrediting IRR as a decision tool to rank projects. I keep saying what I have said during many years: Don’t use IRR to rank projects, HOWEVER, use NPV AND calculate IRR to show the “size” of your return; if your project/firm has a strange behaviour and you can define a constant discount rate to compare with. Those cases of constant discount rates don’t exist for 2 simple reasons: first, in presence of varying inflation even Ku will vary and second, when properly done, D/E will vary in real life (I don’t know real cases of keeping constant D/E or D%, which is not very easy to model) and discount rate will vary as well.


Again, workout the NPV, the IRR and what I call the discounted payback period that is the moment when the firm/project repays all the investment including interest (discount rate).  As follows:

  Imágenes integradas 1

The formulation assumes or shows constant discount rate i, BUT it can be non-constant. There is no (or better, I have no) compact formula to calculate Discounted Payback Period, with constant or non-constant I).

Yes, the problem is not calculating IRR. Usually you can calculate it, but that is not the problem. The problem arises when you calculate IRR = 12.5%, say and your discount rates are 5%, 10% 9%, 13%, 14.5% 10%, whatever. Which discount rate will you use to compare IRR with? A different thing is to calculate the IRR to estimate your avergae return, just that. Will you average your discount rates? If yes, that finally is massaging data. Remember what a econometrist said: If you torture enough your data, they will confess and tell what you need or something like that (

Carlo Alberto Magni, Associate Professor

Department of Economics, University of Modena and Reggio Emilia viale Berengario 51, 41121 Modena

Dear Karnen,

Prof. DeMarzo‘s answer is in line with what he wrote to you in Dec 2015. He seems to dislike rates of return and prefer NPV.

I agree with him that NPV is the gold standard, but practitioners feel the rate-of-return notion is more intuitive, so it is our duty as scholars to provide an NPV-consistent measure of economic efficiency in relative terms

I sometimes work with practitioners and m always requested to provide an alternative to IRR which can be compatible with the NPV notion. AIRR is one such measure.

As for the idea that AIRR “tempts users to rank projects by their returns” my latest paper indeed shows that practitioners may use AIRR for ranking projects, getting the same result as the NPV ranking. You can download the paper:

Chisini Means and Rational Decision Making: Equivalence of Investment Criteria (by Carlo Alberto Magni, University of Modena and Reggio Emilia – Department of Economics; Piero Veronese, Bocconi University – Department of Decision Sciences; Rebecca Graziani, Bocconi University – Department of Policy Analysis and Public Management) Date written September 14, 2017

, where I presnet two different-but-equivalent method.

This is not to say that ranking projects with rates of return is recommended in general. It is only to say that (standardized) rates of return may be reliably employed for ranking projects. Evidently, care is needed to handle these cases and, needless to say, ranking with IRR is NPV-inconsistent.

I am currently working on a monograph presenting an integrated approach to proejct appraisal. Its title is “Project appraisal and the logic of valuation. Linking finance, acconting, and engineering economics” and all these issues will be investigated.


My paper on reinvestment assumption of IRR and NPV, coauthored by Prof. John Martin,  is now available on SSRN, with the title : “The Reinvestment Rate Assumption Fallacy for IRR and NPV“.

Click here to download it.

Best regards

Carlo Alberto


I sent to Prof. Peter DeMarzo, the paper co-written by Carlo Alberto Magni under the title: Chisini means and rational decision making: Equivalence of investment criteria.


Peter’s’ response :  while the math is fine, I have yet to see a practical example where AIRR provides a useful and differentiated insight.  Do you know of one?


Leave a Reply