# Internal Rate of Return: A Cautionary Tale

Tempted by a project with high internal rates of return? Better check those interim cash flows again.

The McKinsey Quarterly, McKinsey & Co.
October 20, 2004

## End the etrnal accounting myopia leading to IRR/NPV debate

Dear all, A Kind reference is drawn to my arguemnts posted in Nov 2009, I recap the relevant portion below: 1. The companies operate in risk/return chunks (e.g those in oil and gas exploration space typically invest in high risk/high gain ventures while those operating in infrastructure or FMCG space have lower risk exposure and comparatively moderate returns) this means that the companies keep on investing in the "reward/risk chunk" they operate in, hence the observation that how can the intermediate money received be assumed as invested at the same rate reflects the thinking of a myopic accountant rather than that of a strategic forward looking CEO. 2. Considering the above as an axiom, the higher rate reflected in IRR is certainly in agreement with the CAPM theory, which is widely accepted. Regards Navneet Jain, IIFT(Indian Institute of Foreign Trade), New Delhi

Posted by Navneet Jain | October 16, 2011 04:25 pm

## youy are wrong

Good point. Tell me more. Why do people insest IRR implies a reinvestment rate? Does IRR consider the future value of money. Is it not a great tool for a equity investor on a single project with an inital investment and a series of positive cash flows My equity investors just want to know how much are thet going to make.

Posted by Dale Williams | February 07, 2010 09:41 am

## The Author of the Article is Right

One of the problems with the IRR is the reinvestment rate assumption in IRR. The prove is fairly simple. Suppose you have the following cash flows for a project: 0: -2,500 1: 500 2: 1,000 3: 1,500 As you know, by definition, IRR is the rate at which NPV = 0. Next step is to calculate all the PVs and make the equation equal zero. The result is: -3,000 + 500/(1+IRR) + 1,000/(1+IRR)^-2 + 1,500/(1+IRR)^-3 = 0 The resulting polynomial equation is one of 3rd degree, which means it has three mathematical solutions. Best way to solve it is using a numerical algorithm. The result is one real and two imaginary solutions. We are interested in the real solution which is 0.0821 or 8.21% return. You can get the same answer using a financial calculator or the function IRR in Excel. Now, the prove that the returns implies reinvestment of IRR. The problem basically says that you invested \$2,500 to got a return of \$3,000 over a period of 3 years. Using the compound interest formula, FV = PV(1+r)^n, you can calculate the real rate of return for the cash flows: PV = \$2,500 FV = \$500 + \$1,000 + \$1,500 = \$3,000 n = 3 years r = ? Isolating r, we get: r = (FV/PV)^(1/n)-1 r = (3000/2500)^(1/3) -1 = 6.27% Someone could argue that some returns were received earlier in the timeline. In this case it doesn't matter because we are trying to prove that the interim cash flows are not reinvested at the IRR. To avoid that and other problems with the IRR the best choice is to use the MIRR, which stands for "Modified Internal Rate of Return". If you need a more detailed explanation, or information on how to use the MIRR don't hesitate to let me know.

Posted by Joseph Todd | January 12, 2010 03:13 am

## The Author of the Article is Right

There are some typos in my previous post, here is the corrected version. ------------------------------ One of the problems with the IRR is the reinvestment rate assumption in IRR. The prove is fairly simple. Suppose you have the following cash flows for a project: 0: -2,500 1: 500 2: 1,000 3: 1,500 As you know, by definition, IRR is the rate at which NPV = 0. Next step is to calculate the PVs of all the cash flows and make the equation equal zero. The result is: -2,500 + 500/(1+IRR) + 1,000/(1+IRR)^2 + 1,500/(1+IRR)^3 = 0 The resulting polynomial equation is one of 3rd degree, which means it has three mathematical solutions. Best way to solve it is using a numerical algorithm. The result is one real, and two imaginary solutions. We are interested in the real solution, which is 0.0821 or 8.21% return. You can get the same answer using a financial calculator, or the function IRR in Excel. Now, the prove that the returns implies reinvestment of IRR. The problem basically says that you invested \$2,500 to get a return of \$3,000 over a period of 3 years. Using the compound interest formula, FV = PV(1+r)^n, you can calculate the real rate of return for the cash flows: PV = \$2,500 FV = \$500 + \$1,000 + \$1,500 = \$3,000 n = 3 years r = ? Isolating r, we get: r = (FV/PV)^(1/n) - 1 r = (3000/2500)^(1/3) -1 = 6.27% In fact, as almost always occurs, IRR overestimated the real return. Someone could argue that some returns were received earlier in the timeline. In this case it doesn't matter because we are trying to prove that the interim cash flows are not reinvested at the IRR. To avoid that and other problems with the IRR the best choice is to use the MIRR, which stands for "Modified Internal Rate of Return". If you need a more detailed explanation, or information on how to use the MIRR don't hesitate to let me know.

Posted by Joseph Todd | January 12, 2010 03:13 am

## The Author of the Article is Right

One of the problems with the IRR is the reinvestment rate assumption in IRR. The prove is fairly simple. Suppose you have the following cash flows for a project: 0: -2,500 1: 500 2: 1,000 3: 1,500 As you know, by definition, IRR is the rate at which NPV = 0. Next step is to calculate all the PVs and make the equation equal zero. The result is: -3,000 + 500/(1+IRR) + 1,000/(1+IRR)^-2 + 1,500/(1+IRR)^-3 = 0 The resulting polynomial equation is one of 3rd degree, which means it has three mathematical solutions. Best way to solve it is using a numerical algorithm. The result is one real and two imaginary solutions. We are interested in the real solution which is 0.0821 or 8.21% return. You can get the same answer using a financial calculator or the function IRR in Excel. Now, the prove that the returns implies reinvestment of IRR. The problem basically says that you invested \$2,500 to got a return of \$3,000 over a period of 3 years. Using the compound interest formula, FV = PV(1+r)^n, you can calculate the real rate of return for the cash flows: PV = \$2,500 FV = \$500 + \$1,000 + \$1,500 = \$3,000 n = 3 years r = ? Isolating r, we get: r = (FV/PV)^(1/n)-1 r = (3000/2500)^(1/3) -1 = 6.27% Someone could argue that some returns were received earlier in the timeline. In this case it doesn't matter because we are trying to prove that the interim cash flows are not reinvested at the IRR. To avoid that and other problems with the IRR the best choice is to use the MIRR, which stands for "Modified Internal Rate of Return". If you need a more detailed explanation, or information on how to use the MIRR don't hesitate to let me know.

Posted by Joseph Todd | January 12, 2010 02:57 am

## mathematical proof of IRR reinvestment

Hi Can anyone explain in simply in mathematical terms how the IRR calculation inherently assumes reinvestment at the IRR. And for that matter, how NPV assumes reinvestment at the discount factor? thanks Lindsey

Posted by Lindsey Byrne | December 07, 2009 03:48 pm

## No reinvestment rate

There is no reinvestment rate assumption in IRR, it is simply the interest rate that makes the NPV equal to zero. In fact, the MIRR makes matters worse since the rate of return is now not internal to the project. Here are 2 questions: 1) Does the YTM calculation of a bond assume reinvestment of the intermediate coupons? No. But the YTM calculation of a bond is nothing more than the IRR of the bond. 2) I have a project that costs \$100 today and returns \$10 in one year and \$110 in 2 years. What is the IRR of the project? It is easy to confirm that the IRR is 10%. Now suppose I spend the \$10 on pizza in one year. Does this change the IRR? No. Now suppose I take the \$10 and reinvest it in another project that returns 2%. Does that change the IRR of the original project? No. The fact that the \$10 was taken out of the original project and invested in another project is irrelevant. The return on the 2nd project has nothing to do with the return on the first project since the IRR only deals with the "return interal to the project" (rearrange to IRR if you wish), not what is done with the intermediate cash flows.

Posted by Joe Smolira | November 16, 2009 05:09 pm

## IRR : Does myopia lead to the Eternal Debate?

Dear All Kindly consider the following: 1. For a long term project is the cost of capital going to remain same?? 2. Isn't cost of capital dependent upon its mix of equity and debt? 3. Does equity not factor in the higher rate for higher risk? 4. 1,2,&3 above inherently build in the higher returns expected from a riskier project (so the room for disparity should not be so huge between the expected returns and the actual returns earned; in case it is so...should one not revisit the RADR assumptions ?? 5. Can the returns be supernormal for a wider spectrum of investment opportunities (..here read "project") If yes lets revise the CAPM and other normalisation models. 6. An industry builds up the investment portfolio across a spectrum of low to high risk areas, with the variation brought in by its cash flow position and strategic positoining. 7. In light of the above when we calculate IRR; we are essentially considering a "risk chunk" and in that particular chunk the industries usually have a turnaround of the cash flows (if we talk of oil and gas industry this "risk chunk" can be exploratory projects in a high risk high gain terrain). Hence the clinical observation of lack of reinvestment opportuinity reflects the view more of a short sighted accountant rather than the view of a strategically forward looking CEO. 8.In addition to the above however, there should be a due dillegence regarding the long term indicators 9. I may be grossly wrong...so dont take it to your heart.. Best regards

Posted by Navneet Jain | September 15, 2009 06:54 am

## IRR

I must say, I agree with Mr McClendon here... If I am not mistaken, IRR just calculate the rate where NPV = 0, so it doesn't actually take into consideration what you do with the positive cashflow you receive from the project.

Posted by Leonard Buntaran | May 03, 2009 11:19 am

## End this debate

Dear Troy The debate on the necessity of reinvestment of cashflows seems to be never ending! This is the 21st Century and yet each day authors still post conflicting statements. Please post your views and matrix and settle this issue. Regards

Posted by Gordon Matthews | August 14, 2008 04:34 pm

## Youy are wrong!

I have been a Senior Instructor for the Commercial Investment Institute (CCIM) for 26 yeaers and I say your statements are wrong and inaccurate. The IRR makes no assumptions about reinvestment of cash flows. Reply and I'll send you a matrix to prove it.

Posted by Troy McClendon | August 09, 2008 08:20 pm