The internal rate of return (IRR) is a widely used investment performance measure in commercial real estate, yet it’s also widely misunderstood. What is IRR exactly? How is it used and what are its limitations? In this article we’ll discuss what IRR is and how it works. We will also identify some common misconceptions and finally clarify these ideas with some relevant examples.

## What is IRR?

First of all, **what is IRR**? Simply stated, the Internal rate of return (IRR) for an investment is the percentage rate earned on each dollar invested for each period it is invested. IRR is also another term people use for interest. Ultimately, IRR gives an investor the means to compare alternative investments based on their yield.

Mathematically, the IRR can be found by setting the Net Present Value (NPV) equation equal to zero (0) and solving for the rate of return (IRR).

If the above equation scares you don’t worry, we will walk through a detailed example next that shows you exactly how IRR works and it will leave you with a solid intuition behind the internal rate of return.

## Step-by-Step Example and Proof of IRR

Memorizing equations is one thing, but truly understanding what’s actually happening with IRR will give you a big advantage. Let’s walk through a detailed example of IRR and show you exactly what it does, step-by-step.

Suppose we are faced with the following series of cash flows:

This is pretty straightforward. An investment of $100,000 made today will be worth $161,051 in 5 years. As shown the IRR calculated is 10%. Now let’s take a look under the hood to see exactly what’s happening to our investment in each of the 5 years:

As shown above in year 1 the total amount we have invested is $100,000 and there is no cash flow received. Since the 10% IRR in year 1 we receive is not paid out to us as an interim cash flow, it is instead added to our outstanding investment amount for year 2. That means in year 2 we no longer have $100,000 invested, but rather we have $100,000 + 10,000, or $110,000 invested.

Now in year 2 this $110,000 earns 10%, which equals $11,000. Again, nothing is paid out in interim cash flows so our $11,000 return is added to our outstanding internal investment amount for year 3. This process of increasing the outstanding “internal” investment amount continues all the way through the end of year 5 when we receive our lump sum return of $161,051. Notice how this lump sum payment includes both the return of our original $100,000 investment, plus the 10% return “on” our investment.

This is much more intuitive than the mathematical (and typical) explanation of IRR as “the discount rate that makes the net present value equal to zero.” While technically correct, that doesn’t exactly help us all that much in understanding what IRR actually means. As shown above, the IRR is clearly the percentage rate earned on each dollar invested for each period it is invested. Once you break it out into its individual components and step through it period by period, this becomes easy to see.

## What IRR is Not

IRR can be a very helpful decision indicator for selecting an investment. However, there is one very important point that must be made about IRR: **it doesn’t always equal the annual compound rate of return on an initial investment**.

Let’s take an example to illustrate. Suppose we have the following series of cash flows that also generates a 10% IRR:

In this example an investment of $100,000 is made today and in exchange we receive $15,000 every year for 5 years, plus we also sell the asset at the end of year 5 for $69,475. The calculated IRR of 10% is exactly the same as our first example above. But let’s examine what’s happening under the hood in order to see why these are two very different investments:

As shown above in year 1 our outstanding investment amount is $100,000, which earns a return on investment of 10% or $10,000. However, our total interim cash flow in year 1 is $15,000, which is $5,000 greater than our $10,000 return “on” investment. That means in year 1 we get our $10,000 return on investment, plus we also get $5,000 of our original initial investment back.

Now, notice what happens to our outstanding internal investment in year 2. It decreases by $5,000 since that is the amount of capital we recovered with the year 1 cash flow (the amount in excess of the return on portion). This process of decreasing the outstanding “internal” investment amount continues all the way through the end of year 5. Again, the reason why our outstanding initial investment decreases is because we are receiving more cash flow each year than is needed to earn the IRR for that year. This extra cash flow results in capital recovery, thus reducing the outstanding amount of capital we have remaining in the investment.

Why does this matter? Let’s take another look at the total cash flow columns in each of the above two charts. Notice that in our first example the total $161,051 while in the second chart the total cash flow was only $144,475. But wait a minute, I thought both of these investments had a 10% IRR?! Well, indeed they did both earn a 10% IRR, as we can see by revisiting the definition or IRR:

The Internal rate of return (IRR) for an investment is the percentage rate earned on each dollar invested for each period it is invested.

The internal rate of return measures the return on the outstanding “internal” investment amount remaining in an investment for each period it is invested. The outstanding internal investment, as demonstrated above, can increase or decrease over the holding period. It says nothing about what happens to capital taken out of the investment. And contrary to popular belief, the IRR does not always measure the return on your initial investment.

## The Myth of The Reinvestment Rate Assumption

One of the most commonly cited limitations of the IRR is the so called “reinvestment rate assumption.” In short, the reinvestment rate assumption says that the IRR assumes interim cash flows are reinvested at the IRR, which of course isn’t always feasible. The idea that the IRR assumes interim cash flows are reinvested is a major misconception that’s unfortunately still taught by many business school professors today.

As shown in the step-by-step approach above, the IRR makes no such assumption. The internal rate of return is a discounting calculation and makes no assumptions about what to do with periodic cash flows received along the way. It can’t because it’s a DISCOUNTING function, which moves money back in time, not forward.

This is not to say the the IRR doesn’t have some limitations, as shown in the examples above. It’s just to say that the “reinvestment rate assumption” is not among them. Should you take into account the yield you can earn on interim cash flows that you reinvest? Absolutely, and there have been various measures introduced over the years to turn the IRR into a measure of return on the initial investment. Some of the more popular approaches include the modified internal rate of return (MIRR), the capital accumulation method, and the external rate of return (ERR). These approaches are beyond the scope of this article, but will be explored in the near future.

### IRR Calculator

Fill out the quick form below and we'll email you our free IRR calculator. You can use our IRR calculator to quickly calculate IRR for any holding period you need. You can also visualize what IRR is doing in each period of the analysis.## Conclusion

The Internal Rate of Return (IRR) is a popular measure of investment performance. While it’s normally explained using its mathematical definition (the discount rate that causes the net present value to equal zero), this article showed step-by-step what the IRR actually does. What is IRR? Once you walk through the examples above this question becomes much easier to answer. It also becomes clear that the IRR isn’t always what people think it is. That is, it isn’t always the compound annual return on the initial investment amount. Understanding what IRR is at an intuitive level will go a long ways towards improving your ability to analyze potential investments.

Gary Schreiber says

Doesn’t the operating agreement determine whether the $5000 in example 2 is considered “return of capital” or “profit”? The IRR would then be different in that case and not 10%

Rob says

What exactly would be the difference in your view?

gary schreiber says

disregard my last comment. The IRR remains 10%. The risk is different in the second case due to earlier return of capital

Zaw Zaw Oo says

Thank you!

Rob says

Sure, glad you found it helpful.

Hank Chang says

I am a retired research engineer and studying/learning finance for my hobby. I found this article not help me but confuse me furthermore. It is your fault. I need to learn finance English more than finance Math. And change my “brain” into finance oriented.

Both examples add up money amounts at different years. I does not make sense to do so since one dollar at different years are not equivalent. Adding apples and oranges together make thing more messy.

Rob says

Can you re-phrase your question and make it more specific? Happy to help, but not sure I can follow your question here.

vostok says

I think the theory of the reinvestment rate refers to deferred cash flows and not cash flows paid current. IRR by its inherent calculation is the discount rate on the cash flows received that set the NPV to 0 for an investment/project. Thus, for cash flows received, the return assumes nothing regarding the reinvestment of those cash flows since those would be outside the current project and into another asset, which would be outside of the scope of the project you’re discounting. For deferred cash flows, it makes 100% reason to assume that an investor would require the same yield on deferred cash flows into a project with respect to the initial investment. A Deferral is in essence a reinvestment into the same project as the project keeps those cash flows and doesn’t return it back to the investor, presumably to reinvest back into the project. Thus, why would you expect a different yield? The only basis I can think of is if the deferred cash some how super-primes the cash flow returns to the initial investment or if they are gauranteed by some 3rd party, in which case they can be considered less risky computed at a different rate, which would drop the IRR. On the other hand, if they are riskier than the initial investment cash flows in any type of subordination, then the IRR would go up, since the yield would be reflective of the repayment of these riskier cash flows. But undoubtedly, deferred cash flows have to be reinvested at some rate and it makes 100% sense that that rate is the same as the initial investment, and thus it must be taken into account when computing IRR.

Rob says

I have an investment where we invested $440,000 up front – a year later we returned to the investors $678,500 – yet the IRR calculation is 54% – it would seem to me the IRR would be infinite as we gave the investors there original money back and $238,500 in “interest” or gain – at the end of year 2 we returned another $50,000 and the IRR dropped to 45% – but I don’t understand what it is calculating against? the investment is returned with a healthy gain a year earlier?

Rob says

Can you post the entire set of cash flows you’re working with? I can gather the following from your comment: [-$440,000, $678,500, $50,000]. Are there any others?

Rob says

the 3rd year the return to investors was $140,000

the project could now be sold in the 4th year for a net return to investors of $2,260,000 – this yields an IRR of ~100%

but the question remains – why would it not be infinite with the return of cash year 1 far exceeding the expected interest rate return of the investors?

Rob says

Remember the definition of IRR – IRR is the percentage rate earned on each dollar invested for each period it is invested. Now, check this chart of your cash flows. Make more sense?

Rob says

Thanks Rob – makes sense – also points out the limitations of IRR as the investors are not expecting an interest rate return of 117% per annum.

Grant Farrell says

I don’t understand how “interest at IRR” is calculated above.

Donna Roberts says

Hmm can I get in on that investment with $25K?

Roy says

Thank you for your article and make me see another point of view of IRR.

Can you advise to calculate IRR if the cashflow was daily/monthly base?

I have some difficulties for calculate the analysis for project in my office which is run for short term. i was in Freight Forwarding company.

Rob says

It would be the same analysis as above, except in your case “Periods” would be daily or monthly rather than yearly. This would also make the resulting IRR daily or monthly since IRR is the percentage rate earned on each dollar invested for each period it is invested.

murali mohan says

Excellent article, which has given me a good insight about IRR. In our business investment decisions, we generally rely on IRR figures and not on Total Cash flows. From the 2 scenarios you have illustrated in the article, which is the best (preferred) investment option and why?

Rob says

I suppose it depends on your goals, but there is no silver bullet, so why not use all of them in making your decision?

Shane says

great article, thank you!

Abhishek says

sir if in your example my required return would have been 20% and cashflows still continue to be 15000 then what would have been my irr ?

Rob says

The IRR would not change. The IRR is what you get and the discount rate (required rate of return) is what you want. The NPV will quantify the difference between what you get and what you want. Take a look at our article on the intuition behind IRR and NPV here:

http://www.propertymetrics.com/blog/intuition-behind-irr-and-npv/

DANISH says

SIMPLY AMAZING!!!! I COULD NOT UNDERSTAND IT BEFORE WATCHED MANY VIDEOS AND RED MANY COLUMNS BUT THE ARTICLE PROPERTYMETRICS.COM SHARED HERE IS SIMPLY MING BOGGLING, IT MADE EASY NOT TO SOLVE PROBLEM ONLY BUT AM ABLE TO UNDERSTAND THE SPIRIT OF IRR.

Villma says

Thanks for your article. One question: if the IRR is the discount rate which mathematically sets the NPV of your investment to zero, then this means that it is the maximum discount rate you could plug in before the investment becomes unworthy. I.e. if you use a higher discount rate the NPV becomes negative, right? Following this logic, the higher the IRR, the better the investment opportunity since it gives you more leeway to a negative NPV, right? Now how does this fit with your definition: the IRR is the percentage rate earned on each dollar invested for each period it is invested?

himal gurung says

really helping,, thankyou so much,but please can you help me further more,, how IRR is calculated??? and IRR is only best choice for investor or rate more than IRR can be the best decision???

Gnv Raghavan says

hi rob – if a loan given out by a bank, is looked at as an investment, the lending rate and the IRR do not match. I’m not able to understand this.

vostok says

I think what b-schools are thinking with respect to re-investment rate of cash flows relates to the carried return, and not actual free cash flows being paid out. Thus, if for ex in 1 period, there is no cash flow that can be paid out but the investor is owed a divident or a payout, then those cash flows are accrued/re-invested into the company/project at the same rate as their required return. This applies primarily to investments senior to equity in the capital structure, namely debt/subordinated and preferred equity.

-Kellogg MBA

Toussaint says

I agree. To illustrate, I had to analyze a preferred equity investment to help fund ground up construction. In the development model the investment would continue to accrue interest on the reinvested basis until there was enough cash flow after debt service to make a distribution to the preferred equity. A similar calculation was used for the subordinated limited partners’ equity with interest building in their capital accounts until enough cash flow occurred to pay off the preferred equity position. To Rob’s point- that scenario, however, differs from a straight equity investment in a property which yields enough after debt service cash flows to enable a distribution which returns a portion of the investors’ initial capital.

Bandit says

Please explain the return of investment of $76795 in the second example. Your examples have given me insight as to why an audit of Federal Government is impossible. Too many interpretations of posted numbers.

Robert says

“Return of” is simply the cash taken out of the investment.

Bandit says

Please explain the return of investment of $76795 in the second example. Your examples have given me insight as to why an audit of Federal Government is impossible. Too many interpretations of posted numbers.

DERICK says

THANKS….. IT IS HELPFUL

Priyank says

Thank you so much for the clear explanation. Can you please tell what is going on in the following two cash flows. With one XIRR is 172.3% and with other is is 9%. Though at the end of both the periods what one earns is 9 points.

31-Dec-14 -100

31-Jan-15 109

30-Dec-15 -109

31-Dec-15 109

XIRR 172.30%

31-Dec-14 -100

31-Jan-15 0

30-Dec-15 0

31-Dec-15 109

XIRR 9.00%

wlievens says

Your assumption at the end is incorrect. You’ve gotten two returns of 109 from the first example. You need to sum the entire flow to get a balance at the end, if you want to compare the end result to the initial investment.

Kannapiran Chinna says

Hi Priyank:

Well, XIRR and MIRR are mathematically inconsistent. See my papers posted here.

If the XIRR or MIRR are real IRR then the NPV must be zero at XIRR or MIRR. But they are not in both cases you furnished. I worked out the details and results furnished below:

case: 1 IRR = 5% when NPVat 5% = 0; With XIRR 172.3% the NPV at 172.3% = -69.3

Case 2: IRR is 3% when NPV at 3% = 0; with XIRR 9% the NPV at 9% = -15.83

Both XIRR and MIRR are mathematically inconsistent and should not be followed. MY next paper (forthcoming) is all about the fun with the socalled XIRR and MIRR).

If you would like to have the spreadsheet with working, -please send me your email ID: mine; ckannapiran@gmail.com

K says

If a 24 month investment gave an IRR of 10,64%, do you have to calculate the annual IRR separately? That is, by calculating the monthly IRR and converting it to annual as follows:

Monthly IRR

= 24√(1+ .1064) – 1

= .004221866

Annual IRR

=(.004221866+1)^12 – 1

= 5,19%

Robert says

Check this:

http://www.propertymetrics.com/blog/time-value-of-money/

Sepik says

Nice and simple to understand. thank you

Suv Sach says

What exactly is return of investment and return on investment?

Robert says

“Return on” is the return on the dollars remaining in the investment. This is what the IRR measures. “Return of” is simply cash distributions which take money out of the investment. The important distinction is that IRR says nothing about what you do with money taken out of the investment. This is what MIRR is used for.

Ellen Mosner says

Good point,

Gene Mack says

I took your whole CRE course and benefited hugely. Working on lender comparisons right now to refinance some properties. Can you help me understand the relationship between IRR and cash-on-cash return? On some loans, the difference between the cash-on-cash return and a five-year IRR is minimal. On others, it’s wildly different. I understand both metrics; I just can’t put my finger on what drives cash-on-cash and IRR to be close on some loans and far apart on others. Thx!

Robert says

The cash on cash only takes into account a single year whereas the IRR takes into account all cash flows. Take a look again at the simple measures of investment performance module in the course.

Ellen Mosner says

Money never sleeps. Gordon Gheko

Hamid says

Hi everyone, I have a question and appreciate it if someone who knows finance would help me.

If I make a contribution every year for $500 for 10 years, I will be returned $1,500 for that specify year but I’m not allowed to touch the money until next 10 years, in other words after 10 years overall I pay $5000 and by year 10 I can take my own money and also $10,500 of return. Is this a good investment to go?

Robert says

Check this:

http://www.propertymetrics.com/blog/time-value-of-money/

Kannapiran Chinna says

IRR will be 3% if the payment is received by you at the end of the 10 years.

propertyguy247365 says

Can you show a similar “underneath the hood” breakdown of the modified internal rate of return (MIRR)?

Robert says

Check this out: http://www.propertymetrics.com/blog/how-to-use-the-modified-internal-rate-of-return-mirr/

Kannapiran Chinna says

Thanks Rob: MIRR is spurious because: a. MIRR is estimated from the Modified NCF (MNCF) that is modified to account for reinvestment of intermediate income. When that assumption is wrong, MNCF and MIRR are spurious; b. MIRR increased without any limit (infinite) with increasing the Investment rate (IR) and when the IR>IRR, the MIRR is higher than IRR but the NCF could not support that higher MIRR (as per Capital Amortization Schedule (CAS) explained in the link); c. MNCF and MIRR are not NCF-consistent, that means the actual NCF will not support the mathematically generated MNCF and the resultant MIRR are therefore they are spurious. CAS reveals that MIRR does not fully utilize the NCF and therefore lower than IRR; when the MIRR is higher than IRR, that MIRR is spurious again as the NCF is not adequate to support that MIRR. MIRR is spurious and therefore not to be used.

Kannapiran Chinna says

also read:. MIRR is a Spurious criterion and should not be used in cost-benefit analysis and investment analysis http://ssrn.com/abstract=2942456

Chris says

Awesome stuff. Question: I’m a buy rent and hold investor. Pay off the mortgage and keep it forever. Do I use Net profit or gross rent?

Eg:

I bought a place for $289K

Gross monthly rent is $2,300. (27600 per year)

But with all the expenses and mortgage I’m only netting $389 per month (4668 per year).

In 15 years mortgage free and still hold on to to property.

What numbers do I use? Gross rent or Net profit?

Eli says

I might be wrong but wouldn’t it be obvious to use your net profit since that is how much you are making?

Kannapiran Chinna says

Chris: I agree with Eli. Net cash flow (NCF) is always used to estimate the IRR and NPV. I remind you..it is cash flow and not funds flow. So do not include depreciation or any sinking fund contribution as expenses. Also interest on home loan is not an expenses for the purpose of the NCF as we are trying to estimate the return of capital and return on invested capital (ROC and ROIC). Interest is a transfer payment in NCF.

Aaron says

The absolute best IRR explanation I have ever heard. I debated with my professor in my MRED course last semester about this for over an hour and this would have cleared it up in 5 minutes. Thank you.

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.LISHA BONTRAGER says

Good article, Thanks!

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Aya says

how do I compute IRR if I only have cash flows and the interest rate? What if the cash flows start positive and then are negative?

Robert says

Check out the MIRR: http://www.propertymetrics.com/blog/how-to-use-the-modified-internal-rate-of-return-mirr/

Kannapiran Chinna says

Please refer the article on MIRR is spurious:. MIRR is a Spurious criterion and should not be used in cost-benefit analysis and investment analysis http://ssrn.com/abstract=2942456

Robert says

Rather than linking to your paper, can you please instead explain in a 1-3 sentences why you assume MIRR is “spurious”?

Kannapiran Chinna says

Hi Aya: I presume you are referring to the non-normal NCF? If so just estimate the sum of all positive and negative cash flows and if the sum will be either zero or negative, then there is no need to estimate IRR! MIRR is not the solution.

Ravi Teja says

the best explanation that can be given to IRR can be nothing more than this. Its just as simple as that and can be understood just by a normal person

Robert says

Thanks Ravi! Glad you found it helpful.

skube says

So of the two examples shown is one

morecorrect than the other? Also, would it be better to always use CAGR when there are no cash flows?wlievens says

If there are no cash flows other than initial investment and final balance, I think the formula is identical to CAGR. Don’t take my word on it though.

Kannapiran Chinna says

Skube:

I agree with wlievens. When there is only initial investment and a lumpsum payment at the date of maturity, the IRR = CGAR.

STFN says

Thank you guys, your articles on investment metrics are amazing! Love ya

Ellen Mosner says

Much thanks for demystifying IRR.

Kannapiran Chinna says

Dear All:

Most of the debates and the solutions to many of your queries could be found in the following papers (links attached). I therefore thought it would be a good idea to share one of my papers entitled:

1. “A New Method to Estimate NPV from the Capital Amortization Schedule and an Insight into Why NPV is Not the Appropriate Criterion for Capital Investment Decision”.

The paper is available in the following link: http://ssrn.com/abstract=2899648

This paper introduces a new method to estimate the NPV based on Capital Amortization Schedule (CAS) and not the conventional DCF method. The new method is more transparent. This paper questions the validity of the NPV as a preferred criterion than IRR. The results also clarify that there is no reinvestment of intermediate income, as CAS does not involve reinvestment. When there is no reinvestment, the MIRR estimate is also redundant.

2. IRR Performs Better than NPV: A Critical Analysis of Cases of Multiple IRR and Mutually Exclusive and Independent Investment Projects.

https://ssrn.com/abstract=2913905

This paper presents evidence to identify the most appropriate investment criterion (IRR vs NPV) with emphasis on the controversial multiple, negative and no IRR, mutually exclusive investment and independent projects. The analysis is based on the estimated return on capital (ROC), return on invested capital (ROIC) and capital amortization schedule (CAS).

3. The Controversial Reinvestment Assumption in IRR and NPV Estimates: New Evidence Against Reinvestment Assumption (February 16, 2017).

https://ssrn.com/abstract=

The analytical results presented in these papers question some of the conventional wisdoms advocated by most finance and economic texts or project analysis guide or publications or teaching materials and therefore the contents will enable the respective authors or organization to revise or update their publications accordingly.

My view is without perfecting the methods any amount of skillful analysis and articulation may not add value and credibility. Please feel free to send me your comments.

Regards

Dr Kannan Arjunan ckannapiran@gmail.com

Robert says

Thanks for sharing these. Can you point out the specific insights in your papers here that would clarify the debates?

Kannapiran Chinna says

Thanks and I must say that Propertymetrics is one of the most useful site and marshal methods and facts well and easy to understand.

Give me sometime to send a consolidated reply to some of the important queries raised here. Regards Kannan

Kannapiran Chinna says

Rob:

I have replied to some of the interesting queries raised by the viewers.

Yash says

Hi Rob,

Thanks for the wonderful article with example. Here are my observations on the article.

1. IRR is in fact assuming the reinvestment of the cash flow. Reinvestment is at the same rate as IRR (10% in this example).

2. Invested 100K becomes 161051 in 5 years, yielding annual rate of 10% (correctly shown in example 1)

3. In example 2, 15k is taken out every year and in the end 69475. The total cash outflow is 144475. The difference is $16576. This is the exact amount one can earn by investing the cash outflow (15k) at the 10% per annum. The calculation is as below shown in column diff. E.g. 15k for next four years at 10 % per annum will give 6961.57 interest.

Period Cash Flow FV Diff

0 -100,000 $0.00

1 15,000 $21,961.57 $6,961.57

2 15,000 $19,965.05 $4,965.05

3 15,000 $18,150.03 $3,150.03

4 15,000 $16,500.01 $1,500.01

5 84,475 $84,475.00 $0.00

$144,475.00 $161,051.67 $16,576.67

4. IRR is equal to CAGR, if I continue to invest the cash outflows at the same IRR (10%).

Robert says

Thanks for your comments and supporting examples. Your examples are valid and could occur as you show. However, notice that you are making these reinvestment rate assumptions, not the IRR function.

IRR can’t make these reinvestment assumptions because it’s a DISCOUNTING function and only moves money backwards in time, not forwards. It says nothing about what you do with money taken out of the investment. Equations don’t make assumptions, but PEOPLE do make assumptions. This is “the myth of the reinvestment rate assumption” in a nutshell.

Kannapiran Chinna says

Dear Yash:

Only when the intermediate income is not paid out but remain invested in the project (similar to reinvestment deposit) or when investment with income is paid only on maturity, the IRR will be equal to DGAR.

See my paper on Capital investment analysis and the controversial reinvestment

assumption in IRR and NPV estimates: Some

new evidence against reinvestment assumption, p13. http://ssrn.com/abstract=2920029

Estimated IRR can also be equivalent to return on total capital invested,

rather than only on the declined balance of capital, if smart investors make

use of the capital divested or released (like external rate of return referred

by Kierulff, 2012). Obviously, such an exogenous reinvestment is confused as an

internal reinvestment. The warning to manager by Kelleher et al. (2004) that

IRR is the return on the declined capital balance must be considered under the

context explained here.

Stan Malikov says

The myth about reinvestment rate doesn’t make sense. Your first example explicitly demonstrates compounded annual growth based on the “reinvestment”principle in the project itself until total payout at year five. The calculation is assuming reinvestment of the funds, because you have no payout in year 1-4, and given that all the funds remain in the original investment until payout, the gain in year one through four is by definition reinvested. The re-invesment principle is not as apparent if you have payouts in year 1-4, because if you have any cash out flow, and it’s not re-invested back into the project or another investment yielding the same return, you essentially throw off the IRR, because the calculation is based on an annual compounded growth. So if you have a payout in year 1-4 and you just put it in the bank or buy bonds that yield 3%, your IRR calculation is no longer valid, because it was assuming the funds are growing at a constant rate.

Robert says

You make a good observation that a project will not always have interim cash flows. However, if a project does not have interim cash flows, that doesn’t therefore mean we assume we reinvest interim cash flows – there aren’t any! It’s the initial investment amount that continues to grow each year at the rate of IRR.

Also, you are right that reinvesting interim cash flows at a different rate than the IRR will change the overall return. This is best captured with the Modified Internal Rate of Return calculation which takes into account reinvestment rate and safe rate assumptions:

https://www.propertymetrics.com/blog/how-to-use-the-modified-internal-rate-of-return-mirr/

However, this doesn’t “throw off” IRR, since IRR only measures the return on the balance of funds remaining in the investment (it says nothing about what you do with funds taken out of the investment). For many reasons this makes the MIRR more compelling, but it certainly doesn’t “throw off” IRR any more than it would throw off the cap rate rate of cash on cash return. They are different measures and as such measure different things.

Stan Malikov says

Robert,

I think we are mostly on the same page, except for one, reinvestment assumption. If a project doesn’t have interim cash flows, the re-investment theory is simply implying that the unrealized gains that remain in the project, are growing at the calculated compounded rate is in-itself the re-investment.

I understand your interpretation, in that technically you are not reinvesting because you don’t have interim cash on hand and the money is compounding automatically, however given that the IRR calculation makes the same assumptions if the payout is interim or at the end, the re-investment notion is simply telling us that,

1) If there are no interim cashflows, and the investment IRR is calculated based on the initial compounded amount, you have no access to the gains until they are realized, and you receive the calculated IRR.

2) If there are interim cashflows, then the IRR calculation is assuming the same thing as the previous calculation, the difference is if you have interim cashflow, it must be re-invested at exactly the same rate in order to get the same exact IRR, other wise the compounding will not yield the same result.

Given that both calculations have the same underlying assumptions, with the first example, you don’t have to physically re-invest, however since you don’t have access to the gains, you have to treat them as if they are re-invested because of opportunity cost. If however you have interim cash flow, the burden of re-investment is on the receiver, because the IRR calculation is assuming your money is still compounded year after year, hence you assume re-investment.

I completely understand your point of view, but given that other investments yields interim cash-flows, you can’t ignore the difference, re-investment assumption reconciles this, where one investment automatically yields the return at the end, and the other one has to be re-invested, therefore we treat the first investment “as-if” it’s re-invested, because once again, you can’t ignore the differences with one investment yielding tangible cashflow, and the other un-realized gains.

Robert says

This really cuts to the heart of the myth of the reinvestment rate assumption. To be clear – the IRR makes no assumptions about what to do with periodic cash flows received along the way. It can’t because it’s a DISCOUNTING function (moves money back in time).

Your consideration of what to do with interim cash flows is important as a decision making tool, and this can be addressed with something like MIRR, but this is not something built into the IRR. Consider this excerpt from an article published in the Journal of Real Estate Portfolio Management:

“Since the late 1950s, most textbooks and many professors have been inadvertently defining the internal rate of return (IRR) of an investment incorrectly vis-à-vis the reinvestment of an investment’s cash flows. The genesis of this unfortunate error can be traced to an article by Renshaw (1957). Most textbooks and many professors have since paraphrased a quotation taken from the Renshaw article that attempted to paraphrase (out of context) the words of Solomon (1956). Both the Renshaw and the Solomon quotations, in their entireties, go on to properly explain the issue of reinvestment vis-à-vis the IRR. However, the paraphrasing of the partial quotation of Renshaw has perpetuated what some now call the ‘reinvestment rate controversy.’ The Renshaw (1957:193) quotation states: ‘…the (net) present value (NPV) approach assumes reinvestment of intermediate cash receipts at the discounting rate, while the internal rate-of return (IRR) approach assumes reinvestment at the internal rate…’

“The fact is that neither approach makes any assumption whatsoever about either the reinvestment of cash flows or the rate of return to be earned if reinvestment were to be considered. The discounted cash flow (DCF) equations for the IRR and the NPV are just that: formal statements of equivalence. Equations do not make assumptions; people make assumptions. A calculation cannot assume; people assume. Hence, reinvestment has nothing to do with the calculation of the IRR. However, reinvestment may be critical to the application of the IRR as an investment decision-making tool.”

Revealing the True Meaning of the IRR via Profiling the IRR and Defining the ERR

Crean, Michael J., Journal of Real Estate Portfolio Management 11. 3 (Sep-Dec 2005): 323-330.

Stan Malikov says

I think we can agree to disagree. The excerpt elaborates on the authors interpretation of re-investment. If the growth rate is not maintained on interim cashflow, your IRR is no longer valid. If at any point you break the chain, and the balance earning the initial IRR changes, you are no longer seeing the initial IRR calculations, which is where MIRR comes in.

Ahmed Surajuddin says

this is absolutely amazing !!!

I have just started studying for the CFA with zero prior knowledge and the concept of IRR was a pain to intuitively grasp, its such an important concept that relates to all of finance …portfolio theory, equity valuation, corporate finance , fixed income…..etc ,not to be properly understood is simply out of the question

and after reading this article it hit me !!

to start with questions such as

why would i even want the NPV to be =0 ?? why not NFV=0

why in portfolio returns the money weighted return (MWR) is by “definition” the IRR.

i Just understood the link between amortization payments – discounting the equal monthly payment to the present value of the loan to solve for the periodic rate -, is exactly the same as finding the IRR for a non-equal cash payments…

bravo sir…bravo

Robert says

Awesome, I’m glad you found it helpful!

Ricardo Sasso says

I’m sorry but I gotta say I don’t find your explanation very coherent.

First of all:

“Why does this matter? Let’s take another look at the total cash flow columns in each of the above two charts. Notice that in our first example the total $161,051 while in the second chart the total cash flow was only $144,475. But wait a minute, I thought both of these investments had a 10% IRR”

Yeah, both have a 10% IRR, but if you consider that the cashflow is reinvested you can easily explain the difference. The difference comes from the fact that you are not considering time value of money from interim cashflow, so 144,475 shoud be different from 161,051 symply because the money is received at different periods. If you reinvest at 10% IRR rate the interim cashflow, you easily arrive at a final value of 161,051.

So, concluding, if I have a 10% IRR, and a 100,00 initial apport, reinvesting the interim cashflow’s I will always arrive at a final value of 161,051, no matter what distribution of cashflow I have for a 10% IRR.

Secondly:

In your MIRR explanation you just admit that in the IRR calculation we are considering a reinvestment at the IRR rate.

“As you can see, the MIRR when using a 10% reinvestment rate is 15.98%. This is less than the 18% IRR we initially calculated above. Intuitively, it’s lower than our original IRR because we are reinvesting the interim cash flows at a rate lower than 18%”

I’m sorry for my english, i’m not exactly fluent

Robert says

It sounds like you have the difference between IRR and MIRR correct. Did you have a specific question?

Ricardo Sasso says

The entire article que author claims that in the IRR calculation there’s no reinvestment assumption correct? But to justify the difference between MIRR and IRR calculation the author makes that assumption.

Robert says

Where is that assumption made exactly? Can you be more specific?

endless says

Does the IRR ever include the anticipated sales price (profit) on a real estate investment where the intent is to hold it for X years and then sell?

Robert says

Yes, take a look at the examples used in this article – they include sales proceeds in the final year of the holding period.

endless says

OK, but can the IRR include the estimated appreciated value of property (i.e., its current assessed value) BEFORE It is sold and the profit taken? I’m asking because the IRR of an investment proposal seems to include the property’s annual appreciation along with the annual income actually received. The net effect in the proposal I’ve been given is to boost the IRR figure considerably.

K.Jones says

This is a great article for people new to understanding these concepts.

But your statement that the issue of re-investment is a “myth” that the entire finance, engineering, and accounting community has got wrong is simply inaccurate.

Of course the IRR discounts and moves backwards. The projected cash flows are a projection of frontwards movement. The equation’s assumption is there in black and white, specifically the greek sum symbol and the variables surrounding it – do in fact assume that those projected future cash flows are re-invested at the internal rate. The equation does not simply ignore those returns. It aggregates them and assumes it is at the same rate.

It is pretty self righteous to think you have the ability to discard an entire academic community that has outlined how and why the IRR makes this assumption.

Disqus makes it easy for sites to delete comments so I am sure this comment will be deleted but I have to put it out there – people may make real jackasses of themselves if they repeat this particular aspect of your article.

An example of why you can’t always regurgitate what you read online.

Robert says

Thank you for your observations and comments. I would, however, disagree with you that these ideas discard the entire academic community. Consider the following excerpt from an academic journal article:

“The notion that the internal rate of return (IRR) and net present value (NPV) have reinvestment rate assumptions built into them has long been settled in the academic finance literature. Specifically, there are no reinvestment rate assumptions built into, or implicit to, the computation and use of either the IRR or NPV.”

Here is another excerpt from an article published in the Journal of Real Estate Portfolio Management:

“Since the late 1950s, most textbooks and many professors have been inadvertently defining the internal rate of return (IRR) of an investment incorrectly vis-à-vis the reinvestment of an investment’s cash flows. The genesis of this unfortunate error can be traced to an article by Renshaw (1957). Most textbooks and many professors have since paraphrased a quotation taken from the Renshaw article that attempted to paraphrase (out of context) the words of Solomon (1956). Both the Renshaw and the Solomon quotations, in their entireties, go on to properly explain the issue of reinvestment vis-à-vis the IRR. However, the paraphrasing of the partial quotation of Renshaw has perpetuated what some now call the ‘reinvestment rate controversy.’ The Renshaw (1957:193) quotation states: ‘…the (net) present value (NPV) approach assumes reinvestment of intermediate cash receipts at the discounting rate, while the internal rate-of return (IRR) approach assumes reinvestment at the internal rate…’

“The fact is that neither approach makes any assumption whatsoever about either the reinvestment of cash flows or the rate of return to be earned if reinvestment were to be considered. The discounted cash flow (DCF) equations for the IRR and the NPV are just that: formal statements of equivalence. Equations do not make assumptions; people make assumptions. A calculation cannot assume; people assume. Hence, reinvestment has nothing to do with the calculation of the IRR. However, reinvestment may be critical to the application of the IRR as an investment decision-making tool.”

Revealing the True Meaning of the IRR via Profiling the IRR and Defining the ERR

Crean, Michael J., Journal of Real Estate Portfolio Management 11. 3 (Sep-Dec 2005): 323-330.