IRR Defined and Explained With Examples

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1/18/13 Internal rate of return (IRR) defined and explained with examples

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Internal Rate of Return (IRR) and Modified Internal Rate of Return (MIRR): Definition, Meaning,and Usage.Ency clopedia of Business Terms and Methods, ISBN 978-1-929500-10-9. Revis ed 2013-01-11.

Internal Rate of Return (IRR ) is a financial metric for cash flow analysis, often used for evaluating proposedinvestments, funding requests, acquisitions, or the results of business case analysis. Like several other cash flowmetrics (such as net present value, payback period, and return on investment), IRR takes an "investment view" of expected financial results. This means, essentially, that the magnitude and timing of cash flow returns are compared tothe magnitude and timing of cash flow costs. Each of these financial metrics compares returns to costs in its own wayand each carries a unique message about the value of the investment.

IRR analysis begins with a cash flow stream, a series of net cash flow figures expected to follow from the investment (or action, acquisition, or business case scenario). The expected cash flow stream for IRR analysis might appear something like this:

Each bar represents the net of inflows and outflows for one two-month period. The complete set of net cash flow events is a cashflow stream . If this cash flow stream represents one investment,another investment might show a different cash flow profile. IRRs for each investment can be compared to help decide which is the better business decision. Other things being equal, the investment with thehigher IRR is viewed as the better choice.

Notice especially the shape , or profile of this example cash flowstream. The figure shows a typical investment curve . Net casoutflows at the outset and net cash inflows in later periods mean that

costs initially exceed incoming returns, but if the investment performs as expected, returns eventually outweigh thecosts. The IRR metric, in fact, "expects" this kind of cash flow profileearly costs and later benefits. When the cashflow stream has this kind of profile, an IRR can probably be found and usefully interpreted. When the cash flow streamdeviates substantially from this profile, however, it may not be possible to find an IRR for the stream. Or, other strangeIRR results may appear, such as multiple IRRs for the same stream, or a negative IRR for the cash flow stream. In suchcase s, the resulting IR Rs are ei ther very difficult to inter pret or meaning les s.

Most people in business have heard of "internal rate of return." Some are required by thier financial officers to producean IRR to support budget requests or action proposals. IRR is in fact a favorite metric of many CFOs, Controllers, andother financial specialists. There is a widespread belief in the financial community, for instance, that IRR is a more"objective" metric than NPV, because NPV depends on an arbitrarily chosen discount rate, whereas IRR is determinedeintirely by the cash flow figures and their timing. Many also believe that IRR allows investment returns to be comparedreadily with inflation, current interest rates, and financial investment alternatives. Many organizations establish an IRRhurdle rate, that is, an IRR rate that must be reached or exceeded by incoming proposals if they are to be approved andfunded.

It should no surprise to learn that most business people who are not in finance have a limited or poor understanding of IRR and its meaning. It may be more surprising, however, to learn that research on professional competencies finds
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consistently that most of the same financial specialists who require IRRs with proposals or funding requests arethemselves largely unaware of IRR's serious deficiences and unprepared to explain its meaning and proper use.

For that reason, this entry puts a special emphasis on understanding IRR meaning and interpretation, as well ascommon misconceptions and misuses of IRR. And, in comparing IRR to other financial metrics, this entry also presentsthe modified internal rate of return (MIRR ), a more easily interpreted alternative to the better known IRR.

Internal rate of return (IRR) defined and illustrated with an example - First IRR Interpretation: IRR as a measure of risk - Finding IRR: Can IRR be calculated?

IRR re-defined: Why is it called internal rate of return? - Second IRR interpretation: Financing costs and reinvestment gains

Modified Internal Rate of Return (MIRR): A better metric? Internal rate of return compared to other financial metrics

- IRR vs. MIRR, NPV, ROI, and Payback Period - IRR disgused as yield to maturity (YTM) for bond investing Lease vs. buy and other problem IRR results

- Lease vs. Buy vs. IRR - Negative, multiple, and impossible IRRs

Internal rate of return (IRR) defined and illustrated with an example

As the word "return" in its name implies, an IRR view of the cash flow stream is essentially an investment view: paid outfunds are compared to returns. The best known IRR definition explains this comparison in terms that call for a basicunderstanding of discounted cash flow concepts present value, net present value (NPV), and the role of the discountrate (interest rate) in determining NPV:

IRR Definition 1 (textbook definition) : The internal rate of return (IRR) for a cash flow stream is the

interest rate (discount rate) that produces a net present value of 0 for the cash flow stream.

That definition, however, can be less than satisfying when first heard. Many ask: "What does that tell me about returnsand costs?

A first interpretation of IRR meaning is illustrated with an example. Consider two proposed investments compting for funding: Case A and Case B. The expected net cash flow streams for A and B are as follows:

Both cases call for an initial cash outlay of $220. Case A brings a net gain of $200 over 7 years while case B brings a net gain of $240 over the same 7 years.Before finding IRRs and other metrics, note in the image below how the two cash flow streams streams differ:

Both streams have the investment curve profile described above. Case A (blue bars) has large early returns but thesediminish year by year. A's profile could represent an investment in an income producing asset that becomes lessproductive or more costly to maintain each year. Case B (yellow bars) has smaller returns at first, but B's returns groweach year. B's profile could result from investing in a product launch that returns greater profits each year. The analystwill thus compare two different kinds of investments with the same metric, IRR, to address questions like these: Whichis the better investment? Which is the better business decision? The net cash flow figures above, when analyzed with aspreadsheet function or another IRR program, show an IRR of 30.6% for Case A and an IRR of 20.8% for Case B.
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The next table and figure below show the result of applying IRRDefinition 1 (the discount rate that brings an NPV of 0). Only one of the above cash flow streams is shown here, Case A. The first tablerow shows net cash flow each year, and the second row shows thediscounted values (present values) of the same cash flows.Discounting is applied using the IRR rate found by the analyst for this cash flow stream, 30.6%.

Notice in the table that this discount rate leads to a total presentvalue (NPV) of 0. The chart at leftalso shows the same thing in visualterms.

Dark blue bars are future value net cash flows for Case A .

Light blue bars are present values of the same cash flows at theIRR discount rate of 30.6%.

You may just be able to see or imagine that the heights of theseven positive (upward pointing) light blue bars starting with Yr 1add up exactly to the height of the one negative (downward pointing)light blue bar at "Now." In other words, at the IRR discount rate, thesum of positive PVs excactly cancels the sum of negative PVs.

First IRR interpretation: IRR as a measure of risk

In the example above, which is the better Investment, Case A or Case B? Case A has an IRR of 30.6%, Case B has an IRR of

20.8%. Other things being equal, and using IRR as the decision criterion, the one with the higher IRR (Case A) isconsidered the better choice. One reason for this conclusion is that a higher IRR indicates less risk. That is, IRRindicates just how high inflation rates or risk probabilities have to rise in order to eliminate the present value of thisinvestment.

For the Investment A cash flow, the prevailing discount rate (which includes an inflation component and a risk

component) would have to rise all the way to 30.6% to drain this investment of present value.

The B investment would lose all present value if the discount rate rose to 20.8%.

Most people, however, find this first interpretation of IRR of little value for evaluting and comparing proposed investments.The sections below, therefore, move torward a second, more useful IRR interpretation, a comparison of IRR with other cash flow metrics, and a description of a more easily interpreted metric, the modified internal rate of return (MIRR).

Finding IRR: Can IRR be calculated?

Other cash flow metrics such as NPV , ROI , and even payback period , can be calculated directly from simple


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formulas, assuming you have the proper input data. However, the verbal IRR Definition 1 above does not lend itself readily to presentation in formula form. The best that can be done for producing an IRR formula is this: Consider theformula for calculating net present value (NPV) for a cash flow stream, using end of period discounting:

The "FVs" in the formula are the net cash flow figures for each year.For Case A and Case B above, n = 7 (7 periods). The person whowants a "formula" for IRR can be handed the above formula along withthe FVs (net cash flow values) and then asked to do the following:

Set NPV equal to 0, then solve the formula for i . i = IRR when NPV = 0.

In fact, there is no easily applied analytic solution to the above request, and it is more accurate to say that IRR is"found" rather than "calculated." IRR can be found either by graphical analysis oras Excel doesby "trial and error"(more precisely, by "successive approximations").

Consider first the plotting data for a graphcial IRR solution:

Using the formula above, NPV was calculated for A and B cash flow streams at 10different discount rates. The table shows the calculated NPV values. These are plotted

below, showing the relationship between discount rate (horizontal axis) and resulting NPV(vertical axis).

As you would expect, increasingdiscount rates brings lower NPVs for both streams. However, Case A'sNPV reaches 0 at a discount rate of 30.6%, while B's NPV reaches 0 at adiscount rate of 20.8%. Therefore,IRR A = 30.6%. and IRR B = 20.8%

Instead of finding a graphical solutionfor IRR, most people use MicrosoftExcel or a pre-programmedcalculator. Either way, thesoftware starts with an arbitrarily

chosen discount rate and finds the NPV for a given cash flow stream. It then keeps changing the rate until it finds a ratethat produces a 0 NPV. This occurs very quickly, so that the IRR result seems to appear as soon as the data areentered. The analyst, for example, might enter the Excel IRR function into a formula like this: =IRR (B3:B10, .1 )

The spreadsheet cell with this formula shows the IRR for a range of cells with net cash flow figures in cells B3 through

B10. These could be, for instance, the eight cash flow values for Case A in the example above The ".1" figure is simplythe analyst's first "guess" at the IRR. The guess is simply a starting discount rate for calculating NPV on the firstiteration and it can be almost anything or even omitted. The analyst will probably format the spreadsheet cell as apercentage, so that the IRR result looks like this: 30.6%

IRR re-defined: Why is it called internal rate of return?

The textbook IRR Definition 1 above explains how to find an IRR but says very little about what it represents. IRR's moreimportant meaning is easier to understand in terms of another definition, one that refers investment financing costs andreinvested returns.


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IRR Definition 2 : The internal rate of return (IRR) for a cash flow stream is based on two assumptions:

1. There is a financing cost (or opportunity cost) for using the funds to make the investment.

Investment costs will be financed across the time until the final cash flow event.

and

2. Incoming returns will be reinvested for the time remaining until the last cash flow event.

IRR is then defined as the single interest rate (for financing costs and for reinvestment earnings) that setsthe total gains exactly equal to total costs.

The example below shows how IRR can be defined as the interest rate that balances these two factors. Cash flowfigures in blue cells are from example Case A above.

In the example, an initial cashoutflow of $220 (at "Now") representsinitial investment costs. Each year

after that, the investment bringspositive net cash flow returns.Excel's IRR function has beenentered in the yellow cell below Year 7 net cash flow and it reports that adiscount rate of 30.63% produces anNPV of 0 for the cash flow stream.The IRR for this cash flow stream isthus 30.63%.

Consider first the interest earned byre-investing the incoming cash flowsfrom Years 1 through 7. If each

incoming return is reinvested with for the remaining years, at an annual interest rate of 30.63%, the total seven year gains would be $1,428.17 (inflows + interest earned).

Now, assume that the initial cash outflow of $220 is borrowed and financed at the same 30.63% annual rate. Theexample shows shows the total cost of repaying this loan is also $1,428.17 (initial cash outflow plus financing). The IRRrate exactly balances total investment costs with total investment gains.

Second IRR Interpretation: Financing costs and reinvestment gains

The second-definition example above should begin to suggest a reason that financial people look to IRR and often trust itas an important decision criterion: IRR has built into it the presumption that investment costs (opportunity costs or borrowing) are financed at a cost, and that incoming returns are reinvested, earning additional gains. This view providesmeaning for another IRR interpretation, namely that the analyst will compare the IRR rate to actual financing rates andactual reinvestment rates. This comparison, however, has to be interpreted carefully. It is easy to overinterpret or misinterpret IRR at this point.

When a proposed investment produces IRR's like those shown above30.6% for examplemany are tempted toreason as follows:


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"For this investment, we will not actually borrow (or pay an opportunity cost) at the IRR rate. Our real costwill be subject to rates closer to our cost of capital, probably less than 10%. Therefore [the reasoninggoes], the investment is a net gain because financing rates will really be under 10%, while returnsrepresent earnings at a much higher rate, something like 30.6%."

In reality, that conclusion may or not be supportable, depending on the company's actual financing cost rates (oropportunity cost rates) and the actual reinvestment rates to be applied. The conclusion is most supportable when theIRR rate is close to actual cost of capital and actual reinvestment rate. The conclusion is most likely misleading or

wrong when IRR is quite different from actual cost of capital and reinvestment rates.

This reasoning can grossly overstate the value of investments like, for example, Case A above. Suppose, for instance,that the company's real reinvestment rate is closer to 8%, even though the cash flow stream has an IRR of 30.6%.Notice especially that cash flow stream A expects large returns in the first and second years of the 7-year investment(that is, Investment A is "front loaded," or "biased" towards the early years). The high IRR assumes year 1 and year 2gains will be reinvested at 30.6% for all the remaining years but in fact, this long term of high-rate reinvestment will beabsent. Cash flow stream B, on the other hand has a lower IRR than A, but when real investment returns are comparedto the reinvestment earnings presumed by the IRR, B in fact has less "missing" returns of this kind than A. IRRoverstates the real value of investment A far more than it overstates the value of investment B for two reasons: First,

stream A is front loaded and stream B is "back loaded," and second, because A's IRR is further from the realreinvestmen rate than B's IRR.

In brief, it is reasonable to view a proposed investment as a net gain when IRR is greater than the company's cost of capital. However, using IRR to assess the magnitude of the net gain is problematic, especially when (a) IRR greatlyexceeds cost of capital and the real reinvestment rate, and (b) when comparing two cash flow streams with differentprofiles as in the example above. The latter point (b) is illustrated again in the discussion below on IRR in "Lease vs.buy" comparisions.

Modified Internal Rate of Return (MIRR): A better metric?

The meaning of IRR magnitude is difficult to interpret, as shown, because IRR can differ from the actual financing andreinvestment rates. It is natural to ask, therefore, "Why not calculate an internal return metric that does reflect the reafinancing cost rate and real reinvestment rate?" In fact, this solution is readily available as the modified internal rateof return (MIRR ) metric.

Input data for MIRR include the same net cash flow figures as IRR, but the MIRR calculation also requires as input afinancing rate and a reinvestment rate . Here for comparison are the IRR and MIRR results for example Cases A andB from above. MIRR for this example is based on a reinvestment rate of 8% and a financing rate of 6%:

Investment Case A: IRR = 30.6% MIRR = 15.1%

Investment Case B: IRR = 20.8% MIRR = 14.7%Notice immediately that Case A also has a higher MIRR value than Case B, but both MIRR values are much closer toeach other than are the two IRR values.

The full meaning of MIRR is easier to explain after showing how the MIRR results are derived. MIRRunlike IRRcanbe computed directly from a formula:

Future values of cash inflows are calculated using the reinvestmentrate. Present values of the cash outflows are calculated using thefinancing rate. The radical sign calls for the nth root of the Future
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value/Present value ratio. n is the number of periods, here 7. Subtracting 1.0 from the resulting root yields MIRR. For example cash flow stream A using a reinvestment rate of 8% . . .

FV (Positive CFs) = $120 (1.08)6 + $100 (1.08)5 + $80 (1.08)4 + $55 (1.08)3 + . . .

. . . +35 (108) 2 + $20 (1.08)1 + 10 (1.08)0

= $190.42 + $146.93 + $108.84 + $69.28 + $40.82 + $21.60$ + $10.00= $587.91

Present values of negative cash flows are also calculated using the financing rate, which is 6% in this case. However, for this particular example there is only one negative cash flow and because that occurs immediately ("Now") its presentvalue shows 0 discounting effect.

PV (Negative CFs) = ($220) (1.06) 0 = ($220.00)

Negative cash outflows will have a negative present value, so the formula preceeds the Present Value sum with a ""making it a poisitive number. Using the above formula, MIRR for Investment Case A is thus:

Here, at last, is an investment result with a clear, easily understood meaning:

If the original $220 investment is simply put on deposit, earning interest at the MIRR annual rate of 15.1%for 7 years, the total value with compound interest would be $587.91. Or, if instead the projected cashinflows from the Investment Case A were reinvested at 8.0%, the total investment value with compoundinterest would be the same $587.91.

Similarly, Case B has a MIRR of 14.7%. If B's initial cash outflow is simply put on deposit for 7 years,earning interest compounded at the MIRR rate, the total investment value would be 573.76. Or, if instead,the projected incoming cash flows from investment B were reinvested at the reinvestment rate of 8.0%, thetotal investment value after 7 years would be the same $573.76.

Note: To check these calculations yourself, use the more precise MIRR A rate of 15.0757% and MIRR B

rate of 14.6732%.

In other words, ma king these investments brings the same result as simply putting the investment costs in thebank and receiving interest at the MIRR rate.

Note that IRR results show a larger advantage for Case A over Case B. The realtive investment advanage of A over B ismuch smaller with MIRR. However, most people can easily compare MIRR results with compound interest growth and

understand the magnitude of the MIRR difference. As shown, understanding the meaning of the IRR difference is moreproblematic.

Incidentally, the MIRR formula is really a geometric meanexactly the same formula used to find comulative averagegrowth rate for figures that grow exponentially, such as compound interest earnings. And, calculations like those abovecan be avoided entirely by simply using Excel's MIRR function. For Case A, whose cash flows are located in cells B3through B10, using a reinvestment rate of 8% and financing rate of 6%, Excel's MIRR function would be: =MIRR(B3:B100.06, 0.08) .

Internal rate of return compared to other financial metrics
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Referring to the example cases above, which is the better business investment: Case A or Case B? In manyorganizations, IRRs may play a role in answering such questions, but generally, such questions should not be answeredon the basis of a single financial metric. The prudent financial specialist, investor, or business analyst will compare bothinvestments with several financial metrics. As shown below, different metrics may suggest different answers to thequestion.

This section compares A and B investmentson the basis of net cash flow, internal rate of return, modified internal rate of return, net present value, return on investment, and payback period. For more coverage of the individual metrics in this

encyclopedia, please follow the links provided with each metric.

IRR vs. Net Cash Flow, MIRR, NPV, ROI, and Payback period

IRR results in the above examples required only the net cash flow figures for each investment each period. In order tocompare investments with a wider range of cash flow metrics, however, the analyst needs to see individual cash inflowsand outflows as well as the net figures:

Before looking at the individualmetrics, notice some of the most apparent differences between Case A and Case B cash flow. First, A's inflows andoutflows are much larger than B's. A actually brings in larger inflows but these come at larger costs. This difference isnot apparent when viewing only the annual net cash flow figures. Secondly, as already noted, A's large returns arriveearly, whereas B's larger returns occur in later years. It will take more than one financial metric to fully develop theimplications of these differences.

Financial metrics results for Case A and Case B based on these data are as follows:

Total Net Cash Flow

The net cash flow metric favors investment B over investment A : A brings a $200 net gain over 7 years, while Bbrings in $240. Case B thus has a $40 (20%) advantage in net cash flow over A. For situations where cash flow and

working capital are problematic, this could be an important advantage for Case B. However, see the discussion onPayback Period, below, for a different view of these cash flow consequences.

Internal Rate of Return (IRR)

Investment A outscores Investment B on the IRR metric , 30.6% to 20.8% . Both IRR figures are very likely above thecompany's cost of capital, and both investments can thus be viewed as net gains. A's larger IRR can be taken as asignal that A provides a better rate of return than B (assuming that incoming cash flows are reinvested). Beyond that,however, the IRR figures themselves do not show the magnitude of A's real rate of return advantage over B. When IRRsare several times larger than cost of capital, or more, the real rate of return difference between two different investmentsdepends heavily on the timing of cash flows, the cash flow stream profiles, and the actual rates available for cost of
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capital and reinvestmentnone of which can be seen in the IRR figures.

Modified Internal Rate of Return (MIRR)

With an 8% real reinvestment rate, investment A slightly outscores investment B on the MIRR metric, 15.1% to14.7%. The MIRR's meaning is easily understood: MIRR essentially compares investment results to the growth of compound interest earnings. Assuming that incoming returns are reinvested at 8%, investment A, for instance gives theinvestor exactly the same result as putting the initial cash outflow on deposit for seven years and receiving compound

interest earnings at the MIRR annual rate, 15.1%.

Net Present Value (NPV)

The better business decisionor preferred investmentA or B, according to net present value depends onthe discount rate . At a 5% discount rate, B's NPV of $155 exceeds A's $149 NPV. However, NPV leadership reversesat higher discount rates. With discounting at 10%, A's NPV of $107 is higher than B's $91. As the discount rate rises,B's large returns in later years suffer greater discounting impact than A's larger returns in the early years. This illustratesone reason some financial specialists prefer IRR to NPV when choosing between competing investments: NPV uses anarbitrarily chosen discount rate, which may determine results of the comparison, as shown. IRR, on the other hand, issometimes seen as more "objective" because it does not rely on an arbitrarily chosen rate. IRR instead uses net cashflow figures themselves to find a rate that satisfies its definition.

Return on Investment (ROI)

According to the ROI metric, it's "no contest!" B's ROI of 52.2% beats A's ROI of 18.7%, hands down . The ROmetric shown here is Simple ROI, sometimes called "cash on cash" analysis (incremental gains divided by investmentcosts). All cash flow metrics above, including ROI, show both investments as net gains for the investor. ROI alone,however, is sensitive to the magnitudes of individual annual inflows and annual outflows. The other metrics derive onlyfrom the net cash flow figures. A's larger total costs ($1,200) are compared directly to A's incremental gain of $200. Bscores much higher on ROI because B has a larger incremental gain ($240) and a much smaller total cost ($460). The

large difference in costs is "masked" or hidden in the other metrics. This could be problematic because investmentcosts must be budgeted and paid for, no matter how large the returns, and the investor may simply have troubleproviding the larger funding costs.

Payback Period

The payback period metric shows that investment A "pays for itself" in 2.0 years, while it takes 3.4 years for B's incoming gains to fully cover investment costs . Investors prefer payback periods that are shorter rather thanlonger for at least two reasons. First, the investment funds are available again for re-use sooner with a shorter paybackperiod. Second, investments with longer payback periods are considered more risky.

Financial Metrics Conclusions

When stating a decision criterion as a general rule, business analysts and finance officers often borrow a phrase that isa favorite of economist: Other things being equal , the investment (or action, or decision, or scenario) with the higher IRRis the better business decision. The different financial metrics comparisons above illustrate the point that IRR is blind tomany "other things" that may differentiate competing investments, these things may have important financialconsequences, and they are very rarely truly equal.

Consequently, it is usually recommended that IRR not be used as a decision criterion when comparing competingmutually exlusive investments or actions. When the investor can or will make only one of the two proposed investments,the choice of one over the other represents so-called constrained financing .
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Finally, In business investingas in gamblinga wise investment (or a good gambling bet) is one where potentialrewards compare favorably with investment risks. None of the metrics above fully measures investment risk, althoughrisk considerations are partially visible in IRR, NPV, and payback period:

An Investment with a high IRR can be viewed as less risky than a low IRR investment. Interest rates for

discounted cash flow include a "risk" component and an "inflation" component. If inflation rates rise during the

investment period, or if the appropriate discount rate for NPV rises because of risk considerations, the high IRRinvestment retains greater NPV than the lower IRR investment.

A longer payback period is considered more risky than a shorter payback, simply because of the longer time it

takes to recover investment funds.

When using the above metrics as decision criteria, however, the prudent investor will attempt to assess the likelihoodthat returns actually appear as projected, as well as the liklihood that other beter and worse results appear.

IRR disgused as yield to maturity (YTM) for bond investing

Another reason that IRR is a favorite metric for people trained in finance, is that IRR (disguised under a different name) isused in evaluating bond investments. If the IRR exercises above remind you of something you have seen beforesolvingan NPV equation for an interest rateit is likely you are already familiar with the concept yield to ma turity ( YTM) usein bond investing. IRR and YTM are mathematically the same concept, with only a slight difference in definition (for amore complete coverage of yield and other bond concepts, see the encyclopedia entry for bond).

Yield to Maturity is the interest rate, i , that satisfies this version of the NPV equation:

Bond Purchase Price = FV 1 / (1+ i )1 + FV 2 / (1+ i )2 + ... + FV n / (1 + i ) n

This definition for YTM can be changed into Definition 1 for IRR, simply by subtracting "Bond Purchase Price" from eachside of the equation. This way, Bond Purchase Price becomes the FV 0 in the NPV equation that is used for IRR. The

formula for IRR then asks for the same i that solves the equation:

0 = FV 0 + FV 1 / (1+ i )1 + FV 2 / (1+ i )2 + ... + FV n / (1 + i ) n

Given the same cash inflows and outflows, the same value of i solves both equations. This is another reason thatfinancial specialists use IRR as a metric for evaluating and comparing potential business investments, even when theinvestments are quite different in nature.

Lease vs. buy and other problem IRR results

IRR can usually be found and usefully interpreted when based on net cash flow streams with the "investment curve"profile" shown above: Net cash outflows appear very early in the stream while net cash inflows follow through the rest of the investment life. This profile is common for some kinds of financial investments, such as bond investments or bankdeposits. The investment curve profile may also characterize some investments in income-producing assets, or sometimes even the financial consequences of projects, programs, product launches, and other actions.

In many organizations, however, IRR is routinely calculatedor requiredto support action proposals, even whenexpected cash flows do not fit the investment curve profile. Those who are called upon to provide IRR support for funding
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requests, project or program proposals, or business case analysis are often dismayed when they find that an IRR doesnot exist for their cash flow stream, or they find multiple IRRs for a single stream, or that there is a negative IRR. Or,they may be at a loss to interpret IRR message when one proposed action shows an IRR 10 or 20 times larger than acompeting option.

The situations described below illustrate the four "commandments" of IRR Usage.

1. Do not use IRR when the net cash flow stream differs substantially from the investment curve profile (early net

cash outflows, later net cash inflows).

2. Do not use IRR to compare competing cash flow streams whose profiles differ substantially from each other,

even if they are both roughly "investment curves."

3. Do not be tempted to over interpret IRR magnitude and suggested return rates when IRR differs substantially

from the real cost of capital and real reinvestment rates.

4. Do not even try to find an IRR when the net cash flow stream is entirely positive or entirely negative. There is no

IRR for such situations.

Lease vs. Buy vs. IRR

In a Lease vs. Buy comparison, the "Buy" option typically has a high initial cash outflow to buy the asset. In theremaining years of the asset's life, there should be cash inflows as the asset earns returns. The "Lease" option for thesame asset starts with a very small initial cash outlay (if any), followed by almost the same net returns projected, butreduced somewhat by the periodic leasing fees (this would be the case for a typical operating lease). These cash flowstreams fit this pattern

The "Buy" stream has an IRR o42.6%. The "Leasing" option has an IRR of 1,400.0% . Imagine choosing between these two options, using IRR as thesole decision criterion!

In fact, when results of this kind appear, most people immediatelly ask: "What's wrong with this picture?" Here are threeof the problems:

The Buy option is properly called an "investment," but the Lease option is better described as a 7-year

commitment to a service contract with periodic fees. The two profiles difffer substantially from each other. The

Lease profile is not an investment curve.

IRR looks only at the net cash flow figures each year. The leasing costs each year are "masked" or hidden from

IRR by the larger positive inflows. When these options are compared with the Simple ROI metric (cash on

cash), which is sensitive to individual periodic cash inflows and outflows, both options have exactly the same

ROI, that is, 226%

Both IRRs are certainly much higher than the company's real cost of capital and real reinvestment rates,

especially the Lease option IRR. The analyst who still insists on taking an "investment" view of both options
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should probably turn instead to the modified internal rate of return. The MIRR for the Buy option is 22.5%, while

MIRR for the Lease option is 99.3% (basing MIRR on an 8% reinvestment rate.

In brief, a Lease vs. Buy decision based on IRR would always choose leasing as the better business decision becauseIRR views the action as a financial investment. However, IRR can be blind to period leasing costs, as shown, and in theLease vs. Buy decision, other factors can be more important, such as the impact on the company's asset base, taxconsequences, and flexibility to upgrade or replace assets.

Negative, multiple, and impossible IRRs

With certain cash flow streams it is m athematically poss ible to produce negative IRRs, or multiple IRRs for the cash s tream(more than one discount rate that leads to a 0 NPV). For other cash flow s treams, there is sim ply no mathematically correct IRRsolution.

Impossible IRR : There is no possible IRR solution when the cash flow Includes only positive or only negative

net cash flows. As shown above, there may be real cash outflows in every period, but when the inflows always

outwiegh outflows, there will be no negative net cash flows. There is no IRR in such cases. Other patterns of negative and positive net cash flows can also have no IRR solution.

Multiple IRRs : A net cash flow stream will have multiple IRRs when it Includes more than one sign change.

When the first cash flow is negative and the second cash flow is positive, that is one sign change and there will

be one IRR for the stream. If another, later, net cash flow event is negative, that makes 2 sign changes. There

will be one IRR for every sign change in the cash flow stream. In such cases, it is probably best to consider the

IRR closest to the real cost of capital as the "true" IRR.

Negative IRR : It is also possible for some net cash flow streams to produce a negative IRR value. This signals

simply that the investment or action should be considered a "net loss." Further quantitative analysis of negative

IRRs is not advised. Negative IRRs should certainly be diregarded when the analyst prepares IRR averages, or

weighted average IRRs for multiple investments.

By Marty Schmidt . Copy right 2004 - 2013 Solution Matrix Limited and Marty Schmidt+ . All Rights Reserved

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