*Interpreting IRR in a way that makes practical sense for investors and decision makers is a challenge.*

## What is internal rate of return?

**Internal Rate of Return **(**IRR**) is a financial metric for cash flow analysis, primarily for evaluating investments, capital acquisitions, project proposals, programs, and business case scenarios.

Like other cash flow metrics—NPV, Payback period, and ROI—the IRR metric takes an **investment view** of expected financial results. As a result, each of these financial metrics compares the magnitudes and timing of cash flow returns to cash flow costs, while each makes the comparison in its own way, and each carries a unique message about the value of the action.

By definition, IRR compares returns to costs by by finding the interest rate that produces a zero NPV for the investment cash flow stream.

Not surprisingly, interpreting IRR results in a way that makes practical sense for investors and decision makers is a challenge.

### IRR is All About a Cash Flow Stream

IRR analysis begins with a cash flow stream, a series of net cash flow results expected from the investment (or action, acquisition, or business case scenario).Consequently, cash flows for IRR analysis might look like the figure below. Note especially In the figure:

- Each bar represents the net of cash inflows and outflows for one two-month period.
- Positive values are net inflows and negative values are net outflows.

The complete set of net cash flow events is a **cash flow stream**.

Notice especially the shape, or *profile* of this example stream. This figure represents a typical investment curve because:

- Net cash outflows at the outset and net cash inflows in later periods mean that costs initially exceed incoming returns.
- Returns eventually outweigh the costs and the investment brings a net gain.

The IRR metric, in fact, "expects" this kind of cash flow profile—early costs and later benefits. As a result, when the cash flow stream has this profile, an interpretable IRR probably exists. When cash flow events have another profile, instead, the stream may not have an IRR. In addition, other strange IRR results may also appear when the profile is something other than an investment curve. As a result, other profiles can lead to multiple IRRs for the same stream, or a negative IRR for the stream. Consequently, in such cases, the resulting IRRs are either very difficult to interpret or meaningless

### Internal Rate of Return: Many Use IRR, Few Really Understand IRR

Most people in business have at least heard of "internal rate of return." This is probably because financial officers often require an IRR estimate to support budget requests or action proposals. IRR is in fact a favorite metric of many CFOs, Controllers, and other financial specialists.

Also, some businesspeople know of IRR because many organizations define a hurdle rate in terms of IRR. They specify, that is, an IRR rate that incoming proposals must reach or exceed to qualify for approval and funding.

Two possible reasons for IRR's popularity may be the following:

- Firstly, many in the financial community see IRR as more "objective" than net present value (NPV). This is because NPV results from arbitrarily chosen discount rates while IRR, by contrast, results entirely from the cash flow figures themselves and their timing.
- Secondly, some also believe that IRR readily compares return rates with inflation, current interest rates, and financial investment alternatives. Note especially in the discussions below that this belief is sometimes supportable and sometimes not.

It should be no surprise to learn that most businesspeople 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 consistently that most financial specialists who require IRRs with proposals or funding requests are themselves largely unaware of IRR's serious deficiencies. And, many are also unable to explain its meaning and proper use.

### Emphasize IRR Meaning and Interpretation

This article emphasizes several IRR issues:

- Firstly, IRR meaning and interpretation.
- Secondly, common misconceptions and misuses of IRR.
- Thirdly, comparing IRR to other financial metrics.
- Fourthly, presenting
**modified internal rate of return**(**MIRR**) as an easy-to-understand alternative to IRR.

## Contents

- What is internal rate of return?
- Defining internal rate of return IRR. Example illustration.
- First IRR Interpretation: IRR as a measure of risk.
- Finding IRR: Can you actually calculate IRR?
- IRR Graphical Solution.
- IRR Spreadsheet Solution.
- Re-defining IRR: Why is it called
*internal*rate of return? - Second IRR interpretation: Financing costs and reinvestment gains.
- Is modified internal rate of return (MIRR) a better metric than IRR?
- How does Internal rate of return compare to NPV, ROI, and other financial metrics?
- Is IRR appropriate for lease vs. buy analysis? What are other problem IRR results?
- Four commandments for using IRR.

## Related Topics

- Is IRR another case of the emperor's new clothes? See the IRR Blog post.
- NPV, ROI, Payback and IRR cash flow metrics. See Financial Metrics.

## Defining Internal Rate of Return IRR

First Textbook Definition Illustrated Example

As the word "return" in its name implies, an IRR view of the cash flow stream is an* investment* view. This means essentially that the metric compares outgoing funds to incoming to the magnitude and timing of incoming returns.

The best known IRR definition explains this comparison in terms that call for a basic understanding of cash flow discounting concepts: present value, net present value (NPV), and the role of the discount rate (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. As a result, many businesspeople ask: What does *that* tell me about returns and costs?

### A First Interpretation of IRR Meaning: IRR As a Measure of Risk.

Consider two investment proposals competing for funding: Case Alpha and Case Beta. The expected net cash flow streams for Alpha (A) and Beta (B) are as follows:

Alpha

Beta

**Total**

**$200**

**$240**

### Different IRRs for Front Loaded vs. Back Loaded Cash Flow Streams

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

- Both streams qualify as investment curve profiles, but one is "front loaded" while the other is "back loaded." These terms refer to the timing of returns in the cash flow stream.
- Case Alpha (dark blue bars) has large early returns but these diminish year by year. Case Alpha is therefore
**front loaded**. Alpha's profile could represent the acquisition of an income producing asset that becomes less productive or more costly to maintain each year. - Case Beta (light yellow bars) has smaller returns at first, but Beta's returns grow each year. Beta is therefore
**back loaded**. Beta's profile could result from funding a product launch that returns greater profits each year.

- Case Alpha (dark blue bars) has large early returns but these diminish year by year. Case Alpha is therefore

The analyst therefore compares two different kinds of investments with the same metric, IRR, to help decide which is the better business decision. Using a spreadsheet or another IRR program to analyze the cash flow figures above, the IRR results are as follows:

**Case Alpha:** IRR_{A} = 30.6%

**Case Beta**: IRR_{B} = 20.8%

### IRR is the Dscount Rate That Brings NPV to Zero

The next table and figure below show the result of applying IRR Definition 1, that is, finding IRR as the discount rate that brings an NPV of 0. Note that this example shows only one of the above cash flow streams, Case Alpha.

**Total**

**$200**

**$0**

The first column shows net cash flow each year, while the second column shows the discounted values (present values) of the same cash flows. Discounting here uses the IRR rate for this cash flow stream, IRR = 30.6%.

In this graph:

- Dark blue bars are future value net cash flows for Case Alpha.
- Light blue bars are present values of the same cash flows at IRR discount rate of 30.6%.

You may just be able to see or imagine that the heights of the seven positive (upward pointing) light blue bars starting with Yr 1 add up exactly to the length of the one negative (downward pointing) light blue bar at "Now." As a result, the IRR definition is satisfied because the sum of positive present values exactly cancels the sum of negative PVs.

## First IRR Interpretation

IRR As a Measure of Risk

Which investment choice should the analyst recommend, Case Alpha or Case Beta? Alpha has an IRR of 30.6%, while Beta has an IRR of 20.8%. Using IRR as the decision criterion—and with other factors being equal—the analyst will recommend the case with the higher IRR (Alpha) as the better choice.

This recommendation may follow partly because a higher IRR indicates less *risk*. That is, IRR shows just how high inflation rates or risk probabilities have to rise in order to eliminate the present value of this investment.

- For Case Alpha, the discount rate would have to reach 30.6% to drain Alpha's results of present value.
- Case Beta would lose all present value if the discount rate rises to 20.8%.

Most people, however, find this first interpretation of IRR of limited value for evaluating and comparing investment proposals. Later sections below, therefore, move to

- Firstly, find another IRR definition and interpretation that is more helpful.
- Secondly, compare IRR results with other cash flow metrics.
- Thirdly, present another cash flow metric that is easier to interpret than IRR. This is the
**modified internal rate of return**(**MIRR**).

## Finding the Internal Rate of Return IRR

Can You Actually Calculate IRR?

Analysts calculate cash flow metrics such as NPV, ROI, and even payback period, directly from formulas. However, the verbal IRR Definition 1 above does not readily lend itself to expression as a formula.

Remember first that IRR Definition 1 *refers to* another metric that *does* calculate from a formula, net present value NPV. The panel below shows the formula for calculating NPV for a cash flow stream using end of period discounting.

Here, the **FV**s in the formula are net cash flow figures for each period, ** i** is the discount rate, while

*is the number of periods. For Case Alpha and Case Beta,*

**n***= 7. That is, the cash flow stream covers 7 periods where each period is one year.*

**n**Consequently, when met with a request for an "IRR formula," about the only response possible is to start with (1) the NPV formula above and (2) the FVs (net cash flow values), and then proceed as follows:

To find IRR:

- Set
**NPV**equal to 0. - Solve the formula for
**i** **IRR**=when**i****NPV**= 0.

In fact, there is no easy analytic solution to this request and most people will be unable to solve the NPV = 0 equation for *i. *As a result, It is more accurate to say that IRR is *found* by indirect means and not *calculated* directly. Consequently, analysts instead find IRR either by (1) graphical analysis or (2) as spreadsheets do, through a series of approximation trials (successive approximations).

## Finding IRR With a Graphical Solution

The graphical solution for IRR is useful for helpful for understanding the first IRR definition, above. Note especially that businesspeople in the pre-computer era relied heavily on the graphical approach for finding IRR. The alternative approach, successive approximations, was simply impractical, given the number of high precision calculations required. Now, of course the reverse is true. Analysts use spreadsheets or other software produce high precision IRRs almost instantly. And, the graphical solution serves simply as a teaching aid.

The graphical approach begins with a table of discount rates and NPV values, such as the table below. This example shows NPVs for Alpha and Beta cash flow streams at 10 different discount rates. Note especially that NPVs here derive from the original net cash flow figures and the NPV formula above. to produce this table, the analyst had to calculate 20 NPV figures.

### Example Graphical Solution

Rate

Beta

The data above are plotted below, showing the relationship between discount rate (horizontal axis) and resulting NPV (vertical axis).

The graph shows that increasing the discount rate lowers NPV for both streams. The graph also shows how cash flow streams with a positive total net cash flow can produce NPVs that are either positive or negative, depending on discount rate.

Regarding IRR, note especially: that Alpha's NPV reaches 0 at a discount rate of 30.6%, while Beta's NPV reaches 0 at a discount rate of 20.8%. From the graph, therefore, the analyst concludes that IRR_{A} = 30.6%. and IRR_{B} = 20.8%.

## Finding IRR With a Spreadsheet Solution

The NPV curves above are useful for helping show exactly the meaning of IRR Definition 1. Today, however, most people in business use spreadsheets or pre-programmed calculators to find IRRs.

Either way, a program starts the approximation process using an arbitrarily chosen discount rate to calculate NPV for a cash flow stream. If the resulting NPV is not 0, the discount rate is adjusted, and the NPV calculation repeats. And the program checks again to see if NPV = 0.

Trials of this kind repeats hundreds or thousands of times until a discount rate that does produce NPV = 0 appears. In Microsoft Excel, for instance, trials continue until Excel finds an IRR accurate to 0.00001 percent. Because trials execute very quickly, however, the IRR result seems to appear immediately with data entry.

The analyst, for example, might enter a Microsoft Excel IRR function into an Excel formula like this:

**=IRR (B3:B10, 0.1)**

The spreadsheet cell with this formula shows the IRR for a worksheet range with net cash flow figures in cells B3 through B10. These cells could, for instance, hold the eight cash flow values for Case Alpha in the example above. Note also that the **0.1** figure is simply a user-provided first guess at the IRR. The guess is just a starting discount rate for calculating NPV on the first iteration and it can be almost anything.

The IRR value itself will appear in the cell holding the above Excel formula. Here, the actual numerical result in cell B13 is **0.306325**. However, the analyst will probably format cell B13 as a percentage so that the IRR result looks like this: **30.6%**

## Re-Defining IRR: Second Definition

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. As a result, IRR's more important meaning is usually easier to understand interms of another IRR definition, one that refers to investment financing costs and reinvested returns.

### IRR Definition 2:

The Internal Rate of Return (IRR) For a Cash Flow Stream is Based on 2 Assumptions:

- There is a financing cost (or opportunity cost) for using funds to make the investment. The calculation assumes that invested funds incur financing costs (or opportunity costs) until the final cash flow event.

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

### IRR is then is the single interest rate for financing costs and for reinvestment earnings that sets the total gains exactly equal to total costs.

The table below shows how IRR achieves this balance:

This example shows how the IRR rate causes total investment returns (cash inflows) to equal total investment costs (cash outflows). First, the spreadsheet finds IRR for the cash flow stream (blue cells). This rate then serves in calculating both returns and investment costs. Using the IRR rate, "Total from inflows reinvested" equals "Financing costs."

### At IRR, Investment Costs Balance Investment Returns

In the example, a cash outflow of $220 (at "Now") represents initial investment costs. Subsequently, each year after that, the investment brings positive cash flow returns. Excel's IRR function result appears in the yellow cell below Year 7 net cash flow, reporting an IRR of 30.63% for the net cash flow stream running from "Now" through Year 7. Therefore, the same IRR value of 30.63% is used for calculating both (1) reinvestment earnings and (2) financing costs.

- Consider first the interest earned by re-investing the incoming cash flows from Years 1 through 7. Because each incoming return is reinvested for the remaining years at an annual interest rate of 30.63%, the total seven year gains are $1,428.17 (inflows + interest earned).
- Now, assume the initial cash outflow of $220 is borrowed and financed at the same 30.63% annual rate. Because the total cost of repaying this loan is also $1,428.17, exactly (initial cash outflow plus financing), the IRR rate exactly balances total costs with total gains.

## Interpreting the Second IRR Definition

Financing Costs and Reinvestment Gains

The second-definition example above should suggest a reason that financial people often trust IRR as an important decision criterion.

- First; remember that the IRR result assumes the use of funds to pay investment costs brings additional costs, either borrowing costs or opportunity costs.
- Second, remember also that incoming returns are reinvested, earning additional gains.

These two assumptions provides meaning for another IRR interpretation, namely that the analyst will compare the IRR rate to actual financing rates and actual rates for reinvesting. These comparisons must be interpreted carefully, however, because It is easy to over interpret or misinterpret IRR at this point.

### Tempting But Faulty IRR Reasoning

When an investment proposal produces IRR's like those shown above—30.6% for example—it is tempting to reason as follows:

"For this investment, we will not actually borrow or pay an opportunity cost at the IRR rate. Therefore, our real financing cost will be subject to a much lower interest rate, closer to our cost of capital, probably less than 10%. Thus, [the reasoning goes], the investment is a

net gainbecause financing rates will really be under 10%, while returns represent earnings at a much higher rate, something like 30.6%."

In reality, that conclusion may or not be supportable, depending on the *actual *financing rates (or opportunity costs) and the *actual* reinvestment rates that apply. The conclusion is arguably valid only when the IRR rate is close to actual cost of capital and actual reinvesting rate. When IRR is quite different from actual rates, however, the same conclusion is more likely misleading or quite wrong.

### Why Does IRR Sometimes Overstate the Value of Returns?

This kind of reasoning can grossly overstate the value of actions like Case Alpha. Suppose, for example, that the real earnings rate on reinvested returns is close to 8%, much lower than the stream's IRR of 30.6%. Notice especially :

- Cash flow stream Alpha has its largest returns in the first and second years of the 7-year analysis period. That is, Case Alpha is "front loaded," or "biased" towards the early years. IRR therefore overstates the real earnings rate because it assumes year 1 and year 2 gains will earn at a 30.6% rate for all the remaining years. In fact, however, this long term of high-rate earnings will be absent.
- Cash flow stream Beta, on the other hand has a lower IRR than Alpha, but when real return earning rates are compared to the earnings that IRR assumes, Beta in fact has less "missing" returns of this kind than Alpha.
- IRR thus overstates the real value of Case Alpha far more than it overstates the value of Case Beta for two reasons:
- First, stream Alpha is front loaded and stream Beta is back loaded.
- Second, Alpha's IRR is further from the real reinvestment rate than Beta's IRR.

In conclusion, one can reasonably view the investment outcome as a net gain when IRR exceeds the organization's cost of capital. Beyond this, however, further assessment of IRR magnitude can be problematic, especially when:

- IRR greatly exceeds cost of capital and the real earnings rate for returns
- Comparing two cash flow streams with different profiles as in the example above. This point appears again in the discussion below on Lease vs. buy comparisons.

## Modified Internal Rate of Return MIRR

Is MIRR a BetterMetric Than IRR? MIRR is Easier to Interpret Than IRR.

The meaning of IRR magnitude is difficult to interpret, as shown, because IRR can differ from the actual financing and earnings rate for returns. It is natural to ask, therefore, "Why not use instead an internal rate of return metric that *does* reflect the real financing cost rate and real earnings rate for returns?"

In fact, exactly this solution is readily available as the **modified internal rate of return **(**MIRR**) metric.

### Example: MIRR Calculates Directly From a Formula

Input data for MIRR include the same net cash flow figures as IRR, but the MIRR calculation also requires a financing rate and a reinvestment rate as input. Here for comparison are the IRR and MIRR results for Cases Alpha and Beta from above. MIRR for this example assumes an 8% earnings rate for reinvested returns, and a 6% financing rate on costs.

**Case Alpha**: IRR_{A} = 30.6% MIRR_{A} = 15.1%

**Case Beta**: IRR_{B} = 20.8% MIRR_{B} = 14.7%

Notice immediately that Case Alpha also has a higher MIRR value than Case Beta, but both MIRR values are much closer to each other than are the two IRR values.

The full meaning of MIRR is easier to explain after showing first how to calculate MIRR. Unlike IRR, MIRR derivces directly from a formula:

To obtain MIRR, the analyst calculates:

- Future values of cash inflows using the reinvestment rate.
- Present values of the cash outflows using the financing rate.

The radical sign calls for the *n*^{th} root of the (Future Value)/(Present Value) ratio.* *The number of periods* *in this case, *n*, is 7. Subtracting 1.0 from the resulting root yields MIRR.

### MIRR Example Calculation

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**·**(1·08)^{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

The anayst also calculates the Net Present Value of the negative cash flows, applying the financing rate (given as 6% in this case). Note, however, that for this simple example there is only one negative cash flow ($220) and because that occurs immediately ("Now"), its present value 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 precedes the Present Value sum with a minus sign ("–") making it a positive number. Using the above formula, MIRR for Investment Case Alpha is thus:

### MIRR Meaning is Easy to Understand

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

- If the original $220 cost is simply put on deposit in the bank, earning interest at the MIRR annual rate of 15.1% for 7 years, the total investment value with compound interest will be $587.91.
- If instead the investment cost of $220 is paid and the projected cash inflows from Case Alpha are reinvested at 8.0%, the total investment value for them with compound interest earnings will be the same $587.91.
- Similarly, Case Beta has a MIRR of 14.7%. If Case Beta's initial cash outflow is simply put on deposit for 7 years, earning interest compounded at the MIRR rate, the total value will be 573.76.
- If instead, the projected incoming cash flows from Case Beta are reinvested at the rate of 8.0%, the total value after 7 years will 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 Simpler terms, taking the actions proposed in Case Alpha and Case Beta brings the same results as simply putting the initial costs in the bank and receiving compound interest earnings at the MIRR rate!

Note that IRR results show a large advantage for Alpha over Beta. The relative advantage of Alpha over Beta is much smaller in the MIRR results. In conclusion, most people can easily compare MIRR results with compound interest growth and understand the magnitude of the MIRR differences. As shown, understanding the meaning of the IRR difference is more problematic.

### Finding MIRR with Excel

Incidentally, the MIRR formula is really a geometric mean, exactly the same formula used to find cumulative average growth rate for figures that grow exponentially such as compound interest earnings. And, calculations like those above can be avoided entirely by simply using Excel's MIRR function. For Case Alpha, whose cash flows are located in cells B3 through B10, using a reinvestment rate of 8% and financing rate of 6%,

Excel's MIRR function for this example would be:

**=MIRR(B3:B10, 0.06, 0.08)**

## How Does IRR Compareto Other Cash Flow Metrics?

IRR Results compare to Results from NPV, ROI, Payback, and Other Metrics

Referring to the example cases above:

Which is the better business decision, Case Alpha or Case Beta?

Financial specialists in many organizations use IRR to address such questions, while recognizing also that important questions deserve more than a single financial metric. The prudent financial specialist, investor, or business analyst will compare both proposed actions with several financial metrics. Examples below show how different metrics may suggest different answers to the question.

This section compares Alpha and Beta cases using **net cash flow**, **internal rate of return**, **modified internal rate of return**, **net present value**, **return on investment**, and **payback period**.

### 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 period (year). In order to compare investments with a wider range of cash flow metrics, however, the analyst must see individual cash inflows and outflows for each period along with the net figures.

**Cash**

Flow

Flow

Flow Alpha

Flow Alpha

Cash

Flow

Alpha

Flow Beta

Flow Beta

Cash

Flow

Beta

**-220**

**-220**

**120**

**50**

**100**

**60**

**80**

**60**

**55**

**65**

**35**

**70**

**20**

**75**

**10**

**80**

**1,400**

**1,200**

**200**

**700**

**-460**

**240**

### Comparing Financial metrics from cash flow data

This section compares different financial metric, calculating all from the set of cash flow figures above.

Before calculating individual metrics, however, notice first some of differences between Alpha and Beta cash flows.

- First, Alpha's inflows and outflows are much larger than Beta's. Alpha (A) actually brings in larger inflows but these come at larger costs. This difference is not apparent when viewing only the annual net cash flow figures.
- Secondly, Alpha's large returns arrive early, whereas Beta's larger returns occur in later years.

It will take more than one financial metric to fully develop the implications of these differences. Below are financial metrics calculated from the Case Alpha and Case Beta cash flow figures above:

**Financial**

Metrics

Metrics

**Case Alpha**

**Case**

Beta

Beta

###

Total Net Cash Flow

The net cash flow metric favors investment Beta over investment Alpha: Alpha brings a $200 net gain over 7 years, while Beta brings in $240. Beta thus has a $40 (20%) advantage in net cash flow over A. For business situations where cash flow and working capital are in short supply, this could be an important advantage for Beta. However, see the discussion on Payback Period, below, for a different view of these cash flow consequences.

### Internal Rate of Return (IRR)

Case Alpha outscores Case Beta on the IRR metric, 30.6% to 20.8%. Both IRR figures are very likely above the company's cost of capital, and both proposals are thus viewed as net gains. Alpha's larger IRR can be taken as a signal that Alpha provides a better rate of return than Beta (assuming reinvestment of incoming cash flows). Beyond that, however, the IRR figures themselves do not show the *magnitude* of Alpha's real rate of return advantage over Beta.

When IRRs are larger than cost of capital, by several times or more, the real rate of return difference between two investments depends heavily on:

- The timing of cash flows.
- The cash flow stream profiles.
- The actual rates available for cost of capital and earnings on returns

The IRR figures say nothing about these factors.

### Modified Internal Rate of Return (MIRR)

With an 8% real earnings rate for returns, Case Alpha slightly outscores Case Beta on the MIRR metric, 15.1% to 14.7%. MIRR's meaning is easily understood: MIRR essentially compares results to the growth of compound interest earnings. Assuming that incoming returns are reinvested at 8%, Alpha, for instance gives the investor 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)

According to the net present valuemetric, the better investment choice ( Alpha or Beta) depends on the discount rate.

- Thus, with a 5% discount rate, Beta's NPV of $155 exceeds Alpha's $149 NPV. However, NPV leadership reverses at higher discount rates.
- With discounting at 10%, Alpha's NPV of $107 is higher than Beta's $91.

As the discount rate rises, Beta's large returns in later years suffer greater discounting impact than Alpha's larger returns in the early years. This illustrates one reason some financial specialists prefer IRR to NPV when choosing between competing proposals:

- The analyst arbitrarily chooses a discount rate to calculate NPV. This may determine results of the comparison as the NPV example here shows.
- IRR, on the other hand, is sometimes seen as more "objective" because it does not rely on an arbitrarily chosen rate. IRR instead uses net cash flow figures themselves to
*find a*rate that satisfies its definition.

### Return on Investment (ROI)

According to the ROI metric, it is "no contest!" Beta's ROI of 52.2% beats Alpha's ROI of 18.7%, hands down. The ROI metric shown here is Simple ROI(the ratio of incremental gains to investment costs).

All cash flow metrics above show both actions as resulting in net gains for the investor. ROI alone, however, is sensitive to the magnitudes of individual annual inflows and annual outflows. By contrast, the other metrics derive only from the *net* cash flow figures.

Alpha's larger total costs ($1,200) are compared directly to Alpha's incremental gain of $200. Beta scores much higher on ROI because Beta has a larger incremental gain ($240) *and* a much smaller total cost ($460). Other metrics that derive from net cash flow are blind to the large differences in costs. This could be problematic because the investor must budget these costs and pay them, no matter how large the returns. The investor may simply have trouble providing the larger funding costs.

### Payback Period

The payback period metric shows that investment Alpha "pays for itself" in 2.0 years, while Beta needs 3.4 years to fully cover investment costs. Investors prefer shorter payback periods over longer payback for at least two reasons.

- First, the investment funds are available again for re-use sooner with a shorter payback period.
- Second, investors see the longer payback period as more risky.

### Financial Metrics Conclusions

When stating a decision criterion as a general rule, business analysts and finance officers often borrow a phrase that is a favorite of economists:

Other things being equal, the investment (or action, or decision, or scenario) with the higher IRR is the better business decision.

The different financial metrics comparisons above show that IRR is blind to many "other things" that may differentiate competing investments, and these things may have important financial consequences. And, they are very rarely truly equal.

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**. Consequently, analysts usually recommend that IRR *not* be used as a decision criterion when comparing competing mutually exclusive investments or actions.

### Rewards Must Compare Favorably to Risks

Finally, In business investing—as in gambling—a wise investment (or a good gambling bet) is one where potential rewards compare favorably with investment risks. None of the metrics above fully measures investment risk, although risk 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 discounting 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 IRR investment retains greater NPV than the lower IRR investment. - Analysts view longer payback periods as more risky than shorter paybacks, 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 likelihood that returns actually appear as projected, as well as the likelihood that other better and worse results appear.

### IRR is the Same Metric as Yield to Maturity (YTM) for Bond Investing

Another reason that IRR is a popular metric for people trained in finance, is that IRR is usually center stage when evaluating bond investments. Note, however, that in bond investment, the sam IRR metric is known as **yield to maturity** (**YTM**).

If the IRR exercises above remind you of something you have seen before—solving an NPV equation for an interest rate—it is likely you are already familiar with the yield to maturity concept in bond investing. IRR and YTM are mathematically the same concept, with only a slight difference in definition.

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 transforms into Definition 1 for IRR, simply by subtracting "Bond Purchase Price" from each side of the equation. Thus, bond purchase price becomes the FV_{0} in the NPV equation used above for IRR Definition 1. Remember that IRR is the interest rate that solves this form of the NPV 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 suggest another reason that IRR is popular with financial specialists. IRR serves evaluate and compare different business investments, even when the investments are quite different in nature.

For more complete coverage of yield and other bond concepts, see the encyclopedia entry Bond.

## Is IRR Appropriate for Lease vs. Buy Decisions?

What Are 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 life. This profile is common for some kinds of financial transactions, such as bond investments or bank deposits. The same 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, the routine practice—or policy—is to require IRR estimate for all action proposals, even when expected cash flows do not fit the investment curve profile. Consequently, those who must provide IRR support for their proposals may react with surprise and dismay when they produce the metric and find any of the following:

- An IRR does not exist for their cash flow stream.
- Multiple IRRs exit for a single stream.
- There is a negative IRR.
- One proposed action shows an IRR 10 or 20 times larger than a competing option.

In such cases, IRR is simply inappropriate for evaluating the investment or action.

### Lease vs. Buy vs. IRR

When the Lease option in a "Lease vs. Buy' comparison involves a typical operating lease,the two net cash flow streams in view have these characteristics:

- The "Buy" option starts with a high initial cash outflow for asset purchase. In the remaining years of the purchased asset's life, it should earn returns, leading to positive net cash flows most periods.
- The "Lease" option for the same asset starts with a very small initial cash outlay (if any) for the first period. Periodic net cash flows for following periods should be almost the same as the net returns under the "Buy" option, but slightly less due to the periodic leasing fees.

These cash flow streams fit this pattern:

**Purchase**

& Lease

Cash Flow

& Lease

Cash Flow

Asset

Net CF

Asset

Net CF

**Total**

**$2485**

**$485**

The "Buy" stream has an IRR of 42.6%. And, the "Leasing" option has an IRR of 1,400.0%. Imagine choosing between these two options using IRR as the sole decision criterion!

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

### What's wrong with this IRR picture?

- The analyst can properly refer to the Buy option as an "investment." A better description of the Lease option is a 7-year "commitment to a service contract with periodic fees." The two profiles differ substantially from each other. Note especially that the Lease profile is not an investment curve.
- IRR looks only at the
*net*cash flow figures each year. The larger positive inflows each year "mask," or "hide" the leasing costs from IRR.

Some analysts may compare these investment options with the Simple ROI metric, in preference to IRR. Simple ROI*is*sensitive to individual periodic cash inflows and outflows. Note especially that 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 earnings rates for returns, especially the Lease option IRR.

The analyst who still insists on taking an "investment" view of both options should probably turn instead to the modified internal rate of return MIRR. Here, MIRR for the Buy option is 22.5%, while MIRR for the Lease option is 99.3% (basing MIRR on an 8% earnings rate for returns).

As a result, "Lease vs. Buy" decisions that turn entirely on IRR values almost always favor leasing as the better business decision. This is because IRR views both actions as financial investments. In the Lease vs. Buy decision, however, decision makers may consider factors other than IRR as more important, such as the impact on the company's asset base, tax consequences, and flexibility to upgrade or replace assets.

### Negative, multiple, and Impossible IRRs

For some cash flow streams it is mathematically possible to produce negative IRRs, or multiple IRRs (more than one discount rate that leads to a 0 NPV for a single cash flow stream). In addition, for other cash flow streams, there is simply no mathematically correct IRR solution.**No IRR Solution**.

- There are many possible cash flow stream profiles that for which there is no IRR solution.
- Firstly, there may very well be real cash outflows and real cash inflows every period, but when the inflows always outweigh outflows, net cash flow is therefore positive every period. Therefore, cash flow streams with positive net cash flows only, do not have an IRR solution.
- Secondly, when cash outflows outweigh cash inflows every period, net cash flow every period is negative. There is also no IRR in such cases.

- Some other cash flow stream profiles that include both negative and positive periodic net cash flows, for which an IRR solution is still impossible.

**Multiple IRRs**.

A net cash flow stream will have multiple IRRs when it Includes more than one

*sign change*in the net cash flow figures.- When the first periodic net cash flow is negative and the second net cash flow is positive, that is one sign change. And, If there are no more sign changes in later periods, there will be no more than one IRR for the stream.
- If another, later, net cash flow event is negative, that makes 2 sign changes. There can be one IRR for every sign change in the cash flow stream. Therefore, a cash flow stream with 4 sign changes may have up to 4 IRR values.

- When multiple IRRs exist for a single stream, analysts may simply 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.- When the stream of net cash flow figures leads to a negative IRR, the analyst may simply consider the investment or action a "net loss."
- Further quantitative analysis of negative IRRs, however, is almost always meaningless.

- Analysts should disregard negative IRRs when preparing IRR averages, or weighted average IRRs for multiple actions.

## Using IRR: The Four Commandments

But How Long is Life?

The IRR outcomes and problems above help explain four "commandments" of IRR usage:

- IRR is not appropriate when the net cash flow stream differs substantially from the profile with early net cash outflows and later net cash inflows.
- Do not use IRR to compare competing cash flow streams whose profiles differ substantially from each other.
- The analyst should not over interpret IRR magnitude and return rates when IRR differs substantially from the real cost of capital and real earnings rates for returns.
- Do not expect to find an IRR when the net cash flow stream is entirely positive or entirely negative, because there
*is*no IRR for such situations.