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# Internal Rate of Return IRR and Modified IRR ExplainedDefinition, Meaning, and Example Calculations

Investors and decision makers use the ROI metric to compare the magnitude and timing of expected gains with the magnitude and timing of costs.

## What is internal rate of return?

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

Like other cash flow metrics (net present value, payback period, and return on investment), IRR takes an "investment view" of expected financial results. This means, essentially, that the magnitudes and timing of cash flow returns are compared to cash flow costs. Each of these financial metrics compares returns to costs in its own way and each carries a unique message about the value of the action.

IRR analysis begins with a cash flow stream, a series of net cash flow figures expected from the investment (or action, acquisition, or business case scenario). The expected cash flows for IRR analysis might appear as shown below. 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.

This stream shows expected results from one action, and another stream for a different action might show a different cash flow profile. IRRs for each stream can be compared and, other things being equal, the proposed action with the higher IRR is viewed as the better choice.

Notice especially the shape, or profile of this example stream. The figure shows a typical "investment curve." Net cash outflows at the outset and net cash inflows in later periods mean that costs initially exceed incoming returns, but if results appear as expected, returns eventually outweigh the costs. The IRR metric, in fact, "expects" this kind of cash flow profile—early costs and later benefits. When the cash flow stream has this profile, an IRR can probably be found and usefully interpreted. When the stream deviates substantially from this profile, however, it may not be possible to find an IRR for the stream. Or, other strange IRR results may appear, such as multiple IRRs for the same stream, or a negative IRR for the stream. In such cases, the resulting IRRs are either very difficult to interpret or meaningless.

Most people in business have heard of "internal rate of return." Some are required by their financial officers to produce an IRR to support budget requests or action proposals. IRR is in fact a favorite metric of many CFOs, Controllers, and other financial specialists. Many in the financial community believe, for instance, that IRR is a more "objective" metric than NPV, because NPV depends on an arbitrarily chosen discount rate, while IRR is determined entirely by the cash flow figures and their timing. Many also believe that IRR allows returns to be compared readily with inflation, current interest rates, and financial investment alternatives. Organizations sometimes establish an IRR hurdle rate, that is, an IRR rate that must be reached or exceeded by incoming proposals before approval and funding.

It should be 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 consistently that most financial specialists who require IRRs with proposals or funding requests are themselves largely unaware of IRR's serious deficiencies and unprepared to explain its meaning and proper use.

This article puts a special emphasis on understanding IRR meaning and interpretation, as well as common misconceptions and misuses of IRR. This article also compares IRR to other financial metrics and presents the modified internal rate of return (MIRR), a more easily interpreted alternative to the better known IRR.

## How is internal rate of return IRR defined? Ilustrated example.

As the word "return" in its name implies, an IRR view of the cash flow stream is essentially an investment view: paid out funds are compared to returns. The best known IRR definition explains this comparison in terms that call for a basic understanding of discounted cash flow 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. Many ask:
What does that tell me about returns and costs?

A first interpretation of IRR meaning is illustrated with an example.

Consider two proposed investments competing for funding: Case Alpha and Case Beta. The expected net cash flow streams for Alpha (A) and Beta (B) are as shown.

Net CF
Alpha
Net CF
Beta
Now
-220
-220
Year 1
120
50
Year 2
100
60
Year 3
80
60
Year 4
55
65
Year 5
35
70
Year 6
20
75
Year 7
10
80
Total
200
240
Both cases call for an initial cash outlay of \$220. 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 in the image how the two cash flow streams differ:

Both streams have the investment curve profile described above. Case Alpha (blue bars) has large early returns but these diminish year by year. 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 (yellow bars) has smaller returns at first, but Beta's returns grow each year. Beta's profile could result from funding a product launch that returns greater profits each year. The analyst will thus compare two different kinds of investments with the same metric, IRR, to help decide which is the better business decision. The net cash flow figures above, when analyzed with a spreadsheet function or another IRR program show:

Case Alpha:  IRRA = 30.6%
Case Beta:    IRRB = 20.8%

The next table and figure below show the result of applying IRR Definition 1 (the discount rate that brings an NPV of 0). Only one of the above cash flow streams is shown here, Case Alpha.

Now
-220
-220
Year 1
120
92
Year 2
100
59
Year 3
80
36
Year 4
55
19
Year 5
35
9
Year 6
20
4
Year 7
10
2
Total
200
0

The first table column shows net cash flow each year, and the second column shows the discounted 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 present value (NPV) of 0. This chart also shows the same thing in visual terms.

• Dark blue bars are future value net cash flows for Case Alpha.
• Lightt blue bars are present values of the same cash flows at the 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." In other words, at the IRR discount rate, the sum of positive present values exactly cancels the sum of negative PVs.

### First IRR interpretation: IRR as a measure of risk

In the example above, which is the recommended choice? Case Alpha or Case Beta? Alpha has an IRR of 30.6%, Beta has an IRR of 20.8%. Other things being equal, and using IRR as the decision criterion, the case with the higher IRR (Alpha) is considered the better choice. One reason for this conclusion is that 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 (which includes an inflation component and a risk component) 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 rose to 20.8%.

Most people, however, find this first interpretation of IRR of little value for evaluating and comparing proposed investments. The sections below, therefore, move to 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 from simple formulas, if you have the proper input data. However, the verbal IRR Definition 1 above does not readily lend itself readily to expression as a formula. The best that can be done for producing an IRR formula is this: Consider instead the formula for calculating net present value 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 Alpha and Case Beta above, = 7 (7 periods).

The person who wants a formula for IRR can be handed the above formula along with the FVs (net cash flow values) and then asked to do the following:

To find IRR:

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

In fact, there is no easily applied analytic solution to this request, and it is more accurate to say that IRR is "found" rather than "calculated." IRR can be found either by graphical analysis or—as Excel does—by "trial and error" (more accurately, by successive approximations).

Discount
Rate
NPV Alpha
NPV
Beta
0%
200.00
240.00
4%
158.15
170.05
8%
123.04
114.73
12%
93.27
70.40
16%
67.80
34.45
20%
45.82
4.98
24%
26.69
-10.44
28%
9.93
-39.88
32%
-4.85
-57.09
36%
-17.97
-71.75

Using the formula above, NPV was calculated for Alpha and Beta cash flow streams at 10 different discount rates. The table shows the calculated NPV values. These are plotted here, showing the relationship between discount rate (horizontal axis) and resulting NPV (vertical axis).

As you would expect, increasing discount rates brings lower NPVs for both streams. However, 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%. Therefore, IRRA = 30.6%. and IRRB = 20.8%

Instead of finding a graphical solution for IRR, most people use Microsoft Excel or a pre-programmed calculator. Either way, the software 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 rate that produces a 0 NPV. This occurs very quickly, so that the IRR result seems to appear as soon as the data are entered. The analyst, for example, might enter the Excel IRR function into a 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 could be, for instance, the eight cash flow values for Case Alpha in the example above. The "0.1" figure is simply a 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 or even omitted. The analyst will probably format the spreadsheet cell as a percentage, 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 more important meaning is easier to understand in terms of another 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 two assumptions:

1. There is a financing cost (or opportunity cost) for using funds to make the investment. Investment costs will be financed (or the opportunity cost will be incurred) until the final cash flow event.
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 sets the 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 flow figures in blue cells are from example Case Alpha above.

In the example, a cash outflow of \$220 (at "Now") represents initial investment costs. Each year after that, the investment brings positive net cash flow returns. Excel's IRR function has been entered in the yellow cell below Year 7 net cash flow and it reports that a discount rate of 30.63% produces an NPV of 0 for the cash flow stream. The IRR for this cash flow stream is thus 30.63%.

Consider first the interest earned by re-investing the incoming cash flows from Years 1 through 7. If each incoming return is reinvested 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 the initial cash outflow of \$220 is borrowed and financed at the same 30.63% annual rate. The example shows the total cost of repaying this loan is also \$1,428.17 (initial cash outflow plus financing). The IRR rate exactly balances total costs with total gains.

### Second IRR Interpretation: Financing costs and reinvestment gains

The second-definition example above should suggest a reason that financial people look to IRR and often trust it as 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 provides meaning for another IRR interpretation, namely that the analyst will compare the IRR rate to actual financing rates and actual reinvestment rates. This comparison, however, has to be interpreted carefully. It is easy to over interpret or misinterpret IRR at this point.

When a proposed investment produces IRR's like those shown above—30.6% for example—many are tempted to reason as follows:

"For this investment, we will not actually borrow or pay an opportunity cost at the IRR rate. Our real cost will be subject to a lower rate, closer to our cost of capital, probably less than 10%. Therefore [the reasoning goes], the investment is a net gain because 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 company's actual financing cost rates (or opportunity costs) and the actual reinvestment rates to be applied. The conclusion is most supportable when the IRR 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 rates.

This 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 that 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. The high IRR assumes year 1 and year 2 gains will earn at a 30.6% rate for all the remaining years but in fact, 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 presumed by the IRR, Beta in fact has less "missing" returns of this kind than Alpha. IRR 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, and second, Alpha's IRR is further from the real reinvestment rate than Beta's IRR.

In brief, it is reasonable to view the results of a proposed action 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

• 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 is illustrated again in the discussion below on  Lease vs. buy comparisons.

## Is Modified Internal Rate of Return (MIRR) a better metric 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 calculate an internal return metric that does reflect the real financing cost rate and real earnings rate for returns?" In fact, this solution is readily available as the modified internal rate of return (MIRR) metric.

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 example Cases Alpha and Beta from above. MIRR for this example is based on an earnings rate for returns of 8% and a financing rate of 6%:

Case Alpha:    IRRA = 30.6%     MIRRA = 15.1%
Case Beta:     IRRB = 20.8%     MIRRB  = 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 how the MIRR results are derived. Unlike IRR, MIRR can be computed directly from a formula:

Future values of cash inflows are calculated using the reinvestment rate. Present values of the cash outflows are calculated using the financing rate. The radical sign calls for the nth 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.

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

Present values of negative cash flows are also calculated, discounted at the financing rate (given as 6% in this case). However, for this example there is only one negative cash flow 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 "–" making it a positive number. Using the above formula, MIRR for Investment Case Alpha is thus:

The meaning of the MIRR results is easily understood by anyone who understands compound interest growth.

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 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 were reinvested at the rate of 8.0%, the total value after 7 years would be the same \$573.76.

Note: To check these calculations yourself, use the more precise MIRRA rate of 15.0757% and MIRRB rate of 14.6732%.

In other words, 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. 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 more problematic.

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 would be:

=MIRR(B3:B10, 0.06, 0.08)

## How does internal rate of return compare to NPV, ROI, and other financial metrics?

Referring to the example cases above, which is the better business decision, Case Alpha or Case Beta? In many organizations, IRRs are used to address such questions, but generally, such questions should not be answered on the basis of a single financial metric. The prudent financial specialist, investor, or business analyst will compare both proposed actions with several financial metrics. As shown below, different metrics may suggest different answers to the question.

This section compares Alpha and Beta cases on the basis of 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 needs to see individual cash inflows and outflows as well as the net figures.

Cash
Flow
Cash In
Flow Alpha
Cash Out
Flow Alpha
Net
Cash
Flow Alpha

Cash In
Flow Beta
Cash Out
Flow Beta
Net
Cash
Flow
Beta
Now

0
-220
-220
0
-220
-220
Yr 1

200
-80
120
100
-50
50
Yr 2

200
-100
100
100
-40
60
Yr 3

200
-120
80
100
-40
60
Yr 4

200
-145
55
100
-35
65
Yr 5

200
-155
35
100
-30
70
Yr 6

200
-180
20
100
-25
75
Yr 7

200
-80
10
100
-20
80
Total

1,400
1,200
200
700
-460
240

Financial metrics compared in this section are all calculated from the set of cash flow figures, above.

Before looking at the individual metrics, notice some of the most apparent 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 six financial metrics calculated from th Case Alpha and Case Beta cash flow figures above:

Financial
Metrics
Case Alpha
Case
Beta
Net CF
200.00
240.00
IRR
30.6%
20.8%
MIRR
15.1%
14.7%
NPV@5%
148.80
155.01
NPV@10%
107.56
91.37
ROI
16.7%
52.2%
Payback
2.0 Yrs
3.4 Yrs
= The better metric result in each row

#### 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 situations where cash flow and working capital are problematic, 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 proposed actions can thus be 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 that incoming cash flows are reinvested). 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 several times larger than cost of capital, or more, the real rate of return difference between two different investments depends heavily on the timing of cash flows, the cash flow stream profiles, and the actual rates available for cost of capital and earnings on returns—none of which can be seen in the IRR figures.

#### 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%. As discussed above, the 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)

The better business decision or preferred choice, Alpha or Beta, according to net present value,depends on the discount rate. At 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: NPV uses an arbitrarily chosen discount rate, which may determine results of the comparison, as shown. 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 (incremental gains divided by investment costs). All cash flow metrics above, including ROI, 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. 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). The large difference in costs is hidden in the other metrics based on net cash flow. This could be problematic because the costs must be budgeted and paid for, no matter how large the returns, and 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 it takes 3.4 years for Beta's incoming gains to fully cover investment costs. Investors prefer shorter payback periods rather than longer payback for at least two reasons. First, the investment funds are available again for re-use sooner with a shorter payback period. 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 is a favorite of economist: 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, these things may have important financial consequences, and they are very rarely truly equal.

Consequently, it is usually recommended that IRR not be used as a decision criterion when comparing competing mutually exclusive 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.

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 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 IRR investment 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 likelihood that returns actually appear as projected, as well as the likelihood that other better and worse results appear.

### IRR disguised 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, is used in evaluating bond investments. 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 concept yield to maturity (YTM) used in bond investing. IRR and YTM are mathematically the same concept, with only a slight difference in definition (for a more 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
= FV1 / (1+ i )1 + FV2 / (1+ i )2 + ... + FVn / (1 + i )n

This definition for YTM can be changed into Definition 1 for IRR, simply by subtracting "Bond Purchase Price" from each side of the equation. This way, Bond Purchase Price becomes the FV0 in the NPV equation that is used for IRR. The formula for IRR then asks for the same i that solves the equation:

0 = FV0 + FV1 / (1+ i )1 + FV2 / (1+ i )2 + ... + FVn / (1 + i )n

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

## Is IRR appropriate for lease vs. buy analysis? 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 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 calculated and to support action proposals, even when expected cash flows do not fit the investment curve profile. Those who are required to provide IRR support for funding requests, project or program proposals, or business case analysis, are often dismayed when they find that an IRR does not 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 a competing 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 profile with early net cash outflows and later net cash inflows.
2. Do not use IRR to compare competing cash flow streams whose profiles differ substantially from each other.
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 earnings rates for returns.
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 the remaining years of the asset's life, there should be cash inflows as the asset earns returns. The "Lease" option for the same asset starts with a very small initial cash outlay (if any), followed by almost the same net returns projected, but reduced somewhat by the periodic leasing fees (this would be the case for a typical operating lease). These cash flow streams fit this pattern:

Purchase
& Lease
Cash Flow
Asset
Lease
Asset

Now
-215
-5
Year 1
100
70
Year 2
100
70
Year 3
100
70
Year 4
100
70
Year 5
100
70
Year 6
100
70
Year 7
100
70
Total
2485
485

The "Buy" stream has an IRR of 42.6%. 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:

• 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 differ 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, 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 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. The 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).

In brief, a Lease vs. Buy decision based on IRR would always choose leasing as the better business decision because IRR views the action as a financial investment. However, IRR can be blind to periodic leasing costs, as shown, and in the Lease vs. Buy decision, other factors can be 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

With certain cash flow streams it is mathematically possible to produce negative IRRs, or multiple IRRs for the cash stream (more than one discount rate that leads to a 0 NPV). For other cash flow streams, there is simply no mathematically correct IRR solution.
• 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 outweigh 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 disregarded when the analyst prepares IRR averages, or weighted average IRRs for multiple actions.

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