perils of the internal rate of return

8/10/2019 Perils of the Internal Rate of Return

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Economics Interactive Tutorial

Perils of the Internal Rate of Return

Copyright 2000 Samuel L. Baker

The two most-used measures for evaluating an investment arethe net present valueand the internal rate of return. (Two earliertutorials discussed these concepts. See thetutorials listfor links totutorials for discounting future income and the internal rate ofreturn.)

It is often assumed that higher is better for both of the net presentvalue and the internal rate of return. In particular, it is usuallystated that investments with higher internal rates of return aremore profitable than investments with lower internal rates ofreturn.

However, this is not necessarily so. In some situations, aninvestment with a lower internal rate of return may be better, evenjudged on narrow financial grounds, than an investment with ahigher internal rate of return. This interactive lecture explores whyand when this reversal takes place.

To review, both the net present value and the internal rate ofreturn require the idea of an income stream, so let's start there.An income streamis a series of amounts of money. Each amount ofmoney comes in or goes out at some specific time, either now or inthe future. The income stream represents the investment; theincome stream is all you need to know for financial evaluationpurposes.

In real life, individuals, charitable institutions, and even for-profit

businesses have social or other goals when selecting investments.For businesses, the benefits of community good will are no less realfor being difficult to measure precisely. For enterprises with socialas well as financial goals, the measures discussed here are stilluseful: They tell you how much it costs you to advance your socialgoals.

Here is an income stream example, from the interactive lectureabout theinternal rate of return.

Year 0 1 2 3 4 5 6

Income amounts -$1000 $200 $200 $200 $200 $200 $200
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Here we see seven points in time and, for each, a dollar inflow oroutflow. At year 0 (now), the income amount isnegative. Negative income is cost, or outgo. In this example,the negative income amount in year 0 represents the cost of buying

and installing the machine.In the future, at years 1 through 6, there will be net income of $200each year.

All of the amounts in the income stream are netincome, meaningthat each is income minus outgo, or revenue minus cost. In year 0,the cost exceeds the revenue by $1000. In years 1 though 6, therevenue will exceed the cost by $200.

This investment evidently has no salvage value. That is, there isnothing that can be sold in year 6, the last year. If there were, theamount that could be realized from the sale would be added to theincome amount for year 6.

For simplicity, all my examples have the incomes and outgoes atone-year intervals. Real-life investments can have income andexpenses at irregular times, but the principles of evaluation are thesame.

Now let's discuss our two measures in connection with this incomestream:

Net Present Value

The net present valueof an income stream is the sum of thepresent values of the individual amounts in the income stream.Each future income amount in the stream is discounted, meaningthat it is divided by a number representing the opportunity cost ofholding capital from now (year 0) until the year when income isreceived or the outgo is spent. The opportunity cost can either behow much you would have earned investing the money someplaceelse, or how much interest you would have had to pay if youborrowed money. See theinteractive lecture on discounting futureincomefor more explanation. That tutorial has a niftyspreadsheetsetup for calculating present valuesthat you can copy and use inyour own spreadsheet.The word "net" in "net present value" indicates that our calculationincludes the initial costs as well as the subsequent profits. It also

remindsus that all the amounts in the income stream are net profits,
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revenuesminus cost. In other words, "net" means the same as "total" here.The net present value of an investment tells you how thisinvestment compares either with your alternative investment or

with borrowing, whichever applies to you. A positive net presentvalue means this investment is better. A negative net present valuemeans your alternative investment, or not borrowing, is better.

Consider again this income stream:

Year 0 1 2 3 4 5 6

Income amounts -$1000 $200 $200 $200 $200 $200 $200

Let's assume that the discount rate (the interest rate that you couldearn elsewhere or at which you could borrow) will not change overthe life of the project. This makes the calculation simpler. Withthis assumption, we can use the usual formula:

Present Valueof any one income amount = (Income amount) / ( (1+ Discount Rate) to the apower)

a is the number of years into the future that the income amountwill be received (or spent, if the income amount is negative).

The net present value (NPV) of a whole income stream is the sum ofthese present values of the individual amounts in the incomestream. If we still assume that income comes or goes in annualbursts and that the discount rate will be constant in the future,then the NPV has this formula:

Varying future interest rates

The future interest rate does not have to be constant for thistheory to apply. The interest rate can vary, but that makes theformulas messier. For example, if r1is the expected interest ratenext year, and r2is the expected interest rate the year after that,then the present value today of I2income in year 2 isI2/(1+r1)(1+r2).

TheI 's are income amounts for each year. The subscripts (whichare also the exponents in the denominators) are the year numbers,


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0. That would seem to imply that projects with higher incomeshave higher internal rates of return.

Similarly, if you lower any of the income amounts in years 1 through6, then a lower discount rate will be needed to bring the netpresent value back up to 0. That would seem to imply thatprojects with lower incomes have lower internal rates of return.

These seeming implications are actually often true, if the projectsbeing compared have about the same shape, with the costs comingearly and the benefits coming late, and if the projects beingcompared switch from net outgo to net income at about the sametime. Otherwise, though, the implications might not be true.

Before we go on to that, a little review:Which of these measures (net present value and internal rate ofreturn) requires you to know the future income and outgoamounts?

Which of the measures requires you to know what the discount ratewill be in the future?

Please do not scroll down past this area until you have answered thequestion.

The textresumeshere:

The internal rate of return does not require you to predict future

discount rates. That would seem to make the internal rate of return


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the more useful (or less uncertain) measure. Sometimes, though,the internal rate of return can fool you.

Contradictory Results

A few years ago, the New England Journal of Medicinepublished astudy that evaluated various types of professional education as ifthey were financial investments.The article is: Weeks, W.B., Wallace, A.E., Wallace, M.M., Welch,H.G., "A Comparison of the Educational Costs and Incomes ofPhysicians and Other Professionals," N Engl J Med, May 5,1994, 330(18), pp. 1280-1286.The idea was to see if doctors were overpaid, by consideringprimary and specialty medical education as investments and

comparing them with investing in education in business, law, anddentistry (but not university professors -- that would have been tooembarassing). Adjustments were made for differences in averageworking hours. The authors found that primary medicine was thepoorest investment of all of these. Specialty medicine did better,but was not out of line with the other professions.

In the results was this oddity: By the criterion of the net presentvalue of lifetime educational costs and income benefits, specialistphysicians tied for highest with attorneys. Both were ahead ofbusiness school graduates. However, by the criterion of theinternal rate of return, specialty physicians, with a 21% averagereturn, were well behind the attorneys' 25% average return, whilethe business school graduates' 29% average return was the highestof all. The present value and the internal rate of return rankedthe alternatives differently!

By the way, since this article's 1994 publication, managed care hasforced specialty physician incomes down by perhaps one-

third. This has sharply lowered the investment value of aspecialty medical education.

The NPV Curve

One way to understand how the net present value and the internalrate of return can give seemingly different advice is to use what Iwill call the net present value curve, or NPV curve. The NPVcurve shows the relationship between the discount rate and the netpresent value for a range of discount rates. The present value at

a given discount rate, such as 5%, and the internal rate of returnare each points on the NPV curve.


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The NPV curve, the relationship between the discount rate and thenet present value has a formula that can be written like this:

This, of course, is the formula we saw already for the net presentvalue, for annualized costs and revenues and a constant discountrate. Each Iis an income amount for a specific year. Thesubscripts (which are also the exponents in the denominators) arethe year numbers, starting with 0, which is this year. Theconstant discount rate is r. The number of years the investmentlasts is n. In Weeks's study of professionals' incomes, nwas about44, because costs and incomes were calculated from age 21 to age65.

We'll use an example with an n of 6, so the formula fits on yourscreen:

This is our machine investment example that we have been using allalong. The NPV is a function of r. Graphed, it looks like this:

The blue curve shows the net present value for discount rates (r)from 0 to 0.1 (0% to 10%). The red dots are the two points we getfrom our measures. The left red dot shows the net present valueat the discount rate of 0.05 (5%). The right red dot shows theinternal rate of return, because it is where the curve crosses thehorizontal line indicating an NPV of 0. That right red dot is

between the 0.05 and 0.06 marks on ther axis, so the internal rate


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say the second investment is better. A look at the graph aboveconfirms that the second investment is better at all discount rates,so it is fair to say that the second investment is unequivocablybetter than the first.

Can You Do Both Investments?

Doing an investment increases your wealth if its net present value isgreater than 0 at the discount rate relevant to you. If yourdiscount rate is less than 5.47%, both NPV curves are in positiveterritory, and you should do both, if you can.

Sometimes, though, the alternative investments are mutuallyexclusive. For example, there may be two ways to build a dam

across a particular river. You can do one or the other, but notboth. There may be several alternative ways to address aworkplace safety problem. There is no point to doing more thanone if any one way solves the problem. Deciding on a professionaleducation involves somewhat mutually exclusive choices. A fewpeople do go to medical school and then law school, but theadditional return from the second degree is not the same as whatsomeone going to law school fresh out of college would expect.

If you can only do one investment, you should choose the one with

the highest net present value at the discount rate appropriate toyou. A problem with that advice, though, is that discount ratescan change with general economic conditions. You are thereforemore confident about choosing one investment over another if yourchosen investment has a higher net present value over a broadrange of possible discount rates. In our example so far, the green-line investment has a higher net present value at all discount rates,so we would choose it with confidence. Regardless of what happensin the future to discount rates, we'll be better off with the green-

line investment than with the blue-line investment.Can NPV Curves Cross?

Yes, they can. If the NPV curves cross, then the choice ofinvestment depends on the discount rate.

To create an example, I'll change the blue line investment so thatits profits come much later. This increases the effect of thediscount rate on the net present value. Below are the two income

streams, now. Also shown are their net present values at a 5%discount rate and their internal rates of return.


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Year 0 1 2 3 4 5 6 NPV at0.05discountrate

Internalrate ofreturn

Green lineinvestment

-$1000

$220 $220 $220 $220 $220 $220 $117 0.086

Blue lineinvestment(modified)

-$1000

$0 $0 $0 $0 $0 $1550 $157 0.076

The green line invesment has the higher internal rate of return, butthe blue line investment has the higher net present value at a 5%discount rate. Our two measures are giving us opposite advice!

The graph shows what's going on, by showing the Net Present Valuecurves for both investments for discount rates between 0% and10%. The curves cross at a discount rate of about 0.064, or 6.4%.

Now, to choose which investment we want to do, assuming wecannot do both, we have to make a guess about what futurediscount rates will be. If we expect discount rates to be less than6.4%, where the curves cross, we choose the blue lineinvestment. For discount rates above 6.4%, but below 8.56% (theinternal rate of return of the green line investment -- the discountrate at which the net present value of the green line investment is$0), we choose the green line investment. At higher discountrates than 8.56%, we don't do either, because the net present

values are below $0 for both investments.


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If Costs Come Later Than Profits

If costs come later than profits, the NPV curve can tilt the otherway, making it even more problematic to use the internal rate ofreturn to compare investments.

Costs can come later than profits if an investment createsenvironmental problems that will have to watched or cleaned uplater. Nuclear power plants are a good example. After about 40years of service (sometimes less than that), they become toocontaminated with radiation to continue in service. They mustthen be closed and either guarded where they are for thousands ofyears or dismantled and moved to a disposal site.

Consider this income stream:

Year 0 1 2 3 4 5 6

Income amounts -$200 $200 $200 $200 $200 $200 -$900

I've reduced the initial cost, but added a big cost at the end. Let'ssee what a difference this makes in how the NPV changes when thediscount rate changes. In the applet below, the starting discountrate 5%. The net present value (NPV) is -$6. That's negative six

dollars, so if your discount rate really were 5%, you would not wantto do this investment.

Try changing the discount rate, by clicking in the discount rate boxand changing the 0.05 to something else. Try 0.04 or 0.03. In theexamples above, the NPV goes up when the discount rate islowered. Is that true for this project? Then try 0.06 or 0.07. Whathappens to the NPV?

(Keep the discount rates reasonably small, like between 0.00,which is 0%, and 0.3, which is 30%.)

The relationship between the discount rate and the NPV is thereverse of what we see with "normal" investments! With this kind ofincome stream, higher discount rates make the net present valuebigger, and lower discount rates make the net present valuesmaller.

Before leaving the applet above, see if you can find the internal

rate of return, the discount rate that makes the net present valueequal to $0.


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Here is the NPV graph:

The left blue dot shows the net present value at a 5% (0.05)

discount rate. It is at -$6 on the net present value scale.The right blue dot is where the curve crosses the discount rate axis,which is where the net present value is $0. The discount ratehere, 0.054 (5.4%), is the internal rate of return.

Or, at least, it fits the standard definition of internal rate of return.However, unlike the usual situation, this project isprofitable atinterest rates abovethis IRR and unprofitable at interest ratesbelow this IRR.

Suppose we have an alternative project which also has this shape,with a big cost at the end, but slightly lower profits in theintermediate years. I'll call the new alternative the "green lineinvestment."

Year 0 1 2 3 4 5 6 NPV at0.05discountrate

Internalrate ofreturn

Red lineinvestment

-$200

$200 $200 $200 $200 $200 -$900

-$6 0.054

Green lineinvestment

-$200

$195 $195 $195 $195 $195 -$900

-$27 0.070

The green line investment has a lower NPV than the red lineinvestment at all discount rates, because it has lower profits inyears 1 through 5, and the same costs in years 0 and 6. In

particular, as the table above indicates, it has a lower NPV at the0.05 discount rate. The graph below shows the NPV curves for both


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investments, with the green line lying below the red line at alldiscount rates.

The green line investment is clearly inferior, but it has the higherinternal rate of return. The green line investment's IRR is0.07. The red line investment's is 0.054.

Thus, for projects with big late costs, the better projects willhave lower internal rates of return, the opposite of the rule fornormal projects that have their costs early and their positivereturns later.

Now let's discover something even more strange. Here's anotherapplet that lets you change the discount rate and see the effect onthe red line investment's value. This one, though, allows you totake the discount rate over 0.3 (30%) and all the way up to 1.0(100%). Those rates are much higher than, hopefully, we will eversee in the U.S., but they are theoretically possible, and they show astrange phenomenon.

Try raising the discount rate to 0.3, and notice what happens to the

net present value. Then, raise the discount rate some more abovethat. In which direction does the NPV move now?

See if you can find the second IRR, where the NPV is zero again!

Please do not scroll down past this area until you have answered thequestion.


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The textresumeshere:

Here's the NPV curve for the red line investment for discount ratesfrom 0% to 100%.

At discount rates below 0.054, the NPV is negative, and thisinvestment is worse than doing nothing.At a discount rate of 0.054, the NPV is 0. The first IRR for thisinvestment is 0.054.If the discount rate rises above 0.054, the NPV turns positive, andthis investment switches to being profitable.At a discount rate of 0.262 (26.2%), the NPV for this investmentreaches its maximum. If the discount rate rises further than that,

the NPV falls.The NPV reaches 0 again at a discount rate of 0.86. This is thesecond IRR for this investment.If the discount rate were rises even more, above 0.86, the NPVturns negative again. This investment reswitches to beingunprofitable.

Lesson: The NPV curve gives better guidance than the IRR alone

The lesson I would like you to get from this is that the internal rate

of return, by itself, can fool you. If the investments you areconsidering have different shapes (that is, very different timing of


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costs and benefits) or if the project has large late cleanup costs,then the higher-IRR-is-better rule can steer you to the wronginvestment. Ideally, you want the NPV curve, if you want toevaluate an investment.

Additional notes

My use of the terms "switch" and "reswitch" refers to thereswitching controversy of the 1960's. This was between economistsin Cambridge, England, and Cambridge, Massachusetts, overwhether capital markets can be analyzed just like other commoditymarkets. The English economists, led by Joan Robinson, arguedthat capital markets were special because of the possibility ofreswitching, which raises basic questions about the standard view

that the return to owning capital is a society's reward for abstainingfrom consumption.

Some economists would say that only the second of our IRR's is thetrue IRR, by defining the IRR as the place where the NPV is0 andwhere the NPV is falling. The problems with that are: (1) thisdistinction is usually lost in practice, and (2) by making the patternof costs and profits more complex, I can make up an investmentthat has multiple discount rates where the NPV is 0 and the NPV isdeclining.

The oldest discussion of this tutorial's issues that I have found in theeconomics literature is Lorie JH, Savage LJ, "Three Problems inCapital Rationing,"Journal of Business, Vol. 28, October 1955.

That's all for now. Thanks for participating! Your comments wouldbe appreciated!Please e-mail me [email protected]
mailto:[email protected]:[email protected]:[email protected]:[email protected]

perils of the internal rate of return

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