IRR Economics Interactive Lecture

8/10/2019 IRR Economics Interactive Lecture

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Economics Interactive Lecture

The Internal Rate of Return

Copyright 1997-2013 Samuel L. Baker

When you are evaluating an investment, a useful number to know isthe internal rate of return.

For some investments, like bank accounts, the internal rate of returnis easy to figure because the bank tells you what it is. For example, a5% simple interest bank account has an internal rate of return of 5%.

For other investments, you have to do some work to calculate theinternal rate of return. This is especially true of investments likebuilding a factory or getting an education. These kinds of investmentsgenerally don't pay money in nice even amounts like a bank accountdoes. Nevertheless, you can calculate an internal rate of return forthese investments, and use it to decide which investments pay best.

To evaluate investments and calculate an internal rate of return, weneed the concept of income stream.

Income Stream

In the grubby world of economic theory, where money is everything,any investment can be expressed as an income stream. An incomestream lists years (or months or whatever) and the amounts ofmoney that flow in orout

An Income Stream Example

Here is the income stream for what you get if you

1. Put $1000 in a 5% simple interest bank account2. Take out the $50 interest each year. ($50 is 5% of $1000.)3. Take all the money out at the end of the sixth year.

Year 0 1 2 3 4 5 6

Income -$1000 $50 $50 $50 $50 $50 $1050


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In Year 0, which represents Now, we put $1000 in the bank. We put anegative number, -$1000 in for Income in Year 0, because the $1000flows out from us.

Year 0 1 2 3 4 5 6

Income -$1000 $50 $50 $50 $50 $50 $1050

In years 1 through 5, we will get paid $50, 5% of $1000. The 50sabove are positive numbers, showing that the money flows to us.

Note: To keep things simple, we imagine that interest is paidannually. Most real life bank accounts pay interest monthly. Also, weimagine that we withdraw each year's interest payment from thebank. We don't leave it in the bank to compound (earn interest on theaccumulated interest) during the following years.

Year 0 1 2 3 4 5 6

Income -$1000 $50 $50 $50 $50 $50 $1050

Imagine that at the end of year 6 we take our $1000 back. Our totalincome in year 6 is $1050, the $1000 principal plus the $50 interest

we get in year 6.

An Alternative

Investment

Now let's consideran investment thatour financial vicepresident hasproposed as analternative to puttingour $1000 in thebank for six years.

This investmentinvolves buying a

machine that will cost $1000. It will give us $200 in operating profitper year for six years, starting the year after we buy it. At the end ofsix years, the machine will have no value, due to wear andobsolescence. There's no lump of money waiting for us at the end, asthere is with the bank account.

This table shows the bank income stream and the machine incomestream.


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Year 0 1 2 3 4 5 6

Bank -$1000 $50 $50 $50 $50 $50 $1050

Machine -$1000 $200 $200 $200 $200 $200 $200

It's not obvious which investment is better, is it? We could try addingup the income streams ...

Bank Account: -1000 + 50 + 50 + 50 + 50 + 50 +1050 = 300

Machine: -1000 +200 +200 +200 +200 +200 + 200 = 200

The income stream from the bank account adds up to $300. Theincome stream from the machine adds up to $200. Does this make

the bank account better?Yes. Why make a simple problem complicated?

No, not necessarily. We need a more complicated calculation.

The key is "present value" concept. This concept is reviewed below,but it is introduced in its ownInteractive Tutorial on DiscountingFuture Income.Please try that tutorial now if the above questionpuzzled you.

Why we need that concept: The bank account income stream paysmore money in total, but most of that income is in the big lump of$1050 in year 6. The machine pays less in total, but it pays moremoney per year in the years that come sooner. Getting the moneysooner may give the machine's income stream a higher present valuethan the bank's.

The Present Value of an Income Stream

The present value of a future amount of incomeis:

Present Value = (Future Value)/(1 + Discount Rate),

where the exponent is the number of years in the future that thefuture value will be received. The discount rate is the same as the

interest rate.

An income stream is a series of future values. The present value ofan income streamis calculated by adding up the present values ofall the items in the income stream.
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To calculate a present value, we need to pick a discount rate. Sinceone of our alternative investments is a 5% per year bank account,let's pick 5% per year as the discount rate.

Year (a

intheformula)

0 1 2 3 4 5 6 Total

Machineincomestream

-$1000

$200 $200 $200 $200 $200 $200

1.05 1.0000 1.0500 1.1025 1.1576 1.2155 1.2763 1.3401

Presentvalues, at a5%discountrate.2nd rowdivided by3rd row

-$1000

$190.48 $181.41 $172.77 $164.54 $156.71 $149.24 $15.14

The total of the present values is $15.14. This is the present value ofthe machine income stream at a 5% discount rate.

(If you check the addition, using the numbers shown in the table,you'll get $15.15. The .01 difference is due to round-off error.)

The tutorial on discounting future income has a niftyspreadsheetsetup for calculating present valuesthat you can copy and use inyour own spreadsheet.

Let's use the same method on the 5%-interest bank account.

Year 0 1 2 3 4 5 6 Total

5% bank accountincome stream

-$1000

$50 $50 $50 $50 $50 $1050

1.05 1.0000 1.0500 1.1025 1.1576 1.2155 1.2763 1.3401

Present values, 5%discount rate2nd row divided

by 3rd row

-$1000

$47.62 $45.35 $43.19 $41.14 $39.18 $783.53 $0.00

These present values add up to $0.

(Actually, they add to $0.01, but that's due to round-off error.)
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The present value of an 5% bank account, evaluated at an 5%discount rate, will always turn out to be $0. Whenever the discountrate equals the interest rate, you will get $0 for the present value.

When we compare, we find that, at a 5% discount rate, the machinehas a higher present value ($15.14) than the 5% bank account ($0).The machine wins!

The primitive method of adding up the income streams ...

Bank Account: -1000 + 50 + 50 + 50 + 50 + 50 +1050 = 300

Machine: -1000 +200 +200 +200 +200 +200 + 200 = 200

... would be valid if the interest rate were 0%. That would be if youcould borrow money and pay it back without any extra for interest.

So, now we have a way to compare investments and choose thebetter ones. A drawback of the method is that we have to specify adiscount rate first and do the calculation second. Another method, theinternal rate of return, lets us calculate a number for the investmentfirst and compare that with our discount rate second.

Calculating the Internal Rate of Return

One way to evaluate an investment is to calculate an internal rate ofreturn. The calculation we just made for the bank account gives us anidea of what we want to do. Recall that the bank account's interestrate (5%) was the discount rate that made the present value of itsincome stream total zero.

That is how we define the internal rate of return. It is the discountrate that makes the net present value of the investment equalzero.

Consider this table. It shows the present value of the machine at twointerest rates, 5% and 6%:

Year 0 1 2 3 4 5 6 Total

Incomestream

-$1000

$200 $200 $200 $200 $200 $200

Presentvalue, at a

5%discount

-$1000

$190.48 $181.41 $172.77 $164.54 $156.71 $149.24 $15.14


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rate

Presentvalue, at a6%

discountrate

-$1000

$188.68 $178.00 $167.92 $158.42 $149.45 $140.99 -$16.54

At a 6% per year discount rate, the machine investment's presentvalue is less than $0. At a 5% discount rate, the present value isgreater than $0.

Mathematically, the present value is a continuous function of thediscount rate. The Intermediate Value Theorem implies that theremust be a discount rate between 5% and 6% at which the present

value is $0.

Let's find that discount rate.

Type a number between .0500 and .0600 in the box. Click the buttonto see what comes out for the present value.

53

Type a number in the box. Then click this. If you type anumber starting with 5, I'll move the decimal point to where it issupposed to be. Don't type a % sign.

Year 0 1 2 3 4 5 6 Total

Incomestream

-$1000

$200 $200 $200 $200 $200 $200

Presentvalue, ata 5.000%discountrate

-$1000

$190.48 $181.41 $172.77 $164.54 $156.71 $149.24 $15.14

Please get to within $0.10 of a $0 total before moving on.

If your total is above $0, try a slightly higher discount rate. If your totalis below $0, try a slightly lower discount rate.


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You can get a present value $0.06 with a discount rate of .0547 or5.47%. That is as close to a $0 total as you can get unless you go tothe next digit after the decimal point.

The internal rate of return for the machine is therefore 5.47%.

Comparing the internal rates of return of the two investments, we seethat the machine's 5.47% internal rate of return is higher than thebank account's 5% internal rate of return. This tells us that themachine is a better-paying investment.

Two cautionary notes:

1. The idea that better investments have higher internal rates of

return is appropriate for comparing investments that have theircosts first and their positive incomes later, and which have


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about the same initial costs. Our imaginary bank account andmachine fit this criterion, so we are OK to use the internal rateof return for comparison. More on this issue at the end of thistutorial.

2. Riskcan complicate the comparison of investments. For thistutorial let us assume that the risks of our alternativeinvestments are the same. In particular, we will assume thatthe machine is just as safe as the bank account.

In real life, investments that offer better payback generallycarry greater risks that the future income won't be paid. If themachine is riskier than the bank account, you may prefer thebank account, even if its internal rate of return is lower. Even

so, the internal rate of return is useful to know. It tells you howmuch caution would cost you, or how much reward there is ifyou choose to assume some risk.

Let's look at the two investments again. Here are the incomestreams and each investment's internal rate of return (IRR):

Year 0 1 2 3 4 5 6 IRR

Machine -$1000 $200 $200 $200 $200 $200 $200 5.47%

5% bank account -$1000 $50 $50 $50 $50 $50 $1050 5.00%

A student once asked: Suppose you don't need any money until year6? Doesn't that make the bank account better? The total of theincome stream (not discounted) is higher for the bank, and it givesyou the money when you need it.

The answer is: Even if the times when you'll need money don't matchwhen the investment pays, you should still go by the internal rate ofreturn. That's especially true if the investment pays youmoney beforeyou need it.

That's because you can use the bank, even if you buy the machine.You can deposit the extra income from the machine into a 5%account. At the end of Year 6, you'll have a bigger lump of moneythan you would have had if you had put your $1000 in the bank.

Here's how it works, in laborious detail, after you buy the machine inyear 0 for $1000:

At the end of Year 1, you get $200. You keep$50 for spending, just like you would do for the

bank account (according to what we assumed).You have $150 left over. You put the extra $150

End of year 1:Starting bank balance is

$0.00.The machine pays you


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into the bank. $200.00.You take $50.00.You add to bank account$150.00.Bank balance is $150.00.

At the end of Year 2, the bank pays you 5%interest on your $150. That makes your bankbalance $157.50. At the same time, you getanother $200 from the machine. You keep $50of that for spending, and put $150 in the bank.Your bank balance is $157.50 + $150 =$307.50.

End of year 2:Bank adds 5% of $150.00,which is $7.50.The machine pays you$200.00.You take $50.00.You add to bank account$150.00.Bank balance is $307.50.

At the end of Year 3, the bank pays you 5%interest on your $307.50. That makes your bankbalance $322.88. The machine pays youanother $200. You keep $50 and put $150 inthe bank. Your bank balance is $322.88 + $150= $472.88.

End of year 3:

Bank adds 5% of $307.50,which is $15.38.The machine pays you$200.00.You take $50.00.You add to bank account$150.00.Bank balance is $472.88.

At the end of Year 4, the bank pays you 5%

interest on your $472.88. That makes your bankbalance $496.52. The machine pays youanother $200. You keep $50 and put $150 inthe bank. Your bank balance is $496.52 + $150= $646.52.

End of year 4:Bank adds 5% of $472.88,

which is $23.64.

The machine pays you$200.00.

You take $50.00.You add to bank account

$150.00.Bank balance is $646.52.

At the end of Year 5, the bank pays you 5%interest on your $646.52. That makes your bankbalance $678.84. The machine pays you

another $200. You keep $50 and put $150 inthe bank. Your bank balance is $678.84 + $150= $828.84.


which is $32.32.The machine pays you

$200.00.You take $50.00.You add to bank account

$150.00.Bank balance is $828.84.

Finally, at the end of Year 6, the bank pays you5% interest on your $828.84 That makes yourbank balance $870.28. The machine pays youits last $200. Your withdraw the $870.28 fromthe bank, and you have $870.28 + $200 =$1070.28. By comparison, at the end of six

years with the bank alone you get $1050. Withthe machine, you're ahead by $20.28. OK, it's


which is $41.44.Bank balance is $870.28.

The machine pays you$200.00.

The total you finish with is$1070.28.


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not that much, but it does show that even if youdon't need most of your money until Year 6, youwind up with more if you buy the machine.

Buy the machine, take out $50 a year from the proceeds, and youfinish with $1070.28. Using the bank alone, you finish with $1050.

So, if you need $50 a year for five years, and then all the money aftersix years, the machine/bank combination is a better investment thanthe bank alone. You are better off buying the machine and then usethe bank to earn interest on the money that you don't need right awayeach year.

If the machine investment pays you money afteryou need it (for

instance, if you need $300 in Year 1) then you should compare theinterest rate you'd pay to borrowmoney with the machine's internalrate of return.

A digression: We keep talking about the "internal" rate ofreturn. Were you wondering if there is such a thing as an "external"rate of return? There is, and the above analysis is an example,because it takes into account the interest that you can earn frommachine's payments in the bank, which is an investment separatefrom, and thus "external" to, the machine investment itself. (Thanks

to H.E.M. for putting me onto this.)

Internal Rate of Return Summary (so far)

The internal rate of return is the interest rate that makes the presentvalue of the investment's income stream -- its costs and payoffs,discounted to the present -- add up to 0.

The internal rate of return is a measure of the worth of an investment.

Ignoring differences of risk, an investment with a higher internal rateof return is a better money-maker than an investment with a lowerinternal rate of return.

That generality has some exceptions. We come back to this at theend of this tutorial.

Some Applications

Let's look at three applications of the internal rate of return.

1. Evaluating a bond sold at a discount.


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2. Detecting an economic shortage.

3. The effect of regulation on innovation.

Application 1. Evaluating a bond sold at a discount.

Here's a common type of investment, a bond sold at a discount.

The Sam's Software Corporation (a fictitious entity) is selling 5-year$1000 bonds that pay 1% interest per year. The bonds are selling at$800.

How do you evaluate the bonds as an investment?

Help! Please explain what a bond is.

Calculate the internal rate of return. That tells you the yield tomaturity.

Why bother with math? The rate of return is 1%, the interest rate.

Leaving space until you answer the above question correctly.


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The first step is to write out the income stream. I'll expand thatcompressed description of the bond:

In year 0, you pay out $800 to buy the bond. In years 1 through 4, you get 1% of $1000 (which is $10) as an

interest payment. In year 5, you get a $10 interest payment, and you get the

$1000 face value of the bond.

Year 0 1 2 3 4 5

Type an income streamnumber in this row:

$-800

$10

$10

$10

$10

$1010

Click the correspondingbutton:

Checkwhat'sabove.

Checkwhat'sabove.

Checkwhat'sabove.

Checkwhat'sabove.

Checkwhat'sabove.

Checkwhat'sabove.

Leaving space while you work on this.

When you have them all, or when you have had enough, click this.


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Here is what you should have above. It is the income stream for a$1000 5-year bond, paying 1% interest annual, selling at $800.

Year 0 1 2 3 4 5

Income stream -$800 $10 $10 $10 $10 $1010

Let's do a net present value calculation with this.

Let's use a discount rate of 1%. That equals the bond's nominal rateof interest, in that it pays 1% of the face value in interest each year.

Year 0 1 2 3 4 5 Total

Income -$800 $10 $10 $10 $10 $1010

... dividedby ...

1.01 1.01 1.01 1.01 1.01 1.01

Incomediscountedat 10%

-$800 $9.90 $9.80 $9.71 $9.61 $960.98 $200.00

The present value of this bond, evaluated at a 1% discount rate, is

$200. Notice that this equals the amount of the initial price discount.The $800 price is $200 less than the $1000 maturity value. It works


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this way whenever the discount rate equals a bond's nominal rate ofinterest.

Let's find the internal rate of return for this bond.

You know that the internal rate of return is bigger than 1%, becausethe present value at a 1% discount rate is bigger than 0.

See how close you can get the total present value to $0. Thediscount rate at that the present value closest to $0 will be ourapproximation of the internal rate of return. As before, make a mentalnote of this discount rate, then go on.

Let's find that discount rate.

Type a number between .010 and .090 (which is way too big) in thebox. Click the button to see what comes out for the present value.

Type a number in the box. Then click this. If you type anumber starting with 1, 2, 3, etc., I'll move the decimal point to whereit is supposed to be. Don't type a % sign.

Year 0 1 2 3 4 5 Total

Income stream -

$800

$10 $10 $10 $10 $1010

Present value, at a 1.00%discount rate

-$800

$9.90 $9.80 $9.71 $9.61 $960.98 $200.00




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The discount rate that zeroes -- approximately -- the total presentvalue is 0.057, or 5.7%. Being able to buy this bond at a discountmade the internal rate of return much higher than this bond's 1%coupon rate.

If a bond is risky, it will sell at a discount like this. That gives thebuyers a bigger return, which rewards them for assuming the risk thatthe bond will default. The perceived riskiness determines the size ofthe discount.

Application 2: Detecting an Economic Shortage

Economists use the term economic shortageto mean a situation inwhich the quantity supplied of some good or service is less than whatit would be if the market were functioning freely and competitively. Inother words, "economic" shortages or surpluses happen when themarket fails to equilibrate.

Shortages can manifest themselves two ways. One is with long linesof buyers and empty store shelves. The other is with prices higher

than cost plus normal profit. (More on "normal profit" later.)


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Detecting the first type of shortage is easy -- you just observe thelines or empty shelves. Detecting the second type is harder.

How do you judge when prices are high? Use the internal rate ofreturn! We'll illustrate this with our machine investment example.

Suppose that there are two investments available to large numbers ofinvestors. One is machines that currently pay off like this:

Year 0 1 2 3 4 5 6 IRR

Income -$1000 $200 $200 $200 $200 $200 $200 5.47%

The other is the 5% bank account.

The machine has a higher internal rate of return, at 5.47%, than thebank's 5%. What happens? Firms like yours take money out of thebank to buy machines.

As more and more firms buy machines, put them to work, and try tosell the machine's products, what would you expect to happen to thetypical machine's income stream?

The income amounts should not change.

The income amounts should go up.

The income amounts should go down.

Leaving space until the above question is answered correctly.


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As the supply of the machine's products expands, the amount ofincome earned in years 2-6 should shrink, on average. That willmake the internal rate of return fall.

For example, suppose that annual income falls by just $1 to $199.

Year 0 1 2 3 4 5 6 IRR

Income -$1000 $199 $199 $199 $199 $199 $199 5.31%

The internal rate of return falls to 5.31%, from the 5.47% it wasbefore, if the expected annual payout falls by $1.

How far would you expect the income amounts to fall?

Until the income amounts are all $0.

Until the internal rate of return is 0%.

Until the internal rate of return matches the bank's 5%.

Leaving space until the above question is answered correctly.


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Market equilibrium income stream for the machine under competition,if banks are paying 5% interest:

Year 0 1 2 3 4 5 6 IRR

Income -$1000 $197.02 $197.02 $197.02 $197.02 $197.02 $197.02 5.00%

Once income falls this far, people will no longer want to take moneyout of the bank to buy machines. The supply of machine products willstop expanding, and the income you can get from a machine will stopfalling. (Given the time lag in decision-making, income may fallfurther than this and then later come back up. Equilibrium will bereached when there's no incentive to shift money either from bank tomachine or from machine to bank.)

How economic shortage fits with this:

If an investment has a high rate of return that persists over a longperiod of time, economists infer that competition must not be workingas it should to lower income and equalize rates of return.

Normal profitis profit equivalent to what you could earn in the bank.No business would persist that didn't earn normal profit. Why putmoney into a business if you can do better in a no-effort bankaccount? But if the return to an investment stays higher than normalfor a long time, that's an indication of an economic shortage, andsome kind of monopoly restriction.

Speaking of above-normal profit and economic shortages, medicalequipment makers are notorious for selling equipment in the U.S. at a

generous markup over cost.

If there is competition in the machines' products' market, andincomes fall to where the incentive to buy more machines is gone,then the demand for machines will now be elastic. A reduction in theprice of the machine to $985 would put the IRR back up to 5.47%.Machine manufacturers would start making money again, until theIRR for the machines' users got back down to 5%.

Looking at this from the opposite direction, if there is no competition

in the market for the machine's product, the price of the machine canstay high.


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Medical school, internship, and residency are, for the individualdoctor, an investment with an IRR. In the past, surgical specialtyeducation had a higher IRR than primary care education. This helpedinduce new M.D.s to go into specialties. If, with more government

payment control, the IRR for specialties drops, that may turn off theincentive that lures doctors into specialties. If doctors' incomes get solow that the IRRs are less than student loan rates, there could beshortages of doctors in the future.

In other countries, they solve this by making medical educationpractically free for the student. Lowering the initial cost raises the IRRfor a medical career. This also helps the medical schools competewith other professions' schools for students.

Application 3: Regulation Delays and the Internal Rate of Return

Changing the timing of future payoffs changes the internal rate ofreturn.

Let's go back to our original machine investment again. Imagine thatwe have a patent to protect us against competitors. Our payoffs looklike this (as above):

Year 0 1 2 3 4 5 6 IRR

Income -$1000 $200 $200 $200 $200 $200 $200 5.47%

Suppose that Congress enacts a new law regulating machine safety.The Food and Drug Administration now requires that themanufacturer prove that the machine is safe. That will take time,causing a delay in the start of the use of the machine, even if you buyit now. The income stream might change to this:

Year 0 1 2 3 4 5 6 7 IRR

Income -$1000 $0 $200 $200 $200 $200 $200 $200 ?????

The amounts in the income stream are the same, but the income isshifted later by one year.

Let's find the internal rate of return for this machine now.

Type a number between .0100 and .0547 in the box. Click the buttonto see what comes out for the present value.

Type a number in the box. Then click this. If you type a


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number starting with 1, 2, 3, etc., I'll move the decimal point to whereit is supposed to be. Don't type a % sign.

Year 0 1 2 3 4 5 6 7 Total

Incomestream

-$1000

$0 $200 $200 $200 $200 $200 $200

Presentvalue, ata 5.47%discountrate

-$1000

$0 $179.79 $170.47 $161.63 $$153.24 $145.30 $137.76 $-51.81




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The discount rate that makes the total present value as close as youcan get to $0 with this number of decimal places is 4.19%. Theregulatory delay drops the internal rate of return to below what youcan get from the bank. No one will invest in one of these machinesnow.

One way this machine's products could come to market would be ifthe prices for the machine's products go up. The prices would haveto rise enough to make the internal rate of return as high as thebank's 5%. In this way, consumers wind up paying for the regulatorydelay in higher prices. Notice that I didn't add any paperwork ortesting cost to the calculations. The higher prices are purely becauseof the delay and the need to compete with other investments. Addingin the paperwork cost of complying with regulations would make theinternal rate of return even lower.

Another way this machine's products could come to market would beif the price of the machine goes down. Like the discounted bond, adiscount in the price of the machine would raise the internal rate ofreturn. In this circumstance, the machine's manufacturer would paythe delay cost of the regulation. The numbers here imply that theprice would have to drop to $967 from $1000 to get the IRR up to justover 5%.

Suppose that the machine manufacturer had been expecting to sell10,000 machines this year. Would it spend $100,000 on lawyers who

would tie up the FDA and prevent the regulations from being writtenfor six years?

You betcha!

They could, "but it would be wrong."

Internal Rate of Return -- Review and Summary

An income stream for an investment is all of the investment's costsand payoffs, along with when each cost and payoff will happen.
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The present value of an investment is the amount of money you'dneed now to be able to duplicate the investment's income stream.The present value is calculated using a discount rate which you set toequal your bank's interest rate or the rate of return of your best

alternative investment.The internal rate of return is the discount rate that makes the presentvalue of the investment's costs and payoffs add up to 0.

From the applications:

The yield to maturity of a discounted bond equals its internal rate ofreturn.

Investments with higher internal rates of return attract money awayfrom investments with lower internal rates of return.

If a kind of investment has a persistently high internal rate of return,something is preventing the market from reaching a competitiveequilibrium.

Regulation can reduce the rate of return to innovation, just bydelaying the payoffs. How to protect the public without stiflinginnovation is a major problem of regulation, in pharmaceuticals, for

example.

Perils of Using the Internal Rate of Return

The internal rate of return is nota good way evaluate an investmentthat has costs later rather than just earlier. An example of that wouldbe an investment that generates an environmental problem that willrequire a cleanup at the end of the income stream. For some suchinvestments, the worseinvestments have the higher internal rates of

return. Please see thenext tutorial.

That's all for now. Thanks for participating!

Comments? Please e-mail me [email protected] the list of tutorials
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