Economics Basics: Internal Rate of Return

When you are evaluating an investment, a useful number to know is the Internal Rate of Return.

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

For other investments, you have to do some work to calculate the internal rate of return. This is especially true of investments like building a factory or getting an education. These kinds of investments generally don't pay money in nice even amounts like a bank account does. Nevertheless, you can calculate an internal rate of return for these investments, and use it to decide which investments pay best.

To evaluate investments and calculate an Internal Rate of Return, we need the Concept of Income Stream.

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

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

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

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

The two income streams viz. the bank income stream and the machine income stream can be represented through a figure as below:

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

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

Not necessarily!; income in distant future is worth less than income in near future. You should not just add up the amounts in each income stream, because the present value of income in the distant future is less than the present value of income in the near future.

The key is "Present Value" Concept. This concept is reviewed below, but it is introduced in its own interactive Tutorial on Discounting Future Income . Please try that tutorial now if the above question puzzled you.

The present value of a future amount of income is:
Present Value = (Future Value)/(1 + Discount Rate)ª,

An income stream is a series of future values. The present value of an income stream is calculated by adding up the present values of all the items in the income stream.

To calculate a present value, we need to pick a discount rate. Since one of our alternative investments is a 5% per year bank account, let's pick 5% per year as the discount rate

Year (a in the formula below)	Machine income stream	1.05ª	Present values, at a 5% discount rate
0	-$1000	1.0000	$1000
1	$200	1.0500	$190.48
2	$200	1.1025	$181.41
3	$200	1.1576	$172.77
4	$200	1.2155	$164.54
5	$200	1.2763	$156.71
6	$200	1.3401	$149.24
Total	.	.	$15.14

Each of the numbers in the Present values row is the number in the Machine income stream row, divided by 1.05ª, where the exponent ª is the year number.

The total of these present values is $15.14. This is the present value of the 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, due to round-off error.)

The fact that this $15.14 total is bigger than $0 is enough to tell us that the machine is a better-paying investment than a 5% interest bank account.

Not convinced yet? Let's find the total present value of the 5% interest bank account.

Year (a in the formula below)	5% account income stream	1.05ª	Present values, at a 5% discount rate
0	-$1000	1.0000	$1000
1	$50	1.0500	$47.62
2	$50	1.1025	$45.35
3	$50	1.1576	$43.19
4	$50	1.2155	$41.14
5	$50	1.2763	$39.18
6	$1050	1.3401	$783.53
Total	.	.	$0.00

These present values add up to $0. (Actually, they add to $0.01, but that's due to round-off error.)

The present value of an X% bank account, evaluated at an X% discount rate, will always turn out to be $0, no matter what X is.

At a 5% discount rate, the machine has a higher present value ($15.14) than the 5% bank account (with its present value of $0), so the machine is the better-paying investment.

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 you could borrow money and pay it back without any extra for interest.

So far, so good, but what if we have other alternative investments? How do we compare them? How do we decide what discount rate to use?

At a 6% per year discount rate, the machine investment's present value is less than $0. At a 5% discount rate, the present value is greater than $0. The Intermediate Value Theorem implies that there is a discount rate between 5% and 6% at which the present value is $0. Let's find that discount rate.

Year (a in the formula below)	Income stream	Present values, at a 5.47% discount rate
0	-$1000	$1000
1	$200	$189.63
2	$200	$179.79
3	$200	$170.74
4	$200	$161.63
5	$200	$153.24
6	$200	$145.30
Total	.	$0.06

A discount rate of 5.47% makes the total present value $0.06, which is as close as you can get to zero without going to the next decimal place. Thus, 5.47% is our best approximation the internal rate of return for the machine investment.

The machine investment's 5.47% internal rate of return is higher than the bank account's 5% rate of return. This is sufficient to tell us that the machine is a better-paying investment.

Two cautionary notes:

The idea that better investments have higher internal rates of return is appropriate for comparing investments that have their costs first and their positive incomes later, and which have about the same initial costs. Our imaginary bank account and machine fit this criterion, so we are OK to use the internal rate of return for comparison. More on this issue at the end of this tutorial.

Risk can complicate the comparison of investments. For this tutorial let us assume that the risks of our alternative investments are the same. In particular, we will assume that the machine is just as safe as the bank account.

In real life, investments that offer better payback generally carry greater risks that the future income won't be paid. If the machine is riskier than the bank account, you may prefer the bank account, even if its internal rate of return is lower. Even so, the internal rate of return is useful to know. It tells you how much caution would cost you, or how much reward there is if you choose to assume some risk.

A student once asked: Suppose you don't need any money until year 6? Doesn't that make the bank account better? The total of the income stream (not discounted) is higher for the bank, and it gives you the money when you need it.

The answer is: Even if the times when you'll need money don't match when the investment pays, you should still go by the internal rate of return. That's especially true if the investment pays you money before you 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 money than you would have had if you had put your $1000 in the bank.

Here's how it works, in laborious detail

You buy the machine 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), and put the extra $150 into the bank	End of year 1: You get $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 bank balance $157.50. At the same time, you get another $200 from the machine. You keep $50 of 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, $7.50. You get $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 bank balance $322.88. The machine pays you another $200. You keep $50 and put $150 in the bank. Your bank balance is $322.88 + $150 = $472.88.	End of year 3: Bank adds 5% of $307.50, $15.38. You get $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 bank balance $496.52. The machine pays you another $200. You keep $50 and put $150 in the bank. Your bank balance is $496.52 + $150 = $646.52.	End of year 4: Bank adds 5% of $472.88, $23.64. You get $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 bank balance $678.84. The machine pays you another $200. You keep $50 and put $150 in the bank. Your bank balance is $678.84 + $150 = $828.84.	End of year 5: Bank adds 5% of $646.52, $32.32. You get $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 you 5% interest on your $828.84 That makes your bank balance $870.28. The machine pays you its last $200. Your withdraw the $870.28 from the bank, and you have $870.28 + $200 = $1070.28. By comparison, at the end of six years with the bank alone you get $1050. With the machine, you're ahead by $20.28. OK, it's not that much, but it does show that even if you don't need most of your money until Year 6, you wind up with more if you buy the machine.	End of year 6: Bank adds 5% of $828.84, $41.44. Bank balance is $870.28. You get $200.00. The total you have is $1070.28.

So, if you need $50 a year for five years, and then all the money after six years, the machine/bank combination is a better investment than the bank alone. The amount you'll get in year 6 will be higher if you buy the machine and then use the bank to earn interest on the money that you don't need right away each year.

If the machine investment pays you money after you need it (for instance, if you need $300 in Year 1) then you should compare the interest rate you'd pay to borrow money with the machine's internal rate of return.

A digression: We keep talking about the "internal" rate of return. 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 from machine's payments in the bank, which is an investment separate from, 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 present value of the investment's income stream -- its costs and payoffs -- add up to 0.

The internal rate of return is a measure of the worth of an investment. If the risks are equal investments with higher internal rates of return pay better.

The internal rate of return is not a good way evaluate an investment that has costs later rather than just earlier. An example of that would be an investment that generates an environmental problem that will require a cleanup at the end of the income stream.

For some such investments, the worse investments have higher internal rates of return. Please see the next tutorial .