## Internal Rates of Return

The internal rate of return measures the profitability of a project whose costs and revenues are spread through time. It is used by Robert Allen in his explanation of why the Industrial Revolution occurred in Britain during the eighteenth century.

Most capital projects have the property that the costs exceed the revenues during the early years (when the project is under development), and the revenues exceed the costs during the later years. The internal rate of return for this kind of project has the following properties:

• If the project is to be funded by borrowing money, it is profitable when the internal rate of return exceeds the market interest rate and unprofitable otherwise.
• Suppose that the investor has money available, and that he can either fund the project or lend the money at the market interest rate. If the internal rate of return is greater than the market interest rate, the investor’s profits are greater when he undertakes the project. If the internal rate of return is less than the market interest rate, his profits are greater when he lends the money.
• A project with a higher internal rate of return is more profitable than one with a lower rate of return.

If you are content with this summary, move on to Allen’s explanation. If you want the details, keep reading.

### The Rate of Return on Simple Investments

The rates of return on simple investments are easy to determine. If you invest $100 this year in order to get back$110 next year, for example, you can calculate that your rate of return is 10% without even putting pen to paper. There are also investments whose rates of return can be calculated with only a couple of lines of basic algebra. Here are two of them.

Suppose that you invest $88 this year in order to get back$104 next year. The amount that you get back (104) is the sum of the investment (88) and the return on the investment ($88r$, where $r$ is the rate of return). Writing this observation as an equation gives

$88(1+r)=104.\qquad\qquad(1)$

Rearranging this equation gives

$r=\frac{104}{88}-1=\frac{2}{11},$

implying a rate of return of about 18%.

Now suppose that you invest $100 this year in order to receive$121 in two years’ time. If $r$ is the rate of return, your investment will be worth $\100(1+r)$ at the end of the first year.1 This amount is invested for the second year. At the end of the second year you receive the amount invested plus the return on that amount, $\100(1+r)(1+r)$. Then:

$100(1+r)^2=121.\qquad\qquad(2)$

$(1+r)^2=\frac{121}{100}.$

$r=\sqrt{\frac{121}{100}}-1.$

Simplifying the last equation shows that the annual rate of return is 10%.

Each of these examples involves a one-time cost and a one-time receipt. If an investment is more complicated than that, you probably can’t rely on your intuition and will need to calculate an internal rate of return.

### The Present Value of an Investment

The first step in calculating an investment’s internal rate of return is to determine its present value. The principle underlying present values is that dollars received today and dollars received at some time in the future are not equally valuable.

Consider a person who can lend or borrow at an interest rate of 25%. (He’s a loan shark who can only borrow from other loan sharks.) He would be indifferent between these two options:

Option A: receive $100 in one year Option B: receive$80 today

If he wants $100 in one year, he can choose option A and wait for the money. Alternatively, he can choose option B and lend out for one year the$80 that he receives. The loan repayment in one year will be $100 ($80 in principal plus $20 in interest). If he wants$80 today, he could choose option B. He could also choose option A and then take out an $80 loan. In one year he would receive$100, which would be exactly what he needs to repay the loan.

For our loan shark, a dollar received in one year is worth $0.80 today. A dollar in two years is worth$0.64 today, and a dollar in three years is worth $0.512 today. These amounts are the present values of dollars received at future times. Let $i$ be the interest rate. The general rule is that the present value of one dollar received $t$ years from now is $\left(\frac{1}{1+i}\right)^t$ because this amount grows to one dollar in $t$ years if lent out at the current interest rate. The present value of, say,$50 received $t$ years from now is found by multiplying $50 by the present value of a single dollar. The payment is then said to be discounted back to the present. Money received at different moments of time has different values, but of course, so does money spent at different moments of time. Future expenditures are discounted back to the present in exactly the same manner as future receipts. A project or investment opportunity can extend over a number of years, with expenditures exceeding revenues in some years and revenues exceeding expenditures in others. Since those expenditures and revenues arise at different times, they are not directly comparable. They are made comparable by discounting them back to the present. Let $\pi_t$ be the project’s profits $t$ years from now. Since profits are simply the difference between revenues and expenditures, they can be either positive or negative. The present value of the project—the value of this stream of profits today—is calculated by discounting every year’s profits back to the present and summing them together: $PV=\pi_0+\pi_1\left(\frac{1}{1+i}\right)+\pi_2\left(\frac{1}{1+i}\right)^2+...+\pi_{T-1}\left(\frac{1}{1+i}\right)^{T-1}+\pi_T\left(\frac{1}{1+i}\right)^T.$ This equation assumes that there are profits in the current year (0 years in the future) and in the next $T$ years, after which the project terminates. Consider again the two examples set out above. The present value of the investment in which$88 was given up today for $104 in one year is $PV=-88+104\left(\frac{1}{1+i}\right). \qquad\qquad(3)$ The present value of the investment in which$100 was given up today for $121 in two years is $PV=-100+121\left(\frac{1}{1+i}\right)^2. \qquad\qquad(4)$ ### The Internal Rate of Return of an Investment An investment’s internal rate of return is the interest rate — the value of $i$ — at which its present value is equal to zero. Why is this an appropriate measure of the rate of return on a project? Imagine that the project is a factory. The investor spends money for several years while the factory is being constructed, and then earns profits for a number of years while the factory is in operation. Finally, $T$ years in the future, the factory falls apart overnight and that’s the end of the story. Imagine as well that the investor has no money of his own, so that he has to borrow money to build the factory and then repay his loans from his profits. If the interest rate that he pays reduces the present value of the project to zero, the project is worthless to the investor: all of the project’s profits go to the repayment of the loan. The only beneficiary of the project is the lender, who earns a stream of interest payments, and we know his rate of return: it’s equal to the interest rate that he charges. The rate of return on the project as a whole is therefore equal to the interest rate that drives the present value to zero. Now, if the lender charges a lower interest rate, the lender and the investor will split the benefits of the project — but that doesn’t change the rate of return on the project as a whole. The rate of return remains equal to the interest rate that drives the present value to zero. Let’s look again at the trade of$88 now for $104 in one year. Its present value is given by equation (3). If the internal rate of return $r$ is the value of $i$ at which the present value is zero, $r$ satisfies the equation $-88+104\left(\frac{1}{1+r}\right)=0.$ This equation is the same as equation (1), which was solved to obtain the rate of return on this investment. Now consider the trade of$100 now for \$121 in two years. Its present value is given by equation (4), so the internal rate of return is the value of $r$ that satisfies the equation

$-100+121\left(\frac{1}{1+r}\right)^2=0.$

This equation is the same as equation (2), which characterizes the rate of return in this problem.

The internal rate of return gives the same answer as our intuition for simple problems. The advantage of the “find the present value and set it equal to zero” methodology is that it can be applied to any problem.

### An Investment Rule

The equation that determines the internal rate of return is almost always a polynomial of high degree, and this kind of equation can have multiple solutions.2 Multiple solutions would be problematic: what would it mean for the rate of return on an investment to be both 3% and 12%? Fortunately, we can sidestep this issue if we confine our attention to investments in plant and equipment. These investments involve a development or construction period during which profits are negative, followed by an operating period during which profits are positive. Descartes’ Rule of Signs then comes to the rescue: the profits stream changes sign only once, so there is exactly one positive and real solution.3

These investments have another useful property. Their present values are positive at every interest rate below the internal rate of return, and they are negative at every interest rate above the internal rate of return. This observation gives rise to a simple investment rule: invest if the prevailing interest rate is below the project’s internal rate of return, and don’t invest otherwise. This rule holds whether or not the project is to be financed with borrowed money. Suppose, for example, that the interest rate is below the internal rate of return, so that the present value of the project is positive. If the investor must borrow to finance the investment, the present value represents the value of the investment net of the cost of the loans. A positive present value means that undertaking the project makes him better off. If the investor has his own money, he must choose between lending out the money at the prevailing interest rate and undertaking the project. For him, the present value measures how much better off he would be if he undertook the project instead of lending out the money. A positive present value means that undertaking the project makes him better off than lending the money.

1. This amount is again the original investment plus the return earned by that investment during the year.
2. The variable of the polynomial is $\frac{1}{1+r}$. Its degree is equal to the largest exponent on the variable ($T$ in our terminology) . The number of solutions is potentially equal to the degree of the polynomial.
3. While $\frac{1}{1+r}$ must be positive, $r$ can be negative. Some projects are just really bad.