Internal Rate of Return

As stated in the text, a $1,000,000 expenditure that yields a one-time added profit of $1,200,000 one year later has an expected return of 20%: ($1,200,000 - $1,000,000)/$1,000,000 = 0.20. But what if the added profit is spread out over many years? Is there a method for determining the "rate of return" under these circumstances? As you might suspect, the answer is "yes," but the solution is not as simple as in this one-period example. To begin, we will need to investigate how the value of money varies with the time at which it is received.

Suppose you have the opportunity to obtain an annual return of r on an investment of $X. At the end of the year, your investment would be worth the principal, X, plus interest, rX, or X(1 + r). Call this amount Z₁, so that Z₁ = X(1 + r). Alternatively, we might ask: How much money would have to be invested now at an interest rate of r, so that at the end of one year, you would have accumulated a total of Z₁? Clearly, this amount is X = . The value X is known as the "present value" of Z₁, and tells us the current "worth" of Z₁ to be received one year from now.

Instead of withdrawing your investment at the end of the first year, however, suppose you "let it ride," accumulating an additional year’s interest. At the end of the second year, your initial investment of X would be worth Z₁ (its value at the end of year 1) plus the second year’s interest, rZ₁. Call this amount Z₂: Z₂ = Z₁(1 + r). Substituting Z₁ = X(1 + r), we find Z₂ = X(1 + r)². Again, we might ask: What is the current value of an amount Z₂ to be received two years from now? Working backwards, we see this is X = , the present value of Z₂. In like manner, we find that the present value of Z_t received t years in the future is .

More generally, suppose we expect to receive a stream of payments equal to Z₀, Z₁, Z₂, ... Z_n, where the subscript references the year the payment is to be received. The present value of this stream of payments is X = Z₀ +  +  + ... + .

The "rate of return" on a given investment is defined as the value of r that equates the present value of its stream of benefits to its stream of costs. That is, r is the solution to the equation:

B₀ +  +  + ... + = C₀ +  +  + ... + .

If we let Z_t be the difference between benefits and costs of the investment in year t, Z_t = B_t - C_t, then we see that the rate of return on the investment is the value of r that equates the present value of net benefits to zero: 0 = (B₀ - C₀) +  +  + ... + = Z₀ +  +  + ... + . If we multiply each term on both sides of this equation by (1 + r)ⁿ, we get an n^th-degree polynomial in 1 + r: 0 = Z₀(1 + r)ⁿ + Z₁(1 + r)^n-1 + Z₂(1 + r)^n-2 + ... + Z_n.

The one-year example in the text is easily seen, then, as a special case of this formula. In that case, n = 1;
Z₀ = B₀ - C₀ = 0 - 1,000,000; Z₁ = B₁ - C₁ = 1,200,000 - 0, and we have r as the solution to:
0 = -1,000,000(1 + r) + 1,200,000 or, r = = 0.20.

To address the question first raised in this note, what if the $1,200,000 is instead received over two years--say, $600,000 at the end of the first and second years? Under these circumstances the rate of return is the solution to:
(0 - 1,000,000)(1 + r)² + (600,000 + 0)(1 + r) + (600,000 + 0) = 0, whose solution is r = .1306623, or approximately 13%.

Why is this rate of return lower than our previous result of 20%? Intuitively, it takes another year to obtain the same number of dollars, $1,200,000. Funds received two years from now are of lower value than funds received only one year from now.