Internal Rate of Return and Yield to Maturity

We now assume that the discount rate is endogenous, in which case we solve for the rate that equates two sets of cash flows. Suppose you make an investment in a business equal to dollars. This investment is expected to yield a stream of cash flows for n periods equal to . The internal rate of return (IRR) is the discount rate, which makes the two streams, the outflow and the present value of the inflows , equivalent. That is, the IRR is the value of r that discounts the following set of cash flows to the initial outlay P₀:

Since is an outlay, hence, a negative cash flow, then the preceding equation is the same as:

Notice that this is a net present value, too, but with the important exception that the discount rate is exogenous in NPV problems. The internal rate of return, on the other hand, is the solution objective. The internal rate of return gets its name from understanding that it is the interest rate implied by the internal cash flow stream to the firm. In a sense, it is the firm's required rate of return necessary to achieve a breakeven level on its investments. It is therefore not a market rate. Firms use the IRR most often to compare alternative investments.

In general, the IRR is difficult to solve because it doesn't have an analytic solution; rather, one must resort to iterative techniques to arrive at a solution. Most software packages’ solvers use some form of Newton's method to solve for the IRR (see the IRR and XIRR functions in Microsoft Excel).

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Example 1.2

Let's put numbers in Example 1.1. Assume the initial outlay is $100 and we expect to receive cash flows in years 1 to 4 equal to ($50, $0, $100, $100). Then the IRR is the rate that solves:

This is equal to:

where . This is a polynomial of order four. The solution is r = IRR = 39 percent, whose details can be found on the chapter spreadsheet.

What does this mean? Well, suppose again that this is your firm. Then this rate discounts your cash flows to a present value equal to your outlay of $100. This is a pretty good rate of return if all other investments generate cash flows with IRRs less than 39 percent. Thinking differently, if your firm requires an annual rate of return over four years on their cash flows equal to 39 percent, then a $100 investment with the stated cash flows will meet that requirement.

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Example 1.3

Suppose the required return is 39 percent on a $100, four-year investment with expected cash flows in each of the four years given by ($0, $0, $100, $150). What is the estimated IRR (r in the denominator of the following equation) and will this investment be undertaken?

Example 1.3 Table

Initial Outlay	–100
Year 1 Cash Flow	0
Year 2 Cash Flow	0
Year 3 Cash Flow	100
Year 4 Cash Flow	150
Required Rate of Return (RRR)	39%
Internal Rate of Return	29%
RRR = IRR	FALSE
Check the IRR Calculation
Initial Outlay	–100
C =	0
C2 =	0
C3 =	46.289
C4 =	53.711
SUM =	0

In this example, the IRR = 29 percent, which is below the required return of 39 percent. Therefore, the investment should not be undertaken.

Now let's briefly jump ahead and look at the similarity between the IRR and what bond analysts call the yield to maturity. Suppose you lend $100 for a period of five years. The borrower promises to pay you $25 in each of those five years to expunge his debt. What is the rate of return that equates the present value of the creditor's payments to the loan amount? We set this up as:

The IRR that solves this problem is 8 percent. As a lender, you therefore receive an 8 percent annual return on your investment—in this case, a loan. This is essentially a bond, and in the world of bonds, the 8 percent is the yield to maturity. The yield to maturity is a return that is equal to an IRR. Thus, bond yields are IRRs.

Go to the companion website for more details.

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