$200 million to purchase investments and distributed $30 million after one year. At the end of the second year, it is being appraised at $180 million. The IRR is a since-inception IRR and is 2.7 %.
Investment C had been in existence three years when it was purchased by BK Fund for $500. In the three years following the purchase, the investment distributed cash flows to the investor of $110, $120, and $130. Now in the fourth year, the investment has been appraised as being worth $400. The IRR is based on realized cash flows and an appraised value. The IRR may be described as a point-to-point IRR and is 15.0 %.
3.4 Problems with Internal Rate of Return
This section begins with two major types of complications in the computation and interpretation of IRRs. In the previous section, IRR was easily computed and interpreted because of the simplified cash flow patterns used and because the investment was being viewed in isolation. The first complication arises when an investment offers a complex cash flow pattern other than the traditional pattern of a cash outflow to initiate an investment, followed only by cash inflows until the investment is terminated. The second complication occurs when investments must be compared to see which is preferred. These two complications are addressed in the first half of this section, followed by a brief discussion of other challenges.
3.4.1 Complex Cash Flow Patterns
For the purposes of this analysis of IRRs, a complex cash flow pattern is an investment involving either borrowing or multiple sign changes. A borrowing type cash flow pattern begins with one or more cash inflows and is followed only by cash outflows. An example of the borrowing pattern is when an investment such as a real estate project is sold and leased back. The divestment generates current cash at the cost of future cash outflows and may be viewed as a form of borrowing. A multiple sign change cash flow pattern is an investment where the cash flows switch over time from inflows to outflows, or from outflows to inflows, more than once. An example of a multiple sign change investment would be a natural resource investment involving (1) negative initial cash flows from purchasing equipment and land to set up an operation such as mining, (2) positive interim cash flows from operations, and (3) negative terminal cash flows from ceasing operation and restoration expenses. Exhibit 3.2 illustrates the complex cash flow patterns.
Exhibit 3.2 Complex Cash Flow Pattern Examples
In the case of borrowing type cash flow patterns, there is a unique solution (i.e., there is only one IRR that solves the equation), but the IRR must be interpreted differently. In borrowing type cash flow patterns, a high IRR is undesirable because the IRR is revealing the cost of borrowing rather than the return on investment. Also, when a trial-and-error search is performed to find the IRR, any increase in the discount rate lowers the present value of the cash outflows rather than lowering the present value of the cash inflows, as would be the case in a simple cash flow pattern. Thus, the trial-and-error process must operate in a reverse direction from the simplified investment cash flow pattern. In other words, if the net value with a given discount rate is positive, the next IRR in the search should be lower rather than higher, as occurs in the case of a simplified cash flow pattern.
In the case of multiple sign change cash flow patterns, the problems are more troublesome. Whenever there is more than one sign change in the cash flow stream, more than one IRR may exist. In other words, two or more answers can probably be found using the IRR formula. In fact, the maximum number of possible IRRs is equal to the number of sign changes. When more than one IRR is calculated, none of the IRRs should be used. There is no easy way for the IRR model to overcome this particular shortcoming.
Consider a derivative deal that ends poorly for Investor A. The derivative required a $5,000 outlay from Investor A to the counterparty to open. In the first period, the derivative generates an $11,500 cash inflow to Investor A from the derivative's counterparty. The derivative then generates a cash outflow of $6,550 from Investor A at the end of the second period, at which point the derivative terminates. The derivative's cash flows from the perspective of Investor A are given in Exhibit 3.3, assigning period 0 to the first nonzero cash flow.
Exhibit 3.3 Cash Flows of Hypothetical Derivative Contract
This cash flow pattern changes signs twice, once from negative to positive and once from positive to negative. There are two IRRs: 3.82 % and 26.20 %. Both 3.82 % and 26.20 % satisfy the definition of the IRR because they set the present value of all cash inflows equal to the present value of all cash outflows. The net value of the present values of the cash inflows and outflows is illustrated in Exhibit 3.4. Note that the line crosses the horizontal axis twice, defining two different IRRs.
Exhibit 3.4 An Example of Multiple IRRs
With the two IRR solutions 3.82 % and 26.20 %, there may be a temptation to think that the two IRRs can be somehow analyzed in unison to generate an intuitive feel for the derivative's attractiveness. But neither number is particularly useful, because the investment is really a combination of investing from period 0 to period 1 and borrowing from period 1 to period 2. In this particular case, the cash flow patterns have a positive net value between the two IRRs, using discount rates between 3.82 % and 26.20 %. But as a derivative, it is obvious that the cash flows to the other side of the derivative (the counterparty) would have the same numbers, but the signs of the cash flows would be reversed. In this case, the cash flows would be +$5,000, –$11,500, and +$6,550. From the counterparty's perspective, the IRR solutions would still be exactly the same at 3.82 % and 26.20 %. However, the deal's graph would appear as a mirror image, with negative net values between the two IRRs. As we would expect with a derivative deal, gains to one side of the contract would equal losses to the other side of the contract. Both sides would view the same IRRs because they used the same cash flows, but they would be looking at opposite cash flows and opposite net values. Therefore, using only the IRRs to decide if the derivative is beneficial is not possible.
3.4.2 Comparing Investments Based on IRRs
The previous section reviewed the difficulties of computing and interpreting IRRs when an investment offers a complex cash flow stream. But even if the investments being analyzed offer simplified cash flow streams (a cash outflow followed only by cash inflows), the IRR method of measuring investment performance has serious challenges. This section details the major challenges of comparing investments based on IRR.
The major challenge with comparing IRRs across investments occurs when investments have scale differences. Scale differences are when investments have unequal sizes and/or timing of their cash flows. When comparing investments with different scales, an investment with a higher IRR may be inferior to an investment with a lower IRR.
The following is a simple example that illustrates the problems that occur when comparing IRRs. Assume that a bank is offering high initial yields on a limited-time basis to induce investors to open a new account. Investors are allowed to open only one account. The example includes three types of accounts, each with the following interest rates and restrictions on time and amount:
■ Account Type A: Receive 100 % annualized interest for the first day on the first $10,000.
■ Account Type B: Receive 100 % annualized interest for the first year on up to $10.
■ Account Type C: Receive 20 %
