cash flow was deposited or withdrawn (CFT).
3.3.2 Computing the IRR
In some simplified cases, such as investments that last only a few periods or investments in which most of the cash flows are identical (i.e., annuities), the IRR may be solved algebraically with a closed-form solution. In cases involving several different cash flows, the solution generally relies on a trial-and-error search performed by an advanced financial calculator or computer.
A simplified example to illustrate the trial-and-error method involves an investment that costs $250 million and lasts three years, generating cash inflows of $150 million, $100 million, and $80 million in years 1, 2, and 3, respectively. The IRR is found as that interest rate that solves the following equation:
The trial-and-error process selects an initial guess for IRR, such as 10 %, and then searches for the correct answer: the IRR that sets the left-hand side of Equation 3.10 to zero. Inserting IRR = 0.10 (10 %) into Equation 3.10 generates a present value of inflows equal to $279.11 million and a value to the entire left-hand side of $29.11 million. The objective is to have the value of the left-hand side of the equation equal to zero. In the case of this investment, a higher discount rate will generate a lower net value. If the next guess is an interest rate of 15 %, the value of the left-hand side of the equation declines to $8.65 million. The process continues with as much precision as required. The IRR of this investment is 17.33 % carried to the nearest basis point.
Advanced calculators and computer spreadsheets perform the trial-and-error process automatically. This solution of 17.33 % for the IRR can be found on most financial calculators by inserting the cash flows (using cash flow mode) and requesting the computation of the IRR or in a spreadsheet with a function designed to compute IRR.
In this example, the trial-and-error process for finding the IRR works well because any increase in the discount rate lowers the present value of the cash inflows, and any decrease in the discount rate raises the present value of the inflows. The solution to the IRR problem is illustrated in Exhibit 3.1.
Exhibit 3.1 The Solution to IRR in a Simplified Investment
Because the IRR is the discount rate that sets the NPV of the investment to zero, the IRR is represented by the point at where the NPV curve crosses through the horizontal axis. This occurs between 17 % and 18 % on the figure, which corresponds to the previous solution of 17.33 %. There is only one solution, and it is quite easily found. If a bank offered an interest rate of 17.33 %, then an investor could deposit $250 million, and withdraw $150 million, $100 million, and $80 million after one, two, and three years, respectively; and the final account balance would be $0, ignoring rounding errors.
3.3.3 Interim Valuations and Four Types of IRRs
The primary reason for using the IRR approach is that regular valuations of the investment, such as daily market prices, are not available. An IRR can be performed on a realized cash flow basis or an expected cash flow basis. A realized cash flow approach uses actual cash flows through the termination of the investment to compute a realized IRR. An expected cash flow approach uses expected cash flows projected throughout the investment's life to compute an anticipated IRR. An IRR may be computed during an investment's life using both realized cash flows and either a current valuation or projections of future cash flows.
There are four types of IRRs based on the time periods for which cash flows are available. Although these terms are not uniformly defined in practice, they are useful for our purposes:
1. LIFETIME IRRS: A lifetime IRR contains all of the cash flows, realized or anticipated, occurring over the investment's entire life, from period 0 to period T. In other words, if in the context of Equation 3.9, time period 0 is the inception of the investment and time period T is the termination of the investment, then the IRR is a lifetime IRR.
2. SINCE-INCEPTION IRRS: A since-inception IRR is commonly used as a measure of fund performance rather than the performance of an individual investment. The cash flows that would then be used in Equation 3.9 are aggregate cash flows of a fund rather than a single portfolio company. The terminal (time period T) cash flow in this case is the appraised value of the fund's portfolio at time T rather than a liquidation cash flow. Interim cash flows represent actual fund-level cash flows from liquidated investments.
3. INTERIM IRRS: The interim IRR is a computation of IRR based on realized cash flows from an investment and its current estimated residual value. The key to an interim IRR is that generally T would not be the termination of the investment; thus, CFT is an estimated value rather than a realized cash flow. The interim IRR can be calculated on an investment purchased subsequent to its inception.
4. POINT-TO-POINT IRRS: A point-to-point IRR is a calculation of performance over part of an investment's life. All cash flows are based on realized or appraised values rather than expected cash flow over the investment's projected life. Although any IRR is calculated from one point in time to another, a point-to-point IRR would typically not be used to refer to a lifetime IRR.
For IRRs computed over a time interval that begins after the investment's inception, the cash flow in time period 0, CF0, would be either the first cash flow paid by an investor to acquire the investment or some valuation after the investment's inception, such as an appraisal. For IRRs computed over a time interval that ends prior to the investment's termination, the cash flow in time period T, CFT, would be a valuation such as an appraisal or the sales proceeds at a date prior to the investment's termination. Three applications follow to illustrate lifetime, since-inception, and point-to-point IRRs.
Investment A is expected to cost $100 and to be followed by cash inflows of $10 after one year and then $120 after the second year, when the project terminates. The IRR is based on anticipated cash flows and is an anticipated lifetime IRR. The IRR of the investment is 14.7 %.
Fund B expended $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