the first year on up to $10,000.
The IRR of alternatives A and B is 100 %, whereas the IRR of alternative C is only 20 %. However, alternative A has very small scale due to a time limitation of one day (timing), and alternative B has very small scale due to a cash flow size limitation of $10 (size). If annualized market interest rates are 5 %, alternative A has a net present value of less than $30, and alternative B has an NPV of less than $10. Alternative C has an NPV of about $1,500, even though its IRR is only one-fifth that of the other two alternatives. The reason for this is that although all three alternatives have favorable IRRs, alternative C has much larger scale.
In this example, it is better to receive a lower rate on a large scale. In actual investing, scale differentials can be complex and subtle. In judging when a larger scale is worth a sacrifice in return, approaches to investments using the NPV method offer substantial potential in evaluating investment opportunities of different scales. But in alternative investments, especially private equity, IRR is the standard methodology, and scale differentials represent a challenge in ranking performance.
3.4.3 IRRs Should Not Be Averaged
Another challenge to using IRRs involves aggregation. Aggregation of IRRs refers to the relationship between the IRRs of individual investments and the IRR of the combined cash flows of the investments. Suppose that one investment earns an IRR of 15 % and another earns an IRR of 20 %. What would the IRR be of a portfolio that contained both investments? In other words, if the cash flows of two investments are combined into a single cash flow pattern, how would the IRR of the combination relate to the IRRs of the individual investments? The answer is not immediately apparent, because the IRR of a portfolio of two investments is not generally equal to a value-weighted average of the IRRs of the constituent investments. If the cash flows from two investments are combined to form a portfolio, the IRR of the portfolio can vary substantially from the average of the IRRs of the two investments.
This section demonstrates the difficulty of aggregating IRRs, and the following extreme example illustrates the challenges vividly. Consider the following three investment alternatives:
The IRRs of the three alternatives are easy to compute because each investment simply offers two cash flows: one at time period 0 and one at time period 1. Using Equation 3.9, the IRR for a one-period investment is found by solving the equation 0 = CF0 + CF1/(1 + IRR), which generates the equation
Inserting the values for Investment A (CF0 = –100, CF1 = +110) generates the IRR of 10 %, shown in the IRR column. Investments B and C both have CF0 = –CF1, so the IRRs of both Investment B and Investment C are 0 %.
One might expect that combining Investment A with either Investment B or Investment C would generate a portfolio with an IRR between 0 % and 10 % because one investment in the portfolio would have a stand-alone IRR of 10 %, as with Investment A, and the other would have a stand-alone IRR of 0 %, as in the case of either Investment B or C. But IRRs can generate unexpected results, as indicated by the following analysis:
The computations simply sum the cash flows of two investments and compute the single-period IRR of the aggregated cash flows. The IRR of combining Investments A and B is –20 %, and the IRR of combining Investments A and C is +20 %. The IRRs of both combinations are well outside the range of the IRRs of the individual investments in each portfolio. What generates the unexpected result in this example is that Investments B and C begin with cash inflows and end with cash outflows (i.e., they are borrowing investments). But in practice, alternative investments, such as commodity or real estate derivatives and private equity, can have cash flow patterns sufficiently erratic to cause serious problems with aggregation of IRRs.
3.4.4 IRR and the Reinvestment Rate Assumption
Even if all the investments have simplified cash flow patterns without borrowing or multiple sign change problems, the IRR does not necessarily rank investments accurately. The use of the IRR to rank investment alternatives is often said to rely on the reinvestment rate assumption. The reinvestment rate assumption refers to the assumption of the rate at which any cash flows not invested in a particular investment or received during the investment's life can be reinvested during the investment's lifetime. If the assumed reinvestment rate is the same rate of return as the investment's IRR, then no ranking problem exists.
Suppose that Investment A offers an attractive IRR of 25 % compared with the 20 % IRR of Investment B. As previously discussed, it is possible that an investor would select Investment B over Investment A if investment B offers larger scale, meaning more money invested for longer periods of time. But if an investor who selects Investment A is able to invest additional funds at a 25 % rate of return and is able to reinvest any cash flows from Investment A at the 25 % rate, then the scale problem vanishes, and IRRs can be used to rank investments effectively. In practice, there would typically be no reason to assume that cash inflows could be reinvested at the same rate throughout the project's life, so ranking remains a problem. The reinvestment rate assumption is addressed by the modified IRR. The modified IRR approach discounts all cash outflows into a present value using a financing rate, compounds all cash inflows into a future value using an assumed reinvestment rate, and calculates the modified IRR as the discount rate that sets the absolute values of the future value and the present value equal to each other.
Extensions of the modified IRR methodology can be adapted to develop realized rates of returns on completed projects or for projects in progress. In the case of a private equity or private real estate investment with known cash flows since inception and with a current estimate of value, a realized or interim IRR can be calculated using the assumption that intervening cash inflows are reinvested at the benchmark rate.
3.4.5 Time-Weighted Returns versus Dollar-Weighted Returns
The purpose of this section is to provide details regarding time-weighted returns versus dollar-weighted returns. Briefly, time-weighted returns are averaged returns that assume that no cash was contributed or withdrawn during the averaging period, meaning after the initial investment. Dollar-weighted returns are averaged returns that are adjusted for and therefore reflect when cash has been contributed or withdrawn during the averaging period. The IRR is the primary method of computing a dollar-weighted return.
When evaluating the return of hedge funds, mutual funds, or any investment, it's important to recognize the distinction between the time-weighted return, which is similar to what is reported on performance charts in marketing literature and client letters, and the dollar-weighted return, which represents what the average investor actually earned; the two can be very different.
Suppose there is a hedge fund that in year 1 starts with $100 million of AUM (assets under management). Let's further suppose that the hedge fund generates an average annual return of 20 % for each of its first three years. With such a performance history, the hedge fund attracts quite a bit of new capital. Let's assume that the hedge fund attracts $200 million in new assets for year 4, another $200 million for year 5, and nothing in year 6. Unfortunately, the new capital does not help the hedge fund manager maintain the fund's stellar performance, and the manager earns 0 % in years 4, 5, and 6. If we use time-weighted returns over this six-year period, the hedge fund manager has an average annual return of 9.5 %:
In effect, the time-weighted return assumes that a single investment (e.g., $1) was made at the beginning
