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 of the period and was allowed to grow with positive returns and decline with negative returns until the end of the measurement period, with no cash withdrawals or additional contributions. The rate that equates the initial value with the accumulated value is the time-weighted average return, and it is somewhat near the arithmetic average annual return (in this case, 10 % per year). The idea is that a single sum of money invested at the start of the first year and allowed to remain in the fund until the end of the last year would accumulate to the same value as if it had been invested at a fixed return of 9.5 % per year, ignoring rounding.
But in practice, investors often contribute additional cash (i.e., make additional investments) or withdraw cash (e.g., liquidate part of the investment or receive cash distributions) during the time period under analysis. Their average returns depend on whether the amount of money invested was highest during the high-performing periods or during the low-performing periods. Dollar-weighted returns adjust the average annual performance for the amount of cash invested each year. In the case of the hedge fund, an investor who had much more cash in the fund in the early years than in the later years would earn more than an investor whose money was primarily invested in the last three years, when the fund generated 0 % returns.
Dollar-weighted returns can be computed for each investor using investors' cash flows into and out of the hedge fund. The total cash flows into and out of the fund for all investors can be used as an indication of the performance of an average investor. The dollar-weighted return that individual investors experience depends on their cash contributions and withdrawals.
When the timing of the aggregated cash flows for the entire hedge fund is taken into account, the bulk of the hedge fund's assets earned a 0 % return in years 4, 5, and 6. The example shows that only the first $100 million earned the great rates of return of the first three years. The $400 million that flowed into the hedge fund in years 4 and 5 earned a 0 % return. When the timing of the aggregated cash flows is taken into account, the dollar-weighted return (solving for the IRR with cash flows reinvested) is only 4.3 %. The IRR is found in this case with CF0 = –100, CF1 = 0, CF2 = 0, CF3 = –200, CF4