9. Internal Rate of Return

With no loss of generality, we assume \(I_0=0\).

This is not strictly true, because the equity capital might in principle have a sign which differs from the sign of the invested capital. We will not dwell on this.

Financial transactions include (but are not limited to) loans. In a generic financial transaction, lending positions and borrowing positions may change over time.

In Chap. 10 we will introduce another relative metric which is genuinely internal. To pinpoint the difference with AIRR, we will switch “average” and “internal” and label it “Internal Average Rate of Return”.

Equation (9.5) may be reframed asThe latter may well be considered an alternative definition of IRR (note that the solutions of this equation are necessarily invariant under changes in r).

See also Magni (2013a) for a compendium of 18 flaws of the IRR (many of which are treated in the next section).

This cash-flow stream was presented in Eschenbach (1995, Sect. 7.6), associated with a mineral extraction project, and later analyzed by Hazen (2003) and Magni (2010b).

This does not exclude that other IRRs exist in the interval \((-1,0)\).

See also Soper (1959), Hicks (1973), Nuti (1974), Sen (1975), Ross et al. (1980), Cuthbert (2018) on truncation.

This is in line with Definition 8.​4 in this book. Indeed, Hazen’s (2003) definition may be interpreted as a special case of it (see Magni 2010b).

Hazen’s adjusted IRR criterion also implies a third result: The IRR’s financial nature is not internal, for it depends on the sign of \(\text {PV}[C(\sigma )]\) which in turn depends on the COC, r (see also Chap. 10).

Complex-valued IRRs are also dealt with in Hazen (2003) and Pierru (2010).

The literature on multiple IRRs is enormous. Authors from various fields advanced methods for choosing the relevant IRR among the multiple ones or methods for avoiding the multiple-IRR issue. To cite just (very) few contributions,(The results found in Hartman and Schafrick 2004; Magni 2010b; and Cuthbert 2018 have been awarded the “Eugene L. Grant” award by ASEE in 2005, 2011, and 2019, respectively.) .

In the previous ROSCA, existence of IRR is guaranteed by the fact thatso the solution \(\sigma =0\) always exists. Conversely, in this case, where \(m^{\prime }<m\),inexistence of IRR may indeed occur.

The distinction between a project and a portfolio of projects is not clear-cut: A project might be itself split into lower-order components (e.g., as we have seen, a project is a portfolio of equity and debt as well as a portfolio of operating assets and non-operating assets). Whether an asset should be considered an individual project or a portfolio is a matter of perspective and, therefore, of convention: Whether several courses of action should be aggregated into a single one or a single course of action should be partitioned into multiple ones depends on the conceptualization/framing of the situation, which may differ depending on several variables such as the domain and the purpose of the analysis, the pieces of information available to the analyst, the easiness and intuitiveness of the conceptual framework, etc.

The equity may be viewed as a portfolio of an investment position (project) and a financing position (debt). See also above on the equity IRR.

In Magni and Marchioni (2019), a more refined example of PhV plant with no IRR is illustrated.

We assume the first meeting will take place in one period. In this respect, it is irrelevant whether the first cash flow occurs at time 1 or at time 0 (no matter what unit of time has been selected): The solution of the IRR equation and the sign of \(\text {PV}[C(\sigma )]\) are not affected.

The debt’s IRR is not necessarily the ROD. The ROD is a single-period concept (the ratio of the interest expense to the BOP debt), while the IRR is a multiperiod concept. ROD and debt’s IRR coincide if and only if the ROD is time-invariant (a rather unrealistic occurrence unless the debt consists of one single liability bearing interest at a constant rate of interest).

In discrete terms, \(\bar{\imath }^{uo}=-10.69\%=-9.71\%\cdot 1.1\) and the economic inefficiency is \(\xi ^{uo}=-20.69\%\) (\(=-10.69\%-10\%\)).

Even assuming that \(r^d_t=r^d\) is constant, the debt-to-equity ratio \((V^d_{t\!-\!1}/V^e_{t\!-\!1})\) will most frequently be time-varying.

The AIRR approach may help technically solve this problem by computing the arithmetic mean of the COCs, weighted by the discounted internal capitals. A net investment is worth undertaking if and only if this mean is smaller than the IRR (see Magni 2013a, pp. 105–108).

See Joy and Bradley (1973) and Whisler (1976) for less recent contributions on this issue.

This paragraph and Example 9.31 are adapted from Magni (2013a), The Internal Rate of Return Approach and the AIRR paradigm: A refutation and a corroboration, The Engineering Economist, 58(2), 73–111 (p. 82 and p. 100).

This paragraph and Example 9.32 are adapted from Magni (2013a), The Internal Rate of Return Approach and the AIRR paradigm: A refutation and a corroboration, The Engineering Economist, 58(2), 73–111 (pp. 79–80).

The answer is: None of them. The project’s rate of return is the weighted mean of the income rates that are generated in the two periods. For example, if the vector of capitals is \(\varvec{C}=(100, 50,0)\), then the vector of incomes is \(\varvec{I}=(0, 70, -50)\) and the holding period rates are \(i_1=70\%\) and \(i_2=-100\%\). This means that, in the first period, \(\$\)100 is invested and the capital grows by \(i_1=70\%\). In the second period, the firm loses the whole capital that remains invested in the project, so the period rate of return is \(i_2=-100\%\). An appropriate rate of return is a mean of these income rates, properly weighted by the respective capitals (i.e., the average ROI in the AIRR approach).

For simplicity, we assume \(I_0=0\).

We do not dwell on it, but, in Chap. 10, we will introduce a class of rates of return which do not incur this difficulty.

Among others, see Solomon (1956), Baldwin (1959), Kirshenbaum (1965), Lin (1976), Athanasopoulos (1978), Chang and Owens (1999), Kierulff (2008).

This method is applicable only if the compounded sum of the cash flows has a different sign from \(F_0\).

More specifically, let t be the last date where a negative cash flow occurs. Then, one discounts back the outflow by one period at rate \(y^{\text {ext}}\) and subtracts it from the preceding cash flow. If the resulting amount is nonnegative, the procedure ends, otherwise it is iterated, until some date \(\tau \), where the resulting amount is nonnegative. At this point, if there is another negative cash flow occurring prior to \(\tau \), the same iterative procedure is reinstantiated, until no negative cash flow occurs after some positive cash flow (first version) or until no negative cash flow occurs after some positive cash flow and an no negative cash flow occurs after the first negative flow (second version). Other variants of the SFM are possible (e.g., later positive cash flows may be discounted to pay off earlier negative cash flows (see Herbst 2002 on SFMs).

For example, consider the SFM in the example above: At time 4 an outflow of \(\$\)4 occurs. Assuming that \(\$\)3.33 is invested for one year in a sinking fund earning 20%, the accumulated amount is \(\$\)4, which zeroes out the outflow. At time 3, the investment of \(\$\)3.33 is subtracted from \(\$\)6 to get a net inflow of \(\$\)2.7.

For example, in the compounding approach, the \(\$\)7 is zeroed out by assuming a reinvestment of \(\$\)7 at 20% for 4 periods, and the \(\$\)6 is zeroed out by assuming a reinvestment of \(\$\)6 at 20% for two periods. At time 5, the accumulated amount of the two reinvestments will be added to the terminal cash flow: \(7(1.2)^4+6(1.2)^2+1.8=25\). In this way, at time 1 and 3, the net cash flow is zero.

For example, in the accumulating approach, the outflow of \(\$\)11 at time 2 is zeroed out by assuming that \(\$\)11 is borrowed at 20%. The accumulated amount is then subtracted from the accumulated amount of the other integrative operations realized at the various dates.

Keane (1979) expressed the same concept with the following words:

If the satellite projects are physically or economically dependent, then the original cash flow specifications are incorrect because they should have included the effects of the entire programme of interdependent projects. (Keane 1979, p. 50)

If a manager measures the NPV of a project’s cash flows without including the reinvestment flows, the only implicit assumption in the decision to do so is that the reinvestment opportunities are not relevant. If they are in fact relevant there is no benefit in omitting the additional flows and speculating about the arithmetic relationship between the NPV of the incomplete set and the NPV that would have obtained had the additional flows been included. (Keane 1979, p. 53).

If \(C_t (i_B, i_L)\ge 0\) for every t, then the project is internally pure and \(i_B\) does not apply: The rate \(i_L\) is the traditional project’s IRR (which exists and is unique by STRMG): \(i_L=\sigma \). If \(C_t (i_B, i_L)\le 0\) for every t, then the project is internally pure and \(i_L\) does not apply: The rate \(i_B\) is the traditional project’s IRR (which exists and is unique by STRMG): \(i_B=\sigma \).

There may however be a cognitive rationale in preferring the investment setting. Practitioners and academics seem to prefer to think in terms of rates of return rather than rates of cost.

The financial nature of an IRR is not genuinely internal: It depends on the cost of capital. See Chap. 10.

An exception is the global (non)additivity of AIRR when the COCs of a portolio’s constituent assets are not equal.