## Abstract

While the internal rate of return (IRR) remains widely used despite its well-documented flaws, considerable progress has been made in the literature on the search for a proper measure of performance for illiquid investments. Most recently, two general approaches have been proposed—namely the average internal rate of return (AIRR) approach and the aggregate return on investment (AROI) approach—and the following cash flow-based metrics have appeared in the literature: the economic AIRR, the index comparison method (ICM)-based AROI, and the direct alpha method. Motivated by the AIRR approach, the author presents a new cash flow-based metric, excess return on time-scaled contributions (ERTC), that is free from all the IRR flaws and represents a solution to the public market equivalent (PME) problem. ERTC is intuitively defined as the ratio of net present value to the aggregate discounted net contributions scaled by the length of time the net contribution is relevant to the investment. The author shows that ERTC can be obtained as a special case of the continuous version of AIRR with an ICM-based assumption on capital growth. Moreover, as part of the definition of ERTC, the author introduces a decomposition procedure that can be adapted to help ensure value consistency (i.e., value additivity and consistency with known market values) for all cash flow-based AIRR or AROI metrics.

The internal rate of return (IRR) is widely used among both academics and practitioners for measuring performance of investments where frequent market values are not readily available, such as private equity, venture capital, and private real asset investments. The IRR is believed to provide a money-weighted return (MWR) that takes into account the impact of cash flows.^{1} Unfortunately, a large body of literature has long documented various flaws with the IRR approach (Magni [2013b] lists 18 of them), among which are (i) multiple IRRs may arise, in which case it is unclear which one is the economically correct rate of return^{2}; (ii) a real-valued IRR may not exist all^{3}; (iii) inconsistency with the net present value (NPV) in project acceptance/rejection or ranking decisions; and (iv) violation of value additivity (Magni [2013b, F9]), which means “the value of any portfolio of assets must be equal to the sum of the asset’s values.”^{4}

Correspondingly, numerous contributions have been devoted to searching for corrective procedures aimed at healing the IRR flaws (see Magni [2010] for a good review of such procedures), among which the average internal rate of return (AIRR) approach introduced by Magni [2010] stands out for the insights it conveys on the problem. The author defines rate of return as aggregate income divided by aggregate capital, both in present value terms to account for the time value of money. The author presents AIRR as “a complete solution to all the IRR problems.” However, the general AIRR measure rests on the assumption that the capital values (or capital stream) can be exogenously obtained; hence, it is not directly applicable to most illiquid investments for which the capital values are not readily available.

Magni [2011] introduces a similar approach named aggregate return on investment (AROI) that is based on undiscounted capital values and cash flows, but also requires the availability of capital values. For estimating interim capital values, Magni [2014, sec. 6] proposes to use an IRR-based approach if there is good empirical reason to support the assumption of an investment’s value growing at a constant rate, or otherwise a market-pricing approach based on discounting prospective cash flows at the market rates (termed “economic AIRR” in Magni [2013b, eq. (24)]). Altshuler and Magni [2015] (and also Magni [2015]) introduce a special AROI that uses the index comparison method (ICM) of Long and Nickels [1996] to generate interim capital values (referred to as the market-investment AROI in Magni [2011], but we shall refer to it as the ICM-based AROI hereafter).

On another front, a series of research has been devoted to finding an appropriate measure of excess return on illiquid investments relative to a public market benchmark (the so-called public market equivalent (PME) analysis). The first in this series is the ICM of Long and Nickels [1996], who consider the IRR on a benchmark portfolio that replicates the investment’s cash flow stream but grows at the benchmark rate. One perceived issue with the ICM approach is that a strong outperformance can result in the benchmark portfolio’s value being negative. To rectify this issue, Rouvinez [2003] proposes the PME+ method and Cambridge Associates [2013] proposes the mPME method. The idea of both methods is to scale the distributions for the benchmark portfolio so that its value stays nonnegative, but with different ways of scaling. More recently, Gredil, Griffiths, and Stucke [2014] introduce the direct alpha method based on the IRR of the discounted cash flows. The direct alpha method stands out in two ways: (i) it directly calculates the excess return of the investment relative to the benchmark, instead of relying on a heuristic difference between the investment’s IRR and the benchmark portfolio’s IRR; and (ii) it uses continuously compounded returns (a.k.a. log returns). However, similar to its predecessors, the direct alpha method is still subject to the IRR flaws mentioned previously. Kaplan and Schoar [2005] introduce a PME measure that does not use the IRR, but it is a measure of total wealth creation relative to the benchmark rather than a measure of the rate of excess return. The ICM-based AROI metric provides a long-needed link between the PME literature and the IRR literature, since the PME problem is simply the IRR problem with variable cost of capital.

This article contributes to the literature in the following two main ways. First, we introduce an ICM-based continuous AIRR, termed excess return on time-scaled contributions (ERTC), as an alternative to the recently proposed cash flow-based metrics: IRR-based AIRR, economic AIRR, ICM-based AROI, and the direct alpha method. ERTC’s main advantage is its simple and intuitive definition: it is defined as the ratio of net present value to the aggregate discounted net contributions (i.e., distributions are considered as negative contributions) scaled by the length of time the net contribution is relevant to the investment. It is also easy to calculate and always exists uniquely.

Second, we show that the decomposition procedure used by ERTC can be adapted to help ensure value consistency (i.e., value additivity and consistency with known market values) for all cash flow–based return metrics. For example, although the IRR violates value additivity (Magni [2013b, F9]), if we adjust the calculation of the IRR of portfolios based on the decomposition procedure, value additivity will be preserved. Similarly, if the economic AIRR or the ICM-based AROI is calculated directly from a portfolio’s cash flows rather than its underlying investment’s cash flows, we show that value additivity can be potentially violated. We then show how the decomposition procedure can help ensure value consistency for these and other cash flow–based AIRR or AROI metrics.

It is worth noting that while we set the context of this article to be illiquid investments performance measurement for ease of discussion, our methodologies are general enough that they apply equally well to other areas such as capital budgeting and project finance, or liquid public investments when it is appropriate to take into account the impact of cash flows.

The rest of the article is organized as follows. The next section defines ERTC and discusses its intuition. The section after that provides some numerical examples to illustrate the calculation of ERTC and to contrast ERTC with existing MWR metrics. The subsequent section discusses the relationship between the prevalent discrete AIRR and the underresearched but more theoretically appealing continuous AIRR. The following section discusses an interesting combinatorics of cash flow–based AIRR or AROI metrics, and the section after that shows how ERTC’s decomposition procedure can be adapted to help ensure value consistency for all cash flow–based AIRR or AROI metrics, which is followed by a concluding section. Lastly, the appendix shows that ERTC can be obtained as a special case of continuous AIRR with ICM-based capital stream.

**EXCESS RETURN ON TIME-SCALED CONTRIBUTIONS**

This section describes the definition of ERTC.

For a given project, let
be the project’s cash flow stream, so
is the cash flow at time *t _{i}
*, with negative values denoting contributions and positive values denoting distributions. Times (

*t*

_{0}, …,

*t*) do not have to be equally spaced, so there is no need to insert zero cash flows in between

_{n}*t*

_{0}and

*t*, but [

_{n}*t*

_{0},

*t*] should either represent the economic or operating life span of the project (Magni [2013b, F10]) or a specific evaluation period one has chosen, in which case should be the project’s net asset value (NAV) at

_{n}*t*plus any actual cash flow at

_{n}*t*, and should be the negative of the project’s NAV at

_{n}*t*

_{0}.) Let be the discount factor between time

*t*and

_{i}*t*

_{0}based on a benchmark. That is, if

*b*is the benchmark return during interval [

_{i}*t*

_{i-1},

*t*

_{i}] for

*i*= 1, 2, …,

*n*, then

We define

2where ATC is the aggregate time-scaled net contributions, discounted to time *t*
_{0}, and **
b
** denotes the benchmark returns. The numerator of ERTC is the well-known net present value (NPV). To calculate the ATC of the project, we use a three-step procedure to decompose the cash flow stream

**into a series of “indivisible” cash flow streams:**

*f*

*f*^{1}, …,

*f*^{m}, so that and we set

Since the first step of the decomposition (see next) is to break portfolios into its underlying assets, Equation (3) ensures value additivity for ATC (and hence ERTC). Let
(1 = *k* = *m*), the ATC of the *k*th indivisible cash flow stream is defined as

(Note that
is the discount factor between time *t*
_{k,i} and *t*
_{0} (not *t*
_{k,0}), so all cash flows are discounted back to the same starting time *t*
_{0}.) The intuition is that the cash flow
is only relevant to the indivisible project during interval [*t*
_{k,i},
], so it is scaled by the length of the interval: (
). (The minus sign at the beginning of the formula is to make the contributions positive since they are treated as negative cash flows, and vice versa for distributions.)

We now explain the procedure of decomposing a project into indivisible projects or cash flow streams. The procedure should be carried out sequentially with the three steps described next, in the order given.

**Step 1 (S1):**If the investment represents a portfolio of economically distinct investments (such as a private real estate fund consisting of multiple property investments), it should be decomposed into the underlying individual investments.**Step 2 (S2):**For ease of notation, let*f**f**t*' is and*t*<_{k}*t*' <*t*_{k+1}for some 0 =*k*<*n*, then*f**t*': 5If

*t*' happens to be the same as*t*, then_{k}*f*The steps should be carried out similarly for

*t*' =*t*_{k+1}. (See Magni [2013b, sec. 6] for a similar treatment.) This splitting should be done for all known capital values, so that if there are*s*known capital values,*f**s*+ 1 sub-cash-flow-streams.**Step 3 (S3):**Again, for ease of notation, let*f**f**t*_{n}is the first time the NPV turns positive, if it turns positive at all. Specifically, if 7exists and

*j*_{0}<*n*, then*f*The same procedure should be applied to

*f*^{2}, and so on, until no more NPVs turn positive before the end of the cash flow stream. The resulting sub-cash-flow-streams are now considered indivisible. As an example, consider*f**f*

*Remark 1*. The purpose of S3 is to ensure that the ICM-based capital values of **
f
** are nonnegative (see the appendix). This is often necessary in the current context, since most private investments have limited liability structures such as limited partnerships or limited liability companies (LLCs), i.e., the investor is liable only to the extent of the amount of money he has invested (so his capital balance can never be negative). In cases where it is reasonable to allow negative capital values (e.g., investments with recourse loans or mining projects), this third step may be skipped. As a result, the ATC of the overall project may end up being negative, in which case the project should be viewed as a net borrowing (Magni [2010]).

*Remark 2*. S2 and S3 are conceptually appealing since they essentially convert the treatment of known interim capital values and positive NAVs into the treatment of portfolios. In addition, S3 can be viewed as a special form of S2 by setting certain capital values to 0. This is evident by noting that Cash Flow Streams (8) is equivalent to setting
in Cash Flow Streams (6).

*Remark 3*. As we will show in the appendix, ERTC represents a weighted average (excess) log return that is a special type of excess continuous AIRR. To quote ERTC as an annual return, we simply need to express all the time-scaling factors in unit of years (see Example 7).

*Remark 4*. If **
f
** is already an indivisible cash flow stream, then Equation (2) and Equation (4) imply

Suppose also **
b
** = 0, i.e., the benchmark returns are constantly 0 and hence all discount factors are equal to 1, Equation (10) reduces to

This can be recognized as the familiar modified Dietz formula (Dietz [1966]), except for the factor
, which converts the modified Dietz holding period return into an average return over the interval [*t*
_{0}, *t _{n}
*]. This interesting coincidence sheds new light on the modified Dietz as an MWR, that is, the modified Dietz formula implicitly assumes a 0 benchmark and should only be applied directly to indivisible cash flow streams. For a general cash flow stream

**, it should probably be redefined as (**

*f**t*-

_{n}*t*

_{0}) × ERTC(

**|0). This solves the problem of the modified Dietz in cases of large cash flows causing the denominator to be 0 or even negative, which is often viewed by practitioners as a major shortcoming of the method.**

*f*
*Remark 5*. Similar to the direct alpha method of Gredil, Griffiths, and Stucke [2014], ERTC is a direct measure of the excess return of an illiquid investment relative to a benchmark, so it is not necessary to derive a PME return and then heuristically subtract it from the investment’s return. Sometimes it might still be useful to have a PME-type return for its own sake, so we provide the following definition of benchmark return on time-scaled contributions (BRTC):

where the same decomposition as in ERTC is applied to **
f
**, and

*b*

_{k,j}is the benchmark return during interval [

*t*

_{k,j-1},

*t*

_{k,j}].

BRTC can be viewed as a weighted average log return. Note

13so that the log return during interval [*t*
_{k,i},
], i.e.,
, is weighted by the corresponding component of ATC:
In the special case that **
b
** represents a constant log return

*b*, i.e.,

it is straightforward to verify that

15Lastly, we also define aggregate return on time-scaled contributions (ARTC) as

16which we will show is a special type of continuous AIRR.

*Remark 6*. If **
f
** represents an individual investment for which capital values are known for all cash flow dates, then according to S2, we need to split

**into the following**

*f**n*one-period cash flow streams:

If there are also known capital values in between the cash flow dates, further splitting is needed and the resulting ATC and ERTC can be different. Therefore, it is important to note that ERTC accounts for all known capital values, not just those that fall on the cash flow dates. For now, let’s assume no more capital values are known. It should be clear that

18If *t*
_{0}, *t*
_{1}, …, *t*
_{n} are equally spaced and each interval represents one unit of time (e.g., monthly cash flows with unit of time being month), then ATC reduces further to
(i.e., aggregate discounted capital values) and

This resembles closely the (excess discrete) AIRR (Magni [2010], Altshuler and Magni [2012]) except for a factor of (1 + *b*) when the benchmark rate is constantly *b* on each interval. However, the difference is noteworthy since ERTC is fundamentally a continuous AIRR rather than a discrete AIRR—that is, ERTC is an average of instantaneous returns rather than the discrete one-period returns.

**NUMERICAL EXAMPLES**

This section provides some numerical examples to illustrate the calculation of ERTC and to contrast ERTC with existing MWR metrics.

**Example 7.** Consider a portfolio of two (limited liability) investments, Investment A and Investment B, with cash flows, benchmark rates,^{5} and discount factors, as shown in Exhibit 1. Suppose the capital value of Investment A is known to be 2 on June 2, 2015. To calculate the ERTC for the portfolio, we first calculate the NPV of the portfolio, which is simply the sum of the products between the portfolio’s cash flows and the discount factors, which yields 4.11. Next, we need to calculate the ATC of the portfolio, for which we need to decompose the portfolio’s cash flow stream into its underlying indivisible cash flow streams.

According to S1, we first decompose the portfolio’s cash flow stream into Investment A’s cash flow stream and Investment B’s cash flow stream. S2 applies to Investment A only because no interim capital values are known for Investment B. As a result, Investment A is decomposed into Cash Flow Stream A1 and Cash Flow Stream A2, as in the left half of Exhibit 2. Since Investment A and Investment B are limited liability investments, S3 applies to Cash Flow Stream A1, Cash Flow Stream A2, and Cash Flow Stream B. Only Cash Flow Stream A1 has NPV turning positive before the end of the cash flow stream (the first two cash flows of Investment A1 has a NPV of 0.84 > 0), so Cash Flow Stream A1 should be further decomposed to Cash Flow Stream A11 and Cash Flow Stream A12, as in the right half of Exhibit 2. All together, we have the following four indivisible cash flow streams: A11, A12, A2 and B.

Exhibit 3 lists the discounted indivisible cash flows along with the time-scaling factor for each cash flow. The ATC for each indivisible cash flow stream is calculated as the sum of the products of the discounted cash flows (multiplied by -1) and the time-scaling factors, and the results are in the bottom row of Exhibit 3. Finally, the ATC for the portfolio is the sum of the ATCs for the indivisible cash flow streams, which is equal to 2628.08. The ERTC for the portfolio is then given by

20Since we expressed the time-scaling factors in unit of days, this should be quoted as a daily (excess) log return. To quote it as a yearly return, we simply need to express the time-scaling factors in a unit of years, which is equivalent to multiplying the earlier result by the number of days in a year. Assuming we use the 365-day convention, the yearly ERTC is 0.16% × 365 = 57.14%.

The next example contrasts ERTC with some of the existing MWR metrics, namely IRR, ICM-based AROI, and economic AIRR.

**Example 8.** Consider an individual (limited liability) investment with cash flow stream **
f
**: (-10, 30, -25, 0). We assume the benchmark rate is constantly 5%.

Since no real-valued IRR exists for **
f
**, the IRR approach clearly fails in this case.

To calculate the ERTC, we first calculate the NPV:

21Next, we calculate the ATC. Note that the first two cash flows of **
f
** has a positive NPV of 18.571, according to S3,

**should be decomposed into**

*f*We have

23So

24Therefore,

25Now we consider the ICM-based AROI and economic AIRR on this investment as well. The assumption on capital values underlying the ICM-based AROI is the same ICM-based capital growth assumption as that underlying the ERTC, except that the ICM-based AROI applies the assumption directly to **
f
** without any decomposition. This means that the ICM-based AROI implied capital values are

The second capital value being negative is inconsistent with the fact that **
f
** represents a limited liability investment, so the ICM-based AROI cannot be directly applied to this investment.

^{6}However, if

**does not represent a limited liability investment and capital values are allowed to be negative, the ICM-based AROI is calculated as**

*f*According to Magni [2010], since the aggregate capital (the denominator above) is negative, the investment should be viewed as a net borrowing.

Similarly, the economic AIRR implied capital values are

28The second capital value is also negative, so the economic AIRR cannot be directly applied to this investment as well if it is a limited liability investment. If not, the economic AIRR is calculated as

29Again, since the aggregate capital (the denominator above) is negative, the investment should be viewed as a net borrowing.

Finally, if the investment is not a limited liability investment, ERTC should be calculated differently, since S3 should not be applied. Instead, we have

30Therefore,

31and the corresponding ARTC is

**DISCRETE AIRR AND CONTINUOUS AIRR**

In this section, we discuss the relationship between the discrete AIRR (which has been the main focus of the literature) and the underresearched but more theoretically appealing continuous AIRR.

Consider the cash flow stream **
f
** of an investment. Let

*c*(

*t*) be the capital value at time

*t*with and

*c*(

*t*) = 0, and

_{n}*r*is the investment’s return during interval [

_{i}*t*

_{i-1},

*t*

_{i}] for

*i*= 1, 2, …,

*n*, then

The discrete AIRR with variable cost of capital is based on the following identity (Magni [2009, eq. (8)]):

33The excess discrete AIRR is defined as the Chisini mean (Chisini [1929]) of the one-period excess returns: (*r _{i}
* –

*b*

_{i}), i.e.,

Note that for the mean to make mathematical sense, the time intervals *t*
_{0}, *t*
_{1}, …, *t*
_{n} have to be equally spaced, which can be done by choosing some small unit of time and filling in with zero cash flows. For example, if cash flows can occur on any day of the month, we need to perform the calculation with daily cash flows and daily capital values. While one has the freedom to insert as many zero cash flows as one wishes and thus potentially perform the calculation with a higher periodicity than that of the cash flows, one must note that the resulting AIRR is specific to the periodicity chosen. For example, if we have yearly cash flows, we can calculate a yearly AIRR, or we can calculate a quarterly AIRR by inserting zero cash flows for quarter-ends that are not a year-end, or we can even calculate a monthly AIRR or daily AIRR in similar ways. However, since the quarterly AIRR uses quarter-end capital values in addition to the year-end capital values, the result can be different from the yearly AIRR. The usual rules of conversion between returns of different periodicities do not apply to AIRR. This can seem quite counterintuitive to a practitioner since he is likely accustomed to the IRR that is not dependent on the periodicity of the calculation. Still, this is not a theoretical flaw of the AIRR since it is an arithmetic return rather than a compounded return. It is, however, a practical inconvenience.

On the other hand, the quarterly AIRR can be considered as more “informative” than the yearly AIRR, since it takes into account of all the quarterly capital values in addition to the yearly capital values. Similarly, the monthly AIRR is more informative than the quarterly AIRR. With this logic, one should choose the smallest unit of time possible for the AIRR. This leads to the continuous AIRR for which the time interval is infinitesimal and hence accounts for all capital values on the interval [*t*
_{0},*t*
_{n}]. A discrete AIRR with a small unit of time can then be viewed as an approximation to the continuous AIRR.

Similar to the discrete AIRR, the continuous AIRR is based on the following continuous version of Equation (33) (Magni [2013a, remark 3.2]):

35where *r*(*t*) and *b*(*t*) are the continuously compounded return at time *t* of the investment and the benchmark, respectively, and *v*(*t*) is the discount factor between *t* and *t*
_{0}.

Similar to the above, the excess continuous AIRR is defined as

36Correspondingly, continuous AIRR is given by

37and the continuous average cost of capital (ACOC) is defined as

38which is the counterpart to the conventional PME measures. It is clear that the following relationship holds:

39**A COMBINATORICS OF CASH FLOW–BASED METRICS**

There is an interesting combinatorics of cash flow-based metrics. Note that we have two general approaches of defining the metrics: AIRR and AROI (the difference between the two is whether to use discounted or undiscounted cash flows and capital values); either can be defined with discrete returns or continuous returns (continuous AROI can be defined the same way as continuous AIRR without the discounting factors); and we have three basic types of assumptions on capital growth: ICM-based, economic, and IRR-based (i.e., constant growth). Therefore, there are 2 × 2 × 3 = 12 types of cash flow–based metrics we can define, and they encompass all the metrics that we consider in this article. For example, ERTC is the ICM-based continuous AIRR, as we will show in the appendix. Exhibit 4 shows how each metric fits in this framework.

*Remark 9*. The three capital growth assumptions represent three distinct patterns: the ICM-based assumption assumes capital values grow at the benchmark rates except for the last period, the economic assumption assumes capital values grow at the benchmark rates except for the first period, and the IRR-based assumption assumes capital values grow at a constant rate (or excess rate). One can presumably explore other assumptions on capital growth rates, such as an excess rate that is a linear function of time, each of which would lead to four new metrics. Development of such assumptions and metrics, while interesting, is beyond the scope of this article.

The two AIRR metrics that are not included in Exhibit 4 are economic continuous AIRR and ICM-based discrete AIRR. The first one has a form similar to ERTC, because of the symmetry between the ICM-based assumption and the economic assumption (see next section). The second one can be viewed as the discrete version of ERTC. We will also consider the IRR-based discrete or continuous AROI in next section. Fully exploring all 12 metrics is beyond the scope of this article.

**VALUE CONSISTENCY**

We first define what we mean by value consistency:

**Definition 10.** A cash flow-based AIRR (or AROI) metric is considered to be *value consistent* if it is both

1. value additive: for any portfolio of investments, the associated capital stream of the portfolio is the sum of the associated capital streams of the investments, and

2. consistent with known market values: for any investment, if the market value is known to be

*m*for any time_{t}*t*, the associated capital value for time*t*is also*m*_{t}.

Clearly, value consistency is a required property for any valid cash flow–based AIRR (or AROI) metric. Explicit application of S1 in calculating capital values ensures value additivity, while explicit application of S2 ensures consistency with known market values, so explicit application of S1 and S2 combined ensures value consistency. It follows that ERTC is value consistent. On the other hand, S1 and S2 are not specific to ERTC, and they can and should be explicitly applied to any cash flow–based AIRR or AROI metric to ensure value consistency. In the following, we use a simple numeric example to show that, similar to the calculation of the IRR, value additivity can be potentially violated if we apply the ICM-based or economic assumptions directly to a portfolio’s cash flows.

**Example 11.** Consider a portfolio of two partially overlapping investments, Investment A and Investment B, with cash flows shown in Exhibit 5,^{7} together with their unique real-valued IRRs. Assume the benchmark rate is 3%. The capital values associated with the IRRs are shown in Exhibit 6.^{8} The sum of the capital values of Investment A and Investment B in Period 2 or 3 are not equal to those of the portfolio.^{9} Moreover, the sum of the aggregate (discounted) capitals of Investment A and Investment B is 38.86, which is also different than the aggregate capital of the portfolio.

The capital values calculated with the ICM-based and economic assumptions are shown in Exhibit 7. For illustrative purposes, the ICM assumption and the economic assumption are directly applied to the portfolio’s cash flows to calculate the capital values on the portfolio. Again, the sum of the aggregate capital values of Investment A and Investment B are not equal to that of the portfolio in each case. A closer look at the individual capital values gives some insight to the cause of the problem. For the ICM-based assumption, the violation of value additivity occurs in Period 3. The ICM-based assumption is that the portfolio’s capital values grow at the benchmark rate of 3% (except for the last period), but that cannot be true for Period 3 since Investment A is liquidated in Period 3 with a large distribution, which implies that Investment A’s capital values grow at a much higher rate in Period 3. In other words, the portfolio’s capital value in Period 3 is understated since it does not take into account the fact that Investment A is realized with a higher-than-benchmark return. Similarly, the economic assumption is that capital values grow at the benchmark rate except for the first period (essentially the “opposite” of the ICM-based assumption), hence the portfolio’s capital value in Period 2 is also understated since it does not take into account the fact that Investment B is realized with a lower-than-benchmark return (or more specifically, a loss in this case) during Periods 2 to 4.

Clearly, in all three cases above, value additivity is preserved if the portfolio’s capital stream is simply calculated as the sum of the underlying investment’s capital streams (Magni [2013a, sec. 4]), or equivalently, S1 is explicitly applied in calculating any portfolio’s capital values. It is trivial to show that S2 is required to ensure consistency with known market values. Therefore, any cash flow-based AIRR or AROI metric should be calculated with explicit application of S1 and S2. That is, to calculate a cash flow–based AIRR or AROI of an investment (which may or may not be a portfolio), we should not directly apply whatever capital growth assumption using the investment’s cash flows, but instead use S1 and S2 to decompose the investment into its underlying “investments” (quotes added to indicate these may not be actual investments since S2 can potentially decompose an actual investment into hypothetical investments), and then apply the capital growth assumption using the underlying investment’s cash flows. The original investment is then viewed as a portfolio of these underlying investments, and its capital stream is the sum of the underlying investment’s capital streams.

Because of the additive nature of capital values and cash flows, a portfolio’s AIRR (or AROI) is a certain type of weighted mean of its underlying investment’s AIRRs (or AROIs). Magni [2013a, sec. 4] lists the various types of such weighted means for AIRRs. Two of the important results are (i) the (excess) AIRR of a portfolio is the arithmetic mean of the underlying investment’s (excess) AIRRs weighted by the investment’s aggregate capitals, discounted to the portfolio’s start time (or any other common time; what is important is that they are the same across investments to ensure additivity); and (ii) the excess AIRR of a portfolio is the harmonic mean of the underlying investment’s excess AIRRs weighted by the investment’s NPVs (again discounted to a common time). Both results are true with either discrete AIRR or continuous AIRR (Magni [2013a, remark 3.2]), so they apply to both the ERTC and economic AIRR. It is straightforward to show that two similar relationships hold for AROI: (i’) The (excess) AROI of a portfolio is the arithmetic mean of the underlying investment’s (excess) AROIs weighted by the investment’s aggregate undiscounted capitals; and (ii’) The AROI of a portfolio is the harmonic mean of the underlying investment’s AROIs weighted by the investment’s net (undiscounted) cash flows (i.e., distributions minus contributions).

The first result for either AIRR or AROI is convenient when the investment’s aggregate capitals can be easily calculated; otherwise, the second result may be more convenient, especially if the AIRR or AROI metric can be calculated without explicit calculation of the aggregate capitals for the underlying investments, such as IRR-based metrics. For example, consider the IRR-based AIRR of an investment with cash flow stream **
f
**, which is decomposed into

*m*underlying investments:

*f*^{1}, …,

*f*^{m}. When the benchmark rate is a constant

*b*, Magni [2013a, proposition 4.4] implies the following quick way of calculating the investment’s IRR-based AIRR:

**Proposition 12.**
*The investment’s IRR-based AIRR with a constant benchmark rate* b *is given by*

*where*
*is the NPV of*
**
f
**

^{k}

*discounted to time t*

_{0}.

Applying Equation (40) to the example above yields the portfolio’s IRR-based AIRR as 7.5%, which is different from the portfolio’s IRR of 7.9%.

When the benchmark rate is variable, Equation (40) still holds, but *b* should be replaced by the average cost of capital, which involves the calculation of the interim capital values. A more convenient approach is to use the IRR-based (discrete) AROI. The calculation is quite simple since it is an “internal” measure (no benchmark rates are involved). Relationship (ii’) implies

**Proposition 13.**
*The investment’s IRR-based* (*discrete*) *AROI is given by*

*where NCF stands for net* (*undiscounted*) *cash flow*.

The IRR-based continuous AROI works similarly, with IRR(**
f
**

^{k}) replaced by log(1 + IRR(

*f*^{k})), since we need to use log returns.

Alternatively, we can consider the IRR-based continuous AIRR, which is the direct alpha (DA) method of Gredil, Griffiths, and Stucke [2014]. The following is the continuous version of Proposition 12:

**Proposition 14.**
*The direct alpha of*
**
f
**

*with respect to a benchmark*

*b**is given by*

*Remark 15*. S3 of the decomposition procedure is not required to ensure value consistency. However, it should be applied to all ICM-based metrics to ensure that capital values are nonnegative in cases of limited liability investments. Because of the symmetry between the ICM-based assumption and the economic assumption, S3 can also be adapted as follows to ensure that capital values are nonnegative for all economic-based metrics:

**Step 3* (S3*):**Let*f**f**t*is the first time the reverse-NPV turns negative, if it turns negative at all. Specifically, if 43_{0}exists and

*j*_{0}> 0, then*f*The same procedure should be applied to

*f*^{1}, and so on, until there are no more reverse-NPVs turning negative before the beginning of the cash flow stream.

For example, to apply the economic AIRR to the limited liability investment in Example 8, note that the last two cash flows (-25,0) have a negative reverse-NPV of -22.676, so similar to the calculation of ERTC, we should also decompose **
f
** into

*f*^{1}and

*f*^{2}as in Cash Flow Streams (22). The economic AIRR implied capital values of

*f*^{1}is (10, 0, 0, 0), while those of

*f*^{2}is (0, 0, 25, 0), so the economic AIRR implied capital values of

**is (10, 0, 25, 0). Therefore, the economic AIRR of the investment should be calculated as**

*f*An interesting case of an economic-based metric is the economic continuous AIRR, i.e., the “economic” counterpart of ERTC. Again, because of the symmetry between the two assumptions, it is straightforward to adapt the proof of Proposition 19 to show that the economic continuous AIRR can be defined very similarly to ERTC, with S3 replaced by S3*, and Equation (4) replaced by

46Quite counterintuitively, the first cash flow of each underlying investment, , which is usually the initial contribution, drops out of the aggregate capital calculation since the time scaling factor is 0.

*Remark 16*. For IRR-based metrics, it would be interesting to see if there is a similar decomposition step (or other procedure) to ensure that capital values are nonnegative (and maybe as importantly, the IRR is unique for underlying investments). This is again beyond the scope of this article.

*Remark 17*. While the main objective of the decomposition procedure is to ensure value consistency, it helps alleviate the IRR’s other flaws as well. The cash flows of underlying investments (after decomposition) are much less likely to have multiple changes in sign than those of the original investment (especially if it is a portfolio), so the problem with multiple or no IRRs is less of an issue since we only need to apply the usual IRR calculation on the underlying investments. In cases where we do have multiple or no IRRs on an underlying investment, it is usually an indication that the assumption of capital growing at a constant rate is not appropriate for that investment, and perhaps the investment should be decomposed further. For example, it may make sense to split a venture investment into two phases where the cutoff is when the venture business starts to gain more traction, so the capital grows at 0 or a low rate during the first phase and at a higher rate during the second phase.

*Remark 18*. Technically speaking, the capital growth assumptions on the underlying investments do not have to be the same, i.e., some can be IRR-based, while others can be ICM-based or economic.^{10} What is important is that the choice of the capital growth assumption for each underlying investment is economically meaningful. That being said, one may prefer to use the same assumption across the underlying investments for the sake of consistency.

**CONCLUDING REMARKS**

Measuring the relative performance of illiquid investments is a topic of great importance to industry practitioners. Despite the well-documented flaws, the IRR is widely used for this purpose owing to the lack of viable alternatives. We present a special case of continuous AIRR, named excess return on time-scaled contributions (ERTC), that is both theoretically appealing and intuitive. It has a closed-form solution and can be easily implemented in a spreadsheet type of program, so the hurdle is low for adopting the new metric.

We also show that there are at least 12 alternative cash flow-based AIRR and AROI metrics (including ERTC) from which to choose. They can all benefit from ERTC’s decomposition procedure to ensure value consistency, i.e., value additivity and consistency with known market values. The decision on choosing a specific metric in part depends on what one can reasonably assume regarding the growth of the capital values, supported by sound economic reasoning and empirical evidence. On the other hand, if economic reasoning and empirical evidence do not lead to a clear preference, we recommend the use of ERTC for its intuitiveness, simplicity, and consistency.

One important insight from the decomposition procedure is that cash flows need to be decomposed before they can be converted to capital values (or returns), whatever the assumption is. This is especially relevant when calculating returns on a portfolio, such as a private equity fund. Just having portfolio or fund level cash flows is not enough. The cash flows on the underlying investments are needed to ensure value additivity.

Lastly, the following areas are interesting and may be worthy of future research:

• Alternative assumptions on capital growth based on cash flows,

• Properties of all cash flow–based AIRR and AROI metrics beyond what is considered in this article, and

• A decomposition step (or other procedure) similar to S3 to ensure that capital values are unique and nonnegative for IRR-based metrics.

**APPENDIX**

**ERTC AS A SPECIAL CASE OF CONTINUOUS AIRR WITH ICM-BASED CAPITAL STREAM**

In this section, we show that ERTC can be obtained as a special case of continuous AIRR with ICM-based capital stream, that is, when capital values are assumed to grow at the benchmark rate. To clarify, we apply only the ICM-type assumption on the growth of the capital stream at the indivisible cash flow stream level. Any level higher is determined by value additivity.

Let **
f
** denote an indivisible cash flow stream. A ICM-based assumption is that

*c*(

*t*) replicates the cash flow stream

**(except ) and grows with the benchmark rate until the end of**

*f***, that is, for any**

*f**t*

_{-1}=

*t*<

*t*

_{i}, we have

and

48We show that with this particular choice of capital values, the excess continuous AIRR becomes ERTC, the continuous ACOC becomes BRTC, and the continuous AIRR becomes ARTC as we defined earlier. It is sufficient to show that the aggregate discounted capital is equal to the ATC, and the numerator of Definition (38) is equal to that of Definition (12). These are the results of the following proposition and value additivity.

**Proposition 19.**
*For an indivisible cash flow stream*
**
f
**,

*with c*(

*t*)

*given as above, the following hold:*

and

50
*Proof.* Note

Similarly,

55 56By applying Equation (48) iteratively we get

57so

58Now from Equation (54) we have

59 60which is Equation (49). Similarly, from Equation (56) we have

61 62 63which is Equation (50).

It is worth noting that the third step of the decomposition procedure (if carried out) does ensure that *c*(*t*) is always nonnegative. This can be seen by noting that Equation (57) implies

Since the NPVs never turn positive before the end of the cash flow stream, *c*(*t _{i}
*) is nonnegative. Equation (47) implies that

*c*(

*t*) is always nonnegative as well.

## ENDNOTES

↵

^{1}The IRR is sometimes even viewed as synonymous with MWR, but in the context of this article, we view MWR as a general class of return measures that accounts for the impact of cash flows, as opposed to time-weighted returns (TWR) that ignore the impact of cash flows.↵

^{2}For example, the cash flow stream (-100, 390, -503, 214.5) has the following three IRRs: 10%, 30%, and 50%. Which one is the relevant rate of return for an investor (Magni [2013b, F1])?↵

^{3}For example, the cash flow stream (-10, 30, -25) has no real-valued IRR (Magni [2013b, F2]).↵

^{4}Because of the different terminology in the literature, we use the terms “asset,” “project,” “investment,” and “cash flow stream” interchangeably. Similarly, we also use the terms “benchmark rate of return,” “market rate of return,” and “cost of capital” interchangeably.↵

^{5}The benchmark rates are the benchmark’s holding period returns between the consecutive dates. For example, 3.0% is the benchmark’s holding period return between March 5, 2015, and June 2, 2015.↵

^{6}We show in a later section how ERTC’s decomposition procedure can extend application of the ICM-based AROI to limited liability investments.↵

^{7}Investment A’s operating life is from Period 1 to Period 3, while Investment B’s operating life is from Period 2 to Period 4.↵

^{8}Technically speaking, these are just one of the infinite number of possible capital streams that share the same aggregate capital (Magni [2010, sec. 4]), but these are the most natural choice that assumes capital grows constantly at the IRR rate. Moreover, the aggregate capital values violate value additivity regardless of the choice of the specific capital stream.↵

^{9}As a reminder, the aggregate capital of Investment A is calculated as 10 (1 + 3%)^{-1}+ 12.25 · (1 + 3%)^{-2}= 21.25, according to the AIRR approach. We can also aggregate capital values on an undiscounted basis as in the AROI approach, and all arguments in this example will still apply.↵

^{10}Magni [2014, footnote 8] makes a similar comment for treating different segments of cash flows between known capital values, which is our decomposition Step 2.

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