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## Abstract

In this article, the authors introduce a methodology to estimate portfolio internal rate of return (IRR) from portfolio constituents when only the IRR and the money multiple of the constituents are available. They call this methodology the reconstructed average time zero (RATZ) IRR. A deviation from the portfolio’s true IRR will remain because not all cash flow information is available. They derive an analytical expression of the difference between the RATZ IRR and the portfolio IRR. The authors provide empirical evidence about the quality of the estimator using a proprietary dataset from a large private equity investor. Furthermore, by applying the RATZ IRR, they present an overview of the portfolio construction and performance of the private equity programs of U.S. pension funds.

Within private equity, and within private markets more generally, it is uncommon to share data on the transactions and performances of funds. The performance of private equity is mainly reported in two metrics: internal rate of return (IRR) and money multiple (MM). The IRR is the discount rate that makes the net present value of the cash flows of a project equal to zero. The MM for fully realized funds is defined as the sum of all distributions divided by the sum of all capital contributions. To calculate the IRR and MM, one needs to have the underlying cash flows of a fund. However, this information is typically only available to investors who have an active relationship with a general partner (GP) or are invested in their fund.

To understand a fund’s performance, it is important that at least both the IRR and the MM are reported. If only the IRR of a fund is available, then it is not known for how long this rate of return was generated, and we therefore do not know the total return of the fund. In this case, it is also unclear exactly when capital was invested and when it was returned. Many cash flow paths could have led to the same IRR. On the other hand, if we only know the MM, we know the amount of capital that was received back for each amount invested, but we do not know for how long it was invested. Therefore, only the combination of the two metrics, MM and IRR, gives insight into the performance of a fund.

More extensive insight into the fund’s performance can be gained if all cash flows are available. Additionally, if intermediate information about the valuation of the portfolio is available, the performance through time can be analyzed. An important and well-known caveat here is that valuations may not reflect true market value, as highlighted by Jenkinson, Sousa, and Stucke [2013].

The IRR and MM are absolute performance measures, and investors naturally are interested in benchmarking the performance of their portfolios. In private equity, investors can compare the performance to other private equity investments or to the public markets. Several methodologies have been described in the literature to compare private equity performance to the public markets, the first being the index comparison method/public market equivalent (ICM/PME) methodology described by Long and Nickels [1996]. Several extensions to the methodology have since been proposed, including PME+ by Rouvinez [2003] and mPME by Cambridge Associates [2013]. Kaplan and Schoar [2005] next introduced the KS-PME, a PME ratio.

Gredil, Griffiths, and Stucke [2014] described the direct alpha methodology and extensively covered the different methodologies of PME and its extensions. KS-PME and the direct alpha are the academically preferred alternatives. A PME calculation compares the cash flows in and out of a private equity vehicle to the equivalent cash flows that could have been made alternatively, had the cash been invested in a public market. The KS-PME represents this metric in MM terms, whereas the direct alpha is an annualized rate of return.

Various private equity data providers, such as Cambridge Associates, Burgiss, and Preqin, gather the performance of private equity funds and construct benchmarks. The resulting benchmarks allow investors to compare the funds to which they consider committing, or predecessor funds of the GP, to other funds in the same vintage year, size, strategy, or region. Several studies have used the data of these providers to assess the performance of private equity funds.

Stucke [2011] identified a problem with the Venture Economics data. Funds were not updated since around 2001 but remained in the database, which likely led to a downward bias in the results from Kaplan and Schoar [2005] and Phalippou and Gottschalg [2009], who concluded that buyout fund investors earn slightly less than the public market.

Harris, Jenkinson, and Kaplan [2014] used the Burgiss database and compared private equity performance to that of public markets and found that buyout funds have consistently outperformed public markets. Additionally, they found that venture capital funds outperformed in the 1990s and underperformed in the 2000s. They also found that the performance of the funds in the Burgiss and Preqin databases show similar performance persistence from one fund to the next for the same GP.

Apart from the performance of individual funds, investors are often interested in the performance of portfolios of funds. They would like to be able to compare their portfolio performance with the performance of their peers to better understand their performance in terms of portfolio construction and fund selection. No methodology is available by which investors can compare their private equity portfolio performance against other privately held portfolios, as described by Meyer [2014]. Understanding how randomly constructed portfolios would have behaved can provide additional insights. To test the behavior of a portfolio, a substantial amount of performance data often is required. Data tend to be available in terms of IRR and MM, but less so in the form of cash flow data.

The lack of fund cash flow data poses a problem for performance measurement on a portfolio level because no consistent benchmarking is possible for portfolios of funds. We can only generate an overall IRR and MM of the portfolio if we have the cash flows of each of the underlying fund investments. The literature does not describe any methods to estimate a portfolio IRR from its constituents when cash flow data are not available for the constituents. When cash flow data are available, one can aggregate cash flows and as such calculate the cash flow of the portfolio to compute the portfolio IRR and MM.

The MM of a portfolio of funds is straightforward to calculate if we know the fractions of capital invested in each of the funds. The portfolio MM is then simply the weighted average by the amount invested of the MM of each of the underlying constituents. However, there is no way to calculate the portfolio IRR when one only knows the underlying IRRs and MMs (and the fractions of invested capital) of the funds making up the portfolio (Phalippou [2009]). Moreover, Phalippou [2009] showed that a simple (weighted) average of the underlying IRRs can lead to biased estimates of the portfolio IRR.

In general, two effects make it impossible to calculate the portfolio IRR from the IRRs and MMs of underlying portfolio constituents. The first effect is the variation in the starting times of the underlying constituents. If we have two funds in a portfolio, and fund 1 started its cash flow pattern sooner than fund 2, the effect of fund 2 on the portfolio IRR will be smaller than the effect of fund 1. Whichever two funds one combines, as the difference in starting times between them increases, the portfolio IRR will converge to the IRR of the fund that started first.

The dominance of the funds that started first is the reason why market practitioners often compare funds’ performances as if they had started at the same time. More sophisticated market practitioners also compare the deals within funds as if they had started at the same time and as such compare performance across funds. This makes sense; if one commits to a private equity fund, the limited partner does not have control over the moment the fund starts investing or which investment is done first. Therefore, a comparison of the deals and the funds as if they had started at the same time is perceived as more valuable information than the realized portfolio IRR. This method of mitigating the effect of different starting times of investments, at the fund or portfolio company level, is called *time zero*. For an extensive discussion of time-zero IRR (TZ IRR), see Kocis et al. [2009] and the references therein.

The second effect that complicates the computation of the portfolio IRR from the underlying IRRs is due to the money weighting at different moments in time. Consider another example of two funds, both now starting at the same time. If fund 1 has an investment pace of half of its committed capital per year and fund 2 has an investment pace of a quarter of its committed capital per year, the return of fund 2 will have less impact on the portfolio return. If we do not have cash flow data, this effect cannot be incorporated in an estimate of the portfolio IRR from the IRRs of the underlying fund investments. We have to accept that there is simply not enough information if we only have the fund IRRs and MMs compared to having the whole cash flow history of the underlying fund.

Being able to say something meaningful about the IRR of a portfolio of funds of which we only know the IRRs and MMs allows us to make use of a much greater pool of available data. For instance, under the Freedom of Information Act, U.S. public pension funds are obliged to share the performances of their fund investments in private equity. These and other fund performances are collected by several institutions, Preqin being one of them. Over the years, Preqin has captured the performance, in terms of IRR and MM, of several thousand private equity, real estate, and infrastructure funds. These data are mainly used for performance comparisons across individual funds; they currently cannot be used for comparison of performances of different portfolios of private equity funds.

In this article, we introduce an estimator for the portfolio TZ IRR of fund investments, of which we only know the underlying fund IRRs and MMs. We call it reconstructed average time-zero internal rate of return (RATZ IRR). We provide an analytical expression for the difference between the RATZ IRR and the TZ IRR. Moreover, using a comprehensive dataset from AlpInvest, a large private equity manager from whom we do have the cash flow data, we show that RATZ IRR estimates are very close to the TZ IRR of the portfolios. We demonstrate the accuracy of the estimator and show that our estimate reaches an R^{2} of 0.99.

Furthermore, we apply the methodology to make a comparison of the fund investment portfolios of several large pension plans in the United States. We collect the underlying private equity funds of all these pension funds’ private equity portfolios, together with their respective IRR and MM, and calculate the portfolio RATZ IRR of each pension fund to compare portfolio returns. We conclude that, by our new metric, the private equity portfolios of U.S. pension funds have performed fairly similarly.

The remainder of the article is organized as follows. In the next section, we introduce the RATZ IRR. Next, we discuss the quality of the RATZ IRR as an estimate of the portfolio IRR. We apply our estimate to the AlpInvest data and compute the estimation error empirically for various portfolios. The fourth section compares the performance of U.S. pension funds’ private equity portfolios using the RATZ approach. The final section concludes. In the Appendixes, we derive an analytical expression for the difference between the RATZ IRR and portfolio IRR.

## RATZ IRR

### Notation

Let us first introduce some notation. Define *C _{ij}* to be cash flow

*j*of fund

*i*.

*C*is negative for investments and positive for distributions. For funds that are not fully realized, we take the current net asset value (NAV) as the final distribution of the fund that takes place on the reporting date of the NAV. Let

_{ij}*t*denote the time at which cash flow

_{ij}*j*of fund

*i*occurs. Let

*r*be the log IRR of fund

_{i}*i*,

*r*= ln(1 + IRR

_{i}_{i}). Now the following net present value identity holds for each fund

*i*:

Similarly, let *r*_{p} be the log IRR of a portfolio of funds:

For each fund *i*, cash flows *j* can be partitioned into investments (*j* ∈ ) and distributions (*j* ∈ ). Using this partition, the MM of fund *i* is given by

and the MM of the portfolio is

4Finally, we define the *holding period* of a fund as the time it takes to grow to the fund’s MM at the fund’s IRR:

or,

6Note that the holding period cannot be calculated in cases when the IRR equals exactly zero (i.e., when the MM equals exactly one). For any fund investment *i* for which this holds, we will have to estimate the RATZ holding period in a different way than outlined below. In these exceptional cases, we propose estimating the holding period by the average holding period of similar funds (e.g., buyouts or venture capital) with nonzero IRRs and from the same vintage year.

### RATZ IRR

We can now create a basic cash flow reconstruction for each fund *i* within a portfolio by assuming that each fund *i* commences at the same time (i.e., time zero, or *t*_{0}). We assume each fund invests a single amount equal to the fund’s aggregated investments, . These investments are then held for a period *h _{i}* and subsequently disbursed as a single amount equal to the fund’s aggregated distributions, . We refer to this simple cash flow pattern as the reconstructed cash flow of fund

*i*.

We collect all the reconstructed cash flows of all funds and generate the reconstructed cash flow for the portfolio. We do this by adding all cash flows on the same day and as such generate one cash flow pattern for the portfolio. If we know the invested capital relative to the commitments of a fund, we can use this as input, or we can estimate the invested capital of each fund to be the committed capital within the portfolio we would like to analyze. Hence, we get one cash flow in at time zero. The total cash flow in will be the sum of all investments, that is, . This procedure also generates for each fund constituent *i* a cash flow out at time *t*_{0} + *h _{i}*. The size of the cash flow out is minus the size of the cash flow in of fund constituent

*i*times the MM of fund constituent

*i*, being , which is equal to the unobserved sum of the cash flows out of fund constituent

*i*, . We can now calculate the IRR of the reconstructed cash flows of the portfolio. We refer to this IRR as the RATZ IRR of the portfolio. Hence, the log RATZ IRR, ρ = ln(1 + IRR

_{RATZ}), is defined as

and the corresponding RATZ holding period of the portfolio *h* as

For an analytical derivation of the difference between the RATZ IRR and the portfolio IRR, we refer to Appendix A. From the appendix, it can be seen that the difference between the RATZ IRR and the portfolio IRR is expressed as a Taylor expansion that is based on the difference between the timing of the cash flows and the RATZ holding period—that is, (*t _{ij}* −

*h*)—weighted in a way that incorporates the sizes of the cash flows. Hence, in most cases, the difference will become smaller if the portfolio IRR is a TZ IRR.

In Appendix B, we show that the difference between the portfolio IRR and the RATZ IRR can be analytically expressed through the difference between the portfolio holding period and the RATZ holding period.

## QUALITY OF THE RATZ IRR

### Empirical Approach for All Vintage Years Together

To assess the quality of the RATZ methodology, we take an empirical approach by making use of an AlpInvest portfolio of more than 400 primary fund investments over the period 2000–2010. These funds are private equity funds that AlpInvest selected and managed on behalf of two large pension fund managers, APG and PGGM. We have daily cash flow data in local currency for these private equity funds.

We first split the portfolio of more than 400 funds into different subsegments by region, size, and strategy, yielding 19 subportfolios. We do this for all vintage years together. We create the portfolios using the lowest underlying level of Preqin classification to avoid including the same fund in multiple portfolios. For example, one portfolio is the European small buyout group for vintages 2000–2010. In this way, all funds are classified into different portfolios. For all 19 subportfolios, we calculate the IRR, weighted-average IRR (weighted toward the invested capital of the underlying funds), TZ IRR (using the actual cash flows), and RATZ IRR. The objective is to compare these four metrics and to assess the quality of different proxies.

To calculate RATZ IRR, we follow the methodology as explained earlier, assuming we do not have the cash flows of the private equity funds. The only information that we use is the composition of the portfolio (how much was invested by each fund), MM, IRR, and vintage year.

We also calculate the TZ IRR of the portfolio using the actual cash flows. The TZ IRR is calculated by letting the underlying funds of the selected portfolio start at the same time (i.e., the first cash flow of each fund is put on the same date). The cash flows are aggregated, and the IRR of the aggregate is calculated. Next, we calculate the weighted-average IRR of the portfolio by averaging over the IRRs, weighting the individual IRRs by the invested capital.

We compare the results with the actual IRR of each portfolio. Exhibit 1 shows the results of the difference between the portfolio IRR and the RATZ IRR, the TZ IRR and the RATZ IRR, and finally the weighted-average IRR and the portfolio IRR.

As can be seen in Exhibits 1 and 2, the RATZ methodology yields a better estimate for the portfolio IRR than averaging over IRRs. Comparing the portfolio TZ IRR with the RATZ IRR for the simulated portfolios shows that the methodology introduced in this article is a qualitatively good estimate of the TZ IRR. Although the difference between the RATZ IRR and the TZ IRR has a 0.48% standard deviation, the RATZ IRR is the best available estimator of the portfolio TZ IRR.

As can be seen in the analytical derivation of the differences in Appendix A, the differences between the RATZ IRR and the portfolio TZ IRR are dependent on the cash flow moments and cash weighting. The RATZ IRR is arguably a more objective measure for portfolio performance because fund investors do not control the variables on which the difference between TZ IRR and RATZ IRR are based. If one uses the RATZ IRR, the weighting of the funds in the performance measure is done more objectively and more consistently than with the IRR performance measure.

Exhibit 2 represents a graphical representation of the fit. The R^{2} is calculated in a simple regression for comparison of RATZ IRR versus portfolio IRR of the 19 subportfolios.

The RATZ IRR can be used as a proxy for the TZ IRR if we do not have the actual cash flows of the underlying funds or as a separate performance measure. In Exhibit 3, we compare the RATZ IRR to the TZ IRR of the 19 subportfolios that we construct. As we can see, the RATZ IRR is a good proxy of the actual TZ IRR with an R^{2} of 0.97.

We can therefore conclude that the RATZ IRR is a proxy for the IRR of a portfolio, a good estimate of the TZ IRR, and therefore a fair approximation of the performance. The RATZ IRR can also be viewed as a separate performance measure of a portfolio of investments.

### Empirical Approach, Individual Vintage Years Separated

In this section, we group the dataset of more than 400 funds to create portfolios using their lowest underlying level of Preqin classification, just as in the previous section. This time, however, we only include funds from the same vintage year in each portfolio. Therefore, the number of portfolio constituents decreases and the number of portfolios increases. We excluded the groups in which only one fund investment remains in a given vintage year for a specific Preqin classification. In this way, we now create 91 subportfolios.

If we do the same exercise for each vintage year separately, we find that the RATZ IRR estimates of the portfolio IRR are improving, as can be seen in Exhibits 4 and 5. This can be explained by the fact that if we estimate the portfolio IRR, the early investments weigh more in the total IRR than later investments. Therefore, if we maximize the difference in starting time to one year, we will find that all investments are weighted more equally in the total. Furthermore, differences in cash flow patterns stemming from macroeconomic influences are minimized if one groups fund investments from the same vintage year together.

The differences between RATZ IRR versus portfolio IRR, RATZ IRR versus TZ IRR, and RATZ IRR versus weighted-average IRR for these 91 subportfolios have the following characteristics: We observe that the estimates become more precise, and the fit of the RATZ IRR versus the TZ IRR increases from 0.97 to 0.997 R^{2} through a simple linear regression. This is represented in Exhibit 6. By comparing the fit for RATZ IRR versus the portfolio IRR, we see a significant increase in the fit of a simple linear regression as the R^{2} increases to 0.99, as represented in Exhibit 5. Therefore, the quality of the RATZ IRR estimation for the portfolio TZ IRR and the portfolio IRR for separate vintage years seems to be good.

## COMPARISON OF U.S. PUBLIC PENSION PLANS’ PRIVATE EQUITY PORTFOLIOS

### Data

Preqin requests the performance of each U.S. public pension fund’s primary fund investments on a quarterly basis. The performance figures stored by Preqin include the IRR, MM, and the distribution to paid in (DPI). The data do not include the cash flows of each of the public pension funds or each of the underlying private equity funds. We can therefore not derive the public pension fund performances from the portfolio constituents because it is not possible to average over IRRs or reconstruct the portfolio on a cash flow basis and calculate an IRR from this. Preqin does not request the performance of part of the portfolios split in vintage years, strategies, and/or sizes. Therefore, a portfolio comparison cannot be made. Studies that compare portfolio performances in private equity are scarce.

This article compares the performance and portfolio characteristics of the U.S. pension funds’ primary programs and provides an estimate of their RATZ IRR.

The literature on the performance of portfolios of private investment is very limited (see Meyer [2014]). Andonov, Hochberg, and Rauh [2016] examined governance and its relationship to the investment performance of the private equity portfolios of public pension funds. Some reports, including the Private Equity Growth Capital Council [2014] report, show the performance of U.S. pension funds on an annualized 10-year return, which does not properly measure the performance of the whole program, nor does it go into any detail on segments of the portfolio. Other industry reports, including a report by Talmor and Vasvari [2014], show the percentage allocated to private equity and the allocation to subcategories of private equity (buyout, venture, debt, or fund of funds) on an aggregated basis but do not include any performance calculations.

Under the U.S. Freedom of Information Act, U.S. public pension plans must share certain data if asked by the public. Preqin collects the performances of the U.S. pension funds’ private equity investments on a quarterly basis, as well as data delivered through different sources, cross-checking the data where possible. Preqin requests information on the size of the private equity funds and the public pension schemes’ committed capital. We use the commitment levels of the public pension funds as proxies for the invested capital.

For this exercise, we use the data provided by Preqin with a valuation date of December 31, 2014. In some cases, Preqin does not report on the level of commitment from the public pension scheme to the private equity fund. In these cases, we estimated the commitment level of the public pension scheme to be the average fund participation of the pension fund for a given vintage year (estimated based on available commitment levels for that vintage year) or, if that is bigger than the biggest fund commitment, the biggest fund commitment for that year. In this way, we fully include the public pension schemes’ portfolio constituents of the primary part of the portfolio.

Furthermore, we excluded the funds for which Preqin did not report either an IRR or an MM. In these cases, we were unable to include them in a portfolio setting to calculate a RATZ IRR because we were unable to derive the holding period.

The performance of the private equity funds in Preqin’s database is denominated in local currency. Because cash flow data are unavailable, the IRRs, MMs, and DPIs cannot be changed into one standardized currency. This is another obstacle to a fair comparison of the performance of the U.S. public pension plans. Assuming the currencies are fully hedged, one can construct the portfolio returns using local currencies. In this way, the cost of hedging is not included. For currency pairs with large interest rate differentials, this does not give an accurate view of the performance that could have been realized. Nevertheless, as can be seen from Exhibit 7, a great majority of the U.S. pension funds commitments are in North America. Most of these and some additional private equity commitments are denominated in USD. For commitments to private equity funds that are not denominated in USD, we convert the commitments to USD commitments, using the exchange rate at the end of the vintage year of the fund.

The performance of the private equity programs of U.S. pension funds might include investments in secondaries and co-investments. We only use the data of the primary funds program of the U.S. public pension funds. We therefore do not have a complete overview of each public pension plan’s private equity program because we have not included the co-investment, direct investments, and secondaries. Because these parts of the portfolio can be significant contributors to a private equity program, we cannot conclude anything about the performance of the total program of each of the public pension funds; we can only make conclusions about the performance, portfolio construction, and selection skills of the primary part of the private equity programs. The U.S. pension funds selected for this study are the 16 plans with the largest private equity exposure in NAV terms.

Because we have too little fund investment data available in the given vintage years, we were unable to reconstruct the primary fund investment program of the Massachusetts Pension Reserves Investment Management Board to a sufficiently granular level to include their program in the analysis. For the funds in which Virginia RS is invested, Preqin does not publish the regional split.

The amounts in Exhibit 8 represent the number of private equity funds committed to by the pension plans (as defined by Preqin, so not including infrastructure, timber, natural resources, or real estate) from the period 2000–2010. We choose the period 2000–2010 to have a representative period in which the portfolio would be sufficiently mature and out of the J curve.

In Exhibit 7, one can see the total USD committed to the private equity funds made by the public pension plans. Furthermore, Preqin has information stored in its database on a large part of the private equity funds on which it reports. The characteristics of these private equity funds include, among other things, fund sizes, gross invested capital, DPI, total value to paid in (which is equal to MM), net IRR, vintage years, fund strategy, regional focus, the fund sector focus, and so on. From this, Preqin can also classify the funds in terms of size and regional split.

### RATZ IRR of U.S. Public Pension Schemes

In this section, we present the performance of the primary part of U.S. pension funds’ private equity portfolio in terms of RATZ IRR. We also present the performance for the different subsegments and regional split.

In Exhibit 9, we present the MM of the different portfolios. In Exhibit 10, we show the results for the RATZ IRR of the portfolios as presented. These RATZ calculations include the 2000–2010 vintage years in one RATZ IRR. In the last line of Exhibits 9 to 11, we show the simple average and the median of each of the columns in the bottom line.

We made the same calculations over the vintage years 2000–2010, but now we have calculated the RATZ IRR in such a way as to reconstruct the cash flows so that the vintage year 2001 started one calendar year later than the 2000 vintage and so on. In this way, we generate a better approximation of the portfolio IRR observed by the pension funds. The difference is the time-zero effect of the vintage years. The results are shown in Exhibit 11.

Through the RATZ IRR, we can also calculate return attributions. We could, for example, equalize all commitments to see the effect of the allocation decisions, or we could equal the commitment sizes to each vintage year to see the effect of the difference in allocations made through the vintage years.

## CONCLUSIONS ON THE PERFORMANCE OF THE U.S. PUBLIC PENSION SCHEMES

Serious concerns over transparency in private equity have been expressed over the past years by practitioners and regulators across the globe. This article contributes to further transparency in the industry by comparing the construction and performance of private equity portfolios of U.S. pension plans. Our methodology allows independent reviewers to assess the quality of the programs in more detail by making use of publicly available information.

With the outcome of the calculations, we can make the following observations on the performance of the U.S. public pension schemes. The dispersion in the performance is low overall, as can be seen in Exhibit 11, with a difference between top and bottom performer of less than 4%. This could be explained by the fact that we look at overall large portfolios with relatively similar sizes of allocations through vintage years, regional diversification, and different styles.

The results presented in Exhibit 10 show less dispersion than the results in Exhibit 11. The difference of the dispersion of the IRRs therefore must come from a higher dispersion in the early vintage years of the programs. This indicates that the differences in return for the U.S. public pension funds over the years might be lower than observed when looking at the IRR of the portfolios in total because part of the difference might stem from deviating weights among the vintage years in the program. It could also be the case that deviations in private equity fund returns have decreased over time.

Buyout and growth capital seem to have positively contributed to the overall return, whereas venture in most cases has contributed negatively.

The majority of the programs have invested most of the capital in buyout. Growth capital allocation has been small. Venture allocation in general was larger than the growth capital allocation, but the average commitment size for venture capital was much lower than for buyout. There is also a significant overlap in the portfolios because several private equity funds have been selected by several pension funds.

In terms of regional allocation, the majority of the capital has been invested in the United States. Furthermore, the performance of the U.S. allocation of the pension schemes seems to be less dispersed, suggesting a higher level of diversification for each of the pension plans. The allocation to European funds is somewhat lower, and the performance for the different pension schemes seems more dispersed, suggesting a lower level of diversification, which can also be concluded from the lower level of funds committed to in Europe. The overall performance of the European funds seems to be in line with the U.S. funds.

For Asia, this is different. The return generated by the pension schemes from commitments made to Asian private equity funds seems to be lagging the European and U.S. returns. Furthermore, the allocation to Asia was much lower than to Europe, which was already lower than the allocation to U.S. funds.

In a recent study, Gottschalg [2016] analyzed the performance of the private equity portfolio of CalPERS and found that the private equity program has been very profitable. In relation to the programs of the other U.S. pension funds, we find that the CalPERS program is, nevertheless, not one of the top performers.

### Future Work and Possible Applications

In this article, we have established the RATZ methodology as an estimator for portfolio IRR and as a performance measure. We have advocated the use of TZ IRR for performance measurement. Because performance data for which only IRR and MM are known are more widely available than cash flow data and because the RATZ IRR is simple to calculate, the methodology opens the door to multiple new applications.

Among the many applications one could consider, we would like to highlight a few. Because we now possess the ability to construct a portfolio return metric from the IRR and MM of its constituents, we can leave out part of the constituents and calculate portfolio attributions for each (group of) constituents. For example, if we leave out all venture capital investments of a private equity portfolio and calculate the difference of the RATZ IRR of the portfolio with and without venture capital funds, we can find the return attribution of venture capital in the portfolio. Another example is to set all commitment levels of each fund to one and calculate the RATZ IRR; comparing this to the RATZ IRR of the portfolio would give an indication of the contribution to the commitment level of decisions made during portfolio construction.

Another class of applications could be benchmarking. If one wants to judge a portfolio outcome, it is now possible to construct randomly a similar portfolio out of benchmark data. Because this might be possible several times (depending on data availability) in a randomized setting, it is now possible to compare the observed outcome within the distribution of randomly generated outcomes. In this way, one can customize the benchmark to the funds selected.

The RATZ technique can also be helpful for portfolio construction analysis. The technique allows for the calculation of diversification effects. By randomly constructing portfolios including different numbers of constituents, it is possible to calculate the reduction in the variability of the portfolio when adding one more portfolio constituent, therefore calculating the diversifying effect of the addition to the portfolio. Hence, the technique allows for the calculation of the diversification effects. In a similar fashion, the correlations between classes of funds (e.g., venture versus buyout or mezzanine) can be calculated using RATZ.

Because the RATZ methodology now allows for a general way to construct a portfolio return measure from its constituents, many more applications might be conceptualized. Because the performance data for private equity funds reported with only IRR and MM are readily available through several sources and because the RATZ methodology is simple to calculate, the applications could be plentiful.

## APPENDIX A

### DIFFERENCE BETWEEN RATZ IRR AND PORTFOLIO IRR

If we wish to express the difference between our RATZ IRR estimate and the portfolio IRR, we can write Equation (2) in the following way:

A-1Now setting Equation (8) equal to Equation (A-1), we get

A-2Now, write down a Taylor series expansion of the right hand side of this equation around *t _{ij}* =

*h*:

The first term of this sum is

A-4Therefore, the total expression becomes

If we multiply both sides by , we get

After some manipulation, we can write the same expression as follows:

We can rewrite this to be

A-5where weights *w*_{ij} are defined as

If we take the log of this and divide by *h*, we get the difference between ρ and *r*_{p}.

## APPENDIX B

### HOLDING PERIOD

We can define the portfolio holding period *h*_{p} as follows:

Given this definition and previous observations, we know the following:

Therefore, the problem of estimating the expected value of the difference between the RATZ IRR and the portfolio IRR is the same as estimating the difference between the ρ and *h _{p}*. The first cash flow will be a capital call, so

*t*

_{i}_{1}= 0 ∀

*i*. We know or have estimated the sum of each fund’s invested and distributed capital separately. We therefore know

*h*for each fund

_{i}*i*, except for funds with IRR equal to zero. We also know Equation (A-5) equals 1 for

*h*=

*h*

_{p}.

## Footnotes

**Disclaimer**The views expressed in this article are the authors’ and do not necessarily reflect those of APG Asset Management or PGGM.

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