Nowcasting Net Asset Values: The Case of Private Equity

Abstract

Valuing illiquid assets accurately is hard but necessary for many critical investment decisions made by institutional investors.¹ Private equity (PE) investments are a prime example. In most cases, secondary markets are undeveloped for private equity and likely reflect the marginal utility to trade of an unrepresentative investor.² Instead, various stakeholders rely heavily on infrequently reported net asset values (NAVs) provided by fund managers. However, investors observe additional information that can be used to generate unbiased and more timely higher-frequency estimates of the true value of a fund. This paper develops a method of how to use other types of data jointly with reported NAVs in a unified framework to (i) learn about risk, return, and reporting quality characteristics of individual funds, and (ii) to estimate unbiased asset values at the relevant data arrival frequency (in our case weekly),—that is, to nowcast PE fund “true” NAVs. Nowcasting is the prediction of the present, the very near future, and the very recent past. We expand on the methods used in the literature, as nowcasting NAVs is more complex than macroeconomic variables like GDP growth (the most studied example). In particular, we show how (i) comparable asset returns and (ii) cash flows between fund limited partners (LPs) and general partners (GPs) can be used to generate nowcasts. Moreover, our method can potentially also be expanded to include other types of information, including characteristics of individual assets held by funds, comparable private transactions, and occasional secondary trades.

We model “true” asset values of a fund and, correspondingly, “true” returns as the latent factor in a sparse-data state space model (SSM) estimated at (i) high frequencies (relative to quarterly fund reporting) and (ii) at an individual PE fund level. The novelty of our approach rests on two key pillars. One is that we use the individual fund-level data on cash flows and reported NAVs simultaneously to identify how NAVs are “smoothed” in order to estimate the fund’s systematic and idiosyncratic risk exposures. This is different from existing work featuring the complete fund-level data that either (i) relies on reported NAVs and attempts to remove autocorrelation induced by smoothing via a distributed lag market-model (Woodward and Hall 2004; Ewens et al. 2013; Goetzmann et al. 2018), or (ii) disregards the NAV information and relies solely on funds’ realized cash flow data (see, for example, Driessen, Lin, and Phalippou 2012; Franzoni, Nowak, and Phalippou 2012; Buchner and Stucke 2014; Ang et al. 2018). The second pillar is that we unsmooth weekly cumulative returns (rather than quarterly returns; see also Ang 2014). This is more consistent with the microfoundations of the smoothing bias whereby appraisers look back for relevant events within the past quarter to arrive at their valuation assessment. We show that NAV smoothing intensity persists by fund manager, as does the risk-taking.

We evaluate the performance of our method not only via simulated data but also using real PE fund data from Burgiss. In particular, we examine out-of-sample realized cash flows versus the levels implied by the model for 2,513 separate funds. For 69|$\%$| to 89|$\%$| (depending on the metric) of these funds, the model outperforms a naïve, but commonly used, approach whereby reported NAVs are extrapolated using cash flows and returns of a comparable public asset. With partial imputation of auxiliary parameters using peer funds, we can estimate fund-level SSMs for 3,911 funds (over 95|$\%$| of the sample) with a 74–94|$\%$| improvement rate relative to the naïve approach (again, depending on the metric). Peer funds are defined as having the same strategy and industry, a vintage within one year, and the same size tercile as the estimand fund.

Our performance assessment metrics are based on the idea that if the fund’s true returns were observed and used to discount fund cash flows (and the true net asset values), the fund to-date public market equivalents (PMEs; Kaplan and Schoar 2005) computed using these series would be equal to one. Furthermore, we show that, given fund-level SSM estimates, the in-sample performance is a good predictor of the out-of-sample performance, yielding useful criteria for model selection. Using our estimate of fund NAVs reduces the out-of-sample forecast error of subsequent cash flows by 59–65|$\%$| relative to the as-reported NAVs of a typical fund and generates filtered weekly returns with near-zero autocorrelation and realistic standard deviations (e.g., about 29|$\%$| per year for a typical buyout fund and 34|$\%$| for a typical venture fund).

In addition to our methodological contributions, we generate important new findings about fund-level risk and return. For example, we find fund-level systematic risk (beta) to be lower than many other recent studies. The average is around 1.0 for buyout funds and 1.40 for venture funds, as opposed to the pooled estimates of 1.25 and 1.80, respectively, in Ang et al. (2018), even though we use their estimates to set our profile grid for betas. Part of this difference results from the fact that most studies do not actually estimate the market beta of a fund over its lifetime but rather the beta of the panel of time-aggregated fund cash flows. The distinction is important when cash flows from funds are clustered in time and contribute greater weight to the panel (e.g., venture funds in the late 1990s). Panel-based estimates generally will not reflect the beta of an average fund in this case. We provide fund cross-section and time-series index analyses to support this claim. We also show that “nudging” the estimation to feature higher systematic risk exposures, as well as ignoring the heterogeneity in other parameters in the cross-section of funds, results in inferior nowcasting performance in-sample and out-of-sample. In other words, higher systematic risk assumptions are less consistent with the fund-level cash flow realizations.

We find notable variation in systematic and idiosyncratic risk exposures across vintage years. For example, we estimate betas in the vicinity of 2.0 for half of venture funds incepted during the late 1990s, but three-quarters of venture funds incepted during 2000–2006 have beta estimates of less than 1.5. We see an increase in systematic risk-taking by venture funds incepted after 2009. We also observe meaningful trends for the level of idiosyncratic risk of venture funds (which we measure as a multiple of volatility of a matched public benchmark). For example, the risk multiple dipped to twice the benchmark level after the dot-com bust before growing to three times the benchmark level around the global financial crisis and then reversing course again for 2012–2014 vintages. For buyout funds, we note a decrease in systematic risk relative to the levels in the early 1990s and a spike in idiosyncratic risk around 2003–2005. Finally, we note that the trends in PE fund abnormal returns are consistent with those documented in prior literature.

Our paper contributes to recent studies that develop “matrix pricing” for portfolios of PE funds using either secondary market transactions coupled with a selection model as in Boyer et al. (2018) or a replication of PE fund returns using holdings matched to comparable public firms as in Stafford (2022). Our methodology effectively nests both approaches while allowing for fund-level heterogeneity in risk exposures and taking a more structural approach to modeling the fund return process during the periods in which the fund-related data are missing. We also complement contemporaneous work by Gupta and Van Nieuwerburgh (2021), who use machine-learning tools to estimate multi-factor time-varying exposures from a panel of PE fund cash flows as opposed to the bottom-up approach that matches on portfolio company characteristics. Their approach can be used to construct the comparable asset in our approach. Our method also supports using round-level valuations once they are adjusted for differences in the contractual rights, as shown by Gornall and Strebulaev (2020, 2021).

As one of many potential applications of our fund-level approach, we demonstrate how asset allocation decisions using nowcasted NAVs would have been improved around the 2008 financial crisis (Section 4.1). We argue this case study to be instructive for the valuation challenges that PE fund investors face during significant swings in market valuations.

1. Intuition on Methodology

In this section, we illustrate the need for nowcasting in PE fund investing and provide an intuitive overview of our methodology.³ Consider an investor who manages a portfolio that includes a specific PE fund. On an ongoing basis the investor needs to make portfolio decisions that rely, in part, on estimates of the PE fund’s value (and potentially other characteristics like risk, etc.). Because the fund is not publicly traded, the investor’s inferences about fund value are confounded by low-frequency reporting (e.g., quarterly) with long delays (4–20 weeks) as well as appraisal errors. For example, suppose the investor seeks to evaluate her overall portfolio situation on March 31, 2020, amid the broad equity indicies being down 23|$\%$| year-to-date. The latest NAV report from the fund arrived in early February and valued the investor’s stake at $100 million as of December 31, 2019. Furthermore, the fund made a distribution on February 7, 2020, of $15 million. This case underscores a typical problem that PE fund investors face—the latest available NAV reports from PE managers are biased and too outdated for real-time portfolio reporting or analysis. Therefore, even if unwittingly, PE fund investors will engage in some sort of nowcasting of the PE fund NAVs.

We use a simulated fund to highlight the contributions of our paper. Panel A of Figure 1 shows a few reported NAVs (dots), cash distributions (bars), and unobservable “true” NAVs (line) for a hypothetical fund around its ninth year of existence. We note some important features of the fund. First, distributions occur periodically and not necessarily at quarter-ends when NAVs are reported. Accordingly, the true fund NAVs drop by the amount of the distribution on the distribution date. Second, the reported NAVs, in general, do not correspond to the true NAVs on the reporting dates and instead will be related to some average of stale portfolio asset appraisals, as well as some mean-zero valuation errors unrelated to past or future returns (henceforth, “reporting noise”).

Figure 1

PE fund’s asset values at weekly frequency

Panels A though C plot a hypothetical PE fund’s distributions; “true” NAVs (line); reported NAVs (dots) in weeks 455, 468, 481, 494, and 507; cash distributions (bars) in weeks 474, 489, 497, and 505; naïve nowcasts (dotted lines in panel B) and SSM-based nowcasts (dotted lines in panel C). Panel D plots |$-1$| plus the ratio of the nowcasts to true NAVs.

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1.1 Naïve nowcasting benchmark

One common approach by practitioners, which we call naïve nowcasting, is to “grow” the latest NAV report using a public market rate of return and subtract (add) the value of distributions (contributions) as they occur. So for our example, the market was up by 4|$\%$| through February 7. Thus, $15 million distribution would be subtracted from $104 million, and the new estimated NAV of $89 million would be multiplied by |$(1-0.23)/(1+0.04)$| to obtain a March 31 nowcasted NAV of $65.9 million.

Panel B of Figure 1 shows what the naïve nowcasts would look like for this fund. The higher-frequency values always tie back to the reported NAVs (which are stale) and evolve according to the return pattern of the comparable asset. In practice, naïve nowcasting is even more challenging given reported NAV values are only observed with substantial delay. Therefore, in practice, the naïve nowcasts continue to depend on the NAV two quarters prior for quite a while after the end of the quarter. While intuitive and simple to calculate, the naïve nowcasting method makes the following strong assumptions:

• A1. The NAV report reflects the true asset value (or at least an unbiased estimate).

• A2. The returns of the benchmark fully describe those of the fund, and therefore the systematic risk of the fund equals that of the selected public benchmark.

Finally, in the Figure 1 example, the reported NAVs happened to be not particularly far from the true values. More specifically, the root mean squared error (RMSE) for the naïve nowcast in the illustrative example is 0.331 (on average during all weeks 455 through 507).

1.2 State space model nowcasting

The core intuition behind our SSM-based approach proposed in this paper is related to that of the naïve nowcast—namely, the asset value nowcast represents a “blend” of fund NAV reports, cash flow realizations, and comparable public asset returns. However, the blend that we propose provides more realistic and unbiased estimates and is an optimal combination of the available information. Several aspects of our method are novel and important:

• S1. The fund cash flow and NAV reports are used jointly as distinct data points.

• S2. We unsmooth asset values (as opposed to periodic returns) at a higher frequency and allow for a time-varying bias.

• S3. There is no pooling of fund-specific series but instead a partial imputation of hard-to-identify parameters from peer funds with better data.

Unlike the literature cited in the introduction, S1 says that we take the fund’s distributions and NAV reports as distinct observations, recognizing they each separately contain relevant information. The fundamental intuition behind our approach is as follows: While NAVs are informative, they are constructed with error using stale data and therefore are predictably biased estimates of the true fund value. On the other hand, distributions, because they represent cash flows from actual transactions, provide accurate information on the true value but represent only part of total value. Consequently, jointly observing distributions and NAVs, allows for the identification of fund cumulative return paths (and hence the asset values) via a mapping of cash flows to reported NAVs. So, for example, suppose an investor knew portfolio-level marks right before a distribution was made. In this case, it is possible to compute that distribution’s fraction of the unsmoothed portfolio value. Comparing this to the reported NAVs identifies the smoothing function and the residual bias variance, and hence produces the unsmoothed path of returns. Consequently, S1 is fundamentally different from A1, which assumes unbaised NAVs.

Turning to S2, we note that a number of papers have proposed methods to unsmooth returns (see, e.g., Geltner 1991, 1993; Getmansky, Lo, and Makarov 2004; and Couts, Gonçalves, and Rossi 2020). These methods rely on fixed-coefficient moving average representations of the observed return process and therefore do not take into account that the smoothing of PE returns changes through time. Our method produces, among other things, unsmoothed periodic returns as well, but we model the smoothing of reported NAVs rather than the smoothing of periodic returns. This is an important distinction that allows us to interpret the reported NAVs as a weighted average of valuation snapshots taken at different points in the past. Most of the weight will come from weeks within the past quarter (rather than from values at the end of previous quarters), corresponding to a realistic assumption about how the actual appraisal of fund assets is performed. We examine weekly valuations, which we find is a good balance between a desire for higher-frquency valuations and computational complexity.

The first equation determines how |$\bar{r}_{0:t}$|⁠, the smoothed log returns from inception until |$t$| (that the reported NAVs map to), relate to the unsmoothed latent log returns |$r_{0:t}$| over the same time span. Thinking for the moment in terms of a constant |$\lambda\in(.01,0.99)$|⁠, the first equation represents an exponentially weighted moving-average scheme. The closer |$\lambda$| is to one, the more stale are the reported NAVs. Conversely, values closer to zero erase the weight of past returns, and the two series, |$\bar{r}_{0:t}$| and |${r}_{0:t},$| become virtually indistinguishable. Note that |$\lambda$| in Equation (1) is actually not a constant as |$\lambda(\cdot)_t$| pertains to |$\lambda$| at time |$t,$| where we use lower-case letters followed by |$(\cdot)$| to refer to a scalar-valued function involving data. As follows from the second equation, in this case the data are |$w_t\in[0,1]$|⁠, which indicate the fraction that the cash flow in week |$t$| comprises in the naïve nowcasts of fund asset values. Therefore, the weeks in which a fund makes new investments or distributes capital back to fund investors, receive higher weight (i.e., are “remembered more”) in the subsequent NAV reports. Two specific procedures advocated by the American Institute of Certified Public Accountants, namely “Backtesting” and “Calibration” (as previously referenced in footnote 6) provide a microfoundation for this modeling choice, which we find to notably improve the nowcasting performance. The fund example appearing in Figure 1 has |$\lambda$| = 0.9, yet the weight on the past returns for week 489 is only 0.524 when the distributions comprise 72|$\%$| of the naïve NAV nowcast corresponding to |$w_t=0.4178$|⁠.⁴

1.3 Comparing the naïve and state space model nowcasts

We also examine the errors for both methods, namely the difference between the true NAVs and, respectively, naïve and SSM nowcasts scaled by the true NAVs. As evident from panel D, at both aforementioned jump occurrences in weeks 481 and 494, the SSM nowcasts move closer to the true values, while exhibiting 53|$\%$| smaller RMSE right before the NAV reports across the weeks plotted. Similarly with distributions, SSM-based nowcasts jump more (e.g., week 474) or less (e.g., week 497) than the amount distributed, depending on how the distributions compare with the previous nowcast, given the expected fraction of assets that the distributions represent.

To better understand why our method is superior to the naïve method we further explore the implications of Assumptions A1–A2. For example, Assumption A1 is particularly strong in light of the well-documented smoothness of appraisal-based values. In PE settings the appraisals are normally derived from comparable transactions over previous months, historical accounting data, and so on, that are by their nature stale.⁶ The lags built into the process result in quarterly NAV estimates which are both predictably stale and systematically biased.

Recall that the RMSE for the naïve nowcast in the example is 0.331 and that of our SSM-based nowcasts is 0.165, which is a 50|$\%$| reduction. From Assumptions A1–A2 we can think of a decomposition along the following lines: (i) the part of the RMSE attributable to the use of as-reported NAVs and (ii) the part of RMSE attributable to the use of comparable asset returns. Let us start with (i) and examine what happens if we use SSM-based NAVs, which means we correct the biases in the NAVs, but maintain the comparable asset returns. This yields an RMSE of 0.292, as opposed to 0.331. Regarding (ii), if we use SSM-based returns instead of comparable asset ones, the RMSE is 0.233, which is closer to 0.165 of the fully SSM-based nowcasts. About 90|$\%$| of this decrease in RMSE is driven by incorporating signals about the idiosyncratic return of the fund relative to the comparable asset (i.e., relaxing Assumption A2), inferred from NAV reports and distributions. Finally, the market |$\beta$| of the fund in Figure 1 is 1.19, while it is 1.0 for the comparable asset, which also contributes to the reduction in RMSE (again relaxing Assumption A2).

1.4 Dealing with real data

Since in real data the true asset values are not observed, we need a method for comparing the quality of our SSM-based nowcasts to the naïve ones. We propose nowcasting performance metrics that (i) take advantage of the fact that we do not need the full history of fund cash flows to estimate the SSM parameters, and (ii) operationalize the intuition that, if discounted at the true returns, fund cash flows must have zero net present value. Put differently, the closer given return series are to the true fund returns, the closer the PME computed with these series should be to 1.0 (as opposed to the market index return series applied in a standard PME calculation). The same logic applies for to-date PMEs (i.e., whereby the last reported NAV value is treated as a terminal distribution in the calculation) where biases in reported NAVs contribute positively to the variation in PME across periods.

The first estimation quality metric we compute is simply the mean squared deviation from 1.0 of the to-date PME computed quarterly for the remainder of a fund’s life. For naïve nowcasts, we use the matched benchmark and the reported NAVs. For SSM nowcasts, we use the model-based return and NAV estimates. Even though we consider to-date PME values only after the parameter estimation window, each PME value utilizes the history of fund cash flows that occurred since fund inception. Hence, we refer to this as a partially out-of-sample metric (henceforth, Hybrid) and argue that it is a sensible metric for unresolved funds. For a fully out-of-sample metric (henceforth, OOS), we compute PMEs for the last period of each fund’s life, as if the fund started right after the estimation period and the first subsequent NAV (nowcasted for SSM and as reported for naïve) was the only capital call made by the fund. Our simulations suggest that both metrics adequately characterize the nowcasting performance and that there are substantial efficiency improvements from using SSM. For the example in Figure 1, the OOS errors for SSM and naïve nowcasts are, respectively, 0.226 and 0.626.

In addition to the Hybrid and OOS RMSE statistics, we examine the nowcasted return series’ properties—for example, correlations, risk loadings in a linear model, and so on. We show that, unlike the naïve nowcasts, SSM-based returns do not obscure risk exposures and that observing autocorrelations (even at a high frequency) close to zero in a nowcasted series does not ensure correct inference about the funds’ risk.

2. The State Space Model of a Private Equity Fund

This section covers the technical details of the SSM representation of a PE fund, with some supporting material appearing in the Internet Appendix, Sections A.1 through A.4. We start with a summary of the variables in our model listed in Table 1. There are three panels, separating the latent variables from the observed high-frequency series and finally the infrequently observed ones. Recall from the previous section that we use capital letters to denote the levels of variables (e.g., |$D_t$| for distribution amount) and lowercase ones for the logarithm thereof.

The variables in the last panel are observed infrequently. They feature the same time index |$t,$| that is, weekly, with |$NaN$| entries for the missing observations and numerical values during the weeks observations take place. We note that |$R_{ct}$|⁠, that is, the average gross return on publicly traded assets comparable to the fund in week |$t,$| is assumed to be scalar, meaning that the relevant asset return is a single index. Accordingly, we use a GARCH(1,1)-filtered conditional variance for the idiosyncratic returns of |$R_{ct}$| (from a linear projection on |$R_{mt}$|⁠) as a proxy for |$h_t$|⁠.⁷

2.1 Model structure and parameters

Lower-case letters followed by |$(\cdot)$| refer to a scalar-valued function, for example, involving data, such as |$\lambda(\cdot)_t$| pertaining to |$\lambda$| at time |$t$| as specified in Equation (9). Note that |$b$| is the OLS slope of the projection of |$R_{ct}$| on |$R_{m}$| at weekly frequency. As with |$h_t$|⁠, we consider |$b$| as inputs to the model (i.e., estimated separately). Table 2 summarizes parameters in our model that collectively are referred to as |$\theta$|⁠.

We are agnostic about the economic interpretation of the parameter |$\alpha.$| For example, it could be compensation for the liquidity risk incurred by the fund investors or excess returns after appropriate assessment of all relevant risks.⁸ Also, since Equation (5) features log returns, to have an arithmetic CAPM |$\alpha$| interpretation, the estimate needs to be adjusted for the returns’ variance (see, e.g., Korteweg and Sorensen 2010; Driessen et al. 2012). Accordingly, we add |$0.5(\tilde{\sigma}^2-\tilde{\beta}(\tilde{\beta}-1)\tilde{\sigma_m}^2)$|⁠, where |$\tilde{\sigma}$|⁠, |$\tilde{\beta}$| and |$\tilde{\sigma_m}^2$| are OLS estimated from SSM-filtered weekly fund returns projected onto the market. Note that from Equation (5), |$\tilde{\beta}$| is by construction asymptotically equal to |$\beta.$|

The economic interpretations for |$\delta$| and |$\sigma_d$| are, respectively, the trend and the noise of the distribution density. Funds with higher |$\delta$| will tend to return capital faster. Overall there is relatively little insight that can be learned from (⁠|$\delta,\,\sigma_d$|⁠), though. These are auxiliary parameters relating distributions—which reveal the true unsmoothed value of the fund portfolio—and the reported NAVs, therefore allowing for inference about |$\lambda$| and |$\sigma_n$|⁠.⁹ Finally, applying log transformations to Equations (5) through (12), except for (9), which is already in log form, yields a state-space model as detailed in the Internet Appendix, Section A.2.

2.2 Other estimands of interest

Given the data observed for week |$t=1,...,T$| and parameters |$\theta$|⁠, we can apply the Kalman filter to obtain estimates of fund returns for each week. We then apply the mapping function |$M_t$| in equation (8) to obtain the estimate of the fund asset values. We use the notation |$\hat{R}_t,$||$\hat{R}_{\tau:t},$| and |$\hat{V}_{t}.$| Although these estimates are functions of a particular |$\theta$| and the ending period |$T,$| we drop these two arguments to simplify the notation. For each fund and |$\theta$| of interest, we compute:

• (i) Variances and autocorrelations of the filtered weekly returns between |$t = 1$| and |$T$| using the standard estimators at weekly and quarterly frequency.

• (ii) |$\text{PME}_{0:t}$| that are the Kaplan and Schoar PMEs on a to-date basis that utilize filtered returns and asset values along with the complete history of fund cash flows realized up to period |$t$|⁠.

• (iii) |$\text{PME}_{\tau:t}$| that are PMEs on the to-date basis in which capital calls up to period |$\tau<t$|⁠, are replaced with |$\hat{V}_{\tau}$| (using |$\theta$| with data up to |$\tau$|⁠) and |$\hat{R}_{0:t}$| are replaced with |$\hat{R}_{\tau:t}$|⁠.

While implausibly high (or low) variance and autocorrelations of filtered return in (i) provide straightforward diagnostics of the model misspecification and the sensitivity to different parameterization, the measures in (ii) and (iii) are the building blocks to appraise nowcasting performance. As discussed in Section 1, these metrics are based on the simple idea that the NPV of cash flows will be zero if true fund returns are used for discounting. Specifically, we define the following three metrics, so that each can be interpreted as a fund-specific pricing error:

1. In-sample RMSE: mean squared difference of (⁠|$\text{PME}_{0:t}-1$|⁠) over the period |$t=\tau_0,...,\tau$| where |$\tau_0$| and |$\tau$| are within the span of data used to estimate |$\theta$|⁠;

2. Out-of-sample (OOS) RMSE: mean squared deviation of (⁠|$\text{PME}_{\tau:T} -1$|⁠) such that no fund-specific data beyond week |$\tau-1$| is used to estimate |$\theta$|⁠;

3. Hybrid RMSE: mean squared deviation of (⁠|$\text{PME}_{0:t} -1$|⁠) over periods |$t=\tau,...,T$| such that no fund-specific data beyond week |$\tau-1$| is used to estimate |$\theta$|⁠. It is a hybrid between in-sample and out-of-sample data because, even though it utilizes the out-of-sample NAVs only, all since-inception cash flows are included.

2.3 Parameter estimation

We estimate the parameter vector |$\theta$| for each fund using a combination of two methods: (i) maximum likelihood (ML) and (ii) partial imputation. In both, we utilize a penalty function in the spirit of Ridge estimators, and an iterative procedure in the spirit of the expectation maximization (EM) algorithm (Dempster, Laird, and Rubin 1977).

Our method simultaneously exploits cash flow and reported NAV information. It allows us to obtain fund-specific estimates of the parameter vector for a majority of funds independent of information from other funds. Nevertheless, the span of distributions and innovations in NAV reports for many funds are simply insufficient to identify all parameters independently. In such cases we resort to imputation. Another reason for adopting this strategy is that some parameters might not vary much across fund groups, so there are efficiency gains from imputing parameters across peer funds or using estimates reported in the existing literature. We also find that a partial imputation approach for parameters dominates the pooling of the data across funds.

In the following subsections we describe key steps and features of the two approaches we consider: (i) fund-specific estimates and (ii) partially imputed methods. In both, we obtain standard error estimates for every parameter in |$\theta$| using the numerical Hessian method as in Miranda and Fackler (2004). We conclude the section with a synopsis of Monte Carlo simulation results that illustrate parameter estimation efficiency and nowcasting performance across different degrees of model misspecification and data quality. In the Internet Appendix, Section A.4, we provide further technical details regarding the estimation procedure.

The important feature of PE fund data is that distributions almost never occur on the day of NAV reports. This fact allows us to identify |$\sigma_d$| from |$\sigma_n$|—the nonpersistent noise parameter on NAV reports. Meanwhile, the sparsity and irregularity in distributions and NAVs mitigates the effect of the autocorrelation in the observed residuals that arises due to the possible misspecification of both |$\delta(\cdot)_t$| and |$\lambda(\cdot)_t.$| We scrutinize this claim in our simulation and empirical analysis.

2.3.1 Fund-specific estimates

It is well established that identifying just |$\alpha$| and |$\beta$| in PE is difficult even when data are pooled across many funds (see, e.g., Ljungqvist and Richardson 2003, Driessen, Lin, and Phalippou 2012, Ang et al. 2018). To estimate the 10-dimensional parameter vector |$\theta$| by fund, we therefore proceed in two steps. The goal of the first step is to find the fund-specific |$\alpha$| and |$\beta$| that fit the SSM well while satisfying an additional NPV-based restriction. We pursue this goal via a profile likelihood method augmented with a penalty function (see, e.g., Ghysels and Qian, 2019, for details). In the second step, we iteratively estimate the remaining eight parameters in |$\theta$| and the value-to-return mapping function |$\hat{m}_t.$|

To implement the first step, we build a 15-by-15 grid of plausible values for |$\alpha$| and |$\beta$| based on a range of estimates obtained from the prior literature. Specifically, in case of a buyout fund, |$\beta$| ranges between 0.668 and 1.831 with a mean of 1.25 and 15 corresponding probability quantiles – 0.01, 0.05, .125, ..., 0.875, 0.95, 0.99, guided by estimates in Ang et al. (2018).¹⁰ We center the |$\alpha$|-range at the annualized estimate of fund-level excess returns from the Kaplan and Schoar (2005) PME method (see Internet Appendix Section A.4 for details). We consider seven equally spaced values up to |$+/-3.5$||$\%$| per year from each side of the |$\alpha$|-range center. For each of the 225 (⁠|$\alpha,\beta$|⁠)-pairs on the grid we (i) obtain the likelihood score corresponding to the MLE estimate of |$(\delta,\lambda,F,\sigma_n,\sigma_d)$| in which (⁠|$\alpha,\beta$|⁠) are kept fixed at the respective grid values and |$r_{ct}$| is dropped from the observations vector; (ii) evaluate the pricing error that the (⁠|$\alpha,\beta$|⁠)-pair implies against the fund cash flows.

In the second step, we estimate the remaining eight parameters (collected in the subvector |$\vartheta$|⁠) via maximum likelihood while keeping |$\alpha$| and |$\beta$| at their optimized profile levels. We use |$R_{ct}$|-interpolated NAVs as the estimate for |$V_t$| (see Internet Appendix Section A.3 for details) to construct the initial asset-to-value mapping. We then iteratively update the mapping function and reestimate |$\vartheta$| using the new mapping until the change from the previous iteration |$\vartheta$| is below the threshold (see Internet Appendix Section A.4 for details).

2.3.2 Partially imputed estimates

For partially imputed estimates we again proceed in several steps. First, we estimate |$\alpha$|⁠, |$\beta$|⁠, and |$\lambda$|⁠. However, now we profile |$\alpha$| and |$\lambda$| on a 15-by-7 grid, whereby the range for |$\alpha$| is determined as before, while |$\lambda$| is centered around that of the peer funds median. Recall that |$\lambda$| is essentially the exponential moving-average weight, constrained to be below one and above zero. We therefore, define the range of |$\lambda$| in terms of its probit function’s z-score distance from the center.

As for |$\beta$|⁠, we treat it as a free parameter in the first step but constrained to vary within just one standard deviation (again using Ang et al. 2018 estimates by fund type) from that of the peer funds (henceforth, |$\beta$|-anchor). We continue applying the penalty function to the profiled likelihood grid. However, now the penalty measures the distance from zero for the autocorrelation of filtered returns (see Section 2.2) obtained from a nowcast with only |$R_{mt}$|⁠. In doing so, we mitigate forcing too high or too low smoothing intensity upon the fund from its peers. We find this approach to be more informative about the (⁠|$\beta$|⁠, |$\lambda$|⁠) pair location than the PME-based penalty when the fund exhibits only several quarters with meaningful distributions.

In the second step, we iterate the return-to-value mapping function and the remaining free parameters until convergence as we do with the fund-specific estimates; however, we always keep |$F$| and |$\sigma_n$| fixed to the median of the peer funds. Therefore, this EM-like procedure involves only five parameters (rather than eight under the fund-specific estimation approach).

In both steps, the peer fund estimates are defined as medians of the respective parameter estimated independently. The next section examines several ways to define peer funds. We include the fund’s own fund-specific estimates to compute the peer group median. For |$\beta,$| we additionally consider setting the center of the distribution to be equal to the estimates in the literature by fund type (Ang et al., 2018).

2.4 Simulation experiment summary

We conduct a simulation study to examine the performance of our method with regard to both parameter estimation and nowcasting performance. One of the focus areas in this simulation experiment is the consequences of the model misspecification relative to the true data-generating process. For example, we introduce discontinuous jumps in the funds’ idiosyncratic return processes (as in Korteweg and Nagel 2020) contrary to the assumption that those are conditionally Gaussian and examine cases in which the assumed distribution function (i.e., Equation (13)) is different from the true one. For brevity, we defer the details to Internet Appendix Section A.6 where we examine a single case step-by-step process and report extensive summaries for a realistic panel of simulated funds. Here we summarize the main takeaways.

1. Key parameters are consistently estimated despite substantial idiosyncratic risk with non-Gaussian jumps and under reasonable misspecification of the distribution function |$\delta(\cdot)$|⁠; cross-sectional variation in |$\alpha$| and |$\beta$| is detectable with realistic sample sizes.

2. Under the assumption that the true parameters are close to those of the peer funds’ estimates, partial imputation may reduce the estimation error on |$\beta$| and |$\lambda$| by a factor of 1.2 to 1.6.

3. The median OOS nowcast error of 0.126 represents a 42|$\%$| reduction relative to the naïve nowcast, yielding an improvement for 68|$\%$| of the funds; partial imputation of parameters also yields a modest improvement in the nowcasting performance.

Overall, the simulations suggest that parameter estimation and nowcasting performance are promising using our methods and that the results are not particularly sensitive to distributional and functional form misspecifications.

3. Empirical Results

In this section, we report results for parameter estimates and nowcasting performance for the sample of private equity funds provided by Burgiss (see Brown et al. 2015 for description). The data include fund-level history of cash flows between each fund and its investors, fund NAVs reports for most quarters, as well as time-invariant data such as fund strategy, vintage year, and industry (based on eight Global Industry Classification sectors). We use the industry designation to select a comparable asset from the Fama-French 12 value-weighted industry benchmarks. We use the CRSP value-weighted index for market returns.

3.1 PE fund sample

We start by describing the general properties of the fund sample. The sample includes all funds that satisfy the following criteria:

• Vintage year is between 1983 and 2014.

• Operated for at least 24 quarters, and reported NAVs for at least 20 quarters.

• Made at least two distributions.

• Have at least two peer funds for which we obtain fund-specific estimates (Section 2.3.1).

Table 3 reports basic summary statistics separately for 2,444 buyout and 1,679 venture funds in panels A and B, respectively.¹¹ This is a subset of the Burgiss Manager Universe as of March 2021, augmented with additional data on fund industry specialization. We focus on relatively mature funds so we can evaluate nowcasting performance. The peers are defined as funds of the same type (i.e., buyout or venture) with vintage years within a year of the fund of interest, and the same money multiple tercile within the strategy. For feasibility of fund-specific estimates, we additionally require at least five quarters during which the distributions exceed 5|$\%$| of fund size and require that the latest reported NAV does not exceed one-third of its cumulative distributions.

The filters drop 225 buyout and 89 venture funds from our sample. We note that the excluded funds tend to have lower returns and smaller size. For example, the excluded venture funds had median size (PME) of $50 million (0.76), as opposed to $173 million (0.88) for the included funds. These differences are also notable for the buyout sample at $180 million (0.97) for excluded and $384 million (1.01) for included. Nevertheless, once adjusted for vintage averages, the differences in performance are statistically insignificant (test statistic of |$-1.27$| for the combined sample). The excluded funds comprise less than 4|$\%$| and 3|$\%$| of capital committed of, respectively, buyout and venture funds.

The number of quarters with meaningful distributions is an important fund characteristic for estimation of the model.¹² The last rows of each panel in Table 3 show that a quarter of buyout funds have five or fewer quarters with distributions that exceed 5|$\%$| of fund size. For venture funds the corresponding number is only two. This reduces the feasibility of fund-specific estimates for the venture sample especially (Internet Appendix Table A.2 reports summary statistics for PE real estate funds).

For each fund in the sample, we match the comparable asset series, |$R_{ct}$|⁠. For 490 funds that Burgiss classifies as generalists or provides no industry information for, we define the comparable asset for venture (growth) funds as the value-weighted small growth (value) portfolio downloaded from Ken French’s website. For 1,892 funds Burgiss provides us with an industry breakdown of the actual investments made by the fund (top three industries), so we value-weight the respective industries. Otherwise we match the closest industry. We utilize industry benchmarks because it is a relatively unexplored dimension in prior research on PE performance (despite the high attention from practitioners) but, most importantly, because our method requires a proxy of idiosyncratic volatility in the public benchmark as well.¹³

3.2 Fund-specific estimates

An important feature of our model is that estimation and nowcasting uses only the data from the individual funds and their matched public benchmarks. However, data limitations can hinder the estimation in some cases. Overall, we are able to obtain fund-specific parameter estimates for 2,513 PE funds or 61|$\%$| of the sample described in Table 3. Table 4 reports summary statistics for fund-specific estimates of SSM parameters and the assessment of their associated nowcasts.

3.2.1 Parameters

Perhaps the most interesting parameters we estimate are the funds’ |$\alpha$|s and |$\beta$|s. Panel A of Table 4 shows that the average (median) |$\alpha$| of PE funds with fund-specific estimates is 4.7 (3.0)|$\%$| per year with an interquartile range of 15|$\%$|⁠. This is not necessarily representative of a typical fund’s risk-adjusted return since these funds are likely to have a greater fraction of successful deals that resulted in large distribution amounts happening at different points of their lives. The loading on the public market return is 1.30 on average, ranging from 0.84 at the 10th percentile to 1.68 at the 90th.

Also of interest is the scale coefficient estimate on our proxy of time-varying idiosyncratic risk. Results in the third row of panel A report show that the standard deviation of idiosyncratic returns for a typical PE fund is 3.5 times the GARCH(1,1)-filtered idiosyncratic volatility of the matched industry index. However, we note the large dispersion for this parameter. While it is equal or smaller than 1.4 for 10|$\%$| of the funds, it is greater than four for almost a quarter of funds. In fact, for about 5|$\%$| of funds it hits the upper bound of 10 that we imposed in estimations. This suggests that the idiosyncratic volatility parameter is hard to identify at the individual fund level. However, it seems plausible to assume |$F$| is similar across funds within the same strategy, vintage, and size cohort, and we utilize this assumption later.

The next two rows of panel A report summaries of parameters that help us understand the errors in and smoothing of reported NAVs. PE-fund NAVs indeed tend to exhibit a very high degree of appraisal smoothing, as follows from the percentile ranks of |$\lambda.$| The median of 0.961 implies that public market valuations from five or more weeks ago comprise 82|$\%$| (= |$0.961^5$|⁠) of a typical fund NAV report. This exponential weight is consistent with the AR(2) coefficients of about (0.5, 0.3) on quarterly return series derived from reported NAVs. Our results suggest substantially less smoothing for the bottom decile of funds, e.g., |$\lambda$| is 0.52 or less, which corresponds to a nondetectable persistence at the quarterly frequency.

We observe that the average (median) NAV noise—the standard deviation in NAVs that the SSM cannot attribute to past or future returns—is 6.4|$\%$| (5.7|$\%$|⁠). For some funds it appears to be implausibly high or low, for example, 14|$\%$| at the 90th percentile and close to the lower bound of 1|$\%$| for 10|$\%$| of funds. Therefore, this parameter also appears hard to identify reliably. Just as with the idiosyncratic risk levels, we subsequently use the fact that these are plausibly similar across peer funds. Naturally they relate to fund investments’ type, portfolio diversification, and operating period. Panel A also reports estimates for |$\delta$| and |$\sigma_d$| that govern the fund distribution process. While auxiliary in nature, they need to be estimated for each fund to reflect the heterogeneity in the distribution magnitudes (given the assumed functional form of |$\delta(\cdot)$|⁠). Here we just note that for the vast majority of funds both parameters are well within the upper (0.9 and 4) or lower (both 0.0001) bounds imposed in the estimation.

Finally, we examine the parameters that govern the mapping of the comparable asset to the latent fund returns at weekly frequency. The last three rows of panel A report the intercept, slope, and variance scale parameters. To gain intuition regarding these parameters, we plug in (the log of) Equation (5) for |$r_t$| into Equation (6) and denote the intercept (residual) from a projection of |$r_{ct}$| on |$r_{mt}$| with |$a_c^m$| (⁠|$\epsilon_{ct}^m$|⁠). It follows then that |$\psi=a_c^m-\beta_c \alpha$| is not particularly informative, while |$\beta_c=\mathrm{cov}(\epsilon_{ct}^m,\eta_t)/\mathrm{var}(\eta_t)$| is the covariance of the comparable asset and fund returns scaled by the variance of the fund returns, which somewhat limits the parameter’s magnitude comparability across funds. Note that for truly comparable assets this covariance should be positive. Therefore, we impose a lower bound of 0.01 on this parameter. The boundary is hit for 27|$\%$| of funds, indicating that the matched benchmark does not help explain those funds’ NAV and distributions data. A disproportionally large share of these are funds for which we do not have the industry weights from Burgiss, especially if classified as generalists (therefore, a size and style benchmark is utilized). However, for a typical fund in the sample |$\beta_c$| estimate is 0.18 and the average test statistic for the parameter is 2.2 (untabulated).

3.2.2 Properties of filtered returns

We now want to compare the time-series properties of different estimates. Specifically, we compare the autocorrelations and variances of fund returns obtained from a naïve nowcast that is based on as-reported NAVs with those from the fund-specific SSM estimates. Panel B of Table 4 reports statistics for series at two frequencies—weekly and quarterly. For the naïve nowcast, the quarterly series correspond to NAV changes adjusted for the value of within-quarter cash flows, discounted at the industry benchmark returns. In contrast, the weekly naïve nowcast assumes the asset values track the industry benchmark on top of the interpolated between-quarter change in PME (see Internet Appendix Section A.3). We see that those series exhibit notably positive autocorrelations, which looks higher at quarterly frequency at 0.227 for a median fund. The results for variances suggests that the annualized standard deviation of naïve returns nowcasts for the middle half of PE funds is between 20|$\%$| and 38|$\%$|⁠.

The SSM-based return nowcasts are obtained via the Kalman filter at weekly frequency, as explained in Section 2.2, and then aggregated to the quarterly frequency by summation of the log returns within each quarter. Panel B shows that the autocorrelation of the SSM-based returns is only 0.09 for a median fund. At the weekly frequency, we observe effectively zero return persistence for at least three quarters of the sample. Meanwhile, the variance of filtered returns is on average 1.2 times higher than for the naïve nowcasts, suggesting the annualized standard deviations are between 25|$\%$| and 45|$\%$| for the middle half of PE funds.

While these results suggest that the SSM-filtered returns are more representative of the true return process than the ones inferred from reported NAVs, we note that for a significant minority of funds the SSM nowcasts exhibit fairly large autocorrelations. For example, one-fifth of the fund sample exhibits autocorrelations of filtered returns in excess of |$+0.44$| or below |$-0.33$| at the quarterly frequency, and among those there is a 10-fold prevalence of funds with either |$\beta_c<<0.02$| or |$F_c\ge 0.999,$| whereby updates to |$r_t$| are driven solely by the distribution and NAV reports, which gives rise to notable persistence when the backward induction method for filtering is utilized. The remaining cases appear to be driven by the model (e.g., |$\lambda(\cdot)_t$| and |$\delta(\cdot)_t$|⁠) or parameter misspecifications, which usually can be addressed on a case-by-case basis in practice. We also note that usage of the backward induction with the Kalman filter likely underestimates the variance of the filtered return series somewhat, and, hence, overestimates the magnitude of the correlations.¹⁵

3.2.3 Nowcasting performance

The out-of-sample RMSEs are similar to the in-sample RMSEs and are positively correlated, as evident from Figure 2. Hence, in-sample pricing errors are a useful predictor for nowcasting performance, as suggested by the simulations discussed in Section 2.4. Panel C of Table 4 reports fund-level nowcasting performance metrics introduced in Section 2.2. The SSM-based RMSEs are notably smaller than those from the naïve nowcast. For a typical fund, the Hybrid RMSE drops from 0.281 to 0.045, while the OSS error drops from 0.233 to 0.12 and is sizable across all percentiles.

Figure 2

Predictability of nowcasting performance

This figure compares the logs of in-sample nowcasting error (x-axis) with those out-of-sample (y-axis) as defined in Section 2.2 using fund-specific estimates summarized in Table 4. A positive slope indicates that that SSM nowcasting precision is more predictable. Table 3 describes the sample.

Open in new tabDownload slide

Figure 3 shows that the improvements are largest for funds with relatively large naïve nowcast errors and that the SSM-based nowcasts are materially inferior to the reported NAVs only when the latter happen to be particularly precise—for example, to the left of log MSE of |$-$|5 on the x-axis, which corresponds to the RMSE of less than 0.082. Across the buyout and venture sample, SSM-based nowcasts with fund-specific parameter estimates result in smaller Hybrid (OOS) RMSE than the naïve nowcast for 90|$\%$| (70|$\%$|⁠) of individual funds. In Section 4 we show that gains are usually higher when a portfolio of funds is concerned because SSM-based nowcast errors tend to be less correlated than those from the naïve approach.

Figure 3

Nowcasting performance assessment

This figure compares nowcasting errors from SSM (y-axis) using fund-specific estimates summarized in Table 4 with those from a naïve method based on as-reported NAVs and matched benchmark returns. The metrics are defined in Section 2.2 and are based on the idea that the NPV of fund cash flows has to be zero if true returns are used to discount them.

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3.3 Partially imputed estimates by fund type

Next, we examine partially imputed parameter estimates as described in Section 2.3.2 and compare the nowcast performance with those generated by fund-specific estimates. Most importantly, the partial imputation enables us to obtain parameter estimates for approximately 95|$\%$| of the sample funds and, therefore, provide a more representative picture of the institutional PE fund universe.

We consider three ways to define peer funds. First, in what we will refer to as peer-imputed we set |$F$|⁠, |$\sigma_n$| as well as the center locations of the |$\beta$|-anchor and |$\lambda$|-grid to the median values of the respective fund-specific parameter values for funds of the same strategy (buyout of venture) incepted in the same or adjacent vintage year. Additionally, if there are three or more peer funds in the same size tercile, we limit the peer group to those peers only. We then attempt to further narrow the peer group to at least three funds in the same industry of specialization. Second, we take the opposite approach and set the four imputed values to the buyout or venture fund average in what we refer to as average-imputed. Finally, we fix the |$\beta$|-values to the strategy-specific estimate from Ang et al. (2018) in what we refer to as literature-imputed, while identifying all other parameters independently, as described in Section 2.3.1.

We then compare the SSM nowcast performance metrics introduced in Section 2.2 under these four different sets of parameters. This exercise sheds light on three important questions: (i) Is there a meaningful time-series variation in PE-fund risk-return profiles? (ii) How big is the noise reduction resulting from restricting the fund-specific parameters? (iii) Are our systemic risk exposure estimates more consistent with the fund-level cash flow observations than in Ang et al. (2018), who encompass the previous literature estimates as priors for their Bayesian MCMC?

3.3.1 Buyout funds

Our findings show that peer-imputed estimates dominate other partially imputed estimates and benefit from a noise suppression relative to the fund-specific estimates for a majority of funds. There appears to be genuine heterogeneity in parameters across many buyout funds. Forcing higher levels of fund systematic risk does not improve the nowcasts.

The evidence for selected parameter estimates and nowcast performance metrics for the three imputation methods alongside the fund-specific estimates are reported in Table 5. The latter are available for 1,598 funds, while the partial imputation increases the coverage to 2,299 funds (94|$\%$| of the sample reported in panel A of Table 3).

We first focus on how key parameter estimates, namely |$\alpha$| and |$\beta$|⁠, vary across estimation methods. We observe that the average and median |$\alpha$| estimates of 4.5 to 5.1|$\%$| are lower for peer- and average-imputed estimates in comparison to the 6.3|$\%$| mean in the sample of fund-specific estimates (panel A). However, these are still notably higher than the near-zero values for the literature-imputed estimates, which also stand out with notably higher |$\beta$|s of 1.25 on average, especially as compared to the range of 0.87 to 1.09 for the middle two quartiles using the peer-imputed method.

The various estimation methods provide consistent results across funds for other model parameters. Specifically, in panel A we examine the smoothing parameter |$\lambda$|⁠, as well as the variance scale on comparable asset parameter, |$F_c$|⁠. The mean and median |$F_c$| are very close across all four methods. Still the the literature-imputed estimates yield lowest levels, which is consistent with an upward bias in the systematic risk estimates. As for the NAV smoothing rate, we observe that the means are in a narrow range of 0.835–0.886 across all four methods. The interquartile is highest for the average-imputed method, possibly reflecting a tighter constraint on the variation of the |$\beta$| parameter relative to that in the peer-imputed method. However, the interquartile range is smallest for the literature-imputed method whereby |$\beta$| is also fixed. The literature-imputed estimates also feature the highest median |$\lambda$| of 0.964 that is nonetheless very close to that of the fund-specific method (0.954) despite a 0.15 difference in the median |$\beta$| estimate. These results indicate that there is no “mechanical relation” between the estimated levels of systematic risk and NAVs’ smoothing intensity.

We also examine results for weekly autocorrelations and quarterly variances of filtered returns and find generally consistent and stable values across the various estimation methods (panel B). The naïve approach results in similar levels of autocorrelation (6.4|$\%$|⁠) as observed for the fund-specific estimates reported in Table 4 (which includes venture funds). The average (median) annualized standard deviation of returns is estimated to be 28 (24)|$\%$| with the naïve approach. The fund-specific returns have very low persistence and return standard deviations of about 34 (29)|$\%$| per year for an average (median) fund. While the average and the percentile readings of the total fund risk per the peer- and average-imputed methods are very close to those with the fund-specific method, they are characterized by somewhat elevated levels of autocorrelations of filtered returns in the right tail of the cross-section. The 75th percentile of 0.13–0.146 is actually closer to that of naïve nowcasts. This is driven by a larger fraction of funds for which we do not have a good match of the comparable asset: in the subset of funds with observed industry weights, the 90th percentile autocorrelation is only 0.06 (untabulated). As for the literature-imputed estimates, the standard deviations of filtered returns are 38|$\%$| on average, which is notably higher than those of other SSM estimates, while the distribution of autocorrelation coefficients is similar to that of the fund-specific method but is somewhat less symmetric and shifted to the negative domain.

Finally, we examine the goodness-of-fit estimates for the various methods. We find that the partially imputed estimates have lower nowcast errors and higher improvement rates relative to the naïve nowcast on both the Hybrid and OOS basis, except for the literature-imputed estimates (panel C of Table 5). Nonetheless, there are notable improvements over the naïve approach from utilizing the SSM method regardless of how systematic risk levels are identified. For example, the median OOS RMSE with literature-imputed |$\beta$| is 0.113, or 28|$\%$| higher than with peer-imputed estimates but nonetheless 55|$\%$| lower than 0.251 for the naïve approach. Therefore, the SSM method is relatively forgiving to reasonable variations in the model parameters.

3.3.2 Venture funds

We repeat the analysis described earlier for venture capital funds and report the results in Table 6. We are able to obtain fund-specific estimates for 859 funds, while the partial imputation increases the coverage to 1,621 funds (96.5|$\%$| of the venture sample).