## Abstract

It is reasonable to expect that changes in private equity valuations should bear some correspondence to public equity performance because private assets and public assets both respond to common influences such as changes in discount rates. But it is also reasonable to recognize that changes in private equity valuations should depart to some extent from public equity performance owing to differences in risk, liquidity, and cash flow expectations. These differences affect the degree of correspondence. In this article, the authors explore an additional influence on private equity valuations that affects not the degree of correspondence, but rather the symmetry of the correspondence. The authors argue that because private equity managers are less constrained than public market participants by the forces of no-arbitrage pricing, they have greater discretion to introduce biases into their valuations. Based on an extensive sample of private equity valuations, the authors find persuasive evidence that private equity managers produce positively biased valuations that appear to be rationalized by information that should not be relevant.

**TOPICS:** Private equity, security analysis and valuation, performance measurement

All Investment managers like to produce good performance because they wish to do well for their clients and because good performance helps to retain assets and to attract new assets. Managers who operate in public markets have three means by which to produce good performance: skill, effort, and luck. Managers who operate in private markets also rely on skill, effort, and luck, but they have a fourth means by which to produce good performance—discretionary valuations. Because managers who operate in private markets are less constrained by limits to arbitrage than managers who invest in public markets, they have some discretion in valuing their investments. Of course, they are not entirely unconstrained. They must abide by fair value rules, and their valuations must appear reasonable to their investors. Nonetheless, managers who operate in private markets have the potential to introduce biases into their valuations. On the basis of an extensive sample of company-level private equity valuations, we find persuasive evidence that private equity managers are asymmetrically influenced by noncontemporaneous public equity performance. Specifically, after accounting for prior and contemporaneous public equity performance, we find that private equity valuation changes co-vary positively with public equity gains that occur after the valuation period but not with public equity losses that occur after the valuation period. We also find, in the case of venture capital, that changes in private equity valuations co-vary more strongly with persistent negative public equity performance than with persistent positive public equity performance. Interestingly, this behavior does not occur, and in some cases is reversed, for fourth quarter valuations that are audited. We make no claim that this behavior is intentional. It is quite plausible that private equity managers subconsciously produce positively biased valuations merely because they are optimistic.

## LITERATURE

There is an extensive literature on the beta of private equity investments with respect to the public equity market. Many of these papers address the estimation challenges posed by private equity data, but there is little consensus on which particular approach to use. Korteweg and Sorensen (2010) treat discount rates as latent variables and derive the maximum likelihood estimate to fit the observed valuations for venture capital. They estimate betas to the public market that are consistently above 2.0. Ang et al. (2018) pursue a similar approach based on observed cash flows. They compute public market betas of roughly 1.6 for venture capital funds and 1.3 for buyout funds. Axelson et al. (2013) consider a static CAPM regression of deals on market returns, as well as a log-CAPM model and a model that allows for jumps in pricing. They estimate betas ranging from 2.2 to above 3 for buyout funds. In contrast, Ewens et al. (2013) find that the average buyout fund in their sample has a beta of 0.72, whereas the average venture capital fund has a beta of 1.24.

Recently, some research has examined whether private equity valuations appear to be manipulated. Jenkinson et al. (2013) analyze a sample of more than 700 funds held by a large institutional investor, and they find that valuations are generally too conservative, except during periods of fundraising and at year-end when valuations are audited. Brown et al. (forthcoming) corroborate the notion that many funds favor conservative valuations. They find that some funds inflate valuations around fundraising, but they are subsequently punished, on average, with weaker fundraising for subsequent funds.

Our article contributes to the literature in three important ways. First, we analyze an extensive data set covering more than half of the global assets in private equity. Second, we distinguish between valuation changes that occur during up and down public markets and document meaningful asymmetric patterns in the relationship between private and public market returns. Third, we investigate the relationship between private equity valuation changes and subsequent public market performance. Our results suggest that subsequent public market performance affects reported valuations. Typically, these effects are obscured by deal-level analysis that pertains only to realized cash flows of private equity funds, or they are ignored in time series analysis.

## DATA

Our analysis is based on company-level valuations from State Street’s GX Private Equity Index (GXPEI) database. The GXPEI comprises data from State Street’s business as a global asset custodian reflecting more than half of all global private equity assets. As such, we expect these data to be highly representative of the overall private equity market. Valuations are released on a quarterly basis and represent calendar quarter-end valuations that are submitted by general partners of private equity funds following each quarter end, typically within 45 to 90 days. Our analysis spans more than 400,000 quarterly valuations from Q4 2000 through Q1 2016 for US investments. We aggregate the quarterly valuations into seven time series of quarterly percentage changes, grouped according to the type of fund that holds each investment.

There are three venture capital fund types, early stage, balanced stage, and late stage, and four buyout fund types, small buyout, mid buyout, large buyout, and mega buyout. Each quarter, for each fund type, we identify all relevant investments for which valuations are available at the current valuation date and also at the valuation date one quarter prior. We sum all of these company valuations as of the current date, and we sum the corresponding valuations as of the prior date. Then, we compute the percentage change in valuation. As a result, the valuation changes are implicitly weighted according to the size of each investment each quarter. We also obtain quarterly price index percentage changes for the S&P 500 index.

Unless other otherwise stated, all data used in this analysis are from the GXPEI database during the period Q4 2000 through Q1 2016.

## METHODOLOGY

We perform regressions to evaluate the impact of public equity valuation changes that occur before, during, and after the current quarter’s private equity valuation change. We measure public equity’s prior return as the quarterly geometric mean return of the prior three quarters.^{1} Throughout this article, we use the term “public equity return” to pertain only to percentage change in price, ignoring dividend income. To allow for the possibility that these relationships differ when public equity rises versus when it falls, we use a piecewise linear regression that splits each explanatory variable into two new variables. The first variable retains all positive values, but sets negative values equal to zero. The second variable retains all negative values but sets all positive values equal to zero. We estimate the regressions using ordinary least squares (OLS) on pooled observations. To account for possible correlation of errors across fund types we apply an adjustment to the standard errors, which we describe in the appendix. We run one pooled regression for venture capital funds, and a separate pooled regression for buyout funds. These regression equations are specified below. For venture capital funds, we include constant terms for each fund type in which Early Stage, Balanced Stage, and Late Stage are binary variables that equal one for valuation changes associated with the specified fund type, and zero otherwise. For buyout funds, we include analogous variables for the four types of buyout funds. Variables with a + superscript retain all positive values for market returns, but set negative values equal to zero. Variables with a − superscript retain all negative values for market returns, but set positive values equal to zero.^{2} The variable *u* represents the error term.

We are interested in testing whether the coefficients for each up-market variable are significantly different from those of the corresponding down-market variable. We compute the test statistic as the difference in betas divided by the standard error of the difference, which is equal to the square root of the variance of the difference, computed as follows:

## RESULTS

Private equity performance is not recorded immediately after the end of the quarter. Instead, it is released during a period of one to three months. This reporting delay raises the possibility that private equity managers might be influenced by the post-valuation period performance of the public equity market. Our results suggest that this is indeed the case. We find that private equity managers react asymmetrically to market gains and losses that occur after the valuation period. They tend to raise valuations if public equity generates gains following the conclusion of the valuation period, but they tend to ignore subsequent public equity performance if it is negative. In the case of venture capital, we find the opposite asymmetry in regard to public equity results that occur prior to the valuation period. Venture capital managers are more inclined to reduce valuations following several periods of persistent negative public equity performance more so than a string of positive results.

This behavior is consistent with the notion of confirmation bias documented in the behavioral finance literature. If managers believe they have made a good investment that is likely to increase in value over time, confirmation bias would imply that they are more likely to seek confirmatory evidence, and to rely on that evidence more heavily than contradictory evidence. Therefore, when the public market appreciates, they may interpret this result as supporting evidence that their investment has also gained in value. Confirmation bias also implies that managers will subconsciously ignore or downplay the importance of contradictory evidence, such as public market losses. Eventually, private equity assets must be sold at exit, at which point they will receive a true market price. Managers’ optimism about short-term price gains must be tempered by occasional decreases in value. These decreases do occur, and they typically align with public equity losses that persist over multiple periods. This is especially true of venture capital investments because early-stage and middle-stage investments are more likely to experience financial distress. The exhibits that follow document those biases.

### Up- and Down-Market Betas

Exhibit 1 presents the up- and down-market betas for venture capital funds and buyout funds for the entire sample of valuations based on the regression equations specified earlier. Exhibit 1 shows that venture capital managers capture contemporaneous public equity performance in their valuations, but no more so for up markets than for down markets. It also reveals that they capture persistent (three quarters) prior period performance more so for down markets than for up markets, but the difference is not significant. It does reveal, however, a significant bias in the way venture capital managers process public equity performance that occurs after the conclusion of the valuation period. If the public market appreciates in the quarter following the valuation period, they capture a significant part of this appreciation in their valuations. However, if the public market depreciates, venture capital valuations co-vary slightly in the opposite direction.^{3}

In the case of buyout funds, we observe the same pattern for contemporaneous and lagged public equity performance, but we find no association between buyout valuations and prior period public equity performance.

Exhibit 2 presents the same information as Exhibit 1, but the sample excludes 2008 and 2009, when the financial crisis occurred. Exhibit 2 reveals the same post-valuation period bias for venture capital and buyout funds, though it is not significant for venture capital funds. It also shows that venture capital managers were influenced by sustained prior period decreases in public equity valuations, and more significantly so than for the full sample results.

Exhibits 1 and 2 present results based on all four quarters of our sample, but there is an important distinction between the first three quarters and the fourth quarter. The valuations for the first three quarters are not audited, whereas the valuations for the fourth quarter are. We therefore show the betas for just the fourth quarter valuations in Exhibit 3 for the full sample.

Exhibit 3 reveals that the post-valuation period bias is reversed for venture capital funds and disappears for buyout funds in the fourth quarter. It also reveals that the asymmetry between the association of venture capital valuations and prior public equity performance is less significant. These results suggest that private equity managers are less inclined to produce biased valuations when they are faced with audits because the involvement of external parties approximates, to some extent, the effect of no-arbitrage pricing. To underscore this fact, in Exhibit 4 we produce the up- and down-market betas derived from the unaudited valuations that occurred during the first three quarters.

As one would expect, the results for the first three quarters align with the results for all four quarters, confirming that private equity managers are sensitive to whether or not their valuations will be audited.

We also consider whether an accounting rule change that occurred during our sample affected the relationship between valuations and public market returns. The Statement of Financial Accounting Standards (FAS) 157, first announced in 2006, provides guidelines for fair value reporting under generally accepted accounting principles (GAAP). As most private equity funds produce financial statements in accordance with GAAP, this statement requires the funds to report investments at fair value. Prior to this rule, investments were commonly held at cost unless there was a “milestone event,” such as a sale or round of financing. FAS 157 became effective for financial statements beginning in late 2007 and was adopted by funds throughout 2008. Because its full adoption coincides with the financial crisis, we define the pre-FAS 157 period as 2000 through 2005 and the post-FAS 157 period as 2010 through 2016.

Exhibit 5 shows that both subsamples exhibit asymmetry in up-market versus down-market betas. The results are broadly consistent with the full sample results shown earlier; venture capital and buyout funds both have larger up-market betas than down-market betas with respect to the public market’s return in the following quarter, and the opposite effect prevails with respect to the public market’s return in the three prior quarters.

### Distribution of Public Equity Capture Ratios

The up- and down-market betas in the previous five exhibits reveal to us the extent to which private equity valuations change per unit of public equity return, on average. But most individual observations differ from the average. Therefore, in Exhibit 6 we isolate the distribution of each actual quarterly outcome. We define each outcome as the ratio of a valuation change to a corresponding public equity return, using the non-year-end regression results.

We proceed as follows. We subtract from each historical quarterly valuation change the estimated values from the fitted regression model based on the contemporaneous and prior period public equity betas. What remains are the quarters’ residuals; that is, variation not explained by contemporaneous and prior returns. We collect the residuals that correspond to positive next-quarter public equity returns and compute the capture ratio for each observation as the residual valuation change divided by the corresponding next-quarter public equity return. We compute the ratios corresponding to negative next-quarter public equity returns in the same way.^{4}

In the case of venture capital funds, the density of post-valuation-period up-market capture ratios is concentrated to the right of the theoretically unbiased (0%) capture ratio. By contrast, the down-market capture ratios are centered slightly to the left of the unbiased ratio with additional concentration in the left tail.

In the case of buyout funds, we see similar concentration to the right of the unbiased estimate along with additional concentration in the right tail for up-market ratios. For down-market capture ratios, we see a similar pattern but in the opposite direction. These results reveal that the biased response to post-valuation-period public equity performance is relatively pervasive and not a function of a few outliers.

### Cumulative Distribution of Up- and Down-Market Valuations

Perhaps we can visualize the post-valuation period bias most effectively by comparing the cumulative frequency distributions of up- and down-market valuations with the unbiased theoretical cumulative distribution, as shown in Exhibit 7.

The diagonal line in Exhibit 7 represents the cumulative probability distribution under the null hypothesis that the capture ratios of post-valuation-period public equity performance are normally distributed around zero. The lighter-shaded plot maps the empirical cumulative frequency of capture ratios corresponding to post-valuation-period down markets, whereas the darker-shaded plot maps the cumulative frequency of capture ratios corresponding to post-valuation-period up markets. The extent to which an observation is above the diagonal line represents the increase in frequency that an observation fell below that value, and the amount that an observation is below the diagonal line represents the decrease in frequency that an observation fell above that value. The cumulative frequency plots reveal that about 70% of the observations fell below the capture ratios when public equity was down compared with about 30% when public equity was up. These pictures clearly reveal a bias in how private equity managers are influenced by public equity performance that occurs after the conclusion of the valuation period.

## THE EFFECT OF BIASED VALUATIONS ON PRIVATE EQUITY PERFORMANCE

We have documented a significant bias in private equity valuations that occurs during the first three quarters of the years but not in the years’ fourth quarters when valuations are independently audited. As such, we should expect private equity to produce, on average, higher returns relative to the public market in the years’ first three quarters than in their fourth quarters. Exhibit 8 starkly confirms this expectation.

## INVESTMENT IMPLICATIONS

The biases we have documented have serious implications for investment in private equity. We have shown that private equity returns reflect public equity performance from the periods that precede and follow the valuation period. Most investors recognize that private equity valuations are serially correlated; hence they apply de-smoothing techniques to derive better estimates of volatility and correlations. However, these lagged relationships are asymmetric; private equity returns react more to public market gains and less to public market losses. This asymmetry may cause an upward bias in correlations measured from up-market returns and a downward bias in correlations measured from down-market returns, which could overstate the diversification properties of private equity. Because these asymmetries tend to wash out over longer horizons, we suggest that private equity investors base their estimates of volatility and correlations on annual returns instead of quarterly returns, or employ techniques that correct for these biases.

## SUMMARY

• It is reasonable to expect that private equity valuations should reflect changes in public equity valuations that occur contemporaneously, because both markets are influenced by common factors such as changes in discount rates. Our results confirm that private equity managers capture contemporaneous public equity performance fairly equally across up and down markets.

• We see no objective reason why private equity valuation changes should reflect public equity performance that occurs before or after the valuation period. Nonetheless, our results suggest that noncontemporaneous public equity performance does influence private equity valuations. Specifically, changes in private equity valuations co-vary positively with favorable public equity performance that occurs in the quarter following the valuation period. It does not, however, co-vary significantly with unfavorable public equity performance that occurs after the valuation period. We also find evidence that venture capital valuation changes co-vary positively with public equity performance that occurs over the three quarters that precede the valuation period, and more so for negative public equity performance than for positive performance.

• We find that these biased associations with noncontemporaneous public equity performance are reversed or disappear during the fourth quarters of our sample when the valuations are independently audited.

• We also show that our results are robust across subsamples in which we partition history into two segments; that period that precedes the introduction of FAS 157, which requires funds to report investments at fair value, and the period that follows the introduction of FAS 157.

• We document these biases in three different ways. We show the betas of changes in private equity valuations with respect to up- and down-market public equity returns, as well as the differences in these conditional betas. We also contrast the distribution of up- and down-market capture ratios with the theoretical distribution of unbiased capture ratios. And finally, to facilitate visualization of these biases, we contrast cumulative distributions of up- and down-market capture ratios with the theoretical unbiased theoretical distribution.

• We should expect these valuation biases to result in higher private equity performance, on average, during the first three quarters of the years than in the years’ fourth quarters. We show this to be true.

• These biases may cause investors who rely on quarterly returns to misestimate private equity volatility and correlations. We suggest that investors use annual returns to estimate private equity risk or apply techniques to correct for the biases we have documented.

## ACKNOWLEDGMENTS

We thank Josh Lerner and Antoinette Schoar for helpful comments.

## APPENDIX

### Standard Error Adjustment

Consider a pooled panel linear regression model with a scalar dependent variable *y*_{n,t} with a total of *T* quarterly valuation changes for each of *N* fund types, a row vector of *M* independent variables *x _{t}*, a column vector of

*M*coefficients β, and a scalar error term

*u*

_{n,t}:

Exhibit A1 illustrates the configuration of the data, along with the estimated residuals , that can be derived after fitting the regression model and that we reference later.

We have effectively appended observations for ** X** that are mere repetitions of its previous values, but with new observations for

**. These additional observations can potentially strengthen the conclusions we draw from the regression. However, we must also consider the possibility that some of the added information is partially redundant. In our empirical example, the**

*y***variables are public market returns for a given date range, and the**

*X***variables with which we form a panel are valuation changes for different subsectors of private equity, such as small, mid, large, and mega buyout funds. If the errors across fund types are positively correlated, we should expect our coefficient estimates to be less precise than they would be if the errors were all uncorrelated.**

*y*As an analogy, suppose we selected a person to interview and asked him a series of questions. If we asked the same person the same set of questions a second time, we would gain little to no incremental information even though the number of data points grows. On the other hand, if we asked a completely different person these questions, we would gain new information and potentially even double the number of useful data points. In reality, the degree to which we learn new information may lie somewhere in between these two extremes.

We estimate the model using standard ordinary least squares (OLS) on a pooled time series of dependent variable observations (an *NT* × 1 vector *y*) and a pooled (repeated) time series of observations for each independent variable (an *NT* × *M* matrix *X*). The coefficient estimates are

With a simple substitution, we can express the coefficient estimates as the true coefficient plus a function of the error terms ** u**:

To evaluate the statistical significance of , we must estimate its variance. Intuitively, if the errors across observations are positively correlated, there is less “diversification” in errors, which increases the variance of the coefficient estimates. The variance of the vector of coefficients is an *M* × *M* covariance matrix with diagonal elements for the variance of each individual coefficient and off-diagonal elements for the covariances between each pair of coefficients. The estimate of the covariance matrix of coefficient estimates is shown below, where *E*[ ]is the expected value.

To build intuition and simplify the matrix algebra, let’s assume for now that every ** X** variable has a mean of zero. Then we can write

**′**

*X***as**

*X**T*Ω, where Ω is the covariance matrix of

**. Assuming that the errors are unbiased,**

*X**E*[

*u*|

_{t}*x*

_{t}] = 0, we have:

Under the common assumption of errors that are independent and identically distributed (IID), the ** Q** matrix would be an

*NT*×

*NT*diagonal matrix with elements equal to the variance of

*u*: . In this case, the formula simplifies considerably:

In our analysis, the volatility of errors does not differ substantially across fund types nor across time, so we retain the assumption that errors are identically distributed. However, it is plausible that errors are correlated across fund types as a result of common factors that affect private equity, which we do not explicitly include in the regression. Therefore, we multiply the standard estimates by the following adjustment factor:

The ** S** matrix that includes this adjustment is sometimes called a “cluster-robust” variance estimate because it controls for correlation across clusters (fund types) in the panel data. The

*NT*×

*NT*matrix

*C*_{u}contains the correlations between error terms

*u*, across fund types. In practice, we estimate this matrix using estimated residuals , and we assume that errors may be correlated across fund types, but not across time.

For those less comfortable with matrix algebra, it may be helpful to express this adjustment process for one given element of , using summations rather than matrices:

We define ω_{i,k} as the *i*th row and *k*th column of the inverse covariance matrix. In this context, we can think of each beta coefficient estimate as a weighted average of the errors *u _{t}*, where

*a*

_{i,t}are the weights. We define ν

_{i,t}=

*a*

_{i,t}

*u*

_{t}to simplify the notation going forward. If all errors are IID, then:

To strengthen the intuition for this approach, consider two extreme cases. First, suppose we start with one set of data such that *N* = 1, and then we double our sample size to *N* = 2 with observations that are not at all redundant to the original observations. In other words, their residuals are uncorrelated. In this case, the formula above simplifies to divided by *NT*. Second, suppose we double our sample size with observations that are exactly identical to the original observations: their correlation is one. In this case, the formula simplifies to divided by *T* (because the double sum contains *N*(*N* − 1) elements each equal to one), and the new data do not reduce the variance of the coefficient estimate at all. The relationship generalizes to the covariance between any pair of coefficient estimates:

We can write the adjustment factor as follows. The adjustments are straightforward to compute based on correlations of observed residuals.

## ENDNOTES

↵

^{1}We use geometric means so that the prior three quarters’ return is in the same units as the other quarterly returns in the regression.↵

^{2}These variables may also be interpreted as a type of interaction term, each comprising the product of a market returns variable and a binary variable that indicates whether the market return is positive or negative.↵

^{3}The reported*p*-values represent one-tailed tests. In other words, they represent the probability under the null hypothesis that the*t*-statistic would be more negative if the estimated statistic is negative, or more positive if the estimated statistic is positive. The bold font denotes results with*p*-values of less than .10. It is also worth noting that though we refer to the test statistics as*t*-statistics, the adjustment for overlapping data no longer ensures that the small-sample statistics are distributed according to a*t*-distribution. They will be asymptotically normally distributed. However, given the number of data points in the sample, the distinction is not material. The probability implied by a*t*-distribution with the relevant degrees of freedom is within .01 of the probability implied by a normal distribution in every case.↵

^{4}Exhibit 6 shows observations that fall within the range of −2 and +2, with frequencies rescaled with respect to observations in this range. There are also some outliers that fall outside this range, mostly due to market returns close to zero that create very large ratios. For venture capital funds, 10 percent of the up-market data and 17 percent of the down-market data are outliers. For buyout funds, 16 percent of the up-market data and 14 percent of the down-market data are outliers.**Disclaimer**The material presented is for informational purposes only. The views expressed in this material are the views of the authors and are subject to change based on market and other conditions and factors; moreover, they do not necessarily represent the official views of Windham Capital Management, State Street Global Exchange, or State Street Corporation and its affiliates.

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