Estimate Idiosyncratic Risks of Private Equity Funds: A Cross-Sectional Method

Estimate Idiosyncratic Risks of Private Equity Funds: A Cross-Sectional Method Master Thesis By: YAN LIU (U174755), Research Master in Finance, Tilburg University Supervised by: Joost Driessen, Professor in Finance, Tilburg University John Renkema, Senior Portfolio Manager, APG Asset Management Rob an Den Goorbergh, Head of Research & Analytics, APG Asset Management 1

Estimate Idiosyncratic Risks of Private Equity Funds: A Cross-Sectional Method ABSTRACT We propose a new method to estimate idiosyncratic risks of private equity funds. In particular, we assume that private equity fund returns are driven by two independent sources of risks, namely, market risks and idiosyncratic risks. We derived the analytical expression of expected cross-sectional variance in fund returns and then estimate idiosyncratic risks through simulations. Our first result is that estimated idiosyncratic risk (σ ε ) for Buyout funds stands at 3.9%, using IRR as the return metric, and 18.4% when return is measured by the money multiple; estimated idiosyncratic risks for enture funds are somewhat larger, 36.% and 5.1%, respectively, using IRR and the money multiple as return metrics. We then decompose cross-sectional variances into three components: alpha component, beta component, and idiosyncratic risk component. And our second result is that the idiosyncratic risk component consistently accounts for a significant part, over 70%, of the total cross-sectional variance; the beta component also contributes to a range of 18% to 34% of the total cross-sectional variance; the alpha component is relatively small and only accounts for around -6% to 5%. Finally, we estimate idiosyncratic risk by vintage year. Our third result is that idiosyncratic risks vary over time and present a cyclical pattern. I. Introduction Private equity (hereafter, PE) assets grow rapidly and have continuously caught investors attention in recent years. According to Preqin (017), a leading data provider on alternative assets, PE assets under management reached $.49 trillion, an all-time high, as of June 016. They report that 84% of investors have a positive perception of private equity and 48% of investors plan to increase their allocations to private equity over the long term.

While many investors use PE assets to generate higher alphas and/or increase the level of diversification to their portfolios, many investors have concerns about the risks of PE assets. In the academic world, scholars have not yet reached consensus on the risks of PE assets because of a lack of data. Due to the illiquidity of PE assets, no continuous market prices exist for these assets. Therefore, traditional risk methods that are applied on time-series return data cannot directly be used on PE assets. Academic researchers often analyze two types of risks of a financial asset: market risk (or systematic risk) and idiosyncratic risk. Market risk influences a large number of assets; investors cannot do much to diversify away this type of risk. Idiosyncratic risk, however, only affects a very small number of assets and can be diversified by holding a large number of uncorrelated assets. While researchers have developed a few new methods to estimate the market risks (market beta) of PE assets, very little work has been done to measure idiosyncratic risks. In order to address this gap, we propose a new method to estimate idiosyncratic risks of PE assets. An improved understanding of idiosyncratic risks enables investors to make better PE asset allocation decisions for three reasons Firstly, a good measurement of idiosyncratic risk helps investors build efficiently diversified PE portfolios. We know that investing in PE funds can be very expensive due to high management fees, due diligence costs, long-term monitoring costs and so on. An investor would want to construct a welldiversified portfolio with the smallest possible number of PE funds. To do this, the investor needs to know what type of PE funds that he should invest in, how to allocate capital to PE asset types, how many funds he should select, and how much marginal diversification benefit he will gain by adding an additional PE fund into his portfolio. In order to answer these questions, the investor needs to measure idiosyncratic risks of each type of PE funds. We use an extreme example to illustrate this. 3

Let us assume that no idiosyncratic risk exists for any PE fund and all PE funds are only exposed to the same market risks. In this case, all PE fund returns will be perfectly correlated. This means that PE funds will always move in the same direction as the market. As a result, investors cannot gain diversification benefits with these PE funds. Any investment in more than one PE funds would be sub-optimal to the investment in only one fund because investors will not gain extra diversification benefits but will pay more if they invest in more than one PE funds. Contrarily, if idiosyncratic risks are large, for example, large enough to dominate market risks, and moreover, idiosyncratic risks are not highly correlated across funds, then funds will neither move with the market nor with each other. But rather, PE funds will move towards various directions. This means that a large number of PE funds will potentially produce significant diversification effects, although the marginal diversification benefit will decrease with the number of funds in the portfolio. The example above demonstrates that the level of idiosyncratic risks affects portfolio choices. We will also show the relationship between idiosyncratic risks and diversification effects using a simple model. Suppose that the following model generates PE fund returns: r it = α + βr ft + ε it, where r it is the return of fund i at time period t; α is the per-period private equity premium; r ft is the return to factor f at time t and ε it represents the idiosyncratic risk to fund i at time t. Under some assumptions 1, the variance of a PE portfolio consisting of N number of equally-weighted individual PE funds can be written as: ar(r N pt ) = 1 ar(r it ) + N 1 N β ar(r ft ) 1 Assume that market risks are orthogonal to idiosyncratic risks; volatilities of market risks are constant over time; and volatilities of idiosyncratic risks are constant across funds and across time. 4

And the marginal effect of an additional PE fund to the variance of a portfolio consisting of N number of funds can be written as: Marginal Effect N N+1 = 1 N(N + 1) ar(r it ) 1 N(N + 1) β ar(r ft ) The marginal effect formula suggests that marginal diversification benefits positively depend on the level of idiosyncratic risk ar(r it ). A large idiosyncratic risk ar(r it ) will produce a large diversification benefit, ceteris paribus. The theoretical optimal number of funds would be reached when the marginal benefits equal to the marginal costs of the additional PE fund. The second reason for why an improved understanding of idiosyncratic risks enables investors to make better PE asset allocation decisions relates to the importance of fund selection. For example, fund managers would not matter much if fund returns are mainly determined by the market conditions. In contrast, fund managers are essential if idiosyncratic risks contribute to a significant part of fund returns. Our empirical results are more in line with the latter case, suggesting that fund selection is important for both Buyout funds and enture funds. The third reason relates to performance attribution. Suppose a fund manager achieves a very good return for his PE fund, measure by IRR and/or the money multiple, and hence claims being a top PE manager. How much credit shall investors give to him? Is this return due to the manager s unique skillsets or due to favorable market conditions during the fund s holding period? ery likely, both the manager and the market play a role in generating the return. But in order to fairly reward the fund manager s contribution, we need to calculate the manager s component, i.e., the idiosyncratic risk component. Our research provides a way to calculate the manager s component at an aggregated level. We estimate idiosyncratic risks at the asset-type (i.e., Buyout and enture) level, but not yet at the fund level. In order to calculate manager s contribution for each fund, further research needs to be done. We will address this issue later in section I. 5

To overcome the challenge of lack of time-series data, our method directly link cross-sectional performances with idiosyncratic risks. So our method only requires summary performance data such as IRRs or money multiples, which are more generally available and more accurate than intermediate valuations such as net asset values (NAs). We assume that only two sets of risk factors drive a PE fund s return: market-wise risk factors and fund-level risk factors. The former affect a large number of funds in the market and the latter impact only the underlying fund. Instances of typical fund-level risk factors are the capability of fund managers, unique fund strategies, and good relationships between fund managers and stakeholders. If funds face identical market risks, then fund-level risk factors explain differences in their returns, if any. In contrast, if funds face different market risks, then both market risk factors and fund-level risk factors will drive differences in fund returns. We develop our method based on the intuition that if we can identify the market component (market-wise risk factors) of a PE fund s return, than we can also identify the idiosyncratic risk component (fund-level risk factors). We consider two variables that determine market risk exposures of a fund: the holding period and the starting period. If the holding and/or starting periods of funds differ, then they will face different market conditions during their nonoverlapping time periods. Suppose that a few funds start from the same time period, but have different holding periods, then funds with relatively longer holding periods will be exposed to market risks after funds with shorter holding periods have exited the market. To accounts for these heterogeneous market risk exposures originated from non-overlapping time periods, we analyze three scenarios. In the first scenario, we assume identical holding periods for all funds that belong to the same asset type and that start from the same vintage year. We call this scenario homogeneous holding period scenario. In this case (Case I), idiosyncratic risks fully explain cross sectional variances in fund returns. In the second case (Case 6

II), we assume heterogeneous holding periods across PE funds. In this case, both market risks and idiosyncratic risks play a role in generating cross-sectional variance in fund returns. The third case (Case III) extends Case II and assumes heterogeneous holding and starting periods for funds that start in the same vintage year. For each of these three cases, we derive the analytical expression for expected cross-sectional variances in fund returns under some regulatory assumptions. In addition, we calibrate idiosyncratic risk estimators for Buyout funds and enture funds and for each case separately. Our base estimation results show that the annualized idiosyncratic risk (σ ε ) for Buyout fund is 3.9%, using IRR as the return metric, and 18.4% when the money multiple is used. For enture fund, estimated idiosyncratic risks are somewhat larger, 36.% and 5.1%, respectively, using IRR and the money multiple as return metrics. Our estimators are fairly robust to different parameters, although the estimators are more robust when IRR is used compared to when the money multiple is used. Using the same parameters that are used in the base estimation, we then decompose the total cross-sectional variance into three components: alpha component, beta component, and idiosyncratic risk component. Our results show that the idiosyncratic risk component consistently contributes to over 70% of the total cross-sectional variance. This result suggests that fund-level factors are the dominant factors in explaining across-fund performance variations. This result supports the hypothesis that fund selection matters. Finally, we estimate idiosyncratic risks by vintage year. Similar to Ang et al. (014), we find cyclicality in idiosyncratic risks estimators in both Buyout funds and enture funds. Our results show that idiosyncratic risks are large in pre-000 years for enture funds and large in years 003 and 004 for Buyout funds. In contrast, the idiosyncratic risks are low in years 007 to 009 for both types of funds. These patterns confirm anecdotes that 1999 was the best year 7

ever for enture funds, as was 003 to Buyout funds, and that 007 to 009 were the worst years for PE funds. The paper proceeds as follows: Section II reviews relevant private equity literature. We then introduce our theoretical framework in Section III. Subsequently, we present data and descriptive statistics in Section I. In Section, we calibrate idiosyncratic risk estimators. Finally, we provide a conclusion in Section I. II. Literature Review Several previous researchers have investigated the abnormal performance (α) and risk (largely, market β) of PE investments. Ljungqvist and Richardson (003) are the first researchers who analyzed return and risk characteristics of PE funds. Using actual cash flows of PE funds raised over years 1981 to 001, they estimate alpha at 5 plus percent per year, relative to the S&P 500 index. Cochrane (005) uses IPO/acquisition data as well as round-to-round financing data from entureone database to estimate alpha and beta of enture Capital (hereafter, C) investments. The author notices that bad performing C investments are underrepresented in the sample as C investments with bad performance are more reluctant to report their returns and are less likely to survive than their well-performing counterparts. To correct for this selection bias, the author assumes a selection function to account for the biased probability of getting new financing or going out of business. The author recovers a log alpha of -7%, which correspond to an annual arithmetic alpha of 3%. The beta estimate for arithmetic returns is.0, however, the estimated beta for log returns is much lower, at only 0.4. Korteweg, and Sorensen (010) correct for the selection bias and survivorship problem through adding a selection process. They find that betas for 8

C investments are consistently above., with an average of.8. They also report a monthly alpha (simple dollar-weighted) of.5% for C investments. This implies a large annually alpha of 30%, close to the estimates of Cochrane (005). Phalippou and Gottschalg (009) use mature PE funds (over 10 years old) and treat funds remaining net asset values (NA) as zero. They also use a different weight scheme to incorporate investment speed. Instead of using the standard capital-committed weighting, they use present values of investments to weight funds. They came to a net-of-fees alpha estimate of -3% per year, using the S&P 500 index as the proxy for the market. The alpha estimate drops to -6% per year after risk adjustment on leverage, industry, and size. Franzoni, Nowak, and Phalippou (01) use the four-factor model of Pastor and Stambaugh (003), which includes a liquidity risk factor on top of Fama- French s three factors. They estimate an alpha of zero and market beta of 1.3 for Buyout investments. Driessen, Lin, and Phalippou (01) use a dynamic IRR approach where discount rates vary over time. This dynamic IRR methodology allows them to estimate risks and abnormal returns of PE funds using only cash flow data. They find that Buyout funds have a net-of-fees market β of 1.3, and a statistically insignificant negative α both before and after fees, and for both CAPM and Fama- French 3-factor model. For enture funds, they estimate an after-fee market beta of.7 and an after-fee abnormal performance of -1% per annum with CAPM. Harris, Jenkinson, and Kaplan (014) use a new data set of investment cash flows from Burgiss, a record keeping and performance monitoring service provider for institutional investors. Using this new data set, they estimate an alpha of more than 3% per year for Buyout funds and a negative alpha (after 1998) for enture funds, relative to the S&P 500 index. Using the authors value of average PME over 000s, 0.91 in their Table III and a typical fund duration of 5 to 10 years, the implied alpha estimate is around -1% to -% per year for enture funds. 9

Jegadeesh, Kraussl, and Pollet (015) use publically traded PE fund-offund (FoF) and publically traded PE funds (LPE) to estimate abnormal return and risk of private equity. Their alpha estimates are between -0.5% and % per year and market beta estimates are around 1.0 for PE investments. Robinson and Sensoy (011a) report that Buyout funds outperform the S&P 500 index by about 18% over funds life time, while enture funds by about 3%, net of fees. Their results imply an annualized alpha estimate of about 3%-4% for Buyout funds and 0.5% for enture funds. Kaplan and Schoar (005) found that the average PMEs to enture funds and Buyout funds are almost the same, at 0.96 and 0.97, respectively. A close-toone PME means that PE funds perform more or less the same as the S&P 500 index. Hence, their implied alpha estimates are close to 0. Peng (011) use financing data of 5,543 enture-backed firms which raise capital during the time periods of 1987 to 1999 and build a enture Capital index. His beta estimate for C investments is 4.66, relative to the NASDAQ index. Axelson, Sorensen, and Stromberg (014) use a deal-level data set for Buyout investments from a large fund-of-funds. They report an alpha of 8+% per year and a beta of greater than for individual Buyout fund investments gross of fees. Higson and Stucke (01) use cash flow data of a large number of U.S. Buyout funds which accounts for 85% of capital. They recover an alpha estimate of about 4.5% to 8% per year, relative to the S&P 500 index. Ewens, Jones, and Rhodes-Kropf (013) test a verified version of principleagent theory on PE assets using a large data set of C fund returns. They find an alpha of 1.6% to.55% per quarter on C funds. Hence, their annualized alpha is between -6.4% and 10.%. The authors estimate beta at 1.4 for enture funds and 0.7 for Buyout funds. 10

Korteweg and Nagel (016) use a stochastic discount factor (SDF) approach to valuate performance of C investment. They choose the SDF which reflects risk-free rates and returns of public equity markets during the sample period and find that the PME approach overstate alpha for over 15%. Moreover, they find that the upward bias is especially strong when public equity return is high and when investments have betas that are far from one. Robinson and Sensoy (016) use a β of 1.3 for Buyout funds and.0 for enture Capital funds in their paper. Table 1 summarizes all estimators of α and β in these literature. Table 1 Estimates From The Literature Literature Buyout enture Buyout enture Axelson, Sorensen, and Stromberg (014)* 8+%.0 Cochrane (005) * 3%.0 Driessen, Lin, and Phalippou (01) 0% -1% 1.3.7 Ewens, Jones, and Rhodes-Kropf (013) [-6.4%, 10.%] 0.7 1. Franzoni, Nowak, and Phalippou (01)* 0% 1.3 Harris, Jenkinson, and Kaplan (014) 3% [-%, -1%] Higson and Stucke (01) [4.5%, 8%] Jegadeesh, Kraussl, and Pollet (015) [-0.5%, %] [-0.5%, %] 1.0 Kaplan and Schoar (005) 0% 0% Korteweg and Sorensen (010) * 30%.8 Ljungqvist and Richardson (003) 5+% 5+% Peng (001)* 4.7 Phalippou and Gottschalg (009) [-6%, -3%] Robinson and Sensoy (011a) [3%, 4%] 0.01 Robinson and Sensoy (016) 1.3.0 Note: Papers marked with * are the ones in which investment-level data are used. Estimators from these papers are estimated gross of fees. Estimators from other papers are estimated net of fees. To roughly match estimates of α in the literature, we use a BO = 3%, a E = 3% in our base estimation. Note that these are logarithmic alphas as we 11

work with a log model. To convert our logarithmic alphas into their corresponding arithmetic alphas, one can use the following formula: α = a + 1 σ ε, where α on the left hand side is the arithmetic alpha, a on the right hand side is the logarithmic alpha, σ ε is the standard deviation of fund idiosyncratic risk. Regarding the β parameters, we follow Robinson and Sensoy (016) and use β BO = 1.3, β BO = in our base estimation. In our robustness tests, we test our estimators against a wider range of values for both α and β parameters. III. Theoretical Framework Suppose that number of PE funds, which all belong to a certain asset type (e.g., Buyout, enture, etc.), start in the same vintage year. Let T i denotes the holding period for fund i. That is, fund i starts in vintage year and liquidates in vintage year + T i. Suppose we use a single-factor model for fund log returns log(1 + r i ). That is, we assume, for each fund i = 1,,,, its log return at time t can be written as: log (1 + r it ) = a + β f t + ε it,where r it is the return for fund i at time t; a is the PE premium which is assumed to be constant at each time period; β is the sensitivity of fund return to market movements. β is assumed to be constant for funds that belong to the same sub-category but can differ for funds that are in different sub-categories; f t is the log market equity premium at time t; ε it is the idiosyncratic risk for fund i at time t. The model can be easily extended into multi-factor models. 1

Then the log money multiple for fund i can be written as follow: +T i +T i +T i log(mm i ) = log(1 + r it ) = a T i + β f t + ε i,t (1) t=+1 t=+1 t=+1 We simulate five Buyout funds to show how fund log returns move, relative to the market movements, over the same time periods,. For each Buyout fund, we randomly draw a holding period from the distribution and then use formula(1) to generate its log money multiple. These five Buyout funds have different holding periods. From Fund 1 to Fund 5, fund holding periods are 1.5, 5.59, 1.65, 6.47, 3.3, respectively. Simulated log money multiples for Fund 1 to Fund 5 are 0.543, 0.167, 0., 0.594, 0.948 and arithmetic money multiples are 1.7, 1.18, 1.5, 1.81,.58, respectively. Over the same period, log money multiple (arithmetic money multiple) are 0.515 (AMM: 1.674) and 0.30 (AMM: 1.377), respectively, for the market with and without private equity premium (3% per year). Simulation results are shown in Graph 1. Note that once a fund exits the market, its log money multiple does not change any more. As a result, its log money multiple curve levels off over the remaining periods. Graph 1: Accumulative Log Returns, PE Funds versus The Market 13

In addition from the log return model, we make the following assumptions: Assump. I: f vector are multivariate-normally distributed f = (f +1,, f +t, f +Maxi (T i )) ~N N (μ f, Σ f ) Assump. II: ε i vector (for any i) are also multivariate-normally distributed ε i = (ε i,+1, ε i,+t,, ε i,+maxi (T i )) ~N N (0, Σ εi ) i Assump. III: f +t is independent from f +s for, t s, Cov(f +t, f +s ) = 0 Assump. I: the variance of f +t is constant for, t, ar(f +t ) = σ f Assump. : systematic risk factors are orthogonal to idiosyncratic risks, Cov(f +t, ε i,+t ) = 0, Cov(f +t, ε i,+s ) = 0, i,, t, s t Assump. I: idiosyncratic risks have homogeneous variances ar(ε i,+t ) = σ ε i,, t Assump. II: idiosyncratic risks are independent across funds and across time Cov(ε i,+t, ε i,+s ) = 0, Cov(ε i,+t, ε j,+t ) = 0, Cov(ε i,+t, ε j,+s ) = 0,, i j, t, s t Based on Assumption III and Assumption I, we can write the covariance matrix of f vector as follows: 0 Σ f = [ ] 0 σ f σ f Based on Assumption I and Assumption II, we can write the covariance matrix of ε i vector as follows: 14

Σ ε = Definition: Cross Sectional ariance [ σ ε 0 Σ εi = [ ], i 0 σ ε σ ε 0 0 0 [ ] [ ] 0 σ ε 0 0 0 0 σ ε 0 [ ] [ 0 0 0 σ ε We define the equally-weighted cross-sectional variance measure CS for funds log money multiples as follows: ] ] CS = 1 ( (log(mm N i ) log(mm)) ) (), where log(mm i ) is the log money multiple for fund i; log(mm) is the simple average of log money multiples of all these number of funds. log(mm) can be written as log(mm) = 1 ( log(mm N i )) (3) Similarly, the value-weighted cross-sectional variance measure CS W for funds log money multiples is defined as: CS W = w i (log(mm i ) log(mm) W ), where w i is the weight for fund i and log(mm) W is defined as follows: log(mm) W = w i log(mm i ) 15

In our theoretical analysis, we work with the equally-weighted crosssectional variance due to simplicity purposes. However, in our empirical exercises, we also simulate results for the value-weighted cross-sectional variance CS W where funds are weighed by fund values (in US Dollar), instead of by 1 in the equally-weighted case. To understand the properties of cross-sectional variances (CS ) in fund returns, we start the analysis by assuming constant homogeneous holding periods across PE funds, i.e., T i T (Case I) and then extend the analysis to more generalized cases of heterogeneous holding periods (Case II) and heterogeneous holding periods & heterogeneous starting periods (Case III). The following three propositions establish links between CS and market risks & idiosyncratic risks under each of the three scenarios. Case I: Constant Homogeneous Holding Periods across PE Funds Proposition 1: under the assumption of constant homogeneous holding periods, the expected cross-sectional variance in log money multiples across number of funds can be written as: E(CS ) = Tσ ε (1 1 ) (5) Proof can be found in Appendix. This result shows that, under the homogeneous holding period assumption, cross-sectional dispersion in log money multiple is purely driven by idiosyncratic risks. Expected cross-sectional variance positively and linearly depend on the variance of idiosyncratic risks. Everything else being equal, as fund holding period T increases, we would expect that the cross-sectional dispersion of log money multiples gets wider. This model also predicts a positive relationship between E(CS ) and the number of funds ( ), although the marginal effect of to E(CS ) is small, in an order of ( 1 ). But nevertheless, this model predicts a relatively big cross-sectional variance in years with large number of PE funds. 16

We find opposite evidence to this prediction. For example, the number of PE funds reached a historical high in years 007 and 008, but the cross-sectional variances in these two years are the lowest among our sample periods. Case II: Heterogeneous Holding Periods across PE Funds In reality, PE funds can differ significantly (for a few years) in holding periods as illustrated below. Five PE funds all start from the same vintage year, while liquidate at different point in time. Hence, these five funds have various holding periods. We call this heterogeneous holding period scenario. If we use a homogeneous constant holding period model for this heterogeneous holding period scenario, we would mistakenly attribute the crosssectional dispersion fully to idiosyncratic risks, while actually a part of this crosssectional variance is originated from market risks. In another word, we would overestimate idiosyncratic risks using Model I. The estimator derived from Model I are the upper bound of the true idiosyncratic risk. Model II correct the estimator by accounting for heterogeneous holding period effects. Proposition : under the assumption of heterogeneous holding period, the expected cross-sectional variance in log money multiples across number of funds can be written as: 17

N N E(CS ) = 1 (a + aβμ N f ) (T i 1 T N j ) j=1 i + 1 [(T N i ( T N s + ( i)t i ) + 1 N ( (( j) + 1) T j ))] β σ f s=1 j=1 + 1 (T N i T N i + 1 N ( T s)) σ ε s=1 (10) Proof can be found in Appendix. Formula (10) shows that, in Case II, the cross-sectional variance comes from three components. Compared to Case I, two additional components arise as a result of the heterogeneous holding period assumption. The first additional component is the alpha component. As we assume that a fund s alpha return linearly depends on time, heterogeneous holding periods implies that total PE premiums (a T i ) would be different across funds. This alpha component would disappear if a = 0 or T i T. The second additional component is the beta component. As beta return linearly depends on the market conditions over a fund s holding period, funds with different holding periods will have different beta returns. The beta effect would vanish when β = 0 or T i T. The idiosyncratic risk component in Case II is very similar to the one in Case I. The coefficient of idiosyncratic risk in Case II would be the same as the one in Case I when T i T. Suppose in Case I, we set T equal to the average holding period: T = 1 N T N i. Formula (5) and (10) implies that for a given value of σ ε, Model II would on average produce a larger cross-sectional variance than Model I, ceteris paribus. Or, to generate the same cross-sectional variance, Model II would require a smaller σ ε than Model I. In the special case of =, E(CS ) = 1 [(a + aβμ f )(T T 1 ) + β (T T 1 )σ F + (T 1 + T )σ ε ]. It is easy to see that both the alpha component and the beta component exist when T 1 T. The alpha component (a + aβμ f )(T T 1 ) captures two parts of variances. The first part of variance stems from 18

different alpha returns and the second part of variance results from interactive effects between the alpha return and the beta return. The beta component β (T T 1 )σ F captures variances stemmed from different beta returns which are results of the market conditions during the T T 1 time period. Case III: Heterogeneous Holding Periods and Heterogeneous Starting Periods Funds that start in the same vintage year often do not start to invest at exactly the same time. For example, suppose both fund A and B start in the same vintage year. Fund A starts to invest in January while fund B starts its investment in December. These two funds actually start from very different time periods, despite they both start in the same vintage year. Since a typical Buyout fund is on average held for 4.1 years in our sample, an eleven-month difference in starting periods accounts for around % of the total holding period (= we use Model II which implicitly assume homogeneous starting period in this 11 1 ). If 4.1 example, we will get an overstated idiosyncratic risk estimator. This is because the market risk stemmed from heterogeneous starting period will be mistakenly regarded as idiosyncratic risk by Model II. In order to correct this upwards biased estimator, Model III assumes heterogeneous starting period. A simple illustration of heterogeneous starting period scenario is shown in the graph below. All funds in the graph start in the same vintage year. 19

Proposition 3: under the assumption of heterogeneous holding period and heterogeneous starting period, the expected cross-sectional variance in log money multiples across number of funds can be written as: E(CS ) = 1 (a + aβμ N f ) (T i 1 T j ) j=1 + { 1 T N i N [ T s + ( i)t i iτ i τ s ] i s=1 s=i+1 + 1 N [(( j) + 1)T j (jτ j + τ s )]} β σ f j=1 + ( T i + 1 N T j) σ ε (13) Proof can be found in Appendix. j=1 s=j+1 The heterogeneous starting period assumption affects the beta component. This is because starting period is one of the two variables which determine the market risk exposure of a fund. For example, a fund that starts three months earlier would be exposed to different market conditions than if it starts three months later. In contrary, starting period does not affect the alpha component or the idiosyncratic risk components. This is because both a and σ ε are constant 0

over time such that the alpha component and the idiosyncratic risk component only depends on the holding period {T i }, i = 1,,. Therefore, the three components in Case III are essentially the same as the ones in Case II, except that the coefficient of the beta component is somewhat different. Heterogeneous staring period {τ i } enters the cross-sectional variance formula in Case III, while it does not in Case II (where τ i 0, i). To summarize, we build our theoretic framework with multi-periods log models because these models can easily account for heterogeneous holding period and heterogeneous starting period. However, log models are difficult to work with if one wants to study idiosyncratic risks at a portfolio level. Instead, one-period models are ideal for portfolio-level analysis. In the Appendix, we include some theoretic analysis using one-period models. I. Data and Descriptive Statistics Data We obtain PE fund data from Preqin. As of June 30, 017, Preqin has 656 unique PE funds. For each fund, Preqin has information on vintage year, status, fund value, asset type, region focus, GP location, net money multiple, net IRR, benchmarks, and industry focus. Graph shows the number of PE funds by asset type. As we can see, Buyout funds, Real Estate funds, and enture funds are the three largest subcategories (in terms of number of funds) of PE funds in our sample. Because the nature of investment deals in Early Stage funds and Expansion / Late Stage funds are fairly close to enture funds, we reclassify these funds - Early Stage funds and Expansion / Late Stage funds - as enture funds. The reclassification 1

turns enture funds into the second largest sub-category. Our empirical analysis mainly focuses on the two largest asset types, Buyout funds and enture funds. Graph : Number of Private Equity Funds by Asset Type Graph 3 shows the number of PE funds by vintage year. The number of PE funds increase rapidly in the late 1990s and since then more than 100 PE funds have been founded each year. Not surprisingly, an all-time high number of PE funds were raised in year 007, right before the global financial crisis. Graph 3: Number of Private Equity Funds by intage Year

Sample Selection We keep a fund in our sample only when the fund has complete performance data in both IRR and the money multiple. Out of 656 funds, 83 funds do not have information on the money multiple and 170 observations miss the IRR information. Moreover, 4 funds have their money multiples equal to zero, and 8 observations have zero IRRs. We have to leave all these observations out as we cannot compute the holding period for these funds. We are left with 4891 funds after the elimination of these funds. Our sample covers a wide range of vintage years from 1969 to 016. We focus on funds started between year 1997 and 01 based on two considerations. Years before the late 1990s contain relatively small number of PE funds. And PE investment markets were quite different in earlier years. Years after 01, however, contain a significant part of funds that are not or not close to liquidations. Return data for these immature funds are not fully reliable. After the truncation on vintage years, our sample contains 3749 fund observations. Finally, we exclude funds with extremely long holding periods (holding period >15 years). Based on practical experiences, it is very rare that a fund is kept for over 15 years. We believe that either these funds are not representative, or their performance data were not correct. Our final sample consists of 968 Buyout funds and 730 enture funds, out from a total of 3738 PE funds. Table presents a summary of our sample. The number of funds fluctuates over the years but it rarely exceeds the threshold of one hundred for both asset type of Buyout and enture. To be consistent with this evidence, we set the number of funds at one hundred in the simulations. A typical Buyout fund in our sample raises or intends to raise 1,138 million USD from its investors and distributes back cash in total of 1.73 times of the initial investment, achieving an IRR of 15.4%, net of fees. A typical enture fund, however, is four times smaller than the average Buyout fund and returns back to 3

its investors a total amount of cash that is 1.49 times of the initial investment, achieving an IRR of 9.5%. A typical enture fund would produce negative return in bad years, for example, years that precede the dotcom bubble. Contrarily, the average Buyout funds are able to deliver consistently positive returns during our sample periods. 4

Total 1698 784 1.61 1.50 0.17 0.104 968 1138 1.73 1.66 0.154 0.145 730 56 1.49 1.3 0.095 0.077 intage No. of Funds Avg. Fund alue (mn $) Table Descriptive Statistics (Funds) Sample Buyout Money Multiples IRR No. of Funds Avg. Fund alue (mn $) Money Multiples IRR No. of Funds Avg. Fund alue (mn $) enture Money Multiples IRR Mean Median Mean Median Mean Median Mean Median Mean Median Mean Median 1997 107 407.1 1.67 0.316 0.133 53 71 1.56 1.50 0.108 0.093 54 113.66 1.94 0.51 0.31 1998 13 530 1.53 1.41 0.101 0.080 69 84 1.55 1.48 0.08 0.091 54 155 1.51 1. 0.15 0.034 1999 113 51 1.38 1.9 0.049 0.049 54 70 1.74 1.73 0.108 0.119 59 39 1.05 0.79-0.005-0.030 000 170 614 1.59 1.44 0.089 0.08 80 905.1.10 0.194 0.03 90 355 1.1 0.99-0.005-0.003 001 97 575 1.68 1.59 0.16 0.093 4 855.17.00 0.48 0.64 55 363 1.3 1. 0.03 0.031 00 74 465 1.65 1.6 0.135 0.109 38 750.01 1.89 0.40 0.06 36 164 1.7 1. 0.06 0.039 003 71 633 1.67 1.55 0.138 0.095 40 997 1.95 1.84 0.36 0.167 31 147 1.30 1.7 0.011 0.036 004 79 557 1.85 1.60 0.110 0.090 47 8.10 1.86 0.186 0.16 3 175 1.48 1.06-0.00 0.01 005 131 765 1.56 1.46 0.097 0.084 83 1076 1.71 1.65 0.130 0.105 48 15 1.30 1.18 0.039 0.034 006 158 148 1.49 1.56 0.068 0.084 101 090 1.60 1.61 0.089 0.096 57 384 1.31 1. 0.031 0.05 007 155 107 1.66 1.61 0.10 0.10 90 1588 1.65 1.63 0.113 0.109 65 6 1.68 1.59 0.088 0.090 008 134 1308 1.60 1.55 0.116 0.111 79 1998 1.65 1.59 0.140 0.135 55 300 1.53 1.37 0.083 0.079 009 54 851 1.58 1.51 0.138 0.133 35 1073 1.58 1.50 0.147 0.131 19 393 1.58 1.65 0.10 0.135 010 57 655 1.55 1.54 0.145 0.143 41 796 1.51 1.54 0.143 0.164 16 93 1.65 1.5 0.148 0.14 011 81 1007 1.51 1.40 0.150 0.139 50 151 1.36 1.34 0.1 0.13 31 09 1.75 1.75 0.195 0.183 01 94 1156 1.40 1.9 0.158 0.138 66 1506 1.41 1.3 0.175 0.148 8 81 1.36 1.3 0.118 0.098 5

As discussed earlier, the holding period is one of the two variables that determine market risk exposures. Graph 4 plots all fund holding periods by year. enture funds generally have longer holding periods than Buyout funds, indicated by the two distributions of fund holding periods. Graph 4 also reveals that funds started in more recently years (009 till 01) have relatively shorter holding periods. We believe that this is because some of these funds are not liquidated yet. Graph 5 plots holding periods, money multiples, and IRRs for funds pooled over all vintage years in our sample but distinguishes between Buyout funds and enture funds. Blue curves in the two holding period plots (two upper plots) represent corresponding normal distributions with the same means and the same standard deviations. The average holding period of Buyout fund in our sample is 4.1 years, with a standard deviation of 1.58 years. enture funds have generally longer and more volatile holding periods. The average holding period for enture is 5.37 years and the standard deviation is.39 years. These numbers are more or less in line with practical experiences that investors often keep a Buyout fund for three to five years while longer for a enture fund. Money multiples (two middle plots) and IRRs (two lower plots) in our sample are in line with the PE literature. PE fund returns generally have long right tails, and enture funds are more right-skewed than Buyout funds. The average money multiple stands at 1.71 and the standard deviation is moderate at 0.71 for Buyout funds. enture funds, however, generate lower money multiples but larger volatilities than Buyout funds. The average money multiple for enture funds is moderate at 1.47, while the standard deviation is high at 1.38. These numbers suggest that a enture fund, which is one standard deviation from the average, would either earn.85 times of its investment or almost lose the whole investment. 6

IRRs show similar pattern as money multiples. Buyout funds on average yield an IRR of 14.47%, versus 8.91% for enture funds. The standard deviation is 16.14% for Buyout funds, while significantly larger (3.8%) for enture funds. Table 3 presents summary statistics for the three variables holding period, money multiple, and IRR for funds pooled over all vintage years in our sample. Graph 4: Distribution of Fund Holding Periods by intage Year Buyout Funds enture Funds The red line in Graph 1 represents the holding period of 1 year. 7

Graph 5: Distribution of Fund Holding Periods and Fund Returns, Pooling Over All intage Years 8

Table 3 Summary Statistics : Fund Holding Periods and Returns Panel A: Buyout Funds Min. 1st Qu. Median Mean 3rd Qu. Max. Std Dev. Holding Periods 0.7 3.04 4.0 4.1 5.04 15.00 1.58 Money Multiple 0.0 1.30 1.6 1.71.03 5.89 0.71 IRR -49.9% 7.% 1.9% 14.5% 0.7% 39.8% 16.1% Panel B: enture Funds Min. 1st Qu. Median Mean 3rd Qu. Max. Std Dev. Holding Periods 0.17 3.68 5.6 5.37 6.80 14.50.39 Money Multiple 0.01 0.78 1.4 1.47 1.77 19.86 1.38 IRR -79.% -4.3% 4.4% 8.9% 13.5% 514.3% 3.8% Table 4 presents the square root of cross-sectional variances (or CS) in fund returns by vintage year. For Buyout funds, the square root of average CS across all vintage years is 0.451 and 0.135, respectively, using the money multiple and IRR as the return metric, in the equally-weighted basis. With a fund-value-weight scheme, the square root of average CS across all vintage years is somewhat lower, 0.359 and 0.101, respectively, using the money multiple and IRR as the return metric. This implies that returns from funds with relatively large fund value are closer to the average return than funds with smaller fund value. As expected, enture funds have larger CSs than Buyout funds, for both return metrics. The square root of average CS across all vintage years is 0.710 and 0.00, respectively, with money multiple and IRR as the return measure in the equally-weighted basis. The square root of average CS is again lower, 0.594 and 0.179, respectively, with money multiple and IRR as the return measure in the fund-value-weighted basis. We use these eight CS values to calibrate our idiosyncratic risk estimators in our base estimation. We want to emphasize that all returns in our paper are by default in log terms. 9

Table 4 Descriptive Statistics (Square Root of Cross Sectional ariances) Panel A: Buyout Funds Equally Weighted alue Weighted intage LogMM LogIRR Log MM Log IRR 1997 0.566 0.145 0.599 0.15 1998 0.593 0.147 0.63 0.150 1999 0.516 0.1 0.466 0.104 000 0.356 0.090 0.8 0.071 001 0.360 0.117 0.69 0.084 00 0.39 0.135 0.83 0.097 003 0.818 0.50 0.339 0.107 004 0.539 0.169 0.375 0.133 005 0.356 0.118 0.88 0.09 006 0.438 0.090 0.394 0.070 007 0.368 0.085 0.305 0.063 008 0.94 0.086 0.09 0.06 009 0.305 0.110 0.7 0.075 010 0.98 0.11 0.10 0.083 011 0.347 0.150 0.73 0.105 01 0.316 0.140 0.1 0.10 Sqrt of Sample Mean CS 0.451 0.135 0.359 0.101 Std. Deviation of CS t 0.143 0.041 0.13 0.08 Panel B: enture Funds Equally Weighted alue Weighted intage LogMM LogIRR Log MM Log IRR 1997 0.990 0.364 0.791 0.353 1998 1.00 0.358 0.875 0.344 1999 1.054 0.13 0.867 0.156 000 0.661 0.114 0.478 0.074 001 0.681 0.105 0.554 0.088 00 0.605 0.156 0.57 0.146 003 0.688 0.134 0.609 0.109 004 0.757 0.30 0.65 0.87 005 0.845 0.176 0.76 0.13 006 0.617 0.115 0.538 0.096 007 0.685 0.147 0.570 0.15 008 0.58 0.146 0.564 0.147 009 0.37 0.108 0.74 0.069 010 0.33 0.104 0.339 0.114 011 0.43 0.173 0.348 0.135 01 0.565 0.195 0.356 0.136 Sqrt of Sample Mean CS 0.710 0.00 0.594 0.179 Std. Deviation of CS t 0.14 0.086 0.185 0.090 Graph 6 plot the square root of CS against vintage year, in which the two black lines represent the square root of the average CS across all vintage years, in log money multiple and in IRR, respectively. 30

Overall, CSs in money multiples show a downwards cyclical pattern for both Buyout funds and enture funds. CSs in money multiple were at a high level in the late 1990s, then dropped significantly in the early 000s, followed by a recover around year 003 and 004 and a historical low around the 007-008 global financial crisis. Motivated by this time-varying cyclical pattern in CSs, we estimate idiosyncratic risks by vintage year and we find a similar cyclical pattern in our idiosyncratic risk estimates for both Buyout funds and enture funds. Graph 6: Sqrt of Sample CSs of Log Returns by intage Year 31

. Estimate Idiosyncratic Risks We estimate idiosyncratic risks σ ε through simulations. For Buyout funds and enture funds, the simulation proceeds as follows: we first assign a value to σ ε and use the following formula to generate log money multiples for number of funds. As discussed earlier in Section I, we choose = 100 in our simulations. 3

+T i +T i log(mm i ) = a T i + β BO f t + ε i,t t=+1 t=+1 Following Renkema et al. (015), we define the holding period T i as follows: T i log(mm i, (1 + IRR i )) Or we can write it as: (1 + IRR i ) T i MM i 1 + R i Taking the log for both sides of the above equation, we would get: log (1 + IRR i ) 1 T i log (MM i ) Therefore, using the holding period T i of the underlying fund, we can calculate the simulated log(irr i ) from the simulated log(mm i ). For each simulation, we observe a CS in both the log money multiple and the log IRR, across these number of funds. We repeat this simulation procedure for a thousand times. The average of these thousand cross-sectional variances is then the simulated expected cross-sectional variance (or E(CS)). We then assign different values to σ ε and repeat the whole simulation process for each value of σ ε. At the end of the simulation, we obtain a simulated value for E(CS) in both log money multiples and log IRRs for each given value of σ ε. Analytical Expression erification To verify whether the analytical expressions (10) & (13) are derived correctly, we plot simulation results against corresponding analytical results in Graph 7. Analytical results are directly computed using formulas (10) & (13) with the same parameters and fund holding periods as the ones used in simulations. The upper graph is the verification for formula (10) and the lower graph is for formula (13). 33

Graph 7: erification: Simulation Results versus Analytical Results Parameters used: = 100, μ f = 0.05, σ f = 0.16, σ ε = [0: 0.5] with intervals of 0.004 plus one additional point of σ ε = 1, a BO = 0.03, β BO = 1.3, repetition=1000. Holding periods T i for these 100 Buyout funds are randomly drawn from the holding period distribution and then sorted increasingly. Starting periods τ i are zero in Case II. In Case III, starting periods τ i are randomly drawn from a uniform distribution U[0,1]. Simulation results (the red curve) closely resemble the analytical results (the green curve), with relatively small discrepancies on the small values of σ ε. 34

We believe that these discrepancies are essentially small simulation errors. These plots positively support that our analytical formulas are correct. Simulation Results Graph 8 and 9 plot simulation results for Buyout funds and enture funds, respectively. We plot results separately for the two return metrics, the money multiple and the IRR, and for the two weight schemes, the equal weight scheme (EW) and the fund-value weight scheme (W). For each plot, the three sets of results obtained from Case I, II, and III are plotted together. Simulation results are in line with our expectations. For example, given σ ε, Model I on general produces smaller CS than both Model II and III, regardless of return metric. This result is intuitive. In Model I, CS is purely driven by random shocks to each of the hundred simulated funds, while in Model II and III, CS is driven by two additional sources: market risks and heterogeneous holding periods (and starting periods). We know from analytical formulas (10) and (13) that the y-intercept in each of these plots represents a combined effects of both the alpha component and the beta component. As we discussed earlier in Section III, no alpha component nor beta component exist in Model I. Hence, the y-intercept for Model I should be always zero, which is indeed true according to our simulation results. In contrast, alpha effects and beta effects are expected to appear in both Model II and III, as results of the heterogeneous holding period assumption. Simulation results confirm that y-intercepts are indeed positive for Case II and III. Additionally, we expect that idiosyncratic risk will play an increasingly dominant role in determining CS as σ ε gets larger. Simulation results are consistent with this expectation. The initial curvature shown when σ ε is small gradually disappears. Eventually, E(CS) becomes linearly dependent on σ ε as σ ε increases. In other words, both alpha component and beta component become negligible when σ ε gets large enough. 35

Graph 8: Simulation Results for Buyout Funds Parameters used for simulations: = 100, T = 4.1, μ f = 0.05, σ f = 0.16, σ ε = [0: 0.5] with intervals of 0.004 plus one additional point of σ ε = 1, a BO = 3% per year, β BO = 1.3, repetition=1000 times. Holding periods T i for both Case II and Case III are randomly drawn from the Buyout fund holding period distribution and then sorted increasingly. In Case III, starting periods τ i are randomly drawn from a uniform distribution U[0,1]. For value-weighted results, we use the fund value (USD) as weights. 36

37

Graph 9: Simulation Results for enture Funds Parameters used for simulations: = 100, T = 5. 37, μ f = 0.05, σ f = 0.16, σ ε = [0: 0.5] with intervals of 0.004 plus one additional point of σ ε = 1, a E = 3% per year, β E =, repetition=1000 times. Holding periods T i for both Case II and Case III are randomly drawn from the enture fund holding period distribution and then sorted increasingly. In Case III, starting periods τ i are randomly drawn from a uniform distribution U[0,1]. For value-weighted results, we use the fund value (USD) as weights. 38

39

Table 5 presents our benchmark estimators. Estimator is obtained if the σ ε generates a simulated CS which is closest to the observed CS in our sample. 40

For example, using Model I, the σ ε of.3% generates a square root of CS in log money multiples that equals to 0.451, the value observed in our sample. Because both Model II and III correct for heterogeneity in fund holding period, these models produce smaller idiosyncratic risk estimators than Model I. After correction, estimated idiosyncratic risks are 13% to 33% smaller in Model III than in Model I. Interestingly, our results show that IRRs consistently produce larger idiosyncratic risk estimators than the money multiples. One possible explanation for this phenomenon is that the performance data in our sample are not totally consistent. For example, in an equally-weighted basis, the square root of average CS in log money multiple is 0.451 for Buyout funds, while the square root of average CS in log IRR is 0.135. The implied average holding period for Buyout funds should be 3.34 years (=0.451/0.135), which is 19% lower than the average holding period of 4.1 years in our sample. If the two return metrics do not produce consistent estimators for idiosyncratic risks, which metrics should we prefer? We believe that IRR is a better measure than money multiple in our setting because IRR is a timeadjusted measure, while the money multiple is not. Moreover, our robustness tests also provide supports for the IRR measure. Estimators obtained using the IRR measure are more stable against different parameter values than using the money multiple measure. Table 6 presents simulated cross-sectional variances for given values of σ ε, using Model I, II, and III, respectively. This table gives us a sense of how different CSs these models generate. For a given value of idiosyncratic risk σ ε, the difference between the two simulated CSs, generated by Model I and by Model II respectively, is the combined contribution from the alpha component and the beta component. Moreover, the difference between the two simulated CSs, generated by Model II and by Model III, represents the heterogeneous starting period effects. Our results show that heterogeneous starting period effects are rather small, even negligible when σ ε is large. 41

Table 5 Estimate Idiosyncratic Risk, Benchmark Result Panel A: Buyout Funds Equally Weighted alue Weighted Metric LogMM LogIRR Log MM Log IRR Sqrt of Sample Mean CS 0.451 0.135 0.359 0.101 Estimators: Model I.3% 7.6% 17.8% 0.6% Model II 18.8% 4.3% 1.7% 18.0% Model III 18.4% 3.9% 11.9% 17.6% Panel B: enture Funds Equally Weighted alue Weighted Metric LogMM LogIRR Log MM Log IRR Sqrt of Sample Mean CS 0.710 0.00 0.594 0.179 Estimators: Model I 30.9% 46.8% 5.7% 41.5% Model II 5.9% 37.0% 18.6% 33.6% Model III 5.1% 36.% 17.4% 33.% We then decompose CS in log money multiple into three components using analytical expressions (10) and (13). The decomposition results are shown in Table 7. In Model I, CSs are purely driven by idiosyncratic risks, hence both alpha effects and beta effects are zero. Decomposition results are very close for Model II and Model III. Our results show that idiosyncratic risk accounts for a significant part, over 70%, of the total CS in both Model II and III and for both Buyout funds and enture funds. Beta component also generates an important part, 18% to 34%, of the total CS in both Model II and III and for both Buyout funds and enture funds. Alpha effects are relatively small, in a magnitude of 5% of the total CS. Note that the negative alpha effects in enture funds are a result of using parameter a = 3% in the simulation. We want to emphasize that these decomposition results are obtained with the same set of parameters used in our base estimation. Moreover, decomposition 4

results are highly sensitive to the parameters we use. For example, if we use a larger value for parameter a, then the alpha effects would naturally become larger. To generate the same CS value, models will produce smaller estimates for idiosyncratic risk. As a result, relative contribution of the three components would be different. Table 6 Square Root of Cross Sectional ariances by Model I, II, and III Panel A: Buyout Funds, Equally Weighted Metric Models Model I LogMM Model II Model III Model I LogIRR Model II Model III Idiosyncratic Risks: σ ε = 1 % 0.368 0.431 0.445 0.0881 0.1056 0.1084 σ ε = 0% 0.407 0.4703 0.4797 0.0977 0.1154 0.1177 σ ε = % 0.4445 0.5049 0.51 0.1079 0.145 0.16 σ ε = 4% 0.4853 0.5467 0.5538 0.1178 0.1340 0.1360 σ ε = 6% 0.537 0.5838 0.5907 0.171 0.144 0.1458 Panel B: Buyout Funds, alue Weighted Metric Models Model I LogMM Model II Model III Model I LogIRR Model II Model III Idiosyncratic Risks: σ ε = 1% σ ε = 14% σ ε = 16% σ ε = 1 % σ ε = 0% 0.439 0.3478 0.3616 0.059 0.0750 0.0789 0.841 0.377 0.3907 0.0690 0.088 0.0867 0.360 0.4091 0.4190 0.0791 0.0919 0.0944 0.3649 0.4455 0.457 0.0886 0.1006 0.1039 0.4059 0.4851 0.4954 0.0985 0.1095 0.111 Panel C: enture Funds, Equally Weighted Metric Models Model I LogMM Model II Model III Model I LogIRR Model II Model III Idiosyncratic Risks: σ ε = 4% 0.550 0.6773 0.6901 0.105 0.1446 0.1483 σ ε = % 0.6466 0.7513 0.7633 0.104 0.1616 0.165 σ ε = 3% 0.7378 0.830 0.8410 0.1374 0.1797 0.188 σ ε = 36% 0.876 0.9144 0.933 0.1541 0.1958 0.1990 σ ε = 40% 0.931 0.993 1.0010 0.1719 0.135 0.159 Panel D: enture Funds, alue Weighted Metric Models Model I LogMM Model II Model III Model I LogIRR Model II Model III Idiosyncratic Risks: σ ε = 1 % 0.4173 0.590 0.603 0.0777 0.1179 0.11 σ ε = % 0.5095 0.6587 0.6700 0.0949 0.1334 0.1367 σ ε = 6% 0.6039 0.7337 0.74 0.115 0.1483 0.1504 σ ε = 30% 0.693 0.837 0.8300 0.189 0.166 0.1658 σ ε = 34% 0.7906 0.9076 0.9179 0.147 0.1800 0.1834 43

Metric Weights Method Sqrt of Sample Mean CS LogMM Equally Weighted Buyout Models Model I Model II Model III Model I Model II Model III Estimators.30% 18.80% 18.40% 30.90% 5.90% 5.10% Decompose CS: a β σ ε Table 7 Decompose Cross Sectional ariances 0.03 (=0.451^) enture 0.5041 (=0.710^) Effects 0.00% 5.17% 5.13% 0.00% -6.13% -6.18% Effects 0.00% 17.67% 1.6% 0.00% 9.51% 33.69% Effects 100.00% 77.16% 73.5% 100.00% 76.6% 7.49% Total 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% Robustness Check We test how sensitive our estimators are to parameters a, μ f, and β. We select different values for these parameters such that they cover a large part of estimators reported in the PE literature. Simulation results of all robustness checks are obtained with Model III. Table 8 reports the robustness check results. Overall, our estimators are reasonably robust under the equal-weight scheme, while less stable under the fund-value-weight scheme. In our opinion, this is because the observed CS is relatively small when funds are value weighted. Our simulation results show that the same σ ε will generate fairly similar CSs under the two weight schemes. To match the moderate CS under the fund-value-weight scheme, our model consistently produces smaller idiosyncratic risk estimators than it does under the equal-weight scheme. Robustness tests also show that estimators obtained using the money multiple are less robust to parameter a than the ones using IRR. Using analytical formula (13), we have verified that for some large values of a, our model generates quite large CSs in money multiples, even when σ ε is set at zero. A large alpha component naturally leads to a small idiosyncratic risk estimator because the total CS is fixed, ceteris paribus. 44

In contrast, CSs in log IRR are insensitive to parameter a. This phenomenon can be explained by the following formula: lo g(1 + IRR i ) 1 T i lo g(mm i ) = a + β BO T i +T i f t + 1 ε i,t T i t=+1 +T i t=+1 We can see that each fund contains the same a component in its log IRR, hence the alpha component will drop out from the CS in log IRR. Estimate σ ε by intage Year Motivated by the time-varying CSs in our sample, we want to test whether the time-variation in CS comes from market risks, or idiosyncratic risks, or both. We first estimate average market risks for both Buyout funds and enture funds in each vintage year. We obtain monthly market return data from Kenneth French s data library. For each vintage year, we take the monthly market returns of the following several years, calculate the volatility of these monthly returns and then multiply it by 1 to convert this monthly volatility into an annualized market risk. We take the following four years monthly return data for Buyout funds and the following five years for enture funds. We chose the year four and five to roughly match the holding periods of Buyout funds and enture funds. We then estimate idiosyncratic risk through simulations with Model III. For each vintage year, we randomly draw (with replacement) one hundred holding periods from the funds that start in the specific vintage year. Table 9 reports idiosyncratic risk estimators by vintage year. 45

Table 8 Robustness Checks Panel A: Buyout Funds Metric Sqrt of Sample Mean CS Base Estimation (Model III) Sensitivity on a: a = 6% a = 0% a = % Sensitivity on β: β = 1 β = Sensitivity on μ f : μ f = 4% μ f = 6% Equally Weighted alue Weighted Log MM Log IRR Log MM Log IRR 0.451 0.135 0.359 0.101 18.4% 3.9% 11.9% 17.6% 19.7% 3.5% 14.0% 17.4% 18.9% 3.4% 13.0% 17.4% 16.8% 3.4% 9.6% 17.4% 19.8% 4.6% 14.% 18.5% 1.8% 0.0% 0.0% 13.0% 18.7% 3.4% 1.6% 17.4% 18.0% 3.4% 11.7% 17.4% Panel B: enture Funds Metric Sqrt of Sample Mean CS Base Estimation (Model III) Sensitivity on a: a = 1% a = 0% a = 15% Sensitivity on β: β = 1 β = 3 Sensitivity on μ f : μ f = 4% μ f = 6% Equally Weighted alue Weighted Log MM Log IRR Log MM Log IRR 0.710 0.00 0.594 0.179 5.1% 36.% 17.4% 33.% 6.0% 36.0% 18.8% 33.0% 4.0% 36.0% 16.0% 33.0% 11% * 36.0% 0.0% 33.1% 9.8% 39.4% 3.7% 36.6% 11.6% 17.% 0.0% 6.8% 5.6% 36.0% 18.4% 33.1% 4.6% 36.0% 16.8% 33.1% 46

Our results show significant time variation in idiosyncratic risk estimators. Similar to CSs, idiosyncratic risks also show a downwards cyclical patterns. This cyclical pattern is easily identifiable in Graph 10 where we plot the eight time-series of idiosyncratic risk estimators. Idiosyncratic risks were generally high for pre-000 years and year 003 & 004, while low for the financial crisis years for Buyout funds. Similar cyclical pattern is observed in enture funds as well. In comparison with fund-value-weight scheme, equal-weight scheme seems to produce more reasonable estimators. The unreasonably low estimators obtained in year 008 and 009 under the fund-value weight scheme do not show up in an equal-weight case. We think that fund value may not be the best variable for weighting because fund value does not reflect the actual capital that a fund eventually obtains. If funds fail to raise as much capital as they intended to, a frequently occurring event in bad years, fund value would be overstated hence distort our estimators in the value-weighted case. Graph 10: Idiosyncratic Risk by intage Year 47

Table 9 Estimate Idiosyncratic Risk by intage Year Panel A: Buyout, Equally Weighted σ f S rt(cs) intage LogMM LogIRR LogMM LogIRR 1997 0.189 0.566 0.145.1% 8.4% 1998 0.198 0.593 0.147 3.% 9.0% 1999 0.180 0.516 0.1 0.6% 4.0% 000 0.171 0.356 0.090 1.6% 14.0% 001 0.154 0.360 0.117 16.9% 19.4% 00 0.15 0.39 0.135 17.8% 3.6% 003 0.09 0.818 0.50 40.% 45.4% 004 0.13 0.539 0.169 3.0% 31.% 005 0.163 0.356 0.118 10.3% 0.4% 006 0.18 0.438 0.090 16.5% 17.% 007 0.193 0.368 0.085 13.0% 14.5% 008 0.195 0.94 0.086 9.0% 9.% 009 0.16 0.305 0.110 13.9% 15.8% 010 0.135 0.98 0.11 14.9% 14.8% 011 0.11 0.347 0.150 1.0%.0% 01 0.108 0.316 0.140 1.0% 17.3% Panel B: enture, Equally Weighted σ f S rt(cs) intage LogMM LogIRR LogMM LogIRR 1997 0.19 0.990 0.364 45.6% 53.6% 1998 0.189 1.00 0.358 36.8% 60.0% 1999 0.168 1.054 0.13 36.8% 50.4% 000 0.160 0.661 0.114 0.4%.8% 001 0.143 0.681 0.105.4%.4% 00 0.10 0.605 0.156 0.5% 34.8% 003 0.134 0.688 0.134 5.% 9.4% 004 0.153 0.757 0.30 7.0% 60.0% 005 0.169 0.845 0.176 31.5% 36.0% 006 0.178 0.617 0.115 1.4%.4% 007 0.18 0.685 0.147 5.4% 8.8% 008 0.184 0.58 0.146 3.3% 4.9% 009 0.15 0.37 0.108 14.% 14.0% 010 0.135 0.33 0.104 1.5% 17.0% 011 0.119 0.43 0.173 3.4% 6.4% 01 0.101 0.565 0.195 8.0% 8.4% σ ε σ ε 48

Table 9 Estimate Idiosyncratic Risk by intage Year (Cont'd) Panel C: Buyout, alue Weighted σ f S rt(cs) intage LogMM LogIRR LogMM LogIRR 1997 0.189 0.599 0.15 5.% 31.3% 1998 0.198 0.63 0.150 4.% 31.6% 1999 0.180 0.466 0.104 18.3% 0.4% 000 0.171 0.8 0.071 10.% 11.4% 001 0.154 0.69 0.084 1.3% 13.4% 00 0.15 0.83 0.097 1.5% 15.3% 003 0.09 0.339 0.107 15.5% 19.% 004 0.13 0.375 0.133 15.0%.3% 005 0.163 0.88 0.09 6.4% 17.1% 006 0.18 0.394 0.070 13.9% 13.9% 007 0.193 0.305 0.063 7.9% 9.8% 008 0.195 0.09 0.06 0.0%.1% 009 0.16 0.7 0.075 8.5% 10.4% 010 0.135 0.10 0.083 8.8% 10.% 011 0.11 0.73 0.105 15.4% 14.8% 01 0.108 0.1 0.10 14.5% 11.6% Panel D: enture, alue Weighted σ f S rt(cs) intage LogMM LogIRR LogMM LogIRR 1997 0.19 0.791 0.353 3.% 5.0% 1998 0.189 0.875 0.344 30.4% 60%+ 1999 0.168 0.867 0.156 8.5% 33.6% 000 0.160 0.478 0.074 10.0% 5.6% 001 0.143 0.554 0.088 17.0% 16.4% 00 0.10 0.57 0.146 16.4% 34.0% 003 0.134 0.609 0.109 1.3% 3.4% 004 0.153 0.65 0.87 1.3% 59.9% 005 0.169 0.76 0.13 6.3%.8% 006 0.178 0.538 0.096 17.9% 17.9% 007 0.18 0.570 0.15 0.4% 3.% 008 0.184 0.564 0.147 3.4% 4.6% 009 0.15 0.74 0.069 7.% 0.% 010 0.135 0.339 0.114 14.1% 18.7% 011 0.119 0.348 0.135 17.% 19.4% 01 0.101 0.356 0.136 16.8% 18.8% σ ε σ ε 49

I. Conclusion An improved understanding on idiosyncratic risks of PE assets helps investors to diversify their PE portfolios more efficiently. To overcome the challenge of a lack of time-series data, we propose a new method which requires only cross-sectional performance data such as the money multiple and IRR. We succeeded in creating a simple, trackable, and easy to work with method, which can also be applied to other illiquid assets such as natural resources, private real estate and infrastructure. However, these benefits come with the cost of having to make restrictive assumptions. For example, our model treats all Buyout (enture) funds identically. Consequently, all Buyout (enture) funds are assumed to have the same sensitivity to market movements. Furthermore, we assume that idiosyncratic risks are independent across funds and across time. This assumption is restrictive and not likely to hold. For example, when a new policy regarding PE investments is introduced, a large number of PE funds are likely to be influenced by this industry-wise random shock. In this case, the assumption Cov(ε i,+t, ε j,+t ) = 0 will be violated. In addition, the assumption Cov(ε i,+t, ε i,+s ) = 0 also does not hold in general. Indeed, we know that a PE fund is usually managed by the same team over a long time period (e.g., five to ten years). Random variables such as fund strategy, investment style, and industry focus are not likely to change very often but these variables will affect fund returns to some extent. Future scholars can try to relax our restrictive assumptions and extend this research in multiple ways. For example, one might desire to estimate idiosyncratic risk for each unique fund. Funds are likely to have different market betas, despite an identical asset type. For example, some funds take relatively higher leverages and/or have narrower industry-focus than the average fund. Consequently, these funds might react differently to market volatilities relative to the average fund. With an idiosyncratic risk estimator for each fund, one can attribute performance more accurately for each fund. 50

One might also want to investigate the correlation between two PE funds. We have briefly explored this arena with a one-period model (see Appendix). Exposing a simple correlation structure, we have shown that one can identify idiosyncratic risks and correlations using PE portfolio data. One can also use a more sophisticated model to generate log money multiples. For example, one can use a multi-factor model to account for industrywise risks as well. Better quality data can also improve idiosyncratic risk estimates. For instance, we have found two inconsistencies within our dataset. That is, the two return metrics yield different estimates and imply a relatively short average holding period. Scholars may want to verify whether these inconsistencies persist with other datasets. References Ang, A., Chen, B., Goetzmann, W.N. and Phalippou, L., 014. Estimating private equity returns from limited partner cash flows. Axelson, U., Sorensen, M., and Stromberg, P., 014. Alpha and beta of buyout deals: a jump CAPM for long-term illiquid investments. Working paper. Cochrane, J.H., 005. The risk and return of venture capital. Journal of financial economics, 75(1), pp.3-5. Driessen, J., Lin, T.C. and Phalippou, L., 01. A new method to estimate risk and return of nontraded assets from cash flows: the case of private equity funds. Journal of Financial and Quantitative Analysis, 47(3), pp.511-535. Ewens, M., Jones, C.M., and Rhodes-Kropf, M., 013. The price of diversifiable risk in venture capital and private equity. The Review of Financial Studies, 6(8), pp. 1854-1889. Franzoni, F., Nowak, E. and Phalippou, L., 01. Private equity performance and liquidity risk. The Journal of Finance, 67(6), pp.341-373. Harris, R.S., Jenkinson, T. and Kaplan, S.N., 014. Private equity performance: What do we know?. The Journal of Finance, 69(5), pp.1851-188. 51

Higson, C., and Stucke, Rudiger., 01. The performance of private equity. Working paper. Jegadeesh, N., Kräussl, R. and Pollet, J.M., 015. Risk and expected returns of private equity investments: evidence based on market prices. The Review of Financial Studies, 8(1), pp.369-330. Korteweg, A. and Nagel, S., 016. Risk Adjusting the Returns to enture Capital. The Journal of Finance, 71(3), pp.1437-1470. Korteweg, A., and Sorensen, M., 010. Risk and return characteristics of venture capital-backed entrepreneurial companies. The Review of Financial Studies, 3(10), pp.3738-377. Kaplan, S.N. and Schoar, A., 005. Private equity performance: Returns, persistence, and capital flows. The Journal of Finance, 60(4), pp.1791-183. Ljungqvist A., and Richardson, M., 003. The cash flow, return and risk characteristics of private equity. NBER Working Paper, (9454). Pastor, L. and Stambaugh, R., 003. Liquidity risk and expected stock returns. Journal of Political Economy, 111(3), pp. 64-685. Peng, L., 001. Building a venture capital index. Yale ICF Working Paper, (00-51). Phalippou, L. and Gottschalg, O., 009. The performance of private equity funds. Review of Financial Studies,, pp.1747-1776. Preqin, 017. Preqin global private equity & venture capital report. Renkema, J., Goorbergh, R..D. and Rivas, C.G., 017. RATZ IRR: Performance measurement in the absence of cash flow data. The Journal of Alternative Investments, 0(), pp.51-63. Robinson, D.T. and Sensoy, B.A., 011a. Private equity in the 1st century: Liquidity, cash flows, and performance from 1984-010. NBER Working Paper, (1748). Robinson, D.T. and Sensoy, B.A., 016. Cyclicality, performance measurement, and cash flow liquidity in private equity. Journal of Financial Economics, 1(3), pp.51-543. 5

Appendix Proof for Proposition 1 Basic Case: Constant Homogeneous Holding Periods across PE Funds Suppose funds have the same holding period, that is, T i T for any i. For funds that start from the same vintage year, these funds experience the same market movements during their full life cycles. Let f be the systematic risks from vintage year to +T : T f = f +t t=1 The equally-weighted average of log money multiples of all individual funds (assume buyout funds) can be written as: T T log(mm) = 1 ( log(mm N i )) = a 1 T + β N BO f +t + 1 ( ε N i,+t ) t=1 t=1 T = α T + β BO f + 1 ( ε N i,+t ) (3 ) t=1 The equally-weighted cross-sectional variance of individual fund log money multiples can be written as: CS = 1 ( (log(mm N i ) log(mm)) T = 1 ( [ ε N i,+t 1 ( ε N i,+t )] ) t=1 = 1 ( [( ε N i,+t ) T T t=1 ) + 1 N ( ε i,+t) ]) (4) t=1 T t=1 T ( ε N i,+t ) ( ε i,+t ) t=1 T t=1 53

Under assumption I and assumption II, the expected CS can be written as: T T T E(CS ) = 1 ( E [( ε N i,+t ) ] E [( ε N i,+t ) ( ε i,+t )] [ t=1 t=1 t=1 N T + 1 N E [( ε i,+t) ] ) = 1 ( [Tσ N ε Tσ N ε + 1 N (Tσ ε )]) t=1 ] = 1 ( [Tσ N ε 1 Tσ N ε ]) = Tσ ε 1 Tσ N ε = Tσ ε (1 1 ) (5) At the limit when approaches infinity, E[CS ] N Tσ ε (6) End of proof for Proposition 1. Proof for Proposition Case II: Heterogeneous Holding Periods across PE Funds Suppose the holding periods for individual funds i = 1,,, are denoted by T 1, T,, T N, respectively. We rank all funds according to their holding periods in an increasing order. That is, T 1 < T < < T N. Then, the equally-weighted cross-sectional variance at individual fund level can be written as: 54

CS = 1 ( (log(mm N i ) log(mm)) = 1 T i ( ( ) a (T i 1 T N j ) + j=1 Denote by S i + ε i,+t 1 ε N j,+t t=1 j=1 t=1 [ Denote by B i ]) ) T j T i T j β f +t β f +t t=1 j=1 t=1 [ = 1 ( (A N i + A i B i + B i + S i + S i A i + S i B i )) Under assumptions, we know that From assumption I, we know that E(A i B i ) = 0 i Denote by A i ] E(A i ) = β T i μ f β (T N 1 + T + + T N ) μ f = βμ f (T i 1 T N i ) From assumption II, we know that We also know that S i is just a coefficient, so Therefore, the expected CS : What does E(A i ) look like? E(B i ) = 0 E(S i ) = S i = a (T i 1 T N j ), j=1 E(S i A i ) = S i E(A i ) = aβμ f (T i 1 T N i ), E(S i B i ) = S i E(B i ) = 0 E(CS ) = 1 ( E(A N i ) + E(B i ) + S i + S i A i ) 55

T i N T j T i N T j E(A i ) = E [(β f +t β f +t ) ] = β E [( f +t 1 f +t ) ] t=1 T i The three elements of E(A i ): j=1 t=1 t=1 j=1 t=1 T i T j N T j = β E [( f +t ) ( f +t ) ( f +t ) + 1 N ( f +t) ] t=1 t=1 j=1 t=1 j=1 t=1 T i E [( f +t ) ] = T i σ f t=1 T i E [( f +t ) ( f +t )] = E [( f +t ) ( f +t )] Therefore, t=1 T j j=1 t=1 T i t=1 T j j=1 t=1 i = (T N 1 + T + T i 1 + T i + T i + + T i )σ f = ( T N s + ( i)t i ) σ f s=1 N T j N T j j E [( 1 f +t ) ] = 1 N E [( f +t) ] = 1 N ( T s + ( j)t j ) j=1 t=1 j=1 t=1 j=1 s=1 = 1 N ( ((N j) + 1) T j ) σ f j=1 i E(A i ) = (T i ( T N s + ( i)t i ) + 1 N (( j) + 1) T j ) β σ f ( ) s=1 j=1 σ f What does E(B i ) look like? T i N T j E(B i ) = E [( ε i,+t 1 ε N j,+t ) ] t=1 j=1 t=1 T i T i T j N T j The three elements of E(B i ): = E [( ε i,+t ) ( ε i,+t ) ( 1 ε N j,+t ) + ( 1 ε N j,+t ) ] t=1 t=1 j=1 t=1 j=1 t=1 56

T i E [( ε i,+t ) t=1 ] = T i σ ε T i T j T i T j E [( ε i,+t ) ( 1 ε N j,+t )] = 1 E [( ε N i,+t ) ( ε j,+t )] = 1 T N i σ ε t=1 j=1 t=1 t=1 j=1 t=1 T j N T j E [( 1 ε N j,+t ) ] = 1 N E [( ε j,+t) ] = 1 N ( T s) σ ε j=1 t=1 j=1 t=1 s=1 Therefore, E(B i ) = (T i T N i + 1 N ( T s)) σ ε (9) s=1 Putting all the elements together, we get the expression for the expected CS : E(CS ) = 1 { [E(A N i ) + E(B i ) + S i + S i E(A i )]} N N = 1 (a + aβμ N f ) (T i 1 T N j ) j=1 i + 1 [(T N i ( T N s + ( i)t i ) + 1 N ( (( j) + 1) T j ))] β σ f s=1 j=1 + 1 (T N i T N i + 1 N ( T s)) σ ε s=1 (10) End of proof for Proposition. Proof for Proposition 3 Case III: Heterogeneous Holding Periods and Heterogeneous Starting Periods Let τ i denote the real start time when fund i starts to draw money from its LPs. Let T i denote the real holding periods for fund i, that is, from the point in time when fund i starts to draw money till the time when fund i liquidates. We assume that τ i follows a uniform distribution τ i ~U(0, Y 1 ),where Y 1 equals to 365 if 57

it is daily data, or equals to 1 if it is monthly data, so on and so forth. (Note that Case II implicitly assumes τ i 0). In addition, we assume that no fund liquidates within one year, that is, τ i + T i > Y 1. The log money multiple for fund i can be written as: τ i +T i τ i +T i log (MM i ) = a T i + β f +t + ε i,+t (1) t=τ i +1 t=τ i +1 If we spit the systematic part of risk around Y 1 and rewrite the log money multiple as follows: Y 1 τ i +T i Y 1 τ i +T i log(mm i ) = a T i + β ( f +t + f +t ) + ε i,+t + ε i,+t (1 ) t=τ i +1 t=y 1 +1 t=τ i +1 t=y 1 +1 The graph below shows the timeline split around year 1. The average log money multiple log(mm) is given by: log(mm) = 1 log (MM i ) Y 1 τ i +T i = α T N i + β ( f +t + f +t ) t=τ i +1 t=y 1 +1 Y 1 τ i +T i + 1 ( ε N i,+t + ε i,+t ) t=τ i +1 t=y 1 +1 The cross-sectional variance CS can be written as: 58