Risk measures in private equity

Transcription

1 Risk measures in private equity The Expected Cumulative Downside Absolute Deviation ( ECDAD ) approach June 2013 ALEXANDRE FALIN Vice President, Private Equity Mandates FARSHID ABDI Student at Ecole Polytechnique Fédérale de Lausanne (EPFL) In this paper, we present a new framework for assessing risk in private equity. Our approach is based on real cash flows rather than interim valuations. In order to overcome irregularities in the available data, we first develop a stochastic model of cash flows, using an unconditional marked point process. We then introduce a new risk measure, the ECDAD, which quantifies the risk that a fund will distribute less or later than expected to investors. Finally, we show simple applications of the ECDAD for risk/return mapping, portfolio diversification and optimal risk-adjusted allocation.

2 Table of contents 1. Objective and approach Commonly used approaches Basis for our risk measure Our approach for implementing a risk measure Stochastic modelling of distributions Appropriate risk measure Risk/return of the different private equity groups Power of diversification in a portfolio allocation context Optimal risk-adjusted allocation Conclusion References Contact Information Important Information Appendix...36 Risk measures in private equity I 2

3 1. Objective and approach 1.1. Objective Measuring risk in private equity is a challenge due to the nature of the asset class. For example, there is no actively traded market for private equity fund investments. Therefore, the valuation of those investments relies on the guidelines of each private equity manager, leading to the same investment being potentially valued differently by two distinct managers. Therefore, quantifying the risk of a private equity fund based on self-reported values might not be the most appropriate way. We endeavour to find a better base for our risk measure. The objective of this paper is to implement an appropriate risk measure that enables us to accurately quantify the risk of a private equity fund and compare the risk of two private equity funds in a straightforward manner Approach We approach the analysis in five main steps, as follows: I. Commonly used approaches: We screen the available research literature in the field of risk in private equity in order to analyse what has been published so far and what type of measures have already been implemented. Then, we provide reasoning on why the current status of the research literature does not entirely fulfil our objectives. II. Basis and approach for our risk measure: We elaborate on the basis we use for our risk measure, i.e. fund cash flows, and we define what risk in private equity means for us. We then outline how we divide the private equity asset class into several groups with similar qualitative characteristics and risk/return profiles. III. Stochastic modelling of distributions using a marked point process: Splitting the private equity asset class into groups leads to a reduction in the sample size of data. Therefore, we implement a proprietary stochastic model in order to overcome the lack of the data and potential irregularities. IV. Appropriate risk measure: We introduce our risk measure, the ECDAD, which measures the downside area between an individual distribution curve and its related expected distribution curve. V. Applications of our risk measure: Using the ECDAD, we first draw a risk/return mapping of all the private equity groups defined previously. Then, we apply the ECDAD to analyse the power of diversification in a portfolio context. Finally, we present the results of an optimal asset allocation done on an ECDAD-adjusted basis, in relation to different return expectations. Risk measures in private equity I 3

4 2. Commonly used approaches 2.1. Research literature Main areas of available research pertaining to risk in private equity entail (i) the correction of private equity market returns to reflect a more appropriate risk, (ii) a quantile approach and power of diversification, (iii) the effect of qualitative factors on fund performance, and (iv) the use of private equity cash flows to estimate risk versus public market Correction of private equity market returns to reflect a more appropriate risk Woodward (2009) describes an approach to account for the stale pricing 1 phenomenon in private equity returns by including lagging public market returns in the standard regression of private equity returns versus public market returns. This way, Woodward attempts to capture the full relatedness of private equity returns to public market returns. Examples for venture capital and buyout portfolios show that the private equity betas (β) to public market are generally more than double of those from naïve measures ignoring lagging market returns Quantile approach and power of diversification Diller and Herger (2010) examine a quantile approach based on the final TVPI 2 of private equity funds. They name the loss quantile icar (invested Capital at Risk). Similar to the well-known Value-at-Risk, the icar is defined as the amount of money such that there is a 99% probability (or confidence level) of the portfolio losing less than that amount over a given period of time. For example, if the worst performing fund of a private equity portfolio made of 100 funds has a final TVPI of 0.15 times capital invested, then the icar will be equal to 85%, i.e. 100% capital invested minus 15% capital recovered. Using a Monte-Carlo historical simulation on funds, Diller and Herger point out that diversification helps reducing the icar significantly. For a portfolio made of 1 fund, the icar is equal to 84% while for a portfolio made of 15 funds in one vintage, the icar falls to 22%. Also, they demonstrate that the diversification over vintages is more important than the diversification within a vintage. Indeed, the icar of a portfolio made of 1 fund per vintage over 15 years has an icar of 0%, which implies that, at a 99% confidence level, there is no risk of losing capital. A similar study about power of diversification in private equity has been made by Weidig and Mathonet (2004). They show the effect of diversification on the distribution of final multiples at two levels: from individual private equity investment level to fund level, and from fund level to fund-of-fund level. On both levels, diversification has proven to be effective in lowering downside risk. 1 Stale pricing is the phenomenon in which the current price of an asset does not reflect the most recent data available 2 TVPI stands for Total Value to Paid-In and is calculated as the sum of Remaining Value to Paid-In and Distributed to Paid-In Risk measures in private equity I 4

5 Effect of qualitative factors on fund performance The effect of qualitative factors such as fund manager track record, geography and segment (e.g. buyout or venture capital) on performance has been quite extensively studied for different datasets. Bernstein et al (2008) and Marquez et al (2011) study performance persistence in private equity. Chen et al (2009) describes the effect of geography on both successful and unsuccessful private equity investments. Woodward (2009) compares the risk of venture capital to the one of buyout based on their return profiles Using private equity cash flows to estimate risk versus public market Driessen et al (2006) attempt to estimate the private equity betas (β) to public market using fund cash flows instead of self-reported values. They come to the conclusion that venture capital funds have a higher beta to public market (1.23) than buyout funds (0.66). Also, they observe that the beta decreases with public stock market performance, which indicates that private equity funds offer a similar risk profile to call options, i.e. a non-linear risk profile. Accounting for this call option risk profile, the betas of venture capital and buyout increase to 1.89 and 1.04 respectively. The call option feature in venture capital is therefore more pronounced than in buyout Does research literature satisfy our needs? The research literature described in section 2.1 is summarised in Exhibit 1. Exhibit 1: Summary of research literature Dataset used Application Use of selfreported values Use of modified self-reported values Use only final TVPI Use of fund cash flows Risk measured against public market Risk measured within private equity Woodward (2009) No Yes No No Yes No Diller and Herger (2010) No No Yes No No Yes Weidig and Mathonet (2004) Yes No No No No Yes Bernstein et al (2008) Yes No No No No Yes Chen et al (2009) Yes No No No No Yes Marquez et al (2011) Yes No No No No Yes Driessen et al (2006) No No No Yes Yes No Dataset used Self-reported values Risk calculations based on self-reported values or time series of quarterly private equity returns have the clear benefit of being relatively easy to perform and similar to those commonly used in public market (e.g. volatility of returns, beta to a benchmark, etc.). However, these calculations significantly understate the true risk because private equity returns are highly autocorrelated. Risk measures in private equity I 5

6 Exhibit 2: Autocorrelation of returns for a representative private equity portfolio Source: Unigestion, based on quarterly returns by segment from 1998 to 2011 as reported by VentureXpert (Thomson Reuters) and based on the following segment allocation: 45% European buyout, 29% US buyout, 22% US mezzanine and 4% US venture capital. Exhibit 3: Autocorrelation of returns for a representative public market portfolio Source: Unigestion, based on quarterly returns from 1998 to 2011 as reported by Bloomberg for the MSCI World Daily Net Total Return Index (USD). Risk measures in private equity I 6

7 The above illustrations are traditional representations of time series autocorrelation. By measuring the correlation between a stream of returns and the same stream of returns displaced in time by an increasing number of quarters (the lag), it is possible to detect a persistence of information through time. In the public markets, this information is exploited by market players and thus largely removed. This is observed in Exhibit 3 where no observation exceeds the confidence intervals depicted by the blue lines. In private equity however, the absence of supply-demand price formation mechanisms prevents an efficient exploitation of information by market players, which results in a statistically significant autocorrelation. This is a general property characterising returns of illiquid assets. It is also the reason why comparing liquid and illiquid assets may results in misleading conclusions. Following Andrew W. Lo s approach, we have established the extent to which autocorrelation impacts volatility in private equity. This has highlighted that, if measured by the annualised volatility of quarterly returns, the traditional annualised volatility of private equity is underestimated by a factor as large as 60%. Thus, considering self-reported values does not give a true representation of risk. Moreover, adjusting the self-reported values to account for autocorrelation is, in our view, not satisfactory as it leads to adding noise and approximation in the time series of returns. Final TVPI Calculations that only consider final TVPI eliminate the influence of interim valuation biases on returns, thus eliminate the negative effect of self-reported values. However, final multiples convey an incomplete representation of the risk of private equity funds over time. Exhibit 4: Development of the TVPI for two different private equity fund examples 2.2x 2.0x 1.8x 1.6x 1.4x 1.2x 1.0x 0.8x Q1 Q5 Q9 Q13 Q17 Q21 Q25 Q29 Q33 Q37 Q41 Q45 Q49 Q53 Fund 1 Fund 2 Risk measures in private equity I 7

8 Exhibit 4 presents two funds that have exactly the same final TVPI but whose developments are highly different. Indeed, Fund 1 is characterised by an early and relatively smooth TVPI development while Fund 2 is characterised by a late and relatively steep TVPI development. Fund 1 does not have the same risk profile as Fund 2, especially for programmes or funds-of-funds that are running over-commitment 3 strategies. Indeed, in the case of overcommitment, a fund like Fund 1 tends to distribute earlier cash proceeds that will then help finance future drawdowns from underlying funds. This is known as the recycling of distributions. Fund 2 will have the exact opposite characteristics, with its associated liquidity risk. Therefore, considering Fund 1 and Fund 2 with the same risk profile is questionable. Cash flows We believe that fund cash flows are a true and objective representation of the fund s development since cash flows are not biased by valuation subjectivity and can be observed throughout the whole life of the fund Application Being a private equity fund-of-fund manager, our focus is to assess the risk/return profile of private equity funds among themselves, not necessarily versus public market. A comparison between private equity and public market is interesting in a global portfolio allocation context but not from a pure private equity allocation perspective Take away Whilst the research literature constitutes a valuable starting point for our study, we do not find a satisfactory risk measure based on objective data that could be applied within private equity specifically. We will therefore develop our own risk measure. 3 Over-commitment is defined as: ( ) ( ) ( ) ( ) Risk measures in private equity I 8

9 3. Basis for our risk measure 3.1. Rationale for working with cash flows of private equity funds In the private equity asset class, data is rather scarce. What can be found in external data sources are mainly i) fund quarterly Net Asset Values ( NAVs ), ii) fund characteristics and qualitative factors (e.g. size, geography, strategy, sector, etc.) and iii) fund cash flows (e.g. contributions and distributions). As mentioned in section 2, self-reported values or NAVs of private equity funds are subject to several biases: NAVs are not publicly traded; NAVs are calculated by the fund manager itself and not by a third party; NAVs are calculated with lagged data and are subject to stale-pricing; NAVs are the aggregate of the portfolio company values, usually calculated by using various accounting methods to reflect their fair value ; these accounting methods, e.g. enterprise value = EBITDA * EBITDA multiple, can vary from one fund manager to another, and thus, yield different valuations for the same company held by two different fund managers; the fair value of a private company is often based on the average of public comparable data, which leads to consider a private company as the average of public comparable companies, resulting in the underestimation of idiosyncratic risks and volatility. Because of all these biases, considering self-reported values is not a viable basis for our risk measure. Fund characteristics and qualitative factors can be valuable data when it comes to testing a risk measure on different private equity subgroups and drawing conclusions on the behaviour of these subgroups: for example, evaluating if venture capital is more risky than buyout or if sector specialist funds are more risky than generalist funds. However, these qualitative factors can hardly serve as the basis for a risk measure due to their nonquantitative nature. Fund cash flows are real-time cash contributions and cash distributions between an investor or Limited Partner ( LP ) and a private equity fund or General Partner ( GP ). They represent an accurate and objective view of a private equity fund development throughout its life. Besides, at the end of a fund s life, the sum of all distributed cash flows over the sum of all contributed cash flows will equate the net multiple of a fund, which is a commonly used metrics to benchmark funds among themselves. Finally, appropriately discounted cash flows represent the Net Present Value ( NPV ) of an investment, an acceptable rationale for investing. Therefore, we consider cash flows to be the most viable basis for our risk measure. Risk measures in private equity I 9

10 3.2. Essentials about private equity cash flows A LP, such as a private equity fund-of-funds, a pension fund or an endowment, invests in GP in the form of a commitment. This committed capital is drawn during the fund investment period, which usually lasts between 3 to 5 years after fund closing. Thus, a commitment is not an upfront cash payment to the GP but, rather a promise to pay an amount equivalent to the commitment in different instalments, over the investment period. During the investment period, the GP acquires companies and notifies the LP about the capital they are going to need: the GP issues a capital call, also known as a contribution or a drawdown. Failing to honour this capital call, or defaulting, is heavily penalised and should therefore not be considered as a viable choice. The sum of all contributions is generally limited to the initial fund commitment. Distributed cash flows or distributions are paid-back capital coming from the realisation of previously invested companies. These realisations are usually made through public markets (i.e. IPO), through sales to strategic buyers or through sales to another private equity fund (i.e. secondary buyout). After deduction of fund management fees and carried interest, these realisations transform into distributions to the LP. Exhibit 5: Sample cash flows at fund level for a commitment of EUR 10 million EUR m Contributions Distributions Q0 Q5 Q10 Q15 Q20 Q25 Q30 Q35 Q40 Q45 Q50 Risk measures in private equity I 10

11 3.3. Available data set Our proprietary data set is composed of the cash flows of 114 primary funds of vintages ranging from 1996 to In order to work with a more significant number of data, we integrate the Preqin data set, which is composed of the cash flows of more than primary funds of vintages ranging from 1979 to We voluntarily exclude funds of vintages older than 1996, for the sake of performance consistency, as well as funds that are not directly linked to private equity, such as real estate funds. Accounting for overlaps between the two data sets, the pooled data set is composed of funds and approximately cash flows (contributions and distributions) spanning from 1996 to Exhibit 5 shows the split of these funds by private equity segment. Exhibit 5: Split of funds by segment Large buyout Mid-market buyout Venture capital Turnaround/Distressed Mezzanine Source: Unigestion, based on Preqin data. One of the shortcomings of cash flows is that young funds have less quarterly cash flows than older ones. If we consider a zero-based cash flow approach meaning that the sum of cash flows occurring in the first quarter of a fund existence will be assigned as Q1, the second as Q2, etc., regardless of its vintage our pooled data has unavoidably a skewed distribution of cash flows towards the first quarters. Therefore, the reliability and significance of our risk measure will be highly dependent on the number of older cash flows. Hence, in the following sections of this paper, the applications of our risk measure will be based on the fund cash flow developments until quarter 32. Exhibit 6 illustrates the maturity of the quarterly cash flows in our pooled data set. Risk measures in private equity I 11

12 Exhibit 6: Maturity of fund cash flows 1'600 1'400 Number of quarterly cash flows 1'200 1' Q1 Q5 Q9 Q13 Q17 Q21 Q25 Q29 Q33 Q37 Q41 Q45 Q49 Q53 Q57 Q61 Source: Unigestion, based on Preqin data A focus on distributions rather than contributions From our data set, we can draw the cumulative contribution and cumulative distribution curves by private equity segment. Exhibit 7: Cumulative contributions average In % of total commitment Exhibit 8: Cumulative distributions average In % of total commitment 200% 180% 160% 140% 120% 100% 200% 180% 160% 140% 120% 100% 80% 80% 60% 60% 40% 40% 20% 20% 0% Q1 Q5 Q9 Q13 Q17 Q21 Q25 Q29 Q33 Q37 Q41 Q45 Q49 0% Q1 Q5 Q9 Q13 Q17 Q21 Q25 Q29 Q33 Q37 Q41 Q45 Q49 Large buyout Mezzanine Venture capital Mid-market buyout Turnaround/Distressed All Large buyout Mezzanine Venture capital Mid-market buyout Turnaround/Distressed All Source: Unigestion, based on Preqin data. Risk measures in private equity I 12

13 We can observe that contributions are on average very similar for all private equity segments and all end logically around 100%; whereas distributions show a significantly different development for all segments, with ending values ranging from 114% to 174% of total commitment. In the following sections of this paper, we will consequently solely focus on distributions and not consider contributions for our risk measure. We can now define what risk in private equity means for us: the risk is that a private equity fund or group of funds does not distribute capital as per expectations. Risk measures in private equity I 13

14 4. Our approach for implementing a risk measure Now that we have defined the basis for our risk measure and that we have gathered a broad dataset of cash flows, we need to know how to apply our risk measure. We have two options: working with our data set as a whole or dividing this data set into groups. If we were to compare the risk of the private equity asset class versus other asset classes such as public equities, commodities or bonds, we would certainly work with the whole data set available. But, because our goal is to implement a risk measure within the private equity asset class that could discriminate between different types of private equity funds and their respective risk/return profiles, we choose the second option. Hence, we need to cluster private equity into different groups that have similar characteristics. As we have not yet defined our risk measure, we deliberately chose to use qualitative factors in order to define those groups, which will also be later called qualitative groups Dividing our data set into groups The relevant qualitative factors for our study when benchmarking funds are the following: segment; geography; sector; team size; track record length; growth of fund size versus previous fund of the same manager. By assigning all funds with their respective qualitative factors, we perform an analysis to gauge which of these factors play a differentiating role regarding fund risk/return profile. To do so, we compare the return and the risk of all combinations of the qualitative factors above, return being defined as cumulative distributions and risk being temporarily defined as the Expected Gainfall 4. 4 Expected Gainfall is similar to the more commonly used Expected Shortfall, but since it is based on downside gains rather than downside losses, we choose to replace the word shortfall by gainfall Risk measures in private equity I 14

15 Similar to the Expected Shortfall, the Expected Gainfall at confidence level is defined as the average of the cumulative distributions of the worst performing 1 funds for each quarter q, or: = ] with = We test the significance of the qualitative factors by performing a pairwise comparison of all possible combinations of qualitative factors and running a t-test (see below). This approach works under the assumptions of independence of cumulative distributions and is based on the central limit theorem, assuming a normal distribution for the average of cumulative cash flows. The outcome of the t-test on both risk and return leads to the qualitative groups in Exhibit 9. H 0: E(Cumulative cash flows) Group1 = E(Cumulative cash flows) Group2 H 1: E(Cumulative cash flows) Group1 E(Cumulative cash flows) Group2 = + H 0 is rejected if > ( = min( 1, 1) ; 1 /2) Exhibit 9 shows that the sector qualitative factor is not significant in terms of risk/return differentiation. This implies that funds with a sector focus have a similar risk/return profile than their generalist counterpart. Also, the significance of qualitative factors depends on private equity segments. For example, the growth of fund size versus previous fund is significant in the case of large buyout and venture capital but not for the other segments. To be noted, we deliberately excluded Asia as a geography due to the lack of data available. We also voluntarily excluded the fund vintage from the relevant qualitative factors because, unlike the aforementioned factors, it rather constitutes a passive factor that cannot be decided upon. Indeed, in the case of a primary evergreen private equity programme that is set to commit regularly on a yearly basis, an investment decision can be made on all aforementioned factors (invest or not invest in a large buyout fund, invest or not invest in Europe, etc ), whereas the vintage cannot be decided upon, it is a given. Risk measures in private equity I 15

16 Exhibit 9: List of qualitative groups presenting different risk/return profiles Level 1 Level 2 Level 3 Level 4 Group Funds per group Large buyout USA Extreme growth 5 Group 1 50 Reasonable growth 5 Group 2 30 Europe Extreme growth 5 Group 3 31 Reasonable growth 5 Group 4 31 Mid-market buyout USA Broad team 6 Group Narrow team 6 Group Europe Broad team 6 Group 7 51 Narrow team 6 Group 8 67 Europe Group 9 52 Extreme growth 5 Group Long track record 7 Venture capital Reasonable growth 5 Group USA Extreme growth 5 Group Short track record 7 Reasonable growth 5 Group Turnaround/Distressed 8 Group Mezzanine 8 Group TOTAL Funds that are less than 1.7 times larger than their predecessor are classified as reasonable growth, while funds that are more than 1.7 times larger than their predecessor are classified as extreme growth 6 The classification of narrow and broad team derives from our qualitative assessment based on the absolute size of the team, the relative size of the team compared to the strategy and the split of the decision power within the team 7 The third and subsequent funds of the same manager are classified as long track record, while the first and second funds of the same manager are classified as short track record 8 Turnaround/Distressed and Mezzanine have not been divided into qualitative groups due to their relatively small data set Risk measures in private equity I 16

17 5. Stochastic modelling of distributions 5.1. Introduction to stochastic modelling Dividing our data set into groups has the clear benefit of enabling us to perform a risk/return analysis on different subsets of the private equity asset class. However, it has the drawback of reducing the sample size of the data set from funds to as little as 15 funds for Group 12. These 15 funds might not all be mature, meaning that a portion of these funds can be of vintages , with not enough data to be meaningful. Also, irregularities in the cash flow curves can be observed as the data set shrinks. Therefore, we need to develop stochastic modelling to overcome those potential irregularities and the lack of data. To illustrate this, we can analyse the average trend of all quarterly distributions as a percentage of total commitment and see that the pattern is erratic; hence the need to smooth it and make it more regular. Exhibit 10: Average trend of all quarterly distributions as a percentage of total commitment 14% 12% 10% 8% 6% 4% 2% 0% Q1 Q5 Q9 Q13 Q17 Q21 Q25 Q29 Q33 Q37 Q41 Q45 Q49 Q53 Q57 Most common time series analyses cannot cope with the discontinuity and the unequally spaced time interval of private equity distributions. Besides, direct assumption of stationarity is not appropriate due to the age-specific behaviour of distributions (lower in value for earlier years, higher in value for later years). An adequate modelling of this behaviour is needed. Therefore, a marked point process is implemented to efficiently model this behaviour. Risk measures in private equity I 17

18 Based on the observation of our data set at fund level, distributions are: discontinuous; unequally spaced; potentially unlimited in single value or sum; more frequent and higher in value during later years of a fund s life First step towards a stochastic cash flow model for private equity Buchner et al (2007) develop the first stochastic model for private equity cash flows without considering selfreported values. They provide models for both private equity fund contributions and distributions. Their model assumptions conflict with the actual behaviour of private equity cash flows. The most important weakness is the assumption of continuously receiving cash flows and then assigning a distribution to these instantaneous cash flows. In reality, cash flows occur on a discrete basis and time intervals are stochastic themselves. This implies that instantaneous cash flows are always equal to zero unless a cash flow occurs. Unfortunately, no continuous density function can absorb this effect. Even considering quarterly cash flows instead of instantaneous cash flows does not help since the probability of receiving a cash flow different from 0 is approximately 50% at fund level. This alone seriously questions the relevance of imposing a density function to quarterly cash flows. Subsequently, Buchner et all (2010) try to empirically analyse their first stochastic model in order to lower the effect of the aforementioned weakness. To improve their first attempt, they increase the time interval from quarterly to annual cash flows. This leads to a reduction in the probability of receiving a cash flow equal to 0 and makes the probability density function more similar to a continuous density function that can be approximated with a known distribution, e.g. log-normal distribution in this case. However, this method has the following drawbacks: even the sum of cash flows in one year has a significant probability of being exactly zero at fund level; the sum of log-normally distributed random variables is not log-normally distributed: if the instantaneous density function is assumed to be log-normal, the sum over one year will not result in a log-normal distribution; the efficiency of estimation is commonly questioned when, instead of considering single observations, their sum is considered as the random variable; by increasing the time interval to one year, observing co-movements of cash flows between funds will also be significantly less efficient. Due to these caveats, and to our knowledge, adequate modelling for private equity fund cash flows has not been achieved yet. In this paper, we present an unconditional model based on a marked point process, and in Appendix 14.1 and 14.2, we improve the unconditional model with a conditional model based on Engle and Russell s Autoregressive Conditional Duration ( ACD ) model. Risk measures in private equity I 18

19 Cash flow modelling using a stochastic marked point process After thorough research, we select a marked point process in order to model both the timing and the value of fund distributions in percentage of total fund commitment. In practice, our modelling is fitted to the different qualitative groups in Exhibit 9, each group representing a homogenous sample of funds from a qualitative perspective. Our future use of this model commands this approach. When a new primary fund commitment is considered, no data from that particular fund exists. The only relevant information for this new fund encompasses previous funds track record and the qualitative information gathered during the due diligence. Consequently, we could assume that the new fund would behave similarly to previous funds with comparable qualitative factors Marked point process Unconditional model Point process, or how to model the time arrival of a distribution The time arrival of a distribution is modelled with a Poisson distribution: ( ) ~ ( ). Given that the intensity of the time arrivals varies with the maturity of the fund, i.e. receiving distributions is more likely during later years of a fund s life, λ(t) should account for that effect. We find that an appropriate parametric choice belongs to the exponential-polynomial families: ( ) =. Exhibit 11 provides a representation of the range of curve shapes that we can obtain by modifying the two parameters 1 and 2. Exhibit 11: Flexibility of the intensity function for different parameter choices Intensity Maturity of funds (quarters) In Exhibit 12, we show the Maximum Likelihood Estimated ( MLE ) fitted intensity function in comparison to the actual intensity function for Group 1, i.e. the group of large buyout USA extreme growth (see Exhibit 9). The next graphs are only shown for Group 1 but the same conclusion applies for all 15 groups (see Appendix 14.3). Risk measures in private equity I 19

20 Exhibit 12: Point process intensity model versus actual intensity for Group 1 Intensity Maturity of funds (quarters) The robust MLE covariance matrix in Exhibit 13 implies that parameters have been precisely estimated and that the modelled intensity function fits well to the actual intensity function. In Appendix 14.6, we show the parameters for all 15 groups. Exhibit 13: Point process parameters (left) and robust MLE covariance matrix (right) for Group E E E E-10 The accurate modelling fit is directly observable in the Q-Q plot below for Group 1. In addition, our point process model shows repeatable results for all groups. Risk measures in private equity I 20

21 Exhibit 14: Point process homogeneous residuals Q-Q plot (left) and autocorrelation (right) for Group 1 Quantiles of sample Sample autocorrelation Quantiles of exponential distribution Lag With a Poisson distribution, residuals 9 should follow an exponential distribution function. In Exhibit 14, the Q-Q plot shows a fatter tail than with an exponential distribution. Also, the autocorrelation graph highlights a serial dependence in the data. In Appendix 14.1, we demonstrate that Engle and Russell s conditional model is an appropriate method to describe these residuals Mark process, or how to model the value of a distribution The value of a distribution is modelled with a Weibull distribution with a constant shape parameter k and a timevarying scale function l t e : ( ) = = Γ 1 + 1,,, > 0, < 0 In Exhibit 15, we show the MLE fitted value function in comparison to the actual value function for Group 1. Value is expressed in percentage of total commitment. The higher variability in later quarters, i.e. after quarter 40, is due to decreasing number of observations. Therefore, estimating later quarters is less precise and more variable. The next graphs are only shown for Group 1 but the same conclusion applies for all 15 groups (see Appendix 14.4). 9 Residuals are modelled according to the following distribution: = ( ) Risk measures in private equity I 21

22 Exhibit 15: Mark process value model versus actual value for Group 1 Value Maturity of funds (quarters) The robust MLE covariance matrix in Exhibit 16 implies that parameters have been precisely estimated and that the modelled value function fits well to the actual value function. In Appendix 14.6, we show the parameters for all 15 groups. Exhibit 16: Mark process parameters (left) and robust MLE covariance matrix (right) for Group E E E E E E E E E-09 The accurate modelling fit is directly observable in the Q-Q plot below for Group 1. In addition, our mark process model shows repeatable results for all groups. Risk measures in private equity I 22

23 Exhibit 18: Mark process homogeneous residuals Q-Q plot (left) and autocorrelation (right) for Group Overall fit of the marked point process: Exhibit 19 shows the overall fit of the marked point process for Group 1. Again, the fit is highly satisfactory both on discrete and cumulative bases. In Appendix 14.5, we show the overall fit for all 15 groups. Exhibit 19: Marked point process distribution model versus actual distributions for Group 1 Cumulative value Quantiles of sample Sample autocorrelation Quantiles of exponential distribution Lag Value Maturity of funds (quarters) Maturity of funds (quarters) Risk measures in private equity I 23

24 Weaknesses and solutions The unconditional marked point process entails three slightly undesirable features: fat tails of point process residuals; significant autocorrelation of point process; significant autocorrelation of mark process. In Appendix 14.1 and 14.2, we improve the unconditional model with a conditional model based on Engle and Russell s Autoregressive Conditional Duration ( ACD ) model and provide solutions to these undesirable features. Risk measures in private equity I 24

25 6. Appropriate risk measure Given that we define risk in private equity as the risk that a private equity fund or group of funds does not distribute capital as per expectations, we implement an innovative risk measure called the ECDAD Expected Cumulative Downside Absolute Deviation ( ECDAD ) ( ) = [ ( ) ] The ECDAD measures the downside area between an individual distribution curve and its related expected distribution curve. Exhibit 20 shows a graphical illustration of the ECDAD. Exhibit 20: Illustration of the ECDAD Cumulative distributions in % of total commitment 220% 200% 180% 160% 140% 120% 100% 80% 60% 40% 20% 0% Expected Individual Q1 Q5 Q9 Q13 Q17 Q21 Q25 Q29 Q33 Q37 Q41 Q45 Q49 Q53 Maturity of funds (quarters) The ECDAD is a number equal to the sum of the areas below the expected curve. There cannot be any positive offset from the areas above the expected curve; these areas are simply ignored. The larger the area, the higher the ECDAD and the higher the risk. It means that, if a fund is late in distributing cash flows and/or distributes much less than expected, then its risk is magnified and the ECDAD, logically, will be high as compared to a fund that distributes according to expectations. Risk measures in private equity I 25

26 The ECDAD has the following advantages: the result of the ECDAD is a number, so the comparison between two funds is straightforward: a fund with an ECDAD of 2 is simply more risky than a fund with an ECDAD of 1.5; the ECDAD measures how later distributions materialise versus expectations; the ECDAD measures how lower distributions materialise versus expectations; direct calculation of the ECDAD is applicable for single funds or group of single funds; the expected curve can be set with a high flexibility. In the next sections of this paper and unless otherwise stated, we will i) base our ECDAD calculations on eight years of data (i.e. until quarter 32) in order to have enough mature funds but less irregularities due to the scarcity of data at the end of funds life, ii) use our stochastic modelling for funds that have less than eight years of data and iii) set the expected curve at a group level as the average cumulative distribution curve of the different constituents of the group Is the ECDAD a coherent risk measure? To conclude that the ECDAD is a coherent risk measure, we analyse if it verifies some desired properties: normalised: risk (0) = 0 (the risk of holding no assets is zero); monotone: if X Y, then risk (X) risk (Y); sub-additive: risk (X+Y) risk (X) + risk (Y); positively homogeneous: if α 0, then risk (α*x) = α*risk(x); translation invariant: risk (X+α) = risk (X) α. The ECDAD verifies all properties except the monotonicity in few cases. This measure is as such similar to the standard deviation that also does not verify the monotonicity property in all cases. To become a monotone risk measure, where all other properties also hold, it would be sufficient to replace the expected curve at a group level, defined as the average cumulative distribution curve of the different constituents of the group, by a deterministic target such as a fixed target defined as the average cumulative distribution curve of the different constituents of all groups. Risk measures in private equity I 26

27 7. Risk/return of the different private equity groups In order to show a risk/return map within private equity, we consider the 15 groups of Exhibit 9. Risk is proxied by the average ECDAD of the different constituents of each group versus the related group expected distributions to which the constituent belongs. Return is proxied by the present value of the cumulated distributions in percentage of total commitment for the group. The present value assumes a cost of capital of 15%, an amount that is deemed appropriate for private equity. Exhibit 21: Risk/return map of our private equity groups Risk measures in private equity I 27

28 Several conclusions can be drawn from Exhibit 21: except for one outlier in venture capital (Group 11), all groups form relatively cohesive clusters that can be regressed by a line; venture capital is mostly located in the lower left part, which indicates it represents the less risky but also the less performing private equity segment. However, Group 11 shows that higher returns are possible in venture capital. In other words, building a well-diversified portfolio of venture capital funds will likely not yield a high return but selecting only few funds (preferably in the USA with a long track record and reasonable growth) may drive the return significantly upwards, obviously with a correspondingly higher risk; venture capital is less risky from an ECDAD perspective than buyouts, mezzanine and turnaround/distressed; this is counter-intuitive but is well explained by the fact that the venture capital groups have low expectations as compared to other segments and deviate less from this expectations. Therefore their ECDAD is lower. On the other hand, buyout groups, for example, have much higher expectations but deviate much more from these expectations, so their ECDAD is higher; buyout groups have a similar risk to mezzanine but a higher return potential; turnaround/distressed is more risky than buyouts and mezzanine; sector specialists funds and generalist funds have similar risk/return potential; funds I and II of the same manager are not necessarily more risky than subsequent funds. This could mean that the risk of an emerging manager is not as high as the risk of an experienced manager, but the vintage effect plays a role here. Indeed, fund I and II have often been raised in older periods, at times when the economy was more robust than today; narrow teams are generally more risky than broad teams; funds that are less than 1.7 times larger than their predecessor ( reasonable growth ) are not more risky than the other ones ( extreme growth ). This can be explained by the fact than extreme growth of a fund size is often linked to the good performance of the previous fund. Risk measures in private equity I 28

29 8. Power of diversification in a portfolio allocation context Based on a Monte Carlo simulation on historical data, random portfolios of either buyout funds or venture capital funds are built. Each point of the surface charts below is the result of a simulation made with iterations and based on randomly selected portfolios at quarter 32. Here, risk is proxied by the average ECDAD of the constituents of the random portfolios versus the expected distributions of all buyout groups (for Exhibit 22) or the expected distributions of all venture groups (Exhibit 23). Exhibit 22: Diversification surface chart for buyout groups (groups 1 to 8) ECDAD Exhibit 23: Diversification surface chart for venture capital groups (groups 9 to 13) ECDAD Risk measures in private equity I 29

30 The diversification surface charts confirm our intuition that diversification in private equity helps reducing the risk of a portfolio. In a sense, this is a no brainer. What is more interesting is to see that over-diversification in the buyout segment does not bring much additional benefits after reaching a portfolio of 20 funds that is composed of 4 funds in 5 consecutive vintages. Thus, for buyouts, it is possible to build a concentrated portfolio with optimised risk features, while having the possibility to generate above market performance through a skilful fund selection. The same conclusion does not apply for venture capital where minimal risk is not achievable unless building a portfolio of 100+ funds, which would imply an optimised risk but an average market performance without any possibility to generate alpha through a skilful fund selection. Another interesting conclusion can be drawn from Exhibits 23 and 24. We can infer that diversification over vintages yields better results in terms of risk optimisation than diversification over a particular vintage. Indeed, the distribution of the ECDAD for 1 fund in 15 consecutive vintages is more skewed towards the low ECDAD range of values than for 15 funds in one single vintage. Exhibit 24: ECDAD distributions for a simulation over random portfolios for buyout groups funds in 1 vintage ECDAD 1 fund in 15 consecutive vintages ' ECDAD Risk measures in private equity I 30

31 9. Optimal risk-adjusted allocation The final consideration of this paper is to find an optimal risk-adjusted allocation based on the 15 groups. In other words, we endeavour to find the most appropriate allocation between these 15 groups in order to reach a target return while keeping the ECDAD at the lowest possible level. The results are shown in Exhibits 25 and 26. Exhibit 25: Risk-adjusted allocation between the 15 groups for a given target return 100% Weight 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Present value of cumulative distributions in % of commitment Group 15 Group 14 Group 13 Group 12 Group 11 Group 10 Group 9 Group 8 Group 7 Group 6 Group 5 Group 4 Group 3 Group 2 Group 1 Exhibit 26: Corresponding ECDAD ECDAD Present value of cumulative distributions in % of commitment Risk measures in private equity I 31

32 10. Conclusion In this paper, we develop an innovative approach for measuring the risk of private equity funds based on a stochastic cash flow modelling. The risk measure, called the ECDAD, derives from our own interpretation of what is risk in private equity, which can be summarised as the risk of a private equity fund not to distribute capital as per expectations. Thus, the ECDAD is equivalent to the downside area between an individual distribution curve and its related expected distribution curve. It is equal to a number, which makes the comparison between different private equity funds or group of funds straightforward. We also present several applications of the ECDAD and, based on historical observations, the following conclusions can be drawn: except for one outlier group in venture capital, all private equity groups form relatively cohesive risk/return clusters that can be regressed by a line. Venture capital is located in the low risk/low return region while buyouts, turnaround and mezzanine are located in the high risk/high return region; in order to optimise risk in a buyout portfolio, over-diversification is not needed, given that a portfolio of 20 funds composed of 4 funds in 5 consecutive vintages already exhibits optimised risk features. This is not true for venture capital, where risk tends to be low only for portfolios made of 100+ funds. Such portfolios would hardly be able to generate alpha due to over-diversification; assuming the same number of funds in a portfolio, we prove that diversification over vintages yields better results in terms of risk optimisation than diversification over a particular vintage; the risk/return profile of a private equity portfolio is highly dependent on its segment allocation: reaching the lowest risk/return portfolio commands the following segment allocation: 34% large buyout, 35% mid-market buyout, 18% venture capital, 3% turnaround/distressed and 10% mezzanine; reaching an intermediate risk/return portfolio commands the following segment allocation: 57% large buyout, 32% mid-market buyout, 0% venture capital, 3% turnaround/distressed and 8% mezzanine; reaching the highest risk/return portfolio commands the following segment allocation: 100% large buyout. Finally, we have voluntarily excluded the fund vintage effect from this analysis for the reason explained in section 4.1. Being aware that cyclicality in private equity can be significant and thus, influence allocation decisions or market timing, we have already initiated a new research project on that subject and will publish a paper next year. Risk measures in private equity I 32

33 11. References Bernstein, Lerner, Sorensen, Strömberg, 2008, Private Equity and Industry Performance, Harvard Business School, working paper. Buchner, Kaserer, Wagner, 2007, Stochastic Modelling of Private Equity An Equilibrium Based Approach to Fund Valuation, Technische Universität München, working paper. Buchner, Kaserer, Wagner, 2010, Modelling the Cash Flow Dynamics of Private Equity Funds: Theory and Empirical Evidence, Journal of Alternative Investments. Chen, Gompers, Kovner, Lerner, 2009, Buy local? The Geography of Successful and Unsuccessful Venture Capital Expansion, Harvard Business School, working paper. Diller, Herger, 2010, Assessing the Risk of Private Equity Fund Investments, Capital Dynamics, working paper. Driessen, Lin, Phalippou, 2006, Estimating the Risk of Private Equity Funds: A New Methodology, Amsterdam Business School, working paper. Engle, Russell, 1998, Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data, Econometrica working paper. Engle, Russell, 2004, Analysis of High Frequency Financial Data, working paper. Filipovic, 2011, Stochastic Calculus for Finance, Ecole Polytechnique Fédérale de Lausanne (EPFL), lecture notes. Lo, 2001, Risk Management for Hedge Funds: Introduction and Overview, MIT Sloan School of Management, working paper. Mancini, 2010, Econometrics, Ecole Polytechnique Fédérale de Lausanne (EPFL), lecture notes. Mancini, 2011, Advanced Topics in Financial Econometrics, Ecole Polytechnique Fédérale de Lausanne (EPFL), lecture notes. Marquez, Nanda, Yavuz, 2011, Performance Persistence in Private Equity Fund Returns, working paper. Weidig, Mathonet, 2004, The Risk Profiles of Private Equity, European Investment Fund, working paper. Woodward, 2009, Measuring Risk for Venture Capital and Private Equity Portfolios, Sand Hill Econometrics, working paper. Risk measures in private equity I 33

34 12. Contact Information If you are an institutional investor: If you are a consultant: If you are a journalist/press agency: [email protected] [email protected] [email protected] Risk measures in private equity I 34

35 13. Important Information This document is addressed to professional investors, as described in the MiFID directive and has therefore not been adapted to retail clients. This document has been prepared for your information only and must not be distributed, published, reproduced or disclosed by recipients to any other person. It is a promotional statement of our investment philosophy and services. It constitutes neither investment advice nor an offer or solicitation to subscribe in the strategies or in the investment vehicles it refers to. Some of the investment strategies described or alluded to herein may be construed as high risk and not readily realisable investments, which may experience substantial and sudden losses including total loss of investment. These are not suitable for all types of investors. The views expressed in this document do not purport to be a complete description of the securities, markets and developments referred to in it. To the extent that this report contains statements about the future, such statements are forward-looking and subject to a number of risks and uncertainties, including, but not limited to, the impact of competitive products, market acceptance risks and other risks. Data and graphical information herein are for information only and may have been derived from third party sources. Unigestion takes reasonable steps to verify, but does not guarantee, the accuracy and completeness of this information. As a result, no representation or warranty, expressed or implied, is or will be made by Unigestion in this respect and no responsibility or liability is or will be accepted. Unless otherwise stated, source is Unigestion. All information provided here is subject to change without notice. It should only be considered current as of the date of publication without regard to the date on which you may access the information. Past performance is not a guide to future performance. You should remember that the value of investments and the income from them may fall as well as rise and are not guaranteed. Rates of exchange may cause the value of investments to go up or down. An investment with Unigestion, like all investments, contains risks, including total loss for the investor. Risk measures in private equity I 35

36 14. Appendix As stated in section 5.2.4, the unconditional marked point process entails three slightly undesirable features: fat tails of point process residuals; significant autocorrelation of point process; significant autocorrelation of mark process. The two first weaknesses can be addressed by describing the autocorrelation in the point process. An Autoregressive Conditional Duration ( ACD ) model as introduced by Engle and Russell in 1998 is an appropriate method that will absorb the autocorrelation, reduce the conditional variance and control the fat tail issue. A similar conditional autoregressive model can be introduced to describe the autocorrelation in the mark process. Since distributed cash flows have a highly asymmetric distribution and have a minimum value of zero, most autoregressive models are not suitable. Indeed, estimated values based on these models may result in negative cash flows. This can be solved by adopting a similar approach than the one used for ACD models. To avoid confusion with the model used for the point process, the name of Autoregressive Conditional Value ( ACV ) is chosen for the mark process ACD model Point process A self-exciting arrival process is fitted to the data. Before fitting an ACD (1,1) model to data, time intervals have been diurnally adjusted. This adjustment, which is done using a cubic spline function, absorbs the predictable part of time arrivals that we represent with φ E[t t ]. We use a non-parametric Kernel smoothing approach to estimate this function. As shown in Exhibit 27, this function shows the desired features: high time intervals in first years, i.e. low intensities and low time intervals in later years, i.e. high intensities. Exhibit 27: Non-parametric estimated function for time interval to next cash flow, fitted to all buyout funds Intensity Maturity of funds (quarters) Risk measures in private equity I 36

37 Now, in order to draw the Q-Q plot of residuals and the autocorrelation with the ACD model, we use the following method: = =, h = ~ (1) = + + The number of lags in the above GARCH approach is the minimum number of lags that provide insignificant autocorrelation in residuals. For Group 1, the estimated parameters are presented in Exhibit 28. Exhibit 28: Point process ACD(1,1) parameters for Group As shown in the Q-Q plot of residuals in Exhibit 29, the model fits well to the data and fatter tails have been reduced as compared to the unconditional model s Q-Q plot in Exhibit 14. Besides, the autocorrelation function is not significant for any lag. This autocorrelation is also not significant for square of residuals or log-residuals, which assures independencies of residuals. Exhibit 29: ACD point process homogeneous residuals Q-Q plot (left) and autocorrelation (right) for Group 1 Quantiles of sample Sample autocorrelation Quantiles of exponential distribution Lag Risk measures in private equity I 37

38 14.2. ACV model Mark process Since values of cash flows show a similar dependence structure and since the distribution of values tends to follow a Weibull distribution, a similar approach to the ACD model is implemented for values of cash flows, namely the ACV model. Exhibit 30: Non-parametric estimated function for values of cash flows, fitted to all buyout funds Value Maturity of funds (quarters) Again, in order to draw the Q-Q plot of residuals and the autocorrelation with the ACV model, we use the following method: = 1 =, h = Γ 1 + 1, = ~ (1, ) = + + The only difference with the ACD model is that residuals are assumed to follow a scaled Weibull distribution. This scaling forces expected values of residuals to 1. The same conclusions for the ACV model apply. As shown in the Q-Q plot of residuals in Exhibit 31, the model fits well to the data and fatter tails have been reduced as compared to the unconditional model s Q-Q plot in Exhibit 18. Besides, the autocorrelation function is not significant for any lag. Risk measures in private equity I 38

39 Exhibit 31: ACV mark process homogeneous residuals Q-Q plot (left) and autocorrelation (right) for Group 1 Quantiles of sample Sample autocorrelation Quantiles of exponential distribution Lag Risk measures in private equity I 39

40 14.3. Unconditional model Point process intensity model versus actual intensity for all 15 groups Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group 10 Group 11 Group 12 Group 13 Group 14 Group 15 Risk measures in private equity I 40

41 14.4. Unconditional model Mark process value model versus actual value for all 15 groups Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group 10 Group 11 Group 12 Group 13 Group 14 Group 15 Risk measures in private equity I 41

42 14.5. Unconditional model Marked point process distribution model versus actual distribution for all 15 groups Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group 10 Group 11 Group 12 Group 13 Group 14 Group 15 Risk measures in private equity I 42

43 14.6. Unconditional model Fitted parameters for all 15 groups Risk measures in private equity I 43