9
Value-at-Risk
The VaR concept has become a standard approach in the toolkit of financial risk managers and regulators. Generally, the VaR is interpreted as the maximum mark-to-market loss a given portfolio can suffer for a given time horizon and a given confidence level. In essence, the VaR aims to calculate the portfolio impact of rare events. For instance, on a 1-year horizon and at a typical confidence level of 99.5%, the estimated loss should occur only once every 200 years, given historical market movements of asset prices (for a general overview of the VaR concept, see Jorion, 2006).
The VaR has been criticized for a number of reasons. For instance, it has been claimed that the VaR gives a false sense of confidence as it ignores tail risk and may even lead to excessive risk taking. Rather than making VaR control the chief concern, it has been claimed that it is far more important for risk managers to worry about portfolio losses that may be suffered if the VaR is exceeded. While we agree with most of the criticism, we believe that the VaR may still provide important insights for a risk manager as long as he is aware of the shortcomings of the approach. We concur with Jorion and Taleb (1997), who argue that “the greatest benefit of VAR [sic] lies in the imposition of a structured methodology for critically thinking about risk. Thus the process of getting to VAR [sic] may be as important as the number itself”.
While we have little to add to this debate, we start from the observation that VaR has become an integral part of financial regulation ever since financial institutions were allowed to use VaR in regulated capital modelling in 1996.1 Given increasing requirements regarding risk reporting and integrated risk management practices, it is probably only a question of time until VaR will be extended beyond traditional asset classes. However, this raises important questions as to how VaR can be estimated for portfolios of limited partnership funds, which are highly illiquid and for which market prices are not observable.
9.1 DEFINITION
For a given portfolio of funds, probability and time horizon, the VaR at time t0 is defined as the loss on the portfolio over a given time horizon [t0, t1] and for a given probability level α, 0 ≤ α ≤ 1. In order to calculate VaR, it is generally assumed that there are normal economic conditions and no changes in the composition of the portfolio of funds.
While regulators require confidence levels of typically 99–99.5%, this may not be meaningful for illiquid assets, given the limited number of data points risk managers typically work with. The danger of putting this threshold too high is that risk parameters cannot be validated empirically. Potential losses may be less extreme but more frequent than those associated with once-in-200-year events. As Mittnik (2011) argues, a focus “on such extreme risks is like asking a medical doctor to determine the right dosage of treatment and providing her with a thermometer that only indicates temperatures of 42 degrees Celsius and higher”.
Generally, the proper measurement of risk of limited partnership funds requires the “fair” valuation of such investments. Thus, risk modelling should fulfil the following criteria:
These criteria raise an important issue – how can we define the value that may be potentially at risk? And how do market fluctuations affect that value?
In principle, there are two methods for valuing assets. The first is the current market valuation as evidenced by prices observed in recent transactions, or an estimate of what that price might be. The second is the present value (PV) of the estimated future cash flows from that asset. In liquid markets, arbitrage ensures that the two methods are closely aligned. However, illiquidity and other market inefficiencies may cause the two methods to diverge, sometimes even substantially.
9.2 VALUE-AT-RISK BASED ON NAV TIME SERIES
In finance, risk is generally measured by the volatility of the price of an asset. Consistent with the standard mean–variance approach, risk in private equity and similar asset classes is usually calculated using (quarterly) changes in the NAV of a pool of funds (e.g., McCrystal and Chakravarty, 2011; for a more detailed discussion on estimating market risk in private equity, see Chapter 5). In estimating private equity returns and their volatility, researchers and investors typically employ time series provided by commercial data vendors (e.g., Preqin, Thomson VentureXpert) or special service providers (Cambridge Associates, Burgiss). Alternatively, market indices of publicly listed private equity are sometimes used.
9.2.1 Calculation
Calculations of returns from NAV time series are typically based on a chained modified Dietz formula. According to this formula, the investor has NAVt - 1 at the beginning of the period, t - 1, which ends at time t. The return Rt - 1t of this period is defined as
An investment in a new company increases the NAV. Conversely, exiting a company decreases the NAV of the fund, even if no changes in the valuation of the remaining portfolio companies take place. Over that period, the fund draws down and distributes capital, resulting in a change in the NAV reported at the end of the period. The market value of a portfolio is assumed to be
The yearly standard deviation can be derived from the standard deviation of the quarterly log-returns, essentially “annualizing forward” (Figure 9.1):
Figure 9.1 Annualizing quarterly returns forward.
Figure 9.2 shows the annual change in NAV of private equity funds between 1980 and 2010 as reported by Thomson Reuters. While the NAV jumped by 59% in 1999, negative changes are reported in the context of the bursting of the tech bubble in 2001–2002 and the recent financial crisis in 2008.2
Figure 9.2 Annual NAV changes based on Thomson Reuters data from 1980 to 2010.
Source: Thomson Reuters.
Changes in the NAV of a portfolio of funds can be used to calculate a return index. An example is Preqin's Private Equity Index, which is updated quarterly for buyout funds and venture capital partnerships. A recent EVCA (2012) study uses a similar sample of funds tracked by Preqin, as well as data provided by Pevara – another data vendor. The total sample includes almost 2000 partnerships, for which quarterly and monthly changes in the NAV are calculated for the period 1980–2010 (Figure 9.3). The quarterly and monthly NAV changes are then employed to calculate the VaR at the 99.5% level of confidence.
Importantly, the EVCA study was prepared in response to Solvency II, a new regulatory initiative for European insurance companies. In determining the solvency capital requirement (SCR), the European Insurance and Occupational Pensions Authority (EIOPA) has proposed a VaR calibrated to a 99.5% confidence level over a 1-year horizon covering all risks faced by an insurer. Employing the LPX 50 index of publicly listed private equity, EIOPA has calculated a stress factor of 49% to be applied to “Other Equity” encompassing, inter alia, private equity, hedge funds, commodities and infrastructure.3 However, as shown by EVCA (2012), the proposed stress factor appeared too high. Instead, using a NAV-based index of private equity funds a shock factor of less than 38% was calculated, with a correlation coefficient between private and public equity of 50–75% depending on the methodology.
Modelling market risk for private equity in a similar way as for public equity has a clear advantage in the sense that it uses the same “language” that risk managers of institutional investors, such as pension funds and insurance firms, as well as regulators are familiar with. Employing NAV-based time series appears straightforward and attractive, as quarterly returns are thought to be easily comparable to public indices, for example with regard to estimating correlations. However, the apparent simplicity of this approach may be deceptive for a number of reasons.
9.2.2 Problems and limitations
To begin with, the frequency of changes in the NAV of a fund or portfolio of partnerships is relatively low. Given that NAVs are usually available only on a quarterly basis, the number of data points is usually very limited. Even in the US private equity market, the market with the longest history, the sample period includes only around 30 years, or around 120 quarterly observations. For less developed alternative asset markets, notably in emerging economies, the sample period is typically substantially shorter. This raises important econometric issues.
Furthermore, NAVs are subjective appraisal-based valuations instead of observable market prices. Given the appraisal value effect, NAV time series tend to understate volatility, an effect that is known as stale pricing, which is closely related to the time lag effect (Emery, 2003; Woodward and Hall, 2003). Since risk is generally measured by the variance of returns, the naive use of NAVs may lead to the underestimation of risk of investing in private equity funds and similar partnership structures in other asset classes (and the overestimation of risk-adjusted returns). In addressing the stale price problem, NAV time series need to be de-smoothed, an issue we have already discussed in Chapter 5 (Geltner et al., 2003). There are various statistical techniques to adjust NAV-based time series for stale pricing, for instance Dimson (1979) or Getmansky et al. (2004).4 However, it is important to recognize that the adjusted time series can only be approximations of the true valuations that are not observable.
Apart from the econometric challenges risk managers face in working with NAV time series, there are several more fundamental reservations against such an approach (Mathonet and Meyer, 2007). Importantly, the NAV ignores a fund's lifecycle characteristics, such as the J-curve effect, the future use of undrawn commitments, the future management fees and the fund manager's value added (or value destroyed). The NAV time series approach aims to project the fund's future development based on a limited history. This may not be an insurmountable problem for relatively mature funds, where value is largely derived from existing portfolio companies and where undrawn commitments can essentially be ignored (EVCA, 2011). However, as far as immature funds are concerned where undrawn commitments are still substantial, using only NAVs violates the criteria for proper risk modelling in the sense that the model is neither “complete” nor “unbiased”.
Moreover, changes in a fund's underlying portfolio do not translate directly into cash flows to and from LPs. While the increase in a portfolio company's valuation may increase the probability of an exit, the decision to sell remains at the discretion of the fund manager. Consequently, the NAV time series approach cannot easily be reconciled with a cash flow scenario of the funds' in- and outflows.
Additionally, the concept of market risk at the portfolio company level can be problematic from the perspective of a fund's LPs. Suppose, for instance, the NAV of a 2-year-old fund drops by 10%. There remain 8 years for the NAV to recover. The situation is fundamentally different for an 8-year-old fund where the relative impact on the remaining LP's share would be significantly stronger, given the shorter remaining life of the fund. Thus, NAV-based risk measures are inconsistent with our criterion that risk in illiquid investing in limited partnerships should be subject to a monotonic decline over the life of a fund. Importantly, therefore, employing a NAV volatility-based modelling approach may result in overestimating risk, potentially leading to a sub-optimal underallocation to private equity and real assets.
Although the modified Dietz formula described above captures cash in- and outflows in the portfolio, it does not differentiate between funds and their underlying positions. Since LPs committing to a primary fund are investing in a “blind pool”, a NAV can only be reported after the fund has already started its investment activity – i.e., after risk management would theoretically have had the strongest impact. A key risk for LPs lies in the funding risk represented by the undrawn commitments, which are essentially ignored. In fact, as we have argued in Chapter 4, limited partnerships serve the purpose of shielding fledgling portfolio companies in their early stages as well as more mature companies in need of being restructured from adverse market developments.
An index can be a suitable proxy for the risk of the portfolio of funds held if the composition of the index is representative of the investor's own portfolio composition and diversification. This is a reasonable assumption for public equity where portfolio managers track an index. However, the composition of a specific limited partner's portfolio is typically significantly different from market indexes calculated on the basis of data provided by standard data vendors or service providers. This will especially be the case for smaller, less diversified LPs who are more exposed to risk.
Finally, since NAVs are appraisal-based and observable market prices do not exist, risk models employing this valuation metric cannot be back-tested in the true sense. By contrast, liquidity events are observable and a model could therefore be tested against a fund's cash flows. In many ways, cash flow projections appear to be a more appropriate approach for building a VaR model as they capture other risk dimensions, notably liquidity and funding risk.
9.3 CASH FLOW VOLATILITY-BASED VALUE-AT-RISK
An alternative to the NAV volatility-based approach is to focus on the variability of outcomes over the full lifetime of the portfolio of funds and base the VaR calculation on cash flow information (Kaserer and Diller, 2004a; Diller, 2007; Diller and Herger, 2008). The important advantage of a cash flow-based approach lies in the fact that undrawn commitments are properly reflected in the VaR.
Instead of employing time series of returns, a cash flow-based approach is interested in the “terminal wealth dispersion”, which relates directly to expected return and volatility levels.5 Since market valuations are not regularly available for limited partnership funds, the time period relevant for risk assessment is the entire lifetime of the portfolio of funds. Ideally, only fully liquidated funds should be considered in the empirical risk analysis of cash flows. However, given the limited number of such funds, it is common to include also mature funds, which are still active but exceed a set threshold for the minimum age in the underlying data sample.6 Calculating their cash inflows and outflows and reflecting the last reported NAV results in the calculation of the total value to paid-in (TVPI). Based on the outcomes for the TVPI for each fund, a probability density function can be determined (see Figure 9.4).
Figure 9.4 Calculation methodology based on the terminal wealth dispersion.
For mature funds, the NAV has a lower weighting, because the investment period is already complete and the first distributions and exits have already occurred. Therefore, this approach not only takes into account the changes in NAV but also reflects the cash flow behaviour. The risk profile for a portfolio of funds is derived from the returns of comparable mature funds (Weidig and Mathonet, 2004). This avoids to a large degree issues related to too few data points, such as autocorrelation and de-smoothing, by assuming that funds in a currently held portfolio will perform like funds in the past.
Suppose an investor at time t0 wants to determine the VaR of the portfolio of funds held for time t1 in the future, typically the end of the year. To do this, the investor needs to determine the probability distribution for the valuation of a portfolio of funds at t1 > t0. A standard approach is a Monte Carlo simulation, randomly drawing returns for mature funds from a database that reflect the characteristics of the portfolio to be modelled.
- The higher the number of runs for the Monte Carlo simulation the more stable the results will be.
- For every fund in the limited partner's portfolio a specific cash flow scenario is generated (this will be discussed in detail in Chapter 11).
- To project cash flows in every simulation run a new set of randomly chosen parameters is generated as input for a fund model.
- Correlations are reflected by constraining the random draws to subsections of the database (e.g., specific vintage years or strategies).
- All individual fund scenarios are aggregated, resulting in a scenario for the entire portfolio of funds' cash flow.
- A suitable discount rate is applied to determine the PV for this portfolio scenario.
- All PVs of the simulated portfolio scenarios are compiled, which provides the distribution function.
- Based on this distribution function, the VaR for confidence level α is determined.
The annual standard deviation of returns can be derived from the terminal wealth dispersion by “annualizing backwards” (see Figure 9.5).
Figure 9.5 Annualizing final returns backwards.
This analysis reflects the risks of an investor in a portfolio of limited partnership funds who has sufficient liquidity to respond to all capital calls and thus is under no pressure to sell his stake in the secondary market under potentially unfavourable market conditions during the portfolio's lifetime. This is a crucial assumption, which appears sensible for most large institutional investors who tend to have a relatively small allocation to private equity and similarly illiquid assets. However, as we have stressed before, there should be no room for complacency, given the experience of even some of the most sophisticated investors during the recent financial crisis. Instead, a funding test is required to confirm that the LP is in such a position.
How can we calculate a portfolio's cash flow-based VaR using the backward annualization technique? We propose two alternative approaches.
Generally, it is important that the risk manager understands the assumption as well as the implications and shortcomings of the various possible methodologies and then chooses the one most appropriate for his objectives.
9.3.1 Time series calculation
The starting point is a single fund with yearly cash flows in periods n = 0 to 10, i.e. the end of its lifetime (see Appendix Table 9.A.1). The fund receives contributions Cn from period n = 0 to 4, i.e. during its investment period, and generates distributions Dn thereafter. The fund returns a multiple of 1.55 times the invested capital with an IRR of 6.0%.
As a first step, we calculate the various PVs of the fund from each period until the end of its lifetime using a discount rate of d = 5% to reflect the opportunity cost for this asset.7 For instance, at the beginning of period 3, we calculate the PV based on all cash flows from period 3 to 10:
In a second step, we calculate the VaR for a portfolio of 10 funds over a period of 10 years with different investment and divestment periods and different returns (see Appendix Table 9.A.2). In this example, we assume for simplicity that all funds start in one single year (this is not a requirement for the analysis, which can also be performed for a portfolio that contains funds with different degrees of maturity).
We adjust the PV with the cash flows between two observation periods. For instance, assuming a time interval of 1 year, the PV of the current period is the PV of the previous period adjusted by the cash flows. For longer time intervals, the cash flow adjustments themselves need to be compounded, too. For instance, for period 3, the cash flow of 50 EUR in the first period will be compounded over two periods, that in the second period (–20 EUR) over one period, etc. Hence, the cash flow adjustment (CFA) is the sum of the cash flows (80 EUR) plus the discount rate of 5% over the last periods:
The annual PV for period t, PVt, is therefore the PV adjusted by the cash flows which took place in the previous periods. Hence, for this example, the annual PVt = 3 in period 3 is the PV of all cash flows from year 3 to 10 adjusted by all compounded cash flows from period 0, 1 and 2. These numbers build the basis for the VaR calculation of the underlying portfolio. Thus, for period 3 the annual PV is
which results from the PV of 94 EUR adjusted by the negative cash flow of 86 EUR at a value of 7.55 EUR. For this, as well as for the other periods, the PV is positive as the fund is returning a higher return of 6% compared with the opportunity cost of 5%. If we assume a higher opportunity cost of 8%, how would it affect the PV calculation? In this case, the annual PVs would be negative (such as funds 4, 9 and 10 in the example; see Appendix Table 9.A.2).
Appendix Table 9.A.3 shows the annual PV as well as the VaR for a 99% confidence level. The VaR is calculated over the cross-section of the various annual PVs of all funds of one period. In our example, the 99th percentile of the annual PV is negative for each of the periods. The relation between the net invested amount and the cumulative invested amount helps us to assess the VaR's relative magnitude. Note, however, that a portfolio with 10 funds is too small to be statistically significant and thus unlikely to reflect the characteristics of a large and well-diversified portfolio.
Figure 9.6 visualizes this approach. Based on the cash flow simulation of each fund, the PV is assessed by computing the future cash flows which will be adjusted by the compounded historical cash flows that have occurred. These annual PVs will be used to derive the annual VaR for the portfolio.
Figure 9.6 Time series calculation approach.
9.3.2 Fund growth calculation
In contrast to the time series approach, the fund growth approach is based on a number of simulation paths for each fund over its remaining lifetime. In addition, the VaR is calculated based on the difference between the PVs of each simulation run and its current value.
The higher uncertainty that results from the use of alternative scenarios should be expected to affect the distribution of the density function. Specifically, the VaR calculation follows five steps.
A simple example may help illustrate this approach. Suppose we want to determine the VaR for 1 year for a single fund. For this fund we run three cash flow scenarios with a given discount rate, resulting in three PVs (Figure 9.7).
Figure 9.7 Fund growth calculation approach.
These three scenarios are based on different cash flow scenarios and, hence, reflect possible real outcomes of the fund. Assuming that they will materialize with equal probability, the fair value of the fund at time 0 is the average of these scenarios:
We calculate the risk of this fund based on these scenarios and the gain/loss over a given time period. Under scenario 1 the fund would gain a value of 4.2 EUR p.a. over its projected lifetime:
Under scenario 2 the gain is 7.7 EUR p.a. while under scenario 3 the fund would lose 16.4 EUR p.a. Based on these results, the density function for the fund's valuation after 1 year can be determined as follows:
- Should scenario 1 materialize, after 1 year the fund would be worth 53.3 EUR + 4.2 EUR = 57.5 EUR.
- Should scenario 2 materialize, after 1 year the fund would be worth 53.3 EUR + 7.7 EUR = 61.0 EUR.
- Should scenario 3 materialize, after 1 year the fund would be worth 53.3 EUR - 16.4 EUR = 36.7 EUR.
These valuations allow us to determine the risk of losing capital on a 1-year horizon. Let us now discuss another example to compare the calculation method to the time series calculation approach described before.
We start by generating a set of scenarios. The first simulation path A results in the cash flow series that has already been shown in Appendix Table 9.A.1. In addition to this simulation, we run additional simulations (in reality, risk managers would typically run between 1, 000 and 100, 000 paths per fund). These results (“A” to “J”) for fund 1 are shown in Appendix Table 9.A.4. After running the simulations for fund 1, we do the same for fund 2, 3, …, 10. For simplicity, we only show the results of the cash flow scenarios for fund 1. However, they would look very similar for the other funds.
Having estimated all cash flow paths for each scenario m, we calculate the PVs for these scenarios. For scenario A, we obtain a PV = 6.85 EUR assuming a discount rate of 5%. Scenario B leads to a PV = –18.47 EUR. In order to determine the current fair value of the fund, we calculate the average of these numbers, which is 19.11 EUR. Now, we are ready to calculate the differences for each scenario. For example, for scenario A, we obtain 6.85 EUR - 19.11 EUR = -12.27 EUR, which represents the fund's total loss over its full life. To determine its annual loss, we need to divide this amount by the projected lifetime of 10 years, resulting in –1.23 EUR p.a.
Similarly, the annual loss under scenario B is estimated at 3.8 EUR, whereas the investor expects an annual gain of 0.12 EUR under scenario C.
Given these results, we can determine the histogram and density function for many scenarios of fund valuations at a future point in time. This forms the basis for determining the VaR over a required projection period (annually or quarterly). While the results become increasingly robust with a larger number of runs, the difference in the results is higher in our example for each new simulation. This also shows that in comparison to the time series-based approach, we have additional volatility through the simulations. This appears to be a better reflection of the true situation and the diversification of the portfolio. Therefore, this example might have a better coverage of the possible outcomes through higher variations but it is also more computation-intensive.
9.3.3 Underlying data
To be able to calculate a VaR we need to use representative market data (see also Box 9.2) or synthetic cash flows and be able to project the funds' future cash flows. In Chapter 11 we discuss in greater detail how such projections can be generated. In the preceding discussion, we have annualized the final returns “backwards” to calculate an annual VaR for a portfolio of limited partnership funds. Depending on an investor's objectives or specific financial regulations, there may be situations where quarterly, rather than annual, VaRs are required. In this case, PVs for each quarter would be calculated until each fund's end of lifetime, applying a quarterly interest rate. In addition, cash flows would need to be adjusted on a quarterly basis and compounded accordingly.
Large institutional investors with a long history of investing in private equity and real assets usually possess proprietary datasets which reflect their experience and investment strategies. By contrast, smaller LPs and those who have begun to invest in limited partnership funds only recently usually have to rely on publicly available data provided by data vendors (e.g., Preqin, Thomson VentureXpert) or specialized service providers (Burgiss, Cambridge Associates, State Street). Cornelius (2011) and Harris et al. (2012) provide a detailed analysis of these datasets and find significant differences in terms of sample periods, sample sizes, geographic coverage, mean returns and the dispersion of returns (see Chapter 5). No dataset is complete in the sense that it includes all funds that have ever been raised, which raises the issue of potential sample biases.
The choice of the dataset that is used to calculate the VaR already involves an important decision by the risk manager. For instance, the Thomson VentureXpert database has the longest sample period going back to the early 1980s, covering several market cycles, such as the first buyout wave in the second half of the 1980s, the tech bubble in the late 1990s and the second buyout wave in the mid-2000s (Figure 9.8). Thus, Thomson VentureXpert is widely used by practitioners as well as in academic research. However, Harris et al. (2012) and Stucke (2011) find evidence that the quality of the Thomson VentureXpert data might have been compromised by infrequent updates of NAVs. Against this background, risk managers are well advised to work with alternative datasets to examine the sensitivity of outcomes with respect to the underlying data.
Furthermore, since rarely will all relevant scenarios be covered by one or even several combined data providers, working with “synthetic” cash flows can be a – and in some situations the only – solution. This can mean adding stress factors, like lower distributions (affecting the performance) as well as faster contributions and slower distributions (affecting the holding period and, hence, having an impact on liquidity management). As we describe in Chapter 15, rating agencies use these kinds of approaches also for deriving the rating for bonds of securitized private equity portfolios.
9.4 DIVERSIFICATION
LPs typically hold portfolios of funds that are diversified across several dimensions, such as investment strategies, underlying assets, fund managers, vintage years, industries, geographies and currencies. Larger diversification over more funds has two important characteristics: first, there is more diversification across fund managers and their applied strategy, which usually has a positive effect. Second, and perhaps even more importantly, a greater degree of diversification at the fund level implies a greater degree of diversification in terms of portfolio companies. A portfolio of 10 private equity funds containing 80 to 150 underlying portfolio companies has different risk characteristics than a portfolio of, say, 150 funds with 1200 to 2250 companies, which have been acquired at different stages of the business cycle.
Let us consider just two dimensions of diversification, i.e. (1) the number of funds in a portfolio and (2) the vintage years during which the funds were raised. We employ data provided by Thomson VentureXpert for a sample of 3183 private equity partnerships. These partnerships include US and European buyout and venture capital funds raised between 1980 and 2006, with valuations as of end-September 2011. Choosing 2006 as a cut-off ensures that the sample includes only fully liquidated funds or those that are already relatively mature and in their divestment phase. For fully liquidated funds, we use their real cash flows (information on DPI); for the rest of the sample, we include the last NAV as a final cash flow similar to the TVPI calculation.
Suppose an investor picks randomly just one fund out of a randomly chosen vintage year in our sample between 1980 and 2006. As shown in Figure 9.9, the highest probability (i.e., the mode) for the distribution is located close to the multiple of 1. In other words, the most probable outcome of a random investment into a single private equity fund is that the investor will get his invested capital back. However, the distribution is clearly skewed to the right with some funds having generated TVPIs of 5 and more. The average TVPI is 1.62, which is higher than the median of 1.24. On the left-hand side of the distribution, losses are limited to a multiple of 0, with a very small number of funds having lost their entire capital (complete write-offs). The tenth percentile in the sample has achieved a multiple of 0.6, the fifth percentile a multiple of 0.43 and the first percentile a multiple of 0.14. This implies that the private equity investor has a confidence level of 99% to lose less than 86% of his investments at the end of the fund's lifetime. This corresponds to an invested capital-at-risk (iCaR) of 84%, a concept introduced by Diller and Herger (2008).
Figure 9.9 TVPI probability density function for randomly picked private equity fund (vintage years 1980 to 2006) – based on Thomson Reuters data.
Let us repeat this analysis using a dataset provided by Preqin. The Preqin sample includes 1892 funds. While the sample is biased towards US funds, the results we obtain are broadly similar (Figure 9.10). Specifically, given a confidence level of 99% the investor is expected to lose less than 92% of his capital by choosing randomly a single fund in the sample.
Figure 9.10 TVPI probability density function for randomly picked private equity fund (vintage years 1980 to 2006) – based on Preqin data.
As Weidig and Mathonet (2004) show, diversified portfolios are significantly less risky than an investment in any single fund, with well-performing funds compensating investors for the losses they may incur by committing capital to “defaulting” funds. While private equity is not unique in terms of diversification benefits, Weidig and Mathonet (2004) and Meyer and Mathonet (2005) find that significant gains can be reaped at relatively low levels of diversification. This is particularly true for investors diversifying across vintage years (Diller and Herger, 2008).
Consequently, the continuous monitoring and management of diversification is an integral component of a limited partner's risk management framework. Diversification reduces the long-term risk of a portfolio of funds. For large portfolios, diversification is found to increase the expected median returns – which, however, comes at the price of a reduced potential to harvest extraordinary returns (Mathonet and Meyer, 2007). Thus, investors who are confident of being able to select the best funds may decide to have a relatively less diversified portfolio. In this context, it is important to recognize that cash flows tend to become highly correlated during market downturns. Therefore, even funds following different strategies or targeting different geographies can become subject to similar degrees of liquidity risk in the short and medium term. The impact of diversification depends also on the interaction between the portfolio of funds and other assets held, for example, through the so-called denominator effect.
Using correlations as a measure of dependence between the funds within a portfolio has significant limitations as it builds on measuring the risks of funds and is thus faced with difficulties similar to those discussed before. Practitioners try to address this issue with different modelling approaches:
- Direct correlation modelling based on performance data observed for funds (or co-investments, as we discuss below) if available from either public or private sources.
- Implied correlation modelling based on systemic factors (i.e., value drivers) which are usually mapped to each fund and/or underlying portfolio company.
Alternatively, the relative dependence or independence of funds within a larger portfolio can be assessed through other tools such as cluster analysis. Cluster analysis is a technique to classify similar objects into relatively homogeneous groups and dissimilar objects into different groups (Lhabitant, 2004). It can be used to analyse the degree to which a portfolio of funds is “clogged”, i.e. tends to form clusters of sub-portfolios of funds with a high degree of interdependence. Funds that belong to the same cluster should be modelled as moving in the same direction. In applying stress scenarios, it should be assumed that portfolios of funds tend to get increasingly clogged. Thus, in constructing portfolios of limited partnerships, investors should pursue highly heterogeneous strategies across a wide range of pre-identified dimensions, such as vintage years, stage focus, industry focus and geographical focus.
As already mentioned, vintage year diversification is generally found to be the most powerful dimension, at least in private equity. Vintage year diversification requires a high degree of discipline, however. Murphy (2007) shows that investors may reach their target allocation to private equity in a relatively short period of time. The downside, however, is that such a strategy inevitably results in highly concentrated portfolios, implying (other things being equal) higher risk. As funds are usually investing over a period of several years, the acquisition multiples, the debt environment and the price levels of these companies tend to vary significantly. While this can be observed at the fund level, it is even more pronounced at the portfolio company level. A larger degree of diversification over time also allows for a wider spread in terms of industry, geography, debt situation and stages in the lifecycle of underlying companies, which as a consequence are little correlated among each other.
In practice, of course, no investor invests in just one fund. Thus, let us come back to our previous example, but this time considering a strategy whereby the investor commits to several funds spread over several vintage years. In particular, we are interested in the relative importance of vintage year diversification as opposed to the benefits investors may obtain by investing in different funds in a single vintage year. Let us consider two investors with a portfolio of 36 randomly chosen funds. Investor A has built his portfolio over 4 years with 9 funds per year, whereas investor B has committed to only 4 funds per year but has allocated his capital over a period of 9 years.
The effects of the two strategies are depicted in Figures 9.11 and 9.12, respectively. Comparing the two strategies suggests that diversifying over time have a significantly larger risk-mitigating effect. In addition to the substantial reduction of risk on the left-hand side of the distribution, there is a significant shift of the entire distribution into the positive area as can be seen. Importantly, there is no risk for the investor to lose capital as the entire distribution lies in the positive multiple range. For investor A, the median of the distribution of multiples is 1.15, with the first percentile portfolio achieving a multiple of 1.06. As far as investor B is concerned, the median multiple is 1.35, with the first percentile portfolio generating a multiple of 1.20. At the same time, the tails of the distribution are flattened to a significant degree. Obviously, the price of pursuing a more diversified investment strategy is a more limited probability of achieving an extraordinarily large return.
Figure 9.11 Investor A's portfolio built over 4 years with 9 funds per year – calculation based on Thomson Reuters data.
Figure 9.12 Investor B's portfolio built over 9 years with 4 funds per year – calculation based on Thomson Reuters data.
Thus far, we have focused exclusively on fund investments. Increasingly, however, investors also pursue co-investment strategies where an investor holds shares in portfolio companies alongside a fund where he is a limited partner. Generally, co-investments where exit decisions are solely taken by the GP can be covered within the model proposed above. For example, Smith et al. (2012) model co-investments in the form of artificial “co-investment funds” with the same structure as the primary private equity funds taken as reference, i.e. containing the same number of investments but with each co-investment from a different fund manager, from the same geography, roughly the same size and invested within 1 year of the reference investment in the primary fund. However, assumptions need to be made regarding the limited partner's selection skills and the resulting range for the multiples.
Generally, there is no contractual or legal obligation to fund additional financing rounds in the case of co-investments; consequently, funding risk is usually not an issue. Nevertheless, there is some funding risk in the case of expansion-related investments (e.g., buy-and-build strategies which might entail the acquisition of a competitor) or equity injections to comply with debt covenants and, in worst-case scenarios, to remain solvent.
Finally, some limited partners also pursue direct investment strategies. However, as in such cases investment and divestment decisions rest with the LP, they require specific risk management approaches that are beyond the scope of this book.8
9.5 FACTORING IN OPPORTUNITY COSTS
Given the time value of capital, investors need to factor in the opportunity costs of committing capital to limited partnerships. This issue is different from the risk that the fund manager might not be able to return the invested capital to the investor. Here, we are concerned with the issue of returns that fail to meet an investor's target return.
Opportunity costs can be viewed from different perspectives. To begin with, investors usually have the choice of investing in various asset classes. Therefore, the opportunity cost of investing in limited partnership funds is the expected return of other asset classes like bonds, stocks, real estate, hedge funds, etc. Benchmarking returns from private equity funds against, for instance, public equity indexes or high-yield bonds, the VaR can be calculated in such a way that it mirrors the risk of not achieving that benchmark. Alternatively, insurance companies and pension funds have guaranteed or fixed interest rates over the lifetime of the life insurance product in some countries.9 In this case, the VaR could reflect the risk of not achieving this return with fund investments in private equity and real assets. Further, investors like insurance companies, banks or industrial enterprises also have a cost of capital for their enterprise.
In order to reflect such opportunity costs, it is necessary to project the cash flows over the portfolio of funds' entire lifetime. The reference base for the risk calculation is then the invested capital of the portfolio compounded by the annual yield of the opportunity cost over the average holding period.10 Consider a randomly chosen portfolio of 50 funds an investor commits to over a period of 10 years (Figure 9.13). Incorporating opportunity costs in each of the Monte Carlo simulations has a non-trivial impact, as depicted in Figure 9.14, with opportunity costs assumed to be 4%: the distribution shifts to the left with an associated increase in VaR. Hence, a larger degree of diversification is necessary to reach the same risk exposure in percentage terms as without opportunity costs.
Figure 9.13 Distributions of TVPIs for 10-year investment period – calculation based on Thomson Reuters data.
Figure 9.14 Impact of opportunity costs – calculation based on Thomson Reuters data.
9.6 CASH-FLOW-AT-RISK
Investors use VaR as a basis for determining their capital adequacy and measuring traded risk. Non-financial firms, however, have found this concept difficult to apply in their risk management as value mainly takes the form of real investments in fixed assets that cannot be monetized easily. Industrial companies tend to look at the cash-flow-at-risk (CFaR) as a more relevant measure for their risk exposures. The CFaR is the maximum deviation between actual cash flows and a set level (e.g., a budget figure) due to changes in the underlying risk factors within a given time period for a given confidence level.
While in the case of tradable assets VaR is usually computed for very short time periods (days or weeks), CFaR relates to longer periods, typically quarters and sometimes even years (see Damodaran, undated). As far as financial firms are concerned, it has been argued that marked-to-market portfolios are convertible into cash at short notice and therefore their VaR is also their CFaR (see Damodaran, undated; Yan et al., 2011). However, this argument does not apply to illiquid assets, which ties into our previous discussion of the cash flow volatility-based approach to calculating the VaR for portfolios of funds.
When looking at the CFaR for a portfolio of funds we focus on variations in cash flow within a given time interval [t1, t2] (Figure 9.15). For a LP, both directions of cash flow are relevant as they play into the funding test: positive cash flows as they will be needed by the LP for new investments or to honour future capital calls, negative cash flows as they expose the LP to liquidity risks because available cash may not be sufficient to meet its financial obligations.
Figure 9.15 Cash flow distributions at two different time intervals to determine CFaR
9.7 CONCLUSIONS
In this chapter, we have presented a VaR approach for investments in limited partnership funds. Intuitively, the purpose of calculating a VaR for such funds is similar to the original idea behind the VaR concept, which has been designed for marketable assets – namely to estimate the maximum loss for a given portfolio, a given confidence level and a given time horizon. To be sure, the VaR concept has been criticized for a number of reasons, not least because it has not prevented a number of financial institutions from failing during periods of financial turmoil, notably the recent global financial crisis, the biggest casualty of which was Lehman Brothers. While investors need to be acutely aware of the shortcomings of VaR, its greatest benefit should be seen in the imposition of a structured methodology for critically thinking about risk (Jorion and Taleb, 1997).
Investors in illiquid funds face additional issues when they apply a VaR approach in order to determine the risk of their portfolios. Specifically, there are no observable market data that can be used as input. So what to do? We have proposed two alternatives: first, an approach based on NAV time series and second, one based on cash flows. While a NAV-based approach is relatively simple to implement, there are a number of important problems that limit its applicability, in particular with regard to relatively immature portfolios. As we have argued in this chapter, immature portfolios are characterized by a significant amount of unfunded commitments, thus favouring the use of cash flow-based VaR modelling.
We know from other asset classes that risk can be significantly reduced through diversification. Investments in private equity and real assets through limited partnerships are no different. However, there are many dimensions along which diversification can be achieved, including vintage years, investment strategies, stages and geographies. Among these, as we have shown, vintage year diversification provides the largest benefits.
While investment risk is generally seen as the probability of losing capital, this perspective ignores opportunity costs. Alternatively, the LP could have invested in a different asset class, such as a public equity index or high-yield bonds. This can be taken into account by a VaR that is adjusted for opportunity costs.
Finally, we have presented briefly a complementary approach, the CFaR. This approach is generally considered to be particularly relevant for non-financial firms, whose real assets cannot easily be liquidated. Although the VaR concept takes a prominent role in financial regulation, for illiquid assets, in fact, the CFaR may be of even higher relevance. Importantly, therefore, the cash flow-based VaR provides a tool to calculate the VaR and the CFaR following the same approach. Thus, the cash flow-based VaR approach ensures that the VaR and the CFaR concepts are reconcilable. This dovetails into our previous discussion in Chapter 8, highlighting the importance of a funding test to ensure that the investor is always in a position to respond to capital calls, which are uncertain in terms of their timing and size.
APPENDIX – EXAMPLES
Table 9.A.1 One-fund scenario, discount rate 5%
Table 9.A.2 Scenario for portfolio of 10 funds
Table 9.A.3 Scenario portfolio of 10 funds – annual VaR calculation
Table 9.A.4 Ten different scenarios “A” to “J” for one fund – annual VaR calculation
1 See for example Hendricks (1996), Duffie and Pan (1997) and Dowd et al. (2004). As the VaR approach was originally designed for investment banks heavily involved in trading, one may ask to what extent it is applicable to illiquid asset classes. However, concepts like credit-VaR demonstrate that related risk measures can be derived in a similar fashion as market risk.
2 The estimated average return for the sample period is 16.3% with a volatility of 15.3%. Similar results are obtained by Kaplan and Schoar (2005), Studer and Wicki (2010) and Kaserer and Diller (2004b).
3 Initially, an even higher stress factor of 59% was proposed.
4 Conroy and Harris (2007) employ the Dimson approach. Diller and Jäckel (2010) follow Getmansky et al. (2004).
5 However, Kaserer et al. (2003) emphasize several challenges. For instance, since the terminal wealth dispersion gives an average rate of return over a longer period of time, no dependency with market movements can be detected. Similarly, the approach does not allow for estimating correlations within the asset class.
6 As far as private equity is concerned, Diller and Herger (2008) define mature funds as those with an age of at least 5 years.
7 With a discount rate of zero, the calculation would give the “default risk” of not returning the capital to the LP.
8 For example, techniques such as the one proposed by Bongaerts and Charlier (2006) could be applied.
9 In Germany, for example, there is a guaranteed but adjustable rate of interest on a life insurance policy. At the time of writing, this rate was 1.75%, down from 3.25% at the beginning of the century. In Japan, the high guaranteed return on life insurance policies in the 1990s caused several insurers to fail and be taken over.
10 In conducting a Monte Carlo simulation, cash flows need to be simulated under the “risk-neutral” probability measure to take account of the risk properties of the cash flows. Simulating cash flows using historical means and variances and then still discounting such flows using a risk-free rate would overstate the value. We thank Per Stromberg for pointing this out.