Conditional analysis - 4 Private equity risk premiums

4 Private equity risk premiums

4.4 Conditional analysis

As explained above, to help mitigate the concern that investment managers manipulate exposure to risk factors as a function of expected market conditions, we interact the factors with conditioning variables. Arguably, these variables must be known at the beginning of the project, so we measure them at the date of portfolio formation. We experimented with popular instruments for conditional asset pricing, such as the default spread (yield diﬀerence between BAA and AAA corporate bonds in the U.S.), the term spread (yield diﬀerence between AAA bonds and treasuries in the U.S.), and the dividend yield for the U.S. market portfolio (as provided by Robert Shiller). In addition, we used the aggregate liquidity level of Pastor and Stambaugh (2003).

The only statistically signiﬁcant results were obtained using the default spread, and we report them in Table 10. To ease interpretation, the instrument is expressed as a z-score (that is, we subtract the mean and divide by the standard deviation).

The first specification shows that market betas have no significant dependence on the default spread for the CAPM specification. In the second specification (three-factor model), in contrast, we find that the loading on the HML factor increases in bad times (i.e. high levels of default spread). This finding resonates with similar evidence for value stocks in public equity markets (see e.g. Lettau and Ludvigson (2001)) and confirms the common belief that private equity investments behave like value stocks. The last two specifications focus on the four-factor model. First, we show that exposure to liquidity risk is significant and depends negatively on the aggregate market conditions when the loadings on other factors are kept constant (specification three). This result remains significant (at the 10%

level) when the loadings on the other factors are also a function of the default spread (speciﬁcation four).

Focusing on the last set of estimates, the economic magnitude of the effect of the conditioning variable is large. A one-standard deviation upward move in the annual default spread decreases the liquidity risk loading by about 0.27, raises the market beta by 0.52, and increases exposure to the HML factor by about 0.78. The direct effect of the default spread in this specification is not significant, nor is the impact on the SMB loading. Overall,

the annual cost of capital increases by roughly 6.6% for a one-standard deviation increase in the default spread.²⁴

The finding that the exposure to the liquidity premium (IML) decreases when credit spreads widen may shed additional light on the determinants of liquidity risk in private equity. Arguably, this result is consistent with the refinancing risk channel. Managers expecting tough credit market conditions may choose to reduce the leverage of their invest-ments and/or to invest in assets that are more collateralizable in bad times. This makes default due to lack of refinancing opportunities less likely. A similar outcome can be im-posed upon private equity firms by their lenders who, in bad times, could decide to fund less risky projects. In either case, the evidence suggests that the exposure of private equity investments to liquidity risk is managed dynamically to protect against any deterioration in liquidity.

5 Conclusions

Using cash flows from 4,403 liquidated private equity investments drawn from a novel and comprehensive dataset, we find a positive and significant relation between returns and liq-uidity risk measures. These results are robust to the use of different measures for liqliq-uidity risk. They also hold after controlling for macroeconomic conditions and investment char-acteristics. Larger investments have higher exposure to liquidity conditions, and so do investments made by older private equity firms. These findings are consistent with three economic channels through which liquidity risk can affect private equity returns: transaction costs, the risk tolerance of long-horizon private equity investors, and leverage. Depending on data availability, future research could further investigate the separate contribution of each of these channels. Finally, using the Pastor and Stambaugh (2003) four-factor model, we find that the liquidity risk premium is about 3% annually, the total risk premium is about 18%, and the alpha (gross-of-fees) is not statistically different from zero.

The results in this paper are relevant for academics, practitioners, and regulators, as

24This estimate results from multiplying the observed changes in factor loadings by the average risk premiums on the factors from Table 8 and annualizing: (-0.27 × 0.37 + 0.52×0.63 + 0.78×0.42)×12 ' 6.6%.

we quantify the systematic risks and the pricing efficiency of an asset class that has gained increasing importance in the financial markets. Our evidence suggests that the apparently high performance (before fees) of private equity investments can be largely explained as compensation for the different risk factors to which returns are exposed, and liquidity risk is one important source of this risk premium.

Our results provide practitioners with a hurdle rate to evaluate private equity. Using such a benchmark, they can assess the NPV of their track record. The cost of capital of about 18% in excess of the risk-free rate that we estimate is in sharp contrast to the widely-used hurdle rate of 8%. In addition, our results may call current compensation practices into question. Fund managers (GPs) and, oftentimes, the private equity team within the investor’s organization, receive performance-based compensation if they achieve returns above 8% per annum, but this hurdle rate seems low in view of our ﬁndings.

Knowing the risk proﬁle of private equity investments is also an important input for portfolio risk management. The Harvard endowment example given in the introduction shows that investors may underestimate the importance of systematic liquidity risk in pri-vate equity. In bad times, these investments may not oﬀer the cushion that investors expect from them.

Regulators may also ﬁnd some useful insights in our results. Solvency II and Basel II require insurance companies and banks to set aside a provision for the risk on their private equity investments (see Bongaerts and Chalier (2009)). As the current method of weighting assets by risk does not reﬂect the large exposure to liquidity risk, this may result in too low a provision.

Finally, for academics, this paper ﬁnds that the liquidity risk factor identiﬁed in public equity is consistently related to private equity performance. This contributes to the recent literature showing the pervasiveness of liquidity risk across asset classes.

Appendix

In this appendix, we provide the explicit derivation of equations (7) and (6) in the text. The reported formulas diﬀer slightly from the formulas in Cochrane (2005), because we have a multifactor model and the factors are not in logarithmic form.

From equation (2), Rⁱ_t+1 is the exponential of a normally distributed variable:

Rⁱ_t+1= R^f_t+1e^γ+δ⁰^f^t+1^+εⁱ^t+1

Also, by assumption, the factors are normal. Hence, the expression of the expected return is

E( Rⁱ_t+1)

= R^f_t+1e^γ+δ⁰^µ^F⁺¹²^δ⁰^σ²^F^δ+¹²^σ² (A-1)

Applying Stein’s lemma, the covariance can be expressed as

Cov(

f_t+1, Rⁱ_t+1)

= Cov(f_t+1, δ⁰f_t+1+ εⁱ_t+1)E( Rⁱ_t+1)

= Cov(

f_t+1, δ⁰f_t+1+ εⁱ_t+1)

R^f_t+1e^γ+δ⁰^µ^F⁺¹²^δ⁰^σ²^F^δ+¹²^σ²

= V ar (f_t+1) δR^f_t+1e^γ+δ⁰^µ^F⁺¹²^δ⁰^σ²^F^δ+¹²^σ²

where, for the last step, we use the fact that εⁱ_t+1and ft+1 are uncorrelated. The expression for beta then follows:

β = V ar (ft+1)⁻¹Cov(

ft+1, Rⁱ_t+1)

= V ar (ft+1)⁻¹V ar (ft+1) δR^f_t+1e^γ+δ⁰^µ^F⁺¹²^δ⁰^σ²^F^δ+¹²^σ² (A-2)

= δR^f_t+1e^γ+δ⁰^µ^F⁺¹²^δ⁰^σ²^F^δ+¹²^σ² (A-3)

To compute alpha we use the standard deﬁnition

α = E( Rⁱ_t+1)

− R^f_t+1− β⁰E (ft+1) (A-4)

where E (f_t+1) = µ_f. Replacing the expressions for the expected return in (A-1) and beta

Although we do not use them in the estimation, it is interesting to derive the continuous time limits for α and β. These are:

β = δ (A-6)

α = γ +1

2δ⁰σ²_fδ +1

2σ². (A-7)

To obtain these formulas, one can start from the continuous time equivalent of equation (2)

d log (V_t) = γdt + r_fdt + δ⁰df_t+ σdZ_t (A-8)

where dft = µ_fdt + σfdZf,t, Zt and Zf,t are independent vectors of standard Brownian motions, and r_f is the instantaneous risk-free rate. Then, apply Ito’s lemma to equation (A-8) to obtain the process for the return in levels

dV_t

Then, from equation (A-9), we obtain beta using the standard deﬁnition

β = V ar (dft)⁻¹Cov

Finally, to obtain equation (A-7), use the deﬁnition of alpha and the result in (A-10)

αdt = E

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Table1:DatacoverageThetablecomparesthecoverageoftheCEPRESdatasettothatofCapitalIQ.Itshowsthenumberof investments,theirtotalsize(equityinvested)andtheiraveragemultiple(size-weighted).Multipleisthesumofallthedividendsdivided bythesumofallinvestments.Statisticsareshownforboththesampleofliquidatedinvestmentsandthesampleofnon-liquidated investments.Statisticsarealsobrokendownbyinvestmentperiods. LiquidatedNon-liquidatedTotalCEPRESCapitalIQFraction NumberSizeMultipleNumberSizeMultipleNumberNumberCEPRESvs Investments($million)Mean(VW)Investments($million)Mean(VW)InvestmentsInvestmentsCapitalIQ 1975-19895924,5813.30102373.566025811.04 1990-19941,32017,3543.03521,5232.201,3721,1021.25 1995-19991,70238,0592.3866528,5691.652,3673,4160.69 2000-200678927,0272.262,068140,0031.662,8578,9120.32 Total4,40387,0212.522,795170,3321.677,19814,0110.51

Table 2: Data representativenessThis table compares the success rates of firms included in the CEPRES dataset and of firms that are not included in CEPRES dataset. The universe of firms and their success ratio comes from Thomson Venture Economics. Successful exit rate is the fraction of investments exited via IPO or M&A over the total number of investments. Only investments made before 2002 and firms with more than 5 investments are considered.

CEPRES TVE (ex-CEPRES) Diﬀerence

(1) (2) (1) minus (2)

Number of ﬁrms 117 535 -418

Successful-exit rate

20th percentile 0.43 0.39 0.04

50th percentile 0.61 0.56 0.05

80th percentile 0.75 0.72 0.03

Mean 0.59 0.55 0.04

Table 3: Cash flows of a typical investment. A typical cash flow stream is shown. The initial investment is normalized to 100. To construct it, we first compute the median number of intermediate cash inflows and outflows. The result is 1 and 3, respectively. Next, we compute the median time at which intermediate cash inflows and outflows occur. The result is 0.25 and 2.5 years respectively. Next, we compute the median size of intermediate cash inflows and outflows (normalized to initial amount invested). The result is 25% and 195% respectively. Then we compute the median size of the final cash outflow and its median time. The result is 135% and 4 years respectively.

Date (in years) Cash ﬂows

0.00 -100

0.25 -25

0.50 0

0.75 0

1.00 0

1.25 0

1.50 20

1.75 0

2.00 0

2.25 0

2.50 20

2.75 0

3.00 0

3.25 0

3.50 20

3.75 0

4.00 200

Table 4: Performance by year and region. The table reports the modiﬁed IRR of a group of investments. Groups are based on the year of investment initiation and the region where the investment is located. Performance is computed on the pooled cash ﬂows of each group. The reinvestment rate is the return on the S&P 500 index.

Panel A: Modiﬁed Internal Rates of Return (S&P as re-investment rate)

1975-1989 1990-1994 1995-1999 2000-2006 1975-2006

US 0.18 0.18 0.19 0.13 0.18

UK 0.17 0.16 0.17 0.20 0.17

Europe (ex-UK) 0.17 0.14 0.25 0.21 0.20

Rest world 0.21 0.15 0.18 0.17 0.17

All countries 0.18 0.17 0.21 0.21 0.19

Panel B: Number of Investments

1975-1989 1990-1994 1995-1999 2000-2006 1975-2006

US 323 533 534 237 1627

UK 172 440 526 139 1277

Europe (ex-UK) 68 269 499 246 1082

Rest world 17 23 121 152 313

All countries 592 1320 1702 789 4403

Table5:Baseregressionanalysis.ThistablereportstheresultofOLSregressions.ThedependentvariableistheModified IRR(S&P500indexasreinvestmentrate)ofindividualinvestments.Explanatoryvariablesinclude(equally-weighted)averageaggregate liquidityinnovationsduringtheinvestment’slifeandinvestmentcharacteristics.Therearethethreeliquiditymeasuresdrawnrespectively fromPastorandStambaugh(P&S),AcharyaandPedersen(A&P)andSadka.Eachmeasureisstandardizedbyremovingthesample averageanddividingbythesamplestandarddeviation.Investmentcharacteristicsincludeinvestmentsize(i.e.equityinvested),firmageat thetimeofinvestmentinitiation,adummyvariablethattakesthevalueoneiftheinvestmentisclassifiedasgrowth(zerootherwise),and adummyvariablethattakesthevalueoneiftheinvestmentislocatedintheU.S.Sizeandfirmagearerelativetosame-yearinvestments andarewinsorized(95thpercentilesofsame-yearinvestments).Countryandindustryfixedeffectsmaybeaddedtothesetofexplanatory variables.Standarderrorsareclusteredattheinvestmentyearlevelandcorrespondingt-statisticsarereportedbeloweachcoefficient betweenparentheses. P&Sliquidityinnovations0.1260.1220.1260.095 (4.410)(4.679)(4.779)(3.130) A&Pliquidityinnovations0.0740.0800.0870.024 (2.255)(2.874)(2.871)(0.856) Sadkaliquidityinnovations0.0670.0620.0640.024 (3.115)(3.245)(3.278)(1.251) Size0.0010.000-0.007-0.006-0.033-0.034-0.024 (0.046)(-0.019)(-0.427)(-0.349)(-1.997)(-1.948)(-1.498) Firmage0.0240.0200.0340.0290.0290.0280.022 (1.654)(1.301)(2.145)(1.799)(1.924)(1.660)(1.348) Growthinvestment-0.171-0.184-0.195-0.211-0.196-0.207-0.209 (-4.016)(-4.356)(-4.459)(-4.694)(-4.362)(-4.612)(-4.578) USinvestment-0.019-0.024-0.026-0.0320.0040.001-0.015 (-0.676)(-0.863)(-0.872)(-1.135)(0.128)(0.017)(-0.495) Countryfixedeffectsnonoyesnonoyesnonoyesyes Industryfixedeffectsnonoyesnonoyesnonoyesyes Adj.R20.0450.0620.0840.0150.0390.0600.0120.0380.0590.088 Nobservations4403440340374403440340374020402036763676

Table 6: Regressions with cross-eﬀects. This table reports the results of OLS regressions.

The dependent variable is the Modified IRR (using the S&P 500 index as the reinvestment rate) of individual investments. Explanatory variables are the same as in Table 5 except for the cross-effects. Standard errors are clustered at the investment year level. T-statistics are reported below the coefficients in parentheses. Results are shown for each liquidity measure separately: Pastor and Stambaugh (Panel A), Acharya and Pedersen (Panel B) and Sadka (Panel C).

Panel A: Pastor and Stambaugh liquidity measure

Liquidity conditions 0.128 0.127 0.108 0.161 0.130

(4.820) (4.706) (3.265) (5.669) (4.417)

Size 0.021 0.007

(1.624) (0.547)

Firm age 0.012 0.022

(1.095) (1.860)

Growth investment -0.164 -0.179

(-3.790) (-4.107)

US investment -0.032 -0.030

(-1.009) (-1.243)

Liquidity * Size 0.021 0.019

(1.739) (1.682)

Liquidity * Firm age 0.028 0.029

(2.221) (2.562)

Liquidity * Growth investment 0.072 0.063

(1.646) (1.607)

Liquidity * US investment 0.023 0.025

(1.601) (2.099)

Country ﬁxed eﬀects no no no no no

Industry ﬁxed eﬀects no no no no no

Adj. R2 0.047 0.048 0.061 0.048 0.072

N observations 4403 4403 4403 4403 4403

Table 6: (continued)

Panel B: Acharya and Pedersen liquidity measure

Liquidity conditions 0.087 0.076 0.096 0.084 0.101

(2.892) (2.305) (3.440) (2.607) (4.313)

Liquidity * Firm age 0.028 0.031

(1.727) (2.043)

Liquidity * Growth investment -0.070 -0.066

(-1.414) (-1.498)

Liquidity * US investment 0.008 0.008

(0.459) (0.634)

Country ﬁxed eﬀects no no no no no

Industry ﬁxed eﬀects no no no no no

Adj. R2 0.026 0.019 0.034 0.016 0.055

N observations 4403 4403 4403 4403 4403

Panel C: Sadka liquidity measure

Liquidity conditions 0.067 0.069 0.050 0.084 0.054

(3.235) (3.202) (2.081) (2.908) (1.938)

Liquidity * Firm age 0.022 0.008

(1.849) (0.527)

Liquidity * Growth investment 0.054 0.071

(1.627) (2.009)

Liquidity * US investment 0.013 0.010

(0.693) (0.728)

Country ﬁxed eﬀects no no no no no

Industry ﬁxed eﬀects no no no no no

Adj. R2 0.016 0.014 0.027 0.014 0.045

Table7:Liquidityandmacroconditions.ThetablereportstheresultsofOLSregressions.Thedependentvariableisthe ModifiedIRR(usingtheS&P500indexasthereinvestmentrate)ofindividualinvestments.Explanatoryvariablesincludeameasure ofaggregateliquidityinnovationsduringtheinvestmentlife,adummyvariablethattakesthevalueoneiftheinvestmentisclassified asgrowth(zerootherwise),stock-marketfactorsandmacro-conditionmeasures.Rm-Rf,HMLandSMBaretheaveragereturnofthese (Fama-French)factorsduringtheinvestment’slife.Macrovariablesaretheaverage(overinvestmentlife)ofmonthly:changeindefault spread,industrialproductiongrowth,changeinU.S.IPOvolumeandchangeinVIX.Eachexplanatoryvariableisexpressedasaz-score (wesubtractthesamplemeananddividebythesamplestandarddeviation).Countryandindustryfixedeffectsmaybeaddedtothe setofexplanatoryvariables.Standarderrorsareclusteredattheinvestmentyearlevel.T-statisticsarereportedbelowthecoefficientsin parentheses.TheliquidityvariableiseitherdrawnfromPastorandStambaugh,AcharyaandPedersenorSadka. P&Sliquidityinnovations0.0940.0870.0690.074 (2.995)(2.932)(2.071)(2.228) A&Pliquidityinnovations0.0610.0520.0100.016 (2.222)(1.630)(0.348)(0.492) Sadkaliquidityinnovations0.0520.0420.0270.022 (2.201)(2.234)(1.563)(1.148) Growthinvestment-0.148-0.120-0.156-0.127-0.164-0.128-0.153-0.132-0.139-0.156 (-3.629)(-2.967)(-3.838)(-3.193)(-3.932)(-3.078)(-3.344)(-3.015)(-3.177)(-3.598) Rm-Rf0.113-0.0190.058-0.0630.086-0.0320.076-0.056-0.085-0.089 (4.787)(-0.440)(1.712)(-1.419)(3.177)(-0.690)(2.356)(-1.203)(-1.872)(-1.938) HML0.0210.0380.0070.0200.0060.028-0.0100.006-0.0030.009 (0.396)(0.947)(0.145)(0.519)(0.108)(0.688)(-0.198)(0.164)(-0.090)(0.229) SMB-0.037-0.047-0.037-0.044-0.045-0.043-0.057-0.060-0.053-0.048 (-1.239)(-1.312)(-1.635)(-1.561)(-1.805)(-1.285)(-1.994)(-1.777)(-1.908)(-1.627) Deltadefaultspread-0.047-0.039-0.025-0.049-0.038-0.036 (-2.906)(-2.612)(-1.205)(-2.747)(-1.660)(-1.435) Industrialproductiongrowth0.1420.1420.1490.1480.1500.165 (3.382)(3.880)(3.561)(3.200)(3.676)(4.198) DeltaIPOvolume0.0450.0180.0320.0320.0110.009 (2.925)(1.466)(2.558)(2.362)(0.800)(0.537) DeltaVIX-0.0537-0.06715-0.05204-0.052-0.064-0.066 (-1.618)(-2.270)(-1.584)(-1.532)(-2.127)(-2.078) Countryfixedeffectsnononononononononoyes Industryfixedeffectsnononononononononoyes Adj.R20.0520.0780.0710.0920.0610.0820.0610.0870.0980.121 Nobservations4403428644034286440342864020393339333600

Table 8: Correlations and distributions of the traded factors. This table shows the correlation matrix and summary statistics for the (time-series of the) four traded risk factors: the illiquid-minus-liquid factor by Pastor and Stambaugh (2003), the excess market return, HML, and SMB. The time period is from October 1975 to December 2007. The frequency is monthly. Returns are in percentages.

IML PS Rm-Rf HML SMB

Correlations:

IML PS 1.000

Rm-Rf -0.100 1.000

HML -0.276 -0.460 1.000

SMB 0.042 0.236 -0.341 1.000

Mean 0.375 0.630 0.417 0.241

St. Deviation 4.138 4.320 3.009 3.166

5th percentile -5.767 -6.410 -3.960 -4.180

Median 0.608 0.940 0.370 0.120

95th percentile 5.530 7.010 5.330 4.800

Table 9: Risk models and the cost of capital. The table reports the results of OLS estimation of three diﬀerent factor models for private equity returns (Panel A) and the resulting alphas and cost of capital (Panel B). In Panel A, the dependent variable is the logarithm of one plus the monthly MIRR minus the log of one plus the risk-free rate. Each observation corresponds to a portfolio of private equity investments formed by the starting date of the investment. Portfolios must contain at least thirty investments. Each observation is weighted by the square root of the investment duration to correct for unequal variance. Explanatory variables include the Fama and French (1993) three factors (excess market return, HML, SMB) and the illiquid minus liquid portfolio (IML PS) by Pastor and Stambaugh (2003). Each explanatory variable is computed by taking its average value during the investment life. All variables are in monthly frequency. All speciﬁcations are run between October 1975 and December 2007. The table also reports the estimate of the residual

In document Private Equity Performance and Liquidity Risk (Page 31-60)