# Private Equity Fund Valuation and Systematic Risk

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1 An Equilibrium Approach and Empirical Evidence Axel Buchner 1, Christoph Kaserer 2, Niklas Wagner 3 Santa Clara University, March 3th 29 1 Munich University of Technology 2 Munich University of Technology 3 Passau University

2 Table of Contents I 1 Motivation 2 3 Assumptions Derivation and Result Numerical Illustration 4 Derivation and Result Numerical Illustration 5

3 General Problem The illiquid character of private equity investments poses particular challenges for private equity (PE) research. As PE investments are not traded on secondary markets, liquidity is low and observable market prices are missing. Lack of Liquidity is a critical issue in evaluating PE investments. Therefore, valuation is a crucial task. The optimal allocation of PE in a investment portfolio typically requires the risk-return characteristics of PE as an asset class.

4 Contents Motivation Model PE as an asset class under the simplifying assumption of a frictionless market and knowledge of investor preferences. Derive a first value assessment (upper boundary) assuming no private valuation information is available. Derive the theoretical (boundary) value of a typical private equity fund based on equilibrium asset pricing considerations. Valuation is based on a stochastic model of a typical fund s cash flows, capital drawdowns and distributions.

5 General Assumptions and Notation Assumption Assumption 1: We consider a Private Equity fund with total (initial) commitments denoted by C. Assumption 2: The PE fund has a total (legal) maturity T l and a commitment period T c, where T l T c must hold. Cumulated capital drawdowns up to t are denoted D t, undrawn committed amounts up to time t are U t. Under these assumptions it must hold: D t = C U t, where D = and U = C.

6 Cumulated Capital Drawdowns I Assumption Assumption 3: Capital drawdowns over the commitment period T c occur in continuous-time. The dynamics of the cumulated drawdowns D t can be described by the ordinary differential equation: dd t = δ t U t 1 { t Tc }dt, where δ t denotes the rate of contribution or simply the fund s drawdown rate at time t. 1 { } is an indicator function.

7 Cumulated Capital Drawdowns II The solution of the ordinary differential equation (1) can be derived by substituting the identity dd t = du t, and using the initial condition U = C. It follows: D t = C C exp dd t = δ t C exp t t δ u du, where t T c, δ u du 1 { t Tc }dt.

8 Stochastic Process Drawdown Rate Assumption Assumption 4: The drawdown rate is modeled by a non-negative stochastic process {δ t, t T c }. In particular, the mean-reverting square root process dδ t = κ(θ δ t )dt + σ δ δt db δ,t is applied, where θ > denotes the long-run mean of the drawdown rate, κ > governs the rate of reversion to this mean and σ δ > reflects the volatility of the drawdown rate. B δ,t is a standard Brownian motion.

9 Simulated Capital Drawdowns I.25 1 Capital drawdowns Cumulated capital drawdowns Figure: Simulated Paths of the Capital Drawdowns (Left) and Cumulated Capital Drawdowns (Right); Parameters: C = 1, κ = 2, θ = 1, δ =.1 and σ δ =.2

10 Simulated Capital Drawdowns II.25 1 Capital drawdowns Cumulated capital drawdowns Figure: Simulated Paths of the Capital Drawdowns (Left) and Cumulated Capital Drawdowns (Right); Parameters: C = 1, κ = 2, θ = 1, δ =.1 and σ δ =.5

11 Simulated Capital Drawdowns III.2 Capital drawdowns Cumulated capital drawdowns Figure: Simulated Paths of the Capital Drawdowns (Left) and Cumulated Capital Drawdowns (Right); Parameters: C = 1, κ =.5, θ = 1, δ =.1 and σ δ =.2

12 Consistency Test: Capital Drawdowns I.18 Cumulated capital drawdowns Capital drawdowns Figure: Full Sample of Liquidated Funds (N = 95); Data set from Venture Economics, January 198 to June 23

13 Consistency Test: Capital Drawdowns II.2 Cumulated capital drawdowns Capital drawdowns Figure: VC-Funds of the Sample of Liquidated Funds (N = 47)

14 Consistency Test: Capital Drawdowns III.2 Cumulated capital drawdowns Capital drawdowns Figure: BO-Funds of the Sample of Liquidated Funds (N = 48)

15 General Assumptions and Notation Assumption Assumption 1: (Non-negative) capital distributions of the PE fund to its investors occur in continuous-time over the fund s legal lifetime T l. Assumption 2: Instantaneous capital distributions p t are assumed to be log-normally distributed according to d lnp t = µ t dt + σ P db P,t, where µ t denotes the time dependent drift and σ P the constant volatility of the stochastic process. B P,t is a standard Brownian motion that is uncorrelated with B δ,t, i.e., Cov t [db P,t, db δ,t ] =.

16 Multiple of the Fund Definition: The funds multiple M t is defined by: M t = t p sds/c. Expected instantaneous capital distributions are related to the fund multiple by the relationship: E t [p t ] = E t [dm t /dt]c. Assumption Assumption 3: The funds expected multiple M t is assumed to follow the ordinary differential equation: E s [dm t ] = α t (m E s [M t ])dt, s t, where m is the multiple s long-run mean and α is the constant speed of reversions to this mean.

17 Stochastic Process Capital Distributions From Assumption 2 it follows: [ t ] p t = p s exp µ u du + σ P (B P,t B P,s ). s If the multiple M t satisfies Assumption 3, it can be shown that the stochastic process of the instantaneous capital distributions is given by: { p t = αt(mc P s )exp 1 } 2 {α(t2 s 2 ) + σp(t 2 s)] + σ P (B P,t B P,s ), where (B P,t B P,s ) = ǫ t t s with ǫt N(, 1).

18 Simulated Capital Distributions I Capital distributions Cumulated capital distributions Figure: Simulated Paths of the Capital Distributions (Left) and Cumulated Capital Distributions (Right); Parameters: C = 1, m = 1.5, α =.5 and σ P =.2

19 Simulated Capital Distributions II Capital distributions Cumulated capital distributions Figure: Simulated Paths of the Capital Distributions (Left) and Cumulated Capital Distributions (Right); Parameters: C = 1, m = 1.5, α =.5 and σ P =.4

20 Simulated Capital Distributions III Capital distributions Cumulated capital distributions Figure: Simulated Paths of the Capital Distributions (Left) and Cumulated Capital Distributions (Right); Parameters: C = 1, m = 1.5, α =.2 and σ P =.2

21 Consistency Test: Capital Distributions I Cumulated capital distributions Capital distributions Figure: Full Sample of Liquidated Funds (N = 95); Data set from Venture Economics, January 198 to June 23

22 Consistency Test: Capital Distributions II Cumulated capital distributions Capital distributions Figure: VC-Funds of the Sample of Liquidated Funds (N = 47)

23 Consistency Test: Capital Distributions III Cumulated capital distributions Capital distributions Figure: BO-Funds of the Sample of Liquidated Funds (N = 48)

24 Simulated Fund Cash Flows Net cash flows Cumulated net cash flows Figure: Simulated Paths of the Net Cash Flows (Left) and Cumulated Net Cash Flows (Right); Parameters: C = 1, κ = 2, θ = 1, δ =.1, σ δ =.2, m = 1.5, α =.5 and σ P =.2

25 Assumptions Derivation and Result Numerical Illustration Assumptions Underlying the Valuation Assumption Assumption 1: PE funds are priced under the risk neutral measure as if they were traded in a frictionless capital market in equilibrium (upper price boundary). Assumption 2: There exists a representative investor with log utility such that in equilibrium expected returns of all assets are generated by a specialized version of the ICAPM. Assumption 3: The drawdown Rate δ t is uncorrelated with returns of the market portfolio. Assumption 4: The covariance σ PM between log capital distributions and returns of the market portfolio is constant.

26 Derivation I Assumptions Derivation and Result Numerical Illustration From Assumption 1 the (upper boundary) market value of a fund Vt F is defined as [ ] [ Tl ] Tl Vt F = E Q e r f (τ t) dp τ F t E Q e r f (τ t) dd τ 1 {t Tc } F t = V P t V D t, t where Vt P (Vt D ) is the present value of all outstanding capital distributions (drawdowns) at time t. Discounting by the riskless rate r f is valid, as all expectations are defined under risk-neutral or equivalent martingale measure Q t

27 Derivation II Assumptions Derivation and Result Numerical Illustration Applying Girsanov s Theorem, it follows that the underlying stochastic processes under the Q-measure are given by: dδ t = [κ (θ δ t ) λ δ σ δ δt ] dt + σ δ δt db Q δ,t, d ln p t = (µ t λ P σ P )dt + σ P db Q P,t, where B Q δ,t and BQ P,t are Q-Brownian motions; λ δ and λ P are market prices of risk, defined by: λ δ µ(δ t, t) r f σ(δ t, t), λ P µ(p t, t) r f σ(p t, t).

28 Result Assumptions Derivation and Result Numerical Illustration From Assumptions 2-4 the market prices of risk are given by λ δ = and λ P = σ PM /σ P. The (upper boundary) value of a private equity fund at any time t [, T l ] during its finite lifetime T l can then be stated as: Tl Vt F =α (m C P t ) e r f (τ t) e C(t,τ) dτ t Tl + U t e r f (τ t) (A (t, τ) B (t, τ)δ t )e A(t,τ) B(t,τ)δt dτ1 {t Tc }, t where A(t, T), B(t, T) and C(t, T) are known deterministic functions.

29 Assumptions Derivation and Result Numerical Illustration Dynamics of the s over Time I 1.5 Value Fund lifetime (in years) Riskless rate.3 Figure: Over the Fund s Lifetime for Varying Values of r f

30 Assumptions Derivation and Result Numerical Illustration Dynamics of the s over Time II Value m Fund lifetime (in years) Figure: Over the Fund s Lifetime for Varying Values of m

31 Assumptions Derivation and Result Numerical Illustration Dynamics of the s over Time III 1.5 Value alpha Fund lifetime (in years) 2 Figure: Over the Fund s Lifetime for Varying Values of α

32 Assumptions Derivation and Result Numerical Illustration Dynamics of the s over Time IV 1.5 Value Fund lifetime (in years) Covariance.1 Figure: Over the Fund s Lifetime for Varying Values of σ PM

33 Assumptions Derivation and Result Numerical Illustration Dynamics of the s over Time V Value Theta Fund lifetime (in years) Figure: Over the Fund s Lifetime for Varying Values of θ

34 Derivation I Derivation and Result Numerical Illustration Same assumptions that were employed to derive the market value (Assumptions 1-4, Section ). The (annualized) instantaneous return R F t of a private equity fund at time t is defined by: R F t = dv F t dt + dpt dt ddt dt Vt F = dv P t dt dv D t dt + dpt dt ddt dt Vt P Vt D. Taking the conditional expectation E P t [ ], the expected instantaneous fund return µ F t is given by: µ F t = E P t [ R F t ] = E P t [ ] dv P t dt Et P E P t [ dv D t dt ] + E P t [ dpt dt [ V P t ] E P t [V D t ] ] [ E P ddt ] t dt.

35 Derivation II Derivation and Result Numerical Illustration From Assumptions 1-4, it must hold that: [ ] [ dv Et P D t = Et P ddt dt dt [ ] [ dv Et P P t = Et P dpt dt dt ] + r f V D t ] + (r f + σ PM )V P t. Remark: Equilibrium where expected returns are generated by the specialized version of the ICAPM (Assumptions 1-2). The first equation holds as drawdowns are assumed to carry zero systematic risk (Assumptions 3). The second equation holds as constant covariance σ PM is assumed for distributions (Assumptions 4).

36 Result I Derivation and Result Numerical Illustration Substituting the previous results, the expected instantaneous fund return is given by: µ F Vt P t = r f + σ PM Vt P Vt D We can also view the expected fund returns from a traditional beta perspective. It follows: µ F t = r f + β F,t (µ M r f ) where µ M is the expected return of the market portfolio and β F,t is the beta coefficient of the fund at time t given by: β F,t = β P V P t Vt P Vt D with β P = σ PM /σ 2 M..

37 Result II Derivation and Result Numerical Illustration Or more generally, when the beta coefficient of the capital drawdowns β D : β F,t = β P V P t Vt P Vt D β D V P V D t t Vt D, i.e., the beta coefficient of the fund returns is the market value weighted average of the betas of the fund s capital distributions and drawdowns. This result implies that the beta coefficient of the fund is non-stationary whenever β P β D holds, i.e., capital distributions and drawdowns carry different levels of systematic risks. Quite similar result to Brennan (1973), Myers and Turnbull (1977) and Turnbull (1977) on the systematic risk of firms.

38 Derivation and Result Numerical Illustration Expected Return and over Time I Expected fund return Figure: Ex-ante Expected Returns Over the Fund s Lifetime Parameters choice: C = 1, T c = T l = 2, r f =.5, κ =.5, θ =.5, σ δ =.1, δ =.5, m = 1.5, α =.25, σ P =.2, σ PM =.5, µ M =.1125 and σ 2 M =.626.

39 Derivation and Result Numerical Illustration Expected Return and over Time II Table: Beta Coefficients Over the Lifetime of the Fund The Table shows the evolution of Vt P, V t P, (V t P/V t P Vt D ) and of the beta coefficient β F,t over the lifetime of the fund. Time t (in years) V P t V D t V P t V P t V D t β F,t

40 Conclusion We perform stochastic modeling and equilibrium pricing of private equity funds as an asset class. The paper provides a solution for the upper boundary market value of private equity funds with typical drawdown and distribution characteristics. Model parameters can be calibrated from a cross-section of historical cash flow data of private equity funds.

41 Outlook Motivation Future work may address the following points: Calibration and estimation for larger (available?) data sets. Pricing results and (short term?) emergence of secondary markets. Incorporate an illiquidity discount in the pricing model.

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