Analytical Valuation of Private Equity Investments

ADIA Lab Research Paper Series

Alexander Lipton & Marcos Lopez de Prado, ADIA Lab (June 2025)

This paper introduces a new mathematical framework for valuing private equity (PE) investments by developing closed-form, analytical solutions to complex valuation problems—especially under conditions that more accurately reflect the reality of PE: illiquidity, nonlinearity, and market incompleteness.

Context

Private equity is increasingly central to institutional portfolios (e.g. sovereign wealth funds, endowments), yet remains difficult to value due to long lock-up periods, irregular cashflows, and lack of market comparables. Traditional valuation methods like Discounted Cash Flow or Public Market Equivalent (PME) metrics fall short under these conditions.

Key Contributions

1. Certainty Equivalent (CE) Pricing via Utility Functions
   The paper models investor behaviour using utility theory—focusing on exponential (CARA), quadratic, and briefly, CRRA utility functions. The certainty-equivalent price (what an investor would pay today for a risky investment) is derived by solving Hamilton-Jacobi-Bellman (HJB) equations.

2. Mathematical Innovation

   • Hopf-Cole Transformation: Used to solve otherwise intractable nonlinear PDEs for CARA utility investors even when assets are not perfectly correlated.

   • Legendre Transformation: Applied to linearise the HJB equations in one-dimensional (liquid asset) cases.

   • Self-similarity and separation of variables: Used for CRRA cases in unleveraged settings.

   • Analytical vs. Numerical: Unlike much of the literature that relies on Monte Carlo or simulation-based approaches, this paper prioritises closed-form solutions.

3. Modelling Illiquidity
   PE's illiquidity is explicitly modelled via investment in assets that cannot be rebalanced continuously. The paper shows that under CARA and quadratic utilities, there exist optimal thresholds for allocating capital to illiquid assets—helping quantify "how much" and "when" PE exposure is worth it.

4. Leverage & Payoff Structure
   The authors also consider leveraged PE investments, incorporating payoff structures resembling financial options (e.g., hockey-stick shaped terminal payoffs with GP/LP sharing rules). These payoffs are simplified to make the valuation analytically tractable.

5. Comparison with Discrete-Time Models
   A notable outcome is the contrast between continuous-time optimisation (e.g. Merton-style models) and single-period or “buy-and-hold” strategies, often used in PE. While continuous optimisation yields higher expected values, discrete strategies may offer more desirable risk profiles (e.g. fatter right tails, thinner left tails).

Who Is This For?

This paper is most useful for:

• Quantitative finance professionals and institutional investors managing alternative assets

• Academics working on asset pricing in incomplete markets

• Actuarial and risk modelling teams at sovereign funds or pension schemes

It assumes familiarity with stochastic calculus, PDEs, and utility theory, but it fills a significant gap by offering closed-form analytical tools in a field dominated by empirical or simulation-heavy methods.