The Nagare Financial Engine: A Stochastic Cohort-Based Projection Model for Private Capital

Authors: Engineering & Quantitative Research Team Version: 3.0

Abstract

We present the mathematical specification and architectural design of the Nagare Financial Engine, a high-performance projection system for private capital funds (Private Equity, Venture Capital, Private Credit, Real Estate). The engine departs from traditional iterative monthly simulations in favor of a Yearly Cohort-Based Model utilizing analytic integration of survival functions (Weibull, Log-Logistic, and Mixture Models) to determine exit timing. This approach achieves a 10-20 $\times$ performance improvement over discrete monthly methods while strictly preserving mass conservation and timing accuracy. We introduce a MOIC-First Calibration framework that solves for exit multiples to match target TVPI outcomes exactly, eliminating the need for iterative growth rate calibration. The system further incorporates a Stochastic Overlay using competing risks and market-modulated pacing to simulate volatility and tail risk.

1. Introduction

Private market forecasting faces unique challenges compared to public markets. Returns are driven by discrete, irregular liquidity events (exits) rather than continuous price discovery. Traditional models often rely on simple cash flow pacing (e.g., the Yale Model or Takahashi-Alexander), which lack the granularity to model idiosyncratic portfolio risks, or fully discrete Monte Carlo simulations, which are computationally expensive.

The Nagare Engine bridges this gap by employing a hybrid Semi-Analytic Framework:

Deterministic Core: Computes expected cash flows using closed-form integrals of survival functions applied to annual vintages (cohorts).
Stochastic Modulation: Overlays market beta, idiosyncratic volatility, and lines-layer granularity on top of the deterministic path.

This paper details the mathematical formulation of these components.

2. Model Specification

2.1. Deployment Dynamics

Capital deployment is modeled as a function of the total commitment $C$ and a normalized deployment density function $d(t)$ .

Let $d_y$ be the target deployment percentage in year $y$ , such that $\sum d_y \le 1$ . The capital call in month $m$ is given by:

$\ Call_m = C \cdot \frac{d_{\lfloor m/12 \rfloor}}{12}$

Carry-Forward Mechanism: To account for real-world under-deployment, we define the cumulative deployment capacity at year $Y$ as $K_Y = C \sum_{y=0}^Y d_y$ . The actual call in any period is constrained by the remaining capacity:

( \text{Call}_m^{\text{planned}}, \quad K_Y - \sum_{i=0}^{m-1} \text{Call}_i^{\text{actual}} )

This ensures strict adherence to commitment limits while allowing catch-up deployment in later periods.

2.2. Exit Timing: Survival Analysis Framework

Exit timing is modeled using Survival Analysis, where $S(t)$ represents the probability that a portfolio asset has not exited by age $t$ . The probability of an asset exiting in year $n$ , given it was deployed in vintage year $v$ , is the integral of the failure density over the deployment window.

Let $w(u)$ be the intra-year deployment density function (where $\int_0^1 w(u) du = 1$ ). The discrete probability of exit in year $n$ for vintage $v$ is:

$P(\text{Exit}_n | \text{Vintage}_v) = \int_0^1 w(u) \left[ S(n - v - u) - S(n - v + 1 - u) \right] du$

This formulation accounts for the continuous aging of assets deployed throughout the year, preventing the "lumpiness" artifacts found in simple annual models.

2.2.1. Parametric Distributions

The engine supports multiple survival distributions tailored to specific asset classes:

A. Weibull Distribution (Buyout / Real Estate) Suitable for assets with increasing hazard rates (predictable exits).

( -\left( \frac{t}{\lambda} \right)^k )

$k \approx 3.0$ for Buyout (steep J-curve).
$k \approx 2.0$ for Real Estate (gradual exits).

B. Log-Logistic Distribution (Growth Equity / Infrastructure) Heavy-tailed distribution for assets that may be held indefinitely.

$S(t) = \frac{1}{1 + (t/\beta)^\alpha}$

C. Mixture Models (Venture Capital) Models the bi-modal nature of VC returns (early "acqui-hires" vs. late "unicorns").

$S(t) = w_{\text{early}} S_{\text{Weibull}}(t; \lambda_1, k_1) + (1 - w_{\text{early}}) S_{\text{Weibull}}(t; \lambda_2, k_2)$

2.3. Calibration

We employ a Two-Point Calibration method to fit distribution parameters $(\lambda, k)$ to observable market benchmarks:

Median Holding Period ( $t_{0.5}$ ): $S(t_{0.5}) = 0.5$
Fund Life Constraint ( $T_{\text{life}}$ ): $S(T_{\text{life}}) = \epsilon$ (typically $0.05$ )

Solving for the Weibull parameters yields:

$k = \frac{\ln(-\ln \epsilon) - \ln(\ln 2)}{\ln(T_{\text{life}}) - \ln(t_{0.5})}$

$\lambda = \frac{t_{0.5}}{(\ln 2)^{1/k}}$

This ensures that 95% of the portfolio exits by the end of the fund's natural life, regardless of the shape parameter.

3. Valuation and Cash Flows

3.1. MOIC-First Valuation

Traditionally, models calibrate an internal growth rate ( $g$ ) to match a target IRR. We adopt a MOIC-First approach, where the primary driver of value is the Multiple on Invested Capital (MOIC) at exit.

$\text{Proceeds}_{v,n} = \text{Deployed}_{v} \cdot P(\text{Exit}_n | v) \cdot \text{MOIC}_v$

The portfolio-level MOIC is solved numerically (using Brent's Method) to satisfy the target Total Value to Paid-In (TVPI) ratio:

$\text{Target TVPI} = \frac{\sum \text{Distributions} + \text{NAV}_{\text{final}}}{\sum \text{Calls}}$

This separates exit timing (driven by $S(t)$ ) from return magnitude (driven by MOIC), providing a stable and orthogonal calibration surface.

3.2. Net Asset Value (NAV) Evolution

The NAV of vintage $v$ at time $t$ evolves recursively:

$\text{NAV}_{v,t} = \text{NAV}_{v,t-1} \cdot (1 + g)^{\Delta t} + \text{Call}_{v,t} - \text{Dist}_{v,t} - \text{Fees}_{v,t}$

Where $g$ is the unrealized appreciation rate. In the MOIC-First framework, $g$ is typically set to 0, as all value creation is captured at the exit event via the MOIC multiplier.

4. Stochastic Extensions

To model risk, the engine extends the deterministic core with stochastic processes.

4.1. Pacing Modulation

Market conditions affect the velocity of capital calls and distributions. We introduce a market state variable $M_t$ (e.g., equity market index) and modulate flows:

( \beta_{\text{call}} \cdot r_M(t) )

( \beta_{\text{dist}} \cdot r_M(t) )

Where $\beta < 0$ for distributions implies that GPs delay exits during market downturns.

4.2. The Lines Layer (Granularity)

For small portfolios ( $N < 50$ ), smooth probabilities are unrealistic. The Lines Layer simulates discrete exit events. The number of exits in period $t$ follows a Beta-Binomial distribution to capture clustering (overdispersion):

( N_{\text{active}}, p_t, \rho )

Where $p_t$ is the deterministic hazard rate and $\rho$ is the correlation coefficient.

5. Implementation & Performance

The engine is implemented in TypeScript, utilizing a vectorized architecture for the Yearly Engine.

5.1. Hybrid State Management

The system supports "mid-flight" projections by blending actual historical data with future forecasts.

Phase A (Actuals): Cohorts are built from realized transaction logs.
Phase B (Forecast): Future cohorts are constructed from the remaining commitment and the deployment curve.
Transition: A "Carry-Forward" logic ensures unused capacity from the historical period is available for the forecast period, preventing discontinuities.

5.2. Performance Benchmarks

Component	Time Complexity	Benchmark (10y Fund)
Monthly Iterative	$O(M^2)$	~8,000 ms
Yearly Analytic	$O(Y^2)$	~400 ms

6. Conclusion

The Nagare Financial Engine represents a state-of-the-art approach to private capital modeling. By combining analytic rigor with flexible stochastic overlays, it provides GPs and LPs with a tool that is both fast enough for real-time interactivity and rigorous enough for institutional risk management.

References

Cox, D.R. (1972). "Regression Models and Life-Tables." Journal of the Royal Statistical Society, Series B (Methodological).
Phalippou, L. & Gottschalg, O. (2009). "The Performance of Private Equity Funds." Review of Financial Studies.
Harris, R.S., Jenkinson, T., & Kaplan, S.N. (2014). "Private Equity Performance: What Do We Know?" Journal of Finance.
Takahashi, A. & Alexander, S. (2002). "Illiquid Alternative Asset Fund Modeling." Journal of Portfolio Management.

The Nagare Financial Engine: A Stochastic Cohort-Based Projection Model