Optimal Capital Deployment Under Stochastic Deal Arrivals: A Continuous-Time ADP Approach

Suppose you are a fund manager with \$100 million to deploy and two years to invest it. A deal comes across your desk that looks appealing but costs \$50 million -- half of your available capital. Should you take it, or wait for something better? The decision hinges on the trade-off between current opportunities and uncertain future arrivals. This work formulates the problem of capital deployment under stochastic deal arrivals as a continuous-time Markov decision process (CTMDP) and solves it numerically via an approximate dynamic programming (ADP) approach. We model deal economics using correlated lognormal distributions for multiples on invested capital (MOIC) and deal sizes, and model arrivals as a nonhomogeneous Poisson process (NHPP). Our approach uses quasi-Monte Carlo (QMC) sampling to efficiently approximate the continuous-time Bellman equation for the value function over a discretized capital grid. We present an interpretable acceptance policy, illustrating how selectivity evolves over time and as capital is consumed. We show in simulation that this policy outperforms a baseline that accepts any affordable deal exceeding a fixed hurdle rate.