Estimating the Alpha and Beta of Private Capital Using State Space Modeling and Bayesian Inference

The authors estimate the alpha and beta of private equity investments using their desmoothed quarterly returns. Although computing the alpha and beta using a lagged linear regression between private and public market returns is straightforward, reconstructing desmoothed private market returns is trickier. A time series of desmoothed private capital returns has application in asset allocation, risk modeling, and return attribution. The authors use a state space modeling framework in which the desmoothing and the parameter estimation are done simultaneously using Bayesian inference. This is likely the first attempt to solve a full Bayesian model that can simultaneously estimate a desmoothed time series of returns as well as perform a lagged linear regression to estimate capital asset pricing model (CAPM) style parameters for private capital. The authors perform extensive model checking, look out for the model's convergence, and resolve any issues leading to overfitting. The estimates are precise with very low Monte Carlo simulation errors and are also in alignment with those in the literature, which validates this model and desmoothing technique.