An Empirical Implementation of the Shadow Riskless Rate

We address the problem of asset pricing in a market where there is no risky asset. Previous work developed a theoretical model for a shadow riskless rate (SRR) for such a market in terms of the drift component of the state-price deflator for that asset universe. Assuming asset prices are modeled by correlated geometric Brownian motion, in this work we develop a computational approach to estimate the SRR from empirical datasets. The approach employs: principal component analysis to model the effects of the individual Brownian motions; singular value decomposition to capture the abrupt changes in condition number of the linear system whose solution provides the SRR values; and a regularization to control the rate of change of the condition number. Among other uses (e.g., for option pricing, developing a term structure of interest rate), the SRR can be employed as an investment discriminator between asset classes. We apply the computational procedure to markets consisting of groups of stocks, varying asset type and number. The theoretical and computational analysis provides not only the drift, but also the total volatility of the state-price deflator. We investigate the time trajectory of these two descriptive components of the state-price deflator for the empirical datasets.