Sigmoidal dynamics of macro-financial leverage

Abstract

Logistic sigmoids due to their flexibility seem to be natural candidates for modelling macrofinancial leverage behavior. The sigmoidal leverage transition towards its stationary value, which was driven by the yield spreads, could have replicated the dynamics of macrofinancial assets, debt and capital. The leverage transition, in its turn, has been a major factor in better balancing macrofinancial liabilities and assets. The sigmoidal leverage trajectories including their inflections and different phases were identified by a nonlinear transition function providing information necessary for steering the process towards its stable state. Solving the stationary Kolmogorov-Fokker-Plank logistic equation revealed that random leverage realizations might follow the gamma distribution. Parameters of its stationary probability density function, as well as the expected and the modal leverage, were dependent on the process variance and the yield spreads. Thus, the stochastic leverage behaviour reproduced a sequence of stylized phases similar to the observed in the US Treasuries market meltdown in 2020. In particular, larger yield spreads and smaller modal leverage signalled a "defensive" market response to sudden volatility increases. In addition, it was shown that the logistic leverage modelling could be helpful in the analysis of debt and money dynamics including some consequences of "minting a one trillion dollars coin".