# The Laws of Motion of the Broker Call Rate in the United States

## TL;DR

This paper analyzes the dynamics of the U.S. broker call rate, deriving stochastic models and theoretical constraints on market risk premia, linking empirical observations with margin loan pricing theories.

## Contribution

It introduces a stochastic differential equation framework for the broker call rate and leverage ratios, integrating empirical data with margin loan pricing theories and arbitrage constraints.

## Key findings

- The broker call rate exhibits mean-reverting behavior.
- Derived stochastic models describe the evolution of margin interest rates.
- Market constraints imply call loans exceed 70% of leveraged portfolio values.

## Abstract

In this paper, which is the third installment of the author's trilogy on margin loan pricing, we analyze $1,367$ monthly observations of the U.S. broker call money rate, which is the interest rate at which stock brokers can borrow to fund their margin loans to retail clients. We describe the basic features and mean-reverting behavior of this series and juxtapose the empirically-derived laws of motion with the author's prior theories of margin loan pricing (Garivaltis 2019a-b). This allows us to derive stochastic differential equations that govern the evolution of the margin loan interest rate and the leverage ratios of sophisticated brokerage clients (namely, continuous time Kelly gamblers). Finally, we apply Merton's (1974) arbitrage theory of corporate liability pricing to study theoretical constraints on the risk premia that could be generated in the market for call money. Apparently, if there is no arbitrage in the U.S. financial markets, the implication is that the total volume of call loans must constitute north of $70\%$ of the value of all leveraged portfolios.

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Source: https://tomesphere.com/paper/1906.00946