# Nash Bargaining Over Margin Loans to Kelly Gamblers

## TL;DR

This paper develops a Nash bargaining model for margin loan negotiations between sophisticated investors and brokers, deriving formulas for optimal interest rates and bet sizing that benefit both parties in high Sharpe asset markets.

## Contribution

It introduces a practical Nash bargaining framework for margin loan negotiations, providing explicit formulas for interest rates and bet sizes that improve outcomes for both investors and brokers.

## Key findings

- Optimal interest rate formula: r_L^*=(3/4)r+(1/4)(
u-rac{\sigma^2}{2})
- Investors increase leverage for lower interest rates and higher growth
- Brokers profit from negotiated agreements exceeding non-cooperative scenarios

## Abstract

I derive practical formulas for optimal arrangements between sophisticated stock market investors (namely, continuous-time Kelly gamblers or, more generally, CRRA investors) and the brokers who lend them cash for leveraged bets on a high Sharpe asset (i.e. the market portfolio). Rather than, say, the broker posting a monopoly price for margin loans, the gambler agrees to use a greater quantity of margin debt than he otherwise would in exchange for an interest rate that is lower than the broker would otherwise post. The gambler thereby attains a higher asymptotic capital growth rate and the broker enjoys a greater rate of intermediation profit than would obtain under non-cooperation. If the threat point represents a vicious breakdown of negotiations (resulting in zero margin loans), then we get an elegant rule of thumb: $r_L^*=(3/4)r+(1/4)(\nu-\sigma^2/2)$, where $r$ is the broker's cost of funds, $\nu$ is the compound-annual growth rate of the market index, and $\sigma$ is the annual volatility. We show that, regardless of the particular threat point, the gambler will negotiate to size his bets as if he himself could borrow at the broker's call rate.