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I derive practical formulas for optimal arrangements between sophisticatedstock market investors (namely, continuous-time Kelly gamblers or, moregenerally, CRRA investors) and the brokers who lend them cash for leveragedbets on a high Sharpe asset (i.e. the market portfolio). Rather than, say, thebroker posting a monopoly price for margin loans, the gambler agrees to use agreater quantity of margin debt than he otherwise would in exchange for aninterest rate that is lower than the broker would otherwise post. The gamblerthereby attains a higher asymptotic capital growth rate and the broker enjoys agreater rate of intermediation profit than would obtain under non-cooperation.If the threat point represents a vicious breakdown of negotiations (resultingin 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 $\sigma$ isthe 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 atthe broker's call rate.

Nash Bargaining Over Margin Loans to Kelly Gamblers