  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.
