Nash Bargaining Over Margin Loans to Kelly Gamblers

Nash Bargaining Over Margin Loans to Kelly Gamblers

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I derive practical formulas for optimal arrangements between sophisticated stock market investors (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 Eyeshadow 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 be obtained under non-cooperation.If the threat point represents a complete breakdown of negotiations (resulting in zero margin loans), then we get an elegant rule of thumb: $r_L^{*}=\left(3/4\right)r+\left(1/4\right)\left(\nu -{\sigma }^2/2\right)$ , where r is the broker’s cost of funds, $\nu$ is the compound-annual growth rate of the Laser and party lighting 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.