Nash Bargaining Over Margin Loans to Kelly Gamblers

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 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: <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi>r</mi> <mi>L</mi> <mo>*</mo> </msubsup> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mn>3</mn> <mo>/</mo> <mn>4</mn> </mfenced> <mi>r</mi> <mo>+</mo> <mfenced separators="" open="(" close=")"> <mn>1</mn> <mo>/</mo> <mn>4</mn> </mfenced> <mfenced separators="" open="(" close=")"> <mi>&#957;</mi> <mo>&#8722;</mo> <msup> <mi>&#963;</mi> <mn>2</mn> </msup> <mo>/</mo> <mn>2</mn> </mfenced> </mrow> </semantics> </math> </inline-formula>, where <i>r</i> is the broker&#8217;s cost of funds, <inline-formula> <math display="inline"> <semantics> <mi>&#957;</mi> </semantics> </math> </inline-formula> is the compound-annual growth rate of the market index, and <inline-formula> <math display="inline"> <semantics> <mi>&#963;</mi> </semantics> </math> </inline-formula> 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&#8217;s call rate.