= Benchmark-relative power-utility portfolio
{title2=$\pi^*=R^{-1}V^{-1}(\mu-r\mathbf1)+(1-R^{-1})q$}
In a Brownian <complete market> with constant coefficients and nonsingular $\sigma$, maximize expected <CRRA utility> of terminal <portfolio wealth> divided by a <geometric stock index>. Writing $p=1/R$ and $q=n^{-1}\mathbf1$, the pointwise marginal condition gives $w_T^*=\lambda^{-p}\zeta_T^{-p}J_T^{1-p}$. Its <state-price budget constraint> determines $\lambda^{-p}=w_0/\mathbb E[(\zeta_TJ_T)^{1-p}]$. Conditional lognormal moments and replication then give the displayed constant dollar fractions. The first term is ordinary optimal risk exposure; the second hedges the random benchmark. The bank holds the remaining fraction, which can be negative when borrowing is allowed.
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