= Constant-proportion portfolio
{title2=$\pi_t=\gamma V_t$}
A <constant-proportion portfolio> holds a fixed fraction $\gamma$ of its <portfolio wealth> in the <stock>. In the <Black-Scholes model> with bank rate $\rho$, initial wealth $V_0$, and no <consumption>, its value is
$$
V_t=V_0(S_t/S_0)^\gamma\exp\left((1-\gamma)(\rho+\tfrac12\gamma\sigma^2)t\right).
$$
The <stock> holding is $\gamma V_t/S_t$ and the remaining value $(1-\gamma)V_t$ is invested in the <bank account>. The <Itô formula> verifies $dV_t/V_t=[\rho+\gamma(\mu-\rho)]dt+\gamma\sigma dW_t$ when the physical stock drift is $\mu$.
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