Local efficient frontier

Since

$$\begin{eqnarray*} \mathsf{E}\mathrm{d}\pi - \pi\,r\mathrm{d}t &=& p^\prime\, U\... ...r}\! (\mathrm{d}\pi) &=& p^\prime\, UVU^\prime\, p\,\mathrm{d}t \end{eqnarray*}$$

the locally efficient portfolio is

$$p^\mathrm{eff} = \arg_{p\left\vert\mu = p^\prime\,U\,\lam... ...mbda}{\lambda^\prime U^\prime (UVU^\prime)^- U\lambda}\quad ,$$ (28)

and the efficient frontier is

$$\frac{\mathsf{var}\! (\mathrm{d}\pi)}{\mathrm{d}t} = \frac{\mu}{\lambda^\prime U^\prime (UVU^\prime)^- U\lambda}\quad ,$$ (29)

i.e.

$$\sigma^\mathrm{eff}(\mu) = \frac{\vert\mu\vert}{\sqrt{\lambda^\prime U^\prime (UVU^\prime)^- U\lambda}}\quad .$$ (30)