= Solution
The centered term in $\rho^*$ has zero mean, giving $\mathbb E\rho^*=1/R$ and hence $\mathbb E[\rho^*B_1]=1=B_0$. For the risky assets, symmetry of the <covariance matrix> gives
$$
\mathbb E[\rho^*S_1]
=\frac\mu R+\mathbb E[(S_1-\mu)(S_1-\mu)^T]V^{-1}h
=\frac\mu R+h=S_0.
$$
Therefore \b[$\rho^*$ reproduces every asset's initial price].
For another square-integrable $\rho$ satisfying the price moment constraints, set $\delta=\rho-\rho^*$. The bank equation and $R\ne0$ give $\mathbb E\delta=0$, while the risky equations give $\mathbb E[\delta S_1]=0$. Since $\rho^*$ is an affine combination of $1$ and the coordinates of $S_1$,
$$
\mathbb E[\delta\rho^*]
=\frac1R\mathbb E\delta+h^TV^{-1}\mathbb E[\delta(S_1-\mu)]=0.
$$
The <Pythagorean theorem in an inner-product space> now gives
$$
\boxed{\mathbb E[\rho^2]=\mathbb E[(\rho^*)^2]+\mathbb E[(\rho-\rho^*)^2]
\geq\mathbb E[(\rho^*)^2].}
$$
Equality holds exactly when $\rho=\rho^*$ almost surely. If a competing weight has infinite second moment, the inequality is immediate in the extended sense. The minimum itself is
$$
\mathbb E[(\rho^*)^2]=\frac1{R^2}+h^TV^{-1}h.
$$
Thus $\rho^*$ is the orthogonal projection of any feasible square-integrable pricing weight onto the linear span of the traded payoffs.
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