Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 39 / 6 / c / Solution

The centered term in ${\rho }^{\ast }$ has zero mean, giving $\mathbb{E}{\rho }^{\ast }=1/R$ and hence $\mathbb{E}[{\rho }^{\ast }{B}_1]=1=B_0$. For the risky assets, symmetry of the covariance matrix givesTherefore ${\rho }^{\ast }$ reproduces every asset's initial price.

For another square-integrable $\rho$ satisfying the price moment constraints, set $\delta =\rho -{\rho }^{\ast }$. The bank equation and $R\ne 0$ give $\mathbb{E}\delta =0$, while the risky equations give $\mathbb{E}[\delta S_1]=0$. Since ${\rho }^{\ast }$ is an affine combination of $1$ and the coordinates of $S_1$,The Pythagorean theorem in an inner-product space now givesEquality holds exactly when $\rho ={\rho }^{\ast }$ almost surely. If a competing weight has infinite second moment, the inequality is immediate in the extended sense. The minimum itself isThus ${\rho }^{\ast }$ is the orthogonal projection of any feasible square-integrable pricing weight onto the linear span of the traded payoffs.