= Solution
The risk-free weight is $1-\mathbf1^tw$, so a zero risk-free weight requires $\mathbf1^tw=1$. Define $H=\mathbf1^t\Sigma^{-1}d$. If $H\ne0$, this condition gives $\lambda=1/H$ and
$$
\boxed{w_T=\frac{\Sigma^{-1}d}{H},\qquad
\mathbb E r_T=R+\frac A H,\qquad \operatorname{Var}(r_T)=\frac A{H^2}.}
$$
These are the requested tangent-portfolio mean and <variance> under the usual condition $H>0$. Then $\lambda>0$, the normalized <portfolio> lies on the upper efficient ray, and its <Sharpe ratio> is $\sqrt A$.
There is a real qualification if the data are allowed to be arbitrary. If $H=0$, the efficient risky direction has zero total risky weight, and no finite multiple makes its risky weights sum to one. If $H<0$, the formal normalized <portfolio> above has negative excess mean and lies on the lower, dominated branch; it is not an efficient tangent <portfolio> as described in the question. The <sign of the normalized tangency portfolio> must therefore be checked, rather than inferred from <positive definiteness> of $\Sigma$ alone.
Back to article page
