I have 4 risky securities (have returns and var-cov matrix for monthly data), and I can lend at 1% per annum, but borrow at 5% per annum. If i wish to obtain the s.d. of 5%, what is the optimal portfolio? How would I go about solving this? I tried finding tangency portfolios for both rates, but not sure how to proceed from there.
In this case the Capital Market Line is composed of three sections: (1) a straight line segment from the 1% point on the y axis that is tangent to the frontier at a point T. A (curved) section of the frontier from T to another point S on the frontier. S is determined as the point of tangency of a line from 5% on the y axis to the frontier. And (3) a ray (half line) that starts at S and is part (the rightmost part) of the tangent mentioned earlier.
So just draw the frontier, the two tangents, the above mentioned three part Capital Market Line (straight, curved, then straight again) and find the point on the CML having a s.d. of 5%.
This figure may help (though unfortunately the points T and S are not clearly labeled as such):