April 2009 Q4 iii

Calculate the efficient frontier:

I calculated the point where the variance was minimum on the efficient frontier for the risky assets.

So V is minimised when x=(Vb-Cab)/(Va+Vb-2Cab)= 16/97

I then calculated the straight line passing through (0,r) and the point in E-sd space given by the above value of x.

...........

In the solution they find x to minimise (E-r)/sd. This gives a different value of x.

Could you explain why my method was wrong?

 

Your method is actually wrong because the efficient frontier won't include the point at which the variance is minimised.

This is because in E-sigma space, the efficient frontier with a risk-free asset is a straight line from the risk-free asset tangential to the efficient frontier with risky assets only. This straight line efficient frontier slopes upwards and so must have a positive but finite gradient at the point of tangency. Consequently, the efficient frontier with risky assets only must also have this same positive but finite gradient at the point of tangency. However, the risky assets only curve must, by definition, be vertical , and hence have an infinite gradient, at the point of minimum variance.

Actually, in the solution, "E-r/sigma" is the gradient of a straight line from the risk-free asset to any point on the risky-assets-only curve. The efficient frontier (with a risk-free asset) is just the straight line from the risk-free asset to any point on the risky-assets-only curve with the highest gradient. So, to find the gradient of the efficient frontier, you need to maximise E-r/sigma, which is what the Examiners' Report does.

 

Hi...
I just have one doubt. The efficienct frontier of risky assets only will be upward sloping. How did you conclude that it is vertical by definition? Which definition guarantees this?