I took some classes in portfolio theory, and learnt the Markowitz Mean-Variance Analysis. If only two risky assets, the efficient frontier would be a hyperbola passing through the two points; now if added another asset has some correlation with them two, the efficient frontier would be pushed to the left of these three risky assets, generally not passing through them; so, if we keep adding more assets, it seems intuitively that the efficient frontier would be pushed further to the left of the assets points; wouldn't the efficient frontier be like tangent to the y-axis(y axis is for return), causing the standard deviation to be almost zero? I also read Pennacchi's book: Theory of Asset Pricing. it derived the equation for the minimum variance set using linear algebra. And found the position for the global minimum standard deviation point. And it's positive. I understand the equation derivation, but didn't get through the problem as I just thought of. Any help? Much appreciated.
In mean-variance analysis, you combine different assets to minimize variance and maximize expected return. The hyperbola is not a function of the number of assets, but of their mean and variance. If the efficient frontier where a tangent to the y-axis (which can't be) or nearly a tangent, that would mean you would have almost zero portfolio-variance, which won't be the case for risky assets, and clearly not so by adding further assets with variance >0.