Editing Modern portfolio theory (section)

==== Markowitz bullet ====
[[File:Mean-variance analysis, quadratic optimization 3D.gif|thumb|The ellipsoid is the contour of constant variance. The <math>x+y+z=1</math> plane is the space of possible portfolios. The other plane is the contour of constant expected return.

The ellipsoid intersects the plane to give an ellipse of portfolios of constant variance. On this ellipse, the point of maximal (or minimal) expected return is the point where it is ''tangent'' to the contour of constant expected return. All these portfolios fall on one line.]]
Since we wish to maximize expected return while minimizing the standard deviation of the return, we are to solve a quadratic optimization problem:<math display="block">\begin{cases}
E[w^T R] = \mu \\
\min \sigma^2 = Var[w^T R ]\\
\sum_i w_i = 1
\end{cases}</math>Portfolios are points in the Euclidean space <math>\R^n</math>. The third equation states that the portfolio should fall on a plane defined by <math>\sum_i w_i = 1</math>. The first equation states that the portfolio should fall on a plane defined by <math>w^T E[R] = \mu</math>. The second condition states that the portfolio should fall on the contour surface for <math>\sum_{ij} w_i \rho_{ij} w_j</math> that is as close to the origin as possible. Since the equation is quadratic, each such contour surface is an ellipsoid (assuming that the covariance matrix <math>\rho_{ij}</math> is invertible). Therefore, we can solve the quadratic optimization graphically by drawing ellipsoidal contours on the plane <math>\sum_i w_i = 1</math>, then intersect the contours with the plane <math>\{w: w^T E[R] = \mu \text{ and } \sum_i w_i =1\}</math>. As the ellipsoidal contours shrink, eventually one of them would become exactly tangent to the plane, before the contours become completely disjoint from the plane. The tangent point is the optimal portfolio at this level of expected return.

As we vary <math>\mu</math>, the tangent point varies as well, but always falling on a single line (this is the '''two mutual funds theorem''').

Let the line be parameterized as <math>\{w + w' t : t \in \R\}</math>. We find that along the line,<math display="block">\begin{cases}
\mu &= (w'^T E[R]) t +  w^T E[R]\\
\sigma^2 &= (w'^T \rho w') t^2 + 2 (w^T \rho w') t + (w^T \rho w)
\end{cases} </math>giving a hyperbola in the <math>(\sigma, \mu)</math> plane. The hyperbola has two branches, symmetric with respect to the <math>\mu</math> axis. However, only the branch with <math>\sigma > 0</math> is meaningful. By symmetry, the two asymptotes of the hyperbola intersect at a point <math>\mu_{MVP}</math> on the <math>\mu</math> axis. The point <math>\mu_{mid}</math> is the height of the leftmost point of the hyperbola, and can be interpreted as the expected return of the '''global minimum-variance portfolio''' (global MVP).