Editing Modern portfolio theory (section)

=== Geometric intuition ===
The efficient frontier can be pictured as a problem in [[Conic section|quadratic curves]].<ref name="Merton" /> On the market, we have the assets <math>R_1, R_2, \dots, R_n</math>. We have some funds, and a portfolio is a way to divide our funds into the assets. Each portfolio can be represented as a vector <math>w_1, w_2, \dots, w_n</math>, such that <math>\sum_i w_i = 1</math>, and we hold the assets according to <math>w^T R = \sum_i w_i R_i </math>.

==== 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).

==== Tangency portfolio ====
[[File:Mean-variance analysis.gif|thumb|Illustration of the effect of changing the risk-free asset return rate. As the risk-free return rate approaches the return rate of the global minimum-variance portfolio, the tangency portfolio escapes to infinity. Animated at source [https://upload.wikimedia.org/wikipedia/commons/f/f4/Mean-variance_analysis.gif].]]The tangency portfolio exists if and only if <math>\mu_{RF} < \mu_{MVP}</math>.

In particular, if the risk-free return is greater or equal to <math>\mu_{MVP}</math>, then the tangent portfolio ''does not exist''. The capital market line (CML) becomes parallel to the upper asymptote line of the hyperbola. Points ''on'' the CML become impossible to achieve, though they can be ''approached'' from below.

It is usually assumed that the risk-free return is less than the return of the global MVP, in order that the tangency portfolio exists. However, even in this case, as <math>\mu_{RF} </math> approaches <math>\mu_{MVP}</math> from below, the tangency portfolio diverges to a portfolio with infinite return and variance. Since there are only finitely many assets in the market, such a portfolio must be shorting some assets heavily while longing some other assets heavily. In practice, such a tangency portfolio would be impossible to achieve, because one cannot short an asset too much due to '''short sale constraints''', and also because of '''price impact''', that is, longing a large amount of an asset would push up its price, breaking the assumption that the asset prices do not depend on the portfolio.

==== Non-invertible covariance matrix ====
If the covariance matrix is not invertible, then there exists some nonzero vector <math>v</math>, such that <math>v^T R</math> is a random variable with zero variance—that is, it is not random at all.

Suppose <math>\sum_i v_i = 0</math> and <math>v^T R = 0</math>, then that means one of the assets can be exactly replicated using the other assets at the same price and the same return. Therefore, there is never a reason to buy that asset, and we can remove it from the market.

Suppose <math>\sum_i v_i = 0</math> and <math>v^T R \neq 0 </math>, then that means there is free money, breaking the ''no arbitrage'' assumption.

Suppose <math>\sum_i v_i \neq 0 </math>, then we can scale the vector to <math>\sum_i v_i = 1</math>. This means that we have constructed a risk-free asset with return <math>v^T R </math>. We can remove each such asset from the market, constructing one risk-free asset for each such asset removed. By the no arbitrage assumption, all their return rates are equal. For the assets that still remain in the market, their covariance matrix is invertible.