# The Efficient Frontier

The concept of efficient frontier arises naturally when asking the portfolio selection question (i.e. What combination of stocks and bonds should I buy?)

The Efficient Frontier is the set of all portfolios which give a minimum risk for a given expected return (ergo, you should always seek to invest in the Efficient Frontier).

The derivation of the Efficient Frontier equation was first published in…. by…

I haven’t found many resources explaining the subject in-depth, beyond the original paper; so here is a derivation of the Efficient Frontier equation, modernized to soothe your vector-notation cravings.

### Prerequisite: Lagrange multipliers

We often find the problem of constrained optimization, which can be stated as:

$minimize f(X)$

Subject to

$\forall_i \: c_i (X) = 0$

Lagrange multipliers reduce this problem to an unconstrained optimization problem. The previous problem is reduced to:
$minimize f+\sigma \lambda_i c_i$
(* A generalization of Lagrange multipliers is given by KKT conditions, in which constraints can be a mix of equalities and inequalities. We will revisit these when we analyze de case of portfolio selection when you can’t hold short positions.)

### Derivation

#### Statement of the problem

We have the option to invest in several assets. Our portfolio is vector $P$, the ratios of our total capital that we invest in each asset. Since they are ratios and we are supposed to invest all our money, they add up to 1:

$1 = \textbf{1}.P$

The return of an asset on a given time interval is a random variable. As there are usually many assets in which we could invest, we are dealing with a vector of random variables $X$. The return of our portfolio will also be a random variable, given by:

$x_p = X . P$

The expected return of an asset, on the other hand, is just a number. Therefore, for a vector of expected returns $\mu = E(X)$, the expected return of our portfolio will be:

$\mu_p = \mu . P$

The risk of a single asset is statistically described by its variance $\sigma^2$. But since we are dealing with a vector of assets, the quantity of interest is its covariance matrix $C = cov(X)$. By the definition of covariance, C will be a symmetric matrix. The risk of our portfolio will be:

$\sigma_p^2 = P^T . C . P$

We want to minimize this risk (or minimize half this risk, which is the same), for a given expected return $\mu_p$. We are now ready to formalize our problem:

$minimize \frac{1}{2} P^T . C . P$

Subject to:

$1 = \textbf{1}.P$
$\mu_p = \mu . P$

#### Solution

By Lagrange multipliers, an equivalent formulation of the above problem is

$minimize \frac{1}{2} P^T . C . P + \lambda_1 (1 - \textbf{1}.P) + \lambda_2 (\mu_p - \mu . P)$

Taking the derivative of this expression with respect to $\lambda_1 , \lambda_2, P$ will yield an equation system that determines the solution of the problem. We get:

$C . P = \lambda_1 \textbf{1} + \lambda_2 . \mu$
$1 = \textbf{1}.P$
$\mu_p = \mu . P$

To simplify define this, let’s define:

$\lambda = (\lambda_1, \lambda_2)$
$U = (\textbf{1}, \mu)$
$U_p = (1, \mu_p)$

With this definitions, the above equations turn into :

$C. P = U. \lambda$
$U_p = U^T . P$

Solving these equations yields:

$\lambda = M^{-1} . U_p$
$P = C^{-1}.U. M^{-1} . U_p$
where $M = U^T . C^{-1} . U$

Note that we finally got our risk minimizing portfolios $P$, as a function of portfolio return $\mu_p$. By the definition of $\sigma_p^2$ in the previous section, the risk of such portfolios is:

$\sigma_p^2 = U_p^T . M^{-1} . U_p$

Note that this gives us the minimum possible risk for a given return $\mu_p$, and it turns out to be a quadratic function.

#### With Risk-Free Asset

Besides investing in risky assets (stocks), we can also invest in a risk-free asset (bonds). This particular case is usually given a separate treatment. Instead, we will just say that for this particular case, matrix $C$ becomes singular, and we shall therefore replace inverses by pseudo-inverses when apropiate in the equations above. That’s all.

### No short-position

#### Statement

It may happen that we are not allowed to hold short-positions in some assets. That means all components of $P$ should be positive. The resulting problem is:

$minimize \frac{1}{2} P^T . C . P$

Subject to:

$1 = \textbf{1}.P$
$\mu_p = \mu . P$
$P>0$

### Implementation

for I in range(10)