3  Markowitz Portfolio Example

Harry Markowitz, one of the pioneers of computational finance and founder of Modern Portfolio Theory, introduced his financial portfolio model in 1952. He defines portfolio items (e.g. stocks) with their risk (usually standard deviation or variance of returns) and reward (mean return). (Trivia: Markowitz is a Nobel Laureate in Economics, 1990)

3.1 Problem Description

There are multiple versions of this problem, but we will model a simple one. Suppose there is an array of investment items with different risk and reward values. We would like form a portfolio to minimize the total risk, given a desired return level. Naturally, risk should be higher with the return.

3.2 Model Building Steps

  1. Define the decision variable \(x_{i}\) as the fraction of our portfolio assigned to investment item (i.e. stock) \(i\).

  2. Define risk parameter as \(\sigma_{i}\) and reward parameter as \(\mu_{i}\) for item \(i\).

  3. Define desired return level parameter as \(q\).

  4. Add constraint of the total portolio should add up to \(1\).

  5. Add constraint of minimum return requirement from the portfolio.

  6. Add objective function of minimum risk to the portfolio.

3.3 Mathematical Model

3.3.1 Decision Variables

  • \(x_{i}\): Fraction of the budget allocated to investment item \(i\).

3.3.2 Parameters

  • \(\sigma_{j}\): Risk parameter of item \(i\).
  • \(\mu_{i}\): Reward (return) parameter of item \(i\).
  • \(q\): Required minimum reward level from the portfolioe.

3.3.3 Model

\[\begin{gather} \min z = \sum_{i} \sigma_{i} x_{i} \label{eq:obj.fun} \\ s.t. \nonumber \\ \sum_i \mu_{i}x_{i} \ge q \label{eq:reward.constraint} \\ \sum_i x_{i} = 1 \label{eq:portfolio.constraint} \\ x_{i} \ge 0, \ \forall_i \label{eq:non-negativity}\\ \end{gather}\]

3.3.4 Constraints

  • (\(\ref{eq:obj.fun}\)) is the objective function to minimize total spending.
  • (\(\ref{eq:reward.constraint}\)) is the minimum reward requirement constraint.
  • (\(\ref{eq:portfolio.constraint}\)) is the constraint to make sure that sum of all portfolio fractions is 1.
  • (\(\ref{eq:non-negativity}\)) Non-negativity constraint. It is not possible to sell negative amount of each toys (i.e. no backorders, no returns etc. in this case).