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What Is Quadratic Programming?

Quadratic programming (QP) is a specialized area within mathematical programming that involves optimizing (minimizing or maximizing) a quadratic objective function subject to linear constraints. As a type of nonlinear programming, quadratic programming plays a significant role in various fields, notably in quantitative finance and portfolio theory. It is particularly useful for problems where the relationship between variables involves quadratic terms, allowing for the modeling of interactions among them. Financial institutions and investors leverage quadratic programming to make informed decisions about asset allocation and risk management, often seeking to balance expected return with a measurable level of risk.

History and Origin

The foundational application of quadratic programming in finance can be largely attributed to Harry Markowitz's seminal work on modern portfolio theory (MPT). In his 1952 paper, "Portfolio Selection," Markowitz introduced a framework for constructing investment portfolios that maximize expected return for a given level of risk or minimize risk for a given expected return. This groundbreaking work mathematically formalized the concepts of expected return and portfolio risk, using the variance of portfolio returns as a measure of risk. Markowitz's innovative use of quadratic programming to solve this portfolio optimization problem was a significant advancement, ushering in a scientific approach to investment decision-making.⁷ Prior to his contributions, portfolio construction often relied on rules of thumb rather than rigorous mathematical optimization.⁶

Key Takeaways

• Quadratic programming (QP) optimizes a quadratic objective function subject to linear constraints.
• It is a core component of modern portfolio theory, enabling the construction of portfolios that balance risk and return.
• QP problems are typically solved using numerical algorithms.
• The output of quadratic programming in portfolio optimization often defines the efficient frontier.
• QP is widely applied in finance, economics, and various operational research problems.

Formula and Calculation

A general quadratic programming problem seeks to find a vector (x) that minimizes the following quadratic objective function:

$\min_{x} \left( \frac{1}{2} x^T Q x + c^T x \right)$

subject to linear constraints:

$A x \le b$
$E x = d$

Where:

• (x): A vector of decision variables (e.g., asset weights in a portfolio).
• (Q): A symmetric matrix representing the quadratic coefficients (e.g., the covariance matrix of asset returns).
• (c): A vector of linear coefficients (e.g., related to expected returns).
• (A), (E): Matrices defining the linear inequality and equality constraints, respectively.
• (b), (d): Vectors defining the right-hand side of the linear inequality and equality constraints.

In portfolio optimization, the (Q) matrix often represents the covariance matrix of asset returns, which captures how different assets' returns move together. This formulation allows for the minimization of portfolio variance (risk) while meeting specified return targets and other investment rules.

Interpreting Quadratic Programming

In the context of finance, interpreting the results of quadratic programming typically involves understanding the optimal allocation of assets within a portfolio. The solution (x) provides the weights or proportions to be invested in each asset to achieve a specific objective, such as minimizing portfolio risk for a desired level of expected return.

The output of a quadratic programming model, when applied to portfolio selection, often yields a point on the efficient frontier. This frontier represents a set of optimal portfolios, each offering the highest possible expected return for a given level of risk, or the lowest possible risk for a given expected return. Investors can then choose a portfolio along this frontier that aligns with their individual risk tolerance and investment objectives. Portfolios falling below this frontier are considered suboptimal, as they either provide less return for the same risk or higher risk for the same return.

Hypothetical Example

Consider an investor who wants to build a diversified portfolio using three assets: Stock A, Stock B, and a Government Bond. The investor aims to minimize the portfolio's total risk while achieving a minimum expected return of 8%.

Inputs:

• Expected Returns: Stock A = 12%, Stock B = 10%, Government Bond = 4%
• Covariance Matrix (Q):
  [
  Q = \begin{pmatrix}
  \sigma_A^2 & \sigma_{AB} & \sigma_{AGB} \
  \sigma_{BA} & \sigma_B^2 & \sigma_{BGB} \
  \sigma_{GBA} & \sigma_{GBB} & \sigma_{GB}^2
  \end{pmatrix}
  ]
  (where (\sigma^2) represents variance and (\sigma_{ij}) represents covariance between assets i and j)
• Assume for this example the calculated covariance matrix is:
  [
  Q = \begin{pmatrix}
  0.04 & 0.01 & -0.005 \
  0.01 & 0.0225 & 0.002 \
  -0.005 & 0.002 & 0.0009
  \end{pmatrix}
  ]
• Minimum Expected Portfolio Return: 8%
• Sum of weights must equal 1 (100% of the investment).
• No short selling (all weights must be non-negative).

Quadratic Programming Formulation:

The objective function to minimize portfolio variance would be: