The Application of fmincon in Financial Modeling: Optimizing Portfolio Returns

# 1. An Introduction to fmincon and the Basics of Optimization Theory

fmincon is a function in MATLAB used for nonlinear constrained optimization. It is based on the Sequential Quadratic Programming (SQP) algorithm and is capable of solving optimization problems with nonlinear objective functions and constraints.

### Basics of Optimization Theory

Optimization theory aims to find the minimum or maximum value of an objective function subject to given constraints. fmincon employs the SQP algorithm, which approximates the optimal solution through an iterative process. During each iteration, the SQP algorithm constructs a quadratic approximation of the objective function and constraints and solves this quadratic problem to obtain a new candidate solution.

### fmincon Function Parameters

The fmincon function accepts multiple parameters, including:

- `fun`: Handle to the objective function.
- `x0`: Initial guess for the solution.
- `Aineq`: Coefficient matrix for linear inequality constraints.
- `bineq`: Right-hand side vector for linear inequality constraints.
- `Aeq`: Coefficient matrix for linear equality constraints.
- `beq`: Right-hand side vector for linear equality constraints.
- `lb`: Lower bounds for the variables.
- `ub`: Upper bounds for the variables.

# 2. Practical Applications of fmincon in Financial Modeling

### 2.1 Building a Portfolio Optimization Model

#### 2.1.1 Portfolio Return and Risk Measurement

In portfolio optimization, return and risk are two key indicators. Return measures the reward of a portfolio over a period, ***

***mon return measures include:

- **Annualized Return:** The average return a portfolio earns in a year.
- **Sharpe Ratio:** The ratio of a portfolio's excess return to its standard deviation, ***

***mon risk measures include:

- **Standard Deviation:** The standard deviation of a portfolio's returns, measuring the volatility of the portfolio.
- **Maximum Drawdown:** The percentage decline from peak to trough in a portfolio, measuring the maximum loss.

#### 2.1.2 Objective Function and Constraints

The objective function in a portfolio optimization model is typically to maximize the portfolio's return or Sharpe ratio. Constraints may include:

- **Risk Constraint:** The portfolio's standard deviation or maximum drawdown should not exceed a certain threshold.
- **Asset Allocation Constraint:** The weights of different asset classes in the portfolio must be within a certain range.
- **Liquidity Constraint:** The proportion of assets that can be quickly liquidated in the portfolio must meet a certain threshold.

### 2.2 Solving the Portfolio Optimization Model with fmincon

#### 2.2.1 Principles of the fmincon Algorithm

fmincon is a function in MATLAB for nonlinear constrained optimization. It uses an interior-point method, an iterative algorithm, to find the optimal solution that satisfies the constraints within the feasible domain.

The working principle of the interior-point method is as follows:

1. Initialize a feasible point.
2. Solve the linearized sub-problem to find the local optimum near the current point.
3. Update the current point to the local optimum.
4. Repeat steps 2 and 3 until the termination conditions are met.

#### 2.2.2 Parameter Settings and Solving Process

When using fmincon to solve the portfolio optimization model, the following parameters need to be set:

- **Objective Function:** The return or Sharpe ratio of the portfolio.
- **Constraints:** Risk constraints, asset allocation constraints, and liquidity constraints.
- **Initial Point:** Initial weights of the portfolio.
- **Algorithm Options:** Including termination conditions, maximum iterations, and step size.

The solving process is as follows:

% Define the objective function
objective = @(x) -sharpe(x, mu, Sigma);

% Define the constraint conditions
A = [1; -1];
b = [1; 0];
Aeq = [];
beq = [];

% Define the initial point
x0 = ones(n, 1) / n;

% Solve the optimization model
options = optimoptions('fmincon', 'Display', 'iter', 'Algorithm', 'interior-point');
[x, fval, exitflag, output] = fmincon(objective, x0, A, b, Aeq, beq, [], [], [], options);

### 2.3 Analysis of Optimization Results and Portfolio Construction

#### 2.3.1 Interpretation of Optimization Results

The optimal solution x obtained by fmincon represents the weights of different assets in the portfolio. These weights can be used to calculate the portfolio's return, risk, a