Portfolio Risk Minimization

Proposal

• Proposal

Source Code

1. Retrieve Data

1-retrieve-data.py

Generates the CSV file stock_data.csv with the daily stock data for the past year. It includes the opening, high, low, closing, and volume of the stock.

2. Compute Returns

2-compute-returns.py

Reads the CSV file stock_data.csv and computes the daily percentage returns of the stock. Adds a new column daily_return to the DataFrame. Generates the CSV file daily_returns.csv.

3. Fit MLE

3-fit-mle.py

Skew-t Distribution

Function neg_loglik_skewt implements the negative log-likelihood function for the skew-t distribution.

Function skewt_pdf implements the probability density function for the skew-t distribution.

Function fit_skewt fits the skew-t distribution to the daily returns of the stock. It returns:

• x: the range of source values to plot the PDF
• pdf_fitted: the PDF of the x values
• sample: a large sample from the fitted skew-t distribution
• xi: the location parameter
• omega: the scale parameter
• alpha: the shape parameter
• nu: the degrees of freedom

Student's t Distribution

Function neg_loglik_studentt implements the negative log-likelihood function for the Student's t distribution.

Function fit_studentt fits the Student's t distribution to the daily returns of the stock. It returns:

• x: the range of source values to plot the PDF
• pdf_fitted: the PDF of the x values
• sample: a large sample from the fitted Student's t distribution
• mu: the location parameter
• sigma: the scale parameter
• nu: the degrees of freedom

4. Multivariate Student-t Distribution

4-multivariate-student.py

Prints the parameters required to calculate the CVaR of a portfolio:

• Mean
• Sigma
• Nu
• Constant C

5. Calculate VaR and CVaR

5-calculate-var-cvar.py

Given a example of portfolio weights, calculate the 1‑day VaR and CVaR

• VaR
• CVaR

6. Plot Pareto Frontier

6-pareto-frontier.py

Solves the multi-objective optimization problem to generate the Pareto frontier for portfolio management. For a range of expected return targets, the script minimizes CVaR and computes the corresponding optimal portfolio weights.

The output is a plot showing the trade-off between expected return and CVaR, also known as the efficient frontier under risk constraints.

• Uses parameters fitted in 4-multivariate-student.py:

• Constant C for CVaR calculation under Student-t distribution

Outputs:

Pareto Frontier graph: Expected Return vs. CVaR