Understanding Value at Risk (VaR) and Expected shortfall (ES) of various assets

This analysis explores various risk management concepts applied to financial instruments. We will utilize a dataset containing historical prices for five stocks (INTC, JPM, AA, PG, MSFT) from 2014 to 2023.

Question 1: Statistical Analysis of an Equally Weighted Portfolio

• We construct an equally weighted portfolio with the five stocks and analyze its return characteristics.
• Value at Risk (VaR) is estimated at different confidence levels using a rolling window approach to assess potential portfolio losses.
• The Kupiec test is employed to evaluate the accuracy of the VaR models.
• We investigate the presence of serial dependence in VaR violations.

Question 2: Risk Parity Portfolio

• This section focuses on the risk parity portfolio, aiming for equal component VaR across all assets.
• The portfolio is constructed using the first half of the data and evaluated on the second half.
• Performance is measured using Sharpe ratio, maximum drawdown, and VaR violations.
• We compare the risk parity portfolio’s performance to the equally weighted portfolio.

Question 4: Option Portfolio VaR and Expected Shortfall

• We estimate the VaR and Expected Shortfall (ES) for a portfolio containing options on four stocks (INTC, JPM, AA, PG).
• The Black-Scholes formula is used to price the options, considering historical volatility and risk-free rate.
• Monte Carlo simulation with at least 10,000 simulations is employed to estimate VaR and ES at a 99% confidence level for a 10-day horizon.
• Marginal VaR and Component VaR/ES are calculated for the non-Gaussian case of option portfolios.

This analysis provides valuable insights into risk management techniques for both traditional stock portfolios and those containing options. By understanding portfolio risk characteristics and potential losses, investors can make more informed decisions.

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