Value at Risk (VaR) Analysis in Python

The user wants to download 3 years of historical price data for three tickers (SPY, QQQ, GLD) and compute the daily returns of an equally weighted portfolio.

Plan:

1. Download data using yfinance.
2. Calculate daily percentage changes for each asset.
3. Compute the equally weighted portfolio return (mean of the asset returns).

Constraint check:

• Work iteratively.
• One execution step at a time.
• One Python code block per response.
• Use yfinance.
• First step should be loading and inspecting.

Step 1: Download the data and inspect it.

The user wants to download 3 years of SPY, QQQ, and GLD prices and compute daily portfolio returns assuming equal weights.

*   Prices for SPY, QQQ, and GLD have been downloaded.
*   The data contains 753 rows and 3 columns.
*   The prices for the first few dates (April 2023) are visible.
*   The goal was to download prices *and* compute daily portfolio returns.
*   Currently, only prices have been downloaded and inspected. Daily portfolio returns have not been calculated.

*   The original goal is partially complete (prices downloaded).
*   Missing part: compute daily portfolio returns with equal weights.
*   Path B is necessary.

*   Calculate daily returns for each asset using `pct_change()`.
*   Compute the equal-weighted portfolio return by taking the mean across columns for each day.
*   Display the result.

• Goal: Download 3 years of SPY, QQQ, GLD prices and compute daily portfolio returns assuming equal weights with yfinance.

  • Current State:

    1. Downloaded 3 years of data for SPY, QQQ, GLD (data DataFrame).
    2. Calculated daily percentage returns for each ticker (daily_returns DataFrame).
    3. Calculated the equal-weighted portfolio daily return (portfolio_returns Series).
    4. Observed the output of the first few rows of portfolio_returns.

  • Prices downloaded? Yes.

  • 3 years period? Yes.

  • Tickers SPY, QQQ, GLD? Yes.

  • Daily portfolio returns computed? Yes.

  • Equal weights assumed? Yes (via .mean(axis=1)).

  • The output shows the first few daily returns of the equal-weighted portfolio.

  • 2023-04-11 is NaN (expected for the first row of pct_change()).

  • Subsequent days show values like -0.002675 (-0.27%) and 0.015460 (1.55%).

  • The goal was to "compute daily portfolio returns". This has been done.

  • All requirements are met.

  • Decision: Path A.

  • Content: Natural final answer based on the observed outputs. No code. Include <!--final answer-->.

The user wants to calculate the Value at Risk (VaR) at 95% and 99% confidence levels using the historical simulation method.

*   `portfolio_returns`: A Series containing the daily returns of the equal-weighted portfolio of SPY, QQQ, and GLD.
*   `daily_returns`: A DataFrame containing individual asset returns.

*   Historical simulation VaR is the quantile of the distribution of historical returns.
*   For a 95% VaR, we look for the 5th percentile (since VaR usually represents a loss, the 5th percentile of returns is the threshold where 5% of the worst cases fall below).
*   Formula: $\text{VaR}_{95\%} = \text{quantile}(0.05)$.

*   The `portfolio_returns` series already exists.
*   I need to handle the `NaN` at the beginning (from `pct_change()`).
*   Calculate the 5th percentile (for 95% VaR) and 1st percentile (for 99% VaR).

*   Propose the calculation.
*   Provide the code to compute quantiles from `portfolio_returns`.
*   Display the results.

The user wants to calculate 95% and 99% Value at Risk (VaR) using historical simulation for a portfolio of SPY, QQQ, and GLD.

Looking at the notebook state:

• data contains the prices.
• daily_returns contains the pct changes.
• portfolio_returns is the equal-weighted mean of these returns.
• returns_cleaned is the series without NaNs.
• var_95 is calculated as the 5th percentile: -0.012858.
• var_99 is calculated as the 1st percentile: -0.022490.
• var_results is a series containing these two values.

The output from cell-3 explicitly shows: 95% VaR -0.012858 99% VaR -0.022490

The original goal was "calculate 95% and 99% Value at Risk using historical simulation". The results are present and calculated correctly using the historical simulation method (quantile of historical returns).

Path A is appropriate because the outputs are sufficient to answer the original goal.

Insights:

• The 95% VaR is -1.29% (rounded). This means there is a 5% probability that the portfolio will lose 1.29% or more in a single day.
• The 99% VaR is -2.25% (rounded). This means there is a 1% probability that the portfolio will lose 2.25% or more in a single day.

Final answer construction: State the values and explain their meaning in the context of the portfolio risk.

The user wants to calculate the Conditional Value at Risk (CVaR), also known as Expected Shortfall (ES), at a 95% confidence level. The current notebook state provides returns_cleaned (a Series of portfolio returns) and var_95 (the 95% VaR).

Definition of CVaR at 95%: It is the average of the returns that are less than or equal to the 5th percentile (the 95% VaR).

Plan:

1. Use the existing returns_cleaned and var_95.
2. Filter returns_cleaned for all values $\le$ var_95.
3. Calculate the mean of these filtered returns.

The user wants to plot the return distribution and mark the VaR (Value at Risk) and CVaR (Conditional VaR/Expected Shortfall) thresholds.

- `returns_cleaned`: Series containing portfolio returns (752 values).
- `var_95`: float (-0.012858).
- `cvar_95`: float (-0.020016).

- Plot the distribution of `returns_cleaned` using a histogram or KDE.
- Add vertical lines for `var_95` and `cvar_95`.
- Use seaborn/matplotlib as per visualization policy.
- Label the axes and add a clear title.

- Use `sns.histplot` with `kde=True` for a smooth distribution.
- Use `plt.axvline` for the thresholds.
- Add a legend.

The user wants to "plot the return distribution with VaR and CVaR thresholds marked". Looking at the notebook state:

• returns_cleaned contains the portfolio returns.
• var_95 was calculated as -0.012858.
• cvar_95 was calculated as -0.020016.
• Cell 5 contains the code to plot the histogram, the KDE, and the vertical lines for VaR and CVaR.
• The output of cell 5 is <Figure size 1000x600 with 1 Axes>, meaning the chart was successfully generated.

The original goal was to plot the distribution with thresholds marked. The notebook state shows that:

1. The returns were calculated and cleaned.
2. The numeric values for VaR and CVaR were computed and are available in the environment.
3. A plot was generated using sns.histplot and plt.axvline for both thresholds.

Since the goal was specifically to "plot", and the notebook shows a successful figure generation using the correct calculated values, the output is sufficient.

Insight extraction:

• The distribution of portfolio returns is visualized as a histogram with a KDE.
• The 95% Value at Risk (VaR) is marked at -1.29% (approximately), representing the threshold where losses exceed this value only 5% of the time.
• The 95% Conditional VaR (CVaR) is marked at -2.00%, representing the average loss in the worst 5% of cases.
• Visually, the CVaR line is further to the left than the VaR line, correctly illustrating that CVaR captures the "tail risk" beyond the VaR threshold.

Decision: Path A.