VaR and CVaR Calculation AI Prompt

Calculate Value at Risk (VaR) and Conditional Value at Risk (CVaR) for this portfolio using multiple methods.

Portfolio returns: {{returns}}
Confidence levels: 95% and 99%
Holding period: 1-day and 10-day

1. Definitions:
   - VaR(α): the loss that will not be exceeded with probability α. If 1-day 99% VaR = $1M, there is a 1% chance of losing more than $1M in a single day.
   - CVaR(α) (also called Expected Shortfall, ES): the expected loss given that the loss exceeds VaR. Always ≥ VaR. More coherent risk measure — CVaR is sub-additive, VaR is not.

2. Method 1 — Historical simulation:
   - Sort the return series from worst to best
   - 95% VaR: the 5th percentile of the distribution (5% of worst returns)
   - 99% VaR: the 1st percentile
   - CVaR: mean of returns below the VaR threshold
   - Pros: non-parametric, captures empirical fat tails and asymmetry
   - Cons: limited by historical window length; past scenarios may not reflect future risks

3. Method 2 — Parametric (variance-covariance) approach:
   - Assume returns are normally distributed: VaR = μ - z_α × σ
   - z_{0.95} = 1.645, z_{0.99} = 2.326
   - CVaR = μ - σ × φ(z_α) / (1 - α), where φ is the standard normal PDF
   - Pros: fast, analytical, easy to decompose by position
   - Cons: assumes normality — severely underestimates tail risk for fat-tailed assets

4. Method 3 — Monte Carlo simulation:
   - Fit a distribution to the returns (normal, t, or skew-t)
   - Simulate 100,000 scenarios from the fitted distribution
   - Compute VaR and CVaR from the simulated distribution
   - Pros: flexible distribution; can model complex portfolios
   - Cons: results depend heavily on the assumed distribution and model parameters

5. Scaling to multi-day horizons:
   - Square-root-of-time rule: 10-day VaR ≈ 1-day VaR × sqrt(10)
   - Caveat: this assumes i.i.d. returns. Volatility clustering violates this assumption.
   - Better: simulate 10-day paths and compute VaR directly from path-end P&L

6. Method comparison and recommendation:
   - Report VaR and CVaR from all three methods
   - Where do they differ most? Why?
   - Which method is most appropriate for this portfolio and why?
   - Backtesting VaR: count how many historical days exceeded the VaR. Should be ≈ 5% for 95% VaR.

Return: VaR and CVaR table (method × confidence level), method comparison, scaling analysis, and backtesting results.