Value at Risk (VaR) & Conditional Value at Risk (CVaR)

📉 Value at Risk (VaR)

🔍 What is VaR?

Value at Risk (VaR) estimates the maximum expected loss of a portfolio or investment over a given time period, at a certain confidence level.

🧮 Formula (Parametric / Variance-Covariance VaR):

VaR=Zα⋅σ⋅t\text{VaR} = Z_{\alpha} \cdot \sigma \cdot \sqrt{t}

Where:

• ZαZ_{\alpha} = Z-score corresponding to the confidence level (e.g. 1.65 for 95%)

• σ\sigma = standard deviation of returns

• tt = time horizon (e.g., 1 day, 10 days)

📌 Example:

A portfolio worth $1 million has a daily standard deviation of 2%. At 95% confidence:

VaR=1.65×0.02×1=3.3%\text{VaR} = 1.65 \times 0.02 \times 1 = 3.3\%

So, there’s a 5% chance of losing more than $33,000 in one day.

✅ Key Features of VaR:

• Expressed in money or % terms

• Common confidence levels: 90%, 95%, 99%

• Timeframes: daily, weekly, monthly

• Methods:

  • Historical Simulation (uses past returns)

  • Variance-Covariance (assumes normal distribution)

  • Monte Carlo Simulation (randomized modeling)

❗ Limitations of VaR:

• Doesn’t show the magnitude of losses beyond VaR

• Assumes normality (which markets often violate)

• Can be misleading during market stress

• Not subadditive (violates risk aggregation logic)

📊 Conditional Value at Risk (CVaR) — Expected Shortfall

🔍 What is CVaR?

Conditional Value at Risk (CVaR), also known as Expected Shortfall, estimates the expected loss in the worst-case tail — i.e., the average loss if the loss exceeds the VaR threshold.

🧠 Why CVaR?

CVaR is designed to address VaR’s weakness: it captures the severity of tail losses, not just the threshold.

📌 Formula (simplified):

CVaRα=E[L∣L>VaRα]\text{CVaR}_{\alpha} = E[L \mid L > \text{VaR}_{\alpha}]

Where LL is the portfolio loss and VaRα\text{VaR}_{\alpha} is the threshold value at the confidence level.

📌 Example:

If VaR at 95% = $1M loss, and losses beyond that average $1.4M, then CVaR = $1.4M — giving a fuller picture of potential extreme risk.

✅ Benefits of CVaR:

• Accounts for tail risk

• Coherent risk measure (subadditive, convex, etc.)

• Preferred in regulatory risk management (Basel III, Solvency II)

🆚 VaR vs CVaR – Comparison Table

📚 Conclusion

• VaR is a basic but useful tool for measuring risk exposure in normal market conditions.

• CVaR provides a more realistic and robust picture by focusing on extreme losses.

• Together, they help build stronger, risk-aware financial models and guide capital allocation, stress testing, and regulatory compliance.