Value at Risk (VaR) is the most widely used risk metric in finance. It tells you the maximum amount you can expect to lose over a specific time period with a given confidence level. Banks, hedge funds, and professional traders rely on VaR to understand their risk exposure and make informed decisions.
What is Value at Risk?
VaR answers a simple question: "What is the worst loss I can expect under normal market conditions?" It provides a single number that summarizes your portfolio's risk exposure.
VaR Definition: A 95% daily VaR of $5,000 means there is only a 5% chance of losing more than $5,000 in a single day under normal market conditions.
VaR has three components:
- Time period: Usually 1 day or 10 days
- Confidence level: Typically 95% or 99%
- Loss amount: The maximum expected loss
Three Methods to Calculate VaR
There are three main approaches to calculating VaR, each with its own strengths and weaknesses.
1. Historical VaR
Uses actual historical returns to estimate potential losses. Simply sort your historical returns from worst to best and find the loss at your confidence percentile.
Calculation: For 95% VaR with 100 days of data, the VaR is the 5th worst daily return.
Pros: No assumptions about distribution, captures actual market behavior
Cons: Limited by available history, assumes future resembles past
2. Parametric (Variance-Covariance) VaR
Assumes returns follow a normal distribution and uses statistical parameters.
Formula: VaR = Portfolio Value x Z-score x Standard Deviation
Pros: Quick to calculate, works well for large portfolios
Cons: Assumes normal distribution, which underestimates tail risk
3. Monte Carlo VaR
Simulates thousands of possible future scenarios using random sampling.
Process: Generate thousands of simulated returns, then find the loss at your confidence percentile.
Pros: Can model complex positions and non-normal distributions
Cons: Computationally intensive, depends on model assumptions
Parametric VaR Calculation Example
Let us calculate VaR for a $100,000 portfolio using the parametric method.
Given:
- Portfolio value: $100,000
- Daily standard deviation: 1.5%
- Confidence level: 95%
- Z-score for 95%: 1.645
Calculation:
- Daily VaR = $100,000 x 1.645 x 0.015
- Daily VaR = $2,467.50
This means there is a 95% probability that the portfolio will not lose more than $2,467.50 in a single day.
Converting VaR Across Time Periods
To convert daily VaR to other time periods, use the square root of time rule:
Time Scaling Formula: VaR(T days) = VaR(1 day) x sqrt(T)
Example: Converting our $2,467.50 daily VaR to other periods:
- Weekly VaR (5 days) = $2,467.50 x sqrt(5) = $5,517.56
- Monthly VaR (21 days) = $2,467.50 x sqrt(21) = $11,308.38
- 10-day VaR = $2,467.50 x sqrt(10) = $7,802.20
Historical VaR Example
Let us calculate historical VaR using 100 days of portfolio returns.
Process:
- Collect 100 daily return percentages
- Sort from worst to best
- For 95% VaR, find the 5th worst return (100 x 0.05 = 5)
- Multiply by portfolio value
Example data (sorted worst returns):
- 1st worst: -4.2%
- 2nd worst: -3.8%
- 3rd worst: -3.1%
- 4th worst: -2.9%
- 5th worst: -2.6%
For a $100,000 portfolio: Historical 95% VaR = $100,000 x 2.6% = $2,600
Understanding Confidence Levels
The confidence level determines how extreme the loss estimate is:
| Confidence Level | Z-Score | Interpretation |
|---|---|---|
| 90% | 1.28 | Exceeded 1 in 10 days |
| 95% | 1.645 | Exceeded 1 in 20 days |
| 99% | 2.33 | Exceeded 2-3 times per year |
VaR for Options Portfolios
Options add complexity because their risk changes as the underlying moves. For options, consider:
- Delta-adjusted VaR: Scale position by delta for directional risk
- Full revaluation: Recalculate option prices at different underlying levels
- Greek sensitivities: Include gamma, vega, and theta effects
Example: You own 10 call options with delta of 0.50 on a $100 stock. Delta-equivalent exposure = 10 x 100 x 0.50 = $500 worth of stock. Apply VaR calculation to this adjusted exposure.
Limitations of VaR
VaR is useful but has important limitations:
- Does not measure worst-case: VaR tells you the threshold, not how bad it can get beyond that threshold
- Assumes normal markets: Breaks down during market crashes and black swan events
- Backward-looking: Based on historical data that may not predict the future
- Can be gamed: Traders might structure positions to minimize VaR while taking hidden risks
Conditional VaR (CVaR)
Conditional VaR, also called Expected Shortfall, addresses a key VaR limitation by measuring the average loss when VaR is exceeded.
CVaR Definition: If 95% VaR is $2,500, CVaR answers: "When we do lose more than $2,500, how much do we lose on average?"
CVaR is always larger than VaR and provides a more complete picture of tail risk.
Practical VaR Application
Here is how to use VaR in your trading:
- Set risk limits: Determine maximum acceptable VaR for your portfolio
- Monitor daily: Recalculate VaR as positions change
- Stress test: Calculate VaR using stressed market periods
- Compare to capital: Ensure VaR is appropriate relative to your account size
Monitor Your Portfolio Risk
Pro Trader Dashboard helps you track your positions and understand your risk exposure across your entire portfolio.
Summary
Value at Risk is an essential tool for understanding portfolio risk. Whether calculated historically, parametrically, or through Monte Carlo simulation, VaR gives you a concrete number representing your maximum expected loss at a given confidence level. While it has limitations, especially during extreme market events, VaR remains the industry standard for risk measurement. Combine it with CVaR and stress testing for a more complete picture of your risk exposure.
Learn more about Monte Carlo simulation or tail risk hedging.