Value at Risk, or VaR, answers one simple question: "How much could I lose?" It puts a number on the worst-case loss over a set time, with a set confidence level. We measure risk because uncertainty is the only constant.
There are three main ways to calculate it. Each has its own flavor, its own strengths, and its own blind spots. Let's break them down side by side.
| Method | Core Idea | Best For | Main Weakness |
|---|---|---|---|
| Parametric(Variance-Covariance) | Assumes returns follow a normal distribution. Uses mean and standard deviation. | Linear, simple portfolios. Quick daily checks. | Totally fails in market crashes. Fat tails are ignored. |
| Historical Simulation | Replays actual past market moves on the current portfolio. | Complex, non-linear portfolios. When you want actual history, not theory. | Cannot predict new crises. The past is just the past. |
| Monte Carlo Simulation | Builds a model and runs thousands of random trials to generate scenarios. | Complex derivatives. Forward-looking stress tests. | Computationally heavy. "Garbage in, garbage out" model risk. |
These methods are not rivals. They are tools. A good risk manager uses all three to see the same risk from different angles.
Parametric is fast but naive. Historical is realistic but stuck in the past. Monte Carlo is powerful but complex.
Never trust a single number. Always sanity-check your VaR calculation.
Parametric Method (Variance-Covariance)
This is the "textbook" method. It assumes that gains and losses are spread out evenly on a bell curve. Once you know the average return and the volatility, you just calculate the Z-score.
The math is simple. For a 95% confidence level, you use -1.65. For 99%, you use -2.33. It’s fast, but markets are not this polite.
Imagine you flip a coin. The Parametric method assumes the coin is perfectly fair every single day. But in real markets, sometimes the coin lands on its edge, or worse, someone steals the coin. The bell curve doesn't catch those rare disasters.
| Component | Symbol | Description |
|---|---|---|
| Mean Return | μ | Average daily or weekly gain. Often assumed to be zero for short periods. |
| Standard Deviation | σ | The volatility of the asset. How wide the price swings are. |
| Z-Score | Z | 1.65 for 95% confidence. 2.33 for 99% confidence. |
| Portfolio Value | V | Current market value of the holding. |
| Formula | VaR | VaR = V * (μ — Z * σ) |
This works perfectly for a single stock. It gets messy fast when you have options. Options have curved payoffs, not straight lines.
Historical Simulation
This method doesn't guess the future. It takes the last 500 days of market data and says: "What if those exact days happened again, right now?" You re-price your portfolio under old shocks.
It is brilliant for capturing real-world panic. But if you never had a crash in your window, you won't see one. It assumes history repeats itself exactly.
Think of a driving test. The Historical Method looks at the last 250 trips on a specific road. If it never snowed in those 250 trips, this method tells you it will never snow. That's a dangerous conclusion to drive with.
| Step | Action | Example |
|---|---|---|
| 1. Collect Data | Gather percentage changes for the last N days (e.g., 250 days). | Stock X fell 2% on day 1. Rose 1% on day 2. |
| 2. Apply Moves | Apply these old changes to today's price. | If you hold $1M, a -2% move means a $20k loss. |
| 3. Rank Results | Sort the hypothetical profits and losses smallest to largest. | The 5th worst loss in a 250-day set is your 95% VaR. |
| 4. Read VaR | Pick the loss at the chosen confidence level. | "We are 95% sure we won't lose more than $X tomorrow." |
Your VaR number changes wildly depending on the look-back window. Using 100 days of calm data gives a tiny VaR. Using 500 days that include a crash gives a huge VaR.
Regulators often demand at least a 1-year look-back period.
Monte Carlo Simulation
This is the heavy artillery. You don't look at the past and you don't assume a perfect bell curve. Instead, you build a mathematical engine that creates random price paths.
You let the computer run 10,000 scenarios. Some paths crash. Some soar. You map the distribution yourself. It handles the weird world of knock-out options and structured products easily.
Imagine you want to test a paper boat. You don't just watch a video of last year's river. You build a machine that generates 10,000 random streams—fast flows, slow flows, rocks, whirlpools. You sail the boat in all of them. That's Monte Carlo.
| Feature | Monte Carlo | Parametric | Historical |
|---|---|---|---|
| Distribution Shape | Custom (can have fat tails) | Forced Normal Curve | Lumpy/Real |
| Forward-Looking? | Yes | No (static) | No (replays) |
| Speed | Slow | Instant | Fast |
| Non-linear Risk | Handles well | Very Poor | Moderate |
| Model Risk | High (if model wrong) | Low | Low |
The danger is the "model risk." If you tell the computer that volatility is constant, it will give you a very confident, very wrong answer. You must stress-test your assumptions.
The quality of a Monte Carlo simulation depends entirely on the engine. If your random number generator or volatility model is bad, your VaR is useless.
Banks run huge clusters of computers overnight to calculate this.
Limitations of VaR
VaR is popular, but it has a dark side. It tells you the boundary of normal loss, but nothing about what lies beyond the boundary. "Beyond VaR" is where total ruin lives.
It also isn't stable. A sudden volatility spike changes the number instantly. You need to pair it with measures like Expected Shortfall, which looks at the average loss when the boundary breaks.
A pilot doesn't just ask "What wind speed knocks me out of the sky?" They also ask "What happens when I fall?" VaR tells you the wind speed. Expected Shortfall checks if you have a parachute.
Key Takeaways
| Key Point | What It Means | Action Item |
|---|---|---|
| Parametric is a quick filter. | Assumes a normal world. Highly misleading in crises. | Use it only for plain vanilla stocks in stable markets. |
| Historical simulation respects reality. | It captures actual fat tails seen in the past. | Always use a look-back period that includes a bear market. |
| Monte Carlo is a what-if machine. | Generates thousands of possible futures, not just one path. | Invest in coding skills. Validate the volatility model heavily. |
| VaR doesn't describe the worst loss. | It only marks the edge of a confidence interval. | Combine VaR with Expected Shortfall and stress testing. |
| No single method is "correct". | Good risk management compares outputs from all three. | Create a dashboard that shows Parametric, Historical, and Monte Carlo VaR side-by-side. |