Calculating and Applying VaR

Welcome to the World of VaR!

Hello there! Today, we are diving into one of the most famous (and sometimes infamous) concepts in finance: Value at Risk (VaR). If you’ve ever wondered how a big bank or a hedge fund answers the question, "What is the most we could lose tomorrow?", you are in the right place. VaR is the industry-standard way to put a single number on risk. Don't worry if you’ve found this chapter intimidating before; we’re going to break it down step-by-step using simple logic and clear examples.

1. What Exactly is VaR?

At its heart, Value at Risk (VaR) is a statement about a potential loss. It tells us the maximum loss we expect to suffer over a specific time period, given a certain level of confidence.

To define VaR, you always need two "ingredients":

1. The Time Horizon (T): This is the period we are looking at (e.g., 1 day, 10 days, or 1 year).
2. The Confidence Level: This is how sure we want to be (e.g., 95% or 99%).

Example: If a portfolio has a 1-day 95% VaR of \$1 million, it means that under normal market conditions, there is only a 5% chance that the portfolio will lose more than \$1 million in a single day.

Quick Review: The Three Ways to Say the Same Thing

Students often get confused by the wording. These three statements mean exactly the same thing:
• We are 95% confident we won't lose more than \$1M.
\n• There is a 5% probability (significance level) that we will lose \$1M or more.
• In a 100-day period, we expect to lose more than \$1M on about 5 days.

Key Takeaway: VaR summarizes the "worst-case" scenario, but it doesn't describe what happens in the absolute extreme tail (the 5% or 1% "crash" zone).

2. Calculating VaR: The Parametric (Delta-Normal) Method

The most common way to calculate VaR in the FRM curriculum is the Parametric Method. It assumes that asset returns follow a Normal Distribution (that classic bell-shaped curve).

The Formula

If you know the mean (\( \mu \)) and the standard deviation (\( \sigma \)) of the portfolio's returns, you can calculate the VaR using Z-scores from the normal distribution table.

\( VaR = [ \mu_{p} - (z \times \sigma_{p}) ] \times Portfolio Value \)

Note: In many FRM problems, we assume the expected return (\( \mu \)) is zero for short horizons (like 1 day) to simplify things.

Z-scores You MUST Memorize

You will use these two numbers constantly. Save them to your "mental desktop":
\n• For 95% Confidence: \( z = 1.645 \) (often rounded to 1.65)
\n• For 99% Confidence: \( z = 2.326 \) (often rounded to 2.33)

Did you know? Even though 99% is a higher confidence level than 95%, the "tail" is smaller (1% vs 5%). This means a 99% VaR will always be a larger dollar amount than a 95% VaR because it’s looking further into the "bad" end of the curve.

3. Scaling VaR: The "Square Root of Time" Rule

Sometimes you have a 1-day VaR, but your boss wants a 10-day VaR. How do you convert it? We use the Square Root of Time Rule. This assumes that returns are independent and identically distributed (i.i.d.) over time.

To scale a 1-day standard deviation to a T-day standard deviation:
\n\( \sigma_{T-day} = \sigma_{1-day} \times \sqrt{T} \)

Therefore:
\n\( VaR_{T-day} = VaR_{1-day} \times \sqrt{T} \)

Common Mistake: Don't multiply by the time! If you want to go from 1 day to 100 days, multiply by \( \sqrt{100} = 10 \), not by 100. If you multiply by 100, you are drastically overestimating your risk!

4. VaR for Linear vs. Non-Linear Derivatives

How we apply VaR depends on what is in the portfolio.

Linear Derivatives (Stocks and Forwards)

For these assets, the change in price is directly proportional to the change in the underlying factor. We use Delta to estimate the change.
\n\( \Delta Portfolio \approx Delta \times \Delta Underlying \)

Non-Linear Derivatives (Options)

Options are tricky! Their price doesn't move in a straight line relative to the stock price. This is due to Gamma (the curvature).
\n• Delta-Normal Approach: This is a simplified method that ignores Gamma. It treats the option as if it were linear. It's fast but inaccurate for large price moves.
\n• Full Revaluation: This is the most accurate. You actually recalculate the option price using a model (like Black-Scholes) for the new "worst-case" stock price.

Analogy: Imagine walking on a flat road (Linear). If you take 10 steps, you go exactly 10 meters. Now imagine walking on a steep hill (Non-Linear). As you go further, the slope gets steeper, and your simple "steps to meters" calculation fails. That "steepness" is Gamma!

5. Historical Simulation vs. Monte Carlo

If the Parametric method (Bell Curve) isn't right for your data, you have two main alternatives:

Historical Simulation (The "Look Back" Method)

You take the actual price changes from the last, say, 500 days and apply them to your current portfolio. You then sort the results from worst to best.
\n• Pros: No need to assume a normal distribution; it captures "fat tails."
\n• Cons: It assumes the future will look exactly like the past.

Monte Carlo Simulation (The "Future Worlds" Method)

You use a computer to generate thousands of possible random price paths for your assets based on certain parameters.
\n• Pros: Extremely flexible; works for any complex derivative.
\n• Cons: Very expensive in terms of computer power and time.

Quick Summary Table:
\n• Parametric: Fast, easy, but assumes "Normal" distribution (bad for options).
\n• Historical: Uses real past data, no distribution assumption, but "backward-looking."
\n• Monte Carlo: Most powerful, handles everything, but slow and complex.

6. Limitations and Risks of VaR

VaR is a great tool, but it’s not perfect. As an FRM candidate, you must know its flaws:

1. The "Tail" Mystery: VaR tells you the threshold of loss, but it says nothing about how much you could lose if you exceed that threshold. If your 95% VaR is \$1M, your actual loss on a bad day could be \$1.1M or \$100M—VaR doesn't know the difference.
2. Non-Subadditivity: In some cases, the VaR of a combined portfolio can be higher than the sum of the VaRs of the individual parts. This contradicts the basic idea of diversification. (Technically, VaR is not a "coherent risk measure" for this reason).
3. Model Risk: If your assumptions (like the Normal distribution) are wrong, your VaR is wrong. Garbage in, garbage out!

Key Takeaway:

VaR is just one tool in the toolbox. Most risk managers use it alongside Stress Testing (which looks at extreme events) and Expected Shortfall (ES) (which calculates the average loss beyond the VaR level).

Final Encouragement

You've just covered the core logic of Calculating and Applying VaR! Remember the Z-scores (1.65 and 2.33) and the square root of time rule, and you’ll be well on your way to mastering this chapter. If the math feels heavy, just keep coming back to the basic question: "What is my worst-case loss at a certain level of confidence?" You've got this!