Welcome to Market Risk: Non-parametric Approaches!
In your FRM Part I journey, you probably spent a lot of time talking about the "Normal Distribution" (that famous bell curve). While the bell curve is elegant, real-world financial markets are often messy—they have "fat tails" and unexpected spikes. Non-parametric approaches are the solution to this messiness. Instead of forcing the data to fit a specific shape or formula, we let the historical data speak for itself. Think of it as "learning from history" rather than "guessing with math."
1. Historical Simulation (HS): The Foundation
Historical Simulation is the simplest non-parametric method. It assumes that what happened in the past is a good indicator of what could happen tomorrow.
How it works (Step-by-Step):
1. Collect a series of past daily returns for your current portfolio (e.g., the last 1,000 days).
2. Apply those historical returns to your current portfolio value to see what your profit or loss (P&L) would have been.
3. Sort these P&L results from the worst (biggest loss) to the best (biggest gain).
4. If you want a 95% Value at Risk (VaR), you look at the 5th percentile of those sorted losses.
The Analogy: Imagine you are a commuter. To guess how long your drive will take tomorrow, you look at your logs for the last 100 days. You don't calculate a complex "commute distribution"; you just look at the 5 worst days you've had. That is Historical Simulation.
Pros of HS:
• No Distributional Assumptions: You don't care if the data is normal, skewed, or has fat tails.
• Simplicity: It is very easy to explain to senior management.
• Coherency: It naturally handles "Expected Shortfall" (ES) better than some parametric models.
Cons of HS:
• The "Ghost Effect": A massive outlier from 2 years ago stays in your VaR calculation until it suddenly "drops off" the data window, causing your VaR to jump even if the market is currently calm.
• The Sample Size Trap: If your window is too short (e.g., 50 days), you might miss a major crash. If it’s too long, old, irrelevant data might dilute current risks.
Quick Review: HS uses the past directly. No formulas, just sorting historical data.
2. Age-Weighted Historical Simulation (BRW Method)
One major flaw of basic HS is that it treats a data point from 2 years ago exactly the same as a data point from yesterday. In reality, yesterday’s market conditions are usually more relevant.
The Boudoukh, Richardson, and Whitelaw (BRW) approach fixes this by assigning weights to the data. Newer data gets a higher weight, and older data gets a lower weight using a decay factor, \(\lambda\) (usually between 0.97 and 0.99).
The Formula:
The weight for a day \(i\) days ago is:
\(w(i) = \frac{\lambda^{i-1}(1-\lambda)}{1-\lambda^n}\)
Why this helps:
• It reduces the Ghost Effect because old outliers fade away gradually rather than disappearing all at once.
• It allows the model to react faster to recent market shocks.
Key Takeaway: Age-weighting makes the "recent past" more important than the "distant past."
3. Volatility-Weighted Historical Simulation (Hull-White)
What if the market was very calm 6 months ago, but is very crazy (volatile) today? Basic HS would use those "calm" returns and underestimate today's risk. Volatility-weighting (proposed by Hull and White) adjusts past returns to reflect current volatility levels.
The Logic:
If a stock moved 1% on a day when volatility was low, that same move would likely be much larger on a day when volatility is high. We "scale up" or "scale down" the past returns.
The Step-by-Step Scaling:
To adjust a past return from day \(t\), we use:
\(r_{t}^* = r_{t} \times \left( \frac{\sigma_{T}}{\sigma_{t}} \right)\)
Where:
• \(r_{t}^*\) is the adjusted return.
• \(r_{t}\) is the actual return from the past.
• \(\sigma_{T}\) is the current volatility (today).
• \(\sigma_{t}\) is the volatility back on that historical day.
Did you know? This approach can actually produce "simulated" losses that are larger than anything ever seen in the historical record if current volatility is at an all-time high!
4. Filtered Historical Simulation (FHS)
Don't let the name scare you! Filtered Historical Simulation is just a "hybrid" approach. It combines the best of parametric models (like GARCH) with the best of non-parametric models (HS).
The Process:
1. Filter: We take past returns and "standardize" them by dividing each return by the volatility of that day. This leaves us with "residuals" (returns with the volatility sucked out).
2. Bootstrap: we randomly pick from these standardized residuals.
3. Forecast: We multiply these picked residuals by a volatility forecast for the future (using a model like GARCH).
4. Result: This gives us a distribution that accounts for current volatility trends but keeps the "non-normal" shape of the historical data.
Analogy: Imagine you have a recording of a singer. Basic HS is just playing the tape. Filtered HS is taking the singer's voice (the return), removing the background noise/volume (the old volatility), and then playing that voice through a modern, high-powered amplifier (current volatility forecast).
5. Limitations and Comparison
While non-parametric approaches are powerful, they aren't perfect. Here is a quick comparison to keep in your "study toolkit":
Common Mistakes to Avoid:
• The "Past is Prologue" Trap: Remember that HS cannot predict a crisis that has never happened before (like a "Black Swan").
• Data Requirements: Non-parametric methods need a lot of data to be reliable. If you only have 100 days of data, your 99% VaR is just the single worst day—not very robust!
Quick Summary Table:
• Basic HS: Easy, but has ghost effects and ignores current trends.
• Age-Weighted: Fixes ghost effects by favoring recent data.
• Vol-Weighted: Fixes the volatility problem by scaling returns.
• FHS: The most sophisticated; handles both volatility clustering and fat tails.
6. The Bootstrap Method
The Bootstrap is a statistical trick used to estimate the accuracy of our VaR. Instead of just taking one sample of 1,000 days, we "resample with replacement."
How it works:
Imagine a hat with 1,000 pieces of paper (daily returns). You pick one, write it down, and put it back in the hat. You do this 1,000 times. You might pick the same "bad day" three times, or not at all. By doing this thousands of times, you can see how much your VaR estimate varies. This helps us understand the standard error of our risk estimate.
Key Takeaway: Bootstrapping is about testing how "lucky" or "unlucky" our specific historical sample might be.
Don't worry if the weighting formulas seem dense! The most important thing for the FRM exam is understanding why we use them (to fix the flaws of basic HS) and the direction of the adjustment (scaling up returns when current vol is high). You've got this!