FRM Part II · FRM Exam Part II

Non-parametric Approaches for FRM Part II

Non-parametric approaches estimate VaR and expected shortfall directly from historical return data, without assuming a distribution such as the normal. You sort past P&L, then read off the loss at the chosen quantile. Variants add resampling (bootstrap), weights (age, volatility) or smoothing (kernels) to fix weaknesses.

What this chapter covers

This chapter covers how to measure market risk using the data itself rather than a fitted distribution. The core method is historical simulation (HS): take a window of past returns, apply them to today's portfolio, sort the resulting P&L, and read VaR from the loss at the chosen percentile. Expected shortfall is the average of the losses beyond that point.

The rest of the chapter improves on basic HS. Bootstrap resamples the data with replacement to get a more stable VaR estimate and a confidence interval. Weighted HS gives recent or high-volatility observations more influence, for example with age weights that decline by a factor λ. Kernel density estimation smooths the empirical distribution so quantiles are less jumpy. The chapter ends by weighing what non-parametric methods do well and badly.

This links to the rest of Market Risk Measurement and Management. You compare these methods with parametric and Monte Carlo approaches, and with volatility models and backtesting. The same ideas appear when you judge model choice in a case question, such as why a bank's VaR jumped after one extreme day left the window.

Questions on this chapter are usually short and applied, so they reward clean understanding over long calculation. You may be asked to read VaR off a sorted set of returns, compute age-weighted probabilities, or pick the method that fixes a stated problem such as slow reaction to volatility. Because the methods are contrasted with each other, mastering this chapter also sharpens your answers on VaR model choice, backtesting and regulatory use elsewhere in the paper. The effort is modest, and most of it is learning exactly what each variant changes.

Non-parametric Approaches: topics in the order to study them

1Historical Simulation Approach to VaRThis is the base method; every other topic in the chapter modifies it, so learn the sorting and quantile steps first.
2Bootstrap Historical SimulationIt keeps the HS logic and adds resampling, so it is the easiest extension to learn next.
3Weighted Historical SimulationOnce you know equal-weight HS, you can see exactly what age and volatility weights change.
4Non-parametric Density Estimation and KernelsSmoothing the distribution is a more abstract idea and makes sense once you have seen the jumpiness of plain HS.
5Strengths and Weaknesses of Non-parametric MethodsStudy it last, so you can tie each strength or weakness to a method you now understand.

How to prepare Non-parametric Approaches

Keep this chapter practical. Know the steps of each method and the one problem each variant is built to solve.

Work one plain HS example by hand: sort 100 P&L values and find the 95% and 99% VaR. Note which observation is used and how interpolation can matter.
Compute expected shortfall from the same data as the average of the losses beyond the VaR cutoff, and be clear it is at least as large as VaR.
Write down how bootstrap works: resample with replacement, compute VaR on each sample, then average the results or use their spread for a confidence interval.
Practise age weighting. Weights decline geometrically: the weight of the observation i days old is proportional to λ^(i−1), scaled so all weights sum to 1. Then cumulate weights on the sorted losses until you reach the tail probability.
Make a one-page table of each method, the problem it fixes and the cost it brings, including volatility-weighted HS and kernels.
Finish with scenario questions: given a symptom such as slow response to a volatility spike or noisy tail estimates, name the method that helps and why.

Common mistakes in Non-parametric Approaches

Saying historical simulation is a model-free method with no assumptions.
Fix: State the key assumption: the past window is representative of the future. Distribution-free is not assumption-free.
Reading VaR from the wrong end of the sorted data or using the wrong cutoff.
Fix: Convert to losses, sort from worst, and count the tail. At 95% with 100 observations, the cutoff sits around the 5th worst loss, subject to the stated convention.
Forgetting to normalise age weights so they sum to 1.
Fix: Divide each by the sum of all weights, then add the weights from the worst loss until you reach the tail probability.
Believing bootstrap creates new information or worse-than-observed losses.
Fix: Bootstrap only reuses existing observations. It improves estimate stability, but cannot go beyond the worst loss in the sample.
Mixing up age weighting and volatility weighting.
Fix: Age weighting changes the probability of each observation by how old it is. Volatility weighting changes the size of past returns to match current volatility.
Listing weaknesses without linking them to a fix.
Fix: Pair each weakness with a remedy: slow reaction with weighting, noisy tails with bootstrap or kernels, and limited extreme data with other methods such as extreme value approaches.

Last-day revision: Non-parametric Approaches

Historical simulation applies past returns to today's portfolio and reads VaR from the sorted P&L.
It assumes no distribution, so fat tails and skew in the data are kept.
Basic HS gives all observations in the window equal weight.
Expected shortfall is the average loss beyond VaR, so it is never below VaR at the same confidence level.
Bootstrap resamples with replacement from the same data and averages the VaR estimates.
Bootstrap can also give a confidence interval for VaR and tends to make the estimate more stable.
Age-weighted HS uses weights that decay by λ, so recent data counts more and ghost effects fall.
Weights must sum to 1 before you find the tail quantile.
Volatility-weighted HS rescales past returns by current volatility relative to volatility at the time.
Kernel methods smooth the empirical distribution so quantiles are less sensitive to single points.
Main weakness: results depend heavily on the window, and nothing beyond the worst observed loss can be estimated.
Main strength: easy to explain and implement, with no distribution or correlation assumptions.

Non-parametric Approaches practice questions

Non-parametric Approaches in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Non-parametric Approaches: frequently asked questions

What is the difference between historical simulation and Monte Carlo VaR?

Historical simulation reuses actual past returns, so it needs no distribution assumption. Monte Carlo generates scenarios from a chosen model and distribution. HS is simpler, while Monte Carlo can produce losses beyond those seen in the data.

Why does age-weighted historical simulation help?

In equal-weight HS, old data counts as much as recent data, and a large loss dropping out of the window can cause a sudden VaR change. Age weights reduce the influence of old data and make VaR react faster to current conditions.

Do I need to calculate bootstrap VaR in the exam?

A full bootstrap is not practical by hand. Expect conceptual questions on what it does, what it improves and what it cannot do, rather than large calculations.

How many marks does this chapter carry in FRM Part II?

GARP does not publish a mark split by chapter. FRM Part II has 80 equally weighted questions across six topics, and this chapter sits within Market Risk Measurement and Management. Prepare it well because it supports other VaR questions.