Comparing Historical, Parametric, and Monte Carlo VaR Methods

# What is the difference between the methods for calculating VaR?


# What is the difference between the methods for calculating VaR?












There are three different commonly used Value at Risk (VaR) methods:

- Historical method

- Variance-Covariance Method

- Monte Carlo

What is the difference between these approaches, and under what circumstances should each be used?

## Answer by TheBridge (score 14)

https://quant.stackexchange.com/a/33

There are many advantages and flaws to each quoted method by Shane(presuming that I understand them properly), the first one has the big main advantage that it doesn't need any evaluation of probability law, it is just some kind of evolved scenario re-playing "as of" today using the history of (usually) one day market evolutions over one or two year.

So once you know how to evaluate your protfolio (no matter how complex it is) you have something that allows you to mechanically know your quantile and so your VaR. The problem with the method is that there are manny econometrics hypothesis that are actually hard to test for this kind of method to be trustworthy.

Second "Variance-Covariance" I think that Shane means by this that risk factors returns obey multinomial normally distributed laws (or log-normally), once again the upside of method is its simlpicity once you econometrically estimate your VarCovar matrix. The truth is that unfortunately one day returns exhibit fat tails and asymmetry so those kind of laws aren't the best, this can though be partially overcome by using some other multinomial laws that are elliptic. Another thing you should keep in mind is that when you have a VaR indicator and a profolio that isn't rebalanced over a day, and when market becomes excited, then you expect (and your management) your VaR to go up even though nothing much has happened in your econometric estimation, so what you really want is then not a VaR which is unconditionnal but some conditionnal VaR and then you resort to some kind of overweighing the recent observations versus the old one (like EWMA methods). Then you get some indicator that is really "alive" instead of some indicator that takes a long time to adjust to market conditions. Let me now explain why I used the elliptic laws (multinomial Gaussain are part of this class), this is because with this assumption (If I remember well) then VaR requalifies as a true Risk Measure as defined by Artzner et al. (which is not true for genral probability laws has it lacks the subadditivity axiom). Another point, is when you have nonlinear positions in your protfolio such as options then what is usually done (and can be really wrong if not given sufficient attention) is that those position are Taylor derived at order one with respect to risk factors and then Linear VaR is calculated. You can extended your approximation to superior orders but if you have a lot of risk factors then the second derivative estimations are already an achievement, so what you do is that you stay linear and estimate periodically what is the error when quadratic terms are taken into account (or resort to MC methods).

Finally, MC methods then they are efficient methods but very time consumming as you are trying to evaluate a Quantile id est a rare event and you then need to get a very large number of simulation to get something good, moreover it is not always clear which law is to be simulated. In particular in portfolios with derivatives, because it is quite tempting tu use Risk Neutral measure in your simulations but they have two differences over historical estimates, first the underlying model is usually calibrated to model risk factors over large period of times when you are only trying to get estimates over the next day so the underlying measures have different purposes.

And second as you may know in theory the difference between Historical and Risk Neutral measures are hidden "in the drift" of the risk factor dynamics (well this is not true unless complete market is assumed but let's go with it) and over a day you can discard this difference with respect to the diffusion term which should be the same for both measures, it happens that almost always Risk Neutral Volatility (i.e. marekt calibrated volatilities) are higher than historical one (or realized ones).

Well here are my two cents

Best Regards

## Answer by John Channing (score 6)

https://quant.stackexchange.com/a/1590

The Historical Method, which I would call Historical Simulation requires that you have a reasonably clean and accurate time series of data for the underlying asset. Essentially, you are using the past performance of the asset to model its likely behaviour over a time frame of typically 1 to 10 days. Choosing and updating your time series data set needs to be thought about carefully as your VaR number can be impacted significantly by extreme events in the time series used.

Where good time series data is not available, it would be appropriate to use the Variance-Covariance method. This is generally considered to be less accurate than Historical Simulation due to assumptions about the distribution of returns that do not hold perfectly true in real markets (fat tail distributions).

Monte Carlo simulation is computationally a lot more expensive than Historical Simulation or V-CV and requires that a large number of asset paths are calculated to get a statistically significant result.

## Answer by Con Fluentsy (score 0)

https://quant.stackexchange.com/a/84150

You have overlooked the most powerfull method built on exact tail data the Extreme Value Theory: Summary of EVT-Based VaR and Expected Shortfall Methods in Quant Finance

Extreme Value Theory (EVT) is used in finance to model tail risk— the behavior of extreme losses — beyond what normal or Student-t assumptions capture. This post summarizes the main EVT-based methods for computing Value-at-Risk (VaR)and Expected Shortfall (ES).

- Classical (Non-EVT) Benchmarks

| Method | Description | Limitation |
| Historical Simulation | Empirical quantile of past losses. | Fails for rare events. |
| Parametric (Variance–Covariance) | Assumes Normal or t-distributed returns. | Underestimates heavy tails. |
| Monte Carlo | Simulated draws from a fitted model. | Tail estimates depend on assumed law. |

- EVT Foundations

EVT models the tails of a distribution using asymptotic limit laws.

Block Maxima → Generalized Extreme Value (GEV) $$ [ F(x) = \exp{-[1+\xi((x-\mu)/\sigma)]^{-1/\xi}} ] $$ Peaks Over Threshold (POT) → Generalized Pareto Distribution (GPD) $$ [ G(y) = 1 - [1+\xi y/\beta]^{-1/\xi}, \quad y>0 ] $$ In finance, POT/GPD is preferred since it uses all exceedances above a threshold (u).

- EVT-Based Risk Measures

Let:

$(n)$ = total sample size $(N_u)$ = number of exceedances above threshold (u) $(\xi,\beta)$ = tail index and scale parameters

Value-at-Risk (quantile) $$ [ \text{VaR}_p = u + \frac{\beta}{\xi}\Big[\Big(\frac{n}{N_u}(1-p)\Big)^{-\xi} - 1\Big] ] $$ Expected Shortfall (conditional tail mean) $$ [ \text{ES}_p = \frac{\text{VaR}_p}{1-\xi} + \frac{\beta - \xi u}{1-\xi}, \quad \xi<1 ] $$ If $(\xi \ge 1)$, ES diverges (infinite mean tail).

- Parameter Estimation

| Estimator | Assumption / Feature | Typical Use |
| Hill (Pareto) | Power-law tail; simple log-ratio. | Heavy-tailed returns. |
| MLE (GPD) | Full likelihood of exceedances. | Standard for Basel/ESMA. |
| PWM / L-moment | Robust for small samples. | Insurance or sparse data. |
| Declustering | Removes volatility clusters before fitting. | Needed for GARCH data. |

- Conditional EVT (Dynamic Volatility)

Tail risk is often estimated on standardized residuals from GARCH-type models.

Steps:

- Fit GARCH on returns $(r_t = \mu_t + \sigma_t z_t)$.

- Extract residuals $(z_t)$.

- Fit GPD to positive tail of $(z_t)$.

- Re-scale: $$ [ \text{VaR}_{t,p} = \sigma_t \cdot \text{VaR}^{(\text{EVT})}_p ] $$ Common frameworks:

GARCH-EVT (McNeil & Frey, 2000) Filtered Historical Simulation (FHS-EVT) AR-GARCH-EVT hybrids Implementation Outline (Pseudo-Code)

```
# Example (R) workflow
library(evir)
fit <- gpd(data, threshold = u)
VaR_p <- u + fit$beta/fit$xi  ((n/fit$nexc(1-p))^(-fit$xi) - 1)
ES_p  <- VaR_p/(1-fit$xi) + (fit$beta - fit$xiu)/(1-fit$xi)
```

```
# Example (Python)
from scipy.stats import genpareto
xi, loc, beta = genpareto.fit(exceedances)
VaR_p = u + beta/xi  ((n/Nu(1-p))(-xi) - 1)
ES_p  = VaR_p/(1-xi) + (beta - xiu)/(1-xi)
```

- Pros vs. Classical Approaches

| Aspect | Classical VaR/ES | EVT-Based VaR/ES |
| Tail accuracy | Poor | Accurate for rare losses |
| Data used | All | Only tail exceedances |
| Assumptions | Normal/t | GPD/GEV |
| Robustness | Low | High in heavy tails |
| Best for | Ordinary volatility | Stress testing / tail events |

- Key References

Embrechts, Klüppelberg & Mikosch (1997), Modelling Extremal Events. Coles (2001), An Introduction to EVT for Finance and Insurance. McNeil & Frey (2000), Estimation of Tail-Related Risk Measures for Heteroscedastic Time Series. Danielsson & de Vries (2000), Value at Risk and Extreme Returns.