Abstract
Value at Risk (VaR) is a key risk management tool used in finance to quantify the potential loss a portfolio might experience over a specific period, given a certain confidence level. This article examines the different types of VaR, their methods of calculation, and their applications in portfolio management. We explore Parametric VaR, Historical VaR, Monte Carlo VaR, and other advanced variations, including Conditional VaR (CVaR), Incremental VaR (IVaR), Marginal VaR (MVaR), and Component VaR (CVaR). The article provides a structured approach to understanding the pros and cons of each type of VaR and discusses their relevance in modern risk management practices.
What VaR Actually Claims
A VaR figure is a quantile of the loss distribution, and stating it precisely matters because the sentence is easy to get wrong. A one-day 99% VaR of £1M means: on 99% of days the loss will not exceed £1M. It does not mean the maximum loss is £1M, and it says nothing whatever about how bad the remaining 1% of days are.
That silence about the tail is the defining limitation. Two portfolios can report identical VaR while one loses slightly more than the threshold on a bad day and the other loses fifty times as much. VaR cannot distinguish them, because it reports where the tail begins and not what is in it.
The Three Main Types of VaR
1. Parametric VaR (Variance-Covariance VaR)
Parametric VaR, also known as analytical VaR, assumes portfolio returns follow a normal distribution and uses the mean and standard deviation to estimate potential losses. It is popular because it is cheap to compute and needs only two estimated quantities.
For a portfolio of value $V$ over horizon $t$:
\[\text{VaR}_\alpha = V \left( z_\alpha \sigma \sqrt{t} - \mu t \right),\]where $z_\alpha$ is the standard normal quantile (1.645 at 95%, 2.326 at 99%), $\sigma$ the per-period volatility, and $\mu$ the expected return. Over short horizons the drift term $\mu t$ is small and often dropped, which is why the formula is frequently written as $z_\alpha \sigma \sqrt{t}$ alone — a simplification, not the definition.
The $\sqrt{t}$ scaling itself assumes returns are independent across periods. Under volatility clustering, which is the normal state of financial markets, this understates risk at longer horizons.
The deeper problem is normality. Financial returns have fat tails: extreme moves occur far more often than a normal distribution allows. A move the model treats as a once-in-a-century event may show up several times a decade, and it is precisely the tail that VaR is supposed to describe.
2. Historical VaR
Historical VaR makes no distributional assumption. It sorts the observed returns and reads off the empirical quantile directly.
This inherits the true shape of the data — fat tails, skew, and all — at the cost of two constraints. It can only produce losses that have already happened, so a portfolio that has never experienced a crash will report a comfortable VaR right up until the first one. And it weights a return from three years ago exactly like yesterday’s, which is wrong when volatility regimes shift.
Sample size binds tightly here. Estimating a 99% quantile from 250 trading days means the estimate rests on roughly the two or three worst observations.
3. Monte Carlo VaR
Monte Carlo VaR simulates many possible future paths from an assumed model and takes the quantile of the simulated losses. It handles non-linear instruments such as options, where the relationship between risk factors and portfolio value is not proportional and the parametric approach breaks down.
Its flexibility is also its weakness: the output is only as good as the assumed process. A Monte Carlo run using normally distributed shocks reproduces exactly the tail failure of parametric VaR, with more computation and a false impression of sophistication.
Computing All Three
import numpy as np
from scipy import stats
rng = np.random.default_rng(0)
# fat-tailed daily returns: Student-t is far closer to real markets than normal
returns = rng.standard_t(df=4, size=2000) * 0.01
V, alpha = 1_000_000, 0.99
# 1. parametric (assumes normality)
mu, sigma = returns.mean(), returns.std(ddof=1)
var_param = V * (stats.norm.ppf(alpha) * sigma - mu)
# 2. historical (empirical quantile)
var_hist = -V * np.percentile(returns, (1 - alpha) * 100)
# 3. Monte Carlo under a fitted normal
sim = rng.normal(mu, sigma, 200_000)
var_mc = -V * np.percentile(sim, (1 - alpha) * 100)
# expected shortfall: the average loss GIVEN the threshold is breached
tail = returns[returns <= np.percentile(returns, (1 - alpha) * 100)]
es = -V * tail.mean()
for name, v in [("parametric", var_param), ("historical", var_hist),
("monte carlo", var_mc), ("expected shortfall", es)]:
print(f"{name:20} {v:12,.0f}")
The parametric and Monte Carlo figures agree closely — unsurprisingly, since both assume normality — while the historical figure, computed from the same fat-tailed data, is materially larger. That gap is the cost of the normality assumption, and expected shortfall is larger still because it averages the tail rather than reporting its edge.
Why VaR Is Not a Coherent Risk Measure
Artzner and co-authors set out four properties any sensible risk measure should satisfy. VaR fails one of them: subadditivity, the requirement that combining two portfolios cannot increase total risk.
\[\rho(A + B) \le \rho(A) + \rho(B).\]VaR can violate this. Two portfolios each holding a different bond with a 3% default probability may each have zero VaR at 95%, because default sits outside the 5% tail. Combined, the probability that at least one defaults is nearly 6%, which pushes a default inside the tail and gives the merged portfolio a positive VaR larger than the sum of its parts.
The consequence is practical, not academic: under VaR, diversification can appear to increase risk, and a risk limit expressed in VaR can be gamed by moving exposure just beyond the quantile.
Expected Shortfall (also called Conditional VaR) fixes this. It is the mean loss conditional on exceeding the VaR threshold:
\[\text{ES}_\alpha = \mathbb{E}\left[L \mid L > \text{VaR}_\alpha\right].\]ES is subadditive, and it is sensitive to how bad the tail actually is rather than only to where it starts. This is why the Basel framework moved from VaR to Expected Shortfall for market risk capital.
Decomposing Portfolio Risk
Three related measures answer questions about where risk sits rather than how much there is.
Marginal VaR is the change in portfolio VaR from a small increase in one position — the derivative of VaR with respect to that holding. Incremental VaR is the change from adding or removing a position entirely, which is the relevant number when deciding on a trade. Component VaR allocates total VaR across positions so the components sum to the whole, which makes it the natural basis for risk budgeting.
The distinction matters when positions are correlated: a holding with modest standalone VaR can carry large component VaR if it moves with everything else, and a hedge can carry negative component VaR.
Backtesting
A VaR model that is never checked is an assumption, not a measurement. Backtesting compares realised breaches against the number the model implies: a 99% daily VaR should be exceeded on about 1% of days, roughly two or three times a year.
Kupiec’s proportion-of-failures test formalises this as a likelihood ratio against the expected breach rate. Counting alone is not enough, though — breaches should also be independent. Several exceedances clustered in one week suggests the model fails exactly when it matters, even if the annual count looks correct.
Note that Expected Shortfall is harder to backtest than VaR, since it is not elicitable in the same way. That tension — the more coherent measure being the harder one to validate — is a live methodological issue rather than a settled one.
Using VaR Sensibly
VaR remains useful as a common language for risk across desks and a convenient input to limits and capital calculations. Its weaknesses are known and manageable provided the number is not treated as a worst case.
The defensible practice is to report VaR alongside Expected Shortfall, state the confidence level and horizon every time, prefer historical or fat-tailed simulation over the normal assumption, backtest continuously and check breach clustering as well as counts, and pair all of it with stress tests of scenarios that have never occurred. VaR describes ordinary bad days. The days that destroy institutions are not ordinary, and no quantile of the historical distribution will anticipate them.
References
- Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203-228.
- Jorion, P. (2006). Value at Risk: The New Benchmark for Managing Financial Risk (3rd ed.). McGraw-Hill.
- Kupiec, P. H. (1995). Techniques for verifying the accuracy of risk measurement models. Journal of Derivatives, 3(2), 73-84.
- McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management (2nd ed.). Princeton University Press.
- Basel Committee on Banking Supervision. (2019). Minimum Capital Requirements for Market Risk. Bank for International Settlements.
