Value at Risk and expected shortfall

Key takeaways

• VaR is a quantile of the loss distribution: the loss exceeded with probability 1 minus the confidence level over the horizon.
• Parametric VaR assumes a distribution, usually normal, and needs only volatility; historical VaR uses actual past returns; Monte Carlo VaR simulates from a model.
• VaR is scaled to longer horizons with the square root of time, which assumes independent returns and an unchanged portfolio.
• Expected shortfall is the average loss in the tail beyond VaR. It captures tail severity and, unlike VaR, always rewards diversification.
• A VaR model is backtested by counting exceptions, days when the loss exceeded VaR, and comparing the count with what the confidence level implies.
• VaR says nothing about the size of losses beyond it, and is only as good as the window and assumptions behind it.

What VaR measures

A VaR figure has three parts: an amount, a horizon and a confidence level. 'One day 95 percent VaR of $197,000' means that, under the model, the portfolio's loss over one day exceeds $197,000 with 5 percent probability, or on about one trading day in twenty. Statistically, VaR is the (1 minus confidence) quantile of the profit and loss distribution, reported as a positive loss.

Choices of horizon and confidence reflect use. Trading desks typically use one day 95 or 99 percent VaR for daily limits. Bank regulation historically used a 10 day 99 percent VaR. Asset managers and allocators often look at monthly or annual horizons for strategic risk budgeting. The same portfolio can produce very different VaR numbers under different choices, so the parameters should always travel with the figure.

VaR is deliberately silent on one thing: what happens on the bad days. A VaR of $197,000 is consistent with the worst day costing $200,000 or $2 million. That silence is the single most important fact about VaR and the reason expected shortfall exists.

Parametric VaR

Parametric VaR assumes returns follow a known distribution, almost always the normal, and reads the quantile off it. Over a short horizon the expected return is small next to volatility and is usually set to zero, so VaR becomes the portfolio value times its volatility over the horizon times the standard normal quantile for the confidence level: 1.645 at 95 percent and 2.326 at 99 percent.

For a multi asset portfolio, the volatility comes from the covariance matrix exactly as in the portfolio construction topic, which is why this approach is also called variance covariance VaR. Its great strength is that it decomposes neatly: each position's contribution to VaR can be computed from its weight and covariances, and contributions add to the total. That makes it the standard tool for risk budgeting across managers or desks.

Its weakness is the normal assumption. Real returns have fatter tails, so parametric VaR tends to understate losses at high confidence levels, exactly where it matters. It also fails for portfolios with options, whose losses are not a linear function of the underlying. Refinements exist, such as using a Student t distribution or the Cornish Fisher adjustment for skew and kurtosis, but each adds assumptions of its own.

Historical and Monte Carlo VaR

Historical simulation drops the distributional assumption. It takes the portfolio as it stands today, applies each day's actual market moves from a look back window (commonly 250 or 500 trading days), and ranks the resulting hypothetical profits and losses. VaR is read directly from the ranked losses. Fat tails, skew and the real co movement of assets in stressed periods are all captured, provided they occurred in the window.

Conventions for picking the quantile from a finite sample differ. A common one, used here, takes the VaR at confidence c from n observations as the k th worst loss where k = n x (1 minus c); others interpolate between neighbours. With 500 days at 99 percent that is the fifth worst loss. With the deliberately tiny 20 day sample below, 90 percent VaR is the second worst loss, and 95 percent VaR would rest on the single worst day, which shows how few observations actually determine a historical VaR.

Monte Carlo VaR simulates instead. The risk team fits a model to the risk factors (for example, a multivariate distribution of equity, rate and currency moves, possibly with fat tails and time varying volatility), draws tens of thousands of scenarios, fully revalues the portfolio in each, and reads VaR off the simulated losses. It is the standard approach for portfolios with options or structured products, where a full revaluation is needed. Its output is only as good as the model: a Monte Carlo built on normal distributions inherits every weakness of parametric VaR at much greater cost.

Scaling VaR to longer horizons

VaR is usually computed for one day and then scaled. Under the same independence assumption that drives annualisation, volatility over h days is the daily volatility times the square root of h, and so is parametric VaR with zero mean. A 10 day 99 percent VaR is the one day figure times sqrt(10), about 3.16.

Two further assumptions hide inside the rule. The portfolio is assumed to stay unchanged over the horizon, with no trading to cut losses or add risk. And the distribution is assumed not to change, so that ten days of normal returns sum to a normal return. Neither holds in a crisis: volatility clusters, so a bad day raises the chance of another, and liquidity dries up, so positions cannot be cut as assumed. Regulators have responded by requiring liquidity horizons that differ by asset class rather than a single scaled number.

For longer horizons, such as a month or a year for an allocator, the expected return also stops being negligible. Over a year an expected return of several percent is comparable to the volatility term, and dropping it overstates VaR. At that horizon, VaR is better computed directly from monthly or annual return distributions or from scenario analysis than scaled up from daily figures.

Expected shortfall

Expected shortfall (ES), also called conditional VaR or CVaR, answers the question VaR leaves open: when the loss does exceed VaR, how large is it on average? It is the mean of the losses beyond the VaR level. Two portfolios with the same VaR can have very different expected shortfalls, and the one with the larger ES has the more dangerous tail, for example a portfolio short deep out of the money options.

Expected shortfall has a theoretical advantage as well. VaR is not subadditive: the VaR of a combined portfolio can exceed the sum of the VaRs of its parts, which means VaR can penalise diversification and can be gamed by splitting positions across books. Expected shortfall is always subadditive, so it is a coherent risk measure in the technical sense. That, together with its sensitivity to tail severity, is why the Basel Committee's revised market risk framework moved bank capital from 99 percent VaR to 97.5 percent expected shortfall.

Under the normal distribution, ES is a fixed multiple of volatility: the normal density at the VaR quantile divided by the tail probability. At 99 percent that multiple is about 2.67 against 2.33 for VaR, so ES is about 15 percent above VaR. For historical simulation, ES is simply the average of the losses at or beyond the VaR observation. Because it averages only a few tail observations, historical ES is noisier than historical VaR and needs a long window.

Backtesting and the limits of VaR

A VaR model makes a testable prediction: at 99 percent confidence, the loss should exceed VaR on about 1 percent of days. A day when it does is called an exception. Over 250 trading days the expected number of exceptions is 2.5, and the count follows a binomial distribution if the model is right. Too many exceptions mean the model understates risk; too few can mean it overstates risk and ties up capital.

Under the Basel market risk framework, a bank's 99 percent VaR model is backtested over the last 250 days and placed in a traffic light zone. Green is 0 to 4 exceptions, yellow is 5 to 9 and red is 10 or more, with the capital multiplier rising through the yellow zone. The zones reflect the binomial odds: a correct model has about an 11 percent chance of landing in yellow by bad luck and a negligible chance of reaching red. A formal alternative is the Kupiec proportion of failures test, a likelihood ratio test that compares the observed exception rate with the expected one.

Backtesting checks the frequency of exceptions, not their size, and it cannot rescue a model from its structural limits. VaR ignores everything beyond the quantile. A historical window that happens to be calm will produce a low VaR just before a crisis. Correlations estimated in normal times break in stressed ones. And VaR assumes positions can be held or exited at market prices, which fails for illiquid assets. That is why risk teams pair VaR with expected shortfall, stress tests built on specific historical or hypothetical scenarios, and position and liquidity limits that do not depend on any model.

import numpy as np
from scipy.stats import norm

value = 10_000_000
daily_vol = 0.012

# Parametric, normal, zero mean (unrounded quantiles)
var95 = value * daily_vol * norm.ppf(0.95)
var99 = value * daily_vol * norm.ppf(0.99)
es99 = value * daily_vol * norm.pdf(norm.ppf(0.99)) / 0.01

# Historical, 20 illustrative daily returns, k th worst loss convention
r = np.array([0.4, -1.1, 0.7, -0.3, 1.2, -2.4, 0.5, 0.9, -0.6, 0.2,
              -1.8, 0.3, 1.0, -0.9, -3.1, 0.6, -0.2, 0.8, -1.4, 0.1]) / 100
losses = np.sort(-r)[::-1]  # largest loss first
k = int(round(len(r) * (1 - 0.90)))
hvar90 = value * losses[k - 1]
hes90 = value * losses[:k].mean()

print(f"parametric 95% VaR  {var95:,.0f}")
print(f"parametric 99% VaR  {var99:,.0f}")
print(f"parametric 99% ES   {es99:,.0f}")
print(f"historical 90% VaR  {hvar90:,.0f}")
print(f"historical 90% ES   {hes90:,.0f}")

parametric 95% VaR  197,382
parametric 99% VaR  279,162
parametric 99% ES   319,826
historical 90% VaR  240,000
historical 90% ES   275,000

Templates for this lesson

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• ExcelPortfolio optimiser and VaRFour asset mean variance with risk contributions and component VaR, minimum variance and tangency portfolios with weight and correlation grids, and parametric and historical VaR.Download
• NotebookVaR and expected shortfall notebookParametric, historical and Monte Carlo VaR and ES for a three asset portfolio, ten day scaling and a Kupiec backtest.Download