The big picture
Value at Risk, VaR, compresses a portfolio loss distribution into a threshold for a stated horizon and confidence level. That makes it useful for comparison and oversight, but dangerous when described as the maximum possible loss. A one-day95% VaR ofPKR50,000 means the model places its95th percentile of one-day loss at that amount; it leaves a modelled5% tail beyond it. It does not establish tomorrow’s true probability, the size of an extreme loss or protection from a crisis. This topic builds VaR from explicit assumptions, compares calculation methods and connects it to expected shortfall, stress testing, liquidity and validation.
Core concepts
Learning objectives
- Interpret a VaR statement without calling it a maximum loss.
- Calculate a simple historical quantile and a normal-model approximation.
- Explain simulation design, data limitations and correlation risk.
- Combine expected shortfall and stress scenarios with VaR.
- Design backtesting, escalation and a documented risk dashboard.
1. Specify the question before calculating a number
1.1 Horizon, confidence and loss convention
Define loss as positive when portfolio value declines. For loss variable L, VaR at confidence c is a c-quantile of the assumed loss distribution. At95%, modelled losses beyond the threshold occupy approximately5% of outcomes, subject to discrete-data conventions. A99% threshold generally lies farther into the same distribution. Neither tells you that a loss beyond it is impossible; neither supplies the average or largest tail loss.
1.2 Positions and the valuation boundary
Specify one day or another horizon, current positions, currency, valuation time and treatment of cash flows. A static-position VaR usually asks what today’s holdings could lose under price changes, rather than mixing tomorrow’s trading profits into the calculation. Include meaningful market factors, derivatives and currency exposures. A figure calculated from closing prices can miss intraday funding demands, illiquid exit costs and issuer default or operational risks outside the chosen model.
Step-by-step method
- Write the horizon, confidence, loss sign and reporting currency.
- Freeze a clearly documented position snapshot and valuation method.
- List included risks and material exclusions beside the result.
A hypothetical report says95% one-day VaR50,000 on aPKR1 million portfolio. The correct reading is a modelled loss threshold equal to5% of portfolio value, not a guarantee that cash losses cannot exceed50,000.
2. Build historical and parametric estimates transparently
2.1 Revalue using historical shocks
Historical simulation applies past risk-factor changes to today’s holdings and sorts resulting hypothetical losses. The window, data adjustments, missing observations, weights and quantile convention affect the result. It preserves relationships found in the selected shocks but cannot reveal an event absent from that sample. Historical portfolio returns are not automatically equivalent if holdings have changed; revaluation of current exposures answers a different, often more relevant question.
2.2 Volatility and covariance assumptions
For a linear portfolio under a normal return model with mean assumed zero, one-day VaR is approximately z×portfolio value×daily portfolio volatility. At95%, z≈1.645. For value1,000,000 and daily volatility1%, the result is16,450. Portfolio volatility needs weights and the full covariance structure, not the sum of individual volatilities. Normality can understate fat tails; a linear approximation can miss nonlinear option behaviour. Square-root-of-time scaling requires strong assumptions and is not automatically valid for longer horizons.
Step-by-step method
- Clean and align adjusted price or risk-factor histories.
- Publish the historical window and percentile convention.
- Compare parametric results with alternative tails and correlation stresses.
A teaching sample of100 equally weighted hypothetical losses has95th sorted loss40,000. Using the nearest-rank95% convention gives VaR40,000; the five larger observations still require separate examination.
3. Design Monte Carlo rather than trusting random numbers
3.1 Joint scenarios and full revaluation
Monte Carlo simulation samples joint risk-factor scenarios from specified assumptions and revalues the portfolio in each. The simulation needs distributions, dependence, drift treatment, volatility, time step and instrument valuation rules. More paths reduce sampling noise for that model, but do not repair a wrong distribution or missing risk factor. A complex-looking simulation is not necessarily more realistic than a transparent historical estimate.
3.2 Parameter and numerical uncertainty
Preserve economically meaningful dependence across interest rates, equities and currencies. For nonlinear instruments, revalue under each scenario rather than multiplying a small sensitivity by a very large shock without checking accuracy. Set reproducible seeds for development and test convergence across path counts or seeds. Separate uncertainty about estimated parameters from uncertainty within the assumed model. Record software version, inputs and validation tests so the estimate can be reproduced.
Step-by-step method
- Specify joint factors, distributions and dependence explicitly.
- Revalue positions and aggregate scenario profit or loss consistently.
- Check convergence and compare with independent simple calculations.
Two hypothetical simulations each use100,000 paths. One assumes independent sector stocks, the other a shared market factor. Their tail results can differ substantially even though both have equally many paths; the dependence assumption matters more than the impressive path count.
4. Look beyond the threshold with tail and stress analysis
4.1 Expected shortfall
Expected shortfall, ES, describes the mean loss in the specified worst tail under the model, with careful treatment of probability mass at the quantile for discrete distributions. It adds information VaR omits but remains dependent on data and assumptions. In the100-scenario example, let the five losses beyond40,000 be50,000,60,000,80,000,110,000 and200,000. The mean of that worst5% is100,000. A40,000 VaR and100,000 tail mean describe very different aspects of the same teaching distribution.
4.2 Liquidity and reverse stress testing
Stress testing applies deliberately adverse combinations that may not have a reliable probability estimate. Include market gaps, higher correlations, currency shocks, wider spreads, trading restrictions and margin increases. Reverse stress testing asks what conditions would exhaust a chosen cash buffer or breach a capital limit. Exit time may lengthen as liquidity disappears, so a one-day mark-to-market estimate cannot establish the amount needed to liquidate a concentrated position.
Step-by-step method
- Report tail severity alongside the selected VaR threshold.
- Build coherent historical-inspired and hypothetical multi-factor stresses.
- Compare cash demands and losses with actual available resources.
A hypothetical portfolio has a modest normal-day VaR but requires immediate additional collateral during a severe common market decline. The analyst reports both economic loss and peak cash demand; owning illiquid assets does not necessarily meet that cash call.
5. Validate, govern and communicate the estimate
5.1 Backtesting and exceptions
Backtesting compares forecast thresholds with subsequent outcomes measured on a consistent basis. Under a correctly calibrated95% daily model, exceptions are expected over time; approximately5% is a long-run model implication, not a guaranteed count in every sample. Investigate clustering, changing volatility, data errors and valuation differences. Few exceptions can reflect an overly conservative model as well as limited data; frequent exceptions do not identify the cause by themselves.
5.2 Margin is not your personal loss budget
NCCPL describes VaR-based scrip margining with liquidity and volatility inputs. Those official requirements and current parameters must be obtained separately; a personal portfolio model is not a substitute for clearing collateral. Establish ownership of inputs, independent checks, limits, escalation and periodic model review. Report exposure, VaR, ES, stress loss, liquidity, assumptions and exclusions together. Do not scale up merely because recent calm prices lowered a model number.
Step-by-step method
- Compare like-for-like predicted and realised losses and log exceptions.
- Investigate breaches and clustered errors before changing the model.
- Link risk reports to exposure, cash and concentration decisions.
A hypothetical desk sees four exceptions in a short turbulent period after months of calm. Rather than explaining them away as rare events, it reviews covariance estimates, missing factors and available cash, and considers reducing exposure while the model is reassessed.
A threshold does not describe the entire tail
| Measure | Value |
|---|---|
| 95% VaR | 40,000 PKR |
| Worst5% mean | 100,000 PKR |
| Largest sample loss | 200,000 PKR |
Reference table
| Tail scenario | Loss PKR | Tail weight |
|---|---|---|
| 1 | 50,000 | 1% |
| 2 | 60,000 | 1% |
| 3 | 80,000 | 1% |
| 4 | 110,000 | 1% |
| 5 | 200,000 | 1% |
Glossary of key terms
- Backtesting
- Comparison of previously issued risk forecasts with subsequent comparable outcomes.
- Covariance
- A measure of how two variables vary together, including scale.
- Exception
- A realised loss exceeding the relevant forecast VaR threshold.
- Expected shortfall
- Average loss in a defined worst tail under the selected model.
- Monte Carlo
- Repeated scenario sampling from an explicitly specified probabilistic model.
- Quantile
- A threshold corresponding to a cumulative probability level.
- Reverse stress test
- Analysis of conditions that would cause a specified failure or limit breach.
- VaR
- A model-based loss quantile for a specified horizon and confidence.
Frequently asked questions
Is VaR the largest possible loss?
No. It is a quantile and leaves losses beyond the threshold.
Does95% confidence mean tomorrow is95% safe?
It describes the assumed model distribution, not independently verified true odds or safety.
Does ES remove model risk?
No. It describes the modelled tail more fully but can still underestimate it.
Can I use personal VaR instead of margin?
No. Actual collateral follows applicable clearing and broker requirements.
Common pitfalls
| Common mistake | A better approach |
|---|---|
| Omitting horizon or confidence | Publish both beside every result. |
| Adding individual VaRs mechanically | Model joint exposures and dependence. |
| Treating more simulation paths as realism | Challenge distributions and missing risks. |
| Increasing leverage after a calm window | Stress regime changes and check actual loss capacity. |
Your learner task
Create100 clearly hypothetical one-day losses with nearest-rank95% VaR40,000 and worst five50,000,60,000,80,000,110,000,200,000. Verify the100,000 tail mean. Add a stressed liquidity and margin scenario, state exclusions, and explain why neither number can guarantee survival.
Sources & verification
Use the linked documents to check the factual and regulatory context. Source dates and limitations are stated where relevant. Educational examples and analytical frameworks are not personalised recommendations.
- JISPL · Knowledge curriculum ↗
Original topic identity; expanded with corrections and explicit assumptions.
- BIS · Market risk terminology ↗
Definitions of VaR and expected shortfall; bank regulation is not a retail capital guarantee.
- BIS · Comparative analyses of ES and VaR ↗
Explains quantile and tail-loss limitations; teaching scenarios are independently hypothetical.
- NCCPL · Capital market risk management ↗
Official published description refers to liquidity and volatility in scrip margining; verify current operative parameters separately.
Editorial review: 22 October 2026