FINM7405 Chap.3 VaR, Expected Shortfall and Stress Tests
VaR, Expected Shortfall and Stress Tests
Value-at-Risk compresses a loss distribution into a threshold at a stated confidence level and horizon. Expected Shortfall answers the next question by averaging losses beyond that cut. This chapter calculates parametric VaR, contrasts it with historical simulation and Monte Carlo loss engines, and explains how model assumptions change the tail.
Backtesting counts realised exceptions, while stress testing imposes severe coherent scenarios that a short sample or smooth distribution may never generate. Together these tools separate routine reporting from tail severity and model challenge.
Parametric VaR is fast because it maps mean and volatility through an assumed distribution.
Historical simulation reprices the current position under observed return vectors, preserving realised skew and dependence but remaining limited to its window. Monte Carlo can generate new combinations and revalue nonlinear positions, but it inherits every choice about processes, jumps, correlation and calibration.
More scenarios reduce sampling noise without repairing a misspecified model.
Backtesting asks whether realised exceptions occur at a frequency consistent with the forecast, while stress testing asks how the portfolio behaves under a severe coherent state. Reverse stress starts from failure and searches for a joint movement that could cause it.
The chapter treats these methods as complementary controls rather than competing labels.
What this chapter covers
- 01
VaR as a loss quantile
- 02
Parametric VaR calculation
- 03
Historical and Monte Carlo simulation
- 04
Expected Shortfall and tail severity
- 05
Backtesting and stress testing
Worked example · free
A one-day normal VaR
The marks used here separate the VaR method and are not an official university marking scheme.
- +1Calculate the tail return: 0.04% + (-2.326 x 1.30%) = -2.9838%.
- +1Quote VaR as a positive loss: 2.9838%.
- +1Multiply by $12 million to obtain $358,056.
- +1State the one-day horizon and 99% confidence.
- +1Explain that larger losses remain possible beyond the threshold.
Key terms
- Value-at-Risk
- A positive loss threshold at a specified confidence and horizon under a chosen model.
- Expected Shortfall
- The average loss among outcomes beyond a selected VaR threshold.
- Backtesting
- Comparison of realised losses with prior model forecasts to count and diagnose exceptions.
- Stress Test
- A severe coherent scenario used to reveal losses that routine probability models may omit.
VaR, Expected Shortfall and Stress Tests FAQ
Is VaR the maximum possible loss?
No. VaR marks a quantile where the bad tail begins. It does not bound the losses beyond that point. Expected Shortfall and stress tests are needed to describe severity and coherent scenarios inside the tail.
When is historical simulation useful?
It preserves observed skew, fat tails and joint factor moves without imposing a normal distribution. Its weakness is the window: it cannot create an event or dependence pattern that the selected history never contained.
What does a VaR exception mean?
An exception occurs when realised loss exceeds the prior VaR forecast. Some exceptions are consistent with the confidence level, but too many or clustered breaches can indicate stale volatility, a thin-tailed model or regime change.
Exam move
For every VaR problem, separate the return quantile from the dollar conversion and perform a magnitude check. Write confidence, horizon and the positive-loss convention beside the result before interpreting it. Compare methods by loss engine rather than acronym: parametric distribution, historical window and Monte Carlo process.
Rebuild the same small loss set into a VaR threshold and an Expected Shortfall average so the conceptual difference becomes numerical. For backtesting, distinguish an individual exception from a pattern of excessive or clustered breaches. For stress testing, state why the selected scenario is coherent for the portfolio and which funding or nonlinear effects it activates.
Learn one sentence each for what VaR, Expected Shortfall, backtesting and stress testing add, then connect them in that order. End every method comparison with a precise blind spot: fat tails, missing history, process misspecification, sparse tail observations or an implausible scenario set. Use a four-line reporting template: method, coordinates, result and limitation.
The method identifies the loss engine; the coordinates give confidence, horizon and units; the result states the threshold or tail mean; the limitation names what the engine cannot represent. When ranking historical losses, write the ordering and quantile convention before selecting an observation so interpolation ambiguity is visible.
When simulating, separate scenario generation from full revaluation and ask which nonlinear positions require repricing rather than delta approximation. Create a miniature backtest table with forecast VaR, realised loss, exception flag and market regime. Look for clustering as well as the total exception count, because serially concentrated breaches can reveal volatility updating failures.
For stress work, start from economic narratives and translate them into mutually consistent factor shocks rather than selecting large independent numbers. Add second-round effects such as margin calls, liquidity discounts and correlation shifts where the portfolio makes them relevant. Reverse the exercise once: begin with a capital or liquidity breach and solve for the combinations that could cause it.
End by recommending how the evidence changes limits, capital, hedging or model review; this links validation to action.
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