Valuation and Risk Models
Valuation models translate market inputs into prices; risk models translate changes in those inputs into possible losses. VaR is a loss threshold, expected shortfall describes tail severity, and stress tests examine specified adverse scenarios.
The topic at a glance
| Concept | How it works | Watch for |
|---|---|---|
| Valuation | Specify cash flows, discount rates and optionality. | Using a price model without checking its inputs. |
| VaR and ES | State confidence, horizon and the loss distribution. | Interpreting VaR as a maximum loss. |
| Stress tests | Revalue positions under a coherent adverse scenario. | Treating a scenario loss as a calibrated probability forecast. |
01Valuation
How it works: Specify cash flows, discount rates and optionality.
Watch for: Using a price model without checking its inputs.
02VaR and ES
How it works: State confidence, horizon and the loss distribution.
Watch for: Interpreting VaR as a maximum loss.
03Stress tests
How it works: Revalue positions under a coherent adverse scenario.
Watch for: Treating a scenario loss as a calibrated probability forecast.
Original worked example
Choose the model output that answers the question
A current portfolio is revalued under 1,000 equally likely historical shocks. The 990th ordered loss is 4.1 million; the worst ten losses sum to 72 million. A separate hypothetical stress loses 16 million.
- STEP 1
Using the nearest-rank convention, 99% historical VaR is the 990th loss: 4.1 million.
- STEP 2
Average the worst 1%: 72 million / 10 = 7.2 million expected shortfall.
- STEP 3
Report the 16 million stress loss separately; this scenario has no assigned probability here.
4.1 million is the threshold, 7.2 million is the tail average, and 16 million is the specified stress result. They are complementary outputs, not interchangeable estimates.
Illustrative assumptions and calculations for learning. Not observed market data or an official exam item.
How the ideas fit together
Before you start: Present value, quantiles and basic probability distributions.
Match the estimator to the modelling choice
Historical simulation applies chosen historical factor shocks to the current portfolio. A parametric model specifies a distribution and estimates its parameters. Monte Carlo generates shocks from a chosen model. All three require a valuation rule. Replaying yesterday’s portfolio losses is not the same as revaluing today’s positions under yesterday’s shocks.
Match the risk measure to the decision
Use a sensitivity to estimate a small local change, a loss quantile for a confidence-level threshold, and a tail average for severity beyond the threshold. A stress scenario can examine a vulnerability absent from the estimation sample. Check market data, factor coverage, valuation approximations and portfolio changes before interpreting differences between model outputs.
Choose an output that answers the question
Use sensitivity for a small local change, full revaluation when nonlinearity matters, and a distributional measure for a probability statement. Historical simulation reflects the selected history; a parametric model reflects distributional assumptions; Monte Carlo reflects the simulated model. More scenarios improve numerical precision but cannot repair an omitted risk factor.
Work through the topic
Follow the chain from position data to valuation, loss scenarios and risk measures. Compare model choices without confusing a precise calculation with a reliable specification.
Build a loss distribution from today’s positions
A risk model needs a portfolio, a horizon, risk factors and a way to turn factor changes into values. Begin by valuing today’s positions under current inputs. Then change the inputs under each scenario and revalue the same positions. Define loss as current value minus scenario value, subject to the stated treatment of cash flows. A consistent sign convention prevents gains from being misclassified as tail losses.
Historical simulation does not ordinarily mean copying the profit-and-loss history of whatever portfolio happened to be held in the past. A changing portfolio can have very different exposures today. In a simple historical revaluation, historical factor changes are applied to current holdings. The chosen sample, return convention and valuation method still matter. The procedure can reproduce historical shocks without claiming that tomorrow must resemble the average day in that sample.
A local sensitivity model replaces full repricing with an approximation. That can be useful for small movements in nearly linear instruments, but options and instruments with changing cash flows can require more detail. Compare a delta-only result with full revaluation in a material stress. An apparent disagreement between risk estimates may originate in the valuation step rather than in the probability model.
Compare historical, parametric and simulated approaches
A historical approach supplies scenarios from observed factor changes. It preserves relationships present in the selected observations, but the usable sample may contain few extreme events and may omit an important regime. A parametric approach imposes a distributional form and estimates its parameters. It can be transparent and efficient, while being sensitive to incorrect tail or dependence assumptions.
Monte Carlo simulation generates scenarios from a specified stochastic model. It can handle many factors and complex valuation rules, but the flexibility shifts responsibility to model design and calibration. Ten times as many simulations can reduce numerical noise without fixing a wrong correlation structure, missing volatility factor or defective valuation function. Separate numerical convergence from economic validity.
Choose the approach by the portfolio and decision, then inspect its vulnerabilities. For a nearly linear portfolio over a short horizon, a simple covariance model can be a useful benchmark. For strongly nonlinear positions, a richer revaluation may be necessary. A benchmark is valuable even when it is not the production method because large unexplained differences can expose data, units or implementation errors.
| Approach | Main modeling choice | Challenge to perform |
|---|---|---|
| Historical simulation | Which observations and scaling rules are used? | Does the history represent today’s important vulnerabilities? |
| Parametric model | Which distribution and parameters are assumed? | Are tails and dependence suitable for the portfolio? |
| Monte Carlo simulation | Which dynamics generate factor paths? | Is the specification sound as well as numerically stable? |
What each scenario engine supplies
- Historical simulation
- Main modeling choice: Which observations and scaling rules are used?
- Challenge to perform: Does the history represent today’s important vulnerabilities?
- Parametric model
- Main modeling choice: Which distribution and parameters are assumed?
- Challenge to perform: Are tails and dependence suitable for the portfolio?
- Monte Carlo simulation
- Main modeling choice: Which dynamics generate factor paths?
- Challenge to perform: Is the specification sound as well as numerically stable?
Read VaR, expected shortfall and stress loss correctly
VaR identifies a loss quantile for a specified horizon and confidence level. It does not cap possible loss. Expected shortfall describes the average loss in the selected worst tail, with careful treatment of probability mass at the boundary. A stress loss describes the result of a specified adverse scenario. Unless a probability model is added, a stress number is not automatically another quantile of the estimated distribution.
In the opening 1,000-scenario example, the nearest-rank 99% VaR is the 990th ordered loss, while the worst ten losses determine the 1% tail average. Those outputs can differ substantially if the largest losses are severe. Changing only the worst loss might leave the chosen VaR order statistic unchanged while moving expected shortfall. That is a property of what the measures summarize, not an arithmetic inconsistency.
Do not infer that one measure alone answers every management question. A small one-day VaR can coexist with a large stress loss, concentrated exposure or an inability to unwind positions. State the distribution, horizon and confidence before comparing two VaR figures. State scenario assumptions before comparing two stress figures. The narrower ES and VaR lessons develop the calculations and failure cases.
Locate the cause of a suspicious risk number
Investigate a surprising output in layers. First reconcile positions and units. Then inspect market inputs, factor mapping, valuation behavior and statistical assumptions. Finally compare the result with realized outcomes or independent benchmarks that fit the model’s purpose. This order prevents a sophisticated statistical explanation from concealing an omitted trade or a basis-point conversion error.
Suppose a portfolio’s reported risk falls after adding options. It might be a genuine hedge, but it could also mean the model mapped only delta and ignored volatility exposure. Reprice the options under an underlying-price move and a volatility move separately. A factor-by-factor comparison shows what the model includes and what it leaves out. The conclusion should follow the diagnosed mechanism rather than the direction of the headline number.
Document the permitted use and the unresolved limitations. A model adequate for a small daily sensitivity report may be unsuitable for a large liquidation decision. Monitoring can reveal deterioration after deployment, so validation is not merely an initial certification. The Part II model-governance lesson develops responsibilities and responses; this overview establishes the chain of modeling choices that those controls must challenge.
Does a higher confidence level always mean a better model?
It changes the part of the distribution being measured. A more extreme quantile is usually harder to estimate because less tail information is available. Confidence level does not repair poor data, wrong factor mapping or a valuation error. Compare models under a common definition and evaluate whether the selected measure supports the decision.
What should accompany a reported risk estimate?
Identify the portfolio date, currency, horizon, confidence or scenario definition, valuation method and material assumptions. Add the limitation that could change the decision. A number with several decimal places is less useful than a clearly defined estimate whose uncertainty and scope are understood.
Try a changed assumption
A model uses ten times as many simulated scenarios but omits the same volatility risk factor. Has that omission been fixed?
Show answer and reasoning
No. Numerical sampling error may fall, but the missing risk channel remains. More simulation paths do not repair the model specification.
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Is Monte Carlo automatically more accurate?
No. It can model complex payoffs, but the simulated factors, dependence and valuation assumptions still determine whether the output is useful.
Sources and scope
Links checked 2026-10-10. GARP establishes exam scope; the technical references support the methods identified below. These are selected study notes and original examples, not complete curriculum coverage or official exam questions.
- GARP FRM study materials and 2026 learning objectives
Exam-domain scope and the official route to the annual learning objectives. This page does not certify coverage of every reading.
- MIT: Value at Risk Models, lecture 7
Loss quantiles, estimation methods and the limitations of VaR.
- Basel Committee: minimum capital requirements for market risk (2019)
Regulatory context for expected shortfall and market-risk model assessment. Local implementation can differ.
- Acerbi and Tasche: On the coherence of Expected Shortfall
Expected shortfall, coherence and the treatment of discrete loss distributions.
- MIT: Factor Modeling, lecture 15
Factor exposures and covariance-based portfolio risk decomposition.
Prepared by MiloPrep editorial team. MiloPrep is independent of GARP. References do not imply endorsement or independent review of these notes.