Simulate without inventing certainty

Learning goals

• Distinguish historical scenarios, bootstrap resampling and model-based simulation.
• Identify calibration, dependence and tail-risk limitations.

Draft · approximately 60 minutes, excluding further revision.

Three different simulation questions

Historical simulation applies observed joint market moves to today’s exposures. It preserves co-movement within the chosen observations but cannot generate shocks absent from the sample. Revalue nonlinear instruments rather than assuming every price response is linear. A chosen lookback window is a model choice: a calm sample can understate current tail risk.

Bootstrap resampling draws observations with replacement to study sampling uncertainty. A resampled set may contain duplicates and omit some original observations. Resample paired asset observations together when preserving cross-asset dependence. Independent resampling destroys serial structure; block bootstrap keeps blocks together but introduces its own block-length choice.

A Monte Carlo result is conditional on a model

Monte Carlo draws random inputs from specified distributions, evolves exposures and evaluates the resulting outcome. Specify the horizon, time step, dependence structure, estimated parameters, cash-flow policy and valuation rule before running paths. A lognormal one-step price model can use P₁=P₀exp[(μ−σ²/2)Δt+σ√Δt Z], where Z is standard normal; this model excludes jumps and cannot make every real market risk disappear.

More draws reduce simulation noise roughly with 1/√N under suitable conditions, but do not remove model misspecification or parameter uncertainty. Use a reproducible seed for comparison, run convergence checks, and test adverse assumptions. Clearly state the quantile convention for loss estimates. A 95% loss quantile is not the average of the worst 5%, and neither is a maximum possible loss.

Distinguish Monte Carlo error from a wrong model

In a fictional simulation,200 of 10,000 paths breach a cash buffer, so estimated breach probability is 2%. Conditional on independent draws from this exact model, approximate simulation standard error is sqrt(0.02 × 0.98/10,000)=0.0014, or 0.14 percentage point. A rough 95% Monte Carlo interval is 2%±0.2744 percentage point. It measures simulation noise, not uncertainty that the chosen distribution describes reality.

Record the random seed, input distributions, correlations, path count, time step and breach rule so another analyst can reproduce the run. More paths reduce numerical noise but do not fix omitted jumps, unstable correlations or survivorship bias. Bootstrap resamples observed data; it cannot invent an unobserved crisis unless a separate stress mechanism is introduced and disclosed.

Formula and assumptions

P₁ = P₀ exp[(μ−σ²/2)Δt + σ√Δt Z]; simulation error ∝ 1/√N; nearest-rank quantile = sorted[ceil(pN)]

A small empirical loss distribution

1. Suppose ten equally likely one-period losses are −4, −2, −1, 0, 1, 2, 3, 4, 8 and 15. Negative losses mean gains.
2. Using the nearest-rank rule, the 90% quantile is observation ceil(0.90×10)=9, or 8. The mean of the worst 10% is 15 under this discrete convention.
3. These results describe this toy distribution only. They do not establish that losses above 15 are impossible in a real portfolio.

Errors to catch

• Resampling each asset separately and accidentally destroying dependence.
• Reporting many paths as evidence that the economic assumptions are correct.

Original knowledge check

Answer every question before checking. Review explanations after submitting; retry starts a new attempt.

Sources and coverage status

These original lessons are drafts. Public outlines are used to check scope; the official learning-outcome text and proprietary question banks are not reproduced here. Follow the source for exam-year requirements and updates.

• Reference · www.cfainstitute.org