Explain historical simulation and how it is used in investing
The idea
Historical simulation estimates the range of possible outcomes by applying risk-factor changes actually observed over a look-back period to today's portfolio. Each past period becomes one scenario, and the resulting profits and losses form an empirical distribution from which value at risk, shortfall probabilities and stress results can be read. Its strength is that it imposes no distributional assumption, so fat tails and the co-movement between assets are kept as they occurred. Its weakness is that it can only reproduce what the window contains: a calm window understates tail risk, and an event that never happened cannot appear. The choice of window therefore drives the answer.
A practice question
Historical simulation generates scenarios for a portfolio by:
- drawing from an assumed normal distributionWhy not: Drawing from an assumed parametric distribution is Monte Carlo simulation.
- applying actual past changes in risk factors — correct
- solving for the portfolio's expected returnWhy not: Solving for an expected return produces one number, not a set of scenarios.
Historical simulation takes the changes actually observed in each risk factor over a look-back period and applies them to today's portfolio, producing one scenario per historical period.
A harder one
A 1,000,000 portfolio is 60% equities and 40% bonds. In one historical scenario, equities fell 12% and bonds rose 3%. The simulated profit or loss for this scenario is:
- -60,000 — correct
- -72,000Why not: Applied the equity loss but left out the offsetting 3% gain on the bond holding.
- -45,000Why not: Averaged the two asset returns equally instead of using the 60/40 weights.
P&L = 1,000,000 x (0.60 x -12% + 0.40 x 3%) = 1,000,000 x (-7.2% + 1.2%) = -60,000.
The hardest tier
An analyst applies 240 historical monthly scenarios to a current portfolio, and 18 of them produce a loss greater than 10%. The estimated probability of such a loss is:
- 8.1%Why not: Divided by the 222 scenarios without the loss rather than by all 240 scenarios.
- 92.5%Why not: Reported the share of scenarios that did not breach the 10% loss.
- 7.5% — correct
Each historical scenario is treated as equally likely, so the estimate is 18 / 240 = 7.5%.
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