Track 16 of 25

Tails and modelling

Why normal distributions understate extremes, what a fat tail is in arithmetic, what a Monte Carlo simulation does and cannot tell you, and the difference between a risk that is priced and a risk that is not.

25 lessons

Mark a lesson complete and it is remembered here.

Why a normal distribution is used at all, and what it buysThe bell curve survives because it makes portfolios addable in closed form, and that convenience outlives every demonstration of its failure.7 min
What the normal distribution says about a five-sigma dayEach extra standard deviation costs more likelihood than the one before it, and that acceleration is the whole mechanism.7 min
How not-normal returns are, stated in counts of daysAt three percent the error is a factor of nine; at seven percent it is hundreds of millions.7 min
Kurtosis, computed rather than describedOne day out of twenty supplies ninety-eight percent of the fourth moment, which is what the statistic is actually measuring.7 min
What "fat" means arithmetically in a fat tailUnder a bell curve a move twice as large is a million times rarer. Under a power law, seven times.7 min
Power-law behaviour, and the scaling that identifies itCounting past three thresholds gives two independent estimates of one exponent; their agreement is the evidence, not the plot.7 min
Why a sample maximum keeps growing with the sampleTen years of data implies a worst day of ten percent and a century implies twenty-two, from one identical process.7 min
Why a standard deviation understates a fat-tailed riskThe estimate is lowest exactly when the window has been calm, which is exactly before the day that needs it.7 min
Value at risk: what it states, and what it is silent aboutOne number, one horizon, one probability — and nothing at all about how bad the bad days actually get.7 min
Expected shortfall, and the question it answers insteadAveraging the losses past the line separates two distributions that share an identical threshold to the dollar.7 min
Why a 99th percentile is not a worst caseA level crossed two and a half times a year on average is a design specification, not a limit.7 min
Aggregation: why two risk numbers do not addUnder a bell curve two exposures combine to less than their sum. Add a jump and they combine to more.7 min
What a simulation inherits from its input distributionClip the random draws at two and a half sigma and every headline number survives; only the disaster disappears.6 min
Why the correlation input moves the answer more than the volatility inputThree years of data pins volatility to within a point. Nothing pins the number that supplies most of the variance.7 min
The same setup simulated normal and fat-tailedMatching the mean and the variance makes the heavy-tailed run quieter in the middle and much worse at the edge.6 min
Why a leveraged path cannot be summarised by its endpointThe units sold at the bottom do not participate in the recovery, so two identical return sequences end differently.7 min
What more runs buy, quantified, and what they do notFour times the work halves the noise, and the noise was never the part of the answer that was wrong.7 min
Historical replay, and its sample-size problemSix episodes in a century supports a frequency anywhere between one year in eight and one year in forty-five.7 min
Stress testing with no probability attachedShock three variables one at a time, add the results, and you will be short by the interaction term.7 min
Reverse stress testing: starting from the failure and working backFix the breach and solve for the inputs, and the answer is not a scenario but a curve.8 min
A risk that is priced, and a risk that is notSomebody quantified the haircut and charged for it; nobody quantified the right to change it, and that one is free.7 min
Uncertainty that has no distribution to put on itWhen the only input is a guess at the probability, the output range is just the range of the guess.7 min
Model risk: the error that is not in the dataThree errors sit inside every simulated number, and which one dominates depends on how far into the tail you look.7 min
Backtesting, overfitting, and what validation cannot establishThree years of daily data accepts a model that is fifty percent wrong about three times out of four.7 min
What a model is for, and exactly where it stopsThe conversion from assumptions to consequences is exact. The assumptions are the part that was never measured.7 min

Educational only. Nothing in this track is investment, legal or tax advice, or a recommendation to enter into any transaction. Completion is stored in this browser and nowhere else.