This is the capstone. Six lessons built the toolkit for the part of the distribution that actually bankrupts people. You learned why the bell curve lies in the tail (its exp(−x²/2) decay buries extremes that real markets produce every few years); how a power law decays politely as x^(−α) and how the tail index α governs both heaviness and which moments even exist; how Extreme Value Theory delivers two universal tail families — GEV for block maxima, GPD for peaks-over-threshold — both governed by the shape parameter ξ; why expected shortfall sees the tail depth that VaR is blind to, and why it’s coherent where VaR isn’t; how copulas separate marginals from dependence and why a t copula captures the crash clustering a Gaussian copula structurally can’t; and how stress testing probes the unprecedented that no fitted model contains. No formula sheet, no hints, no take-backs: every answer locks the instant you submit, the wrong options are the exact traps that fool real desks, and your score stays hidden until the end.
Big picture
Extreme Value & Tails — the whole ladder
- Extreme Value & Tails
- Fat tails vs Gaussian
- Normal tail decays like exp(−x²/2): too fast
- Kurtosis (4th moment) sniffs out fat tails
- Matching VaR ≠ matching the tail
- Power laws & tail index
- P(X > x) ≈ C·x^(−α); smaller α = fatter
- k-th moment finite only if α > k
- Hill estimator reads α; log-log line of slope −α
- Extreme Value Theory
- Block maxima → GEV
- Peaks-over-threshold → GPD
- Shape ξ: >0 heavy, =0 exponential, <0 bounded
- Expected shortfall
- ES = E[Loss | Loss ≥ VaR], always ≥ VaR
- Coherent; VaR can violate subadditivity
- GPD: ES/VaR ≈ 1/(1−ξ)
- Copulas & tail dependence
- Sklar: joint = marginals + copula
- Gaussian copula λ_L = 0; t copula λ_L > 0
- Same correlation, different crash clustering
- Stress testing
- Scenario, historical, sensitivity, reverse
- Reverse: solve backwards from ruin
- Fuse with EVT/copulas; honest limits
- Fat tails vs Gaussian
How this exam works
This is a graded exam. Questions arrive one at a time. Once you submit an answer it is final — there is no going back, no second try, and a wrong answer simply fails that question. Your score stays hidden until the very end, where you need 70% to pass. Read every option before you commit.
Why does a normal distribution assign absurdly tiny probabilities to extreme moves?
Select an answer to continue.
Passed? Here's what you now own
You can read a tail the way a risk practitioner does: recognize when the Gaussian’s exp(−x²/2) tail is lying, sniff out fat tails with kurtosis, read a tail index α off a log-log line or a Hill plot, fit a GPD via peaks-over-threshold and extrapolate a far-tail VaR, replace VaR with the coherent, tail-aware expected shortfall, capture crash clustering with a t copula instead of a blind Gaussian one, and probe the unprecedented with scenario and reverse stress tests — all while knowing exactly where each tool fails.
That’s the extreme-value-and-tails toolkit, end to end — the math of the part of the distribution that decides who is solvent after the crisis. The middle of the distribution was never the problem. You now own the part that is.