This is the capstone. Six lessons built the risk engine from one sentence outward — what VaR actually claims about a horizon, a confidence level and a loss; how to read it off sorted history, off a normal curve, or off a Monte Carlo swarm; how losses scale with the square root of time; why Expected Shortfall stares past the cliff edge that VaR refuses to look over; and how backtesting and the Basel traffic light keep a model honest. No formula sheet, no hints, no take-backs: every answer locks the instant you submit, the wrong options are the exact traps that blow up real risk desks, and your score stays hidden until the end.
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.
A risk report states: '1-day 99% VaR = $4.2M'. What is the correct reading of that line?
Select an answer to continue.
Whatever the score reads, the chain you just stress-tested — the VaR sentence, the three estimation methods, the square-root-of-time rule, Expected Shortfall, and the backtesting that keeps the whole thing honest — is the literacy every risk manager and regulator leans on. Here is the entire topic in one glance.
Big picture
The Value-at-Risk Toolkit
- Value at Risk
- What VaR is
- The sentence: horizon + confidence + loss amount
- VaR is a quantile of the loss distribution
- It does NOT tell you how bad the tail gets
- Read '1-day 99% VaR = $4.2M' = breach on worst 1% of days
- Historical simulation
- Sort realized P&L, read the percentile directly
- No distribution assumed — fat tails baked in
- Window trade-off: stable vs. responsive
- Can't exceed worst observed day; ghost/echo effect
- Parametric VaR
- VaR = z times sigma times value
- z = 1.645 (95%), 2.326 (99%), one-tailed
- Square-root-of-time: 10-day = 3.16 times 1-day
- Two-asset VaR + diversification benefit (corr < 1)
- Normality underestimates fat tails; bad for options
- Monte Carlo VaR
- Simulate, revalue, read the percentile
- Full revaluation handles nonlinear/option books
- Error shrinks like one over root M (4x for half)
- Garbage in, garbage out — model-dependent
- Expected Shortfall
- Average loss beyond VaR (CVaR)
- ES is always greater than or equal to VaR
- Coherent: subadditive where VaR can fail
- Basel FRTB uses 97.5% Expected Shortfall
- Backtesting & limits
- Expected exceptions = (1 minus c) times N
- Kupiec test: is the exception count consistent?
- Basel traffic light: green 0-4 / yellow 5-9 / red 10+
- Fat tails, procyclicality, stress testing
- What VaR is