Unit 11 of 14

Risk measurement and model uncertainty

  • 8 lessons
  • Videos 81 to 88

Risk measures summarize the losses a portfolio may suffer. This unit covers attribution, value at risk, expected shortfall, extreme values, stress tests, liquidity, backtesting and model uncertainty.

A Taylor expansion splits a change in value into carry, market moves and a residual. Value at risk is a loss quantile under a stated convention and can fail to be subadditive, while expected shortfall, a quantile integral, is a coherent risk measure. Extreme value theory extrapolates tail quantiles from threshold excesses, with explicit uncertainty. Stress tests revalue a portfolio under joint scenarios, liquidation adds spread, impact and horizon risk, and backtests check exception counts and their timing. Parameter uncertainty and model ambiguity give ranges of values rather than one price.

Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Attribution and tail risk measures

  1. Profit and loss attribution and sensitivities

    How a Taylor expansion splits a change in portfolio value into market move terms from sensitivities, carry from the passage of time and a residual, and why an attribution convention decides where joint effects go.

    Video 81

  2. Value at risk and quantile estimation

    Defines value at risk as a lower quantile of the loss distribution, the empirical quantile convention behind historical VaR, an exact failure of subadditivity and the sampling uncertainty of quantile estimates.

    Video 82

  3. Expected shortfall and coherent risk

    Defines expected shortfall as a quantile integral that handles atoms, derives its threshold and dual envelope representations, proves that it is a coherent risk measure and explains why tail estimates remain uncertain.

    Video 83

  4. Extreme value theory and tail models

    How the maximum domain of attraction and the Pickands-Balkema-de Haan theorem lead to generalized Pareto tails, and how the tail index, high quantiles, expected shortfall and threshold stability follow.

    Video 84

Stress testing and liquidity risk

  1. Stress testing and scenario aggregation

    How a stress test revalues a portfolio under a scenario vector, why a joint stress scenario needs nonlinear revaluation, when scenario aggregation is exact, and how a reverse stress test finds the least extreme scenario.

    Video 85

  2. Liquidity risk and closeout horizons

    How a liquidation path turns a position into spread cost, liquidation impact cost and inventory variance, how the liquidation horizon balances them, and what a liquidity adjustment to a mark or a risk figure measures.

    Video 86

Backtesting and model uncertainty

  1. Risk backtesting and statistical tests

    How VaR exceptions become Bernoulli observations, why their count is binomial, and how the unconditional coverage, independence and conditional coverage tests are derived and read.

    Video 87

  2. Model uncertainty and robust valuation

    How parameter uncertainty differs from model ambiguity, why positive vega makes a volatility envelope exact at its endpoints, what the dual norm bound gives locally, and how robust valuation bounds are defined.

    Video 88