Stress testing and scenario aggregation

Definition A stress test fixes a portfolio, a valuation date and a horizon, and computes the loss

of the portfolio under one or more specified scenarios $s$. Here $V(s)$ is the value of the portfolio, in currency units, after its risk factors change by $s$, and $V(0)$ is its current value. The loss is positive for an adverse outcome, so a value change $\Delta V = V(s) - V(0)$ corresponds to the loss $-\Delta V$. Positions are held fixed during the scenario unless the test states a rule for trading, collateral or contract events. A stress test needs a value function and a scenario, and no probability. It reports what the portfolio would lose if the stated state occurred, not how likely that state is, whereas a quantile such as value at risk needs a distribution over states. Historical scenarios replay the factor moves of a past period on today’s positions, hypothetical scenarios are chosen to probe a weakness, and a sensitivity test is a scenario that moves one factor alone.

Definition A scenario vector $s = (s_1, \dots, s_d)^{\top}$ lists one change for each of the $d$ risk factors that the valuation uses, such as prices, rates, implied volatilities, credit spreads and exchange rates. Each component has stated units, for example basis points, log returns or absolute changes in volatility points, and a stated type, absolute or relative. A complete scenario also fixes the dates, the horizon, the portfolio, and the liquidity, collateral and contract-state assumptions needed to revalue it, such as whether a barrier has been crossed. The name of a historical crisis does not supply these coordinates by itself. An admissible set $S \subseteq \mathbb{R}^d$, written as bounds or joint restrictions on $s$, records which vectors the test treats as possible. A scenario carries no probability unless a distribution over scenario vectors is specified separately.

Definition Nonlinear revaluation, also called full revaluation, computes $L(s) = V(0) - V(s)$ by pricing every position again at the shocked factor values with its pricing model, instead of approximating the change. A sensitivity approximation uses a Taylor expansion at the current factors. If $V$ is twice continuously differentiable near $0$, with gradient $g = \nabla V(0)$ and Hessian $\Gamma = \nabla^2 V(0)$, then

with $R(s)/\lVert s \rVert^2 \to 0$ as $s \to 0$. The first-order loss $-g^{\top} s$ and the second-order loss $-g^{\top} s - \tfrac12 s^{\top}\Gamma s$ are approximations whose error depends on the size of $s$, and stress scenarios are large by design. Positions with nonlinear payoffs, such as options, callable bonds and barrier products, can have a large remainder exactly in the scenarios that matter. When $V$ is a quadratic polynomial, $R = 0$ and the second-order expansion is exact.

Definition A joint stress scenario moves several risk factors at once, and the portfolio is revalued once at the full vector $s$. Its components should be jointly meaningful, meaning that the moves can occur together under the stated narrative or model, for example because of an economic constraint, a funding link or an assumed dependence. For two factors and a twice continuously differentiable $V$, the second-order loss at $s = (x, y)^{\top}$ contains the cross term $-\Gamma_{12}\,xy$, which depends on both moves and is missed by every scenario that moves only one factor. The dependence that a joint scenario expresses is an assumption of the test design. It is an estimate of a correlation only if a model was fitted to data.

Definition Scenario aggregation combines scenario results into portfolio or summary figures, and each rule has its own hypotheses. Across positions, the portfolio value is the sum of the position values, $V = \sum_i V_i$, so under one common scenario $L(s) = \sum_i L_i(s)$ exactly. Adding losses of positions computed under different scenarios has no such meaning. Across factors, the sum of isolated one-factor losses equals the joint loss for every scenario exactly when $V$ is additively separable, $V(x, y) = f(x) + h(y)$. For a twice continuously differentiable $V$ with mixed partial derivative $V_{12} = \partial^2 V/\partial x\,\partial y$, the gap is

Across scenarios, the worst-case loss $\max_{s \in S} L(s)$ over a finite set $S$ summarizes the set without probabilities. Viewed as a function of the loss profile on $S$, it is monotone, positively homogeneous and subadditive, and adding a cash amount $m$ to the portfolio lowers it by $m$, so it is a coherent risk measure in the sense of Expected shortfall and coherent risk, with the scenarios in $S$ as the states. A probability-weighted average or a quantile over scenarios needs a distribution on $S$, which the scenarios themselves do not provide.

Definition A reverse stress test fixes an unacceptable loss $\ell > 0$ and searches for the scenarios that reach it. To single one out, it needs an admissible set $S$ and a cost $c(s)$ that measures how extreme a scenario is, and it solves

Common costs are the squared Euclidean norm $s^{\top}s$, a covariance-weighted norm $s^{\top}\Sigma^{-1}s$ for an invertible covariance matrix $\Sigma$ of factor changes, and penalties on the number of factors moved. Each gives a different geometry. A low cost does not make a scenario likely unless a distribution is specified. For normally distributed factor changes with mean $0$ and invertible $\Sigma$, the covariance-weighted cost orders scenarios by their density, and when $\Sigma$ is singular the scenarios must be restricted to its range, where a pseudo-inverse defines the cost. For a linear surrogate loss $a^{\top}s$ with $a \ne 0$, $S = \mathbb{R}^d$ and the Euclidean cost, the Cauchy-Schwarz inequality $a^{\top}s \le \lVert a \rVert\,\lVert s \rVert$ shows that $a^{\top}s \ge \ell$ forces $\lVert s \rVert \ge \ell/\lVert a \rVert$, with equality only at

so this is the unique minimizer. A scenario found with a surrogate has to be revalued with the full model. If $V$ is convex, the feasible set $\{s : V(s) \le V(0) - \ell\}$ is convex, so the full problem with Euclidean cost and $S = \mathbb{R}^d$ has a unique solution whenever it is feasible, and the largest attainable loss is $V(0) - \min V$ when that minimum exists.