Basic Prerequisites: VaR & Expected Shortfall and Stress Testing & Scenario Analysis. (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)
1. Intuition & Practical Objective
Every risk measure in this pillar is idiosyncratic: it asks “how much could this one desk / this one bank lose?” Systemic risk is the opposite question, and it is not a bigger version of VaR. It asks: how much damage can the interaction of institutions do to the system - and how does the system’s distress feed back onto each of its members?
Two ideas that are often conflated:
- Systemic risk - the risk that the functioning of the financial system itself is impaired (BIS: a disruption that severs credit supply and breaks the payments/clearing machinery, not merely an unusual loss at one firm).
- Risk aggregation - the problem of how a single institution (or a supervisor) combines market + credit + liquidity + operational losses into one number, despite the fact that the four risk types are measured in different units, on different horizons, and are tail-dependent in a way you cannot observe.
This folder is the hub: it (a) gives the fast formula & measure lookup below, and (b) routes to six sub-pages that walk from raw intuition, through network contagion and the CoVaR/MES/SRISK measures, to the (im)possibility of cross-type aggregation and its macroprudential practice.
The one-sentence essence. “A system’s risk is not the sum of its members’ risks: it is an emergent property of the network and of common tail dependence, and it has no correct formula - only better and worse modelling choices, all of which fail in dangerous directions.”
2. Mathematical Ground Truth & Derivations
Quick-Reference Lookup (job #1). All formulas below were re-executed (stdlib Python) in this folder’s sub-pages; the check column reports the exact reproduced numbers. Notation: = system loss, = firm loss, firm return, market/system return, confidence (used as a tail probability, e.g. ).
| Quantity | Formula | Verified check (§3 of each page) |
|---|---|---|
| MES (marginal expected shortfall) | for bank 1 | |
| CoVaR (system-tests-firm) | ||
| ΔCoVaR (marginal contribution) | ||
| SRISK (capital shortfall) | - (conceptual, §3) | |
| ES of sum (Gaussian, corr ) | naive sum 63.97 → 45.85 / 55.65 / 63.97 as | |
| Naive aggregate = sum of ES | (subadditivity gap) | |
| Co-exceedance (tail dependence) | Gaussian 0.0013 vs t-4 0.0030 at 1% |
Corners to keep straight. ΔCoVaR is not a derivative in the calculus sense - it is a difference of two conditional quantiles (Adrian–Brunnermeier). MES conditions on the system’s tail; CoVaR conditions on the firm’s tail. The two measure different objects and answer different policy questions.
3. Computational Implementation - the measure engine
This folder’s six sub-pages each carry a runnable stdlib-only Python example (no numpy/scipy). They reproduce every number in §2 exactly. A representative aggregation engine:
4. Failure Modes & First-Principles Breakdowns
Hub signposts - the folder’s failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:
- Unmeasurable tail dependence - co-exceedance in the extreme tail can’t be estimated from historical data (you have ~zero joint tail observations), so copula choice silently dominates the answer. Gaussian copulas assume zero tail dependence and systematically understate joint crashes.
- The aggregation fallacy - there is no neutral way to add market + credit + liquidity + operational risk: summing with no correlation overstates, summing as perfectly correlated overstates in a different way, and a Gaussian/independent model understates. Every choice smuggles in an assumption.
- Procyclicality as a feedback loop - VaR/ES-based leverage and capital triggers amplify crises: stress → vol↑ → VaR↑ → forced deleveraging → asset sales → more stress. The “safe” per-firm rule makes the system less safe.
5. References
- McNeil, Frey & Embrechts, Quantitative Risk Management (2015)
- Bellini, Stress Testing and Risk Integration in Banks (2016)
- Adrian & Brunnermeier, CoVaR, AER 106(7) (2016)
- Acharya, Pedersen, Philippon & Richardson, Measuring Systemic Risk (RFS 2017)
- Brunnermeier & Pedersen, Market Liquidity and Funding Liquidity, RFS 22(6) (2009)
- BCBS, Principles for Sound Stress Testing Practices and Supervision (2009)
- Gai & Kapadia, Contagion in Financial Networks, Proc. R. Soc. A 466 (2010); Allen & Gale, Financial Contagion, JPE 108 (2000)
6. Connected Graph Bridges
- Foundational base: VaR & Expected Shortfall · Parametric, Historical & Monte Carlo VaR · EVT & Fat Tails
- Liquidity spirals (the contagion mechanism): Margin & Funding Spirals
- Counterparty contagion (daisy-chain risk): Counterparty Risk · Failure Modes
- Stress testing (the macroprudential engine): Stress Testing & Scenario Analysis · Basel & Regulation