Structured credit and market risk

Regime-Switching CPDO Risk Model

A simulation of a composite Constant Proportion Debt Obligation referencing Siemens, Allianz and LVMH five-year CDS. The framework combines empirical regime calibration, jump diffusion and dynamic leverage.

Reference portfolio3 CDS names

Simulation horizon5 years

Leverage range0–25×

Spread simulation3,000 paths

Executive finding

Combined stress exposes the leverage feedback loop.

The baseline default probability is 0.33%, consistent with a BBB estimate under the project’s internal rating map. When spread widening, volatility, jump intensity and crisis persistence occur together, default probability rises to 17.33% and the indicative rating falls to CCC.

Baseline PD0.33%BBB estimate

Combined-stress PD17.33%CCC estimate

Risk amplification52×relative to baseline

Project mandate

Design a CPDO, estimate its default probability and challenge the rating assumptions.

The model treats the product as a bank-structured credit vehicle. It defines the reference portfolio, cash flows, leverage rule, capital barriers and rating logic, then tests the structure against simulated CDS paths.

Build the product

Specify the CDS portfolio, leverage limits, funding terms and cash-in and cash-out conditions.

Model the market

Estimate credit-spread dynamics from observed normal and crisis periods.

Measure default risk

Simulate capital paths and map barrier breaches to an indicative rating.

Test model risk

Assess how the rating changes when several adverse drivers occur at the same time.

Data and regime classification

Six CDS histories anchor calm and crisis behaviour.

The calibration combines Siemens, Allianz and LVMH CDS data from 2001–2010 and 2020–2025. A day enters the crisis regime when either the spread level or 21-day realised volatility exceeds its rolling 252-day 90th percentile. The classification is lagged one day to avoid look-ahead bias.

Rolling 21-day CDS volatility for Siemens, Allianz and LVMH across recent and historical windows, with entity-specific thresholds.

Spread model

Mean reversion, regimes and jumps are estimated separately.

Regime-specific Ornstein–Uhlenbeck parameters capture mean reversion. A hybrid jump filter identifies extreme returns, while Gaussian mixture models estimate jump size and frequency. An empirical Markov matrix governs movement between normal and crisis states.

Empirical transition matrix

From / to	Normal	Crisis
Normal	95.00%	5.00%
Crisis	10.96%	89.04%

Weighted jump correlation: 0.3263
Normal jump frequency: 33.60 / year
Crisis jump frequency: 49.06 / year
Composite duration: 4.7383 years

Distribution of the fraction of simulated days spent in the crisis regime. — Crisis exposure varies materially across simulated paths.

Average share of simulations in the crisis regime across the five-year horizon. — The simulated crisis share stabilises near 30% after the initial transition.

Sample simulated composite CDS spread paths over 1,260 trading days. — Jump events create occasional sharp spread dislocations.

Normal and crisis regime return distributions compared with a Gaussian benchmark. — The crisis distribution carries substantially heavier tails than the Gaussian benchmark.

Technical analysis

The capital engine links market states directly to leverage and P&L.

The framework separates spread generation from product mechanics. Regime and jump parameters produce CDS paths. Those paths then drive leverage, carry, mark-to-market losses, funding cost and barrier events.

Regime-specific spread returnΔXₜ = θᵣ(μᵣ − Xₜ)Δt + σᵣ√Δt·εₜ + Jₜ

OU parameters vary between normal and crisis regimes. The jump term uses regime-specific frequency and size estimates.

Target leverageL*ₜ = sₜ / (γᵣ · σ²ₜ)

Higher spreads increase target exposure. Higher variance or a larger regime-specific gamma reduces it. Leverage remains between 0× and 25×.

Mark-to-market impactMtMₜ = −0.5 · Dₜ · Δsₜ · Lₜ · Cₜ₋₁

Spread widening reduces capital. Effective duration increases with the spread level but stays within the configured floor and cap.

Capital updateCₜ = Cₜ₋₁ + carry + MtM + yield − funding − costs

The path terminates at the cash-in target, the capital floor or five-year maturity.

Normal gamma1.4721

Crisis gamma2.0202

Rebalancing threshold20%

Trading days1,260

Composite CPDO design

Leverage responds to spreads, volatility and remaining capital.

The vehicle sells CDS protection to earn carry. When spreads move or volatility changes, the model recalculates target leverage. The daily capital account includes carry, mark-to-market impact, funding cost, capital yield and transaction cost.

1Cash accountInitial capital of 100 earns the capital yield.

2CDS portfolioThree corporate CDS positions generate spread carry.

3Daily leverage ruleExposure changes with spread, variance and regime.

4Capital triggerCash in at 110 or breach the floor at 50.

Parameter	Setting	Purpose
Initial leverage	7×	Starting CDS exposure
Maximum leverage	25×	Upper exposure constraint
Cash-in target	110	Successful early termination
Capital floor	50	Default trigger
Barrier trigger	55	20% leverage reduction near the floor
Coupon rate	6.0%	Funding outflow
Capital yield	3.0%	Return on the cash account
Transaction cost	3 bps	Cost of leverage changes
Rebalancing band	20%	Limits unnecessary turnover

Stress testing

Spread widening and combined stress drive the largest rating deterioration.

Each scenario changes one or more calibrated inputs and reruns the spread and capital simulation. The combined case applies broader spread widening, higher volatility, greater jump intensity and more persistent crisis dynamics.

Scenario	Probability of default	Capital breaches	Cash-in events	Indicative rating
Baseline	0.33%	1	0	BBB
Spread widening	8.33%	25	6	B
High volatility	0.67%	2	0	BBB
Higher jump risk	0.00%	0	0	AAA
Greater crisis persistence	0.33%	1	0	BBB
Combined stress	17.33%	52	16	CCC

Interpretation

Isolated jump risk has little effect in this run. Losses become nonlinear when wider spreads, volatility and persistent crisis conditions interact with the leverage rule.

Rating methodology comparison

The main difference lies in how each framework treats persistence and tail risk.

Pre-crisis rating methods relied heavily on mean reversion, low correlations and calm calibration periods. The project framework introduces explicit crisis regimes and empirical jumps, then tests several risk drivers together.

Model dimension	Moody’s pre-crisis approach	S&P pre-crisis approach	Project framework
Spread process	Mean-reverting CEV process with capped volatility	Mean reversion with low spread volatility	Regime-switching OU process in log space
Tail events	No explicit discrete jump process	Extreme spread moves materially understated	Hybrid jump detection, Gaussian mixtures and Student-t diffusion
Dependence	One-factor Gaussian copula with low fixed correlation	Low correlation within a separate default model	Empirical jump correlation and common regime transitions
Calibration	Primarily calm 2003–2006 investment-grade data	Historical inputs with benign volatility assumptions	Three CDS names across 2001–2010 and 2020–2025
Leverage mechanics	Trading rule and NAV simulated within the rating model	CPDO NAV and cash-out probability simulated	Variance-sensitive leverage with cap, barrier de-risking and transaction costs
Rating measure	Expected loss compared with an Aaa threshold	Cash-out probability compared with rating thresholds	Simulated capital-floor breach rate mapped to indicative rating bands

Model validation

The technical result needs stronger statistical controls before production use.

Stationarity

Most regime-level ADF tests do not reject a unit root at the 5% level. This weakens the empirical support for a stationary OU specification.

Parameter stability

Crisis gamma estimates differ sharply between the two historical windows. The weighted estimate reduces that instability but does not eliminate it.

Monte Carlo precision

Rare-event ratings require larger path counts, fixed seeds and confidence intervals. A small number of barrier breaches creates wide sampling uncertainty.

Backtesting

The framework needs out-of-sample spread and capital-path tests, together with benchmark comparisons against simpler specifications.

Dependence

Average jump correlation cannot fully represent state-dependent contagion or joint spread-default dynamics during a systemic event.

Implementation controls

Scenario inputs should remain immutable across runs, with independent validation of leverage, barrier and rating logic.

Model-risk lessons

Calm-period assumptions can support ratings that fail under persistent stress.

The historical CPDO experience shows how mean reversion, correlation and leverage assumptions can suppress estimated tail risk. A credible rating framework must allow adverse market drivers to reinforce one another.

Mean reversion

Fast reversion pulls spreads back too quickly and understates prolonged credit deterioration.

Correlation

Low fixed correlations miss the rise in systemic co-movement during market stress.

Tail behaviour

Gaussian spread changes and volatility caps reduce the frequency and severity of extreme outcomes.

Leverage feedback

Increasing exposure after losses can turn a temporary spread shock into a capital breach.

Joint dynamics

Separate spread and default models fail to capture the dependence that emerges in a crisis.

Validation

Independent code review, fixed seeds and transparent scenarios are necessary before assigning a rating.

Methodology and code

The complete notebook and model documentation are available on GitHub.

The repository contains the regime classification, OU calibration, jump estimation, spread simulation, CPDO capital engine and stress tests. Review the model →