Methodology

Methodology

Detailed description of the modeling approach.

Problem Formulation

Given a portfolio of loans with features X, we aim to generate:

  1. Macro paths M_{1:T}: Correlated macroeconomic time series
  2. State sequences S_{1:T}: Loan state trajectories
  3. Continuous values V_{1:T}: Payments, balances, losses
  4. Portfolio outcomes L: Loss distribution, tranche returns

Hierarchical Generative Model

Level 1: Macro Scenario Generation

We use a Conditional Variational Autoencoder (CVAE) to generate macro paths.

Encoder:

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q_phi(z | M, s) = N(mu_phi(M, s), sigma_phi(M, s))

Decoder:

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p_theta(M | z, s) = Prod_t N(m_t | g_theta(z, s, m_{<t}))

Loss:

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L_macro = E_q[||M - M_hat||^2] + beta * KL(q_phi || p(z))

The scenario label s enables conditional generation under different economic regimes.

Level 2: Cohort Transitions

A Transformer encoder predicts time-varying transition matrices:

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P(t) = f_psi(cohort_features, M_{1:t})

Cross-attention allows the macro path to modulate transition dynamics:

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Attention(Q_cohort, K_macro, V_macro)

Loss:

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L_trans = -sum_{t,i,j} n_ij(t) * log(P_ij(t))

Level 3: Loan Trajectories

An autoregressive transformer generates individual loan paths:

State Generation:

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s_t ~ Cat(softmax(h_t * W_s + P_cohort(s_{t-1}, :)))

Continuous Values (Diffusion):

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v_t = Denoise(epsilon_t, s_t, x_loan)

The diffusion head captures complex distributions of payments and recoveries.

Level 4: Portfolio Aggregation

Cashflows are aggregated and distributed via waterfall rules:

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Collections = sum_i (payment_i)
Losses = sum_i (loss_i)
Tranche_CF = Waterfall(Collections, Losses, Rules)

Differentiable via soft approximations:

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gate(x, threshold) = sigmoid((x - threshold) / temperature)

Correlation Structure

Correlation is induced at multiple levels:

Source Mechanism Magnitude
Macro Shared macro path ~60% of total
Cohort Vintage/asset class grouping ~25% of total
Factor Latent industry/geography ~10% of total
Idiosyncratic Diffusion noise ~5% of total

The hierarchical structure naturally captures:

  • Systematic risk: All loans exposed to same macro
  • Concentrated risk: Cohort-level clustering
  • Diversification: Residual idiosyncratic variation

Training Strategy

Stage 1: Component Pre-training

Train each component separately:

Component Objective Data
Macro VAE Reconstruction + KL Historical macro series
Transition Transformer Cross-entropy on transitions Cohort transition counts
Loan Trajectory State + diffusion loss Loan-month panel

Stage 2: End-to-End Fine-tuning

Joint training with portfolio objectives:

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L_total = L_macro + L_trans + L_traj + lambda * L_portfolio

Where L_portfolio matches:

  • Historical loss rates
  • Tranche return distributions
  • Tail risk measures

Stage 3: Calibration

Final calibration to match:

  • Observed default rates by cohort
  • Historical macro correlations
  • Recovery rate distributions

Scenario Conditioning

Standard Scenarios

Scenario GDP Shift Unemp. Shift Spread Mult.
Baseline 0% 0% 1.0x
Adverse -3% +3% 2.0x
Severely Adverse -6% +8% 4.0x
Stagflation -2% +3% 2.5x

Custom Conditioning

Condition on specific outcomes:

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model.generate_conditional({
    'gdp_growth_yoy': {'month': 12, 'value': -0.04},
    'unemployment_rate': {'month': 24, 'value': 0.10}
})

Evaluation Metrics

Generation Quality

Metric Target
Macro reconstruction RMSE < 0.5%
Transition accuracy > 85%
State sequence accuracy > 80%
Payment RMSE < $100

Portfolio Metrics

Metric Validation
Expected Loss Within 10% of historical
VaR 99% Conservative vs historical
Scenario ordering Severe > Adverse > Baseline
Tranche attachment Consistent with ratings

Computational Considerations

Memory

Component Memory (10k loans, 60 months)
Macro VAE ~100 MB
Transition Transformer ~500 MB
Loan Trajectory ~2 GB
Full simulation (10k sims) ~8 GB

Runtime

Operation Time (GPU)
Train Macro VAE (200 epochs) ~10 min
Train Transitions (100 epochs) ~30 min
Train Trajectories (50 epochs) ~2 hours
Monte Carlo (10k sims) ~5 min

Limitations

  1. Data requirements: Needs historical loan-level data
  2. Stationarity: Assumes stable regime (may need retraining)
  3. Tail estimation: Limited by simulation sample size
  4. Model risk: Deep learning opacity

Future Directions

  • Continuous-time models: Replace discrete monthly steps
  • Graph neural networks: Explicit loan relationship modeling
  • Online learning: Adapt to new data without full retraining
  • Explainability: Attribution of losses to factors

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