Estimating CDOs Expected Losses via a Monte Carlo Simulation

Default Model:
PD_i = 1 - exp(-λ_i T),   λ_i = Spread / (1 - Recovery)

Gaussian Copula:
X_i = √ρ · F + √(1-ρ) · ε_i
F, ε_i ~ N(0,1)

Default Trigger:
Default if X_i < Φ⁻¹(PD_i)

Portfolio Loss:
L = Σ (LGD_i · Notional_i · 1_{default}) / Total Notional

Tranche Loss:
L_tranche = min(max((L - A)/(D - A), 0), 1)