5-Year CDO Tranche Default Probability Analysis

CDX.NA.IG · 10 Companies · Independent Defaults (ρ = 0) · Hull Table 24.1 · No Copula

1. Portfolio Setup

10 Randomly Selected CDX.NA.IG Companies

#	Company	Ticker	S&P Rating	Hull 24.1 — 5-yr Cum. PD (%)	Weight

* Ratings reflect current S&P long-term issuer ratings at time of selection from CDX.NA.IG series.

Model Assumptions

Recovery rate $R = 40\%$ → Loss Given Default $\text{LGD} = 60\%$
Each company has equal weight = $\frac{1}{10} = 10\%$ of portfolio notional
Loss per default: $10\% \times 60\% = \mathbf{6\%}$ of total portfolio notional
Default correlation: $\rho = 0$ (all defaults independent)
Method: Poisson-Binomial distribution via dynamic programming (no copula)
Horizon: 5 years

2. Hull Table 24.1 — Average Cumulative Default Rates (%)

Source: Hull, Options, Futures, and Other Derivatives, Table 24.1 (S&P historical data)

Rating	1 yr	2 yr	3 yr	4 yr	5 yr ← used
AAA	0.000	0.013	0.013	0.037	0.104
AA	0.022	0.068	0.136	0.260	0.410
A	0.062	0.199	0.434	0.679	0.958
BBB	0.174	0.504	0.906	1.373	1.862
BB	1.110	3.071	5.371	7.839	10.065
B	5.187	11.388	17.022	21.281	24.994
CCC	19.476	30.494	39.717	46.904	52.622

3. CDO Tranche Structure

The portfolio notional is divided into four tranches. Losses are absorbed from the bottom up.

Equity
0–3%

Mezz 1
3–6%

Mezz 2
6–10%

Senior 10–100%

Tranche	Attachment	Detachment	Width	Loss-per-default to reach attachment
■ Equity	0%	3%	3%	$>0$ defaults (any loss)
■ Mezzanine 1	3%	6%	3%	$>3\%$ portfolio loss → $k \ge 1$
■ Mezzanine 2	6%	10%	4%	$>6\%$ portfolio loss → $k \ge 2$
■ Senior	10%	100%	90%	$>10\%$ portfolio loss → $k \ge 2$

Key Observation: With loss-per-default = 6%, a single default ($k=1$) produces $L = 6\%$, which simultaneously and completely wipes out both the Equity (0–3%) and Mezzanine 1 (3–6%) tranches. The first default jumps directly over the entire $[0\%,6\%]$ loss region. This is the jump-to-default effect in discrete equal-weight portfolios.

Loss Mapping (k defaults → portfolio loss → tranche loss fraction)

4. Methodology: Poisson-Binomial Distribution

Let $p_i$ be the 5-year default probability of company $i$. Since defaults are mutually independent, the number of defaults $K$ follows a Poisson-Binomial distribution:

$$P(K = k) = \!\!\sum_{\substack{S \subseteq \{1,\ldots,n\} \\ |S|=k}} \prod_{i \in S} p_i \prod_{j \notin S}(1-p_j)$$

This generalises the binomial to allow heterogeneous $p_i$ values. For $n = 10$ companies and at most 10 defaults, it is computed exactly by dynamic programming:

Initialise: $\text{dp}[0] = 1,\quad \text{dp}[k] = 0 \ (k>0)$

For each company $i = 1, \ldots, n$:

$$\text{dp}[k] \;\leftarrow\; (1-p_i)\cdot\text{dp}[k] \;+\; p_i\cdot\text{dp}[k-1], \quad k = n,\ldots,1$$ $$\text{dp}[0] \;\leftarrow\; (1-p_i)\cdot\text{dp}[0]$$

Tranche loss fraction for portfolio loss $L$ and tranche $[\alpha, \beta]$:

$$\ell(\alpha,\beta;L) = \frac{\min\bigl(\max(L-\alpha,\,0),\;\beta-\alpha\bigr)}{\beta - \alpha}$$

Expected tranche loss and tranche default probability:

$$\text{EL}_{\text{tranche}} = \sum_{k=0}^{n} P(K=k)\cdot \ell\!\left(\alpha,\beta;\,k \cdot \text{LPD}\right)$$ $$P_{\text{default}} = \sum_{k \,:\, \ell(\alpha,\beta;\,k\cdot\text{LPD})>0} P(K=k)$$

where LPD = loss per default = 6%.

5. Default Count Distribution $P(K = k)$

6. Tranche Loss Analysis

Tranche 1: Equity [0%–3%]

Portfolio loss $L = k \times 6\%$.
Equity suffers loss whenever $L > 0\%$, i.e. $k \ge 1$.
Since $k=1 \Rightarrow L=6\% \ge 3\%$, a single default fully wipes out the equity tranche. No partial loss is possible: the equity tranche is either intact ($k=0$) or 100% lost ($k\ge 1$).

$$P_{\text{default}}^{\text{Eq}} = P(K \ge 1) = 1 - P(K=0)$$ $$\text{EL}^{\text{Eq}} = P(K \ge 1) \times 100\% = P_{\text{default}}^{\text{Eq}}$$

Tranche 2: Mezzanine 1 [3%–6%]

Mezzanine 1 absorbs losses between 3% and 6%.
With LPD = 6%, the first default delivers exactly $L = 6\%$, which hits the full 3% width of this tranche.
Therefore Mezzanine 1 has identical default probability and expected loss as the Equity tranche.
This is a structural consequence of the discrete jump from $L=0\%$ to $L=6\%$.

$$P_{\text{default}}^{\text{M1}} = P(K \ge 1) \quad \text{(same as Equity)}$$ $$\text{EL}^{\text{M1}} = P(K \ge 1) \times 100\%$$

Tranche 3: Mezzanine 2 [6%–10%]

Mezzanine 2 absorbs losses between 6% and 10%.
$k=1 \Rightarrow L=6\%$: loss reaches exactly the attachment point — no loss to this tranche.
$k=2 \Rightarrow L=12\% > 10\%$: Mezzanine 2 is fully wiped out.
Therefore this tranche is either intact ($k \le 1$) or 100% lost ($k \ge 2$). No partial loss possible.

$$P_{\text{default}}^{\text{M2}} = P(K \ge 2) = 1 - P(K=0) - P(K=1)$$ $$\text{EL}^{\text{M2}} = P(K \ge 2) \times 100\%$$

7. Results Summary

Tranche	Attachment	Detachment	Width	Threshold $k$	Default Prob. $P_{\text{def}}$	Expected Loss (EL)	WA Credit Rating ≈

8. Key Observations & Insights

Jump-to-Default Effect: With equal weights and $R=40\%$, each default creates a 6% portfolio loss. This means losses jump discontinuously through the tranche boundaries. No tranche experiences a partial loss — each is either fully intact or fully wiped out. This is a direct consequence of the granularity of the portfolio (only 10 names).
Equity ≡ Mezzanine 1 Risk: Both tranches share identical default probability and expected loss. An investor in Mezzanine 1 receives no extra subordination benefit from the equity tranche — the first default eliminates both simultaneously. This is structurally unusual and highlights the limitations of coarse portfolios in CDO design.
Mezzanine 2 is protected by two defaults: The 6%-per-default granularity means Mezzanine 2 can only be harmed if at least 2 companies default. With investment-grade names (avg PD ≈ 1.3%), this joint probability is very small.
Independence Assumption (ρ = 0): Real CDO tranches exhibit much higher correlation via the Gaussian copula or factor models. Under ρ = 0, the default distribution is highly concentrated near 0 defaults, severely understating tail risk. The senior tranche in reality faces much more risk than the independent-default model suggests.
Comparison with Copula: A one-factor Gaussian copula with $\rho > 0$ would shift probability mass toward both very few and very many defaults (fat tails), substantially increasing the default probability of Mezzanine 2 and the Senior tranche. The 0-correlation result here therefore represents a lower bound on tranche risk for the upper tranches and an upper bound on risk for the equity tranche (relative to the correlated case).