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 Global Ratings Research and S&P Global Market Intelligence's CreditPro®, 1981–2019)

Rating	1 yr	2 yr	3 yr	4 yr	5 yr ← used	7 yr	10 yr	15 yr
AAA	0.00	0.03	0.13	0.24	0.35	0.51	0.70	0.91
AA	0.02	0.06	0.12	0.21	0.31	0.50	0.72	1.02
A	0.05	0.14	0.23	0.35	0.47	0.79	1.24	1.89
BBB	0.16	0.45	0.78	1.17	1.58	2.33	3.32	4.69
BB	0.61	1.92	3.48	5.05	6.52	9.01	11.78	14.67
B	3.33	7.71	11.55	14.58	16.93	20.36	23.74	27.12
CCC/C	27.08	36.64	41.41	44.10	46.19	48.26	50.38	52.59

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).