4.4.6 Advanced Extensions

Basic Prerequisites: 02 · Structural Model and 05 · Failure Modes.

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1. Intuition & Practical Objective

Every single-name model answers “will this firm default?” The question that actually matters for a bank or a fund is “how much do we lose when several default together?” - a joint question. This page is the launchpad from one obligor to a portfolio:

1. The one-factor (Vasicek) model - put every firm’s asset return on a single common “state of the economy” load plus an idiosyncratic shock. It yields the asymptotic loss distribution in closed form, and it is the model behind the Basel IRB capital formula.
2. The Gaussian copula - the same dependence structure expressed as a copula over default times; the market’s standard model for CDOs and basket CDS.
3. Ratings & transition matrices - the industry’s discrete, non-market alternative (CreditMetrics): a firm migrates between rating states, and PD is read off the default column.

The unifying idea. Portfolio credit risk is entirely a correlation problem. With $\rho =0$ a large portfolio is nearly default-free (diversification); with $\rho =1$ it is one giant obligor. Both the Vasicek closed form and the Basel capital formula are statements about that single number $\rho$.

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2. Mathematical Ground Truth & Derivations

2.1 The one-factor Gaussian model (Vašíček 1987/1991; Hull eq. 24.10)

Let obligor $i$‘s standardised asset return decompose as

$$X_i=\sqrt{\rho }Y+\sqrt{1-\rho }{Z}_i,Y,Z_i\stackrel{i.i.d.}{\sim }N(0,1),$$

where $Y$ is the common systematic factor and $Z_i$ the idiosyncratic shock. Obligor $i$ defaults if $X_i<N^{-1}(p_i)$ (its default threshold). Conditional on the factor $Y=x$, defaults are independent with

$$p_i(x)=N\left(\frac{N^{-1}(p_i)-\sqrt{\rho }x}{\sqrt{1-\rho }}\right).$$

For a large, homogeneous, equal-weighted portfolio (infinite granularity), the portfolio loss fraction equals the conditional default rate $L=p(x)$, so the stochastic factor becomes the loss. Inverting for the loss CDF gives Vašíček’s closed form:

$$\boxed{F(\theta )=\mathrm{Pr}[L\le \theta ]=N\left(\frac{\sqrt{1-\rho }{N}^{-1}(\theta )-N^{-1}(p)}{\sqrt{\rho }}\right)}$$

and the loss quantile at confidence $q$ (this is the Basel IRB form):

$$\boxed{{\theta }_q=N\left(\frac{N^{-1}(p)+\sqrt{\rho }{N}^{-1}(q)}{\sqrt{1-\rho }}\right).}$$

The unexpected-loss capital per unit EAD is ${\theta }_{0.999}-\mathrm{EL}$ with $\mathrm{EL}=p\mathrm{LGD}$ (set $\mathrm{LGD}=1$ above). Hull eq. (24.10) is exactly ${\theta }_q$; the Basel corporate asset correlation is a decreasing function of PD, ${\rho }_{\text{Basel}}(p)=0.12w+0.24(1-w)$, $w=(1-e^{-50p})/(1-e^{-50})$.

2.2 Credit ratings & transition matrices (CreditMetrics 1997; Hull §24.9)

Ratings give a discrete credit state. A one-year transition matrix $M$ collects $\mathrm{Pr}(\text{state }j\mid \text{state }i)$; the default column is the one-year PD for each rating. Because default is an absorbing state, multi-year cumulative default probabilities are read from matrix powers:

$$\mathbb{P}(\text{default by year }n\mid i)=(M^n{)}_{i,\text{Default}}.$$

CreditMetrics simulates each obligor’s asset return against rating-specific thresholds (calibrated so the threshold probabilities reproduce the matrix), correlates the returns with a factor model, and reads the portfolio loss distribution off the simulated rating migrations. It is the discrete, simulation-based sibling of the Vasicek formula.

2.3 CDO tranching and the correlation smile

A CDO splits the portfolio loss into tranches (equity $[0,a]$, mezzanine $[a,b]$, senior $[b,1]$). Each tranche’s value depends on $\mathrm{Pr}[L>a]$ etc., i.e. entirely on $\rho$: as $\rho \uparrow$, more mass moves to both tails, making junior tranches safer and senior tranches riskier (Hull §25.9). Pricing each tranche implies a different $\rho$ - the correlation smile/skew - which is a direct sign that the Gaussian copula is a calibration device, not a structural description.

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3. Computational Implementation - Vasicek closed form vs Monte Carlo, and the transition matrix

A. Vasicek asymptotic loss distribution, verified against a finite-$n$ Monte Carlo. Stdlib only.

The Monte Carlo mean recovers the input PD ($1.997\%\approx 2\%$), and its $99.9\%$ quantile ($18.3\%$) is close to but slightly above the asymptotic Vasicek value ($17.6\%$) - the small granularity premium of a finite 1000-name portfolio. Note the scale of the risk: a $2\%$ PD portfolio loses $17.6\%$ of EAD at the $99.9\%$ level - correlation, not the mean, is what makes the capital.

B. Cumulative default from a rating transition matrix (CreditMetrics / S&P). Stdlib only.

The one-year numbers reproduce the matrix’s default column exactly (BBB $0.18\%$, BB $1.06\%$, B $5.20\%$, CCC $19.79\%$); matrix powers give the multi-year cumulative default rates (BBB $\to 6.6\%$ over ten years). This is the discrete, ratings-based route to PD - the CreditMetrics alternative to the continuous distance-to-default of page 03.

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4. Failure Modes & First-Principles Breakdowns

1. Correlation is the whole game and the least reliable input. $\rho$ drives the $99.9\%$ quantile almost entirely (with $p$ fixed, $\rho$ from $0.15$ to $0.30$ roughly doubles senior-tranche risk), yet asset correlation is estimated with wide error bands and varies through the cycle.
2. The Gaussian copula has zero tail dependence. Two firms’ default times are asymptotically independent in a Gaussian copula, so it understates joint extreme default clusters - the 2008 failure mode. Student-$t$ and other copulas with tail dependence are the standard repair, but the correlation is then no longer a single number.
3. Infinite granularity is an idealisation. The Vasicek closed form assumes a large homogeneous portfolio; concentrated or small portfolios need the finite-$n$ distribution (the MC above shows the size of the granularity gap).
4. Transition matrices are historical and through-the-cycle. They ignore the current state of the economy, contain rating-migration serial dependence, and their default column is estimated from a finite history - applying the matrix unchanged in a crisis understates default (Hull §24.9; CreditMetrics §6 warns against naive matrix multiplication over long horizons).
5. Ratings are coarse and lag. A rating move is a lagging, bucketed proxy for continuous changes in asset value and spread - the discrete model throws away exactly the information DD exploits.

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5. References

• Vašíček, Oldřich - Probability of Loss on Loan Portfolio (KMV, 1987) and Limiting Loan Loss Probability Distribution (1991): the one-factor conditional PD and the asymptotic loss CDF.
• Hull, Options, Futures, and Other Derivatives
• Gupton, Finger & Bhatia (J.P. Morgan) - *CreditMetrics™
• Bluhm, Overbeck & Wagner - Introduction to Credit Risk Modeling
• BCBS - Basel III: Finalising Post-Crisis Reforms (2017)

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6. Connected Graph Bridges

• Back: 05 · Failure Modes · 04 · Reduced-Form & CDS · Index Hub
• Siblings: VaR & Expected Shortfall (the quantile ${\theta }_q$ is a credit VaR) · EVT & Fat Tails (the joint-tail failure of the Gaussian copula) · Stress Testing & Scenario Analysis
• Base: Black–Scholes–Merton · Index Hub