Credit Models: How Default Risk Becomes a Price and a Portfolio Loss

Credit risk combines the possibility of default with the loss suffered when default occurs. A credit spread also reflects pricing premia, liquidity and contractual features. Treating any one of these quantities as a complete description of the others can distort both valuation and risk assessment.

This reference consolidates structural models, default-intensity methods, credit-derivative valuation and portfolio dependence. It is a modelling reference and contains no individual issuer recommendation or current credit-market forecast.

Expected loss is a useful starting point

For a simplified exposure, expected credit loss is

$$\operatorname{EL}=\operatorname{PD}\times\operatorname{LGD}\times\operatorname{EAD},$$

In words: Expected loss multiplies the probability of default, the share lost and the amount exposed.

where PD is default probability over a stated horizon, LGD is loss given default as a fraction, and EAD is exposure at default. The simple product assumes compatible definitions and, when quantities are random, an appropriate treatment of dependence.

For an illustrative exposure of 100 units, one-year PD of 2% and LGD of 60%, expected loss is 1.2 units. It is not the maximum loss or a tail-capital requirement. A portfolio with the same expected loss can have very different concentration and systemic risk.

Recovery can deteriorate when defaults increase. The historical teaching material emphasises this cyclicality: adverse economic conditions may simultaneously raise default frequencies and reduce recoveries. Treating PD and LGD as independent constants can understate stress losses.

Migration adds more states than default alone

A transition matrix describes probabilities of moving among credit states over a specified horizon. Rows should sum to one, and a default state is often treated as absorbing within the model. Multiplying a one-period matrix can generate longer-horizon transitions under a time-homogeneous Markov assumption.

That multiplication assumes the transition structure remains appropriate and that the current state sufficiently summarises relevant history. Ratings can lag changing conditions; different issuers within one rating category can have different risks. A transition matrix is therefore a structured approximation, not a complete credit assessment.

Changes in spread can also create mark-to-market losses without default or formal rating migration. A portfolio analysis should specify whether it measures default loss, migration loss, spread risk or a combination.

Structural models connect default to balance-sheet value

In the basic Merton model, firm asset value $A_t$ follows a diffusion and a single debt payment $D$ is due at maturity $T$. Equity payoff is

$$E_T=\max(A_T-D,0),$$

while debt payoff is $\min(A_T,D)$. Equity therefore resembles a call option on firm assets, and debt resembles risk-free debt minus a put on those assets.

The original maturity-default model defaults when $A_T<D$ at the debt maturity. A first-passage model that defaults when assets cross a barrier before maturity is a distinct extension. A diagram showing an earlier crossing should not silently change the model’s default rule.

Structural intuition is useful: more leverage or greater asset volatility can increase default risk and alter the value transferred between debt and equity. Implementation is harder because firm asset value and volatility are not directly observed in the same way as a traded equity price.

Physical default probability differs from priced default risk

Under a simple lognormal asset model, a physical-measure default probability depends on the asset drift, volatility, debt threshold and horizon. A risk-neutral default probability replaces the relevant drift according to the pricing measure and is used in valuation.

Market-implied default probabilities can exceed historical default frequencies because they include compensation for bearing risk and other priced effects. The difference is not necessarily a data error. Comparing the two requires a consistent horizon, recovery assumption, instrument and liquidity treatment.

Distance-to-default measures standardise the gap between asset value and a default threshold. They can rank modelled vulnerability, but turning that distance into a probability requires a distributional mapping. An empirical mapping and a normal-tail mapping need not give the same answer.

Intensity models specify the timing of default

Let $\tau$ be default time and $\lambda(t)$ a deterministic hazard rate. Survival probability is

$$Q(t)=\mathbb P(\tau>t)=\exp\left(-\int_0^t\lambda(u)du\right).$$

In words: Survival probability falls exponentially with the total default-arrival rate accumulated over time.

For a constant intensity $\lambda$, default probability by $T$ is $1-e^{-\lambda T}$. The intensity is an instantaneous conditional rate, not a probability that can be read directly without a horizon.

With stochastic intensity, survival involves an expectation of the exponential of integrated intensity under the appropriate measure. Replacing this with the exponential of expected intensity can miss convexity and dependence.

The approximation $s\approx\lambda(1-R)$, where $s$ is spread and $R$ recovery, is useful under simplified continuous-premium assumptions. It is not an exact universal conversion from every observed bond or CDS spread to default probability.

A CDS equates premium and protection legs

A credit default swap exchanges premium payments for protection against defined credit events. In a simplified deterministic-rate and intensity setup, the protection-leg value for notional $N$ is

$$N(1-R)\int_0^T D(t)\lambda(t)Q(t)dt.$$

For scheduled premium dates $t_i$, accrual fractions $\Delta_i$ and running spread $s$, the scheduled premium leg is approximately

$$Ns\sum_i\Delta_iD(t_i)Q(t_i),$$

with accrual-on-default and precise settlement conventions added as required. A par spread makes the two legs equal under those assumptions.

Bootstrapping an intensity curve proceeds through maturities, choosing piecewise parameters that reproduce observed par spreads while maintaining the chosen recovery and discounting conventions. It should not treat each maturity as an unrelated flat hazard estimate if the goal is a coherent survival curve.

The historical exercises include a 1,350-basis-point spread scenario and a separate contract valuation exercise dated 19 June 2003 for maturity 20 September 2007. These are retained historical teaching contexts, not live quotes. Complete valuation requires the referenced curve, coupon and recovery inputs; a spread alone is insufficient to reconstruct the original mark-to-market.

Risky annuity determines spread sensitivity

The risky premium annuity is the discounted expected accrual paid while the reference entity survives, including the relevant default-accrual treatment. Multiplying it by one basis point gives a risky PV01 under the chosen definition and notional.

For a small change in spread and approximately fixed annuity, the value change is proportional to notional, spread change and risky annuity, with sign depending on protection bought or sold. Large spread moves alter survival probabilities and the annuity itself, so full revaluation becomes important.

A distressed spread can therefore produce nonlinear sensitivity. Reporting only a linear spread duration can understate how the valuation changes as default becomes more imminent.

Portfolio credit depends on joint defaults

A basket contract can pay when the first, second or another ordered default occurs. Marginal default curves are insufficient: the joint timing of defaults determines the payoff. A copula links marginal distributions through a dependence structure, but correlation alone does not uniquely determine tail dependence.

Gaussian and heavy-tailed copulas can share similar ordinary correlations while producing different joint-extreme probabilities. Calibration to limited instruments can leave substantial model uncertainty in unobserved tranches or maturities.

For a tranche with attachment $A$, detachment $D$ and portfolio loss $L$, tranche loss is

$$\min\{\max(L-A,0),D-A\}.$$

Increasing dependence can move probability from moderate losses toward very small and very large losses, changing senior and junior tranche values differently. There is no single rule that higher correlation uniformly increases every credit product’s value.

Systematic factors create concentration beyond names

A one-factor latent-variable model represents each obligor through a shared systematic factor and an idiosyncratic component. Conditional on the common factor, defaults may be modelled as independent. Unconditionally, the shared driver creates dependence.

Granularity assumptions matter. A large homogeneous portfolio approximation is different from a concentrated book with a few major obligors. Sector, country, collateral and counterparty links can create concentrations that issuer counts alone conceal.

Counterparty exposure adds another dependency: a derivative may become more valuable to the holder precisely when its counterparty becomes less creditworthy. This wrong-way risk prevents a simple product of average exposure and average default probability from capturing the full loss distribution.

The implementation record should state default definitions, measure, recovery convention, discounting, marginal calibration, dependence structure and concentration assumptions. Credit modelling becomes useful when those choices are visible enough to explain both a price and the circumstances under which it may fail.

A constant-intensity example shows the horizon effect

Take an explicitly illustrative annual intensity of 2%. One-year default probability is $1-e^{-0.02}$, approximately 1.98%; five-year default probability is $1-e^{-0.10}$, approximately 9.52%. Multiplying the one-year probability by five gives only an approximation because default removes the name from the surviving population.

With recovery 40%, the continuous-premium spread approximation gives $0.02(1-0.40)=1.2\%$, or 120 basis points. This result assumes constant intensity, a compatible recovery convention and the simplified premium/protection timing. Real contract valuation should use its scheduled payments and accrual-on-default treatment.

If recovery is lowered while an observed spread is held fixed, the inferred intensity may fall because each default now produces a larger protection payment. That calibration relationship can look counterintuitive if one changes recovery without considering what else is held constant.

Recovery assumptions affect both level and sensitivity

Recovery conventions describe different cash flows

These conventions lead to different valuation equations and cannot be represented by one unexplained percentage.

A recovery assumption also interacts with dependence. In a systemic downturn, low recoveries may coincide with many defaults and elevated exposure. A portfolio scenario engine should preserve such relationships if they are part of the risk being assessed.

Sensitivity to recovery should therefore be reported alongside spread and default-intensity sensitivity. Recalibrating the hazard curve after changing recovery answers a different question from holding hazard fixed. Both exercises can be useful if their purpose is clear.

Credit hedges can leave basis risk

A bond and a CDS referencing the same issuer do not have identical cash flows, liquidity, financing or contractual protection. Their spreads can differ because of deliverability, documentation, funding, demand and other effects. Buying protection does not automatically eliminate every risk in the cash bond.

Maturity mismatch leaves exposure to the shape of the credit curve. Currency mismatch can add exchange-rate and cross-currency default effects. A hedge calibrated to a small spread change can also respond differently during a jump to default.

The correct hedge analysis compares scenario cash flows and valuations under consistent default events, recoveries and settlement timing. A common issuer label is an insufficient basis for declaring the position fully hedged.

Separate calibration evidence from validation evidence

Reproducing the instruments used to fit a model establishes calibration consistency. It does not independently validate predictions for different maturities, basket structures or stress conditions. Those outputs depend on the model’s extrapolation and dependence assumptions.

Validation can compare alternative model families, examine parameter stability and test simpler contracts with independently implemented methods. For portfolio models, sensitivity to correlation and tail dependence should be visible. For structural models, sensitivity to asset-value estimation and the default barrier deserves equal attention.

A clear credit report can retain a model’s useful quantitative results without pretending that every input is observable or every probability is physical. The distinction between historical frequency, pricing-implied probability and explicitly chosen stress is what makes the resulting numbers interpretable.