Reduced-form credit risk model: Definition & Examples

A reduced-form credit risk model specifies the timing of default directly, usually through a non-negative default intensity or hazard rate, rather than deriving default from the firm's asset value crossing a debt threshold. It turns a view of default timing into survival probabilities that can be combined with contractual payments and discounting to value credit instruments.

The modelling choice matters: a probability of default alone does not determine a bond price. The valuation also needs a probability measure, interest rates, the promised cash flows and what is paid if default occurs.

Model the clock, then price the payment

Write $\tau$ for the random default time. In the simplest deterministic-intensity model, with time measured in years and intensity $\lambda^{\mathbb{Q}}(t)$ measured per year, the risk-neutral survival probability is

S^{\mathbb{Q}}(T)=\mathbb{Q}(\tau>T) =\exp\!\left(-\int_0^T\lambda^{\mathbb{Q}}(u)\,du\right).

The intensity is a conditional rate per unit time. For a short interval of length

h

, conditional default probability is approximately

\lambda^{\mathbb{Q}}(t)h

given survival to

t

. The probability over a full horizon is

1-S^{\mathbb{Q}}(T)

, not the intensity itself.

A structural credit risk model starts from a different mechanism: assets and liabilities determine when the firm defaults. Reduced-form modelling can still let observable economic or firm characteristics influence intensity; it does not require default risk to ignore the economy. Its primitive is the default-time process rather than a balance-sheet default boundary.

A USD100 claim with only one surviving payment

Consider a wholly synthetic claim valued at time zero, before default. Stipulate:

USD100 face value, paid exactly one year from today only if the issuer survives beyond that date;
zero coupons and zero recovery on default;
constant pricing intensity $\lambda^{\mathbb{Q}}=0.02$ per year;
deterministic default-free rate $r=0.03$ per year, continuously compounded;
no fees, taxes, liquidity adjustment or other cash flows.

Survival probability is $e^{-0.02}=98.0199\%$ , so default probability is $1.9801\%$ . The expected maturity payment under $\mathbb{Q}$ is USD98.0199. Its value today is lower:

\begin{aligned} V_0 &=\mathbb{E}^{\mathbb{Q}}\!\left[e^{-rT}\,100\,\mathbf{1}_{\{\tau>T\}}\right] \\ &=100e^{-0.03}e^{-0.02} \\ &=100e^{-0.05}\approx\text{USD }95.1229. \end{aligned}

The default-free claim paying USD100 at the same date is worth $100e^{-0.03}\approx\text{USD }97.0446$ . The credit-related price reduction in this stipulated model is therefore USD1.9216, using unrounded prices. It is a difference in present values, not USD100 multiplied by the annual intensity.

Synthetic one-year USD100 survival claim: survival98.0199% pays100, default1.9801% pays0; expected maturity payment98.0199 discounts to95.1229 today, versus97.0446 default-free.

Synthetic zero-recovery model, valued today. Branch probabilities are under $\mathbb{Q}$ ; only the survivor receives USD100 at one year. The expectation is a maturity amount, and the price is its discounted value.

Under these assumptions the claim's continuously compounded yield is 5%, a 2 percentage point spread over the 3% default-free rate. That equality between spread and intensity belongs to this zero-recovery survival claim; it is not a general identity for coupon bonds or CDS spreads.

What the simple price leaves open

Recovery requires a payoff rule. A recovery fraction paid on face value at default has a different present value from a fraction of the corresponding default-free bond delivered at default. State the recovery basis and timing before adding a recovery term. Simply replacing the intensity by intensity times loss given default is not a universal bond-pricing rule.

Pricing probabilities are not default forecasts. A curve calibrated to credit prices under

\mathbb{Q}

is an input to relative valuation. A physical default probability under

\mathbb{P}

answers a forecasting question. The example stipulates its pricing intensity; it does not estimate an issuer's real-world default risk. Calibration also depends on recovery and contract assumptions, so a fitted curve does not validate those assumptions by itself.

Random intensity requires an expectation. In a Cox model, conditional on the intensity path, survival has the exponential form. Before that path is known, unconditional survival from time zero is

S^{\mathbb{Q}}(T)=\mathbb{E}^{\mathbb{Q}}\!\left[\exp\!\left(-\int_0^T\lambda_u^{\mathbb{Q}}\,du\right)\right].

This generally differs from exponentiating minus the expected integrated intensity. If rates are random too, pricing must retain their joint dependence with default. The precise information assumptions behind these reductions are explained in Grasselli and Hurd's credit-risk notes, §§5.1–5.2.3 and 5.6.

For a longer derivation, Reduced-Form Models: Hazard Rates and Risk-Neutral Default Probabilities develops deterministic hazard curves, recovery models and simplified CDS valuation. The full lesson requires membership; its public preview shows the scope and prerequisites.

Sources

McNeil, Frey and Embrechts, Quantitative Risk Management: Concepts, Techniques and Tools, revised edition (Princeton University Press, 2015), §§10.4.1–10.4.3, pp.391–401; §10.5.1, pp.406–409. The survival-claim formula is (10.37); the Cox construction is Definition10.10 and Lemma10.11.
Grasselli and Hurd, Math 774 — Credit Risk Modeling, 3 January 2010, §§5.1–5.2.3, 5.6 and 6.1: information sets, random intensity, pricing measure and distinct recovery mechanisms.

Model the clock, then price the payment

A USD100 claim with only one surviving payment

What the simple price leaves open

Sources

Related terms

Credit default swap premium

Default

Hazard rate

Recovery rate

Structural credit risk model