Reduced-Form Models: Default as a Surprise

The structural models of Module 6 tell a story about why firms default: assets drift down until they hit the debt. It’s a good story with one embarrassing implication — in a diffusion world, a healthy firm cannot default tomorrow. Asset value would have to travel too far, too fast. The model therefore predicts near-zero spreads at short maturities, and the market flatly disagrees: even overnight, nothing trades at the risk-free rate.

Reduced-form models respond by giving up the story. Default is not the endpoint of a visible mechanism; it is a surprise — a jump that can happen at any instant, governed by an intensity. Where structural models ask “how far are the assets from the barrier?”, reduced-form models ask only “at what rate do surprises like this arrive?” — and then read that rate out of market prices.

The mathematical object: a hazard rate

Let $\tau$ be the (random) default time. The hazard rate — or default intensity — is defined by

In words: given survival up to $t$, the instantaneous rate at which default strikes. If $\lambda = 2\%$ per year, a firm alive today has roughly a 2% chance of defaulting over the next year. It’s the same mathematics as radioactive decay or mortality tables — credit risk borrowed the machinery wholesale from actuarial science.

From the hazard rate, everything follows:

Play with the machinery — level, slope, and recovery:

The shape selector encodes a real stylized fact. Investment-grade names have upward-sloping hazards: if nothing bad has happened yet, the passage of time mostly adds opportunities for deterioration. Distressed names slope down: if a CCC credit survives the next two years, it probably refinanced or restructured, and its hazard falls. You’ll see exactly this pattern in rating-transition data in Module 8.

Jarrow-Turnbull: the founding paper

Jarrow and Turnbull (1995) built the first complete arbitrage-free framework on this idea: default arrives as the first jump of a Poisson process with intensity $\lambda$, recovery is an exogenous parameter, and — the crucial move — the intensity is calibrated to market prices, not estimated from balance sheets. The firm’s fundamentals appear nowhere. If bonds of firm X trade 150 bps over Treasuries, that spread is the datum; the model exists to interpolate it consistently across maturities and instruments.

A pricing consequence you can carry around: under independence of rates and default, a risky zero-coupon bond is worth

— the risk-free price times “survive and get paid in full, or default and get the recovery.” Everything else in reduced-form pricing is elaborations of this line.

The original paper’s second half extended the idea to a ratings-driven version (intensities for moving between rating classes, not just to default) — machinery we’ll reuse in Module 8 when we get to transition matrices and CreditMetrics.

Duffie-Singleton: the elegant discounting trick

Duffie and Singleton (1999) contributed the formulation practitioners actually compute with. Suppose that upon default the claim loses fraction $L$ of its pre-default market value (recovery of market value, rather than recovery of face). Then defaultable cash flows can be priced by discounting at an adjusted rate:

That’s the whole theorem, and it’s a beautiful one: credit risk enters as a spread on the discount curve. The entire toolkit built for risk-free term structures — affine models, HJM, every bond-math identity — carries over to defaultable debt by swapping the short rate for $r + \lambda L$. It also formalizes the credit triangle: the instantaneous spread literally equals intensity times loss severity.

The practical fine print: recovery-of-market-value makes $\lambda$ and $L$ enter prices only through the product $\lambda L$, so spreads alone cannot separate them. You either fix recovery by convention (the market’s answer — 40% for senior unsecured is the standing default) or bring in instruments with different recovery exposure to identify the split.

CDS: the cleanest read on credit

A credit default swap is an insurance contract on default: the protection buyer pays a running premium (the spread, quarterly, on the notional); upon a defined credit event, the seller pays the loss — in the standard cash-settled form, $(1 - R)$ times notional, with $R$ set by an auction of the defaulted bonds.

Pricing is the reduced-form model in its purest form. Two legs, valued by expectation:

• Premium leg: $\;s \times \sum_i \Delta_i\, P(0, t_i)\, S(t_i)$ — each spread payment discounted and weighted by the probability the name is still alive to pay it.
• Protection leg: $\;(1-R) \times \sum_i P(0, t_i)\,\big[S(t_{i-1}) - S(t_i)\big]$ — the loss payment weighted by the probability default lands in each period.

The fair spread equates the legs. Run the machine forward (given $\lambda$, price the CDS) or backward (given the quoted spread, bootstrap the implied hazard curve maturity by maturity — 1y quote pins $\lambda$ over year one, the 3y quote then pins years two–three, and so on). The downloadable Excel template implements both directions with every formula visible.

Why call CDS the cleanest read on credit? Compare the alternatives. Bond spreads are polluted by liquidity, coupon effects, embedded options, and the choice of risk-free benchmark. Ratings (Module 8) update slowly and deliberately. Equity-implied PDs (Module 6) inherit the stock market’s noise and need a model to translate. A CDS is a purpose-built instrument whose only subject is default: standardized terms, an auction-determined recovery, and a spread that moves the moment opinions move. It has real imperfections — counterparty risk (wrong-way risk at that: your protection seller is likeliest to fail exactly when protection pays — Module 11’s subject), liquidity concentrated in large names, occasional squeezes — but as a signal, nothing else is as direct.

Structural vs. reduced-form: the scorecard

The synthesis view: structural models are for understanding credit, reduced-form models are for trading it. Serious shops run both and treat persistent disagreement between them as information.

Where this connects

• The hazard-rate machinery is identical to survival analysis in the data science track — the same estimators price CDS and model loan prepayment.
• Transition matrices in Module 8 are the discrete-time, multi-state version of intensities: a generator matrix is a hazard rate with a rating dimension attached.
• CVA and wrong-way risk (Module 11) are built almost entirely on reduced-form technology — you’ll see these survival-weighted expectations again with a counterparty attached.

Try it yourself

The Excel template below prices a 5-year CDS end to end: a hazard-curve sheet building survival probabilities from piecewise-constant intensities, discounted premium and protection legs term by term, the fair-spread solver, and a bootstrap sheet that recovers the hazard curve from a strip of market quotes (1y through 10y). Change the recovery assumption from 40% to 20% and watch the implied hazards reprice — the credit triangle, live in the grid.

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