Market-implied default risk · RiskAtoms Academy

By the end you will be able to

1Turn a credit spread into a hazard rate and a survival curve, with Q(t) = e^(−λt).
2Derive the credit triangle λ ≈ s/(1 − R) and bootstrap default probabilities from a CDS curve.
3Explain why risk-neutral PDs from markets are higher than real-world PDs from ratings.
4Choose the right kind of PD for pricing, CVA, provisions and capital.

Why it matters

RiskAtoms Bank rates Bramwick Stores plc grade 6: a one-year PD of 1.13%. This morning its five-year CDS trades at 260bp, which says the market thinks its chance of default within five years is about 20%. The internal grade says 5.5%. Both numbers are right, for different jobs. Knowing which to use, and what the gap between them means, is part of every credit risk manager's toolkit.

Core explanation

Reading default risk from prices

Illustrative dataRiskAtoms Bank plc's credit book

The module's lending book: twelve invented corporate borrowers (exposure at default £587m, expected loss £2.81m), retail mortgages, personal loans and cards, a ten-grade master scale, personal-loan vintages and twenty past workouts. Every firm and figure is made up for teaching, in pounds, at one date; none is real data.

Bramwick Stores plc (C02) is in the book at grade 6. Its CDS curve is an extra for this lesson: 1y 150bp, 3y 210bp, 5y 260bp, 7y 285bp, 10y 300bp, with 40% recovery and a 4% risk-free rate (continuously compounded).

Internal ratings change a few times a year. Credit spreads change every second. L1.7.10: Credit default swaps introduced the rule of thumb behind them: a CDS seller who expects to lose $1 - R$ on default, with a yearly default probability PD, needs a spread of about $\text{PD}\times(1-R)$ . L3.2.07: Credit spread risk used the same idea to price a CDS hedge. Here we make it precise and use it to read PDs from the market.

Hazard rates and survival

Think of default as an event that can strike at any moment. The hazard rate λ is its intensity: a firm that is still alive has a chance of about λΔt of defaulting in the next short interval Δt. If λ is constant, the survival probability decays exponentially, exactly like continuous discounting (M-A05: The number e and continuous growth):

Q(t) = e^{-\lambda t}, \qquad \text{PD}(0,t) = 1 - e^{-\lambda t}

With the credit triangle, λ ≈ s/(1 − R). Bramwick's five-year spread gives λ = 260bp / 0.6 = 4.33% a year, and a five-year survival of 80.5%. A grade's one-year PD converts the other way: λ = −ln(1 − PD), so grade 6's 1.13% is a hazard rate of 1.136%.

A curve, not a number

CDS quotes come at several maturities, and spreads usually rise with maturity for a weaker name: the market expects more risk later. Bootstrapping turns the quotes into a hazard rate for each stretch of time: the 1-year quote fixes λ for year 1, the 3-year quote then fixes λ for years 1 to 3, and so on. The worked example does it for Bramwick.

Risk-neutral is not real-world

The 20% from the CDS is a risk-neutral PD: the probability that makes the price fair if all cash flows are discounted at the risk-free rate. Investors also want paying for bearing default risk, for not knowing the recovery, and for holding something hard to sell. All of that is in the spread, so the market-implied PD is higher than the real-world PD. For Bramwick the ratio of hazard rates is 3.8×. Of the 260bp spread, only 68bp is expected loss at the real-world PD; 192bp is risk premium. Hull, Predescu and White found that for a seven-year A-rated bond the default probability implied by its price was almost ten times the one from historical data. As a ratio, the gap is largest for the safest names.

So what is the market PD good for? Two things. It is the right input wherever a value must agree with market prices: CDS pricing, fair values, and the credit valuation adjustment on derivatives (M3.4: Counterparty credit risk). And it is a fast early-warning signal: a spread that widens much faster than its peers' is news the internal grade has not caught up with yet (L3.3.17: Early warning, watchlists and workout).

Sort itRisk-neutral or real-world PD?

Pricing a 5-year CDS for a client.
Calculating the CVA on an interest rate swap with a corporate.
The IFRS 9 expected credit loss on a loan.
IRB capital for a corporate loan.
Setting the expected-loss part of a loan's margin.

The formal version

The credit triangle. Take a CDS with continuous premium $s$ to maturity $T$ , a constant hazard $λ$ , a risk-free rate $r$ and recovery $R$ . The premium is paid while the name survives; the protection pays $1 - R$ at the moment of default, whose density at $t$ is $\lambda e^{-\lambda t}$ :

\begin{aligned} \text{PV(premium leg)} &= \int_0^T s\,e^{-rt}\,e^{-\lambda t}\,dt = s\,\frac{1 - e^{-(r+\lambda)T}}{r+\lambda} \\ \text{PV(protection leg)} &= \int_0^T (1-R)\,\lambda e^{-\lambda t}\,e^{-rt}\,dt = (1-R)\,\lambda\,\frac{1 - e^{-(r+\lambda)T}}{r+\lambda} \\ \text{par: premium} = \text{protection} &\;\Rightarrow\; s = \lambda\,(1-R) \;\Rightarrow\; \lambda = \frac{s}{1-R} \end{aligned}

With continuous payments and a flat hazard, the triangle is exact for every maturity. Real contracts pay quarterly, with accrued premium on default, and hazards vary with time, so markets solve the leg equation numerically. On a quarterly grid, with $t_q = q/4$ :

\begin{aligned} \text{RPV01} &= \sum_{q=1}^{4T} \tfrac14\,e^{-r t_q}\Big[Q(t_q) + \tfrac12\big(Q(t_{q-1}) - Q(t_q)\big)\Big], \qquad \text{Prot} = (1-R)\sum_{q=1}^{4T} e^{-r t_q}\big(Q(t_{q-1}) - Q(t_q)\big) \\ s_{\text{par}} &= \frac{\text{Prot}}{\text{RPV01}}, \qquad Q(t) = \exp\!\Big(-\int_0^t \lambda(u)\,du\Big) \;\text{ with } \lambda \text{ constant between quoted maturities} \end{aligned}

For Bramwick at five years: $\text{RPV01} = 4.1257$ , $\text{Prot} = 0.10727$ , and $0.10727 / 4.1257 = 2.60\%$ , the quoted spread.

Convention: this quarterly-grid model is a teaching version of the market's standard model. It ignores day counts and business-day rules and uses a flat risk-free rate.

Try it

Start at Bramwick's 260bp and grade 6. Raise the recovery rate: the same spread now implies a higher PD, because each default costs the seller less. Then pick grade 3 and a spread of 60bp: the ratio of the two hazard rates grows as credit quality improves.

Try itFrom a CDS spread to default probabilities

CDS spread260 bp

Recovery rate R40%

Real-world comparison: internal grade6 (PD 1.13%)

Horizon5 years

Survival probability over time. Blue: risk-neutral, from the CDS spread. Second colour: real-world, from the internal grade's PD.

Market-implied (risk-neutral)

Hazard rate λ = s / (1 − R)4.33%

Survival to 5 years80.5%

PD within 5 years19.5%

Internal grade (real-world)

Hazard rate −ln(1 − PD)1.14%

PD within 5 years5.5%

Ratio of hazard rates3.8×

Worked example

Bootstrapping Bramwick Stores plc's CDS curve

From five quotes to a term structure of PDs

Step 1 of 5

1
Year 1
The 1-year spread is 150bp. Solve for the λ that makes the 1-year CDS fair: 2.50%, the same as the triangle's 2.50% to four decimals. Survival to year 1: $e^{-0.0250} = 97.53\%$ .

What the risk team notes

Bramwick: 5y CDS 260bp, market-implied 5y PD 20.1% against 5.5% from grade 6. Ratio 3.6×, no action on the level alone. Add to the spread monitor with a trigger at 400bp or a widening of 100bp relative to retail peers in a month.

Regulatory lens

Since ISDA's 2009 “Big Bang” and “Small Bang” changes, single-name CDS trade as standard contracts: fixed coupons (100 or 500bp in North America; 25, 100, 500 or 1,000bp in Europe) with an upfront payment, quarterly payment dates on 20 March, June, September and December, credit events decided by ISDA's Determinations Committees, and settlement by auction. Prudential rules use both kinds of PD: the standardised approach to CVA capital works from PDs implied by market credit spreads (the basic approach uses supervisory risk weights instead; see M4.3: Market risk capital (FRTB) and CVA), while IRB capital uses long-run real-world PDs (M4.2: Credit risk capital) and IFRS 9 provisions use real-world, forward-looking ones (M4.7: Accounting for risk). Hazard-rate (reduced-form) and structural credit models are developed in M6.4: Credit risk modelling.

Spark Atoms

Spark AtomEight and five-eighths cents

Lehman Brothers filed for bankruptcy on 15 September 2008. The CDS auction held on 10 October 2008 set the final price of its bonds at 8.625 per cent of face value, so protection sellers paid 91.375% of notional: about as close to zero recovery as a large firm gets, and far from the 40% market convention.

Spark AtomAn actuary got there first

In 1825 the actuary Benjamin Gompertz described the “intensity of mortality” and proposed that it rises in geometric progression with age. A hazard rate is that idea applied to companies: actuaries call it the force of mortality, credit analysts the default intensity, and the survival curve e^(−λt) is the same mathematics.

Key takeaways

A hazard rate λ is the default intensity; with λ constant, survival is Q(t) = e^(−λt) and PD(0,t) = 1 − e^(−λt).
Credit triangle: s ≈ λ(1 − R), exact for a flat hazard with continuous payments; Bramwick's 260bp at 40% recovery gives λ = 4.33%.
Bootstrapping turns a CDS curve into hazard rates for each stretch of time and a term structure of PDs.
Market-implied PDs are risk-neutral and well above real-world PDs; the gap is a risk and liquidity premium.
Use market PDs for prices, fair values and CVA; real-world PDs for expected loss, provisions and capital; watch spreads as an early warning.

Reaction

Check what stuck

Question 1 · objective 1

A company's CDS spread is 300bp and the market assumes 40% recovery. What hazard rate does the credit triangle give, in %?

%

Question 2 · objective 1

With a constant hazard rate of 5% a year, what is the probability of surviving two years, in % (two decimals)?

%

Question 3 · objective 2

A CDS trades at 120bp with 40% recovery. What is the market-implied probability of default within 5 years, in % (two decimals)?

%

Question 4 · objective 3

The PD implied by a firm's CDS is about four times the PD of its internal grade. What is the best explanation?

AThe internal rating is wrong and must be cut by two gradesBThe CDS PD is risk-neutral: the spread also pays investors for bearing default risk and illiquidity, so it overstates the real chance of defaultCCDS spreads ignore recoveryDReal-world PDs are always higher than risk-neutral ones

Question 5 · objective 4

Which use calls for the market-implied (risk-neutral) PD rather than the internal grade's PD?

AThe IFRS 9 provision on a loanBThe IRB capital on a corporate loanCPricing a CDS or valuing the counterparty credit risk (CVA) on a derivativeDThe expected loss used to set a loan's margin

Glossary

Hazard rate: Hazard rate (default intensity) λ: the probability per unit of time of defaulting in the next instant, for a borrower that has not defaulted yet.
Survival probability: Survival probability Q(t): the probability that the borrower has not defaulted by time t.
Credit triangle: The approximation s ≈ λ(1 − R) linking a credit spread s, the hazard rate λ and the recovery rate R.
Bootstrapping: Building a curve one maturity at a time: each new CDS quote fixes the hazard rate for the next stretch of time, given the stretches already fixed.
Risk-neutral PD: Risk-neutral PD: the default probability that prices a bond or CDS when every payoff is discounted at the risk-free rate. It includes compensation for risk.
Real-world PD: Real-world (physical) PD: the actual frequency of default, estimated from history and ratings. Used for expected loss, provisions and capital.
Credit risk premium: The part of a credit spread above the expected loss: pay for bearing default risk, uncertainty about recovery, and illiquidity.
Upfront: With a standard fixed coupon, the cash paid at the start of a CDS to make up the difference between the coupon and the market (par) spread.
RPV01: Risky PV01: the value of receiving 1 a year (paid quarterly) until maturity or default. A CDS premium leg is the spread times RPV01.

Deep Dives & Origins

Deep Dives

J. C. Hull, Options, Futures, and Other Derivatives, Pearson, the chapter on credit risk (default probabilities from bond prices and CDS). Intermediate.
D. O'Kane, Modelling Single-name and Multi-name Credit Derivatives, Wiley, 2008. Advanced.

Origins

J. Hull, M. Predescu and A. White, “Bond prices, default probabilities and risk premiums”, Journal of Credit Risk 1(2), 2005, pp. 53–60.
ISDA, FAQ: CDS standardized products (standard coupons and quarterly roll dates), December 2012.
Creditex and Markit, Lehman Brothers credit event auction final results, 10 October 2008.
B. Gompertz, “On the nature of the function expressive of the law of human mortality”, Philosophical Transactions of the Royal Society of London 115, 1825, pp. 513–583.

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