Credit default swaps (CDSs) are described in Chapter 19. They can be valued using (risk-neutral) default probability estimates.

Suppose that the hazard rate of a reference entity is 2% per annum for five years. Table K.1 shows survival probabilities and unconditional default probabilities (that is, default probabilities as seen at time zero) for each of the five years. From equation (19.2), the probability of survival to time *t* is *e*^{− 0.02t}. The probability of default during a year is the probability of survival to the beginning of the year minus the probability of survival to the end of the year. For example, the probability of survival to time 2 years is *e*^{− 0.02 × 2} = 0.9608 and the probability of survival to time 3 years is *e*^{− 0.02 × 3} = 0.9418. The probability of default during the third year is 0.9608 − 0.9418 = 0.0190.

**TABLE K.1** Unconditional Default Probabilities and Survival Probabilities

Time (years) | Probability of Surviving to Year End | Probability of Default During Year |

1 | 0.9802 | 0.0198 |

2 | 0.9608 | 0.0194 |

3 | 0.9418 | 0.0190 |

4 | 0.9321 | 0.0186 |

5 | 0.9048 | 0.0183 |

We will assume that defaults always happen halfway through a year and that payments on a five-year credit default swap are made once a year, at the end of each year. We also assume that the risk-free interest rate is 5% per annum with continuous compounding and the recovery rate is 40%. There are three parts to the calculation. These are shown in Tables K.2 ...

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