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Quantitative Finance · शब्दावली

Hazard rate, survival probability क्या है?

अन्य नाम: hazard rate · survival probability

Definition 23.3 Markets II: Rates, FX and Credit · अध्याय 23 — Credit Default Swaps

The hazard rate $\lambda(t)$ of a reference entity is its instantaneous rate of default at time $t$ given survival to $t$ : the probability of default in $[t, t + dt]$ , given no default before $t$ , is $\lambda(t)\,dt$ . The survival probability to $T$ is $Q(T) = \exp\bigl(-\int_0^T \lambda(u)\,du\bigr)$ , which is $e^{-\lambda T}$ for a flat hazard rate.

Upfront payment of a five-year contract, in per cent of notional, against the quoted spread, for the two standard coupons (flat rate 4%, recovery 40%). The upfront is zero where the spread equals the coupon, and negative, paid to the buyer, below it; the dots are . Illustrative; data: the chapter’s tutorial. — Figure 23.2. Upfront payment of a five-year contract, in per cent of notional, against the quoted spread, for the two standard coupons (flat rate 4%, recovery 40%). The upfront is zero where the spread equals the coupon, and negative, paid to the buyer, below it; the dots are Example 23.6. Illustrative; data: the chapter’s tutorial.

Figure 23.3. Survival probabilities implied by five-year spreads of 100, 300 and 800 basis points with a flat hazard rate and a recovery of 40%. The hazard rates are close to spread divided by

1 - R

: about 1.7%, 5% and 13% a year. These are risk-neutral probabilities, which include a premium for bearing default risk. Illustrative; data: the chapter’s tutorial.

उदाहरण

Example 23.6 (Two quotes, two conventions)

With a flat rate of 4% and a recovery of 40%, the five-year risky annuity is 4.332 at a spread of 100 basis points and 4.165 at 200. An investment-grade name quoted at 200 basis points, on the 100 basis point coupon, costs the buyer $1\% \times 4.165 = 4.16\%$ upfront, USD 416 000 on USD 10 million. A high-yield name quoted at 300 basis points on the 500 coupon pays the buyer 8.01% upfront; at 800 the buyer pays 9.98%, and at 1 200, 20.28%. Investment-grade names are quoted in spread, high-yield names in points upfront.

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