RMBI 4210 Quantitative Methods for Risk Management Tutorial 5 – Implied probability of default
Notations
𝑦 𝑇 : yield on a T-year corporate zero-coupon bond
𝑦∗(𝑇): yield on a T-year risk-free zero-coupon bond
𝑄 𝑇 : probability that corporation will default between time zero and time T, 1 − 𝑆 𝑇
𝑆 𝑇 : cumulative probability of the corporation surviving to time T, 𝑒+ , – +,∗ – –
𝑅: recovery rate
𝜆 𝑡 ∆𝑡: probability of default between time t and 𝑡 + ∆𝑡 conditional on no earlier default(default intensity/ hazard rate)
𝜆(𝑡):average default intensity between time 0 and time t, 𝜆(𝑡) = , – +,∗ – 4+5
𝑞(𝑡)∆𝑡: probability of default that occurs between t and 𝑡 + ∆𝑡 (unconditional default probability density) , 𝑞 𝑡 = 𝑆(𝑡)𝜆 𝑡 (clarify)
Concepts
Credit spreads: compensate investor for the risk of default, ∆𝑦 = 𝑦 𝑇 Increase with maturity, increase as the rating declines
Probability of default: 𝑄 𝑇 =4+78 9 : 89∗ : : 4+5
Default intensities (hazard rates): 𝜆 𝑡 / 𝜆(𝑡) HW Q9-10
VaR reference
http://www.columbia.edu/~mh2078/QRM/BasicConceptsMasterSlides.pdf
Spring, 2021 Wang Jingjing
− 𝑦∗(𝑇)
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