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Lecture1DefaultRiskStructureInteresRates

Lecture1DefaultRiskStructureInteresRates - Columbia...

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Columbia University M.S. in Financial Engineering IEOR 4731: Credit Derivatives Instructor: Rama CONT Teaching Assistants: Yu Hang KAN [email protected] Lecture 1: Default risk, credit risk and the risk structure of interest rates. 1
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Outline: The term structure of interest rates. Default risk and credit risk. Credit events The risk structure of interest rates. Credit spreads Credit ratings Loss given default and recovery rates 2
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The term structure of interest rates Discount factor 𝐵 ( 𝑡, 𝑇 ): value at 𝑡 of 1$ paid at 𝑇 . Time value of money: 𝐵 ( 𝑡, 𝑇 ) < 1 and decreases with 𝑇 . Yield to maturity 𝑌 0 ( 𝑡, 𝑇 ): defined by 𝐵 ( 𝑡, 𝑇 ) = 1 (1 + 𝑌 0 ( 𝑡, 𝑇 )) 𝑇 𝑡 = exp[ ( 𝑇 𝑡 ) 𝑅 0 ( 𝑡, 𝑇 )] (1) 𝑅 0 ( 𝑡, 𝑇 ): continuously compounded yield Forward rate term structure 𝑓 ( 𝑡, 𝑇 ): defined by 𝑅 0 ( 𝑡, 𝑇 ) = 1 𝑇 𝑡 𝑇 𝑡 𝑓 0 ( 𝑡, 𝑢 ) 𝑑𝑢 𝑓 0 ( 𝑡, 𝑇 ) = ∂𝑇 ln 𝐵 ( 𝑡, 𝑇 ) 𝐵 ( 𝑡, 𝑇 ) = exp[ 𝑇 𝑡 𝑓 0 ( 𝑡, 𝑢 ) 𝑑𝑢 ] (2) 𝑓 0 ( 𝑡, 𝑇 ) represents the interest rate prevailing today for borrowing between 𝑇 and 𝑇 + Δ 𝑇 . 3
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Discount factors can be used to value bonds: if a bond pays coupons 𝐶 𝑘 at payment dates 𝑇 𝑘 then its current value is 𝑉 𝑡 = 𝐾 𝑘 =1 𝐶 𝑘 𝐵 ( 𝑡, 𝑇 𝑘 ) (3) This valuation implicitly assumes that payments are certain . This is the case if the debt backed/guaranteed by an authority such as the Federal Reserve, OECD Central Bank, monetary authority,...assumed to be solvent at maturity But, in general, if the issuer of the bond is a corporate entity, it may default – fail to honor– some of the payments. In this case 𝑉 𝑡 < 𝐾 𝑘 =1 𝐶 𝑘 𝐵 ( 𝑡, 𝑇 𝑘 ) (4) Due to the possibility of default even a simple bond becomes an instruments with uncertain cash flows i.e. a contingent claim.
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