MBAX-6270 Credit Default Swaps Credit Default Swaps (CDS) CDS is a form of insurance against a firm defaulting on the bonds they issued CDS are used also as a way to express a bearish view on a company Before the 2008 crisis, CDS were traded more liquidly (5-8 times) than bonds Simplified Mechanics: You pay me a running fee (each quarter) If the reference entity defaults, you deliver the bond to me and I will pay you $100. Fees stop. Typical maturity is 5 years
CDS Structure Example Default Protection Buyer, A 120 bps per year Payoff if there is a default by reference entity=100(1-r) Default Protection Seller, B Recovery rate, R, is the ratio of the value of the bond issued by reference entity immediately after default to the face value of the bond Other Details Payments are usually made quarterly in arrears In the event of default there is a final accrual payment by the buyer Settlement can be specified as delivery of the bonds or (more usually) in cash An auction process usually determines the payoff Suppose payments are made quarterly in the example just considered. What are the cash flows if there is a default after 3 years and 1 month and recovery rate is 40%?
Attractions of the CDS Market Allows credit risks to be traded in the same way as other market risks Can be used to transfer credit risks to a third party Can be used to diversify credit risks Can be used to protect against counterparty risk (remember, an ISDA gives you the same rights as bond holders in bankruptcy court, if you are owed money by the bankrupt party) CDS Valuation Consider a simplified CDS: Pay up-front X If the reference name defaults in the next year then receive $100 in 1 years time. Assume: Probability of default over next year = 10% Risk-free interest rate = 5% What is the fair value X of this CDS? Approximately: %
Credit Some concepts Default is described as an exogenous jump process Intensity Models are easy to calibrate to credit spreads from Credit Default Swaps (CDS) or corporate bonds Default time is the first jump time of a Poisson process Default intensity is a hazard rate and represents an instantaneous credit spread, and is the probability that default first occurs in the time slice : Prob, Initially assume a constant for simplicity, but it can have a term structure and can even be stochastic (if modeling credit spread volatility) Credit Some formulas Let = cumulative distribution function of default time : Pr and 0 0 Also, the derivative of is the default intensity function The survival is the probability of survival to Pr 1 Default rate at time is the hazard rate function = lim Pr Q Q
Credit Some formulas (cont.) The cumulative hazard function is defined as: ln ln Then the survival function can be expressed as a function of the hazard function: e e We will initially assume a deterministic and time homogeneous Poisson Process (simpler). That means: Will have a constant default intensity Formulas simplify immediate link to credit spreads CDS Simplified formulas Given a CDS market quoted with a constant credit spread, and a recovery rate we have: Instantaneous default protection = premium paid: Hence default intensity (or hazard rate) Survival probability to time Probability(default during period )
Survival Curve 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Cpty's Survival curve s 500bp R60% Maturity20 years s/1 0 0 2 4 6 8 10 12 14 16 18 20 SNAC CDS In 2009 ISDA Created the Standard North American Corporate CDS - SNAC Definitions CDS Dates: 20th of Mar/Jun/Sep/Dec Business Day Count: Actual/360 (see 2003 ISDA Credit Derivative Definitions) Business Day Convention Following (see 2003 ISDA Credit Derivative Definitions) Contract Specifications With respect to trade date T: Maturity Date: a CDS Date, unadjusted Coupon Rate: 100bp or 500bp Protection Leg Legal Protection Effective Date: (today 60 days) for credit events and (today 90 days) for succession events, unadjusted Protection Maturity Date: Maturity Date Protection Payoff: Par Minus Recovery
SNAC CDS (cont.) Premium Leg Payment Frequency: quarterly Daycount Basis: Actual/360 Pay Accrued On Default: True Business Day Calendar: currency dependent Adjusted CDS Dates: CDS Dates, business day adjusted Following First Coupon Payment Date: earliest Adjusted CDS Date after T+1 calendar Accrual Begin Date: latest Adjusted CDS Date on or before T+1 calendar Accrual Dates: CDS dates business day adjusted Following except for the last accrual date (Maturity Date) which remains unadjusted Accrual Periods: From previous accrual date, inclusive, to the next accrual date, exclusive, except for the last accrual period where the accrual end date (Maturity Date) is included Payment Dates: CDS dates, business day adjusted Following including the last payment day (Maturity Date). CDS Valuation Steps Build curve of discount factors Build term structure of credit spreads Calibrate Survival Probability from and Price CDS = Protection leg minus Coupon Leg 1 If there is a credit spread term structure a bootstrap technique is necessary to build the survival curve
CDS tree to calibrate probabilities Build a tree of survival / default events Determine cash flows in each case Compute overall PV of protection / fee legs Equate PV of fee (or income) leg to protection leg in order to solve for probabilities Repeat calculations from left to right (bootstrapping) S 1 S 2 S k-1 S k D k S k+1 1 D k+1 D 1 D 2 1 t 0 t 1 t 2 t k-1 t k t k+1 CDS Calibration (bootstrap) Assume $1 notional PV of Cumulative Income (unit 1 credit spread) PV of the Cumulative Protection Notice the 1 1 This is for a credit spread of 1. We will need to multiply this by the actual spread s for that tenor. NOTE: with a credit spread term structure a bootstrap technique is necessary, hence the use a unit credit spread in the calculations.
CDS Calibration (bootstrap) (cont.) Assume $1 notional for a CDS to time t. Equate Cumulative PVs to last period to compute the probability of default in last period : Solve for Compute probability of survival from above NOTE: With a credit spread term structure a bootstrap technique is necessary, hence the use a unit credit spread in the calculations. Think of as the credit equivalent of a term interest rate. CDS - Calibration implementation 1 PV of the Cumul. Income from unit credit spread PV of the Cumul. Protection to period Equate PV of Protection to PV of Income: and solve for Marginal Prob. of dft in period Compute probability of survival from above 1 Assumes $1 notional. Multiply by actual notional.
CDS Valuation Assume $1 notional 1 PV of the Protection Leg PV of the Coupon Leg from CDS coupon Add up over time and compute CDS value. From the protection buyer perspective: Notice the 1 1 Multiply by actual notional to obtain the full dollar figure.