APPENDIX TO CHAPTER 14
Corporate Debt and Credit Default Swaps

A14.1 CUMULATIVE DEFAULT AND SURVIVAL RATES

Proposition: If the hazard rate is constant at normal lamda, then the cumulative survival probability to time t, upper C upper S left-parenthesis t right-parenthesis, is e Superscript minus normal lamda t, and the cumulative default probability, upper C upper D left-parenthesis t right-parenthesis, is 1 minus e Superscript minus normal lamda t.

Proof: The probability of no default to time t plus normal upper Delta t, upper C upper S left-parenthesis t plus normal upper Delta t right-parenthesis, is the probability that there is no default to time t and no default from then to time t plus normal upper Delta t. Mathematically,

(A14.1)StartLayout 1st Row 1st Column upper C upper S left-parenthesis t plus normal upper Delta t right-parenthesis 2nd Column equals upper C upper S left-parenthesis t right-parenthesis times left-parenthesis 1 minus normal lamda normal upper Delta t right-parenthesis EndLayout

Taking the limit of the right‐hand side of (A14.2) as normal upper Delta t approaches zero is the derivative of upper C upper S left-parenthesis t right-parenthesis, denoted upper C upper S prime left-parenthesis t right-parenthesis. Therefore,

where Equation (A14.4) is the solution to Equation (A14.3). The cumulative default probability is just one minus the cumulative survival probability, which gives upper C upper D left-parenthesis t right-parenthesis equals 1 minus e Superscript minus normal lamda t.

A14.2 UPFRONT AMOUNTS

Continue with the notation of the previous section, and add the following:

  • s Superscript upper C upper D upper S: CDS spread
  • c Superscript upper C upper D upper S: CDS coupon
  • upper T: maturity of the CDS, in years
  • n: number of CDS premium payments per year
  • t Subscript i: time of CDS premium payment i, i equals 1 comma ellipsis comma n upper T, with t 0 identical-to 0
  • d left-parenthesis t Subscript i Baseline right-parenthesis: discount factor at time t Subscript i, at risk‐free or benchmark rates
  • upper R: recovery rate
  • upper U upper F: upfront amount

Then, following the logic described in the text, with the detail that premium payments follow the actual/360 day‐count convention, the expected discounted value of the fee leg, upper V Superscript f e e, is,

The value of the contingent leg, upper V Superscript c o n t, again following the logic described in the text, is,

A CDS with a premium of s Superscript upper C upper D upper S is fair if the hazard rate is such that the values of the fee and contingent legs, as given in Equations (A14.5) and (A14.6), are equal.

The upfront amount, following the logic of the text, equals the expected discount value of paying the CDS coupon as a premium payment rather than the CDS spread. Essentially, the upfront amount is computed like the fee leg, with minus left-parenthesis c Superscript upper C upper D upper S Baseline minus s Superscript upper C upper D upper S Baseline right-parenthesis or left-parenthesis s Superscript upper C upper D upper S Baseline minus c Superscript upper C upper D upper S Baseline right-parenthesis replacing s Superscript upper C upper D upper S. Hence,

(A14.7)StartLayout 1st Row 1st Column upper U upper F 2nd Column equals left-parenthesis s Superscript upper C upper D upper S Baseline minus c Superscript upper C upper D upper S Baseline right-parenthesis sigma-summation Underscript i equals 1 Overscript n upper T Endscripts StartFraction t Subscript i Baseline minus t Subscript i minus 1 Baseline Over 360 EndFraction upper C upper S left-parenthesis t Subscript i Baseline right-parenthesis d left-parenthesis t Subscript i Baseline right-parenthesis 2nd Row 1st Column Blank 2nd Column plus one half left-parenthesis s Superscript upper C upper D upper S Baseline minus c Superscript upper C upper D upper S Baseline right-parenthesis sigma-summation Underscript i equals 1 Overscript n upper T Endscripts StartFraction t Subscript i Baseline minus t Subscript i minus 1 Baseline Over 360 EndFraction left-bracket upper C upper S left-parenthesis t Subscript i minus 1 Baseline right-parenthesis minus upper C upper S left-parenthesis t Subscript i Baseline right-parenthesis right-bracket d left-parenthesis t Subscript i Baseline right-parenthesis EndLayout

A14.3 AN APPROXIMATION FOR CDS SPREADS

For the purposes of this section, assume that premium payments of the CDS are all normal upper Delta t years apart, that is, t Subscript i Baseline minus t Subscript i minus 1 Baseline equals normal upper Delta t for all t Subscript i. Substituting that relationship into Equations (A14.5) and (A14.6) and setting the two equations equal, gives the following,

Substituting in from Equation (A14.4), each term of (A14.8) can be simplified as follows,

Furthermore, because t Subscript i Baseline minus t Subscript i minus 1 Baseline equals normal upper Delta t for all t Subscript i, it is easy to show that, if Equation (A14.9) holds for t Subscript i, then it also holds for t Subscript i plus 1. Hence, in this special case, s Superscript upper C upper D upper S can be solved from any one date. Proceeding then from Equation (A14.9), multiply through by e Superscript minus normal lamda t Super Subscript i and simplify to see that,

(A14.10)s Superscript upper C upper D upper S Baseline normal upper Delta t left-bracket 1 plus one half left-parenthesis e Superscript normal lamda normal upper Delta t Baseline minus 1 right-parenthesis right-bracket equals left-parenthesis 1 minus upper R right-parenthesis left-bracket e Superscript normal lamda normal upper Delta t Baseline minus 1 right-bracket
(A14.11)s Superscript upper C upper D upper S Baseline normal upper Delta t StartFraction left-bracket 1 plus e Superscript normal lamda normal upper Delta t Baseline right-bracket Over 2 EndFraction equals left-parenthesis 1 minus upper R right-parenthesis left-bracket e Superscript normal lamda normal upper Delta t Baseline minus 1 right-bracket

Finally, take the limit as normal upper Delta t approaches zero of the terms in the hard bracket on the right‐hand side of (A14.12) to obtain,

(A14.13)s Superscript upper C upper D upper S Baseline almost-equals normal lamda left-parenthesis 1 minus upper R right-parenthesis

A14.4 CDS‐EQUIVALENT BOND SPREADS

Following the logic of the text and this appendix, the expected discounted value of a bond's coupons can be computed along the lines of the fee leg of a CDS. Similarly, the expected discounted value of a bond's principal payment can be computed along the lines of the contingent leg, except that the payment from the bond upon default is upper R times the principal amount, while the payment due to the buyer of protection upon default is 1 minus upper R times the principal amount. Therefore, using Equations (A14.5) and (A14.6) – simplified here to assume that coupon payments are exactly one‐half years apart – the price of a bond, upper P, with a coupon rate, c, given a hazard rate, is given by,

To solve for the CDS‐equivalent bond spread, find the hazard rate that solves Equation (A14.14). Then, using that hazard rate, find the s Superscript upper C upper D upper S that sets the fee leg in Equation (A14.5) equal to the contingent leg in Equation (A14.6).

A14.5 BOND SPREAD WITH MARKET RECOVERY

From the definition of bond spread, if the bond does not default (and rates do not change), then the return on the bond equals the risk‐free or benchmark rate, r, plus the spread. If the bond defaults and recovers upper R Superscript m of its market price, then the return over the moment of default is left-parenthesis upper R Superscript m Baseline upper P minus upper P right-parenthesis slash upper P or left-parenthesis upper R Superscript m Baseline minus 1 right-parenthesis. Therefore, over a short time interval d t years, over which the probability of no default and default are 1 minus normal lamda d t and normal lamda d t, respectively, the expected return on the bond is,

where the approximation follows from ignoring the very small terms, that is, those that are multiplied by left-parenthesis d t right-parenthesis squared.

Assuming that investors are risk neutral or that the hazard rate is a risk‐neutral pricing rate, investors are indifferent between buying a corporate bond and buying a bond without default risk if the expected return of the former, given in Equation (A14.15), is equal to the risk‐free rate. Mathematically, then,

(A14.16)StartLayout 1st Row 1st Column r d t 2nd Column equals left-parenthesis r plus s right-parenthesis d t plus normal lamda d t left-parenthesis upper R Superscript m Baseline minus 1 right-parenthesis 2nd Row 1st Column s 2nd Column equals normal lamda left-parenthesis 1 minus upper R Superscript m Baseline right-parenthesis EndLayout
..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.
Reset
13.58.6.202