1. Credit Derivatives Advanced Methods of Risk Management Umberto Cherubini

2. Learning objectives In this lecture you will learn 1.The difference between product specific and issuer specific credit derivatives 2.How to compute the asset swap spread of a bond 3.How to boostrap the probability of default of an issuer using credit default swaps

3. Credit Derivatives Asset Swap (ASW): swap of cash flows of a security against floating payments plus spread. Credit default swap (CDS): purchase and sale of insurance against default of a issuer Total rate of return swap (TRORS): swap of the overall return (coupons + appreciation/ depreciation) against fixed/float premium Credit spread options (CSO): options to buy or sell bonds at a given spread over others.

4. Hedging credit exposure Assume you buy a defaultable security and buy protection in a credit derivative (CD) Avoiding arbitrage requires Titolo Defaultable = Risk-free – CD Notice: –Default amounts to a position in a derivative and makes every product a structured product. –Default risk can be synthetically built independently of “paper” issued by an obligor.

5. Swap contracts Swaps are standard tools to transfer risk from one party to the other. The idea is simply to exchange flows of payments. Each flow is called “leg” of the contract. In the market everything can be swapped: typical examples are –Fixed vs floating payments plus spread (plain vanilla swap) –Payments denominated in different currencies (currency swap) –Floating payments in same currency but indexed to yield curves of different currencies (quanto swap) –Asset swap, total return swap, credit default swap…

6. Interest rate swaps (plain vanilla) In a plain vanilla IRS –The party which is long pays a cash flow of fixed payments equal to a percentage c of the notional value, expressed in annual basis –The short party pays a cash flow which is indexed to a market rate (typically Euribor, Libor) Value of the fixed leg: Value of the float leg:

7. The parameters in a swap contract The value of a swap contract can be expressed as : – Net-present-value (NPV), or upfront; it is the difference between the present value of cash flows that will be received and those that will be paid. –Running premium (swap rate): the value of a fixed leg payment such that the present value of the fixed be equal to the value of the float leg. –Spread: value of a fixed percentage payment (on a annual basis) to be added to (o subtracted from) the floating payments to equate the present value of the float leg to the present value of the fixed leg.

8. Indexed coupons An index coupon is determined based on an index, typically an interest rate, observed at a date , defined reset date The typical case, denoted natural time lag consists of a coupon with – accrual period from time  to time T – reset date  and payment date T – reference rate for the determination of the coupon i( ,T) (T –  ) = 1/v ( ,T) – 1 Notice that typically the interest rate used is with simple compounding, for the market convention that uses this compound rule for payments with less than one year maturity.

9. Replicating portfolio What is the replicating portfolio of an indexed coupon? Notice that at time  the value of the coupon, determined at time  and paid at time T, given by v ( ,T) i( ,T) (T –  ) = 1 – v ( ,T) The replicating portfolio which is natural to select is –A long position (investiment) of one unit of cash available at time  –A short position (funding) for one unit of cash available at time T

10. Cash flows of a indexed coupon Using the argument above, a indexed coupon is actually a derivative. A derivative is actually a porfolio of long and short positions. In this case the long position is on a zero coupon bond expiring on the reset date and the short position is on a zero coupon bond expiring at time T. An indexed coupon then hides a debt position (leverage)

11. In remembrance of swap rate In a fixed/float swap it is market convention that at origin Value of fixed leg = Value of floating leg

12. Swap rate: definitions 1.Fixed running payment, expressed in percentage of the notional value and in annual terms, equivalent to a flow of indexed payments. 2.A weighted average of forward rates, with weigths given by the discount factors. 3.The internal rate of return of a coupon bond issued at par (par yield curve)

13. Bootstrapping procedure Assume that at time t the market is structured on m periods with maturities t k = t + k, k=1....m, and swap rates are observed on these maturities. Since swap rates corresponds to par yields, and that for par yields coupon rates are equal to the yield, we recover the same bootstrapping procedure defined for fixed rate bonds.

14. Asset Swap (ASW) L’asset swap is a package made up by –A bond –A swap contract The two parties of the contract pay –The cash flows of the bond plus the difference between par and the market value of the bond, if the different is positive positive –Floating payments plus a spread (that may be positive or negative) and the difference between market value and par, it the difference is positive.

15. Asset Swap (ASW) Asset Swap on bond DP(t,T;c) Fixed leg value: Float leg value:

16. Asset Swap (ASW) Spread Spread is recovered equating the value of legs Notice that spread is zero iff

18. Credit Default Swap A credit default swap is an exchange contract in which the protection buyer receives insurance against loss on a set of bonds issued by a reference entity, called a “name” in jargon, against a flow of fixed payments, typically on a running basis. The flow of payments stops at the maturity of the contract or the date of default of the name, whatever comes first. The value of fixed payment is determined in such a way as to equate the value of the contract to zero at the origin of the contract.

19. Credit Default Swap The underlying asset of a CDS is not a bond, like for ASW, but a “name”, that is the issuer. The payment can be done either way between: cash settlement or physical delivery. The latter is the rule rather than the exception and implies the presence of a delivery option. Default isd defined among a set of credit events, specified in the ISDA standars –Bankrupcy –Obligation acceleration –Obligation default –Failure to pay –Moratorium/repudiation –Restructuring The protection seller pays: (1  > t(i-1) – 1  > t(i) ) LGD The protection buyer pays: bp premium * 1  > t(i-1).

20. Payment arrangements The evaluation of a CDS is based on survival probability Q(T). We can assume that the payment of the premium for period t occurred in full if the name defaults between time t – 1 an t or that it did not happen at all

21. Accrued premium payment at default The most common payment structure is that, in case of default at time , the protection buyer pays accrued premium until that date and the protection seller pays the LGD at the time of default. In the case of a 1 year we have

22. A CDS for N years The N year generalization of a CDS is given by The simplified versions with payments at reset dates represent good approximations.

23. Default probability boostrap Credit default swap su Fiat 25/01/2002 MaturityBidAsk 11.451.55 31.701.75 51.801.83 101.952.15

24. Bootstrap 1 Assume that the premium be paid in toto at the end of the default period T = 1: Q(1) = 1 – c 1 /LGD T = 2: Q(2) = T = N: Q(N) =

25. Bootstrap 2

26. Italy

27. Spain

28. Portugal

29. Ireland

30. Greece

31. 5 Year CDS