Scuola Normale Superiore, Pisa, 23-11-11 CDS and the City Umberto Cherubini Scuola Normale Superiore, Pisa, 23-11-11

Outline CDS and credit derivatives: terms of the contract and valuation principles (including adjustment for credit) Use for hedging and for structured products Single name and multi-name credit derivatives CDS and implied information on credit risk in the government crisis Spigolature…

Credit risk Credit risk is a payoff of the kind min(B,V()) that can be decomposed as B – max(B – V(),0) Generally, speaking, B is the exposure, V(t) is the collateral and  is the time of default. The price of defaultable bond embeds a put option. The value of the exposure is then the risk-free value less a put option. Equity is defined as the difference between the value of collateral and that of debt.

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.

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.

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…

Asset Swap (ASW) 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.

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

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

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.

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).

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

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

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.

Vulnerable default put option CDS are negotiated on Over The Counter (OTC) markets, that is in bilateral transactions. This means that in case of default of the counterparty the other is exposed to credit risk. The risk is due to the cost of substitution of the contract. Assume to buy protection from counterparty A on a bond issued by B. Assume the payment is made upfront. Upfront quotes are made for names that are in financial crisis (example Greece)

Credit Valuation Adjustment Vulnerable Default Put = = default put – CVA CVA = =P(t,T)LgdA LgdA CLL[ELA/ LgdA, ELB/ LgdB] Notice: CLL[x,y] is a copula function ELi/Lgdi = Probability of default of i = A,B.

Credit Linked Note CDS DEALER TRUST INVESTORS COLLATERAL Coupon + premium premium CDS DEALER TRUST INVESTORS protection funds funds coupon COLLATERAL

Synthetic CDO Senior Tranche Originator Junior 1 Tranche SPV Protection Sale Special Purpose Vehicle SPV Junior 2 Tranche CDS Premia Interest Investment … Tranche Collateral AAA Equity Tranche

Standard synthetic CDOs iTraxx (Europe) and CDX (US) are standardized CDO deals. The underlying portfolio of credit exposures is a set of 125 CDS deals on primary names, same nominal exposure, same maturity. The tranches of the standard CDO are 5, 7 and 10 year CDS to buy/sell protection on the losses on the underlying portfolio higher than a given level (attachment) up to another level (detachment) on a nominal value equal to the difference between the two levels.

i-Traxx and CDX quotes, 5 year, September 27th 2005   i-Traxx and CDX quotes, 5 year, September 27th 2005 i-Traxx CDX Tranche Bid Ask 0-3% 23.5* 24.5* 44.5* 45* 3-6% 71 73 3-7% 113 117 6-9% 19 22 7-10% 25 30 9-12% 8.5 10.5 10-15% 13 16 12-22% 4.5 5.5 15-30% (*) Amount to be paid “up-front” plus 500 bp on a running basis Source: Lehman Brothers, Correlation Monitor, September 28th 2005.  

Arbitrage relationship Assume we know the expected loss of two equity tranches 0-% and 0-% ( > ) : what is the price of the mezz tranche -%? It is not difficult to see that to avoid arbitrage opportuniy we must have EL(0-%) – EL(0- %) = EL( -%) Notice that it is the an arbitrage relationship similar to that determining the call spread.

Delta A position in a tranche in a so-called bespoke CDO can be done with tranches of standard CDO with a basket of CDS onn the “names” of the CDO With a subset of CDS The hedge position is called delta

Delta

Delta hedged equity… Over the last years hedge funds have developed the so called “delta hedged equity”, made up by A long position in equity A hedge position The hedge positoon is made : With a short position in the index (delta around 14) With a short position in mezzanine (delta around 2) Goal of the transaction: carry, i.e. difference between returns on assets and liabilities.

…blues Between 5 and 10 may 2005 Index CDX S4 5y lost 9 bp (68 to 59) Equity tranche increased 11.5% (upfront), while it should have increased 6.7% (9 x 16.7 (delta) x 0.045 pv01). Surprise effect was 4.8% Mezz tranche lost 47 bp (running), (while it was expected an increase of 9 x 7 (delta) = 63 bp). The surprise effect was 47 + 63 = 110 bp, corresponding to 4.95% upfront.

Bootstrap 1 Assume that the premium be paid in toto at the end of the default period T = 1: Q(1) = 1 – c1/LGD T = 2: Q(2) = T = N: Q(N) =

Bootstrap 2

Italy

Spain

Portugal

Ireland

Greece

5 Year CDS

CDS term structure

Term structure of default intensity