Credit Risk: Risk Premia and Hedging

Credit Risk: Risk Premia and Hedging

Prof. Dr. Holger Kraft

Goethe University Frankfurt

House of Finance

Holger Kraft Credit Risk: Risk Premia and Hedging 1/49

Agenda

1 Introduction

2 Measuring Credit Risk and Risk Premia

3 Credit Derivatives

4 Hedging of CDO Contracts

5 Conclusion


Agenda

1 Introduction




5 Conclusion


Talk

This talk is based on the following papers:

Bick, Hirsch, Kraft, Yildirim: Default and Idiosyncratic RiskAnomalies Revisited, Working Paper, Frankfurt and Syracuse.

Bick, Kraft: Hedging Structured Credit Products during theCredit Crunch: A Horse Race of 10 Models, Working Paper,Frankfurt.


Topics

This talk focuses on measuring and hedging of credit risk.

First, we discuss the estimation of physical and risk-neutraldefault probabilities and the calculation of default premia.

In our research, we studied whether default risk is correctlypriced in the Fama-French model.

This will only be a side-aspect in this talk.

Second, we ask the question of whether CDO tranches couldbe hedged during the financial crisis.


Agenda

1 Introduction




5 Conclusion


Types of Risks

A firm is exposed to several types of risks. The most importantones are

Market Risk: Risk related to movements in market variables suchas stock indices, interest rates, exchange rates etc.

Credit Risk: Risk that a loss occurs because of a default by acounterparty.

Both are very different since

default is a 0-1-event

default risk is harder to measure

default risk cannot be hedged away by a market index


How to Quantify Credit Risk?

There are two important dimensions of credit risk:

1 How likely is a default?

2 How big is the loss if a default occurs?

These dimensions are captured by the

1 default probability (PD),

2 loss given default (LGD).

Note that LGD = 1 - recovery rate.


Pricing of Defaultable Bonds

Payoff of a Defaultable Zero

pd(T ,T ) =

{1 if firm is solvent at TR = 1− L if firm is insolvent at T

,

where R is the recovery rate and L the loss rate.

Key Insight

Prices are expected discounted payoffs under the risk-neutralmeasure.

Use this insight and calculate the price of a defaultable zero:

Price of a Defaultable Zero

pd(0,T ) = (SPQ · 1 + PDQ · R) · e−rT = (1− L · PDQ) · p(0,T )

SPQ and PDQ are risk-neutral survival and default probabilities.


Risk-neutral Default Probabilities

You can use bond data to back out risk-neutral defaultprobabilities:

PDQ =p(0,T )− pd(0,T )

L · p(0,T )

Example. Prices of zeros with maturity one year

p(0, 1) = 0.97, pd(0, 1) = 0.955

Rule of Thumb: Loss rate is 60%.

=⇒ PDQ =p(t,T )− pd(t,T )

L · p(t,T )=

0.97− 0.955

0.6 · 0.97= 0.0258,

i.e. the 1-year default probability of firm XY is 2.58% and thus the1-year survival probability is 97.42%.


Risk-neutral vs. Physical Default Probabilities

In practice, we often need real (=physical) probabilities

PD = Prob(“default”),

instead of the risk-neutral probability

PDQ = ProbQ(“default”).

Rule

For pricing purposes, we use risk-neutral probabilities and, for riskmanagement purposes, we use physical probabilities.

Crucial Q: How are these two probabilities related?

Empirical evidence shows that PDQ > PD.


Risk-neutral vs. Physical Default Probabilities

250

lam_P lam_Q drp

200

150

100

50

0

09 09 09 09 09 09 10 10 10 10 10 10 10 10 10

1‐Jul‐0

1‐Au

g‐0

1‐Sep‐0

1‐Oct‐0

1‐Nov‐0

1‐De

c‐0

1‐Jan‐1

1‐Feb‐1

1‐Mar‐1

1‐Ap

r‐1

1‐May‐1

1‐Jun‐1

1‐Jul‐1

1‐Au

g‐1

1‐Sep‐1

Estimated actual and risk-neutral default intensities for KraftFoods Inc.


Physical Default Probabilities

As we have seen, from bond prices we can back out risk-neutraldefault probabilities.

Q: How can we estimate the default probability of a firm?

Example “Unbalanced Coin”: To estimate the probability ofhead, one can toss a coin several times (e.g. 100) and count thenumbers of heads (e.g. 40).

=⇒ Prob(“head”) = 40%

Q: Can we apply this idea to estimate a firms’s default probability?

Example “General Electric”: For more than 100 years, we donot observe any default of GE.

=⇒ Prob(“default GE”) = 0% ???

This is obviously not true.Holger Kraft Credit Risk: Risk Premia and Hedging 13/49

Physical Default Probabilities

Idea: Group firms into classes of firms with the same creditqualities and count the number of defaults in every class.

This leads to the concept of ratings.

Formally, a rating is nothing else than an estimate of a firm’sphysical default probability disguised in shortcuts like AA or B.


Estimating Physical Default Probabilities

To estimate actual default probabilities, we apply a dynamiclogit approach.

We assume that the marginal probability of default over thenext period is given by a logistic distribution

PD(t − 1, t) = Probt−1(Dit = 1) =1

1 + exp(−α− βxi ,t−1)

Dit is a dummy variable equal to one if the firm defaultswithin month t and zero otherwise.

xi ,t−1 represents a vector of explanatory variables.


Default information from Moodys‘s default risk servicedatabase (1986-2009): Rated Non-financial Firms

Year Active Firms Defaults (%)1986 868 9 1.031987 912 10 1.091988 856 10 1.161989 799 10 1.251990 745 19 2.541991 705 12 1.701992 747 7 0.931993 820 6 0.731994 906 4 0.441995 950 5 0.521996 1050 7 0.661997 1164 10 0.851998 1282 19 1.481999 1357 24 1.762000 1361 25 1.832001 1332 44 3.302002 1301 27 2.072003 1268 17 1.332004 1293 7 0.542005 1289 10 0.772006 1265 2 0.152007 1223 2 0.162008 1182 11 0.922009 1286 27 2.09


Variables Explaining Actual Defaults

Variable Mean Median Std. Dev. p25 p75Panel A: Entire sample

Rel. Net Income (NIMTAAVG) .0032 .0056 .0133 .0009 .0101Leverage (TLMTA) .5676 .5654 .2401 .3818 .7775Rel. Stock Return (EXRETAVG) -.0029 .0001 .0384 -.0212 .0192Stock Vol. (SIGMA) .4186 .3399 .2670 .2393 .5028Rel. Size (RSIZE) -9.2308 -9.0282 1.5105 -10.1686 -7.8763Rel. Cash (CASHMTA) .0603 .0300 .07849 .0103 .0782Market-to-book (MB) 2.0621 1.8068 1.4875 1.1084 2.2757Trunc. Log Share (PRICE) 2.4743 2.7080 .5535 2.6119 2.7080

Panel B: Default subgroupRel. Net Income (NIMTAAVG) -.02856 -.02370 .0243 -.0482 -.0091Leverage (TLMTA) .8863 .9216 .0813 .8872 .9216Rel. Stock Return (EXRETAVG) -.0842 -.0871 .0567 -.1260 -.0427Stock Vol. (SIGMA) 1.2003 1.326 .3232 .9765 1.4661Rel. Size (RSIZE) -12.3357 -12.5299 1.4814 -13.6115 -11.4249Rel. Cash (CASHMTA) .0570 .0311 .0701 .0127 .0728Market-to-book (MB) 2.2531 1.0040 2.4860 .4483 2.2757Trunc. Log Share (PRICE) .2359 .2271 1.0631 -.4259 .9530


Default Prediction

Explanatory variable Coefficient (z-value)Rel. Net Income (NIMTAAVG) -16.66***

(-6.42)Leverage (TLMTA) 5.89***

(7.44)Rel. Stock Return (EXRETAVG) -4.51***

(-4.02)Stock Vol. (SIGMA) 1.97***

(8.97)Rel. Size (RSIZE) -.099*

(-1.95)Rel. Cash (CASHMTA) -4.51***

(-5.41)Market-to-book (MB) .001

(0.05)Trunc. Log Share (PRICE) -.431***

(-5.80)Constant -13.28***

(-15.41)Observations 311,436Defaults 324Pseudo-R2 0.3644


Calculating Default Premia

Our estimated logit model can be used to generate a timeseries of physical default probabilities.From these default probabilities, one can calculate thephysical default intensity λP

PD(t,T ) = 1− eλP(T−t) ≈ λP(T − t).

Fancier models for λP are possible!Eventually, we wish to calculate default risk premia:

χ =λQ

λP,

Therefore, we need to estimate the risk-neutral defaultintensity λQ .Under the assumption of constant intensities, CDS quotesgive an easy access to these intensities:

CDS = `λQ ,

Again fancier models are possible!Holger Kraft Credit Risk: Risk Premia and Hedging 19/49

Price Data: CDS Quotes from Markit

Year Mean Std. Dev. p25 p50 p75 Observations

2001 141 224 48 83 149 45,5762002 216 384 58 97 218 83,5572003 153 282 36 61 146 97,6012004 130 280 32 53 126 125,1012005 136 384 29 48 117 133,1552006 107 186 25 44 112 131,3802007 121 177 28 50 140 149,7402008 308 614 68 128 320 123,5872009 418 1177 77 155 372 117,0972010 209 479 66 112 206 89,288

Total 193 524 39 75 185 1,096,082


Annual Default Risk Premia

Year Mean Std. Dev. p25 p50 p75 Observations

2001 64.04 107.45 10.01 24.81 66.20 249572002 99.80 183.89 11.35 31.65 94.92 484562003 78.84 158.49 10.80 27.47 77.36 562042004 93.14 183.43 15.88 42.15 103.18 722972005 107.20 220.79 18.75 50.23 122.88 766392006 107.27 227.14 18.00 47.03 109.28 751022007 127.11 263.51 21.22 55.41 134.45 861422008 140.92 362.50 18.59 58.94 144.87 737642009 78.46 155.33 10.92 33.39 85.15 865822010 120.92 154.23 29.71 71.80 148.39 73377

Total 105.24 222.02 15.85 45.05 113.87 673520


Fama-French Regressions: Portfolios Sorted on LAM P

PF1 PF2 PF3 PF4 PF5 PF5 − PF1mktrf 0.9863*** 0.9552*** 1.0026*** 1.0195*** 1.1004***

(77.9184) (55.3096) (43.5272) (41.5917) (47.3288)smb -0.1627*** -0.0774*** -0.0582* -0.1217*** 0.1655***

(-8.8456) (-3.9018) (-1.7144) (-4.3627) (3.8725)hml -0.3868*** 0.0870 0.2958*** 0.2263*** 0.6994***

(-11.8191) (1.5729) (5.4369) (3.6903) (7.6093)umd 0.1263*** 0.0426 0.1419*** -0.0773** -0.3039***

(3.6763) (1.0333) (2.9310) (-2.4275) (-5.9120)alpha 0.0113 -0.03105 0.04025 -0.0220 -0.102** -0.1133**

(0.7918) (-1.1371) (1.6099) (-0.6319) (-2.1201) (-2.1050)size 16.69 16.10 15.85 15.53 14.61average return 0.0634 0.0581 0.1201 0.0871 0.0664

This is the so-called default anomaly.


Cumulated Alphas

0.01

lam_P lam_Q dp

0.005

0

2001 2002 2003 2005 2006 2007 2008 2010‐0.005

2001 2002 2003 2005 2006 2007 2008 2010

0 015

‐0.01

‐0.02

‐0.015

‐0.025


Agenda

1 Introduction




5 Conclusion


The Market for Credit Derivatives

70000

Credit FX Equity Commodity

60000

50000

30000

40000

20000

30000

10000

0

Dec.2004 Dec.2005 Dec.2006 Dec.2007 Dec.2008 Dec.2009 Dec.2010

Amounts outstanding of over-the-counter (OTC) derivatives, inbillions of US dollars notional amounts outstanding(Source: BIS Quarterly Review: September 2011)


What is a CDS Contract?

Credit Default Swap

The buyer of a CDS acquires protection from the seller against adefault by a particular company or country (reference entity)

A CDS contract can be interpreted as an insurance contractagainst the default of a third party.

A credit default swap is an agreement between twocounterparties that allows one counterparty to “long” athird-party credit risk, and the other counterparty to “short”the credit risk.

The most controversial area of the specification of thecontract is definition of the credit event (filing forbankruptcy, failure to pay, etc.)


CDS Mechanics

No Default

Markit Credit Indices: A Primer

Copyright © 2009, Markit Group Limited. All rights reserved. www.markit.com 4

Scope This document aims to outline the different credit indices owned and managed by Markit, their characteristics and differences, and how they trade. We focus on synthetic indices backed by single name bonds CDS (senior unsecured) and single name loans CDS (senior secured): the Markit CDX and Markit iTraxx for bonds, and the Markit iTraxx LevX and Markit LCDX for loans. We purposely do not cover synthetic structured indices, such as the ABX and the CMBX, as their functioning is quite different.

Section 1 – Credit Default Swaps Definition A Credit Default Swap (CDS) is a contract between two parties, a protection buyer who makes fixed periodic payments, and a protection seller, who collects the premium in exchange for making the protection buyer whole in case of default. In general trades are between institutional investors and dealers. CDS are over-the-counter (OTC) transactions. They are similar to buying/selling insurance contracts on a corporation or sovereign entity’s debt, without being regulated by insurance regulators (unlike insurance, it is not necessary to own the underlying debt to buy protection using CDS). Before trading, institutional investors and dealers enter into an ISDA Master Agreement, setting up the legal framework for trading. Each contract is defined by:

A Reference Entity (the underlying entity on which one is buying/selling protection on); A Reference Obligation (the bond or loan that is being “insured” - although it doesn’t have to be the

deliverable instrument in a default situation and doesn’t have to have the same maturity as the CDS, it designates the lowest seniority of bonds that can be delivered in case of default);

A Term/Tenor (5 years are the most liquid contracts); A Notional Principal; Credit Events (the specific events triggering the protection seller to pay the protection buyer – The defined

events are bankruptcy, failure to pay, debt restructuring, and the rare obligation default, obligation acceleration, and repudiation/moratorium).

Markit Reference Entity Database (RED) is the market standard that confirms the legal relationship between reference entities that trade in the credit default swap market and their associated reference obligations, known as “pairs”. Each entity is identified with a unique 6-digit alphanumeric code, and a 9-digit code identifies the pair. RED codes are widely and successfully used by CDS market participants to electronically match and confirm CDS transactions. The RED “preferred reference obligation” is the default reference obligation for CDS trades based on liquidity criteria. In case of a credit event, under physical settlement the protection buyer has to deliver a bond of seniority at least equal to that of the reference obligation – if there are multiple bonds deliverable, the protection buyer will most likely deliver the cheapest bond to the protection seller. We can represent the life of a CDS with the following cash flows: From initiation of trade to maturity if there is no Credit Event:

Default

Protection Buyer

Default Protection

Seller

Fixed Quarterly Cash Payments (basis points per annum)

Default



In the case of a Credit Event:

Note that the standard settlement mechanism for Loan CDS and Corporate CDS is cash settlement via a credit event auction. The difference with the example above is that instead of exchanging a cash instrument for par, the exchange is par minus the recovery set in the auction. More details will be covered under Auctions later.

Types The different types of CDS contracts traded:

CDS: indicates that the underlying reference entities and obligations are senior unsecured bonds, issued by corporate or sovereign issuers

LCDS: Loan-only CDS refers to contracts where protection is bought and sold on syndicated secured leveraged loans. These are higher in the capital structure (and with higher recovery rates) than CDS.

MCDS: The reference entity is a municipality, and the reference obligation a municipal bond. ABCDS: CDS on structured securities (Asset Backed Securities typically) Preferred CDS: CDS on Preferreds

Uses Hedging

CDS allow capital or credit exposure constrained businesses (banks for example) to free up capacity to facilitate doing more business.

CDS can be short credit positioning vehicle. It is easier to buy credit protection than short bonds. For LCDS, counterparties can assign credit risk of bank loans without requiring consent of lender (assigning

bank loans often requires borrower consent/notification), therefore CDS reduce bank exposure to credit risk without disturbing client relationships.

CDS may allow users to avoid triggering tax/accounting implications that arise from sale of assets Investing

Investors take a view on deterioration or improvement of credit quality of a reference credit CDS offer the opportunity to take a view purely on credit CDS offer access to hard to find credit (limited supply of bonds, small syndicate) CDS allows investors to invest in foreign credits without bearing unwanted currency risk Investors can tailor their credit exposure to maturity requirements, as well as desired seniority in capital

structure CDS require no cash outlay and therefore creates leverage

Default

Protection Buyer

Default Protection

Seller

Reference Obligation at x% of par +

Accrued CDS Interest

100% of par of the underlying Reference Obligation


Example: AT&T

On 01/29/10 two counterparties A and B enter into a 5 yearCDS on a notional value of USD 100m in which A pays B82bps (=0.82%) per year, i.e. 820,000 dollars.

The reference entity is AT&T Inc.

If AT&T defaults at any time within that five years, then Awill get an insurance payment from B.


Credit Indices: CDX and iTraxx

The CDX-NAIG index is an equally weighted portfolio of 125investment grade North American companies.

The CDX-NAHY index is an equally weighted portfolio of100 below investment grade North American companies.

The iTraxx Europe is an equally weighted portfolio of 125investment grade European companies.

If the five-year CDX index is bid 96 offer 97, then a portfolioof 125 CDS contracts on the CDX companies can be boughtfor 97bps, e.g., USD 800,000 of 5-year protection on eachname (total USD 100m) could be purchased for USD 970,000per year. When a company defaults the annual payment isreduced by 1/125.

On 01/29/10 the spread on the CDX-NAHY was 575bps.


CDX-NAIG: Some Current Members

Markit CDX Indices

Constituents for CDX.NA.IG

Summary Constituents

Index Constituents

Short Name Entity CLIP Reference Obligation Av Rating Sector Weight

ACE Ltd 0A4848 ACE-INAHldgs 8.875 15Aug29 A Financial 0.800%

Aetna Inc. 0A8985 AET 6.625 15Jun36 BondCall A Financial 0.800%

Alcoa Inc. 014B98 AA 5.72 23Feb19 BBB Materials 0.800%

Altria Gp Inc 0C4291 MO 9.7 10Nov18 BBB Consumer Stable 0.800%

Amern Elec Pwr Co Inc 027A8A AEP 5.25 01Jun15 BBB Utilities 0.800%

Amern Express Co 027D97 AXP 4.875 15Jul13 A Financial 0.800%

Amern Intl Gp Inc 028EFB AIG 6.25 01May36 Struc BBB Financial 0.800%

Amgen Inc. 0D4278 AMGN 4.85 18Nov14 BondCall A Consumer Stable 0.800%

Anadarko Pete Corp 0A3576 APC 5.95 15Sep16 BondCall BBB Energy 0.800%

Arrow Electrs Inc 0E69A8 ARW 6.875 01Jun18 BBB Industrial 0.800%

AT&T Inc 0A226X ATTINC 5.1 15Sep14 A Communications and Technology 0.800%

AT&T Mobility LLC 0A232K ATTINC-ML (4) 7.125 15Dec31 A Communications and Technology 0.800%

Autozone Inc 0F8665 AZO 5.5 15Nov15 Struc BBB Consumer Cyclical 0.800%

Avnet, Inc. 058B87 AVT 6 01Sep15 BondCall BBB Industrial 0.800%

Barrick Gold Corp 06DG91 ABX 5.8 15Nov34 BBB Materials 0.800%

Baxter Intl Inc 0H8994 BAX 6.625 15Feb28 A Consumer Stable 0.800%

Boeing Cap Corp 09G715 BA-CapCorp 5.8 15Jan13 A Financial 0.800%

Boston Pptys Ltd Partnership 1B123T BPLP 6.25 15Jan13 BondCall BBB 0.800%

Bristol Myers Squibb Co 1C1134 BMY 6.8 15Nov26 A Consumer Stable 0.800%

Burlington Nthn Santa Fe Corp 1D39H2 BNI 4.3 01Jul13 BBB Industrial 0.800%

Campbell Soup Co 1E786B CPB 4.875 01Oct13 A Consumer Stable 0.800%

Cdn Nat Res Ltd 1E99BD CNQ 5.45 01Oct12 BBB Energy 0.800%

Cap One Bk USA Natl Assn 1F445B COF-BNKNA 5.125 15Feb14 A Financial 0.800%

Cardinal Health Inc 1F55D7 CAH 5.85 15Dec17 BondCall BBB Consumer Cyclical 0.800%

Carnival Corp 1F79BD CCL 6.65 15Jan28 A Industrial 0.800%

Caterpillar Inc 15DA35 CAT 5.7 15Aug16 BondCall A Consumer Cyclical 0.800%

CBS Corp 136CDC CBSCOR 4.625 15May18 BBB Consumer Cyclical 0.800%

CenturyTel Inc 16BD70 CTL 6 01Apr17 BondCall BBB Communications and Technology 0.800%

Cigna Corp 137A59 CI 7.875 15May27 BBB Financial 0.800%

Cisco Sys Inc 1I99CW CSCO 5.5 22Feb16 BondCall A Communications and Technology 0.800%

Comcast Corp 2C033N CMCSA 5.3 15Jan14 BBB Communications and Technology 0.800%

Computer Sciences Corp 2C5899 CSC 5 15Feb13 BBB Communications and Technology 0.800%

ConAgra Foods Inc 225DGF CAG 7 01Oct28 BBB Consumer Cyclical 0.800%

ConocoPhillips 228A7H COP 5.9 15Oct32 (2) A Energy 0.800%

Constellation Engy Gp Inc 2D13A8 CEG 4.55 15Jun15 BBB Utilities 0.800%

Cox Comms Inc 2E6448 COX-CommInc 6.8 01Aug28 BBB Communications and Technology 0.800%

CSX Corp 138A48 CSX 5.3 15Feb14 Struc BBB Industrial 0.800%

CVS Caremark Corp 138CAK CVSCRM (3) 4.875 15Sep14 BBB Consumer Cyclical 0.800%

Darden Restaurants Inc 25A8AD DRI 6 15Aug35 BondCall BBB Consumer Cyclical 0.800%

Deere & Co 2G85AI DE 6.95 25Apr14 A Consumer Cyclical 0.800%

Dell Inc 26B72T DELLN 7.1 15Apr28 A Communications and Technology 0.800%

Devon Engy Corp 2H68GV DVN 7.95 15Apr32 BBB Energy 0.800%

DIRECTV Hldgs LLC 2H99EY DTV-Hldgs (2) 6.375 15Jun15 BondCall BBB Communications and Technology 0.800%

Markit CDX Indices http://www.markit.com/en/products/data/indices/credit-and-loan-indices/...

1 of 1 1/31/2010 8:08 PM


CDO

The underlying of a CDO is a pool of loans, CDS contractsetc.

The pool is sliced into tranches.

The cash flows generated by the pool are used to service thetranches according to the seniority (waterfall).

Therefore, the holder of a tranche receives interest payments,but has to cover losses that are attributed to the tranche.

Like Credit Defaut Swaps, a CDO tranche can thus be split upinto two legs:

Fee leg: Interest rate paymentsProtection leg: Payments upon default


CDO Structuring



Section 4 – Tranches Some of the credit indices are also available in a tranched format, which allows investors to gain exposure on a particular portion of the index loss distribution. Tranches are defined by attachment and detachment points. Defaults affect the tranches according to the seniority of the tranche in the capital structure. Example of the CDX.NA.HY tranches: Tranche Mechanics: The protection buyer of a tranche makes quarterly coupon payments to the protection seller and receives a payment in case there is a credit event in the underlying portfolio. Upfront payments are made at initiation and close of the trade to reflect the change in price. Coupon payments (500bps or 100bps per annum) are made until the notional amount of the tranche gets fully written down due to a series of credit events or until maturity whichever is earlier. Payments are made by the protection seller as long as the losses are greater than the attachment point and less than the detachment point for that tranche. Once the total loss reaches the detachment point, that tranche notional is fully written down. The premium payments are made on the reduced notional after each credit event. Example: An index has 100 equal weighted names, and has the following tranches: 0-5, 5-8, 8-12. 12-15, 15-100 (in this case the 5-8 tranche has an attachment point of 5 and detachment point of 8). Investor B bought protection on the 0-5% tranche with a notional of $10 million. One name defaults – Recovery is set at 65% (35% Loss Given Default – LGD). The payout from the protection seller is: (Notional * LGD * Weighting) / Tranche Size Or $700,000 to Investor B. The 0-5% tranche is adjusted for the reduced notional (0.35 based on LGD) and 4.65% of the notional remains. The new detachment point has to be adjusted for the number of remaining names in the index, using a factor of 0.99 (the 0-5 tranche for new trades now becomes a 0-4.69 tranche). The original principal of the other tranches is unaffected but now has a smaller cushion protecting them against further losses, except that of the super senior tranche which notional is adjusted for the recovery. The detachment point doesn’t change, but the notional is adjusted for the recovery rate. The loss goes to the equity tranche, the recovery to the super senior. Tranche Accruals: After June 22nd 09, tranches mimic indices with an upfront at the trade date, the seller paying the buyer the accrued up to trade date, and the buyer paying full coupon at next payment date. So no matter when a trade is entered, the coupon legs always accrue from the same date and make unwinds, transfers and collapses very easy.

100 Equally Weighted

CDS

Super Senior

Junior Senior

Senior Mezz

Junior Mezz

Equity

Detachment Point – 15%

Attachment Point – 10% 0-10%

10-15%

15-25%

25-35%

35-100%

Source: Markit


CDO: Protection Legs

T_1 T_2 T_3 T_4 T_50

K2

K3

1

Time

Loss

Lt

Tranche Swap Default Leg

Loss covered byProtection seller

Super Senior

Senior

Senior Junior

Mezzanine Senior

Mezzanine Junior

Equity


CDO: Loss Process

At the default time τi of firm i , there is a loss of`i = (1− Ri )Ni (expressed in dollars).

The total loss until time t is given by

Lt ≡I∑

i=1

`i1{τi≤t}

We are also interested in the tranche loss of tranche m attime t.

Relative Tranche Loss

The relative loss of the tranche [Km−1,Km] can be written as

Lmt ≡max{Lt − Km−1; 0} −max{Lt − Km; 0}

Km − Km−1.


Absolute Tranche Loss: Mezzanine

K_2 K_3

F_3=K_3-K_2

Portfolio Loss Lt

Tra

nche

Los

sTranche Loss


CDO Legs

Value of Fee Leg

V f ,mt = δ

∑tk>t

p(t, tk)(1− Et [Lmtk

]).

Value of Protection Leg

V p,mt =

∑tk>t

p(t,

tk−1+tk2

)(Et [L

mtk

]− Et [Lmtk−1

]).

The fair spread is given by smt = V p,mt

V f ,mt

Challenges. Calculate the expected percentage tranche lossesEt [L

mt ].


Agenda

1 Introduction




5 Conclusion


Hedging CDO Tranches

Practitioners like to hedge CDO tranches by CDS contracts(single name or index).

Given a model one can (numerically) calculate deltas oftranches with respect to CDS contracts.

This idea is similar to Black-Scholes delta hedging.

We perform a horse race of 10 models.

Question

Did delta hedging work during the crisis?


10 Models

Goal: Compute Default Probabilities

Bottom-up modelsDefault times are modeled using an intensity model.Aggregation of single-name default probabilities via model forcorrelation structure (e.g. Copula)

Top-down modelsLoss process L is modeled directly.Contagion effects can be addressed in a convenient way.


Top-down Models

Longstaff-Rajan model

Lt = 1− e−γ1N1te−γ2N2te−γ3N3t

Self-exciting model: Intensity of loss process

dλt = κ(θ − λt) dt + δ dLt

Dynamical Generalized Poisson Loss Model: Defaultprocess is given by (Mi Poisson)

Nt = max{Zt , I} with Zt =I∑

i=1

αiMit


Calibration

To Do: Calibrate all models to market data (CDO, CDSindex, CDS)

Bottom-up: Each tranche is represented by one correlation(implied correlation, base correlation)

Top-down: Calibrate several parameters(∑m

(CDOm

t −Modmt (Θ)

CDOmt

)2

+

(Indext −Mod ind

t (Θ)

Indext

)2)1/2


Calibration

Model [0− 3%] [3− 6%] [6− 9%] [9− 12%] [12− 22%]

Gau 0 0 0 0.01 0.26Clay 0 0 0 0 0

DT (4/4) 0 0 0 0.01 2.49DT (6/4) 0 0 0 0.01 2.03DT (6/6) 0 0 0 0.01 1.85T (4) 0 0 0 0 0.72T (8) 0 0 0 0.01 0.85T (12) 0 0 0 0.01 0.73

Average calibration error in percent implied by base correlations. The timeperiod is 09/01/08 to 09/22/08.

Model [0− 3%] [3− 6%] [6− 9%] [9− 12%] [12− 22%]

SE 0 16.16 2.83 0.94 13.06DGPL 0.01 0.85 1.47 6.46 3.49LR 0.06 0.08 0.06 0.12 0.07

Average calibration error in percent across all top-down models. The timeperiod is 09/01/08 to 09/22/08.


Hedging of CDO Tranches

Calculate deltas ψm,index/CDS

P&L analysis

PLt = ∆MTMmt − ψ

m,index/CDSt−1 ∆MTM

CDS/Indext

where MTM mark-to-market value.

Hedging Error

Average absolute P&L of the hedged position

Average absolute P&L of the unhedged position

Vola Residual

Volatiltiy of the P&L of the hedged position

Volatiltiy of the P&L of the unhedged position


Hedging of Tranches with CDS Contracts

Model [0 − 3%] [3 − 6%] [6 − 9%] [9 − 12%] [12 − 22%]

IndexGau 83.7 (75.27) 82.01 (76.35) 84.08 (77.28) 108.65 (95.31) 113.48 (118.23)T(4) 82.5 (76.66) 83.39 (77.76) 79.64 (74.83) 96.83 (86.67) 94.02 (85.07)SE 230.32 (223.69) 140.09 (131.43) 108.26 (102.53) 126.61 (117.22) 118.8 (108.53)

DGPL 165.74 (153.33) 91.66 (86.22) 84.5 (82.84) 104.42 (96.45) 106.5 (96.09)

Portfolio of three single-name CDSGau 75.08 (70.61) 69.89 (68.63) 71.92 (66.31) 114.87 (112.78) 110.4 (101.96)T(4) 77.81 (74.37) 72.18 (69.55) 70.04 (68.99) 90.96 (78.94) 91.09 (89.14)SE 194.11 (198.56) 111.04 (105.29) 83.65 (81.07) 96.61 (93.29) 94.19 (89.13)

DGPL 142.47 (138.93) 75.63 (72.53) 72.63 (70.32) 88.51 (81.71) 89.75 (83.74)

Hedging tranches with index CDS or a portfolio of threesingle-name CDS contracts in September 2008.


Empirical Dependence of Tranches and CDS Index

0

0.005

0.01

0.015

‐0.1 ‐0.08 ‐0.06 ‐0.04 ‐0.02 0 0.02 0.04 0.06 0.08urn on index

‐0.015

‐0.01

‐0.005Retu

Return on [0, 3%] tranche

0

0.005

0.01

0.015

‐0.015 ‐0.01 ‐0.005 0 0.005 0.01urn on index

‐0.015

‐0.01

‐0.005Retu


Daily index returns versus daily tranche returns.


Hedging of Tranches with Tranches

T(4)Hedging instrument [0 − 3%] [3 − 6%] [6 − 9%] [9 − 12%] [12 − 22%]

[0 − 3%] 0 (0) 26.35 (27.21) 29.92 (33.42) 60.3 (61.34) 52.99 (52.99)[3 − 6%] 26.3 (25.74) 0 (0) 22.27 (20.46) 48.85 (45.68) 46.18 (43.26)[6 − 9%] 29.86 (33.86) 21.91 (20.88) 0 (0) 49.69 (52.03) 45.49 (41.74)

[9 − 12%] 98.46 (104.55) 80.89 (81.94) 87.18 (93.41) 0 (0) 46.65 (46.74)[12 − 22%] 82.23 (85.82) 67.11 (67.04) 71.54 (71.37) 42.06 (40.13) 0 (0)

DGPLHedging instrument [0 − 3%] [3 − 6%] [6 − 9%] [9 − 12%] [12 − 22%]

[0 − 3%] 0 (0) 57.31 (53.01) 59.2 (56.64) 59.3 (56.42) 61.07 (57.93)[3 − 6%] 111.47 (105.3) 0 (0) 18.75 (16.6) 36.71 (31.04) 48.03 (41.64)[6 − 9%] 116.59 (113.83) 19.24 (17.13) 0 (0) 29.51 (29.29) 40.95 (37.59)

[9 − 12%] 106.17 (100.43) 34.23 (28.32) 26.86 (25.52) 0 (0) 20.92 (21.03)[12 − 22%] 108.31 (109.24) 42.41 (37) 34.94 (31.88) 19.66 (20.52) 0 (0)

Hedging tranches with tranches in September 2008.


Empirical Dependence of Tranches

‐0.02

0

0.02

0.04

0.06

‐0.1 ‐0.08 ‐0.06 ‐0.04 ‐0.02 0 0.02 0.04 0.06 0.08

6%] tran

che

‐0.08

‐0.06

‐0.04

Return on [3,


‐0.002

0

0.002

0.004

0.006

0.008

0.01

‐0.03 ‐0.02 ‐0.01 0 0.01 0.02

22%] tran

che

‐0.012

‐0.01

‐0.008

‐0.006

‐0.004

Return on [12, 2


Daily tranche returns versus daily tranche returns.


Agenda

1 Introduction




5 Conclusion


Conclusion

We have discussed the estimation of physical and risk-neutraldefault probabilities.

Prices can be used to back out implied risk-neutral defaultprobabilities.

A database with actual defaults is needed to estimate physicaldefault probabilities.

Hedging of CDO tranches with CDS contracts seems to beinvolved.

During the crisis the relation between both was loose.

Hedging of tranches with tranches works better, but such ahedge is hard to implement.

Liquidity issues were not considered and make it even harderto set up hedges.


CDO Structuring: Mortgage-Backed Securities

Sample SuprimeRMBS Structure

Subprime RMBS 101

Mortgage RMBSREMICMortgagePools

RMBSBonds

REMICTrustIndividual Mortgages

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10

M11 M12 M13 M14 M15 M16 M17 M18 M19 M20

2/28Hybrid ARM

MortgagePool

‘AAA’RMBS

Special

M21 M22 M23 M24 M25 M26 M27 M28 M29 M30

M31 M32 M33 M34 M35 M36 M37 M38 M39 M40

M41 M42 M43 M44 M45 M46 M47 M48 M49 M50

M51 M52 M53 M54 M55 M56 M57 M58 M59 M60

‘AA’RMBS

‘A’RMBS

SpecialPurposeVehicle(RMBSTrust)

M51 M52 M53 M54 M55 M56 M57 M58 M59 M60

M61 M62 M63 M64 M65 M66 M67 M68 M69 M70

M71 M72 M73 M74 M75 M76 M77 M78 . . . M2000

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10

Fixed RateMortgage

RMBS

‘BBB’RMBS

‘BBB-’RMBS

Residual

M11 M12 M13 M14 M15 M16 M17 M18 M19 M20

M21 M22 M23 M24 M25 M26 M27 M28 M29 M30

M31 M32 M33 M34 M35 M36 M37 M38 . . .M

www.derivativefitch.com 9

Residual1000

Source: Fitch, RMBS: Residential Mortgage-Backed Securities,

REMIC: Real Estate Mortgage Investment ConduitHolger Kraft Credit Risk: Risk Premia and Hedging 50/49

CDO: Waterfall

Subprime RMBS 101Sample Subprime RMBS Payments

REMICMonthly Mortgage Interest Principal$ I

$ P

AccountsREMICTrust

Monthly MortgagePayments

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10

M11 M12 M13 M14 M15 M16 M17 M18 M19 M20

InterestPayments

PrincipalPayments

ScheduledPrincipal

&Prepayments

Interest

‘AAA’L + % or Net WAC

M21 M22 M23 M24 M25 M26 M27 M28 M29 M30

M31 M32 M33 M34 M35 M36 M37 M38 M39 M40

M41 M42 M43 M44 M45 M46 M47 M48 M49 M50

M51 M52 M53 M54 M55 M56 M57 M58 M59 M60

$

$ I‘AAA’

‘AA’L + % or Net WAC

‘A’L + % or Net WAC

Servicer

M51 M52 M53 M54 M55 M56 M57 M58 M59 M60

M61 M62 M63 M64 M65 M66 M67 M68 M69 M70

M71 M72 M73 M74 M75 M76 M77 M78 . . . M2000

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10

‘AA’

‘A’

ScheduledPrincipal

&P t

L + % or Net WAC

‘BBB’L + % or Net WAC

‘BBB-’L + % or Net WAC

Residual

M11 M12 M13 M14 M15 M16 M17 M18 M19 M20

M21 M22 M23 M24 M25 M26 M27 M28 M29 M30

M31 M32 M33 M34 M35 M36 M37 M38 . . .M

$$ P

‘BBB’

‘BBB-’

Residual

www.derivativefitch.com 10

Prepayments Excess Interest1000Residual

Source: Fitch


Modeling Default Counter and Default Stopping Times

We consider a portfolio consisting of I entities.The default stopping times of the entities are denoted by

τ1, τ2, . . . , τI

The number of defaults is counted by the default process(bottom up)

Nt ≡ 1{τ1≤t} + 1{τ2≤t} + · · ·+ 1{τI≤t}

The k-th jump time of N is denoted by Tk .Therefore, we can also write (top down)

Nt =∑k≥1

1{Tk≤t}

The corresponding loss process reads

Lt =∑k≥1

1{Tk≤t}`k ,

where `k is the loss associated with the k-th loss.Holger Kraft Credit Risk: Risk Premia and Hedging 52/49

Loss Process

T_0 T_1 T_2 T_3 T_4 T_5 T_60

0.5

1

1.5

2

2.5

3

3.5

4

Time

Loss

Loss at 3rd default

Losses are assumed to be 0.2, 0.6 or 1.0


Documents

Credit Risk: Risk Premia and Hedging