Modelling Credit Migration and Default Probabilities ...

Conference on Credit Derivatives London Guildhall University, 10 April 2002

Modelling Credit Migration and Default Probabilities for Pricing and Hedging M A H Dempster Centre for Financial Research Judge Institute of Management University of Cambridge & Cambridge Systems Associates Limited Co-worker: R G Smith © 2002 Centre for Financial Research, Judge Institute of Management, University of Cambridge www-cfr.jims.cam.ac.uk

Outline • • • •

Introduction Examples of credit derivatives Basic applications Pricing of credit-risky contingent claims – – – –

Value-of-firm models Reduced-form models Term structure models Market models

© 2002 Centre for Financial Research, Judge Institute of Management, University of Cambridge www-cfr.jims.cam.ac.uk

Introduction • Formally, a credit derivative is a bilateral contract that isolates specific aspects of credit risk from an underlying instrument and transfers the risk between the two parties • Allows replication, hedging and transfer of credit risk • Demand created by – Specific types of credit risk can be added to a portfolio without acquiring the credit asset itself – Credit risks of instruments can be managed – Broader range of investors can deal in credit – Regulatory capital arbitrage


Defining Credit Derivatives • Financial contracts with a payout linked to: – – – –

Loan or bond values Default or credit events Credit spreads Credit ratings

• With cash settlement or delivery of the relevant underlying asset or portfolio, if appropriate • On single name, baskets, indices • Delivery as notes or OTC contracts • Delivery as swaps or options


Brief History of Credit Derivatives 1992: First use of the phrase “credit derivative” by ISDA 1993-5: Market does not take off – S&P refuse to rate credit derivative products – Doubt among practitioners whether credit derivative deals that had been done would be completed

1996: Vast range of applications for credit derivatives realised 1997: First synthetic securitisation (JP Morgan’s Bistro deal) 1999: ISDA issued series of definitions, including credit events, obligations and settlement (physical or cash) – By end of year, 84% of structures were based on these definitions

1999-: Period of sharp growth in credit derivative market © 2002 Centre for Financial Research, Judge Institute of Management, University of Cambridge www-cfr.jims.cam.ac.uk

Small but growing market

Foreign exchange contracts (Single-currency) Interest rate contracts Equity-linked contracts Commodity contracts Credit Derivatives Other derivatives

Notional amounts ($bn) End-June End-June 1998 2001 22,055 20,434

Gross market values ($bn) End-June End-June 1998 2001 982 967

48,124

75,890

1,354

1,748

1,341

2,039

201

220

506

674

39

88

108 10

694 23

4 0

21 1

Source: Bank of International Settlements © 2002 Centre for Financial Research, Judge Institute of Management, University of Cambridge www-cfr.jims.cam.ac.uk

Examples of Credit Derivatives • First generation – – – –

Total return swap Credit default swap Default digital Spread options

• Second generation – – – –

Floating rate asset derivatives Fixed coupon asset derivatives Rating option Bankruptcy swap

• Third generation exotics © 2002 Centre for Financial Research, Judge Institute of Management, University of Cambridge www-cfr.jims.cam.ac.uk

Total Return Swap • Party A pays any total positive returns on the underlying asset (including interest and capital appreciation) • Party B pays a funding payment (LIBOR + margin) and any capital depreciation Total positive returns

Party B

Party A LIBOR + margin + capital depreciation

• Party A transfers all the credit risk of the underlying asset to Party B without the selling the asset © 2002 Centre for Financial Research, Judge Institute of Management, University of Cambridge www-cfr.jims.cam.ac.uk

Credit Default Swap • Party A pays a fee of x basis points until default or maturity • Party B pays an agreed notional amount upon credit event (relative to an underlying reference loan or security)

Party A

Fees until default or maturity (whichever comes first) Contingent payment upon credit event before maturity; otherwise zero

Party B

• Party A can take cost of swap into account in pricing the loan and remove credit default risk of perhaps a valued customer © 2002 Centre for Financial Research, Judge Institute of Management, University of Cambridge www-cfr.jims.cam.ac.uk

Credit Spread Derivatives • Credit spread is primarily used as compensation for the possibility of default • Two general formats of credit spread – Absolute spread: the credit spread relative to a benchmark that is regarded as default risk free – Relative spread: the spread between two credit-risky assets

• Forwards and options on the credit spread • Call options give the purchaser the right to buy the spread – Seller benefits from a decreasing spread

• Allows the trading of credit spreads as an isolated variable without being exposed to interest rate risk © 2002 Centre for Financial Research, Judge Institute of Management, University of Cambridge www-cfr.jims.cam.ac.uk

Credit Linked Notes • Credit derivative embedded in a fixed-income security • Structure of a credit-default note (where the embedded derivative is a credit-default swap) Principal at start of arrangement

Investor

Interest

Issuer

On redemption, principal if no credit event has occurred; otherwise, principal – default payment

• Both total return swaps (total rate of return credit-linked notes) and credit-spread forwards (credit-spread notes) are also often used as the embedded derivative © 2002 Centre for Financial Research, Judge Institute of Management, University of Cambridge www-cfr.jims.cam.ac.uk

• According to ISDA, credit events are – – – – – –

Bankruptcy Obligation acceleration Obligation default Failure to pay Repudiation/Moratorium Restructuring

• Bankruptcy swap: same structure but only pays out upon bankruptcy; was offered by EnronCredit.com • Rating option: same structure but event that triggers payment is an upgrade or downgrade by a rating agency • Can be extended to portfolios – First to default pays out when one credit in a protected portfolio defaults – “m to default” swap © 2002 Centre for Financial Research, Judge Institute of Management, University of Cambridge www-cfr.jims.cam.ac.uk

Breakdown of Credit Derivative Market in 2000 (based on notional values) Single name Credit Default Swaps

23%

Credit Linked Notes

50% 8%

Credit Spread Options Baskets

6%

Portfolios / CLOs 13%

Source: British Bankers Association © 2002 Centre for Financial Research, Judge Institute of Management, University of Cambridge www-cfr.jims.cam.ac.uk

Some Applications of Credit Derivatives • Transfer credit risk of valued customer to another institution • Dynamic management of credit risk • Gain exposure to restricted markets • Yield enhancement • Regulatory capital arbitrage


Dynamic Management of Credit Risk

Exposure or Risk

Tolerance level

Sell credit derivative or loan

Time © 2002 Centre for Financial Research, Judge Institute of Management, University of Cambridge www-cfr.jims.cam.ac.uk

Gain Exposure to Restricted Markets / Yield Enhancement • Fund wants high yield investment – Emerging markets

• Problem of investment restrictions – Typical: cannot invest below BBB grade debt

Fixed Coupon 5%

Portfolio Manager

Bank Korean loan yield 6.25%


Regulatory Capital Arbitrage • OECD Banks A and B compete to give a loan of £10m to Company X • Risk-weighting is 100% since counterparty is a corporate • Regulatory capital is therefore £800,000 • Banks must raise £9.2m from third-parties BANK A

£10m loan to Company X

BANK B

Funding of 5.50%

£9.2m from other institutions

5.75%

Funding of 5.00%

5.75%


£10m loan to Company X


• Banks now co-operate by entering into a credit default swap • Bank A’s counterparty is now an OECD bank (Bank B): risk-weighting 20% • Regulatory capital is therefore £160,000 • Must raise £9.84m from third-parties • Bank B’s counterparty remains Company X and so must still hold £800,000 of regulatory capital Funding of 5.00%

BANK A

Credit default swap fee of xbps Payment of £10m on default

BANK B

5.75%


£10m loan to Company X © 2002 Centre for Financial Research, Judge Institute of Management, University of Cambridge www-cfr.jims.cam.ac.uk

• Bank A’s income is reduced as it is now paying a credit default swap fee of x basis points per annum • Bank A’s return on capital (net income / regulatory capital) increases under the credit default swap arrangement for values for x < 60bp • Bank B’s net income (and thus return on capital) increases for values for x > 33bp • If feasible then regulatory capital arbitrage can occur and the return on capital for both banks can increase by entering a credit default swap with each other


Models for Credit-Risky Contingent Claims • Value-of-firm models • Reduced-form models • Term structure models • Market models


Value-of-firm Models • Also known as structural models • Model a process that is usually taken to be the value of the assets of the firm that issued the credit • Used as the basis for KMV Corporation models • Black & Scholes (1973) and Merton (1974) took the value of the firm V to be a log-normal process • All claims on the firm taken to be contingent claims with V as the underlying process • Originally the contingent claims were taken to be European options


Development of Value-of-firm Models • Observation that firms are more likely to default when V reaches a value between amount of short-term debt and total liabilities • Built into model by Black & Cox (1976) who introduced a knock-out feature to European option • Consequence that default could occur before maturity of European option • Longstaff & Schwartz (1995) modelled default-free interest rates using Vasicek (1977) model and interest rates correlated with V


Development of Value-of-firm Models • The time of default is a predictable stopping time for all the above models. This led to a problem of unrealistically low short-term credit spreads • This was solved by Schönbucher (1996) and Zhou (1997), who both introduced jumps into the underlying process V • Since the value of a firm’s assets is not observable very difficult to calibrate the models to market prices • The firm’s cash flow was made the underlying process in Epstein et al (1998) as this variable is easier to observe


Reduced-form Models • Do not model the fundamentals of the firm • The two underlying risk components of the default event, the time of default and the recovery rate, are assumed to be exogenous random variables • Used by the CreditRisk+ internal credit model developed by Credit Suisse Financial Products • Jarrow & Turnbull (1995) – Recovery rate fixed and known – Time of default modelled by first jump of a Poisson process with constant intensity


Development of Reduced-form Models • Madan & Unal (1998) – Recovery rate modelled using a two-parameter beta distribution – Time of default modelled by the first jump of a Cox process with intensity driven by underlying stochastic process thought of as firm’s value

• Extended to a rating-transition framework • Continuous-time models created by Lando (1994) and Jarrow, Lando & Turnbull (1997) – Only Markov chain dynamics of the ratings incorporated


Development of Reduced-form Models • Lando (1998) – Allowed stochastic credit spreads within each ratings class – Assumed credit spreads of all ratings classes driven by same factor

• Das & Tufano (1996) assumed default intensities are constant but recovery rates are stochastic – Credit spreads bounded above by the value given when the recovery rate is zero – Empirical evidence far more supportive of converse situation of stochastic default intensities and constant recovery rates


Term Structure Models • Can be thought of as reduced-form models since fundamentals of firm are not being modelled • Term-structure of default-free bonds and defaultable bonds are taken as fundamentals • Model that incorporates both is developed and then conditions are set out that ensure the framework is arbitragefree • Duffie & Singleton (1999) – assumes the payoff at default is delivered in cash equivalent to the fraction (1-q) of the value of defaultable security just prior to default


Term Structure Models • Schönbucher (2000a) – Uses a Heath-Jarrow-Morton approach to model defaultable bonds – Allows a firm to default a multiple number of times. Assumes that upon default debt is restructured and continues to be traded but at a reduced value


Market Models • Default-free market models are based on observable market rates such as LIBOR rates or swap rates • Brace, Gatarek, Musiela (1997) and Miltersen, Sandmann, Sondermann (1997) both developed LIBOR market model • Jamshidian (1997) produced swap market model • Calibration is very simple • Extended to multicurrency framework by Schlögl (1999) • Extended to framework with default risk by Schönbucher (2000b)


Schönbucher’s LIBOR Market Model with Default Risk (2000) • Tenor structure: T0 = 0 < T1 < … < TK • Define dk = Tk+1 – Tk • I(t) is the survival indicator function; if t is the default time, then I(t) = 1{t > t} • Time-t price of default-free zero-coupon bond that matures at Tk is Bk(t) • Time-t price of defaultable zero-coupon bond that matures at Tk is I (t ) Bk (t ) © 2002 Centre for Financial Research, Judge Institute of Management, University of Cambridge www-cfr.jims.cam.ac.uk

Fundamental Quantities • Time-t default-free and defaultable simple-compounded dkperiod forward interest rate from Tk are respectively

ö 1 æ Bk (t ) Fk (t ) = ç - 1÷ d k è Bk +1 (t ) ø

Fk (t ) =

ö 1 æ Bk (t ) 1 ÷ ç d k è Bk +1 (t ) ø

• Default-risk factor at time t for maturity Tk is Bk (t ) Dk (t ) = Bk (t ) • The dk-period discrete-tenor forward default intensity from time Tk as seen at time t is ö 1 æ Dk (t ) H k (t ) := ç - 1÷ d k è Dk +1 (t ) ø © 2002 Centre for Financial Research, Judge Institute of Management, University of Cambridge www-cfr.jims.cam.ac.uk

Assumptions • Default-free forward interest rates follow a lognormal process dFk = m kF dt + s kF dW Fk • Either 1. Discrete default intensities follow a lognormal process dH k = m kH dt + s kH dW Hk 2. Credit spreads follow a lognormal process dS k = m kS dt + s kS dW Sk where Sk (t ) := Fk (t ) - Fk (t ) © 2002 Centre for Financial Research, Judge Institute of Management, University of Cambridge www-cfr.jims.cam.ac.uk

Measures • Tk-forward measure: numeraire is Bk(t) – Under this measure the default-risk factor Dk(0) is the probability of survival from time 0 to time Tk

• Tk-survival measure: numeraire is Bk (t ) – Only attaches probability to survival events – Measure that is reached when Tk-forward measure is conditioned on survival until Tk

• Brownian motions under these two measures differ by

dWk (t ) = dWk (t ) + a kD (t )dt

where a kD (t ) is minus the volatility of Dk © 2002 Centre for Financial Research, Judge Institute of Management, University of Cambridge www-cfr.jims.cam.ac.uk

• Under the Tk+1-forward measure

Bk is a martingale Bk +1

– Fk is thus also a martingale and satisfies

dFk (t ) = s kF Fk (t )dWk +1 (t )

• Under the Tk+1-survival measure – Fk is a martingale and satisfies

Bk is a martingale Bk +1

dFk (t ) = s kF Fk (t )dWk +1 (t )

– Discrete default intensities Hk satisfy Fks kF dH k = (1 + d k H k )a kD+1 - d k H ks kH dt + H ks kH dWk +1 1 + d k Fk

[

]


Recovery of Par Model • If a defaultable coupon bond defaults in the time interval (Tk, Tk+1] then its recovery is composed of the recovery rate p times the sum of the notional of the bond and the accrued interest over (Tk, Tk+1]. • Recovery payoffs occur in cash at the next tenor date, i.e. Tk+1 if the default was in the interval (Tk , Tk+1]. • Assume that all claims of the same seniority have the same recovery rate • Define ek (t ) to be the time-t value of receiving 1 at Tk+1 if and only if a default has occurred in (Tk , Tk+1] © 2002 Centre for Financial Research, Judge Institute of Management, University of Cambridge www-cfr.jims.cam.ac.uk

Example: Defaultable Fixed-Coupon Bond • Notional of 1 • N fixed payments of size c at times Tk k = 1,…, N • Has time-0 price of N

BN (0) + å [cBk (0) + p (1 + c )ek -1 (0)] k =1

Time-0 value of par (par received if no default)

Time-0 value of Tk-coupon

Time-0 value of recovery if default in (Tk-1, Tk]

• Remains problem of calculating ek(t) © 2002 Centre for Financial Research, Judge Institute of Management, University of Cambridge www-cfr.jims.cam.ac.uk

Fixed Payments at Default • If defaults H are assumed to be stochastically independent from default-free interest rates F then ek = Bk +1d k H k

• If H and F are not assumed to be independent then the expression becomes more complicated ek where

d k Fk » Bk +1d k H k - Bk 1 + d k Fk

é æ σ kF AkD,k ö ù ÷ - 1ú êexpçç ÷ êë è 1 - d k Fk ø úû

H H s AkD,m := å l l l Tl Ùm l =0 1 + d l H l k -1d


Example: Credit Default Swap • Gives protection from TM to TN • Party A pays fee of s at time Tk k = M+1,…,N – Tk-payment has time-0 value of sBk (0)

• Receives (1-p) on default from Party B – Time-0 value of recovery if default occurs between (Tk, Tk+1] is (1-p)ek(0)

• CDS has time-0 value (to Party A) of (1 - p )

N -1

N

k =M

k = M +1

å ek (0) - s å Bk (0)


• The value of the fee so that the CDS is fairly priced is (1 - p ) s=

N -1

N -1

å e k ( 0)

k =M

å Bk +1 (0)

k =M

• If it is assumed that F and H are independent then N -1 Bk +1 (0) s := (1 - p ) åw k d k H k where w k := N -1 k =M å B j +1 (0) j=M

– s is a weighted average of the default intensities Hk

• If F and H are not assumed to be independent then s » (1 - p )

D F σ k ) - 1}] åw k [d k H k + (1 + d k H k ){exp([1-Fk (0)] Ak,k

N -1 k =M


Pricing Swaptions The value of a European call option to enter at time TM a credit default swap with maturity TN and strike default swap rate s* which is knocked out at defaults before TM is æ N -1 ö C (0) = ç å Bk (0) ÷[s (0)F ( d1 ) - s * F ( d 2 )] è k =M ø

where d1 and d2 are given by

d1;2 =

ln( s / s*) ± 12 s 2TM s TM

s is the volatility of s in survival © 2002 Centre for Financial Research, Judge Institute of Management, University of Cambridge www-cfr.jims.cam.ac.uk

Summary • Credit derivatives fastest growing sector of derivatives market • No universal or robust method for pricing credit derivatives in the way that Black-Scholes model is used for many options • Value-of-firm models conceptually appealing but calibration to market difficult • Market model is easier to calibrate but lack of liquidity may prevent information from market being accurate enough to input directly into model © 2002 Centre for Financial Research, Judge Institute of Management, University of Cambridge www-cfr.jims.cam.ac.uk

References Bank of International Settlements (2001). The global OTC derivatives market at end-June 2001. Triennial central bank survey of foreign exchange and derivatives market F Black & J C Cox (1976). Valuing corporate securities: some effects of bond indenture provisions. Journal of Finance 31(2) 351-367 F Black & M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy 81(3) 637-654 A Brace, D Gatarek & M Musiela (1997). The market model of interest rate dynamics. Mathematical Finance 7(2) 127-155 British Bankers’ Association (2000). Credit Derivatives Report 1999/2000 Credit Suisse Financial Products (1997). CreditRisk+: a credit risk framework S R Das & P Tufano (1996). Pricing credit-sensitive debt when interest rates, credit ratings and credit spreads are stochastic. Journal of Financial Engineering 5 161-198 ed. S R Das (1998). Credit Derivatives: Trading & Management of Credit & Default Risk. Wiley D Duffie & K J Singleton (1999). Modeling term structure of defaultable bonds. Review of Financial Studies 12(4) 687-720


D Epstein, N Mayor, P J Schönbucher, A E Whalley & P Wilmott (1998). The valuation of a firm advertising optimally. Quarterly Review of Economics and Finance 38(2) 149-166 International Swaps and Derivatives Association (1999). 1999 ISDA Credit Derivatives Definitions F Jamshidian (1997). LIBOR and swap market models and measures. Finance and Stochastics 1(4) 293-330 R A Jarrow, D Lando & S M Turnbull (1997). A Markov model for the term structure of credit risk spreads. Review of Financial Studies 10(2) 481-523 R A Jarrow & S M Turnbull (1995). Pricing derivatives on financial securities subject to credit risk. Journal of Finance 50(1) 53-85 D Lando (1994). Three Essays on Contingent Claims Pricing. PhD Thesis, Graduate School of Management, Cornell University D Lando (1998). On Cox processes and credit risky securities. Review of Derivatives Research 2(2/3) 99-120 F A Longstaff & E S Schwartz (1995). A simple approach to valuing risky fixed and floating rate debt. Journal of Finance 50(3) 789-819 D Madan & H Unal (1998). Pricing the risk of default. Review of Derivatives Research 2(2/3) 121-160


R C Merton (1974). On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance 29(2) 449-470 K R Miltersen, K Sandmann & D Sondermann (1997). Closed form solutions for term structure derivatives with log-normal interest rates. Journal of Finance 52(1) 409-430 E Schlögl (1999). A multicurrency extension of the lognormal interest rate market models. Working Paper, School of Mathematical Sciences, University of Technology, Sydney P J Schönbucher (1996). Valuation of securities subject to credit risk. Working Paper, Department of Statistics, University of Bonn P J Schönbucher (2000a). Credit Risk Modelling and Credit Derivatives. PhD Thesis, Department of Statistics, University of Bonn P J Schönbucher (2000b). A Libor market model with default risk. Working Paper, Department of Statistics, University of Bonn O A Vasicek (1977). An equilibrium characterization of the term structure. Journal of Financial Economics 5 177-188 M Wong & S Song (1997). A loan in isolation. AsiaRisk, June, 21-23 C Zhou (1997). A jump-diffusion approach to modeling credit risk and valuing defaultable securities. Finance and Economics Discussion Series 1997/15, The Federal Reserve Board


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