Modelling Credit Spread Behaviour. FIRST Credit, Insurance and Risk. Angelo Arvanitis, Jon Gregory, Jean-Paul Laurent

Transcription

1 Modelling Credit Spread Behaviour Insurance and Angelo Arvanitis, Jon Gregory, Jean-Paul Laurent ICBI Counterparty & Default Forum 29 September 1999, Paris

2 Overview Part I Need for Credit Models Part II Simple Binomial Model page 2 Part III Jump-Diffusion Model Part IV Credit Migration Model Part V Estimating Credit Spread Volatilities

3 Part I page 3 Need for credit spread models

4 Need For Credit Models (I) - Credit derivatives market - Active management of loan portfolios page 4 Why? Growth of emerging markets Active management of counterparty risk in standard derivatives portfolios

5 Need For Credit Models (II) Valuing credit derivatives, options on risky bonds, vulnerable derivatives page 5 What for? Assessing the credit risk of portfolios - spread and event risk - Optimising portfolio risk / return profile - Relative value analysis

6 Need For Credit Models (III) Estimate the current risk free and risky term structures How? Model the evolution of the risk free rate and the credit spread page 6 Calibrate to observed bond and option prices Yield Credit spread y -free Maturity

7 Credit Data Limited / crude data available on credit Moody s historical data (annual) Default probability 0 25% p i page 7 Pairwise default correlation Credit migration Loss given default 0 5% ρ ij 0 20% q kl 0 100% l i Default correlation and recovery rate difficult to estimate Credit crashes - high default correlation

8 Credit spread for an AA bond Jump page

9 Properties of Credit Spreads more volatile Jump Component - Discrete change in default probability - Credit migration page 9 Credit spread mean reversion downgrade Time Continuous Component - Mean reverting - Change in market price of risk - risk premia

10 Modelling Credit Spread r risky = r + ~ λ risk free Credit Spread page 10 Constant Simple binomial model (Part II) Model underlying credit migration process (Part IV) Continuous and jump components Jump-diffusion model (Part III)

11 Part II page 11 Simple Binomial Model

12 Simple Binomial Model (I) - Constant risk free term structure - Constant recovery rate - Constant credit spread if no default - Jump in credit spread if default occurs page 12 - Derive risk neutral default probabilities from risky and risk- free bond prices neutral default probabilities - Actual default probabilities - premia - Liquidity - Uncertainty over recovery rate

13 Simple Binomial Model (II) risky bond price v( T ) 1 q ~ ( 0, T) 1 promised amount no default page 13 risk free bond price q ~ ( 0, T) probability of default ( ) δ recovery rate [ δ ] vt ( ) = pt ( ) q~ (, T ) + q~ (, T ) ~ 1 vt ( )/ pt ( ) q ( 0, T ) = ( 1 δ ) default - price any product with payoff contingent on default event

14 Part III page 14 Jump-Diffusion Model

15 Jump-Diffusion Model Continuous component - Positive and mean reverting - Correlated with interest rates Jump Component - Jumps of random size occur at random times - Jumps in only one direction page 15 - Standard implementation and calibration - Standard numerical pricing algorithms can be used -free interest rate -Continuous and mean reverting

16 Free Term Structure (I) page 16 Assumptions on the future evolution of the instantaneous risk free rate Volatility σ r r( t) (normal, lognormal, square root, ) Drift / mean reversion Long term mean r( t) Rate of mean reversion t ( ) k ( r t ) r( t) = r( 0) + k ( t) r( t) r( t) dt + σ r( t) dw ( t) r 0 0 t r r

17 Free Term Structure (II) σ r σ r = 01., k r = 10 = 01., k = 2 r page 17 σ r σ r = 01., k r = 2 = 02., k = 2 r

18 Credit Spread Term Structure (I) ~ λ() t ρr() t x() t = + corr( r t x t ) (), () = 0 determines correlation - Uncorrelated with interest rates ~ between λ () t and r() t - Continuous and jump component page 18 Random jump size z, exponentially distributed θe θ z, z > 0 Random number of jumps - follows Poisson process λτ n e ( λτ ) / n! n = 012,,,... τ = time interval

19 Credit Spread Term Structure (II) Credit spread component uncorrelated with the risk free interest rate page 19 t ( ) xt () = x() 0 + kx () s xs () xs () ds t + σ x s x 0 0 () x() s dw () s + Zi () i; τ ( i) t

20 Credit Spread Term Structure (III) page 20 more frequent and larger jumps

21 Part IV page 21 Credit Migration Model

22 Credit Migration Model -Jumpsmodelled as changes in credit ratings and defaults - Continuous part modelled as continually changing risk premia - Model jointly assets in various credit classes - Portfolio management and risk analysis page 22 - Calibration - incorporate economic and historical information - Flexible in terms of data requirements and number of states

23 Markov Chains - Generator Matrix (I) page 23 Continuous time Markov chain Discrete state space ~ Λ= constant over time 1 2 K-1 K 1 2 K-1 K ~ ~ ~ ~ λ1 λ12. λ1, K 1 λ1 ~ ~ ~ ~ λ λ. λ λ..... ~ ~ ~ ~ λ λ. λ λ , K 1 2K K 11, K 12, K 1 K 1, K absorbing state (default) K

24 Markov Chains - Generator Matrix (II) I + Λ ~ dt, transition matrix over short period dt λ ij 0, non-negative transition probabilities page 24 K ~ ~ λ = λ, sum of all probabilities equals 1 i K j i i = 1 j i ij ~ ~ λ λ, ik, k i+ ij j k i+ 1, j 1 A state i+1 is always more risky than state i

25 Markov Chains - Transition Matrix Transition matrix for the period t to T Explicit computation ~ Λ = Σ 1 D Σ ~ QtT (, ) = Σ 1 exp DT ( t ) [ ] Σ page 25 ~ QtT (, ) q~ (, ). ~ (, ) ~ 1 t T q1, K 1 t T q1k (, t T ) ~ (, ). ~ (, ) ~ q21 t T q2, K 1 t T q2 K (, t T ) =.... q~ (, ). ~ (, ) ~ K 11, t T qk 1 t T qk 1, K(, t T)

26 Model Structure (I) States : uniquely determine default probability Credit ratings - can incorporate past credit rating transitions - non-markovian model page 26 ~ Λ ~ Λ - neutral generator matrix - constant jump in credit spread due to downgrade

27 Model Structure (II) Incorporate stochastic risk premia ~ ~ Λ = Λ stochastic U() t continuous process for risk premia page 27 ~ Λ stochastic jump in credit spread due to downgrade

28 Stochastic Generator Matrix Stochastic generator matrix arises from randomly changing risk premia ~ ~ Λ = Λ U() t stochastic page 28 t ( ) U() t = U( 0) + a ku() t dt + σ U() t dw 0 0 t Stochastic component t Closed form formulae for bond prices Mean reverting process

29 Stochastic Premia If eigenvectors are constant, can pose Λ(, tt) = Σ 1 D() tσ Possible evolution of eigenvalues K dx j = ( a j b jx j) dt + σ jdw, D ( t) = diag( X j( t)) j= 1 page 29 Pricing equation is now modified to K T 1 qik (, t T) = ( ) ij exp X j () s ds () t Σ E t D Σ j = 1 Expectation has closed (algebraic) form depends on parameters a bσ and on D(t) jk

30 Calibration (I) Prices of risky bonds for various credit classes and maturities B i ( 0, T) - Least squares estimation - Adjust historical generator matrix to fit market prices - Achieve fit closest to historical data page 30 Λ ~ - Historical generator matrix (estimated from one year transition matrix) - Credit spread historical time series ΛΛ stochastic (, akσ,, ) Simulate Credit Credit Spread Spread Price exotic structures

31 Calibration (II) Least squares fit to match directly observed coupon bond prices (any number) page 31 i min ( ) ( ; ~ ~ K J T 2 K i i i ~ Pj Fj h v h Λ) + Λ i j h = 1 = 1 = 1 i, j= 1 βij market price of coupon at coupon bond for date h prior class i generator matrix ( λ ) ij λij 2 confidence level Obtain solution closest to the historical generator matrix Λ - stable calibration

32 Calibration - Emerging Markets (II) page states 5 states Actual

33 Calibration - Corporate Market (II) 1400 page 33 Credit spread (bp) AAA AA A BBB BB B CCC Maturity (years)

34 Calibration - US Industrials (I) AAA page 34 Credit Spread (bp s) AA A Maturity (years)

35 Calibration - US Industrials (II) 1200 page 35 Credit Spread (bp s) BBB BB B CCC Maturity (years)

36 Part V page 36 Estimating Credit Spread Volatility

37 Credit spread volatilities estimates 74 Bonds Investment grade (Baa and above) US Industrial bonds 7 Speculative grade (Ba and below) Emerging Market bonds page Model states 1 Moody s Aaa Aaa 2 Moody s Aa1 Aa1 --Aa3 Aa3 3 Moody s A1 A1 --A3 A3 4 Moody s Baa1 Baa1 --Baa3 Baa3 5 Moody s Ba1 Ba1 --Ba3 Ba3 6 Moody s B1 B1 --B3 B3 7 Moody s CCC CCC 8 Default Default Historical generator matrix from Moody s average 1y 1y transition matrix

38 Estimated short term spreads 0.2 page Credit ratings AA A BBB BB B CCC Lower ratings have higher spread higher volatility

39 0 Eigenvalues of generator matrix Eigenvalues 0.05 Principal components page first 3 principal components account for 99% of the variance

40 Advanced Modelling Issues Stochastic Recovery Rates - Recovery rates are random with high variance - Exogenous - Endogenous - depend on the severity of default Credit Events Correlated with Interest Rates - Credit migration and defaults depend on interest rates - Joint state variables for interest rates and credit spreads - Incorporate business cycles page 40 Non - Markovian Bankruptcy Process - Autocorrelated migration process - Markovian in state space augmented with lagged values Second Generation Products - Basket options - credit spread, default correlations - Multiple Currencies -Quantos