Merton-model Approach to Valuing Correlation Products

Transcription

1 Merton-model Approach to Valung Correlaton Products Vral Acharya & Stephen M Schaefer NYU-Stern and London Busness School, London Busness School Credt Rsk Electve Sprng 2009 Acharya & Schaefer: Merton model approach for correlaton products 1

2 Bnomal wth Merton Model Important method for calculatng dstrbuton of loan losses: wdely used n bankng used by Basel II regulatons to set bank captal requrements Lnked to dstance-to-default analyss Acharya & Schaefer: Merton model approach for correlaton products 2

3 Mxed Bnomal: Usng Merton s Models as Mxng Dstrbuton In Merton model value of rsky debt depends on frm value and default rsk s correlated because frm values are correlated (e.g., va common dependence on market factor). Value of frm at tme T: V = V exp( ( µ (1/ 2) 2 ) ) where ~ (0,1), σ T + σ T % ε %, V, ε N T V surprse n R C expected value of R C We wll assume that correlaton between frm values arses because of correlaton between surprse n ndvdual frm value (ε ι ) and market factor (m) Acharya & Schaefer: Merton model approach for correlaton products 3

4 Mxed Bnomal: Usng Merton s Models as Mxng Dstrbuton Suppose correlaton between each frm s value and the market factor s the same and equal to sqrt( ). Ths means that we may model correlaton between the ε s as ε = m + 1 v, = 1, K N and corr( ε, ε ) = j Where m and v are ndependent N(0,1) random varables and s common to all frms Notce that f v ~ N(0,1) and m ~ N(0,1) then ε ~ N(0,1) Acharya & Schaefer: Merton model approach for correlaton products 4

5 Structural Approach, contd. From our analyss of dstance-to-default, we know that under the Merton Model a frm defaults when: 1 2 ε R ( µ σ ) T / σ T where = R ln( B / V ),,,, 2 V D V D The uncondtonal probablty of default, p, s therefore: p < Prob ε R ( µ σ ) ( µ σ ) = N σ T σ T D, 2 V, RD 2 V, V, In ths model we assume that the default probablty, p, s constant across frms Acharya & Schaefer: Merton model approach for correlaton products 5

6 Structural Approach to Correlaton the Idea Workng out the dstrbuton of portfolo losses drectly when the ε s are correlated s not easy But, f we work out the dstrbuton condtonal on the market shock, m, then we can explot the fact that the remanng shocks are ndependent and work out the portfolo loss dstrbuton Acharya & Schaefer: Merton model approach for correlaton products 6

7 Structural Approach, contd. The shock to the return, ε, s related to the common and dosyncratc shocks by: Default occurs when: 1 2 RD, ( µ 2 σv, ) ε = m + 1 v < = N ( p) σ T or v < N ( p) m 1 ε = m + 1 v V, Acharya & Schaefer: Merton model approach for correlaton products 7

8 The Default Condton v < N 1 ( p) m 1 A large value of m means a good shock to the market (hgh asset values) The larger the value of m the more negatve the dosyncratc shock, v, has to be to trgger default The hgher the correlaton,, between the frm shocks, the larger the mpact of m on the crtcal value of v. Acharya & Schaefer: Merton model approach for correlaton products 8

9 Structural Approach, contd. Condtonal on the realsaton of the common shock, m, the probablty of default s therefore: Prob(default m)= Prob v < N ( p) m 1 N ( p) m = N = θ ( m), say 1 N ( p) m N = and therfore = ( θ ) 1 Acharya & Schaefer: Merton model approach for correlaton products 9

10 The relaton between m and θ For a gven market shock, m, θ gves the condtonal probablty of default on an ndvdual loan 0.25 Hgh Corr p = 6.0% rho = 25.0% Theta (condtonal default prob) Low Corr p = 6.0% rho = 5.0% market shock (m) Acharya & Schaefer: Merton model approach for correlaton products 10

11 Implcatons of Condtonal Independence For a gven value of m, as the number of loans n the portfolo, the proporton of loans n the portfolo that default converges to the probablty θ The probablty that the loan-loss proporton, L, s < θ s therefore: N ( p) m Prob( L = θ ) Prob N ( θ ) 1 1 = Prob m N ( p) N ( ) 1 ( ) θ 1 = Prob m 1 N ( ) N ( p) ( ) θ 1 = N 1 ( ) ( ) ( ) N θ N p Acharya & Schaefer: Merton model approach for correlaton products 11

12 Loan Loss Dstrbuton Structural Model Ths result gves the dstrbuton of the fracton of loans that default n a well dversfed homogeneous portfolo where the correlaton n default comes from dependence on a common factor Homogenety means that each loan has: the same default probablty, p (mplctly) the same loss-gven-default the same correlaton,, across dfferent loans The dstrbuton has two parameters 1 Prob( L = θ ) N 1 N ( θ ) N ( p) default probablty, p correlaton, ( ) Acharya & Schaefer: Merton model approach for correlaton products 12

13 Loan Loss Dstrbuton wth p = 1% and = 12% and 0.6% 0.08 p = 1.5% rho = 12.0% 0.07 p = 1.5% rho = 0.6% % 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% Portfolo Loan Loss (%( Acharya & Schaefer: Merton model approach for correlaton products 13

14 Example of Vascek formula Appled to Bank Portfolo Source: Vascek Acharya & Schaefer: Merton model approach for correlaton products 14