Vasicek s Model of Distribution of Losses in a Large, Homogeneous Portfolio

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1 Vascek s Model of Dstrbuton of Losses n a Large, Homogeneous Portfolo Stephen M Schaefer London Busness School Credt Rsk Electve Summer 2012 Vascek s Model Important method for calculatng dstrbuton of loan losses: wdely used n bankng used n Basel II regulatons to set bank captal requrements Motvaton lnked to dstance-to-default analyss But, model of dependence s Gaussan Copula agan Key assumptons (apart from Gaussan dependence) homogeneous portfolo (equal nvestment n each credt) very large number of credts Merton-model Approach to Dstrbuton of Portfolo Losses 2

2 Motvaton: Merton s Model 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) Merton-model Approach to Dstrbuton of Portfolo Losses 3 Correlaton structure: Gaussan Copula 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, KN 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) Merton-model Approach to Dstrbuton of Portfolo Losses 4

3 Structural Approach, contd. From our analyss of dstance-to-default, we know that under the Merton Model a frm defaults when: 1 ε, µ σ, σ,, = 2 V 2 R ( ) / where ln( / ) D T V T RD B V The uncondtonal (natural) probablty of default, p, s therefore: RD, ( µ 2 σ V, ) T RD, ( µ 2 σv, ) T p Prob ε < N = σv, T σ T In ths model we assume that the default probablty, p, s constant across frms Merton-model Approach to Dstrbuton of Portfolo Losses 5 Idea: Sngle Common Factor and Large Homogeneous Portfolo 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 Merton-model Approach to Dstrbuton of Portfolo Losses 6

4 Structural Approach, contd. The shock to the return, ε, s related to the common and dosyncratc shocks by: ε = m + 1 v Default occurs when: 1 2 RD, ( µ 2 σv, ) ε = m + 1 v < = N ( p) σ T or v < ( ) N p m 1 V, Merton-model Approach to Dstrbuton of Portfolo Losses 7 Vascek and the Intensty Model We ll see later that the Vascek model s essentally the same as the ntensty model when: the ntensty s the same for all the names; and the number of names becomes large equal nvestment n each name we use the Gaussan copula Merton-model Approach to Dstrbuton of Portfolo Losses 8

5 The Default Condton n Vascek 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 market shock 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. Merton-model Approach to Dstrbuton of Portfolo Losses 9 Condtonal Default Probablty 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 m and therfore = = ( θ ( )) 1 Merton-model Approach to Dstrbuton of Portfolo Losses 10

6 The relaton between θ(m) and m For a gven market shock, m, θ(m) gves the condtonal probablty of default on an ndvdual loan H corr Lo corr Merton-model Approach to Dstrbuton of Portfolo Losses 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 actually default converges to the probablty θ (m) <<<< KEY IDEA In the chart, f the market shock s m* then the ACTUAL proporton of defaults n the portfolo converges to 15% as # loans Merton-model Approach to Dstrbuton of Portfolo Losses 12

7 The crtcal value of m For a gven actual frequency of loss, θ, we can calculate the correspondng value of the market shock, m(θ) that wll produce exactly that level of loss: N N 1 ( p) m( θ ) θ = N 1 ( θ ) = m( θ ) = ( ) ( θ ) N p m 1 N ( p) 1 N ( θ ) Merton-model Approach to Dstrbuton of Portfolo Losses 13 The dstrbuton of portfolo loss Snce the proporton of portfolo losses decreases wth m, the probablty that the proporton of loans that default (L) s less than θ s: Prob ( L < θ ) = prob ( m > m( θ )) = prob m > Prob ( L θ ) N ( p ) 1 N = N 1 1 ( θ ) ( θ ) 1 N N ( p) < = N N ( p) 1 N ( θ ) Merton-model Approach to Dstrbuton of Portfolo Losses 14

8 Prob Loan Loss Dstrbuton ( L θ ) ( θ ) 1 N N ( p) < = N Ths result gves the cumulatve 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 default probablty, p correlaton, Merton-model Approach to Dstrbuton of Portfolo Losses 15 Loan Loss Dstrbuton wth p = 1% and = 12% and 0.6% p = 1.5% rho = 12.0% p = 1.5% rho = 0.6% % 1.00% 2.00% 3.00% 4.00% 5.00% 6.00% Portfolo Loan Loss (%( Merton-model Approach to Dstrbuton of Portfolo Losses 16

9 Example of Vascek formula Appled to Bank Portfolo Source: Vascek Merton-model Approach to Dstrbuton of Portfolo Losses 17 Relatonshp between the Vascek model and the ntensty model wth the Gaussan Copula Merton-model Approach to Dstrbuton of Portfolo Losses 18

10 Fundamentally, Vascek model gves same results Intensty model and Gaussan copula (!) Default condton n Vascek model: 1 2 RD, ( µ 2 σv, ) ε = m + 1 v < = N ( p) σ T In other words, whether a normally dstrbuted N(0,1) varable s larger or smaller than a gven fxed number, N 1 ( p) V, Merton-model Approach to Dstrbuton of Portfolo Losses 19 and.. n ntensty model (wth the Gaussan copula).. the same (!) In the ntensty model default occurs when 1 τ = ln(1 U ) τ * where U = N ( ε ) λ.e., when the default tme τ s smaller than the maturty τ* Defne 1 τ* = ln(1 U*) and U* = N( ε*) λ Then default occurs when ε ε * Merton-model Approach to Dstrbuton of Portfolo Losses 20

11 Or.. n pctures.. If the value of e that we draw s smaller than the crtcal value ε ε * Then τ s less than τ* and we have a default 1 τ = ln(1 U ) τ * where U = N ( ε ) λ Merton-model Approach to Dstrbuton of Portfolo Losses 21 Intensty model wth 1000 names and equal ntensty and Vascek model wth equal default probablty and correlaton Example Merton-model Approach to Dstrbuton of Portfolo Losses 22

12 The bottom lne.. The Vascek model s the same as the ntensty model wth a Gaussan copula, dentcal default probabltes and a large number of names. Merton-model Approach to Dstrbuton of Portfolo Losses 23 Applcatons Vascek s obtans a formula for the dstrbuton of losses wth: sngle common factor homogeneous portfolo large number of credts But the approach can be generalsed to a much more realstc (mult-factor) correlaton structure and granularty n the portfolo holdngs qute wdely used n bankng for management of rsk of loan portfolo Merton-model Approach to Dstrbuton of Portfolo Losses 24

13 Takeaways Vascek s formula gves useful quck method for generatng dstrbuton of losses n large portfolo n one-factor verson fundamentally the same as Gaussan copula explotaton of condtonal ndependence s useful dea Applcatons tend to be n rsk management of actual loan losses (natural dstrbuton) rather than prcng (rsk-neutral dstrbuton) less evdence of poor performance n natural dstrbuton (same story about structural model agan) Merton-model Approach to Dstrbuton of Portfolo Losses 25

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