Probability of Loss on Loan Portfolio

Transcription

1 Probablty of Loss o Loa Portfolo

2 KMV Corporato COPYRIGHT 1987, KMV CORPORATION, SAN FRANCISCO, CALIFORNIA, USA. All rghts reserved. Documet Number: Revso Ths s a hghly cofdetal documet that cotas formato that s the property of KMV Corporato or Kealhofer, McQuow, Vasce Developmet, L.P. (collectvely, KMV ). Ths documet s beg provded to you uder the cofdetalty agreemet that exsts betwee your compay ad KMV. Ths documet should oly be shared o a eed to ow bass wth other employees of your busess, excludg depedet cotractors, cosultats or other agets. By acceptg ths documet, you agree to abde by these restrctos; otherwse you should mmedately retur the documet to KMV. Ay other actos are a volato of the ower strade secret, copyrght ad other propretary rghts. Ay other actos are also a volato of the prevously metoed cofdetalty agreemet. KMV retas all trade secret, copyrght ad other propretary rghts ths documet. KMV Corporato ad the KMV Logo are regstered trademars of KMV Corporato. Portfolo Maager, Credt Motor, Global Correlato Model, GCorr, Prvate Frm Model, EDF Calculator, EDFCalc, Expected Default Frequecy ad EDF are trademars of KMV Corporato. All other trademars are the property of ther respectve owers. Publshed by: KMV Corporato 160 Motgomery Street, Sute 140 Sa Fracsco, CA U.S.A. Phoe: FAX: emal: support@mv.com webste: http: // Authors: Oldrch Alfos Vasce Cofdetal Release Date: 1-February-1987

3 Probablty of Loss o Loa Portfolo PROBABILITY OF LOSS ON LOAN PORTFOLIO Oldrch Vasce, /1/87 Cosder a portfolo cosstg of loas equal dollar amouts. Let the probablty of default o ay oe loa be p, ad assume that the values of the borrowg compaes assets are correlated wth a coeffcet ρ for ay two compaes. We wsh to calculate the probablty dstrbuto of the percetage gross loss L o the portfolo, that s, P = P L=, = 0,1,, Let A t be the value of the -th compay s assets, descrbed by a logarthmc Weer process da = radt +σ Adz where z t, =1,,, are Weer processes wth E ( dz ) = dt ( )( j) E dz dz = ρ dt, j The compay defaults o ts loa f the value of ts assets drops below the cotractual value of ts oblgatos D payable at tme T. Wethushave where [ ] ( c ) p= P A T < D = N 1 c = A D+ rt σ T σ T 1 ( log 0 log ) ad N s the cumulatve ormal dstrbuto fucto. Because of the jot ormalty ad the equal correlatos, the processes z ca be represeted as where z = bx+ aε, = 1,,, Cofdetal 1 Release Date: 1-February-1987

4 KMV Corporato ad b= ρ, a= 1 ρ E ( dx) = dt ( dε ) = dt ( dx)( d ) E E ε = 0 ( )( j) E dε dε = 0, j The term bx ca be terpreted as the -th compay exposure to a commo factor x (such as the state of the ecoomy) ad the term aε represets the compay s specfc rss. The P = P L= = < < ( ) P [ A1 T D1,, AT D, A+ 1T D+ 1,, AT D] ( ) P [ A1 T D1,, AT D, A+ 1T D+ 1,, AT D xt u] dp[ xt u] = < < = < ( ) = P c1 T + bxt + aε 1T < 0,..., c T + bxt + aε T < 0, c+ 1 T + bxt + aε+ 1T 0, [ ], c T + bxt + aεt 0 xt = u] dp xt < u c+ bu c+ bu = a a ( ) N 1 N dn( u) I terms of the orgal parameters p ad ρ,wehave P = ( ) N ( N ( p) ρu) 1 N ( N ( p) ρ u) dn( u), = 0,1,..., 1 ρ 1 ρ Note that the tegrad s the codtoal probablty dstrbuto of the portfolo loss gve the state of the ecoomy, as measured by the maret crease or decle terms of ts stadard devatos. Cofdetal Release Date: 1-February-1987

5 Probablty of Loss o Loa Portfolo LIMITING LOAN LOSS PROBABILITY DISTRIBUTION Oldrch Vasce, 8/9/91 The cumulatve probablty that the percetage loss o a portfolo of loas does ot exceed θ s F ( ) [ θ] θ = = 0 P where P are gve by a tegral expresso Oldrch Vasce s memo, Probablty of Loss o Loa Portfolo, /1/87. The substtuto 1 1 s = N ( N ( p) ρu ) 1 ρ the tegral gves F ( θ) where [ θ] 1 as ( θ ) = ( ) ( 1 ) ( ) F s s dw s = W s N N s N p ρ ( ) 1 1 ( ) = 1 ρ ( ) ( ) By the law of large umbers, [ θ] ( ) ( ) lm 1 0 f s s = θ< s = 0 = 1 f θ> s ad therefore the cumulatve dstrbuto fucto of loa losses o a very large portfolo s ( ) W( ) F θ = θ Ths s a hghly sewed dstrbuto. Its desty s Cofdetal 3 Release Date: 1-February-1987

6 KMV Corporato ( ) ( ) 1 ρ 1 1 f ( θ ) = exp 1 ρn ( θ) N ( p) + N θ ρ ρ ( ) Its mea, meda ad mode are gve by θ= p 1 1 θ med = N N ( p ) 1 ρ 1 ρ θ mode = N N ( p) for ρ < 1 ρ 1 1 Cofdetal 4 Release Date: 1-February-1987