Correlation across latent variables in credit risk models: a direct inference from default rates

Transcription

1 Coelation acoss latent vaiables in cedit isk models: a diect infeence fom default ates Fenando Moeia Univesity of Edinbugh Cedit Scoing and Cedit Contol XIII confeence 8 th 30 th August 013

2 Intoduction Popula cedit isk models use coelation acoss latent vaiables that (supposedly) dive defaults Linea coelation (poduct-moment) is used Example: Basel Accod (estimation of capital equied to cove unexpected cedit losses) Diffeent coelations defined fo each cedit class (e.g. cedit cad and motgage)

3 Coelation Motivation (cont d) Limitation: cuent method (Basel Accods) does not captue potentially highe coelation acoss latent vaiables when default ates (o pobabilities of default = PD) ae eithe extemely low o extemely high coelation etail coelation copoate coelation cedit cad coelation motgage Assume that high-isk (high-pd) loans default due to idiosyncatic (specific) factos and theefoe pesent low coelation PD 3

4 Coelation Motivation We should expect a elationship like the following: Intuition: highest coelation is elated to BOTH low and high default ates (o PDs) because latent vaiables (asset etuns, e.g.) ae moe associated not only when default ates (o PDs) ae low but also when they ae high (latent vaiables decease at the same time egadless of the cause common o specific). PD o default ates Think in tems of potfolios. Intepetation: PD = (expected) default ate 4 in potfolio

5 Empiical estimates Basel Accod (II) Cedit Segment Coelation Copoate, soveeign, bank exposues, SMEs between 0.1 and 0.4 High-volatility commecial eal estate between 0.1 and 0.30 Residential motgages 0.15 Revolving etail cedit 0.04 Othe etail exposues between 0.03 and 0.16 The method and the data used wee not disclosed Academic papes Autho(s) Segment Technique Coelation Rösch (003 and 005), Hamele et al. (003), Hamele and Rösch (006) Copoate MLE to Lopez (004) Copoate Min diffeence models to 0.35 Rösch and Scheule (004) Retail Regessions to De Andade and Thomas (007) Retail Co acoss beh scoes to Cook and Bellotti (01) Cedit Cads MLE to

6 An altenative measue: Tetachoic Coelation Measues the coelation between two (atificially) dichotomized continuous vaiables that follow a bivaiate nomal distibution It is an estimate of the coelation between the vaiables if they wee not atificially dichotomized 6

7 Tetachoic Coelation: An example Relationship between opeational condition and hous of opeation of machines If we obseve dichotomized vaiables: opeational condition (as good o bad ) and hous of opeation (as shot o long ) We don t have data on the actual values, we cannot calculate the linea coelation 7

8 Tetachoic Coelation: An example Calculation based on a contingency table: Hous of opeation Shot (Y=0) Long (Y=1) Total Opeational condition Good (X=1) a b a+b Poo (X=0) c d c+d Total a+c b+d a+b+c+d = N Define two vaiables : 1 ( a c) ( b d) N ( a b) ( c d) N 8

9 Tetachoic Coelation: An example Use 1 and to find two new vaiables h and k: h ! ! Define as: 17 7! N... ad k ( h) ( k) bc ! 7 5 5! 17 7!... whee a, b, c, d, and N ae defined in the contingency table and is the pdf of the nomal distibution Find the tetachoic coelation,, in:! hk 3 3! ( h 1)( k 1) 4 4! hk( h 3)( k 3) 5 5! ( h 4 6h 3)( k 4 6k 3)... Since 1 and the denominatos n! go to, the tems n /n! appoach zeo. Theefoe, the fist few tems ae sufficient to detemine. The tetachoic coefficient can be estimated in, e.g., SAS and STATA. 9

10 As seen befoe, in: Tetachoic coelation: A simplified fomula! hk 3 3! ( h 1)( k 1) 4 4! hk( h 3)( k 3) 5 5! ( h 4 6h 3)( k 4 6k 3)... Since 1 and the denominatos n! go to, the tems n /n! appoach zeo. Theefoe, the fist few tems ae sufficient to detemine. Fom this, we can use the simplified fomula esticted to the fist tem above, (at the expense of some bias): ad bc N ( h) ( k) Fom the pevious slide 10

11 Tetachoic coelation in cedit isk models We obseve continuous latent vaiables (asset etuns of obligos) dichotomized into statuses default o non-default and assumed to be nomally distibuted Thus, since we do not have data on the actual values of the latent vaiables, we can use the tetachoic coelation to estimate thei linea coelation 11

12 Tetachoic coelation in cedit isk models We need to define the contingency table fo this case Use the numbe of pais of loans (joint occuences) with espect to thei statuses (default and non-default) This tem, e.g., gives the numbe of pais in which both loans defaulted (given the default ate in the peiod = i.e. numbe of defaults/total numbe of loans) Default status (loan i) default non-default Default status (loan j) default comb(d*l,) [(L (d*l)) * (d*l)]/ non-default [(L (d*l)) * (d*l)]/ comb((l (d*l)),) whee comb(x,y) stands fo the combination of x elements taken y a time; d is default ate, and L is the numbe of loans in the potfolio. 1

13 tetachoic coelation Estimates based on the tetachoic coelation Estimated asset coelations (i.e. acoss latent vaiables) fo potfolios composed of 100, 1000, and loans (conditional on seveal default ates). Example: 1000 loans 3.5 x default ate Lowest coelation = (fo d = 0.0) Highest coelation = (fo d = 0.05 and 0.95) Intuition: fo eithe vey low o vey high default ates, asset etuns of obligos tend to move togethe (debtos ae in simila situation), so coelation is highe Fo intemediate values of default ates, coelations ae lowe Howeve 13

14 tetachoic coelation Estimates based on the tetachoic coelation Howeve, cannot explain the coelations found fo the exteme values of default ates, i.e., why do they decease? 3.5 x default ate 14

15 tetachoic coelation Estimates based on the tetachoic coelation Theefoe, I popose an atificial adjustment (until a solution is found).8 x default ate 15

16 Limitations of this pape What I am not doing in this pape: I am not elaxing the assumption of nomality (egading the latent vaiables in these models) I am not elaxing the assumption of homogenous coelation (equal value fo all pais) in a paticula potfolio 16

17 Conclusions Theoetically, the tetachoic coelation is moe compatible (than the linea coelation) with taditional cedit isk models (based on latent vaiables) Moe ealistic since it eflects the cuent situation of loan potfolios such that low o high default ates indicate highe coelation in contast to Basel which specifies a monotonic deceasing elationship between coelation and pobability of default (i.e. default ates) 17

18 Conclusions Instead of focusing on cedit segments (as Basel does), I focus on default ates Can be used as time-vaying model (updated accoding to the most ecent default ates) Some wok is still needed to estimate the coelation egading exteme values of default ates 18