TU Braunschweg - Insttut für Wrtschaftswssenschaften Lehrstuhl Fnanzwrtschaft Maturty Effect on Rsk Measure n a Ratngs-Based Default-Mode Model Marc Gürtler and Drk Hethecker Fnancal Modellng Workshop Ulm, 21.09.2005 1
Agenda 1. Introducton: Maturty Effects n CRM 2. Key Issues for a Default Mode Model 3. Maturty Effects n the Merton/Vascek-Model 3.1 The Captal to Maturty Approach 3.2 The Captal for one Perod Approach 4. Emprcal Analyss from Ratng Data 4.1 The Captal to Maturty Approach 4.2 The Captal for one perod Approach 5. Concluson 2
Introducton: Maturty Effects n CRM [1] Today, n order to measure the rsk arsng from credt portfolos, merely two state dscrete tme models are common both n academc research as well as n practse. Well known models are CredtPortfoloVew TM, CredtRsk+ TM, and CredtMetrcs TM, that determne the full PDF of the portfolo on a oneyear tme horzon (rsk horzon) and analyse t wth respect to mean, standard devaton, and Value at Rsk or Unexpected Loss. Commonly, the Value at Rsk s used as a measure for the economc captal, that the bank should hold aganst future losses. Especally nvestment loans often have a tme to maturty longer than the (one-year) tme horzon of the model, but credt portfolo models do not account for ths msspecfcaton and the potental rsk arsng from ths fact. Only few papers deal wth the effect of (longer) tme to maturty on rsk measure/economc captal. 3
Introducton: Maturty Effects n CRM [2] Mark-to-Market (MTM) Models (e.g. CredtMetrcs TM ) Modellng the market value of the credts (and the portfolo), that depends on nterest rates, term structure of credt spreads and the ratng mgraton matrx. Rsk horzon s fxed (one year) and maturty nfluences on the rsk measure, snce the dfferent tmes to maturty lead to dfferent credt spreads. related papers: 4
Introducton: Maturty Effects n CRM [3] Grundke (2003) The Term Structure of Credt Spreads as a Determnant of the Maturty Effect on Credt Rsk Cap. Analytc determnaton of portfolo values usng dfferent term structures of credt spreads Term structure of credt spreads va Merton(1974)-Model Term structure s one of the most mportant determnants for the maturty effect. The Basel II-maturty-adjustment s explanable. Kalkbrener/Overbeck (2001/2002) Maturty as a factor for credt rsk captal/the maturty effect on credt rsk captal Smulaton based analyss usng market credt spreads (US ndustral bonds 1997/2001) The Basel II-maturty-adjustment s very conservatve. Barco (2004) Brngng Credt Portfolo Modellng to Maturty Analytc determnaton usng sattle pont technque. The Basel II-maturty-adjustment s very conservatve. 5
Introducton: Maturty Effects n CRM [4] All these models drectly lnk ther result to the Basel II adjustment formula, but suffer from the problem that they use a MTM approach. In a Default Mode (DM) model (e.g. CredtRsk+ TM, Basel II-IRB- Model) changes n the market value of credts are not of nterest. Lterature on ncorporatng possble rsk, that arses from longer tme to maturtes n a DM framework are scare. L/Song/Ong (1999) Maturty Msmatch Models for credts, that mature before the rsk horzon. Gordy/Hetfeld (2001) Maturty effects n a class of mult-perod default mode models unpublshed 6
Agenda 1. Introducton: Maturty Effects n CRM 2. Key Issues for a Default Mode Model 3. Maturty Effects n the Merton/Vascek-Model 3.1 The Captal to Maturty Approach 3.2 The Captal for one Perod Approach 4. Emprcal Analyss from Ratng Data 4.1 The Captal to Maturty Approach 4.2 The Captal for one perod Approach 5. Concluson 7
Key Issues for a Default Mode Model [1] Some assumptons due to the key ssues on default mode models from BIS (1999: Credt Rsk Modellng: Current Practse and Applcatons, Bank for Internatonal Settlements The exposure s ntended to be held to maturty (buy and hold). IAS 39 (loans and recevables shall be measured at amortsed cost ) The Far value s not observable (IAS 39.46/47) Losses only occur, f there s objectve evdence that an mparment loss has been ncurred (IAS 39.63) Ths especally s vald for so called held-to-maturty nvestment (IAS 39.9) Due to lmted markets, the credt could not be traded before maturty. 8
Key Issues for a Default Mode Model [2] Lqudaton Perod approach: each faclty s assocated wth a unque nterval concdng wth nstrument s maturty. Our approach: Captal to Maturty For the model a tme horzon equal to the maturty of the credts s consdered. Constant Tme Horzon for all Asset Classes approach: a one year tme horzon for all facltes s adopted. Addtonally: New economc captal could not be rased for the followng perod. The bank could not hedge perfectly future potental losses. Our approach: Captal for one Perod Rsk of credts wth long maturty arses from a ncreasng proballty of default (-> ncreasng economc captal) of non-defaulted loans. 9
Agenda 1. Introducton: Maturty Effects n CRM 2. Key Issues for a Default Mode Model 3. Maturty Effects n the Merton/Vascek-Model 3.1 The Captal to Maturty Approach 3.2 The Captal for one Perod Approach 4. Emprcal Analyss from Ratng Data 4.1 The Captal to Maturty Approach 4.2 The Captal for one perod Approach 5. Concluson 10
Maturty Effects n the Merton/Vascek-Model Assets of the borrower are log normally dstrbuted and follow a geometrc Brownan moton. The labltes grow wth a constant rate. (T) (0) (T) (T) (T) ln A = ln A (T) (0) +µ +σ a B = B exp(r T) Default occurs, f the assets at t=t falls short of the outsde labltes. ( ) b ln( B A ) PD = P A < B = N(b ) (T) (T) (T) (T) The credt portfolo s nfntely homogeneous and assets follow a one-factor approach. a = ρ x + 1 ρ ε The (portfolo nvarant) credt rsk contrbuton s quantfed by the dfference (Unexpected Loss) between Expected Loss and VaR of the potental gross loss rate (T) (T) (T) UL( ): = VaR ( ) E( ) = µ σ (T) (0) (0) (T) (T),eff A, (T) (T) (T) (T) (T) ε z E( ) = PD (T) (T) x, ~ N(0,1) ( ) 1 q VaR ( ) N N (PD ) x 1 1 z (T) (T) (T) z = ρ ρ 11
Agenda 1. Introducton: Maturty Effects n CRM 2. Key Issues for a Default Mode Model 3. Maturty Effects n the Merton/Vascek-Model 3.1 The Captal to Maturty Approach 3.2 The Captal for one Perod Approach 4. Emprcal Analyss from Ratng Data 4.1 The Captal to Maturty Approach 4.2 The Captal for one perod Approach 5. Concluson 12
The Captal to Maturty Approach [1] In the Captal to Maturty approach the tme horzon t = T s set to the maturty of the loan. The effect of ncreasng maturty (t = m T) on the probablty of default due to Merton s (here m = 2) (T) (2 T) (2 T) (2 T) 1 (T),eff PD = N( b ) b µ = b (T) 2 σa, The probablty of default rses, f ( ) µ σ < 2 1 b (T) (T) (T),eff A, Snce ths bound ncreases wth lower probabltes of default (for PD < 0,5), so t seems to be more lkely, that the default probablty rses wth shftng to hgher maturty, when probablty ntally s low. 13
The Captal to Maturty Approach [2] In order to calculate the unexpected loss contrbuton we use the probablty of default at maturty t = m T. UL( ): = VaR ( ) E( ) (m T) (m T) (m T) z E( ) = PD (m T) (m T) ( ) 1 q VaR ( ) = N N (PD ) ρ x 1 ρ 1 z (m T) (m T) (m T) z The probablty of default at t = m T s a functon of the probablty of default at t = T ( 1 ) PD = N(b ) = f N (PD ), µ σ,m (m T) (m T) (T) (T) (T),eff A, The maturty adjustment specfes the functon, that lnks the maturty adjustment at maturty t = T to maturty t = m T ( ) UL( ) = UL( ) g PD, µ σ,m, ρ (m T) (T) (T) (T) (T) CtM,eff A, Snce n a ratng based model most parameters are not abservable, the maturty adjustment functon should estmated emprcally. 14
Agenda 1. Introducton: Maturty Effects n CRM 2. Key Issues for a Default Mode Model 3. Maturty Effects n the Merton/Vascek-Model 3.1 The Captal to Maturty Approach 3.2 The Captal for one Perod Approach 4. Emprcal Analyss from Ratng Data 4.1 The Captal to Maturty Approach 4.2 The Captal for one perod Approach 5. Concluson 15
The Captal for one Perod Approach [1] In the Captal to Maturty approach the tme horzon t = T s constant, but s has to be taken nto account, that the probablty of default of a loan possbly rses over tme. The (expected) probablty of default of a loan wth maturty m > 1 for the perod (T, m T) s (here m = 2) (T,2T) (T,2T) (2T) (T) (T) PD PD = PD ( A > B ) = (T) 1 PD The probablty of default wth respect to the frst perod rses, f PD > 0,5 0,25 PD (T) (T,2T) PD < 0,25 (T,2T) The margnal probablty of default s (T) (T) (T) (T) (T,2T) b µ,eff 2 (T) b µ,eff 1 PD = N N b (T), ; (T) 2 2 σa, 2 2 σa, 2 Therefore, a hgh probablty of default n the frst perod leads to low margnal probabltes n the second perod. 16
The Captal for one Perod Approach [2] In order to calculate the unexpected loss contrbuton only for one perod UL( ) = VaR ( ) E( ) (m T) (m T) (m T) z (m T) (m T) E( ) = PD q 1 z We use the hghest expected one-perod probablty of default untl maturty t = m T The maturty adjustment specfes the functon, that lnks the maturty adjustment at maturty t = T to maturty t = m T The functon s ftted emprcally. ( (m T) ) (m T) 1 (m T) VaR z( ) = N N (PD ) ρ x 1 ρ (m1t,mt) ( ) (m T) (T) (T,2T) (2T,3T) ( ) PD = max PD,PD,PD,...,PD ( ) UL( ) = UL( ) g PD, µ σ,m, ρ (m T) (T) (T) (T) (T) CoP,eff A, 17
Agenda 1. Introducton: Maturty Effects n CRM 2. Key Issues for a Default Mode Model 3. Maturty Effects n the Merton/Vascek-Model 3.1 The Captal to Maturty Approach 3.2 The Captal for one Perod Approach 4. Emprcal Analyss from Ratng Data 4.1 The Captal to Maturty Approach 4.2 The Captal for one perod Approach 5. Concluson 18
Emprcal Analyss from Ratng Data [1] We analysed average cumulatve default rates of ratng data on a yearly bass from Standard & Poors (cumulatve default rates/average transton rates, up to 15 years) Moody s (cumulatve default rates/average mgraton rates, up to 20 years) here: Mood s cumulatve default rates, 7 classes Ftch (cumulatve default rates, 5 years) Credtreform Ratng AG (average mgraton rates, 5 years) In the Captal to Maturty -Approach we used the cumulatve default rates as a estmator for the probablty of default (m T) (m T) PD = DR In the Captal for One Perod -Approach we used the condtonal default rates as an estmator for the condtnal (perodcal) probablty of default. ( ) ( ) (T,T+ 1) (T+ 1) (T) (T) (T,T+ 1) PD = DR DR 1 DR = : DR 19
Emprcal Analyss from Ratng Data [2] In order to derve the unexpected loss contrbuton we used the asset correlaton due to the calbraton formula n Basel II ( ) ( (1 year) ) 1 exp 50 PD (1 year) ρ = f PD = 0.12 1 exp 50 ( ) ( (1 year) ) 1 exp 50 PD + 0.24 1 1 exp( 50) As a functon for the maturty adjustment we used the functon of the form lke n Basel II ( (1 year) ) g PD,m ( (1 year) ) ( (1 year) ) 2 ( ) 1+ m 2.5 a b ln(pd ) = 1 1.5 a b ln(pd ) The functon was ftted to the emprcal UL contrbuton usng leased squares. 2 20
Emprcal Analyss from Ratng Data [3] The maturty adjustment functon seems to be a good choce snce only two parameters has to be analysed where the parameter a especally controls the slope wth respect to m, The parameter b especally controlls the slope wth respect to PD 09/2005 Drk Hethecker 21
Agenda 1. Introducton: Maturty Effects n CRM 2. Key Issues for a Default Mode Model 3. Maturty Effects n the Merton/Vascek-Model 3.1 The Captal to Maturty Approach 3.2 The Captal for one Perod Approach 4. Emprcal Analyss from Ratng Data 4.1 The Captal to Maturty Approach 4.2 The Captal for one perod Approach 5. Concluson 22
Analyss of the cumulatve default rate The Captal to Maturty Approach [1] The cumulatve default rate has a concave characterstc for speculatve grades and a convex/lnear characterstc for nvestment grades. 23
Analyss of the UL contrbuton The Captal to Maturty Approach [2] The UL contrbuton rses rapdly for nvestment grades and wth decreasng slope / s at a stretch declnng for speculatve grades. 24
The Captal to Maturty Approach [3] Estmatons for parameter a for a maturty of up to 5 years Our estmates range from 0.06 to 0.22 n comparson to 0.12 n Basel II. 25
The Captal to Maturty Approach [3] Estmatons for parameter b for a maturty of up to 5 years Our estmates range from 0.06 to 0.1 n comparson to 0.05 n Basel II. 26
Comparson of the maturty adjustments: The Captal to Maturty Approach [4] Our adjustment especally fts wth the emprcal values at speculatve grade and overestmates the effect f compared to Basel II. 27
The Captal to Maturty Approach [4] Comparson of the maturty adjustments usng maturtes up to 10 years: The result does not change: the BaselII formula underestmates the effect n the Captal to Maturty approach. 28
Agenda 1. Introducton: Maturty Effects n CRM 2. Key Issues for a Default Mode Model 3. Maturty Effects n the Merton/Vascek-Model 3.1 The Captal to Maturty Approach 3.2 The Captal for one Perod Approach 4. Emprcal Analyss from Ratng Data 4.1 The Captal to Maturty Approach 4.2 The Captal for one perod Approach 5. Concluson 29
The Captal for one perod Approach [1] Analyss of the condtonal one-year default rate The condtonal default declnes for the lower speculatve grades and rses / stays equal for the nvestment grades. 30
Analyss of the UL contrbuton The Captal for one perod Approach [2] The UL contrbuton rses lnear for nvestment grades and stays at an equal level for speculatve grades. 31
The Captal for one perod Approach [3] Estmatons for parameter a for a maturty of up to 5 years Our estmates range from 0.00 to 0.07 n comparson to 0.12 n Basel II. 32
The Captal for one perod Approach [3] Estmatons for parameter b for a maturty of up to 5 years Our estmates range from 0.06 to 0.08 n comparson to 0.05 n Basel II. 33
Comparson of the maturty adjustments: The Captal for one perod Approach [3] Our adjustment especally fts wth the emprcal values at speculatve grade and s very close to the results from Basel II. 34
The Captal for one perod Approach [3] Comparson of the maturty adjustments usng maturtes up to 10 years: For longer maturtes the adjustment declnes and therefore the Basel II formula overestmates the effect n the Captal for one Perod approach. 35
Agenda 1. Introducton: Maturty Effects n CRM 2. Key Issues for a Default Mode Model 3. Maturty Effects n the Merton/Vascek-Model 3.1 The Captal to Maturty Approach 3.2 The Captal for one Perod Approach 4. Emprcal Analyss from Ratng Data 4.1 The Captal to Maturty Approach 4.2 The Captal for one perod Approach 5. Concluson 36
Concluson We have nvestgated the effect of the tme to maturty on economc captal n the CredtMetrcs TM -one-factor default mode model. Therefore, we motvated two approaches based on key ssues n credt rsk modellng: the Captal to Maturty Approach the Captal for one perod Approach. The qualtatve mpact of both approaches was examned usng the Merton/Vascek-model for credt rsk. Addtonally, our Approach were mplemented on emprcal data. The emprcal data shows characterstcs as expected from the model. Our results for the maturty adjustment (especally under the Captal for one perod approach) s close to the result n Basel II. 37