Granularity Adjustment in a General Factor Model

Transcription

1 Graularity Adjustmet i a Geeral Factor Model Has Rau-Bredow Uiversity of Cologe, Uiversity of Wuerzburg has.rau-bredow@mail.ui-wuerzburg.de May 30, 2005 Abstract The graularity adjustmet techique is embedded ito a geeral multi-factor model. This allows a very simple statemet of the coditios uder which the impact of usystematic risk factors asymptotically vaishes. It has always bee take for grated that the graularity adjustmet must be positive. I this paper, a couter-example with egative value of the graularity adjustmet is give for the well-kow Vasicek (2002) model. This meas a discout i terms of capital reserves for a less diversified credit portfolio. A i-depth aalyses of the aalytical formula of the graularity adjustmet reveals that such egative values are possible if the coditioal variace is higher for less favourable values of the systematic risk factors. The reaso is that this implies a relatively high survival probability i bad states of ature. JEL classificatio: D 81, G 21, G 28 1

2 1. Itroductio From the classical capital asset pricig model, the distictio betwee systematic ad usystematic risk is well kow. Ulike systematic risk, usystematic or idiosycratic risk ca be elimiated through diversificatio. Exactly the same also applies to a credit portfolio, where the ifluece of borrower specific risk vaishes completely i a sufficietly diversified portfolio. This ca formally be show by a simple applicatio of the law of large umbers to a geeral factor model. Because the ifluece of a idividual borrower o a large credit portfolio is very small, the aggregated loss will become o-stochastic if the values of the systematic risk factors are take as give. The loss the equals the coditioal mea of the aggregated loss, which depeds oly o the more or less favourable realizatio of the systematic risk factors. However, o real-world credit portfolio is perfectly diversified. I calculatig Value at Risk (VaR) for a credit portfolio, a correctio has therefore to be made i order to accout for the remaiig usystematic risk. The so-called graularity adjustmet techique was itroduced by Gordy (2003), (2004). A closed-form expressio for the adjustmet has bee developed by Wilde (2001), Marti ad Wilde (2002) ad Emmer/Tasche (2005). The graularity adjustmet was also part of a earlier versio of the ew capital accord (Basel II, see BCBS (2004)). Although the graularity adjustmet had bee later dropped from the fial versio of the capital accord, the cocept is oetheless iterestig from a theoretical poit of view. The key idea is that istead of calculatig both types of risks simultaeously, a two step model ca be developed with a add-up for usystematic risk. I this paper, the graularity adjustmet techique is embedded ito a geeral multi-factor model. This allows a very simple statemet of the coditios uder which the impact of usystematic risk factors asymptotically vaishes. I the existig literature, the aalytical formula for the graularity adjustmet has oly bee stated for the oe-factor model. Pykhti (2004) provided a differet applicatio of the graularity adjustmet techique to the multi-factor case. He develops a approximatio to a multi-factor model via a comparable oe-factor model. With icreasig factor correlatio, the respective adjustmet term coverges to zero. The graularity adjustmet has mostly bee cosidered as a techical tool which could be useful i calculatig capital requiremets. But it may also provide theoretical isights. For example, it has always bee take for grated that the graularity adjustmet must be positive. I this paper, a couter-example with egative value of the graularity adjustmet is give for the well-kow 2

3 Vasicek (2002) model. This meas a discout i terms of capital reserves for a less diversified credit portfolio. A i-depth aalyses of the aalytical formula of the graularity adjustmet reveals that such egative values are possible if the coditioal variace is higher for less favourable values of the systematic risk factors. The reaso is that this implies a relatively high survival probability i bad states of ature. The paper is orgaized as follows: Sectio 2 develops the geeral factor model ad describes its asymptotic behaviour. Sectio 3 is devoted to the derivatio ad aalysis of the graularity adjustmet. A example where the graularity adjustmet is egative is give together with a theoretical aalyses of this pheomea. Some fial remarks are give i Sectio Geeral Factor Model 2.1 Basic Assumptios Cosider a portfolio of loas with exposure sizes A 1,.., A. As a percetage of exposure size, the differece betwee the curret value of each loa ad the value at the ed of the plaig horizo (e.g. oe year) is described by a radom loss variable L i. Formally, the relative loss L i of the value of the loa could be positive as well as egative. It is therefore irrelevat whether losses are defied o a book-value or a mark-to-market basis. For example, if a mark-to-market model is used, a upgradig will result i a gai i market value ad cosequetly implies a egative value of the loss variable L i. Let each L i = L i (X,ε i ) be give as a fuctio of some systematic risk factors X = (X 1,...,X k ) ad a usystematic risk factor ε i. The systematic risk factors may also be called backgroud factors ad reflect the state of the busiess cycle i the differet idustry sectors. Each systematic risk factor ca be thought of beig assiged to a certai sector of the ecoomy. The systematic risk factors geerally have a ifluece o more tha oe borrower i the portfolio ad are the reaso why default evets are stochastic depedet. O the other had, each usystematic risk factor ε i has a ifluece o oly oe specific borrower. Ulike for the systematic risk factors, which may or may ot be correlated, usystematic risk factors are always assumed to be pairwise idepedet. May credit risk models ca be see as special cases of this simple but very geeral approach. Structural models such as the Merto (1974) model or CreditMetrics (1997) assume that default 3

4 evets or ratig chages are drive by the evolutio of the value of the firm assets, which i tur deped o the realizatio of some systematic ad usystematic risk factors. The risk factors therefore idirectly determie the potetial loss L i =L i (X,ε i ) of each loa. Of course, the cocrete fuctioal relatioship depeds o how the particular model is specified, which however is ot relevat for the geeral aalysis. A well-kow example for a itesity or default rate model is CreditRisk+ (1997). This model assumes that default probabilities p i = p i (X) are ot costat, but a fuctio of certai backgroud factors X = (X 1,...,X k ). I order to match this ito the above framework, assume that to each borrower there is assiged a additioal usystematic risk factor ε i ad the defie: L i X, ε i ={ LGD i,if ε i N 1 [ p i X ] 0 otherwise (1) Here, it is assumed that the ε i are stadard ormal distributed ad N -1 is the iverse of the cumulative ormal distributio fuctio. LGD i is the loss give default which will arise with probability p i = p i (X). 2.2 Diversificatio Havig the geeral factor model stated, it is ow possible to clarify the role of diversificatio. As a percetage of total exposure, the radom loss of the etire portfolio at the ed of the risk horizo is A i=1 i L i L P = i=1 A i (2) Now assume that the realizatios of the systematic risk factors X = (X 1,...,X k ) occur before the realizatios of the usystematic risk factors ε i. With give values of the systematic risk factors, L P is the sum of stochastically idepedet radom variables. Thus, the cetral limit theorem ca be applied. Coditioal o X, the portfolio loss variable L P is asymptotically ormal-distributed with mea 4

5 A i=1 i μ L i X μ L P X = i=1 A i (3) ad variace A 2 σ 2 i=1 i σ 2 L i X L P X = i=1 A i 2 (4) However, it is easy to show that if 0<A mi <A i <A max ad σ 2 (L i X)<σ 2 max for all i with fiite boudaries A max ad σ 2 max, the σ 2 (L P X) 0 as for every give realizatio of X. For sufficietly large, the coditioal variace teds to zero ad the probability for a arbitrary small deviatio of L P from the coditioal mea μ(l P X) gets arbitrary small. Therefore, as a cosequece of the law of large umbers, the coditioal portfolio loss becomes o-stochastic i a very large, ifiitely fie-graied credit portfolio. This is the mathematical formulatio of the fact how borrower-specific or usystematic risk ca be elimiated through diversificatio. The oly risk that remais is systematic risk, that is the risk that the actual values of the systematic risk factors X = (X 1,...,X k ) result i a higher or lower value of the coditioal mea μ(l P X). If systematic risk factors are varyig, the portfolio loss, cosidered as a percetage of total exposure, fluctuates respectively. If some lumpy credit risk remais withi the portfolio, the the o-zero coditioal variace σ 2 (L P X) is a atural measure for the amout of usystematic risk iheret to the credit portfolio. The coditioal variace will therefore play a promiet role i the formula for the graularity adjustmet to be developed later. Note that the coditioal variace σ 2 (L P X) depeds o the realizatio of the systematic risk factors. I the give cotext, the values of σ 2 (L P X) i those scearios where the realizatio of the systematic risk factors give rise to high losses are of particular importace. Two additioal remarks cocerig the coditioal variace ca be made. First, as a direct cosequece of the so-called law of coditioal variace, the average coditioal variace over all possible scearios for the systematic risk factors equals the differece betwee the variace of L P ad the variace of μ(l P X): 5

6 μ[σ 2 (L P X)] = σ 2 (L P ) - σ 2 [μ(l P X)] (5) That is, the expectatio of σ 2 (L P X) is that part of the portfolio variace that is caused by usystematic risk. Secod, the similarities betwee the coditioal variace ad the Herfidahl idex are obvious. The Herfidahl idex H = i=1 i=1 A i 2 A i 2 (6) is a ofte used measure to quatify the degree of cocetratio i credit portfolios. It is proportioal to coditioal variace σ 2 (L P X) if it is assumed that for each borrower i, the coditioal variaces σ 2 (L i X) of the idividual loa loss variables L i are the same. This implies that differeces regardig the distributio of potetial losses betwee the differet borrowers ca be eglected. Cocetratio risks ca the oly arise from differeces regardig the exposure sizes A i. However, if loas ot oly differ with respect to exposure sizes, but also with respect to e.g. default probabilities or losses give default, the the Herfidahl idex might be a to simple measure of cocetratio risks. 2.3 Oe-factor model As a illustratio, cosider a oe factor model based o the followig assumptios: 1) the loss variable L i is a decreasig fuctio of oly oe systematic risk factor X, i.e. X is a scalar 2) usystematic risk is perfectly diversified away, i.e. L p = μ(l P X ) I this case, a explicit expressio for portfolio VaR with cofidece level 1-α ca be give. Because of α = Prob[L P > VaR 1-α (L P )] = Prob[ μ(l P X ) > VaR 1-α (L P ) ] = Prob(X < x α ) (7) 6

7 where x α is the respective quatile of the systematic risk factor X, portfolio VaR, cosidered as a percetage of total exposure, is give as: A i=1 i μ L i X =x α VaR 1-α (L P ) = μ(l i X=x α ) = i=1 A i (8) A direct decompositio of portfolio VaR is obviously possible: If the bak wats to survive with a probability of at least 1-α, the amout of capital that must be reserved for each Euro borrowed to borrower i is exactly give as μ(l i X=x α ). Margial VaR i a oe factor model is the give as the expected loss coditioal o X=x α.. A special versio of the oe-factor model is attributed to Vasicek (2002). It assumes that borrower i defaults if the retur of the firms assets falls below a certai threshold D i : r i = ρ X 1 ρ ε i D i (9) Here, ρ is the correlatio coefficiet of the asset returs ad X, ε i are idepedet stadard ormal distributed radom variables with mea zero ad variace oe. The coefficiets are chose so that r i is also stadard ormal distributed. The relatioship betwee default threshold ad probability of default is PD i =N -1 (D i ), where N is the cumulative stadard ormal distributio fuctio. With default resultig i a loss give default LGD i (as a percetage of the exposure A i ), oe gets the followig formula for margial VaR: μ(l i X=x α ) = LGD Prob ε ND 1 PD i ρ x α i i 1 ρ (10) = LGD i N ND 1 PD i ρ x α 1 ρ This type of model has also bee adopted by the Basel Committee for Bakig Supervisio i its proposals for the ew Capital Accord. For example, i the cosultative paper from Jauary 2001, formula (12) with x 0,5% = -2,57 ad ρ = 0,2 for a corporate loa portfolio was used to calculate the 7

8 capital charge for a loa with default probability PD i. Some modificatios have bee made i the fial versio of the Capital Accord, but the mai idea was preserved. The assumptio of oly oe systematic risk factor has bee made by the Basel Committee because this is the oly case where capital charges are idepedet from portfolio compositio. This is a problematic assumptio because it assumes a completely parallel developmet of busiess cycles i all coutries ad idustries. It is therefore irrelevat whether all borrowers of the bak belog to the same sector or ot. Cotrary to that, exposure to systematic risk ca be reduced i reality if loas are well distributed over differet idustry ad coutry sectors. But, ulike usystematic risk, exposure to systematic risk does ether completely vaish eve i a perfectly diversified portfolio. I a multi-factor model, a exact decompositio of VaR ivariat from portfolio compositio is o loger possible. However, as it has bee show that the coditioal mea is geerally the derivative of VaR, 1 the coditioal mea remais to be a first order approximatio of VaR cotributios also i a multi-factor model. The expectatio is the to be calculated coditioal o L P = VaR 1-α (L P ). Because this ca o loger be simplified to X=x α if X is ot a scalar, margial VaR is o loger portfolio ivariat. Margial capital requiremets for a additioal loa e.g. to a tech firm will the ot oly deped o the idividual default probability, but regularly also o the overall exposure of the existig credit portfolio to the tech sector. 3 Graularity Adjustmet 3.1 Formula for Graularity Adjustmet It has bee show that i a perfectly diversified credit loa portfolio, the radom variable L P ca be replaced by the radom variable μ(l P X). If the portfolio is ot perfectly diversified, a adjustmet for usystematic risk has to be made. The so called graularity adjustmet is the differece of VaR for μ(l P X ) ad VaR for L P. The graularity adjustmet ca be calculated via a sesitivity aalysis of VaR. However, because the first derivative of VaR equals the coditioal mea of the margial risk, the first order approximatio of the error term i such a sesitivity aalysis is zero. It is therefore ecessary to also kow the secod derivative of VaR. With the techical details left to appedix A, oe fially gets 1 I the appedix, it is show that VaR(Y+hZ) VaR(Y) + h μ[z Y=VaR(Y)] if h 0 for cotiuous distributed radom variables Y ad Z. See also Gourieroux, Lauret & Scaillet (2000). 8

9 the followig closed-ed formula: VaR 1 α L P = VaR 1 α [ μ L P X L P μ L P X ] (11) VaR 1 α [ μ L P X ] 1 2 [ δσ 2 [ L P μ L P X =s] δs σ 2 [ L P μ L P X =s] δl f μ L P X s ] δs s=var 1 α [ μ L P X ] Here, f μ LP X s deotes the desity of the radom variable μ(l P X ). Note that μ(l P X ) is a scalar defied as a fuctio of oe or more systematic risk factors X = (X 1,...,X k ). Cotrary to the results preseted i the literature, this formula for the graularity adjustmet is ot restricted to the oe factor case. A illustratio with a very simple example may be useful. Cosider a oe-factor model for a completely homogeeous credit portfolio with A i =1 for all i ad L i ={ 1 with probability p X 0 otherwise (12) where p(x) is a mootoe decreasig fuctio of the systematic risk factor X. The: μ L P X = p X (13) σ 2 [ L P μ L P X =s]= s 1 s Formula (11) the simplifies to VaR L P p x α 1 2 p x α p x α 1 p x α 2 2 δl f p X s δs s= p xα (14) 9

10 I this case, the graularity adjustmet is iversely proportioal to the umber of loas ad coverges to zero as. 3.2 Sig of the graularity adjustmet It has always bee take for grated i the existig literature that the graularity adjustmet is positive. However, if oes looks to the aalytical formula for the graularity adjustmet give by equatio (11), it is ot immediately clear whether this is ideed always the case. I order to develop a couterexample, cosider the Vasicek model preseted above i chapter 2.3. For a completely homogeeous loa portfolio, the coditioal default probability is: p X =N N 1 PD ρ X 1 ρ (15) I appedix B, it is show that this implies δl f p X s δs s= p xα = N 1 [ p x α ] 2 ρ 1 N 1 PD 1 ρ ρ [ N 1 p x α ] (16) where () deotes the desity of the stadard ormal distributio. With α=30%, PD=20% ad ρ=0,95, oe has p(x α ) = p(-0,52400) = 0,06957 ad equatios (14) the equals VaR 70% L P 0, ,04311 (17) If all loas have default a probability of 20% ad the bak wats to survive with 70% probability, the capital charge i a perfectly diversified portfolio would be 6,957%. However, i this example, a ot perfectly diversified loa portfolio requires a slightly lower (!) capital charge. Although the give choice of the parameters may ot be very realistic 2, the example shows that a egative graularity adjustmet is ideed possible, at least theoretically. 2 Less realistic seem ot to exit, as a itesive umerical aalysis has show. Note that i this example, the 20% default probability of the loas is lower tha the target 30% survival probability of the bak. However, if more ad more loas are added to the portfolio, almost certaily some of these loas will default ad some capital reserves are required to cover these losses. 10

11 From the aalytical formula for the graularity adjustmet, a explaatio is possible how the lack of diversificatio could, i certai cases, result i a lower VaR. First ote that with perfect diversificatio, the bak collapses if μ(l P X) > VaR 1-α (L P ) ad survives if μ(l P X) < VaR 1-α (L P ). If the credit loa portfolio is ot perfectly diversified, the bak could also collapse if μ(l P X) < VaR 1-α (L P ), ad a additioal capital buffer is therefore ecessary to cover usystematic risk. However, oe should also ote that for a ot perfectly diversified bak it is also possible to survive eve though μ(l P X) > VaR 1-α (L P ). I the later case, i which all perfectly diversified baks would collapse, the lack of diversificatio is obviously a advatage. Which of these two cases has greater impact depeds o the amout of usystematic risk i the respective scearios, which is expressed by the value of the coditioal variace, ad also the occurrece probabilities of these scearios. If the coditioal variace σ 2 (L P μ(l P X)=s) is a icreasig fuctio of s, the probability that the bak survives eve though the realizatio of the systematic factors is such that μ(l P X) > VaR is relatively higher tha the risk of collapse give a sceario with μ(l P X) < VaR. I this case, the first summad -δσ 2 /δs withi the formula for the graularity adjustmet is egative. The secod summad -(1/2)σ 2 δl(f μ )/δs of the graularity adjustmet is positive if the desity of the radom variable μ=μ(l P X) slopes dowwards i the right tail, which will usually be the case. The occurrece probability of a sceario where the coditioal mea is above VaR - i which case all perfectly diversified baks would survive - is the higher tha the probability of the opposite. However, as the above example shows, there are certai cases where a egative first summad withi the graularity adjustmet outweighs a positive secod summad. 4. Coclusio The aalytical formula for the graularity adjustmet has bee embedded ito a geeral multi-factor model. This multi-factor model allows to distiguish betwee the ucoditioal ad the coditioal world. I the later case, i which values of the systematic risk factors are take as give, the aggregated portfolio loss is the sum of stochastic idepedet radom variables ad therefore coverges, if cosidered as a percetage of overall exposure, to the respective coditioal mea. However, oly i a oe-factor model this ca be exploit for a decompositio of the portfolio loss. Although the coditioal mea is the geeral expressio for margial VaR, i a multi-factor framework it depeds o the overall compositio of the portfolio which realizatios of the 11

12 systematic risk factors are particularly bad. The coditio for which the coditioal mea is to be calculated ca the ot be stated idepedet from the portfolio. I additio to the coditioal mea, the coditioal variace of the portfolio loss is also a useful variable i aalyzig the riskiess of the portfolio. It is give as the weighted average of the coditioal variaces of the idividual loas, with the weights depedig o exposure sizes. If all these idividual loas have the same coditioal variace, the coditioal variace of the portfolio would be proportioal to the Herfidahl idex. But if loas do ot oly have differet exposure sizes, but also differ with respect to their default probabilities or the amout of loss i the evet of a default, the Herfidahl idex may be a iappropriate measure of cocetratio risks. I geeral, the coditioal variace depeds o the value of the systematic risk factors, ad its value for particularly bad realizatios of these factors are of special iterest. A high coditioal variace i a very bad state of ature implies a relatively high chace that the actual portfolio loss is lower tha its coditioal mea. Formally, this is the case the the coditioal variace icreases together with the coditioal mea. If the divergece betwee actual loss ad its coditioal mea prevets a collapse of the bak, the lack of diversificatio would be a advatage. I this paper, it has bee show that this ideed implies the existece of umerical examples with egative graularity adjustmet. The possibility of a egative graularity adjustmet should ot be very surprisig as Artzer et al. (1999) have show that VaR is ot sub-additive ad therefore does ot always accout correctly for diversificatio. Though it seems that a egative graularity adjustmet is a rare evet which oly occurs for very uusual parameter values, it gives aother hit that VaR is a problematic measure of risk. 12

13 Appedix A: Graularity Adjustmet First ote that VaR 1 α L P = VaR 1 α [ μ L P X L P μ L P X ] VaR 1 α [ μ L P X ] δvar 1 α[ μ L P X h L P μ L P X ] h=0 1 2 δ 2 VaR 1 α [ μ L P X h L P μ L P X ] 2 h=0 Formula (11) for the graularity adjustmet the is a immediate cosequece of the followig Lemma: δvar 1 α Y hz =μ[zy hz =VaR 1 α Y hz ] ad δ 2 VaR 1 α Y hz 2 = 1 2 [ δσ 2 ZY hz =s δs σ 2 ZY hz =s δl f s Y hz ] δs s=var 1 α Y hz Proof: With abbreviatio VaR=VaR 1 α Y hz oe has 0 = δ Prob Y hz VaR = δ = VaR hz f y, z dy dz δvar z f VaR hz, z dz Because of 13

14 f VaR hz, z = f Z zy hz =VaR f Y hz VaR the result for the first derivative of VaR follows by dividig through by f Y+hZ (VaR). To get the secod derivative, oe proceeds as follows: 0 = δ = = = δ2 VaR δ 2 h = δ2 VaR δ 2 h = δ2 VaR δ 2 h δvar z f VaR hz, z dz δ 2 VaR δ 2 h δ 2 VaR δ 2 h f VaR hz, z δvar f VaR hz, z δvar z 2 δf VaR hz, z z dz δf s hz, z δs s=var dz 2 δvar f Y hz VaR z δ [ f Z zy hz =s f Y hz s ] δs s=var dz f Y hz VaR δ δs 2 δvar μ[ 2 δvar z z2 Y hz f Y hz VaR =s]s=var + μ[ δvar z 2 Y hz =VaR] f Y hz VaR δ [0 μ2 ZY hz =s μ Z 2 Y hz =s ] δs s=var + μ[ μ ZY hz =VaR z 2 Y hz =VaR] δf Y hz s δs s=var f Y hz VaR δf Y hz s δs s=var = f Y hz VaR [ δ2 VaR δ 2 h δσ 2 ZY hz =s σ 2 ZY hz =VaR δlf Y hz s ] δs δs s=var q.e.d. 14

15 Appedix B If X is a stadard ormal radom variable ad f p(x) (s) deotes the desity of p X =N N 1 PD ρ X 1 ρ The: δl f p X s δs s= p xα = N 1 [ p x α ] 2 ρ 1 N 1 PD 1 ρ ρ [ N 1 p x α ] Proof: Prob p X s = D ρ X Prob[ N 1 ρ s] = Prob[ X D N 1 s 1 ρ ] ρ = 1 Prob[ X D N 1 s 1 ρ ] ρ = 1 N [ D N 1 s 1 ρ ] ρ with D=N 1 PD. The desity of p(x) is the give as: f p X s = d ds Prob p X s =[ D N 1 s 1 ρ 1 ρ ] ρ N 1 s ρ It follows that 15

16 l f p X s = 1 2 [ D N 1 s 1 ρ ] 2 N 1 s 2 l 1 ρ ρ 2 ρ ad: δl f p X s δs = D N 1 s 1 ρ ρ 1 ρ ρ 1 N 1 s N 1 s N 1 s Refereces = N 1 s 2 ρ 1 D 1 ρ ρ 1 N 1 s q.e.d. Artzer, Ph., F. Delbae, J.-M. Eber, ad D. Heath (1999), Coheret Risk Measures, Mathematical Fiace 9, pp BCBS - Basel Committee o Bakig Supervisio (2004): Iteratioal Covergece of Capital Measuremet ad Capital Stadards, A Revised Framework. CreditMetrics (1997): Techical Documet. J.P. Morga. CreditRisk+ (1997): Techical Documet. Credit Suisse Fiacial Products. Gordy, M. (2003): A risk-factor model foudatio for ratigs-based bak capital rules, Joural of Fiacial Itermediatio 12(3), pp Emmer, S. ad D. Tasche (2005): Calculatig credit risk capital charges with the oe-factor model, Joural of Risk 7(2), pp Gordy, M. (2004): Graularity adjustmet i portfolio credit risk measuremet, i: G Szegö (ed), Risk Measures for the 21st Cetury, Joh Wiley. Gouriéroux, C., J. P. Lauret ad O. Scaillet (2000): Sesitivity aalysis of values at risk, Joural of Empirical Fiace 7, pp Marti, R. ad T. Wilde (2002): Usystematic credit risk, Risk Magazie 15(11), pp Merto, Robert C. (1974): O the pricig of corporate debt: the risk structure of iterest rates, Joural of Fiace, 29, pp Pykhti, M. (2004): Multi-factor adjustmet, Risk Magazie 17(3), pp Vasicek, O. (2002): Loa Portfolio Value, Risk Magazie 15(12), pp Wilde, T. (2001): Probig graularity, Risk Magazie 14(8), pp