On the computation of the capital multiplier in the Fortis Credit Economic Capital model

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1 On the computaton of the captal multpler n the Forts Cret Economc Captal moel Jan Dhaene 1, Steven Vuffel 2, Marc Goovaerts 1, Ruben Oleslagers 3 Robert Koch 3 Abstract One of the key parameters n the computaton of Cret Economc Captal s the so calle captal multpler In the lght of an exstng varance-covarance framework we propose a methoology rather than usng a benchmark number The paper proves an algorthm n ong so 1 Introucton motvaton Cret Rsk s the rsk that a borrower wll be unable to pay back hs loan For any nvual contract, the future loss n a one year pero s rom, e unknown n avance The sum of all these losses s calle the Portfolo Cret Loss Forts uantfes ts cret rsk through the measurement of the varablty of ths portfolo cret loss captal s hel to protect aganst ths rsk The amount of ths captal has been calbrate to acheve the Forts target of a S&P ratng of AA, meanng that the reure captal correspons to a 3 bp or less efault probablty over a one-year tme horzon In orer to calculate ths captal, the current cret rsk framework wthn Forts focuses on 2 measures: Expecte Loss Unexpecte Loss Expecte Loss EL s the expecte annual level of cret losses Actual losses for any gven year wll vary from the EL, but EL s the amount that Forts shoul expect to lose on average Expecte Loss shoul be vewe as a cost of ong busness rather than as a rsk tself The real rsk arses from the volatlty n loss levels Ths volatlty s calle Unexpecte Loss UL UL s efne statstcally as the star evaton of the cret loss strbuton Once these two measures are calculate, Forts etermnes the Economc Captal as a multple of the Unexpecte Loss Ths multple s calle the Captal Multpler: EconomcCaptal Captal Multpler x Unexpecte Loss Forts has moels that enable the calculaton of the portfolo Unexpecte Loss, specfcally for ts portfolo For the Captal Multpler however a benchmark number s beng use Hstorcal ata analyss seems to ncate that the loss rate for a large portfolo follows a Beta strbuton base on ths observaton, the captal multpler for a typcal large bank portfolo, such as Forts, has been etermne 1 Katholeke Unverstet Leuven, Belgum, Unverstet van Amsteram, the Netherls 2 Katholeke Unverstet Leuven, Belgum 3 Forts Central Rsk Management Ths rases the followng uestons: 1 What s a typcal bank portfolo, e what are the Expecte Unexpecte Loss that are mplctly assume n orer to obtan ths captal multpler? 2 Is the assumpton of a Beta strbuton correct n all stuatons, eg what f the portfolo s not large enough? 3 What methoology shoul be use f one eals wth small /or a-typcally versfe portfolos? In close collaboraton wth the Actuaral Research Group of KULeuven, Forts evelope a moel that takes nto account for each loan the nvual rsk parameters: exposure, ratng, loss gven efault efault correlatons Ths new methoology uses the same parameters as Forts s usng now for computng the portfolo unexpecte loss Hence t proves a full bottom-up approach as compare to the current approach where a bottom-up moel s combne wth an external benchmark number Gven the exact strbuton functon of the Aggregate Loss S expresse as a percentage of the Aggregate-Exposure-At- Default, t s straghtforwar to etermne ts 1 ɛ percentle hence the multpler K ɛ whch s efne as F 1 S 1 ɛ ES+K ɛ σ S Here S enotes the Loss rom varable, ES ts expecton, σ S ts star evaton F 1 S ts uantle functon The rom varable S s the sum of the losses on the nvual polces Hence, S s a sum of postve epenent rom varables In the moel that we wll present, we assume that we know the strbuton functons of the nvual losses, as well as the correlatons between these nvual losses It s mportant to note that ths nformaton s not enough to etermne the strbuton of S exactly In fact, knowlege of the whole multvarate strbuton s neee n orer to be able to etermne the strbuton functon of the sum Only n the case of a multvarate normal strbuton, the margnal strbutons together wth the correlaton matrx completely etermnes the strbuton functon of the sum In ths ocument, we propose an approprate approxmaton for the strbuton functon of S Ths approxmaton wll take nto account all the avalable nformaton margnal strbuton functons correlaton matrx We wll escrbe an algorthm that can be use to calculate an approxmaton for the strbuton functon of the Loss Let S be a rom varable havng ths approxmate strbuton functon, then, we propose to etermne the multpler K ɛ by F 1 S 1 ɛ ES +K ɛ σ S c BELGIAN ACTUARIAL BULLETIN, Vol 3, No 1, 2003

2 2 Prelmnary theoretcal results 21 The Gamma strbuton If Y Gammaα,, wth α>0 >0, then the probablty ensty functon pf of Y s gven by f Y y We also have that E [Y ] α, Var[Y ] α 2, α Γα yα 1 e y, y >0 E [ Y k] Γα + k α α +1 α + k 1 Γα k m Y t E [ e ty ] α, t <, t where m Y t enotes the moment generatng functon of Y evaluate at t 22 The Posson strbuton If N Possonλ, wth λ>0, then the probablty functon of N s gven by We have that Pr [N x] e λ λx, x 0, 1, 2, x! E [N] Var[N] λ, m N t E [ e tn ] exp [ λ e t 1 ] 23 The Negatve Bnomal strbuton If N NBr, p, wth r>0 0 <p 1, then the probablty functon of N s gven by r + x 1 Pr[N x] p r 1 p x, x 0, 1, x The frst two moments the moment generatng functon are gven by r1 p E [N] p r1 p Var[N] p 2, m N t E [ e tn ] r p 1 1 p e t 24 The Beta strbuton If Y Betaa, b, wth a>0 b>0, then the probablty ensty functon of Y s gven by f Y y Γa + b ΓaΓb ya 1 1 y b 1, 0 <y<1 The frst two moments the moment generatng functon are gven by E [Y ] Var[Y ] E [ Y k] a a + b, ab a + b +1a + b 2, a a +1 a + k 1 a + ba + b +1 a + b + k 1 25 The Negatve Bnomal strbuton as a Posson-Gamma mxture We assume that the rom varable N Λλ has a Posson strbuton wth parameter λ: N Λλ Posson λ Further, we assume that the mxng rom varable Λ has a Gamma strbuton wth parameters α : Λ Gamma α, The rom varable Λ s also calle the structure varable We fn m N t E [ e tn ] E Λ [E [ e tn Λ ] wth p E Λ [e Λet 1 ] m Λ e t 1 e t 1 p 1 1 pe t +1 We can conclue that N NB α, +1 α α 26 The moment generatng functon of a compoun strbuton Conser a collectve moel S N X 51

3 where the severtes X are nepenent of the freuency N Let Then we fn X X, 1, 2,,n m S t E [ e ts] E N E [e t ] N X N ] E N [m X t N N [ E e N ln mx t] Hence, m S t m N ln m X t It s easy to show that ES EN EX VarS EN VarX+[EX] 2 VarN 27 The sum of nepenent Compoun Posson strbutons Conser the sum of n nepenent Compoun Posson strbutons: N 1 N n S X 1j + + X nj Hence, we assume that the n rom varables N X j are mutually nepenent For each, we assume that the X j are mutually nepenent nepenent of N We also assume that Then we fn X j X N Posson λ m S t E [ e ts] N E exp t X j Π n E exp N t X j Π n m N ln m X t Π n exp [λ m X t 1] [ n ] exp λ m X t 1 exp[λ m X t 1] wth λ λ m X t λ m X t We can conclue that the sum of the n mutually nepenent Compoun Posson strbute rom varables s agan Compoun Posson strbute wth Posson parameter λ severty strbuton F X gven by F X x λ λ λ F X x 28 The sum of epenent Compoun Negatve Bnomal strbutons Conser the compoun strbute rom varables N X j, 1, 2,,n We assume that the rom varables N Λλ have a Posson strbuton wth parameter λ: N Λλ Posson λ We also assume that the mxng rom varable Λ has a Gamma strbuton wth parameters α : Λ Gamma α, Ths mples that the rom varables N are negatve bnomal strbute, but not nepenent Further, we assume that for each, the severtes X j, j 1, 2,, are mutually nepenent: X j X, j 1, 2,, also that the X j are nepenent of the mxng rom varable Λ, of N Λλ Fnally, we assume that the compoun sums N Λλ X j, 1, 2,,n are mutually nepenent Uner these assumptons, the sum S efne by N 1 N n S X 1j + + X nj, s a sum of Compoun Negatve Bnomal strbute rom varables At frst sght, etermnng the strbuton functon of the sum S s not a trval task, as the rom varables N X j, 1,,n are not mutually nepenent, the epenency cause by the common mxng rom varable Λ For a general approach of approxmatng sums of epenent rom varables, we refer to Dhaene, Denut, Kaas, Goovaerts & Vyncke 2002 a, b In ths partcular case however, one can prove that the strbuton functon of the combne portfolo s Compoun Negatve Bnomal strbute 52

4 Inee, we have that the moment generatng functon of S s gven by m S t E [ e ts] N E exp t Now, let X j N E Πn exp t X j E Λ E N Πn exp t X j Λ N E Λ Πn E exp t X j Λ { E Λ Π n m N Λ ln m X t } E Λ {Π n exp [ Λm X t 1]} { n } m Λ m X t 1 [ m X t 1 let X be a rom varable wth moment generatng functon gven by m X t m X t Then we fn [ ] α m S t m X t 1 [ ] α p ln[mx 1 1 p e t] m N ln m X t wth p gven by p + We can conclue that S s Compoun Negatve Bnomal strbute: N S Y wth N NBα, + where the Y Y are nepenent of N, wth the moment generaton functon of the Y gven by m Y t m X t, ] α or euvalently, F Y x F X x Note that the strbuton functon of S can also be etermne from the results of the prevous secton on sums of nepenent compoun Posson strbtons 3 Descrpton of the moel Conser a portfolo of n cret rsks Let I be efne as the ncator varable whch euals 1 f rsk leas to falure n the next pero, 0 otherwse The probablty that rsk leas to a falure s enote by : Pr[I 1] Further, let EAD enote the Exposure-At-Default LGD the Loss-Gven-Default of rsk The Exposure- At-Default s the maxmal amount of loss on rsk, gven efault occurs The Loss-Gven-Default s the percentage of the loss on polcy, gven efault occurs The Aggregate Portfolo Loss the Loss for short urng the reference pero s then gven by Loss I EAD LGD We wll assume that the EAD the LGD are etermnstc We are ntereste n the rom varable escrbng the Loss as a percentage of the Aggregate-Exposure-at-Default, where the Aggregate-Exposure-at-Default s gven by Aggregate-Exposure-at-Default EAD whch s etermnstc because of the assumptons mae above Hence, we are ntereste n etermnng the strbuton functon of S I EAD LGD n EAD j So that the rom varable of nterest can be wrtten as S I c wth EAD c n EAD LGD j Note that we can wrte S as a sum of n compoun Bernoull rom varables: S I c, 53

5 where, by conventon, 0 0 The Aggregate Loss S s the sum of the relatve losses on the nvual cret rsks In orer to compute the strbuton functon S exactly, knowlege of the multvarate strbuton functon s reure We wll assume however that our nformaton about the strbuton functon of S s not complete To be more precse, we assume that we know the margnal strbuton functons nvolve, e we assume that the efault probabltes are gven Furthermore, we assume that the parwse efault correlatons corr [I,I j ], or euvalently, the parwse correlatons between the margnal rsks n the sum, are gven In ths paper, we wll not scuss how to choose or bul a moel of efault correlatons Note that atonal assumptons nee to be mae concernng the epenency structure between the terms n the sum S n orer to be able to etermne approxmatons for ts percentles eg 4 Approxmaton for the strbuton functon of S The rom varable S as efne above can be nterprete as the aggregate clams n an nvual rsk moel, see eg Kaas, Dhaene, Goovaerts & Denut 2001 We wll approxmate ths nvual rsk moel by a collectve rsk moel One major problem n ths respect s the fact that S s the sum of mutually epenent rom varables Inee, n any realstc moel, we wll have that the ncator varables I all wll be postve epenent n some sense, where the postve epenence s cause by a common factor whch escrbes the global state of the economy We propose to approxmate each I by a rom varable N In orer to ntrouce the epenency, we wll conser a Baysan approach Therefore, let us assume that there exsts a rom varable Λ such that, contonally gven Λ λ, the rom varables N are mutually nepenent: N Λλ are mutually nepenent We further assume that, contonaly gven Λλ, the rom varables N are Posson strbute wth parameters λ: N Λλ Posson λ Furthermore, we assume that the rom varable Λ has a Gamma strbuton wth parameters α We wll enote ths as Λ Gamma α, In orer to etermne the strbuton functon of N,we wll etermne ts moment generaton functon We fn E [ e tn] E [ E [ e tn Λ ]] E [ exp Λ e t 1 ] m Λ e t 1 α e t e t whch mples that N NB α, + To summarze, we propose to approxmate the strbuton functon of S : I S c, by the strbuton functon of S : S N c 5 Choce of the parameters α In ths secton, we wll explan how to choose the parameters α such that the strbuton functons of S S are as alke as possble, gven the lmte nformaton on the rom vector I 1,I 2,,I n 51 Determnaton of α A frst reurement for our approxmaton to perform well s that the strbuton functons of I N are as alke as possble In orer to have that E [N ]E [I ], we have to choose α eual to : α - Remark 1: Ths choce mples that Var[I ] 1 Var[N ] 1+ It can be proven that I cx N, where cx sts for smaller n the convex-orer sense Ths means that t s a safe strategy to replace I by N,n the sense that any rsk-averse ecson-maker woul prefer clam-numbers I to N, for more etals see Kaas, Goovaerts, Dhaene & Denut Remark 2: Uner the choce α, the strbutons of I N wll α, 54

6 be close to each other prove s small enough such that hgher orer terms can be neglecte Inee, we have that Pr [N 0] 1 Pr[I 0], + whle Pr [N 1] 1 Pr[I 1] Remark 3: It s straghtforwar to verfy that the choce α mples E [S] E [S ] 52 Determnaton of It remans to etermne an explcte value for the parameter Frst note that for j we have that Covar[N,N j ] E [E [N N j Λ]] E [N ] E [N j ] E [E [N Λ] E [N j Λ]] j { [ j E Λ 2 ] 1 } j Var[Λ] j, whle Var[N ] 1+ Hence, for j, the parwse correlatons n the approxmate moel are gven by j 1 corr [N,N j ] j j In orer to fx the parameter we reure that the secon moments of S S coïnce We have that Var[S] c c j covar [I,I j ], whch s assume to be known On the other h, we have that Var[S ] c c j covar [N,N j ] c 2 + c 2 Hence, the conton VarS VarS wll be fulflle f s chosen as follows: c 2 VarS c2 Note that n the moel that we propose, n fact we replace the known correlatons corr [I,I j ] by corr [N,N j ] j for j Hence, our approxmaton wll perform the best f the exact correlatons corr [I,I j ] are approxmately eual to j Ths correlaton structure seems to be consstent wth many realstc correlaton moels 6 How to compute the f of S? From the Secton Prelmnary Theoretcal Results, we fn that S has a Compoun Negatve Bnomal strbuton: N S Y, where N NB, + where the Y are nepenent of N, wth the moment generaton functon of the Y gven by m Y t m c t Wthout loss of generalty we can assume all the c to be fferent In ths case, we fn that the rom varables Y have the followng probablty functon: Pr [Y c ], 1, 2,,n It remans to present an algorthm wth enables to compute the strbuton functon of a Compoun Negatve Bnomal strbuton Ths can be performe by a well-known recurson n actuaral scences, calle Panjer s recurson, see eg Kaas, Goovaerts, Dhaene & Denut 2001 We have that Pr [S 0]Pr[N 0] also, for x 1, 2,, x Pr [S x] a + bk Pr [N k]pr[s x k], x where k1 a + b a 1 7 Assymptotc behavour of the propose approxmaton In ths secton we wll show that, uner certan assumptons, the strbuton functon of the aggregate loss S tens to a Beta strbuton when the sze of the portfolo becomes suffcently large 55

7 Assume that all EAD LGD are eual to 1, that all efault probabltes are eual to These assumptons mply that all c are eual to 1 n In ths case S has the followng Compoun Negatve Bnomal strbuton: S N n where N NB, + n The moment generaton functon of S s then gven by t m S t m N, n +n 1 1 +n e t/n + n ne t/n If the number of contracts n reaches nfnty, we fn / lm m S n t / t Ths means that f the number of contracts becomes very large, the propose approxmaton S for the aggregate loss wll be approxmately Gamma strbute: S Gamma,/ for n suffcently large Note that the true outcomes of S are n the regon [0, 1], whle the above mentone Gamma approxmaton leas to outcomes n the range [0, In practce however the probablty of exceeng 1, compute wth the Gamma,/ strbuton s almost eual to 0 For eg, ths probablty euals e It s nterestng to compare ths Gamma strbuton wth a Betaa, b strbuton In orer to match the frst 2 moments of the Beta strbuton wth the Gamma strbuton, we must have that a a + b ab a + b +1a + b 2 2 Hence, a 1, b 1 1 a Let us now compare the moments of X Gamma,/ Y Betaa, b where the parameters are connecte as above We fn k EX k +1 + k 1 EY k k k 2 After some straghtforwar computatons we fn EY k k k k 2 k k 1 + k 1 1 k+2 For small, also +1 +k 1 s small On the other h, the factor k+2 s ncreasng n k Hence, for k 3, 4, EX k EY k 1 prove that k +2 s small enough We can conclue that for a large portfolo where all rsks have the same small efault probabltes, the strbuton functon of the approxmaton S tens to be close to a Beta strbuton 8 Concluson The Aggregate Loss, expresse as a percentage of the Aggregate-Exposure-at Default can be wrtten as S I c Here the rom varables I are Bernoull strbutons wth gven efault probabltes Pr I 1 We assume that the covarances CovarI,I j are gven The c are assume to be etermnstc amounts efne by EAD c n EAD LGD j We propose to approxmate the strbuton functon of S by the strbuton functon of S, where N S c, 56

8 wth N NB, + c 2 VarS c2 For ths approxmaton, we have that the strbuton functons of the N the I are very close to each other Moreover, the frst the secon moments of S are eual to the corresponng moments of S The strbuton functon of S can easly be compute by Panjer s recurson If we n aton have that the real correlatons are such that j corr [I,I j ], then also the correlaton structure of S approxmately coïnces wth the correlaton structure of S Gven the strbuton functon of S, the captal multpler K ɛ, corresponng to the 1 ɛ-percentle can then be etermne approxmately by K ɛ F 1 1 ɛ ES S σ S For a large portfolo where all rsks have the same small efault probabltes, the strbuton functon of the approxmaton S tens to be close to a Beta strbuton ACKNOWLEDGEMENTS Jan Dhaene, Marc Goovaerts Steven Vuffel acknowlege the fnancal support of Forts Central Rsk Management REFERENCES [1] Kaas, R; Goovaerts, MJ, Dhaene, J; Denut, M 2001 Moern Actuaral Rsk Theory, Kluwer Acaemc Publshers, pp 328 [2] Dhaene, J; Denut, M; Goovaerts, MJ; Kaas, R; Vyncke, D 2002a The concept of comonotoncty n actuaral scence fnance: Theory, Insurance: Mathematcs & Economcs, vol 311, 3 33 [3] Dhaene, J; Denut, M; Goovaerts, MJ; Kaas, R; Vyncke, D 2002b The concept of comonotoncty n actuaral scence fnance: Applcatons, Insurance: Mathematcs & Economcs, vol 312,