Mixed-Integer Credit Portfolio Optimization: an application to Italian segregated funds

Transcription

1 Mxed-Integer Credt Portfolo Optmzaton: an applcaton to Italan segregated funds L. Passalacqua Unverstà d Roma La Sapenza e-mal: Luca.Passalacqua@Unroma1.It phone: + [39] (06) fax: + [39] (06) Keywords: Credt Rsk, Optmzaton, Smulated Annealng Abstract We have solved the mxed-nteger optmzaton problem of fndng the effcent fronter n the {rsk captal, current yeld} plane for a typcal Italan lfe nsurance segregated fund portfolo of medum sze (90 assets) wth a mnmum number of assets and a fxed duraton constrant. The soluton s obtaned usng a metaheurstc technque known as adaptve smulated annealng. A comparson of the smplfed quadratc programmng problem wth a standard hll-clmbng algorthm has also been performed.

2 1 Introducton The mpressve reducton of nterest rates wtnessed n contnental Europe n the last few years has drven the attenton of the management of lfe nsurance segregated funds towards corporate bonds. Therefore portfolos tradtonally desgned to meet asset-allocaton requrements have now to be optmsed both wth respect to varatons of nterest rates and wth respect to credt rsk, as measured by some knd of captal requrements. In ths work we have nvestgated the asset-allocaton problem for a typcal medum sze Italan lfe nsurance segregated fund portfolo mposng realstc semcontnuous constrants, such as a mnmal number of assets composng the optmal portfolo. The optmzaton s performed n the {rsk captal, current yeld} plane where Rsk Captal s defned as the dfference between the loss at the α = 99% confdence level and the expected (average) loss and current yeld s defned as the rato of the coupon to the current prce. Snce we consder the reducton of credt rsk a second order requrement wth respect to nterest rate rsk control, an addtonal constrant of constant portfolo duraton s also mposed. 2 Credt Rsk modellng Among dfferent models of credt rsk we have decded to mplement the CredtRsk + [1] model of Credt Susse Fnancal Products. CredtRsk + s a pure default model: at the end of the valuaton perod each of the N oblgors can be found ether n the default or non-default state. In case of default, for each contract j ssued by the defaulted oblgor, the lender suffers a fxed loss, the exposure, that s determned as a 1 rr j fracton of the value of the contract, rr j beng the recovery rate assgned on the bass of contractual guarantees (senorty and securty). The uncondtonal default probablty p of the -th oblgor s assumed to be known. Correlated defaults are descrbed followng a typcal actuaral approach: t s assumed that, condtonal on the values of K auxlary varables (x 1,...,x K ), the defaults are ndependent and thus bnomally dstrbuted. Each rsk factor x k s dstrbuted accordng to a Gamma dstrbuton Γ(α k, β k ) wth mean µ k and standard devaton σ k. These parameters are calbrated n order to recover, on average, the uncondtonal probablty p wth standard devaton σ. The dstrbuton of defaults s found by convolvng the Gamma dstrbutons wth the Posson dstrbuton approxmatng the true Bnomal dstrbuton. Ths Posson approxmaton ntroduces some dstortons, notceably the fact that each oblgor can default many tmes, whch become more mportant as p ncreases. The loss dstrbuton s derved from the dstrbuton of defaults after roundng the exposures n unts of an arbtrary unt scale U. Ths roundng allows to lower the combnatoral complexty of the problem snce the loss suffered n case of default of n out of m contracts havng the same dscrete exposure ν s smply n ν. The effects of ths second approxmaton are generally easer to keep under control than those of the Posson approxmaton. The convolutons needed to obtan the portfolo loss dstrbuton are performed by 1

3 standard Z (.e. dscrete Laplace) transforms, obtanng the portfolo loss probablty generatng functon G(z) (z beng the auxlary varable of the transform): [ N ] K G(z) = Exp ω 0, p z ν µ 0 =1 k=1 1 δ k µ k k 1 δ k α (1) N ω k, p z ν where δ k = β k /(1 + β k ), ν s the exposure of the -th ssuer, ω k s the weght of the k-th rsk drver x k on the default of the -th ssuer and ω 0 s the weght of the dosyncratc component of rsk, whch s clearly ndentfable as the source of the frst term n eq. (1). The loss dstrbuton s obtaned from G(z) performng the nverse transform by mean of a numercal algorthm orgnally due to Panjer [2]. As ths algorthm s potentally exposed to large numercal errors, after the orgnal formulaton of the model, other algorthms have been proposed [3]. However, as dscussed by Gordy n [4], t s possble to compute the moments of the loss dstrbuton drectly from G(z) wthout the Panjer nverson and use them as benchmarks. Moreover these errors are less mportant when a sngle sector s consdered. From the loss dstrbuton dfferent measure of rsk can be derved, the most popular beng the α-quantle Value-at-Rsk V ar α, the expected shortfall E α and the rsk captal RC α, where the usual caveats on the lack of sub-addtvty of V ar α apply [5]. The auxlary varables x k can be nterpreted as market factors drvng credt rsk n dfferent ndependent ndustry sectors. The possblty of allocatng each ttle to several sectors by mean of the wegths ω allows to mplement a complex structure of correlatons. The lmtng cases of the model are those n whch there s a sngle market factor, and thus correlatons are maxmal, and that n whch there s only dosyncratc rsk,.e. the defaults are ndependent. In ths analyss we have mplemented the model wth maxmal correlatons (.e. wth a sngle ndustry sector and no dosyncratc rsk), and chosen U as 1/1000 of the largest exposure. The default probabltes p and the correspondng standard devatons σ have been taken from the hstorcal determnatons provded by Moody s on the bass of the ratng of the oblgor [6]. Smlarly the same source has been used to determne the values of the recovery rates. Rsk Captal s defned as the dfference between the loss at the α = 99% confdence level and the expected (average) loss. =1 3 Portfolo Optmsaton The asset allocaton problem we shall address s the followng: gven a portfolo ntally composed of N assets and havng defned an addtonal basket of M assets, we look for the confguraton whch mnmze the Rsk Captal, subject to constrants on the total portfolo value, duraton and current yeld. In addton, for the sake of dversfcaton, we shall also mpose that the fracton of assets whose ssuers belong to a gven ndustry sector s lower than a fxed threshold and that the optmal portfolo s composed by -at least- a mnmum number of assets N mn. Thus the problem can be formally wrtten as: 2

4 Pb1 mnmze Rsk Captal subject to N+M ω = 1 N+M ω d = D N+M ω r = R 0 s j s max s j = N+M S j ω M+N 0 f ω = 0 I N mn I = 1 f ω > 0 ω {0} [ω mn, ω max ] where D and R are the (Macaulay) duraton and the current yeld of the portfolo, ω, d and r are respectvely the weght, duraton and current yeld of the -th asset, and fnally S j s the set of assets whose ssuers belong to the j-th ndustry sector. Notce that the last requrement cannot be enforced n a lnear or quadratc programmng approach and snce ω s a semcontnuous varable, the problem can be classfed as a mxed-nteger optmsaton problem. Ths knd of problem cannot be solved by most commercal packages (e.g. lke NAG, IMSL, MatLab, etc.) but requres ether ad hoc solutons or the use of metaheurstc technques. Other dffcultes derve from the fact that the objectve functon, the Rsk Captal, and ts dervatves can only assume nteger values whch, n addton, have to be computed numercally, and from the large dfference n the value of assets between the defaulted and non-defaulted states, whch s lkely to create deep local mnma from whch t could be dffcult to escape. Metaheurstc technques as genetc algorthms, smulated annealng and tabu search are generally ndcated [8] as the most suted to solve mxed-nteger optmsaton problems. In fact n [7] a genetc algorthm has been mplemented for a rsk-return analyss, although wth the use of dervatves and wthout the numerosty constrant. Dfferently, we have addressed our attenton to numercal methods that do not requre dervatves, and n partcular to smulated annealng. 4 Smulated Annealng Smulated annealng s a random-search optmsaton technque developed n 1983 [9] whch explots the analogy between the way n whch a molten metal cools and freezes nto a mnmum energy crystallne structure (the annealng process) and the search for a mnmum n a more general system: The two pllars of the method are the dea that transton from favourable to unfavourable states (that s from lower to hgher values of energy, whch s the objectve functon) can take place when the temperature of the system s suffcently hgh (thus escapng from local mnma n search of the global mnmum) and that - lowerng the temperature n the approprate way - all possble mnma are attaned. More precsely a smulated annealng algorthm can be descrbed as composed by the followng steps: 3

5 1. startng from the D-dmensonal ntal pont x k a new pont y k+1 s sampled from the next canddate dstrbuton D(x k ); 2. the new pont s randomly accepted or rejected: the acceptance crteron s defned by the acceptance functon A(x k, y k+1, t k ): y k+1 f u A(x k, y k+1, t k ) x k+1 = otherwse x k where u s a random number unformly dstrbuted n [0, 1] and t k s an addtonal parameter whch, due to the analogy wth physcs, s called the temperature; 3. update the temperature usng the coolng schedule C(F k ): t k+1 = C(F k ) where F k s the nformaton collected up to teraton k; 4. check the stoppng crteron and f t fals set k = k +1 and go back to the frst step. Snce the evoluton from x k to x k+1 s a random process ths knd of algorthm can be vewed as the mplementaton of a dscrete-parameter fnte Markov chan. In fact n 1984 [10] t was proven that smulated annealng could statstcally fnd the best mnmum, although n an nfnte tme. As expected the choce of D(x k ), A(x k, y k+1, t k ), C(F k ) and of the stoppng crteron play a decsve role n determnng the speed of convergence. Partcularly mportant s the relatonshp between D(x k ) and C(F k ) snce the convergence s assured only when a suffcently large doman s sampled before lowerng the temperature. The analogy wth physcs had orgnally suggested the choces of the so-called Metropols-Boltzmann annealng: D(x, y, T) = (2πT) D/2 e x2 /(2T) x = x y A(x, y, T) = mn{1, e [E k+1 E k ]/T } = mn{1, e E/T } C(F k ) slower than T k = T 0 ln k where -tradtonally- E k = E(x k ) ndcates the value n x k of the cost functon f(x). Notce that n ths case also D(x) s a functon of the temperature. Snce samplng s fundamental t s not surprsng that a next canddate functon wth heaver tals than those of the Boltzmann dstrbuton s able to assure a faster convergence. In fact n 1984 a method of fast annealng [11] was developed, whch permtted to lower the temperature exponentally faster and guaranteed the convergence n a fnte tme. Shortly thereafter an even faster method [12], orgnally called very fast re-annealng and now known as adaptve smulated annealng (ASA) [13] was developed. In ths case one has: D(x, y, T) = A(x, y, T) = D 1 2( x + T ) ln(1 + 1/T ) e E k+1t e E k+1t + e = 1 E kt 1 + e ET C(F k ) slower than T k = T 0 e k1/d 4

6 where now there are D+1 temperatures (one for each parameter and one for the acceptance functon). The ASA code s probably one of the most tested codes n the lterature and applcatons n a large varety of dscplnes have been reported [14]. Unfortunately, whlst smulated annealng has no need for dervatves and can mplement very easly combnatoral requrements, one of ts major drawbacks s that t could be panful to mplement equalty constrants n hgh-dmensonal problems. Ths s because when the search s lmted to a fnte subspace the ht-or-mss rato can expand the computng tme beyond reasonable lmts. In ths work we have adapted the ASA code to cope wth the lnear constrants of our problem as explaned n the next paragraph. 5 The optmsaton procedure Instead of changng the next canddate generaton n order to satsfy the equalty constrant, we have developed a smple yet effectve procedure to project the ASA canddate pont onto the surface representng the equalty constrants. Thus there wll be two (n fact three) parameters arrays evolvng n parallel, one accordng to the prncples of smulated annealng, and another obtaned from the frst wth smple algebra. In practce the optmsaton algorthm s composed of the followng steps: 1. gven the ntal parameter array ω provded by the ASA generaton mechansm two new arrays are computed: Ω, whch s ω normalzed to 1 n such a way that every component s ether null or belongng to the nterval [ω mn, ω max ] and η whch s obtaned by changng any three components of Ω n such a way to satsfy the three lnear constrants of the problem: η = N+M η = ) 1 d(η) = N+M r(η) = N+M η η ( d = 1 ( D) r = 1 R 2. f η exsts and satsfy all the other requrements, the rsk captal s computed usng a portfolo defned by η and ths value s returned to ASA; otherwse the procedure returns to ASA a penalty functon of the type { P = c Abs(1 Ω k ) + Abs(1 d(ω k )) + Abs(1 r(ω k )) + mn(s(ω k )/s max 1, 0) where c s an arbtrary large constant (e.g ) and s(ω k ) s the largest of the contrbutons to a gven ndustry sector; notce that to mpose that at least N mn components of η are larger than ω mn s trval; 3. ndependently of η, for all the k components of Ω belongng to [ω mn, ω max ], the correspondng components of ω are changed on ext to those of Ω, that s: Ω k f Ω k [ω mn, ω max ] ω k = otherwse ω k } 5

7 ths s done to speed up the convergence of ω to a normalzed array, assurng at the same tme the contnuty of the components that can fall below the ω mn threshold; 4. the algorthm s stopped after 500 teratons f no mprovements have been found n the last 100, otherwse other 500 teratons are allowed. The code s wrtten n ANSI C and Fortan90 and run on a Pentum IV IBM laptop wth the wndows XP/cygwn operatng system, wthout optmsaton. The determnaton (wth companon dagnostcs) of a sngle pont takes about 15 mnutes (real tme). 6 Results For a numercal nvestgaton we have consdered an deal portfolo mmckng the typcal composton of Italan lfe nsurance segregated fund portfolos. The portfolo s composed by 80 bonds dentfed by ther ssuer, the ndustry sector of the ssuer, the Moody s ratng of the ssuer, the Macaulay duraton and the current yeld. The addtonal basket of opportuntes s composed by other 11 bonds. The fgures of mert of the ntal composton of the portfolo are descrbed n table 1, together wth the analyss constrants. Notce that the ntal confguraton s dversfed n a roughly unform way over four ndustry sectors (fnance, ndustral, etc.), wth a small addton of bonds n a ffth one; fnally, to further ncrease dversfcaton, the basket of new opportuntes contans bonds from a sxth sector. Snce the total value of the portfolo s rrelevant for the analyss and for the sake of comparson, the rsk captal has been scaled to 1. The loss dstrbuton s shown n fgure 1. The loss at the 99% confdence level correspond to about 5.56 tmes the standard devaton of the loss dstrbuton, whle the probablty of no losses s about 75%. Intal Value Constrant Current Yeld 4.00% - Rsk Captal 1 a.u. - Duraton 5 years 5 years max(ω ) 8.25% 8.68% mn(ω ) 0.33% 0.32% Sector % 50% Sector 2 3.8% 50% Sector % 50% Sector % 50% Sector % 50% Sector 6 0.0% 50% N mn - 64 Table 1: Fgures of the ntal portfolo confguraton and analyss constrants; notce that the rsk captal has been scaled to 1 and thus t s expressed n arbtrary unts. Before solvng the mxed-nteger optmsaton problem we have performed a check on 6

8 the ablty of procedure to solve the smplfed problem (Pb 2): Pb2 mnmze Rsk Captal subject to N+M ω = 1 N+M ω d = D ω r = R N+M 0 s j s max s j = N+M S j ω [ω mn, ω max ] whch s Pb 1 for the case N mn = N + M. As a benchmark test we wll compare the results obtaned wth our technque wth those obtaned wth a standard gradent-drven approach, namely the DCONF algorthm of the IMSL lbrary [15], whch mplements a modfed verson of the well-known TOLMIN algorthm [16]. Current Yeld 3.65% 3.80% 4.00% 4.20% 4.40% 4.50% 4.60% 4.70% #ω = max(ω ) 8.54% 6.39% 6.79% 8.32% 7.43% 7.86% 8.49% 8.30% Sector % 48.4% 47.7% 44.5% 45.0% 37.6% 38.2% 36.0% Sector 2 4.3% 4.1% 3.7% 4.0% 3.5% 3.7% 4.4% 4.8% Sector % 11.2% 10.1% 10.8% 16.6% 21.4% 20.4% 22.0% Sector % 20.2% 18.2% 19.4% 17.2% 18.3% 19.7% 18.5% Sector % 14.8% 19.1% 20.3% 16.7% 18.0% 16.6% 17.9% Sector 6 1.2% 1.1% 1.0% 1.1% 1.0% 1.1% 0.7% 0.8% Table 2: Fgures of the effcent fronter ponts of Pb1: the table reports the number of assets dscarded, the largest asset weght and the contrbutons of each of the sx ndustry sectors. Fgure 2 resumes the results of the three analyses. Table 2 reports the composton of the portfolos on the effcent fronter of Pb 1. As one can see there s a substancal smlarty n the fronters obtaned wth the two methods for Pb2. On the contrary the fronter of Pb 1 s sgnfcantly dfferent from that of Pb 2. ω 7 Conclusons We have successfully nvestgated the ablty of adaptve smulated annealng n addressng the portfolo optmsaton problem n presence of standard lnear constrants and addtonal numerosty constrants for a typcal medum sze lfe nsurance segregated fund portfolos. The presence of a constrant on portfolo duraton ensures that the mnmal rsk portfolos also meet the requrements of a classcal asset-lablty management. Extensons of the method to nclude other constrants should be straghtforward. Relevant dfferences are found when the number of assets n the portfolo s allowed to vary. 7

9 References [1] CredtRsk + techncal document, Credt Susse Fnancal Products, avalable at [2] Insurance Rsk Models, H. H. Panjer, G. E. Wlmott, Socety of Actuares, Schaumberg (1992) [3] Numercally stable computaton of CredtRsk +, H.Haaf et al., n Credt Rsk + n the Bankng Industry, Sprnger-Verlag (2004) [4] Saddlepont approxmaton of CredtRsk +, M.Gordy, Journal of Bankng & Fnance 26, (2002) [5] Coherent measures of rsk, P.Artzner et al., Mathematcal Fnance 9, 3, (1999) [6] Default and Recovery Rates of Corporate Bond Issuers, , D. T. Hamlton et al., Moody s Investor Servce (2005) avalable at [7] Rsk-return analyss of credt portfolos, F.Schlottmann et al., n Credt Rsk + n the Bankng Industry, Sprnger-Verlag (2004) [8] Heurstcs for cardnalty constraned portfolo optmzaton, T.J.Chang et al., Computers and Operatons Research, 27, (2000) [9] Optmzaton by smulated annealng, S.Krkpatrck et al., Scence 220 (4598), (1983) [10] Stochastc relaxaton, Gbbs dstrbuton and the Bayesan restoraton n mages, S. Geman et al., IEEE Trans. on Pattern Analyss and Machne Intell. 6, (1984) [11] Fast smulated annealng, H.Szu et al., Physcs Letters A 122, (1984) [12] Very fast smulated re-annealng, L. Ingber, Mathematcal Computer Modellng 12, (1989) [13] The ASA code s avalable at [14] Smulated Annealng: practce versus theory, L. Ingber, Mathematcal Computer Modellng 18, (1993) [15] [16] TOLMIN: A Fortran Package for Lnearly Constraned Optmzaton Calculaton, DAMTP, 1989/NA2, (1989). 8

10 Fgure 1: Fractonal loss (loss dvded total portfolo value) dstrbuton and ts cumulatve for the ntal portfolo confguraton ASA Pb1 parabola IMSL Pb2 parabola Intal ASA P2 parabola Current Yeld Rsk Captal Fgure 2: Effcent fronter n {rsk captal, current yeld} plane for Pb1 (squares) and Pb2 (trangles and dots for smulated annealng and IMSL respectvely); lnes are parabolas drawn to gude the eye. 9