A TWO-LEVEL LOAN PORTFOLIO OPTIMIZATION PROBLEM

Proceedngs of he 2010 Wner Sulaon Conference B. Johansson, S. Jan, J. Monoya-Torres, J. Hugan, and E. Yücesan, eds. A TWO-LEVEL LOAN PORTFOLIO OPTIMIZATION PROBLEM JanQang Hu Jun Tong School of Manageen Fudan Unversy Shangha, CHINA Te Lu Rongzeng Cao Analycs and Opzaon Deparen IBM Research - Chna Bejng, CHINA Bo Yang Cred Revew Deparen Indusral Bank Co., LTD. Shangha, CHINA ABSTRACT In hs paper, we sudy a wo-level loan porfolo opzaon proble, a proble ovaed by our work for soe coercal banks n Chna. In hs proble, here are wo levels of decsons: a he hgher level, he headquarer of he bank needs o decde how o allocae s overall capal aong s branches based on s rsk preference, and a he lower level, each branch of he bank needs o decde s loan porfolo based on s own rsk preference and allocaed capal budge. We forulae hs proble as a wo-level porfolo opzaon proble and hen propose a Mone Carlo based ehod o solve. Nuercal resuls are ncluded o valdae he ehod. 1 INTRODUCTION Snce he senal work by Markowz (Markowz 1952), porfolo selecon has becoe one of he pllars of oday s fnance research and pracce. The an concern of an nvesor n porfolo selecon s o balance he expeced reurn and he rsk of possble loss. In hs paper, we sudy a loan porfolo selecon proble ovaed by he pracce a soe coercal banks n Chna. Dfferen fro radonal porfolo selecon proble, n hs proble, here are wo levels of decsons. A he hgher level, he headquarer of he bank needs o decde how o allocae s overall budge aong s branches based on s rsk preference, and a he lower level, each branch of he bank needs o decde s loan porfolo based on s own rsk preference and allocaed capal budge. We forulae hs proble as a wo-level porfolo opzaon proble. In hs paper, we wll use Mone Carlo ehods o solve he porfolo selecon proble. There are wo dfferen Mone Carlo ehods ha have been proposed o solve he porfolo selecon proble: one s based on graden ehod (Hong and Lu 2009) and he oher s o conver he orgnal proble no a saple-based lnear prograng progra (Rockafellar and Uryasev 2000). Our ehod proposed n hs paper s anly based on he cobnaon of (Hong and Lu 2009) and Lagrangan relaxaon ehod. The reander of hs paper s organzed as follows. In Secon 2, we presen he forulaon of he wo-level loan porfolo opzaon proble. A Mone Carlo based soluon procedure s proposed n Secon 3. In Secon 4, we provde several nuercal exaples o valdae our proposed ehod. Fnally, a concluson s provded n Secon 5. 2 PROBLEM FORMULATION In hs secon, we presen he forulaon for he wo-level loan selecon proble. Suppose ha he bank has branches and akes loans o n dfferen ypes of cusoers (ndusres). Le x j be he aoun of loan ha branch akes o cusoer j and p j () be he un value of he loan ade o cusoer j by branch a e ( = 1,...,, j = 1,...,n). We only consder wo perods,.e., = 0,1. Le x = {x 1,...,x n } and p () = {p 1 (),..., p n ()}. The expeced reurn of he loan porfolo for branch s expressed as E [(p (1) p (0)) x ], where we assue ha each branch has s own dsnc belefs on he reurn of s loan porfolo. So, our odel s heerogeneous for branches. We use CVaR as a rsk easure for he porfolo and le C α be he upper l loss for branch, hough our forulaon can be exended o oher rsk easures as well, such as varance and VaR. Therefore, he loan porfolo 978-1-4244-9864-2/10/$26.00 2010 IEEE 2614

Hu, Tong, Lu, Cao and Yang selecon proble for branch can be forulaed as follows: (L ) ax E [(p (1) p (0)) x ] s.. p (0) x w CVaR α ([p (0) p (1)] x ) C α p j (0)x j c j, j = 1,,n x j 0, j = 1,,n, where w s he capal allocaed o branch by he bank and c j s he upper l se by he bank for he oal loan len o ype j cusoers. A he hgher level, he headquarer has a oal capal of w, and s objecve s o axze he oal expeced reurns of all branches wh an upper l C α on he overall rsk. Therefore, he correspondng loan porfolo selecon proble can be forulaed as: (H) ax s.. =1 =1 E [(p (1) p (0)) x ] p j (0)x j c j, CVaR α ( =1 =1 w = w w 0, j = 1,,n, [p (0) p (1)] x ) C α = 1,,, where x s he opal soluon of (L ), whch depends on w. Togeher, (L ) and (H) for a wo-level loan porfolo selecon proble, whch n general s que dffcul o solve. In he nex secon, we wll propose a procedure o solve hs wo-level opzaon proble. We should pon ou ha an alernave one-level forulaon s also proposed n Hu e al. (2010) where nuercal resuls are provded o copare wh he wo-level forulaon. In fac, her nuercal resuls show ha he one-level forulaon s a good approxaon for he wo-level forulaon. 3 A PROCEDURE FOR SOLVING THE TWO-LEVEL PROBLEM In hs secon, we propose a nuercal ehod o solve he wo-level loan porfolo opzaon proble presened n he prevous secon. Le us frs consder he lower level proble L. To solve L, we use he graden-based sulaon ehod proposed n (Hong and Lu 2009). For ease of exposon, le f (x ) = E [(p (1) p (0)) x ]. Defne he Lagrange funcon L(x,λ, µ,τ j,γ j ) = f (x )+λ [p (0) x w ]+ µ [CVaR α ([p (0) p (1)] x ) C α ]+ n =1 τ j (p j (0)x j c j )+ where λ, µ, τ j, γ j are Lagrange ulplers. Then, accordng o he dualy heory, we have f (x ) w n =1 γ j ( x j ) = λ, where x s he opal soluon of L (w ) and λ s he correspondng Lagrange ulpler. We noce ha he objecve funcon of he upper level proble (H) s he su of all branches,.e, f(x) = =1 f (x (w )), where x (x 1 (w 1 ),,x (w )). So he negave graden of f(x) can be expressed as w f(x ) λ = (λ 1,,λ ), whch s a descen drecon of f(x) n space R. To solve he opzaon proble (H), we use he followng graden-base sochasc approxaon algorh o updae w: ehod can be generally fored as w (k+1) = w (k) + α (k) 2615

Hu, Tong, Lu, Cao and Yang where s a descen drecon and α (k) 0 s he sep sze. Snce =1 w = w, we us have =1 d(k) = 0. Hence, we need o projec he descen drecon λ on he hyperplane =1 d(k) = 0 and oban a new drecon d = λ (λ,e) e 2 e 2 where e = (1,,1) 1. Once he descen drecon s deerned, we need o choose an approprae sep-sze α (k). To do ha, we sar wh an nal sep-sze α (k), f he correspondng w (k+1) s a feasble soluon, hen we could ncrease he sep sze, oherwse, we would have o decrease unl he correspondng w (k+1) becoes feasble or we need o choose a dfferen descen drecon. We noe ha 0 w (k+1) = w (k) + α (k) w ( = 1,,), herefore 0 α (k) w w(k) 0 α (k) w(k) f > 0; f < 0. Le α (k) denoe he -h odfcaon of sep sze α (k) and β (k) In order o nsure ha w (k+1) s feasble, we odfy α (k) α (k) = n β(k) {, n >0 Based on wha we dscussed above, we propose he followng algorh: denoe he nu value for he -h odfcaon. as follows: } w w (k) (k), n w j j <0. j 1. For k = 0 and a gven nal feasble soluon w (k) = (w (0) 1,,w(0) ), we solve each lower level proble L (w (0) ) and oban he opal soluon x (0), opal value f (0), and s correspondng Lagrange ulpler λ (0) 2. Le λ (k) = (λ (k) 1,,λ(k) λ (k) (λ(k), e (k) ) e (k). e (k) 2 2 ). In order o ensure =1 w = w and we updae he descen drecon as = 3. In hs sep, we copue an approprae sep-sze α (k) along he descen drecon. The procedure works as follows: a. For = 0, le w (k) = w (k), β (k) = 1. Pre-specfy η > 1, σ < 1, and ε 1 << 1. b. Se If β (k) > α (k) α (k) = n or α (k) < ε 1, sop. β(k) {, n >0 } w w (k), n j <0 w c. Le w (k) = w (k) 1 + α(k) and solve each lower level proble L (w (k), and he correspondng Lagrange ulplers λ (k),. d. Le x (k) = (x (k),1,,x(k), ), λ (k) = (λ (k),1,,λ(k),), and copue +1 sop and goo Sep 4, oherwse, consder he followng hree cases: If boh x (k) If x (k) If x (k) 4. Se α (k) = α (k) soluon x (k+1) 5. If =1 f (k+1) Sep 2. (k) j j.. ) o oban s opal soluon x(k) as n Sep 2. If (d(k) +1,d(k) ) 0, and x (k) (k) 1 are feasble for (H), we ncrease he sep-sze by a facor of η (.e., β(k) +1 = ηβ ). s feasble bu x (k) 1 s no, sop and goo Sep 4. s nfeasble for H, we se β (k) +1 = σβ(k). Le + 1 and goo 2. and w (k+1) = w (k) + α (k). We solve each lower level proble L (w (k+1) ) and oban opal, opal value f (k+1) and he correspondng Lagrange ulpler λ (k+1) =1 f (k). < ε 2 (ε 2 s anoher pre-specfed consan), sop; oherwse, se k k+ 1 and goo, 2616

Hu, Tong, Lu, Cao and Yang 4 NUMERICAL EXAMPLES To es he ehod we proposed n Secon 3, n hs secon, we presen hree nuercal exaples. 4.1 Exaple 1 In hs exaple, we have wo branches ( = 2) and hree ypes of cusoers (n = 3). The values of varous paraeers are gven n he followng able: α α 1 α 2 C α C α1 C α2 p (0) c 1 c 2 c 3 w 99% 95% 97% 0.2 0.25 0.15 1 1 1 1 1 1 1 We assue ha {p (0) p (1), = 1,2} has a ulvarae noral dsrbuon whose ean and covarance arx are gven by (he nubers are generaed randoly): µ = (µ 1, µ 2 ) = ( 0.1730, 0.2714, 0.8757, 0.9797, 0.2523, 0.7373) Ω = 1.4846, 0.3227, 0.0456, 0.1748, 0.1060, 0.8760 0.3227, 0.8009, 0.3184, 0.2192, 0.1618, 0.0131 0.0456, 0.3184, 0.5031, 0.0649, 0.2677, 0.0470 0.1748, 0.2192, 0.0649, 0.6676, 0.1124, 0.0584 0.1060, 0.1618, 0.2677, 0.1124, 0.5729, 0.1221 0.8760, 0.0131, 0.0470, 0.0584, 0.1221, 0.6983 We frs solve he wo-level proble (L ) + (H) usng he enuerave ehod and hen copare he resuls wh our ehod proposed n Secon 3. The resuls are provded n he followng able: w 1 w 2 op. value Enuerave ehod 0.36404 0.63596 0.7975 Our ehod 0.36403 0.63597 0.7980 Fro he above able, s clear ha our ehod works very well and he resuls obaned based on are very close o hose obaned fro he enuerave ehod. 4.2 Exaple 2 Ths exaple s slar o Exaple 1 and have he followng paraeer values: α α 1 α 2 C α C α1 C α2 p (0) c 1 c 2 c 3 w 99% 96% 97% 0.3 0.25 0.2 1 1 1 1 1 1 1 µ = (µ 1, µ 2 ) = ( 0.3475, 0.4972, 0.3097, 0.3246, 0.2535, 0.3609) Ω = 0.7252, 0.2385, 0.1930, 0.0334, 0.1955, 0.3863 0.2385, 1.0934, 0.1405, 0.4391, 0.1905, 0.4216 0.1930, 0.1405, 0.3709, 0.0822, 0.1194, 0.1824 0.0334, 0.4391, 0.0822, 0.7109, 0.0981, 0.5308 0.1955, 0.1905, 0.1194, 0.0981, 1.2288, 0.1146 0.3863, 0.4216, 0.1824, 0.5308, 0.1146, 1.1934 The resuls are provded n he followng able: w 1 w 2 op. value Enuerave ehod 0.12471 0.87529 0.174095 Our ehod 0.124924 0.875076 0.174217 The resuls are slar o hose for Exaple 1. 2617

Hu, Tong, Lu, Cao and Yang 4.3 Exaple 3 In hs exaple, we have hree branches ( = 3) and hree ypes of cusoers (n = 3). The values of varous paraeers are gven n he followng able: α α 1 α 2 α 3 C α C α1 C α2 C α3 p (0) c 1 c 2 c 3 w 99% 95% 97% 96% 0.4 0.25 0.15 0.2 1 1 1 1 1 1 1 Agan, we assue ha {p (0) p (1), = 1,2,3} has a ulvarae noral dsrbuon whose ean and covarance arx are gven by (he nubers are generaed randoly): µ = (µ 1, µ 2, µ 3 ) = ( 0.2981, 0.3639, 0.2428, 0.3765, 0.2080, 0.2692, 0.2115, 0.2243, 0.4059) Ω = 0.7012, 0.1651, 0.3435, 0.5460, 0.0630, 0.5162, 0.3282, 0.2355, 0.4400 0.1651, 0.9495, 0.0253, 0.0931, 0.2481, 0.4407, 0.4844, 0.3813, 0.7256 0.3435, 0.0253, 0.7095, 0.5138, 0.0368, 0.1124, 0.2512, 0.2611, 0.8672 0.5460, 0.0931, 0.5138, 1.4692, 0.2575, 0.0575, 0.4911, 0.5253, 0.5154 0.0630, 0.2481, 0.0368, 0.2575, 0.8545, 0.2114, 0.0087, 0.6932, 0.0146 0.5162, 0.4407, 0.1124, 0.0575, 0.2114, 1.2193, 0.0456, 0.0686, 0.2094 0.3282, 0.4844, 0.2512, 0.4911, 0.0087, 0.0456, 0.7636, 0.1572, 1.0078 0.2355, 0.3813, 0.2611, 0.5253, 0.6932, 0.0686, 0.1572, 1.4252, 0.6000 0.4400, 0.7256, 0.8672, 0.5154, 0.0146, 0.2094, 1.0078, 0.6000, 2.1854 The resuls are provded n he followng able: w 1 w 2 w 3 op. value Enuerave ehod 0.562 0.23 0.208 0.241466 Our ehod 0.57989 0.22739 0.19271 0.24147 Slar o Exaples 1 and 2, our ehod produces very good he soluons. 5 CONCLUSIONS In hs paper, we suded a wo-level loan porfolo selecon proble and proposed a nuercal ehod o solve he proble. Nuercal exaples are provded o valdae he ehod. We plan o nvesgae he convergence properes of he ehod n our fuure research. ACKNOWLEDGMENTS The work repored n hs paper s suppored n par by a gran fro IBM Chna Research Laboraory, by he Naonal Naural Scence Foundaon of Chna under Gran NSFC No. 70832002, and by he Shangha Scence and Technology PuJang Funds under Gran 09PJ1401500. REFERENCES Hong, L., and G. Lu. 2009. Sulang sensves of condonal value a rsk. Manageen Scence 55:281 293. Hu, J. Q., J. Tong, T. Lu, R. Z. Cao, and B. Yang. 2010. A wo-level load porfolo opzaon proble. Techncal repor, Fudan Unversy. Markowz, H. 1952. Porfolo selecon. The Journal of Fnance 7:77 91. Rockafellar, R., and S. Uryasev. 2000. Opzaon of condonal value-a-rsk. Journal of Rsk 2:21 42. AUTHOR BIOGRAPHIES JIANQIANG HU s a Professor wh he Deparen of Manageen Scence, School of Manageen, Fudan Unversy. Before jonng Fudan Unversy, he was an Assocae Professor wh he Deparen of Mechancal Engneerng and he Dvson of Syses Engneerng a Boson Unversy. He receved hs B.S. degree n appled aheacs fro Fudan Unversy, Chna, and M.S. and Ph.D. degrees n appled aheacs fro Harvard Unversy. He s an assocae edor of Auoaca and a anagng edor of OR Transacons, and was a pas assocae edor of Operaon Research and IIE Transacon on Desgn and Manufacurng. Hs research neress nclude dscree-even sochasc syses, sulaon, queung nework heory, sochasc conrol heory, wh applcaons owards supply chan anageen, rsk 2618

Hu, Tong, Lu, Cao and Yang anageen n fnancal arkes and dervaves, councaon neworks, and flexble anufacurng and producon syses. He s a co-auhor of he book, Condonal Mone Carlo: Graden Esaon and Opzaon Applcaons (Kluwer Acadec Publshers, 1997), whch won he 1998 Ousandng Sulaon Publcaon Award fro INFORMS College on Sulaon. Hs eal address s <hujq@fudan.edu.cn>. JUN TONG s a graduae suden wh he Deparen of Manageen Scence, School of Manageen, Fudan Unversy. He receved hs B.S. degree n appled aheacs fro Shangha Unversy, Chna. Hs eal address s <082025027@fudan.edu.cn>. TIE LIU s a saff researcher wh he Analycs and Opzaon Deparen, IBM Research - Chna. Currenly, he works on fnancal rsk anageen, especally on rsk odelng and calculaon. Hs research neress also nclude achne learnng, paern recognon, daa analyss and nng, and fnancal copung. He receved hs BS, MS and PhD degrees fro Xan Jaoong Unversy, n 2001, 2004, and 2007, respecvely. Hs eal address s <lule@cn.b.co>. RONG ZENG CAO a research saff eber a IBM Research - Chna. He serves as co-char of Servce Scence Professonal Ineres Couny (PIC) of IBM Research o enhance he neracons beween IBM researchers and research professonals n he wder world. Hs area of experse s operaon research, wh a parcular neres n he use of analycs and opzaon for busness probles, such as logscs opzaon, prcng, forecasng, sulaon and rsk anageen. Hs eal address s <caorongz@cn.b.co>. BO YANG s a vce general anager of he Cred Revew Deparen - Shangha Cener, Indusral Bank Co., LTD. She has been workng n he fnancal ndusry for over 20 years and has been a loan porfolo anager for several Chnese fnancal nsuons snce 1996. Her eal address s <yb@cb.co.cn>. 2619