European Journal of Business and Management ISSN (Paper) ISSN (Online) Vol.5, No.6, 2013

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1 European Journal of Busness and Management ISSN (Paper) ISSN (Onlne) Portfolo Optmzaton of Commercal Banks- An Applcaton of Genetc Algorthm Dr. A.K.Msra Vnod Gupta School of Management, Indan Insttute of Technology Kharagpur, Inda E-mal: Dr. V. J.Sebastan (Correspondng author) Insttute of Management Technology Gazabad, Delh, Inda E-mal: Abstract Portfolo optmzaton, n case of fnance, s the trade- off between rsk and return to maxmze proft or return from the portfolo. Fnancal regulatons are country specfc and t depends upon the economc condtons prevalng n the country. The portfolo of a commercal bank can be constraned by regulatory prescrpton of exposure lmts, rsk weghts and returns from each category of assets. Hence, optmzaton of return, n case of the loan portfolo, presents a challengng problem due to ts large set of local extremes. In ths context, Genetc Algorthm s used as a possble soluton to optmze the rsk-return trade-off and acheve an deal soluton for portfolo optmzaton. Keywords: Portfolo Management, Rsk-Return Trade Off, Commercal Bankng 1. Introducton The man goal of nvestors s to acheve optmal allocaton of funds among varous fnancal assets. Searchng for an optmal portfolo, characterzed by random future returns, seems to be a dffcult task and s usually formalzed as a rsk-mnmzaton problem. Commercal banks are fnancal ntermedares that accept deposts and channel those deposts nto lendng actvtes. Banks are a fundamental component of the fnancal system, and are also actve players n fnancal markets. The essental role of a bank s to connect those who have funds (such as nvestors or depostors), wth those who seek funds. Bankng ndustry s hghly regulated, and government restrctons on fnancal actvtes of banks have vared over tme. The current set of global standards s called Basel II. Basel II s the second of the Basel Accords, whch are recommendatons on bankng laws and regulatons ssued by the Basel Commttee on Bankng Supervson. The purpose of Basel II, whch was ntally publshed n June 2004, s to create an nternatonal standard that bankng regulators can mplement whle creatng regulatons about how much captal banks need to put asde to guard aganst the varous types of fnancal and operatonal rsks banks face. Bank earn through plethora of nvestments made n loans and equty nvestments. Each category of loans and nvestments has ts own rsk weght and return and t s necessary to combne varous rsk categores of assets wth ther returns n relaton to the avalable captal so as to maxmze the rsk-adjusted return and optmze the utlzaton of captal. A genetc algorthm (GA) s a search technque used n computng to fnd exact or approxmate solutons to optmzaton and search problems. Genetc algorthms are categorzed as global search heurstcs. Genetc algorthms are a partcular class of evolutonary algorthms (EA) that use technques nspred by evolutonary bology such as nhertance, mutaton, selecton, and crossover. 2. Lterature Revew Modern portfolo theory provdes a well-developed paradgm to form a portfolo wth the hghest expected return for a gven level of rsk tolerance. Markowtz (1952, 1959) orgnally formulated the fundamental theorem of mean varance portfolo framework, whch explans the trade-off between mean and varance each representng expected returns and rsk of a portfolo, respectvely. Although Markowtz's theory uses only mean and varance to descrbe the characterstcs of return, hs theory about the structures of a portfolo became a cornerstone of modern portfolo theory (Fama, 1970, Hakansson, 1970, Hakansson, 1974, Merton, 1990 and Mossn, 1969). Genetc algorthm s a stochastc optmzaton technque nvented by Holland (1975) and a search algorthm based on survval of the fttest among strng structures (Goldberg, 1989). They appled the dea from bology research to gude the search to an (near-) optmal soluton (Wong & Tan, 1994). The general dea was to mantan 120

2 European Journal of Busness and Management ISSN (Paper) ISSN (Onlne) an artfcal ecosystem, consstng of a populaton of chromosomes. Each chromosome represents the weght of ndvdual stock of portfolo and s optmzed to reach a possble soluton. Attached to each chromosome s a ftness value, whch defnes how good a soluton the chromosome represents. By usng mutaton, crossover values, and natural selecton, the populaton wll converge to only one chromosomes wth good ftness (Adel & Hung, 1995). Recently, GA attracts much attenton n portfolo formulatons (Orto et al., 2003 and Xa et al., 2000). In the feld of model solvng, Arnone (Arnone et al., 1993) presented a Genetc Algorthm for an unconstraned portfolo optmzaton problem. However, Shoaf (Shoaf, & Foster, 1996), appled genetc algorthm, frst tme, to Markowtz s model. Rolland utlzed Tabu Search (TS) to solve Markowtz prncple (Rolland, 1997). Later, to corroborate the necessty and desrablty of heurstc algorthms, Mansn and Speranza proved that the portfolo selecton problem wth mnmum transacton lots s an NP-complete problem. Subsequently, they proposed three heurstc algorthms to fgure out the MAD model of Konno (Mansn, & Speranza, 1999). Afterwards, they (wth Kellerer) extended ther model to factor fxed transacton costs (Kellerer, Mansn, & Speranza, 1999). Snce late 1990s, a number of nnovatve quanttatve approaches to portfolo credt rsk modelng have been developed (Gupton et al. 1997, Wlson 1997, Kealhofer 1998). Moreover, trade n fnancal nstruments for transferrng credt rsk lke credt default swaps, asset backed transactons, etc. have ncreased sgnfcantly durng the last decade (Ferry, 2002). Basel Commttee on Bankng Supervson has declared new norms on captal regulatons for Banks exposures to credt rsk. These developments have nfluenced the proft-related consderatons; and there s an ncreasng demand for constraned optmzaton of credt portfolos of Banks. Majorty of studes on portfolo selecton focused on equty portfolo optmzaton (Elton and Gruber, 1995) as per the methods developed by Markowtz (1952) Dueck and Wnker (1992), Chang et al. (2000), Gll and Këllez (2002) for dfferent heurstc approaches whch s sgnfcantly dfferent from credt portfolo optmzaton. Andersson et al. (2001) proposed the use of smplex algorthms n a portfolo credt rsk smulaton model framework whle Lehrbass (1999) proposed the use of Kuhn-Tucker optmalty constrants n an analytcal portfolo credt rsk model. The artcle has used Evolutonary Algorthms for solvng credt portfolo optmzaton problems. 3. Portfolo Optmzaton A Theoretcal Perspectve Captal Asset Prcng Model (CAPM) s used to determne a theoretcally approprate requred rate of return of an asset, f that asset s to be added to an already well-dversfed portfolo, gven the non-dversfable rsk of the asset. The model takes nto account the asset senstvty to non-dversfable rsk (also known as systematc rsk or market rsk), often represented by the market beta (β) as well as the expected return of the market and the expected return of a theoretcal rsk-free asset. R = α + β R + e (1) m Ths model makes followng assumptons: a) E ( e ) = 0 b) Cov( R m, e ) = 0 c) E( e, e j ) = 0 Ths lead to: E( R ) = α + β E( R ) (2) m Var( R ) = β σ + σ (3) m e 2 j ββ jσm Cov( R, R ) = (4) 121

3 European Journal of Busness and Management ISSN (Paper) ISSN (Onlne) The above method drastcally reduces the number of estmates to be made hence reduces both computaton tme and complexty of the problem. The Sharpe rato or Sharpe ndex s a measure of the excess return (or Rsk Premum) per unt of rsk n an nvestment asset or a tradng strategy, named after Wllam Forsyth Sharpe. Snce ts revson by the orgnal author n 1994, t s defned as: S E( R) RFR ( ) σ = (5) where R s the return from the asset, R f s the return on a benchmark asset, such as the rsk free rate of return, E[R R f ] s the expected value of the excess of the asset return over the benchmark return, and σ s the standard devaton of the asset. Wth the help of above results we can form effcent fronter as well as fnd the optmal portfolo through Sharpe rato. Process s as below: E( Rp) = we ( R ) (6) 2 σ = wwσσ ρ (7) p j j j j Maxmzaton of the Sharpe Rato from the above two nputs provdes the effcent fronter. 3.1 Genetc Algorthm Specfcatons Genetc algorthms are mplemented n a computer smulaton envronment n whch a populaton of abstract representatons (called chromosomes or the genotype of the genome) of canddate solutons (called ndvduals, creatures, or phenotypes) to an optmzaton problem evolves toward better solutons. Tradtonally, solutons are represented n bnary as strngs of 0s and 1s, but other encodngs are also possble. The evoluton usually starts from a populaton of randomly generated ndvduals and happens n generatons. In each generaton, the ftness of every ndvdual n the populaton s evaluated, multple ndvduals are stochastcally selected from the current populaton (based on ther ftness), and modfed (recombned and possbly randomly mutated) to form a new populaton. The new populaton s then used n the next teraton of the algorthm. Commonly, the algorthm termnates when ether a maxmum number of generatons has been produced, or a satsfactory ftness level has been reached for the populaton. If the algorthm has termnated due to a maxmum number of generatons, a satsfactory soluton may or may not have been reached Ftness Functon A ftness functon s a partcular type of objectve functon that prescrbes the optmalty of a soluton (that s, a chromosome) n a genetc algorthm so that that partcular chromosome may be ranked aganst all the other chromosomes. Optmal chromosomes, or at least chromosomes whch are more optmal, are allowed to breed and mx ther datasets by any of several technques, producng a new generaton that wll (hopefully) be even better Encodng of a Chromosome The chromosome should n some way contan nformaton about soluton whch t represents. The most used way of encodng s a bnary strng. The chromosome then could look lke followng pattern: 122

4 European Journal of Busness and Management ISSN (Paper) ISSN (Onlne) Chromosome Chromosome Each chromosome has one bnary strng. Each bt n ths strng can represent some characterstc of the soluton, or the whole strng can represent a number. The artcle has used followng Mappng Rule: Where: l u l l x = x + ( x x ) / (2 1) (8) x = -th chromosome or soluton l x = lower bound for x u x = Upper Bound for x l = length or resoluton for -th chromosome 3.4. Crossover Crossover selects genes from parent chromosomes and creates a new offsprng. The smplest way to do ths s to choose randomly some crossover pont and everythng before ths pont copy from a frst parent and then everythng after a crossover pont copy from the second parent. The Crossover would be as follows: Chromosome Chromosome Offsprng Offsprng The crossover chosen here s scatter whch means that mutaton pont wll be randomly chosen nsde a chromosome Mutaton After a crossover s performed, mutaton takes place to prevent fallng all solutons n populaton nto a local optmum of solved problem. Mutaton changes randomly the new offsprng. For bnary encodng one can swtch a few randomly chosen bts from 1 to 0 or from 0 to 1. The mutaton depends on the encodng as well as the crossover. Mutaton can take the followng shape: Orgnal offsprng Orgnal offsprng Mutated offsprng Mutated offsprng Roulette Wheel Selecton 4. Parents are selected accordng to ther ftness. The better the chromosomes are, the more chances to be 123

5 European Journal of Busness and Management ISSN (Paper) ISSN (Onlne) selected they have. The algorthm for roulette wheel selecton s a) [Sum] Calculate sum of all chromosome ft nesses n populaton - sum S. b) [Select] Generate random number from nterval (0,S) - r. c) [Loop] Go through the populaton and sum ft nesses from 0 - sum s. When the sum s s greater then r, stop and return the chromosome. 4. Emprcal Desgn A typcal Indan bank holds a portfolo of loans and equty nvestments. In Inda banks have an oblgaton to provde loans to regulated sectors such as agrculture, housng, small and medum enterprses, commercal real estate etc.( Table:1). As per the concentraton rsk, the Bankng sector regulator (Reserve Bank of Inda) has gven dfferent celng lmt for each category of loans. These asset classes have dfferent rsk weghts and returns. Each credt class s generally assocated wth a return. Investment Types Rsk Weght AAA (%) Rsk-Weght AA (%) Table: 1 Return (%) AAA Ratng Book-Value (%) Regulatory Loan Requrement SME W1 Mnmum 12% Commercal Real W2 No lmt Estate Large Corporaton W3 No Lmt Resdental Mnmum 10% W4 Property Consumer Credt W5 No Lmt Regulatory Retal W6 Mnmum 18% Equty Investment W7 Maxmum 5% Soveregn W8 Mnmum 25% Banks W9 No Lmt PSE W10 No Lmt Assets are dvded nto dfferent credt classes as defned above. The returns n the table are for AAA credt ratng class whch s the best credt class for each segment. The portfolo allocaton s to be restraned for the frst two credt class n each segment.e. AAA and AA bonds-loans. The mutaton depends on the encodng as well as the crossover. For adjustng rsk of each asset class the formulaton used s: AR = R CC * RW (9) Where: AR = Adjusted Return R = Return on -th asset class CC = Cost of captal RW =Rsk Weght 124

6 European Journal of Busness and Management ISSN (Paper) ISSN (Onlne) The paper has used AR n place of expected return to account for the specal case of Banks. Rsk of the combned portfolo s calculated as per the followng E( R ) = α + β E( R ) (10) m Var( R ) = β σ + σ (11) m e 2 j ββ jσm Cov( R, R ) = (12) Betas requred n the above equaton have been calculated through regresson from hstorcal data. The return on market has been replaced by Prme Lendng Rate takng nto account the specal case of bank portfolo. Outputs from equatons (10), (11) and (12) are used to calculate Portfolo rsk accordng to the equaton: 2 σ = wwσσ ρ (13) p j j j j From the values of expected portfolo return (Adjusted return) as calculated from equaton (10) and Portfolo rsk calculated from equaton (13) one can create ftness functon and constrants needed n Genetc optmzaton model. Ftness functon: ( ) F x AR nt opt = (14) σ Where: AR = Adjusted Return nt = Interest cost to bank on deposts opt =Operatng cost of bank Constrants are formulated as: 125 w = 1 w1 + w w w 1 Each credt class s generally assocated wth a gven rate of return and rsk level. For dfferent book value.e. W1,W2 W3 etc., one can get dfferent rsk-return portfolo. Each weght can be between 0 to 100%. Each credt class, the weght wll be decded as per regulatory gudelnes ( f t s prescrbed) or decded by the optmzaton technque. These asset classes s agan dvded nto dfferent credt classes as defned above. The returns for each asset class as gven n the table are for AAA credt ratng whch s the best credt class for each segment. The portfolo allocaton s to be restraned for the frst two credt class n each segment.e. AAA and AA bonds-loans Optmzaton Model: Genetc Algorthm Specfcatons Populaton sze of 30 chromosomes was taken. Each chromosome was bnary encoded wth strng length equalng 10 to cover the range of weghts from 0-100%. Eltsm was set at top 3 fttest chromosomes. Eltsm s a method, where the best chromosomes (or a few best chromosomes) are coped to new populaton. Eltsm can very rapdly ncrease performance of GA, because t prevents losng the best found soluton. Crossover probablty s set to 0.4 as crossover s the man crteron for the genetc algorthm to evolve. Mutaton probablty s kept low wth so as not to destroy better chromosomes already found. Mutaton method used here s adaptve, as t randomly generates drectons that are adaptve wth respect to the last successful or unsuccessful generaton. The feasble regon s bounded by the constrants and nequalty constrants. A step length s chosen along each

7 European Journal of Busness and Management ISSN (Paper) ISSN (Onlne) drecton so that lnear constrants and bounds are satsfed. Stoppng crteron s ether 100 generaton reached or the best chromosome ftness worst chromosome ftness s less than 10-6, whchever crteron s reached frst. Outlne of basc Genetc Algorthm s: 1. [Start] Generate random populaton of n chromosomes (sutable solutons for the problem) 2. [Ftness] Evaluate the ftness f(x) of each chromosome x n the populaton 3. [New populaton] Create a new populaton by repeatng followng steps untl the new populaton s complete a) [Selecton] Select two parent chromosomes from a populaton accordng to ther ftness (the better ftness, the bgger chance to be selected) b) [Crossover] Wth a crossover probablty cross over the parents to form a new offsprng (chldren). If no crossover was performed, offsprng s an exact copy of parents. c) [Mutaton] Wth a mutaton probablty mutate new offsprng at each locus (poston n chromosome). d) [Acceptng] Place new offsprng n a new populaton 4. [Replace] Use new generated populaton for a further run of algorthm 5. [Test] If the end condton s satsfed, stop, and return the best soluton n current populaton 6. [Loop] Go to step 2 5. Results & Dscussons The artcle used the data of a leadng publc sector bank of Inda to calculate the weghts they have currently nvested n Asset classes. The calculate weghts and Rsk-Return for ther current portfolo s provded n the Table: 2. On the same bank s data the artcle used the GA technque as dscussed n the artcle. Asset classes are dvded nto AAA and AA and on both case scenaros, as gven n Table 1, the GA technque was used. Effcent fronter was created n both cases and genetc algorthm was appled on both the effcent fronter to fnd the optmal portfolo weghts. Table:2 Asset Class Weght Asset Class Weght SME 8.80% Equty Investment 6.60% Commercal Real estate 10.00% Regulatory Retal 11.30% Large corporaton 20.00% Soveregn 5.00% Resdental Property 10.30% Banks 7.10% Consumer Credt 9.80% PSE 11.10% Optmal Portfolo Return 9.31% Optmal Portfolo Rsk 27.06% As seen n Fgure-1 wth ncreasng rsk, return of the portfolo also ncreases. The portfolo rsk ncreases from 0%, when all the asset value s nvested n soveregn bonds, to 60%, when whole portfolo s nvested n Equty nvestments. 126

8 European Journal of Busness and Management ISSN (Paper) ISSN (Onlne) Fg 1: Effcent Fronter (Asset Class -AAA) Optmal Portfolo accordng to genetc algorthm s: Table:3 Asset Class Weght Asset Class Weght SME 14.95% Commercal Real estate 4.80% Large corporaton 6.05% Resdental Property 5.10% Consumer Credt 10.80% Regulatory Retal 18.00% Equty Investment 8.22% Soveregn 9.05% Banks 9.78% PSE 13.25% Optmal Portfolo Rsk 13.41% Optmal Portfolo Return 11.89% Smlarly, effcent fronter, when all asset classes are AA. Fg 2: Effcent Fronter (Asset Class -AA) The Fgure-2 s steeper than Fgure-1 ndcatng declnng return wth ncreasng rsk. Optmal Portfolo accordng to genetc algorthm s: Table: 4 Asset Class Weght Asset Class Weght SME 13.61% Commercal Real estate 4.18% Large corporaton 6.00% Resdental Property 5.21% Consumer Credt 9.82% Regulatory Retal 18.00% Equty Investment 7.71% Soveregn 21.56% Banks 4.51% PSE 9.40% Optmal Portfolo Rsk 14.61% Optmal Portfolo Return 11.05% 127

9 European Journal of Busness and Management ISSN (Paper) ISSN (Onlne) A comparson has been carred between current portfolo (practce by the bank) and the portfolos that have been created by GA through the standard method of Sharpe rato (Table:5). For estmaton of Sharpe rato, the artcle has used Yeld on 1-Year Government Securty as rsk free nterest rate. Scenaro Rsk Free Rate Table: 5 Portfolo Rsk 128 Portfolo Return Sharpe Rato Portfolo s AAA 6.65% 13.41% 11.89% 39.1% Portfolo s AA 6.65% 14.61% 11.05% 30.1% Actual Portfolo 6.65% 27.06% 9.31% 9.8% The Sharpe rato of current portfolo s least than the portfolos created by GA. If the bank mantaned AAA credt ratng for ts portfolo, the Sharpe rato would be 39.10% and f the bank mantaned AA credt ratng for ts portfolo, the Sharpe rato would be 30.10%. Wth the down gradaton of credt ratng, portfolo rsk s ncreasng along wth declnng return on the portfolo. Wth the ncreasng portfolo rsk, the bank needs to keep more captal to mantan regulatory captal adequacy and the cost of more captal reduce the portfolo return. 5. Concluson Banks are hghly regulated ndustry wth plethora of regulatory prescrptons whch governed ther day-to-day functonng. Regulatory gudelnes on asset concentraton, credt allocaton, credt ratng and captal adequacy nfluence banks portfolo rsk and return. Wth multple constrans optmzaton of banks rsk-return s a challengng task. In ths context, Genetc Algorthm provdes deal soluton. The artcle has bult portfolo wth mean-varance domnatng for both AAA ratng and AA ratng. The GA technque appled to a leadng bank of Inda. Portfolo desgned as per Indan Bankng Regulatons has been outperformed the current portfolo of the bank. Ths model can be further mproved f optmzaton s also done nsde each asset class takng nto account all the credt class of each asset. References Adel, H, & S. Hung, (1995), Machne learnng: neural networks, genetc algorthms, and fuzzy systems, Wley, New York Aello, S & N Cheffe,(1999), Internatonal Index Funds and the Investment Portfolo, Fnancal Servces Revew, 8, Bogle, J.C, (1998), The mplcatons of style analyss for mutual fund performance, Journal of Portfolo Management, 24 (4), Chang, K.P, (2004), Evaluatng mutual fund performance: An applcaton of mnmum convex nput requrement set approach, Computers and Operatons Research, 3, Fama, E.F,(1970), Mult perod consumpton nvestment decsons, Amercan Economc Revew 60, Ferry, J (2002), New players, New Rules, Rsk, 15 (5), Gll,M & E. Këllez,(2002), Portfolo Optmzaton wth VaR and expected shortfall. In: Kontoghorghes, E., Rustem, B. and Sokos, S., Computatonal Methods n Decson-makng, Economcs and Fnance, Kluwer, Dordrecht, Goldberg, D.E,(1989), Genetc algorthms n search, optmzaton and machne learnng, Addson-Wesley, New York Hakansson, N, (1970), Optmal Investment and Consumpton Strateges under Rsk for a class of Utlty Functons, Econometrc, 38, Hakansson, N, (1974), Convergence n mult perod portfolo Choce, Journal of Fnancal Economcs, 1, Holland, J.H,(1975), Adaptaton n natural and artfcal systems: An ntroductory analyss wth applcatons to

10 European Journal of Busness and Management ISSN (Paper) ISSN (Onlne) bology, control and artfcal ntellgence, Unversty of Mchgan Press Kealhofer,S, (1998), Portfolo Management of Default Rsk, KMV Corporaton, San Francsco Lensberg, T, Elfsen, A and T. McKee, (2006), Bankruptcy Theory Development and Classfcaton va Genetc Programmng, European Journal of Operatonal Research, 169 (2), Markowtz, H, (1952), Portfolo Selecton, Journal of Fnance, 7, Markowtz, H, (1959), Portfolo Selecton: Effcent Dversfcaton of Investments, Wley, New York Merton, R.C,(1990), Contnuous tme fnance, Basl Blackwell, Oxford Mossn, J, (1969), optmal mult perod portfolo polces, Journal of Busness, 41, Orto,Y, Yamamotod, H & G Yamazak (2003), Index fund selectons wth genetc algorthms and heurstc classfcatons, Computers and Industral Engneerng, 45, Sharpe, W,(1971), A lnear programmng approxmaton for the general portfolo analyss problem, Journal of Fnancal and Quanttatve Analyss, 6, Sharpe,W, (1966), Mutual fund Performance, Journal of Busness, 39, Speranza, M.G, (1993), Lnear Programmng Models for Portfolo Optmzaton, Fnance, 14, Speranza,M.G, (1996), A Heurstc Algorthm for a Portfolo Optmzaton model Appled to the Mlan Stock Market, Computers and Operatons Research, 23, Wlson, T, (1997), Portfolo Credt Rsk, Rsk 10 (10), Wlson, T, (1997), Portfolo Credt Rsk, Rsk, 10 (9), Wong, F & C. Tan, (1994), Hybrd Neural, Genetc, and Fuzzy Systems In: G.J. Deboeck, (Edtor), Tradng on the edge, Wley, New York,

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