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Rational Decisions, Random Matrices and Spin Glasses Stefano Galluccio†, Jean-Philippe Bouchaud†,∗ and Marc Potters† † Science & Finance, 109-111 rue Victor Hugo, 92523 Levallois Cedex, FRANCE ∗ Service de Physique de l’E´tat Condensé, Centre d’études de Saclay, Orme des Merisiers, 91191 Gif-sur-Yvette Cédex, FRANCE (February 1, 2008) 8 9 We consider the problem of rational decision making in the presence of nonlinear constraints. 9 By using tools borrowed from spin glass and random matrix theory, we focus on the portfolio 1 optimisation problem. Weshowthatthenumberof“optimal”solutionsisgenerically exponentially n large: rationality is thusde facto of limited use. In addition, this problem is related to spin glasses a withLévy-like(long-ranged)couplings,forwhichweshowthatthegroundstateisnotexponentially J degenerate. 1 2 ] It is often hoped that science can help in choosing the cedure, or portfolio selection, as investigated in its sim- n right decision. A rational decision is usually thought of plestversionby Markowitz[3], results in a setof allpos- n asthe solutionwhichoptimizesacertainutility (orcost) sible“optimal”couples R ,σ2 whichalllieonacurve s- function, which is supposed to quantify the performance called the efficient fron{tierP(efP)}. For a given degree of i d of a givenstrategy. A simple example is that of portfolio risk, there is thus a unique selection of assets p∗ N { i}i=1 . optimization: given a set of financial assets, character- which maximizes the expected return. On the opposite, t a ized by their average return and their risk, what is the wewillshowthatinsomecasesthis simplescenariodoes m optimal weight of each asset, such that the overall port- not hold: a very large number of compositions p N - folio provides the best return for a fixed level of risk, or can be “optimal”, in the sense defined above. F{urit}hie=r1- d n conversely,thesmallestriskforagivenoverallreturn? It more, these optimal compositions can be very different o isfrequentlythecasethattheoptimalsolutionisunique, from another. c so that ‘rational’operators should choose this particular In order to illustrate this rather general scenario with [ one, unless they are inefficient. In the case of portfolio an explicit example, we shall investigate the problem of 1 selection, the only free parameter is the proportion of portfolio selection in the case one can buy, but also to v the “riskless asset”; all portfolios should thus look very shortsellstocks,currencies,commoditiesandotherkinds 9 much alike. This is at the heart of the so called Capital of financial assets. This is the case of ‘futures’ markets 0 Asset Pricing Model, (CAPM), one of the cornerstones or margin accounts. The only requirement is to leave a 2 1 of modern theoretical finance [1]. certaindeposit(margin) proportionalto the value of the 0 InthisLetterwewillshowthat,surprisingly,thereare underlyingasset[4]. Thismeansthattheoverallposition 8 cases where the number of “rational” decisions is expo- on these markets is limited by a constraint of the form: 9 nentially large (for example in the number of assets) – / N at much like the number of ground states in spin glasses or γWi pi = , (1) m other disordered systems [2]. This means that there is i=1 | | W X an irreducible component of randomness in the final de- - where W is the price of asset i, the total wealth of d cision: the degeneracy between this very large number i W the operator, and γ the fraction defining the margin re- n of possibilities can only be lifted by small, ‘irrational’, quirement. Here, p is the number of contracts on asset o effects. i c More precisely, the average return R of a portfolio i which are bought (pi >0) or sold (pi <0). It is worth : P stressing that it is the nonlinear form of the above con- v P of N assets, is defined as R = N p R , where p i P i=1 i i i straint that makes the problem interesting, as we shall X (i = 1, ,N) is the amount of total capital invested ··· P seebelow. Atthis pointwesimply notethattheoptimal in the asset i, and R are the expected returns of the r { i} portfolio p∗, which – say– minimizes the risk for a given a individual assets. Similarly, the risk on a portfolio can i return, can be obtained using two Lagrange multipliers be associated to the total variance σ2 = N p2σ2 + P i=1 i i (ν,µ): N p p σ ,whereσ2 arethesingleassetvariancesand i6=j i j ij i P σ aretheinter-assetcovariances. Alternatively,onecan N N N ij ∂ 1 dPefine σ2 = N p C p where C is the covariance pjCjkpk ν pjRj µ Wj pj =0. P i,j=1 i ij j ij ∂pi 2 − − | | matrix. jX,k=1 Xj=1 Xj=1 Generally sPpeaking, the goal of each rational investor   (2) consists in maximizing the portfolio return and/or in Without loss of generality, one can always set W 1. minimizing its variance (or risk). This optimization pro- i ≡ Defining S to be the sign of p∗, one gets: i i 1 N N others. The matrix C can thus be written as C = M p∗i =ν Ci−j1Rj +µ Ci−j1Sj, (3) MT + D, where D is aijpositive diagonalmatrix,andM Xj=1 Xj=1 a N K rectangular matrix. The above representation where C−1 is the matrix inverseofC.Takingthe signof ensur×es that all eigenvalues of C are positive. In order to simplify the problem, we shall make in the following this lastequationleads to anequationfor the S ’s which i theassumptionthatDisproportionaltotheidentityma- is identicalto those defining the locally stable configura- trixandthatthecoefficientsM arecompletelyrandom. tions in a spin-glass [2]: iα For K = N, C is a member of the so-called Exponential N Orthogonal Ensemble (eoe) [6], whichis a maximumen- S =sign H + J S , (4) tropy (least information) ensemble. For any N K ma- i  i Xj=1 ij j otrfixCMis,e(xwaictthlyNkn≥owKn)i,nththeedleinmsiittyNof eigen,vKalu×es ρCa(nλd) where H ν NC−1R and J µC−1 are the ana- Q = K/N fixed [9]. In the limit Q =→1∞(which→w∞e con- i ≡ j ij j ij ≡ ij logue of the “local field” and of the spin interaction ma- sider from now on) the normalized eigenvalue density of trix, respectivePly. Once the S ’s are known, the p∗ are the (square)matrix Mis the wellknownsemi-circlelaw, i i determined by Eq. (3); ν and µ are fixed, as usual, so fromwhichthe density of eigenvaluesofC andJ=µC−1 as to satisfy the constraints. Let us consider, for sim- areeasilydeduced. Inparticular,forthesituationweare plicity, the case where one is interested in the minimum interested in, one finds risk portfolio, which corresponds to ν = 0. [By analogy 1 τ(4+a) λ with spin glasses, the following results are not expected ρ (λ)= − , λ ]aτ,(4+a)τ]. (6) C tochangequalitativelyifν =0[5]]. Inthiscase,onesees 2πτ p √λ aτ ∈ 6 − thatonly µ isfixedbythe constraint. Furthermore,the where a measures the relative amplitude of the diago- | | minimum risk is given by R = µ2 Nj,kSjCj−k1Sk, which nal contribution D and τ the width of the distribution. is the analogue of the energy of the spin glass. It is now The distribution of eigenvalues of J=µC−1, ρ (x), can well known that if J is a randomPmatrix, the so called be easily calculated from Eq.(6). Note that inJthe case “tap” equation (4), defining the metastable states of a where a = 0, ρ (x) has a power-law tail decaying as J spin glass at zero temperature, generically has an expo- ρ (x) x−3/2. For finite a, however, the maximum J nentiallylarge(inN)numberofsolutions[2]. Notethat, eigenva∝lue of J is 1/a. as stated above, the multiplicity of solutions is a direct ∝ consequence of the nonlinear constraint (1). 3000 In what respect can the correlation matrix C be con- sidered as random? Here we take a step in complete analogywiththe originalWignerandDyson’sideaofre- Historical Theory (EOE) placing the Hamiltonian of a deterministic but complex 2000 systembyarandommatrix[7]. Moreprecisely,theypro- posed to study the properties of one generic member of ρ(λ) a statistical ensemble [8] which shares the symmetries of the original Hamiltonian. Similarly, in the present sit- 1000 uation, we would like to see C−1 as a random matrix whose elements are distributed according to a given ensemble, compatible with some general properties. In our case, for instance, we must select C−1 from an ensem- 0 0 0.0002 0.0004 0.0006 0.0008 0.001 ble of positive definite matrices, as all the eigenvalues of λ the covariance matrix are > 0. The choice of a suitable FIG.1. Smoothed distribution of eigenvalues of C, where statistical ensemble is guided by the following observa- thecorrelation matrix C is extracted from N =406 assets of tion. The (daily) fluctuations of the asset i, δWi, can be the S&P500 during the years 1991-1996. For comparison we decomposed as: haveplottedthebestfitobtainedwiththetheoreticaldensity (6) with a ≃ .34,τ ≃ 1.8×10−4. Results are qualitatively K similar inthecase oftheLondonorZurichStockExchanges. δW = M δE +δW , (5) i iα α i0 α=1 X We have studied numerically the density of eigenval- where the E are K independent factors which affects ues of the correlation matrix of 406 assets of the New- α the assets differently, and δW is the part of the fluctu- York Stock Exchange (nyse), based on daily variations i0 ation which is specific to asset i (and thus independent duringthe years1991-96,for a totalof1500days. More- of the E ). The E contain all information on how the over,byrepeatingouranalysisondifferentStockMarkets α α stochastic evolution of a given asset is correlated to the (e.g. LondonandZurich)wehavecheckedtherobustness 2 and universality of our results. The observed density of Once G (x) is known,one can write the averagenum- a eigenvalues is shown in Fig.1 together with the best fit ber of solutions of the tap equations for large N as N obtainedby assuming Q=1. The fitting parametersare exp Nf(a) ,wheref(a)isobtainedfromasteepest N ∼ { } a and the width of the semi-circle distribution τ, which descentapproximation. Indeed, from Eq.(4), one derives arefoundtobe,respectively,a .34andτ 1.8 10−4. the following result, valid for large N [10]: ≃ ≃ × The deviationfrom the theoretical curve is partially due to finite N effects which are expected to smear out the N N singularities at λ = aτ and (4+a)τ. It might also be = θ Si JijSj (9) that D has unequal diagonal elements (this would also N *X{S}iY=1  Xj=1 + smear out the singularities of ρJ(λ)), or let the number max exp[ xZ yW+Ga(x+√y) K of“explicativefactors”be lessthanN (aratherlikely ≃x,y,W,Z{ − − situation). Note that the empirical ρ can be rather Z C +G (x √y)+ln erfc ] (10) a well fitted by a log-normal distribution, suggesting that − −2√W } (cid:18) (cid:18) (cid:19)(cid:19) the relevantensemble has the additionalsymmetry C eNf(a). (11) C−1,althoughwehavenotbeenabletojustifythispro↔p- ≡ erty. where θ(u) is the usual Heavyside function of its argu- The point, however, is not to claim that the density ment. From(8),wefindthatf(a)veryrapidlyconverges of eigenvalues is precisely described by the above model, toln2(themaximumallowednumberofsolutions)when butthatitrepresentsareasonableapproximationforour a is large: purposes. Oncethedensityofeigenstatesρ (λ)isknown, J thenumberofsolutionsoftheoptimizationequation,av- f(a)=ln2 e−a2/2 1 +o 1 , (12) eragedoverthe matrixensemble,canbecomputedusing − √2πa a2 (cid:18) (cid:18) (cid:19)(cid:19) toolsborrowedfromspinglassesandrandommatrixthe- aresultwhichwehavenumericallyconfirmedbyextract- ory. As discussed in [10], the main ingredient is the gen- ing J from the proper statistical ensemble. To give an erating function G(x), defined as [11]: idea, one finds f(a = 1) .686 .003, which is already ≃ ± [J]exp Tr (JA/2) exp N TrG(A/N) (7) quite close to ln2=.693147.... D { }N≃≫1 { } Forsmalla,calculationsbecomeratherinvolved,asthe Z where A is any symmetric N N matrix of finite extremizing set {x∗,y∗,W∗,Z∗} corresponds to a min- rank, and [ J] = d[J]P([J]) is×the probability mea- imum and actually lies on the domain’s border. The D final result is that y∗ = x∗2, W∗ = 1/(8√2x∗3) and sure overthe matrix ensemble. The above formula holds for general complex hermitian matrices J [11]. For ex- Z∗ = 1/(4√2x∗), which leads to f(a) ∝ a1/4 for a ≪ 1. ample, in the simplest possible case, where J is ex- We have confirmed this result numerically (see Fig 2). tracted from a Gaussian Ensemble, one has P([J]) = Therefore, for a = 0, f(a = 0) = 0, i.e. the number of exp Tr(J2/4σ2) /Z andthus G(x)=x2/4. Formally, metastablesolutionsdoesnotgrowexponentiallywithN. {− } This result means that the presence of very large eigen- G(x) can be computed in general from ρ (λ) using a se- J values prevents the existence of many metastable states. ries of rather involved transformations [12], which can This has an interesting consequence in the physical con- be somewhat simplified by the use of a Hilbert integral text, which was conjectured in [15]: a spin glass with transform of the eigenvalue density [13]. a broad distribution of couplings cannot sustain many In the specific case (6), we have been able to compute ground states; one can thus expect that the low temper- G (x) exactlyfora=0,withthe resultG (x)= √ x, a 0 − − ature phase of these models is rather different from the and perturbatively in the limits a 1 and a 1. Leav- ≪ ≫ ‘standard picture’ [2]. We can show that this is true for ing aside all mathematical details [13], the results are: allβ <1[13];wefindinparticularthatf(a) K(β)aβ/2 x2 forsmalla,withaprefactorK(β)algebraical≃lydiverging G (x) , a 1, a ≃ 4a4 ≫ for β 1. For β >1, finally, f(a=0)>0. x Let→us now come back to our main theme. In what G (x) √ag , a 1, (8) a ≃− −a ≪ respect is the above picture a valid description of the (cid:16) (cid:17) portfolio optimization problem? In the real situation, of wherethescalingfunctiong(u)behavesasg(u) √ufor ≃ course,C is givenby the “historical”correlationmatrix. u 1 and as g(u) u for u 1. As a side remark, one ≫ ∝ ≪ If our model is correct,then one should expect the num- canshow that for Lévy randommatrices [14]whereρ(λ) decays as λ−1−β for large λ, the characteristic function ber of optimal solutions to be exponentially large, even behaves as G(x) ( x)β for small x’s, and for all for small a’s. By extracting true correlation matrices ≃ − − of sizes up to N = 20 from the available 406 assets of β <1. Consequently,theproblemweareconcernedwith the nyse (see above), we have indeed numerically found here(β =1/2)canbeseenasthestudyofthemetastable statesinspinglasseswithlong-rangeinteractions[15,13]. that NY exp(fNYN)withfNY =.68 .01. Thisresult N ∼ ± 3 doesnotqualitativelychangeinthecaseoftheLondonor Furthermore, these solutions are usually ‘chaotic’ in the Zurich markets. Despite of the rather low dimensional- sense that a small change of the matrix J, or the addi- ity of the involved matrices, with unavoidable finite-size tionofanextraasset,completelyshufflestheorderofthe effects, the above result is quite well reproduced by our solutions (in terms of their risk). Some solutions might theoretical model (which is valid at N 1). In fact, even disappear, while other appear. It is worth pointing ≫ by using the fitting (a,τ) parametersas inFig.1,we find out that the above scenario is not restricted to portfolio a theoretical value fTH = .63 0.02, not too far from theory, but is germane to a variety of situations. For ex- the real one. Similarly by extra±cting C from 172 stocks ample,itiswellknowninGameTheorythateachplayer of the London Stock Exchange (LSE) we have found an has a different utility function he must maximize in or- empirical fLSE = .43 .09 and a corresponding theoret- der to get the best profit. When, in addition, nonlinear ± ical value fTH = .32 .07. Remarkably, our admittedly contraints are present, we expect a similar proliferation ± approximate description of the density of states of the of solutions. As emphasized above, the existence of an matrixJseemsto accountsatisfactorilyforthe observed exponentially large number of solutions forces one to re- number of “optimal portfolios”. think the very concept of rational decision making. Aknowledgements: We want to thank J.P. Aguilar, R. 0.8 Cont and L. Laloux for discussions. The lse and Zurich 0.7 data were obtained from the Financial Times and the Numerical simulation Best fit: f(a)=0.84 a1/4 nyse from S&P Compustat. 0.6 0.5 f(a)0.4 0.3 0.2 [1] E.J. Elton and M.J. Gruber, Modern Portfolio Theory and Investment Analysis (J.Wiley and Sons, New York, 0.1 1995). [2] M. Mézard, G. Parisi and M.A. Visrasoro, Spin Glass 0 1e-05 0.0001 0.001 0.01 0.1 1 a Theory and Beyond (World Scientific, Singapore, 1987). [3] H. Markowitz, Portfolio Selection: Efficient Diversifi- FIG.2. Thetheoreticalexponentf(a)plottedvsa(a≪1) cation of Investments (J.Wiley and Sons, New York, whenCisextractedfrom theEOE,usinganexactenumera- 1959). See also: J.P. Bouchaud and M. Potters, Theory tionproceduretofindthenumbersofsolutionsuptoN =27. of Financial Risk,Aléa-Saclay,Eyrolles(Paris, 1997) (in Finitesizeeffectsareclearlypresentforthesmallestvaluesof french). a. Thetheoretical line Ka1/4 is plotted forcomparison, with [4] For a recent account, see J. Hull, Futures, Options and K ∼.84. Other Derivatives (Prentice Hall, NewYork, 1997). [5] D.S. Dean, J. Phys. A. 27 L23, 1994 Bymeansofacombinationofanalyticalandnumerical [6] B. V.Bronk, J. Math. Phys.6, 228 (1965). arguments, we have thus shown that the number of op- [7] Forareview,see: O.Bohigas,M.J.Giannoni,Mathemat- ical and computational methods in nuclear physics, Lec- timal portfolios in futures markets(where the constraint ture Notes in Physics, Vol.209, Springer-Verlag(1983) on the weights is non linear) grows exponentially with [8] M. Mehta, Random Matrices (Academic Press, New the number of assets. On the example of U.S. stocks, York, 1995). we find that an optimal portfolio with 100 assets can [9] A.M. Sengupta and P.P. Mitra Distribution of Singular be composed in 1029 different ways ! Of course, the Values for Some Random Matrices, cond-mat/9709283 ∼ above calculation counts all local optima, disregarding preprint. the associated value of the residual risk . One could [10] G. Parisi and M. Potters, J. Phys. A, 28, 5267 (1995). extend the above calculations to obtain Rthe number of [11] C. Itzykson and J.-B. Zuber, J. Math. Phys. 21, 411 solutions for a given . Again, in analogy with [10], (1980). we expect this numberRto grow as expNf( ,a), where [12] E.Marinari, G.ParisiandF.Ritort,J.Phys.A277647 f( ,a)hasacertainparabolicshape,whichRgoestozero (1994). forRacertain‘minimal’risk ∗. Butforanysmallinterval [13] S.Galluccio,J.P.BouchaudandM.Potters,unpublished. R [14] P. Cizeau and J.P. Bouchaud, Phys. Rev. E 50 1810 around ∗, there will already be an exponentially large R (1994). number of solutions. The most interesting feature, with [15] P.CizeauandJ.P.Bouchaud,J.Phys.A,26L187(1993). particular reference to applications in economy, finance or socialsciences, is that all these quasi-degeneratesolu- tions can be very far from each other: in other words, it can be rational to follow two totally different strategies! 4