We propose a new method of valuation of portfolios and their respective
investing strategies. To this end we define a canonical ensemble of portfolios
that allows to use the formalism thermodynamics.Comment: 8 pages, LaTeX, appolb.cls is require

Piotrowski, E. W.

Sladkowski, J.

Acta Physica Polonica B

English

arXiv

We propose a new method of valuation of portfolios and their respective
investing strategies. To this end we define a canonical ensemble of portfolios
th a t allows to use the formalism of thermodynamics

Piotrowski, E. W

Sładkowski, Jan

Repozytorium Uniwersytetu Śląskiego RE-BUŚ

             Title: The thermodynamics of portfolios  Author: E. W. Piotrowski, Jan Sładkowski  Citation style: Piotrowski E. W., Sładkowski Jan. (2001). The thermodynamics of portfolios. "Acta Physica Polonica. B" (2001, no. 2, s. 597-604). T H E  T H E R M O D Y N A M IC S O F P O R T F O L IO S*E .W . P i o t r o w s k iInstitute of Theoretical Physics, University of Białystok Lipowa 41, 15-424 Białystok, Poland e-mail: epQalpha. uwb. edu. plAND J .  SŁADKOWSKIInstitute of Physics, University of Silesia Uniwersytecka 4, 40-007 Katowice, Poland e-mail: sladkQus.edu.pl(Received November 10, 2000)We propose a new method of valuation of portfolios and their respective investing strategies. To this end we define a canonical ensemble of portfolios th a t allows to  use the formalism of thermodynamics.PACS num bers: 02.50.-r, 02.50.Le, 05.70.-a, 05.90 + m1. IntroductionPhysicist, con tra ry  to  m athem atic ians, have only occasionally investi­gated  economic system s. Recently, however, a growing num ber of papers of relevance to  economics is being published in physics jou rnals  and  con­ferences proceedings [1]. M oreover, H. M arkowitz encouraged outsiders to  engage in research on economics in his Nobel P rize Lecture [2]: ”1 believe that microscopic market simulations have an important role to play in eco­nomics and finance. I f  it takes people from  outside economics and finance -  perhaps physicists -  to demonstrate this role it won’t be fo r  the first time that outsiders have made substantial contribution to these fields". Physi­cists have good com m and of stochastic  processes and s ta tis tica l physics so it is hard ly  surprising  th a t  th ey  successfully use th e ir m inds to  analysing* Presented a t the XXIV International School of Theoretical Physics “Transport Phenom ena from Q uantum  to  Classical Regimes”, Ustroń, Poland, September 25- October 1, 2000.economic system s. O ther physical concepts probab ly  have analogue in eco­nom ics, e.g. such notion  as gauge sym m etry  can be identified in financial m arkets [3]. A nother im p o rtan t and  exciting branch  of finances is th e  p o rt­folio theo ry  [4]. Here various physical concepts have direct analogies. U ltra- m etric itv  [5] com m only used in spin glasses can be  used to  describe d istance betw een stocks [6 ]. Spin glasses seem to  have lo ts in com m on w ith  portfolio  theo ry  [7,8]. We would like to  propose a m ethod  of valuation of portfolios and  investing strateg ies th a t  stem s from  therm odynam ics. Investors and portfolio  m anagers set up  th e ir portfolio  according to  th e  m arket inform a­tion  available and  th e ir lore. T hey  lose or gain. It is easy to  gain when all prices are raising and  lose when everything is falling down. B u t which p a r t of th e  gain is due to  skill and  qualities of th e  investor? W hich p a r t is sim ple a resu lt of th e  ac tua l s ta te  of th e  m arket? O ur m odel allows for a tem p era tu re  like p aram eter th a t  m easures th e  quality  and  professionalism  of th e  investor.A portfo lio  is a package of various assets (shares, bonds, derivative in­s trum en ts, etc.) th a t  can be exchanged on th e  m arket. If we denote by ai, i = 0 ,1 , . . .  N  th e  u n it of th e  T th  asset th en  th e  portfolio  P  asw here cy € Z  is th e  num ber of u n its  of th e  *-th asset in th e  portfolio  P . T he coefficients a* can be  negative because th e  stock exchange regulations allow for selling assets th a t  th e  portfolio  owner do no t possesses (short selling). O ne usually  supposes th a t  one asset, say «o, in P  can be exchanged w ith  any o ther asset a t any tim e (m oney). An ex ternal observer describes th e  moves perform ed by th e  portfo lio  m anager as a draw  in th e  following lo ttery . Let P i ,  i =  1 ,..., N  be  th e  p robab ility  of th e  purchase of th e  W{ u n its  of th e  *-th asset. T h e  weights W{ are given before th e  lo tte ry  come in to  operation . T he value of th e  portfo lio  (re tu rn ), —cqqk , fulfils th e  following balance equation  a t an a rb itra ry  m om ent after th e  draw:2. The portfolio descriptionN(1)NCook ( c 0 , • • • ,  c j v ) +  Y 2  lK =  k ] c k w k =  0  fc=0(2)w here k denotes th e  random  variable tak ing  values from  0 ,N ,  th e  assets num bers. T h e  expression [sentence ] takes value 0 or 1 if th e  sentence  is false or true , respectively (Iverson convention) [9]. T h e  coefficient c* denotes thepresent relative price of a u n it of th e  asset a*, c* =  =f w here u* is th e  present price of th e  *-th asset and  u; its price a t th e  m om ent of draw ing. A t the  m om ent of draw ing th e  balance equation  takes th e  form:Nc00k (1, • • •, 1) +  =  k\ u)k =  0 , (3)fc=0If «o represents th e  basic currency (m oney) used to  define th e  financial value of th e  portfolio  th en  we have Co =  1 during  considered period. One can show th a t  th e  expecta tion  value of wq [k =  0] is th e  Legendre transfo rm  of th e  m ean value of th e  portfolio.3. The canonical portfolioLet us consider a portfolio  defined by th e  weights w; and  th e  uniform  d is trib u tio n  of probabilities po = p i = . . .  = p n  =  at+T- T his portfolio  has th e  sam e expecta tion  value of th e  re tu rn  E(cqqk ) as th e  uniform  portfolio  w ith  weights ^ 7 7 , • • •, (every asset is included). I t is in teresting  to  find o u t how m any of th e  portfolio  owners (m anagers) are in th is  s itua tion . To th is  end we consider two types of investors. T he first group, called zombies, consists of investors whose moves on th e  m arket are fully determ inistic . T hey  react according to  th e  m arket condition d is trib u tin g  th e ir cap ita l into various assets or no t and  ac t in th e  sam e way in th e  sam e situations. T he second group, called gamblers, are indeterm inistic . T heir moves can be  different in analogous m arket situations. T his does no t m ean th a t  gam blers often change th e ir strateg ies. We sim ply canno t p red ic t th e ir moves because p a rt of th e ir knowledge and p ast experiences are no t available for us as ex ternal observers. Is th ere  a qualita tive  way of m easuring inform ation on investo r’s behaviour? We do no t consider here m echanism s leading to  disclosure of in form ation and  its  consequences. It seems reasonable th a t  th e  m easure S (p o ,. . .  ,p n )  we would like to  use for m easuring th e  investor behaviour (or m ore precisely th e  lo tte ry  defined by th e  probabilities pfi be additive in the  following sense. I t should no t m a tte r  if th e  portfolio  is construc ted  in one draw  w ith  th e  probabilities p o ,p i,p 2, ■ ■ ■ ,P n  or w ith  two subsequent draws w ith  probabilities poPYPl 4 p0+pi anc  ^ ^ 0 X PI1P2, ■ ■ ■ iPn- T his leads to  the  equation:S  (p0,p i , .  ■ ■ ,p n ) = S ( p Q + p i , p 2, . . . , p N )+ {Po + P i ) S  ^ ^ (4) \Po +  Pl Po + P l Jwhere th e  argum ents describe probabilities of draw ing (th a t is buying) of th e  assets. We will look for a solution fulfillingS  ( p 0 , P i ,  • • • , P n ) =  S  ( p 0 ) +  S  ( p i , p 2 ,  • • • , P n ) • (5)E quations (4) and  (5) after som e s tan d a rd  algebraic m anipu lations lead to  th e  following solution for th e  function  S,  see [1 0 ]Ns  (po,Pi,  • • • ,Pn ) = -  Y Pk lnpk  •k=0For obvious reasons, we will call th is  function  entropy. T h e  num ber K  := es  gives th e  effective num ber of assets in th e  portfolio: K  is equal M  if the  weights are d is trib u ted  uniform ly betw een M  assets, and  is 1 if th e  portfolio  contain  a single asset.Let us now consider th e  “C artesian  p ro d u c t” of two s ta tis tica lly  indepen­den t portfolios consisting of N  +  1 and  M  +  1 assets, respectively. T henM  NS ( M + l ) x ( N + l )  =  ~  Y  T .P k P ' l  {PkP'l)k= 0 1=0N  N=  - Y p k l n (pk) T t p '1 l n =  S m +i +  S n +ifc=0 1=0so th e  en tropy is additive. We would like to  com pare re tu rn s  (“achieve­m ents”) of portfolio  m anagers (owners). T herefore we classify th e  investors according to  th e  value of th e ir portfolio. T his would allows us to  divide the  ap p ro p ria te  re tu rn s  into two p a rts  corresponding to  m an ag er’s lore and tide of th e  m arket. We will choose th e  portfolios th a t  m axim ize th e  en tropy  to  represen t th e  classes of investors. An “ex ternal” observer would know only th e  expecta tion  value of re tu rn s . (We suppose th a t  draw s are no t correlated  and  th e  m arket is efficient.) Perhaps, it would be  b e tte r  to  assum e th a t  the  values of th e  random  variable k  correspond to  whole sectors of th e  m arkets (e.g. oil com panies) ra th e r th en  to  sep ara te  assets [1,11]. T his m eans th a t we are looking for a conditional ex trem a of th e  function  S( po , . . .  , p n )- T he conditions are:NY pk =  1 (8a)k =0and( NY  [k =  fc] ckwk k = 0We have weakened th e  balance condition (2) to  be fulfilled only on th e  ex­p ec ta tio n  values level. T he Lagrange m ethod  of finding conditional ex trem a requires th e  following differential form  to  vanish( N  \  N^  [k =  fc] Ckw k j +  yd  p k =  0 , (9)fc=0 /  fc=0w here ¡3 and  y  are Lagrange m ultipliers. T h e  su b stitu tio n  of equation  (6 ) leads to  th e  condition-  ln p k -  1 +  (3ckw k + 7  =  0  ( 1 0 )which gives explicit dependence of th e  probabilities p k characterizing the  m axim al en tropy portfo lio  on th e  prices ckp k =  exp (/3ckw k +  7  +  1) • (U )T h e  m ultip lier y  can be  elim inated by explicit norm alization  and th e  elimi­nation  leads to  G ibbs-like p robab ility  d istribu tionexp (/3ckw k)p k (c0, . . .  , cN ) = — --------- —------- - .  ( 1 2 )Z f^c=0 exp (pckw k)Now we are in a position  to  define the canonical statistical ensemble of in­vestors which consists of all investors th a t  have got th e  sam e re tu rn . T he canonical ensem ble describes all s trateg ies (zombies and  gam blers) leading to  th e  sam e re tu rn . I t can be represen ted  by th e  portfo lio  m axim izing th e  en­tropy  (canonical portfolio). C hoosing any o ther represen ta tive would m ean lower en tropy and  as a resu lt would give a bias to  one s tra teg y  (knowl­edge). If one recalls th a t  K  =  exp S' gives th e  effective num ber of assetsin th e  portfo lio  th en  one im m ediately  gets th a t  th e  en tropy  S  m easures also th e  diversification of th e  portfolios. T h e  canonical portfolio  is the  safest one in th e  class of th e  sam e re tu rn . We will also define th e  tem ­p era tu re  T  of th e  canonical ensem ble as T  := ^  and  th e  s ta tis tica l sum  Zas Z  (co, • • • ,cjv) :=  Zik=oexP(Pckwk)- If we keeP th e  weights w k constan tth en  th e  changes of th e  prices ck im ply app ro p ria te  changes of th e  value ofth e  portfolio  —coo (re tu rn ). T h e  expected  infinitesim al change of coo is given by—dcoo (co, • • •, cjv) =  d,E (cKw K) . (13)Having in m ind th a t  S  =  — Y2k=oPk n^ Pk In % -  ¡3E (cKwK) we can w rited S  =  l'l3wkCk)dCkz= /3 ( ^ ^ E  ([k = k ) w k) dck -  d E  (cKwK)^j . (14)T his im plies th a t  if we tre a t S  as an  independen t variableNcIcqq (co, . . . ,  cj\T; S)  +  )  ] w^dck = T d S , (15)fc=0where ui^ denotes th e  m ean conten t of th e  fc-th asset in th e  portfolio. Now, th e  tem p era tu re  of th e  portfolio  T  is equal torp dcooT  =  1 ) S '  ( 1 6 )T his tem p era tu re  m easures th e  change of th e  portfo lio  value caused by its en tropy  change. As in classical therm odynam ics, we can form ulate two p rin ­ciples.The I principle of the canonical ensemble: The change o f value o f a canonical portfolio ^coo consists o f two parts. The first one is equal to the change the investors knowledge 8Q and the second is equal to change o f the values o f content o f the portfolio:Ndcpo +  8Q +  y )  w kdck =  0 . (17)fc=0The II principle of the canonical ensemble: The value o f lost investors knowledge —8Q is proportional to the increase in the entropy o f its canonical ensemble:SQ + T d S  =  0 . (18)N ote th a t  we have w ritten  8Q in stead  of th e  obvious dQ  because, in general, such a function  Q does no t exist. Following th e  developm ent of th erm ody­nam ics we will define th e  free value —F  of th e  canonical portfolio  as:F  (co, . . . ,  c j v , T )  :=  coo (co, • • •, cat,  S) —T S .  (19)T his allows for th e  form ulation of th e  two principles of th e  canonical ensem ­ble in a balance-like equation:Nd F ( c Q , . . . , c N , T )  + S d T  + Y ^ w kdck =  0 (20)fc=0which com bined w ith  th e  definition of th e  entropy  S  gives—F  = T \ n Z  (2 1 a)andZ  , n i l .- c °° -  • (21b)T h e  nam e free value can be  justified  as follows. Let us suppose th a t  during th e  m arket evolution (changes of th e  prices ck) th e  class of th e  investor does no t change (T  =  const). T hen  th e  changes of values of th e  assets in the  portfolio, are m easured by changes of th e  p o ten tia l F.  Sucha process can be referred to  as iso therm al. — T  can be in terp re ted  as price of a u n it of th e  en tropy  S.  T he en tropy of a canonical portfo lio  increases when th e  abso lu te  value of th e  p aram ete r T  increases. T his m eans th a t  an am ateu r pays m ore th an  a professional investor for errors of th e  sam e order. B u t if th e  tem p era tu re  is negative an am ateu r gets m ore from  erroneous decisions. A portfo lio  w ith  T  < 0 and sm all en tropy  can only be construc ted  by a specialist who uses his knowledge in reverse. A fter changing signs of the  weights w k (short position) such a portfo lio  gives excellent re tu rn s . N ote th a t  due to  th e  ad d itiv ity  of entropy  th e  tem p era tu re  of th e  portfo lio  construc ted  by m erging two portfolios w ith  tem p era tu re  T  b u t bu ilt in different m arket sectors equals T.It would be  helpful to  give som e flesh to  th e  above consideration by working ou t a num erical exam ple. Let us consider two investors Alice and Bob. B oth  have th e  sam e in itia l cap ita l, say $1 and, besides money, there  are only two assets available on th e  m arket, a\  and  a2. Bob (a zombie) divided his cap ita l in to  two equal p a rts  and  spen t them  on b o th  assets. Alice (a zombie, a pro?) spen t a q u arte r of her cap ita l on a\  and  kept th e  rest. Suppose now th a t  after som e tim e th e  price of th e  asset a\  went down by 20% and  th e  price of th e  asset a2 increased by 30% th a t  is the  relative prices are now c\  =  0.8 and c2 =  1.3. T his m eans th a t  A lice’s and B ob’s portfolios gave th e  re tu rn s  Cqq =  0.95 and cf0 =  1.05, respectively (we have neglected th e  in terest ra tes). T he tem p era tu res  are T A =  —0.46 and  T B =  2.55, respectively (only two decim al positions are kep t). Bob has got b e tte r  re tu rn  th en  Alice even though  T B >  T A. T his is because Bob alone m akes profit from  his knowledge (T B is positive). B u t th e  au thors would prefer Alice to  Bob as an investm ent adviser because w hen one listen to  her advises and acts con trad ic to ry  to  them  one gets b e tte r  re tu rn  then  B ob’s. A lice’s knowledge is g reater th a n  B ob’s because if | T B |> | T A | then  S B >  S A4. ConclusionsWe have proposed a m ethod  th a t  allows num erically  m easure investors qualities. Inspired by therm odynam ics, we were able to  define canonical ensem bles of portfolios, th e  tem p era tu re  of portfolios and, possibly, various therm odynam ics-like po ten tials. We have used th e  relative prices Ck of the  assets th a t  have direct in terp re ta tio n . T h e  theo ry  of financial m arket would prefer a covariant descrip tion  of th e  portfolio. T his can be easily achieved by replacing th e  param eters  coo, c i , . . . ,  cjv by th e ir n a tu ra l logarithm s (th a t is integrals of th e  in stan taneous in terest ra tes)We im agine th a t  analogous m ethods can be  developed for valuation  of cred it repaym ent scenarios or bonds under th e  stochastic  behaviour of in­te re st ra tes.[1] R.N. Mantegna, H.E. Stanley, An Introduction to Econophysics, Cambridge University Press, Cambridge 2000.[2] S. Moss de Oliveira, P.M.C. de Oliveira, D. Stauffer, Evolution, Money, War and Computers, Teubner, S tu ttgart 1999, p. 120.[3] K. Ilinski, J. Phys. A 33, L5 (2000).[4] H. Markowitz, Portfolio Selection: Efficient Diversification of Investment, J. Wiley, New York 1959.[5] R. Rammal, G. Toulouse, M.A. Virasoro, Rev. Mod. Phys. 58, 765 (1986).[6 ] R. Mantegna, Eur. Phys. J. B l l ,  193 (1999).[7] S. Galluccio, J.P. Bouchaud, J.P. Potters, Physica A 259, 449 (1998).[8 ] A. Gabor, I. Kondor, Physica A 274, 222 (1999).[9] R.L. Graham, D.E. Knuth, O. Patashnik, Concrete Mathematics, Addison- Wesley, Reading 1994.[10] C.E. Shannon, Bell System Techn. J. 27, 379 and 623 (1948).[11] R.N. Mantegna, Eur. Phys. J. B l l ,  193 (1999); cond-mat/9802256.’ 00 I ^  Coo ! Cfc I ->  Cfc , r Wk  - »  Wk  ■Coo I(2 2 a)(2 2 b)(22c)R E FE R E N C E S

The thermodynamics of portfolios

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The thermodynamics of portfolios

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