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Analysis of universal portfolios with and without side information Crouspeyre, Gilles 2007

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Analysis of Universal Portfolios With and Without Side Information by Gilles Crouspeyre Diplome d'Ingenieur, Ecole Internationale des Sciences du Traitement de l'lnformation, 2005 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in The Faculty of Graduate Studies (Mathematics) The University of British Columbia August, 2007 © Gilles Crouspeyre, 2007 Abstract In this paper, we want to analyze the concept of Universal Portfolio and to try it on some realistic data. In order to do so, we need first to write a complete version of one of the main results of T. Cover, Universal Portfolio's creator. Then, we study the constant rebalanced portfolio in a general case of N assets. This portfolio is usually the target the Universal Portfolio is aiming at. Once this is done, we look at how to construct the Universal Portfolio for the same N assets, and think about the optimal Dirichlet distribution the proportions of wealth invested should follow, in order to reach the optimal wealth. Finally, we proceed to an extensive numerical analysis and show the limits of the model. i i Table of Contents Abstract i i Table of Contents i i i List of Tables iv List of Figures v 1 Introduction 1 2 Complete version of the main result from Universal Portfolio with Side Information 6 3 The Best Constant Rebalanced Portfolio 11 4 Universal Portfolio Computation for M Assets and a General Dirichlet Distribution 15 5 Numerical Analysis 21 5.1 Simulated Data 21 5.1.1 Exponential Growth Rate of Wealth 21 5.1.2 Optimal Dirichlet Distribution 22 5.1.3 3 Assets Analysis 26 5.2 Real Data 28 5.2.1 Exponential Growth Rate of Wealth 28 5.2.2 Optimal Dirichlet Distribution 28 5.2.3 Universal Portfolio Without Side Information vs With Side Information 31 5.3 Discussion 33 6 Conclusion 35 Bibliography 36 i i i List of Tables 5.1 Choice of the Optimal Dirichlet Distribution 24 5.2 Computation times for 2 and 3 assets 30 iv List of Figures 5.1 Wealth obtained using the Universal Portfolio or the Best Constant Rebalanced Portfolio. Portfolio composed of 1 simulated stock and 1 simulated bond. . . . 22 5.2 Convergence of the Universal Portfolio's exponential growth rate of wealth to-ward the best constant rebalanced portfolio's one. Portfolio composed of 1 simu-lated stock and 1 simulated bond 23 5.3 Wealth obtained using the Universal Portfolio or the Best Constant Rebalanced Portfolio. Portfolio composed of 2 simulated stocks 23 5.4 Convergence of the Universal Portfolio's exponential growth rate of wealth to-ward the best constant rebalanced portfolio's one. Portfolio composed of 2 simu-lated stocks 24 5.5 Optimal Dirichlet distribution choice for portfolio composed of 2 simulated stocks. 25 5.6 Wealth obtained using the Universal Portfolio or the Best Constant Rebalanced Portfolio. Portfolio composed of 2 real stocks: IBM and MS (Morgan Stanley). . 29 ! 5.7 Exponential growth rate of wealth of Universal Portfolio and of best constant rebalanced portfolio. Portfolio composed of 2 real stocks: IBM and MS (Morgan Stanley) 29 5.8 Convergence of the Universal Portfolio's exponential growth rate of wealth to-ward the best constant rebalanced portfolio's one. Portfolio composed of 2 real stocks: IBM and MS (Morgan Stanley) 30 5.9 Optimal Dirichlet distribution choice for portfolio composed of 2 real stocks: IBM and MS (Morgan Stanley) 31 5.10 Wealth in log scale obtained by using a Universal Portfolio with or without side information 32 v Chapter 1 Introduction In his 1952 paper, A New Interpretation of Information Rate [1], Kelly studies the optimal investment problem applied on games. He gives two major results. First, he links the problem to the Information Theory and the data compression problem. Second, he gives a closed form to the optimal proportion of wealth one should bet on a gamble where the probability distribution is known, such as horse races or blackjack. As Kelly did, Thomas Cover treats the problem of adaptive investment, trying to get rid of the distributional approach, in two main papers, Universal Portfolios (1991) [2] and Universal Portfolios with Side Information (with Erik Ordentlich, 1996) [3]. More precisely, he would like to build a dynamic portfolio which gives asymptoti-cally the same wealth as the best state-constant rebalanced portfolio determined in hindsight without having to assume any distribution over the returns. Let's define first the best constant rebalanced portfolio. It is the portfolio which keeps the allocation of wealth constant over every time interval and gives the best final wealth. As T. Cover showed and as we will show later, when we add some side information, we observe some states where the information is the same. Hence, the best state-constant rebalanced portfolio allows the allocations to depend on the state of the information (for a more complete definition, see Universal Portfolio with Side Information, T. Cover). He chooses to compare his Universal Portfolios to the best state-constant rebal-anced portfolio because, supposing that relatives prices of stocks are i.i.d., this portfolio is supposed to achieve the optimal growth rate of wealth (Algoet and Cover(1988) [5], Bell and Cover (1980) [6]). Therefore, he supposes that there is still some optimality studying this portfolio in their setting. By finding a portfolio approaching this best state constant rebalanced portfolio, one could simply use the returns published at time t to compute the optimal port-folio allocations at time t+1! To do so, he introduces the concept of Universal Portfolio designed to achieve the 1 Chapter 1. Introduction same wealth as the best state-constant rebalanced portfolio's one to the first order of exponent, i.e. its exponential growth rate of wealth and the one from the target portfolio are converging asymptotically. In his first paper, T. Cover wants simply to find a portfolio that outperforms the best stock in the market. He decides to choose the best constant rebalanced portfolio as his ultimate goal since, as he proves it, this portfolio can beat the best stock, the Dow Jones average and the value line index at time t. To approach this portfolio as close as possible, he introduces for the first time the concept of Universal Portfolio. The Universal Portfolio, for m assets with relative prices X{, is defined by » i = ( - , - , . . . , - ) mm m where the wealth at time k Sk is 1=1 k Sk(b) = TTb'x, 1=1 Following this definition, the total wealth resulting from the Universal Portfolio i s *=i The main result of his paper first paper is the following: Supposing that for m assets, which relative prices e [a,c]m (a and c strictly positive constants), the exponential growth rate Wn(b) converges toward W(b) strictly con-cave, that the third derivative of this W is bounded on B, that this same W achieves its maximum at b* in the interior of B, and that J* and b* converge respectively toward J* and b*. Then This tells us that the wealth from the best constant rebalanced portfolio at time n, S*, can be approached by the one from his Universal Portfolios within a factor. Therefore, asymptotically, the two exponential growth rate are the same! 2 Chapter 1. Introduction In their 1996 paper, T. Cover and E. Ordentlich study again this Universal Port-folio adding some side information. While writing their paper they noticed once again the close link between the optimal portfolio selection and the data compres-sion theory. They used this time a very different approach. While, in his 91 paper, T. Cover used Laplace's method of integration and supposed that the relative prices were strictly positive, this new paper doesn't assume anything about the stock vectors. Furthermore, the Universal Portfolio becomes the jU-weighted Universal Portfolio defined by b,- = b,-(xI-_i) = - r - — j r , i=l,2... where S is defined as in his first paper and dn(b) = 1. One can see that this portfolio depends strongly on the distribution d\i. Two cases are actually studied. The first case is when fx follows a uniform Dirich-let distribution, meaning Dirichlet(l,l,...,l). The second case uses a distribution Dirichlet(i,i,...,i). T. Cover and E. Ordentlich find similar results as in Cover's 91 paper. For the uniform Dirichlet distribution and the side information yn, they obtain f ^ < n ( » ^ ) + i ) ( , - i ) <(n + l ) i ( m _ 1 ) , Where nr(yn) is the number of relative prices in information state r. While for the Dirichlet(|,|,...,j) and the same side information, they find bn{Xn\yn) r=\ <2*(n+l)* (m-1) /2. Since our goal is to get as close as possible of the wealth induced by the best . constant rebalanced portfolio, and if possible higher, it seems clear that the Dirich-let distribution Dirichlet^,5,...,j) gives better result than the uniform one. While those two distributions have been chosen because of their link with the data com-pression theory, we can see that there might be some optimality in choosing Dirich-let distribution Dirichlet(a,...,a). 3 Chapter 1. Introduction There are, however, a few issues in those two papers. First of all, as in many theoretical finance papers, and due to many practical reasons, the trading costs are not included. As one can imagine, the effect of trading costs on the final wealth can be catastrophic, especially on a rebalanced strategy! There are a few intuitive ways to deal with those costs as seen in Blum and Kalai (1997)[3], although they don't study the case of the DirichletCj,^,...,^). It seems, for instance, possible to subtract a percentage of each trade. Another extension would be to calibrate the same model to find the optimal data frequency. Clearly, intraday data wouldn't work on this algorithm but daily or monthly could be reasonable. There is, however, a bigger issue. The convergence of the Universal Portfolio to-ward the best constant rebalanced portfolio is only asymptotic. Therefore, one should study how fast it could be approached. From equation (8), we can see that 5„(x„|yn) > 5 " ( X " W 2 * n ^ , K ( v „ ) + i ) ( " 1 - 1 ) / 2 For instance, for a > 1 and letting X i , X 2 , . . . = (*1 ,1 ,*1 ,2) , (*2,1,*2,2), (*3,1.*3,2), ( * 4 , 1 , x 4 , * ) i — =(l,a),(l,i),(l,a),(l,-),... a a be the sequence of stock market vectors, where xn represents the bank and xa a simple stock jumping from a to 1/a every day. Using the following 2 states side information: v = J ! . n!=i*/2<nj=,*n y ' 12, n | = i * / 2 > r & n y , = l . We obtain the side information y = 1,2,1,2,... which tells us when the running price of the stock exceeds the cash. Applying now equation (9) to this type of side information, we get 5„(x„|y„) - 2 2 ( " + 1 ) 2 ( 2 - l ) / 2 5 " ( X " l ; y " ) 1 S*n(xn\yn) 4(f + l) 4 Chapter 1. Introduction Unfortunately, we can see that the convergence rate is not really good. In a more general case, the wealth of the Universal Portfolio would be only increased approx-imately by y/nkl<m~^ when adding k states of information. Finally, by adding other assets or more periods the convergence will get also better but the computational time will explode. In this paper, we will first write the complete proof of the main theorem in T. Cover and Ordentlich's paper, Theorem 3. In chapter 3, we will discuss the best constant rebalanced portfolio and show how to compute it in a theoretical framework. The chapter 4 will be dedicated to the generalization of the method to compute the Uni-versal Portfolio for a number of assets bigger than 2 and for a general Dirichlet distribution. Finally, we will proceed to an extensive numerical analysis in chap-ter 5. We will look at the results of the Universal Portfolios with both simulated and real data, we will compare them to the best constant rebalanced portfolio, we will look at the case where there is side information, and finally we will give an answer to the question of the optimal Dirichlet distribution. The conclusion will summarize the results and the issues. 5 Chapter 2 Complete version of the main result from Universal Portfolio with Side Information. While reading "Universal Portfolios with Side Information", we noticed that some results were given without proof. Because some of those results didn't look obvi-ous to us and because they are crucial for later analysis, we decided to write a more complete version of those proofs. In this section, we will write the complete proof of theorem 3. Doing so will actually help us understanding how to built this Universal Portfolio and how to generalize the concept for any number of assets and for any Dirichlet distribution as we will do in Chapter 4. Theorem 3 : For p equal to the Dirichlet(j,^,...,^), the Universal Portfolio with Side Information satisfies for all n, x„ G yn G {1,2, ...,k}n and where nr(yn) is the number of times yj = r in the sequence of yn. Before to start the proof, we want to make a brief recall of some of the Lemma's proved completely in Universal Portfolio with Side Information. We will only use Lemma 2 and 5. Lemma 1 is used to prove Lemma 2 and Lemma 3 and 4 are used for the proof of Lemma 5. s;(xn\yn) Sn{x„\yn) <2kY\{nr(yn) + \)^ <2\n+\) r=l 6 Chapter 2. Complete version of the main result from Universal Portfolio with Side Information. Lemma 2 : For the ju-weighted Universal Portfolio, S*({xi:yi = r}) ^ W=ibj, < max • — §n({xi:yi = r})- J" / f i nLlMM (b) where the maximum is over the set of sequences of indices j" € { l , . . . ,m}" and b* = (bi, ...,bm)' is the best constant rebalanced portfolio for the sequence x" with side information y,- = r. Lemma 5 : For all non-negative integers n\,...,nm, YI?=\ nr = Hf) n m r(n f+j) - r(m)r(n+l) r7ji%rui-f^) K2' K 2' where H(v,,...,vM) = H(v1(;"),...,vm(/)) = , -v r (;")log(v r (/)) and vr = * Remark: We noticed a small notation mistake in the proof of Lemma 5. It should be written [y] instead of [^ J in equation 101. We can now write the proof of Theorem 3. Complete Proof of Theorem 3 : First of all, we can express the wealth obtained by the Universal Portfolio with Side Information as * r Sn(xn\yn) = Y[ Sn(b\r)dp(b) r=l J b where JB Sn (b| r)djj. (b) is the wealth acquired by a Universal Portfolio operating on the subsequences of x" corresponding to the times i where the state of information y, is r. This means that we can look at each state of side information independently. Hence, we can write S*n(*n\yn) A Sn(b*\r) Sn(Xn\yn) l\ fBSn(b\r)dp(b) ^f[S*({xi:yi = r}) l\Sn({xi:yi = r}) where S*({xt: y, = r}) and Sn({xi: y,- = r}) represent the wealth obtained by re-spectively the best constant rebalanced portfolio and the Universal Portfolio on the 7 Chapter 2. Complete version of the main result from Universal Portfolio with Side Information. subsequence of stock vectors where the side information state is r. Furthermore, we can apply Lemma 2 on every sub-Universal Portfolio, and we can write s*n({xi:yi=r}) < m a : c rau*; U{* ••* = '})- i" / * I l ? = i M M ( b ) From Elements of Information Theory by T. Cover and A . Thomas (p.281) and Information Theory: Coding Systems for Discrete Memory-less Systems by I. Csiszar and J. Korner, we can prove that the numerator can be upper-bounded by f\b*. < 2 - « « ( v i . - . v - ) where, again, H(v 1 , . . . ,v m ) = H(v1(/),...,vM(/)) = Z?=x -v r ( / ) l og(v r(/)) and Now, we have to look at the denominator, let's call it D , and this is where a re-sult is given which, we believe, should be explained in more detail: /n TY —) m FK(/) + j) r(n+?)jL-i r(i) ' i = l * V' ' 2' r=l 1 \2' First of all, in order to compute this integral, we must choose the Dirichlet distribution. For now, we wil l look at the one from Cover and Ordentlich's paper, ie the Dirichlet(j,..., |).The distribution has a density given by where Y is still the gamma function. Hence, the integral D becomes Let jn e {1, ...,m} and n,- be the quantity of occurrences of i in j". We can then rewrite D as db 8 Chapter 2. Complete version of the main result from Universal Portfolio with Side Information. We need now to look at the end of the integral. To do so, let's write explicitly D r(f) I I — I r 1 i l rr-avr JB 1 "-1 M[r(i)] Now, by replacing bmby \ — b\ — — &m-i> and by making the integrals explicit, we obtain D = L b" -J bn::\-Hi-bl-...-bm-ly»-ubl...dbm-l The next step is to see a Beta function. To do so, we need to make the change of variable \ - b \ - ...-bm-i' Then D becomes .((1 -0(1 - * i - ...-bm-2))n*-*(\-bx - ...-bm.2)dbx...dbm-2dt r ( f ) r\ l r l — b\ —,..—bm-'i 1 / b"^... / C-T 5(l-^--"-^-2)"m- 1 + , ! m^l...^m-2 [rG)]m Jo The Beta function appears now clearly. Hence, applying the property ff-\\-t) Jo ^ d t = r(*)r(y) n*+y)' we can rewrite D as T(f) r ( n m + l ) r ( n m - 1 + i ) yi m_> [r(i)]m r(«m+«m_, + i) 7o 1 •" j ^ - - - b ^ b n ^ - \ [ x _ h _ _ K _2)nm-s+nmdbx d K _ 2 To obtain the final answer, the same steps have to be done m-3 times. One will change of variable to make the new Beta function appears and will replace the integral by the Gamma functions. The product of the Gamma functions r(n, + 5) will appear while the denominator will simplify after every iteration until the last 9 Chapter 2. Complete version of the main result from Universal Portfolio with Side Information, one. Finally, one wi l l obtain the following result: T ( f ) Y[%xT{nr + \) [r( i)] m r(E- = 1 n r+f) • This gives the final result: r ( f ) firK + i ) r ( n + f ) H r ( i ) Once we have D , we can go back to the first inequality 5 ^ : * = ^ < m a x £,({*,• :y,- = r}) ~ f / f i IT? = 1 Mf i (b ) 2-nH{v\,~,vm) <max-~~ j" D 2~nH{vl,...,vm) = max > " HT ) rrm I>r+^) r ^ + f y l l r = i r(i) Using Lemma 5, we obtain s*({xr.yi = r}) ^r&nn + i) §n({xt:yt = r})-r(*)nn+b <2(nr(y„) + l ) ^ <2(n+])mr~ since nr(yn) < n. Hence, we obtain the final result s ^ l y , ) =As;({xr.yi = r}) Sn{xn\yn) JJ £,({*,•:>,• = /•}) < n 2 K ( y „ ) + l ) ^ r=l < 2 * ( n + l ) ^ 10 Chapter 3 The Best Constant Rebalanced Portfolio, As seen in the introduction, the purpose of the Universal Portfolio is to approach the same exponential growth rate of wealth as the best constant rebalanced portfo-lio. Hence, we need to show how to compute this portfolio. It seems clear that for real data it is impossible to find an analytical formula for the proportions of wealth to invest. However, in the case of simulated data, we can try to find a closed form using simple Ito's calculus and Kuhn Tucker theorem. We will consider in this section a portfolio of N stocks following N independent geometric Brownian motions such that: where u, and o", are constants and z,)f are N independent standard Brownian mo-tions. Although we will give the result for N stocks, it can be easily changed for any number of stocks or bonds. Let St ( b \ , . . . , bN) be the wealth obtained by one dollar invested at time 0 in the con-stant rebalanced portfolio {b\, ...,bN), where bi represents the proportion of wealth invested in stock i. Then, the wealth process is dXi = fXjdt + Gidzi,t d^ S, = (61 u i + ... + bNliN )dt + b\G\ dzi, / + ... + bNcNdiN,t-Let us now consider the standard brownian motion Z, dZ, = b 10"i dz\,, + ... + bN ONdzNj 11 Chapter 3. The Best Constant Rebalanced Portfolio. Applying Ito's Lemma, we obtain df, = d{ln(St)) = ((fciMi + ... + bNHn) - \ib\a\ + - + b2Nol)dt + ^b 2<J 2 + ... + b2No%dZ, = bi(nidt + Oidzi,() + ••• + bN(nNdt + oNdzN,t) - ^{b2o? +... + b2Nol)dt = ^ l i +... + bNd-^L - Ub2a2 + ... + b2<y2)dt Hence, 1 X 1 X I S, = S0exp(-- {b\o2 + ...+b2No%)t+bi (log-^- + -<j2t) + ...+bN (log-^ + -ofc)) 2 A^o 2 A/v,0 2 We are now left with a simple maximization problem with 2*N+1 constraints. N maxS,(&I,...,£>JV) s.t. £ b , = l andO<bt<l. Since we are going to use the Kuhn-Tucker method, we need to transform our problem into a minimization N min -St(t)(bi,...,bN) s.t. ^ b i - \ =0 and bi>0 and l-&,>0, i=l....N. i=l The Kuhn-Tucker equation can be written as follows 1 X 1 X 1 KT = - S o exp(--(b\oi + ... + b2No%)t + bi(log-± + —of?) + ... + bN(log-^- + ~o2Nt)) 2 A\fi 2 AAf,0 2 N+N — ^ hg{i) — p{b\ + ••• + b^ — 1). g(i) being the inequality constraint #i. i=i Differentiating with respect to any fr,-, we obtain the following first order con-dition: dKT X' 1 — = (-haft + log-?- + -a2t)St + Xi- XN+i + ii = 0 dbj Xito 2 Let's now look at the possibilities for the Kuhn-Tucker coefficients. We need to find the combination of coefficients which wi l l verify all the Kuhn-Tucker con-12 Chapter 3. The Best Constant Rebalanced Portfolio. straints for i from 1 to N: bt>0 1 - bt > 0 N £ 6 , - 1 = 0 A,&, = 0 and Ajv+,-(l — bi) = 0 Xi > 0 and X^+i > 0 . In the case where A,- > 0 and X^+i > 0, we end up with bj = 1 — bi = 0, which is not possible. In the case where A,' = 0 and Ajv+i > 0 , we obtain that b,; = 1 , which leads to J^=l bj — l =N—l, which contradicts another constraints. It is the same problem when A, > 0 and AAF+, = 0. The only possibility left is A, = 0 and A#+,• = 0. Re-writing the first order conditions, we can get rid of the Kuhn Tucker coeffi-cients and we obtain: ^={-biaft+iog§L+yt)St+p=o dbj Xito 2. Hence, we obtain the equality biaft-log—1-X\fi 1 2 . 2 . 1 2 -aft = biaft - log-2- --aft 2 X-ifl Which gives us the form of bi as a function of b\. 1 aft hoh + log?±-log^+l-(o?-o?)t But since the sum of the fe,s equals 1 , then N i frl0f.+/0g^-/0g^-rJ(0f-0?). N <-2 " / i=2 ° i ' - / o g — + - (O- - CT, )t X-ifi X\fi I log^-log^ + \{af-a^)t 1 + I f = 2 ^ 1 3 Chapter 3. The Best Constant Rebalanced Portfolio. Hence b\=-b* =—j-1 YN 1 1 ~ *-«'=2 ~Sft or b\oh + log^-log^ + \{oi-ot)t Clearly, the sum of those b, wi l l be equal to one, since we used the constraint to compute them. However, one should check that the Kuhn Tucker constraints bi > 0 and 1 — fr, > 0 are true. This is, unfortunately, not necessary true and de-pends largely on the price values of the assets and their volatility. With a little bit of work, one could give an interval for log(X^) as functions of log(j^) so that the formulas work, but we won't write them in this paper. In the case of two assets, this result is still useful. One just have to take the maxi-mum between zero and the minimum between 1 and the proportion (max(0,min(l ,b\))). However, in the case of 3 and more assets, it is impossible to do this. Hence, i f the conditions on the assets prices are not fulfilled, one possibility would be to use computational strength by trying as many combinations of fe, as possible which is, unfortunately very cumbersome. 14 Chapter 4 Universal Portfolio Computation for M Assets and a General Dirichlet Distribution. Now that we know how to compute the best constant rebalanced portfolio, let's try to find how to compute the Universal Portfolio. In their paper, Cover and Or-dentlich give the result for 2 assets and for a DirichletC^,...,^). We would like to generalize the result for M assets, but we would also like not to give any precise Dirichlet distribution for now in order to be able to find out i f there is an optimal one. The wealth implied by the constant rebalanced portfolio can be written as n Sn(xn,b) =Y[b'xi i = i n = Y[(blXi,\ + - + bmXi,m) i=l ;'"e{l,...,m}"i=l n n—li n - / i — . . . - / m _ 2 n = E I - I M ,-*t Ii«r' ,-- t- ,( E / i =o/2=o / m - i=o y""Gr«(/i / m - i ) ' = i where b is still the wealth proportions invested in the M assets, x,- is the stock price relatives vector at time i , and Tn(l\,...,lm-\) is the set of all sequences jn with l\ l's,..., lm-\ m-l ' s and n —1\ — ... — / m _ i m's. Let's now defineX n(l\,...,lm-\) such that n Xn(h,...,lm-\) = Y\X'Ji y " e r „ ( / i , . . . , / m _ i ) ' = i 15 Chapter 4. Universal Portfolio Computation for M Assets and a General Dirichlet Distribution. Hence, n n-li n-l\-...-lm-2 s„(xn jb) = E E £ 6 / 1 ' . . . ^ : 1 1 ^ - ' ' - - * - ( x B ( / i , . . . , / m - i ) ) /, =Ofa=0 /m_i=0 We can obtain the wealth obtained by the Universal Portfolio at time n by integrat-ing Sn(Xn J b) = j Sn(xn,b)dp(b) /n n-l\ n-l\-...-lm-2 E E - I & / 1 ' . . .&i : ' 1 6sr , ' - -*"-»(x f l ( / , > . . . ) / M - i))rfu(b) /l=0/2=0 /m_i=0 du(b) /i=0/2= 71 n - / i n-/i-...-/m_2 = E I - E xn{iu...,im^) b\...bl--_\bi /! =0/2=0 /m-l=0 7 n n - / i n-/i-...-/ra_2 = E E " • H • X n ( ^ l , - - - , ' m - l ) C „ ( / l , . . . , / m _ l ) /, =0/2=0 /m_,=0 where Cn (l\,..., /m_ 1) is defined as C„(/,,...,/ m _,) = / & / 1 1 . . ; ^ : , , ^ - / ' - - * " - ' d M ( b ) By definition of the wealth proportion invested in the Universal Portfolio, we can write B"=!—7 T / f ib ' x .bdM^)) 5„_,(x._,) [ jri^lb'xibmdp({b)) Let's look now at the first term in the vector, let's call it v. /n—\n-\-l\ 7 1 — 1 — / 1 — / m _ 2 T i - l E E - E I I W * ( b ) ' •=n ; ,=n / r a _ 1 = 0 ^ - l g ^ , i = i :-/l-...-/m_l /,=0 Z2=0 71-1 71-1-/] 7 1 - l - / l - . . . - / m _ 2 = E E I /,=0 /2=0 /ra_,=0 71-1 71-1-/] n—1—/l —... — /m-2 = E E ••• E ^ 7 i - l ( A , - . - , / m - l ) C n ( ' l + l , . . . , / m _ i ) /,=0 /2=0 /m_i=0 h-----'m-2 r E Xn_x{lu...,lm_x) b\+\..bl-\bnm • ,=0 y : - l - / , - . . . - / m _ , </u(b) 16 Chapter 4. Universal Portfolio Computation for M Assets and a General Dirichlet Distribution. Similarly, we compute all the members of the vector and finally obtain Ln - l ^n-\-l\ i - i n - l - i /,=0 2-/2=o -2-/m_,=o b„ = 1 5 n - i ( x „ . Ln - l r - n - l - / i w i - l - / i /,=o2-/2=o •••Lim_i=o ''Xn-\(l\,...,lm-\)Cn(l\,...,lm-\ + I) E" 1 =0^" 2=0 •••L/m_,=o 2 A n - l ( ' l ) - - - ) ' m - l ) C n ( / l , . - . , / m - l ) We are now looking for some recurrence formulas for the terms C„ from the vector. More precisely, we would like to write Cn(l\,...,lm-\) as function of ~ l,h,—dm-\) Cn-\(h,h — l , - - - , ' m - l ) Cn-l(h,h,—dm-1 — 1) C „ _ l ( / l , / 2 , - - ) ' m - l ) Let us now compute Cn(l\,...,lm-\). We are going to use the same approach as in Section 2. However, this time, we are in the case of a distribution Dirichlet(a,...,a) but still M assets. In Section 2, we have shown that r « Y\ — ) m » = / f l n M M ( b ) = ^ n r(f) A r ( « r ( . m * ) However, this result is true when p follows a distribution Dirichlet(^,...,j). By using the same proof but letting ju following a Dirichlet(a,...,a), we can show that d = j n M M ( b ) = f ( — j n r ( a ) But, D=f flMM(b) ^1=1 = Jb\K..b%z\b£ h--"- l~- ldti(b) = C „ ( / i , . . . , / m _ i ) supposing again there are l\ l ' s , . . . , / m _ i m - l ' s a n d n —1\ -lm-i m's. 17 Chapter 4. Universal Portfolio Computation for M Assets and a General Dirichlet Distribution. Hence, nam) " r (n r ( j") + a) c - ( / i - " . / - i ) = r ( n + am)JLi^ r(5) T(am) "p^n/i + a) r ( / i - / i - . . . - / m _ i ) r ( n + a m ) v Ai r(a) ' r(o) Now, we want to compute the other C's and try to write them as a function of Cn(h, •••Jm-\)- We wil l only have to compute one to see how all the others can be written. Let's look at Cn-\(l\ — l , . . . , / m _ i ) . By definition C„_, ( . , - l , . . , / m _ i ) = Jblry.-bt\bmh~--lm-ldli(b) The index of C represents the sum of all the exponents of the b's, n-1. We apply, once again, the same result as before and obtain the following: r n-i i ^- r(am) r( / t + a - 1 ) ^ 1 r(/,-+ a ) , r ( w - / ! - . . . - / „ , - , ; c „ _ u . i ^-^-x)-T{n + a m _ x ) r ( a ) {[[ r ( a ) ) r ( a ) n+am-l Hence, Cn(h,---,lm-l) = —. rC„_l ( / l - l , / 2 , . . . , / m _ l ) n + am — 1 Similarly, we obtain the following formulas: Cn(h,---,lm-i) = _ T - rC„_](/i,/2 - l , - - , / m - l ) n + am — 1 C „ ( / i , . . . , / m - i ) = m 1 — C„_i(/i , /2, . . . , / m -l - 1) n + am - 1 Cn(h,---,lm-\) = ' , „ "* C„_i(/ i , /2 , . . . , /m-l) n + am - 1 Once we know how to get the C's, we would like to know how to compute the X ' s . For Xn(/],..., lm-1) we can find the recursion Xn(h, •••,lm-\) =Xn,\Xn-\(l\ — 1, •••Jm-l) + ••• + Xn,m-\Xn-\{h, •••Jm-l ~ 1) +Xn,mXn-1 ( ' l , • • •, lm-1) 18 Chapter 4. Universal Portfolio Computation for M Assets and a General Dirichlet Distribution. when 1 < /, < n — 1. However, at the borders, we obtain XB(0>...,0)=*n^_i(0,...,0) X„ («,..., 0) =xn,\Xn^(n- 1,...,0) X„(0,...,«) =^„,m_iX„_i(0,... ,n- 1) Let us now define Qn(h,---,lm-1) as Qn{h,---,lm-l) = X„ ( / ] , . . . , / m _ i ) C „ ( / i , . . . , / m _ i ) . We can now express the wealth from the Universal Portfolio as n-\n-l-k n-\-l\-...-lm-2 §n=Z' E - E Qn(h,...,lm-l). ;,=o i2=o /m_,=o We can also use the previous recursions to express the proportion of wealth b's as sums of Q's. Hence, the only thing one need to find when trying to compute the portfolio is Q. The portfolio can be now written as bn 5 „ _ i ( x „ _ i ) Ln-l v n - l - / i v n - l - < i - . . . - f m - 2 l,+g n /, i \ /,=02-/2=0 •••Llm_)=0 n + B m - l ^ - H ' l ' - ^ m - i ; En-1 v " _ 1 _ ' l v i - l - l | - - - l i » - 2 <m-l+g f\ (j i \ /,=o2-/2=o — 2-/m_,=0' n + a m - l ^ - H ' l ' - . ' m - U Lrt-1 r>n-l-/i /, =0^/2=0 • "M m _,=0 n-\-l\-...-lm-2 n-l\ -lm-l+0t Qn-l(hj---Jm-l) /i+am-1 Finally, using the recursions both from C's and X's, we can write the recursion for 19 Chapter 4. Universal Portfolio Computation for M Assets and a General Dirichlet Distribution. Q: Qn(h,---,lm-l) =-*n , l—7- 7Qn-l(h ~ 1, • • • , l m - \ ) + n + am — 1 lm-\ + a -1 . . + xn,m-l — 7-(2n-l{h,--,lm-\ ~ ij n + am — 1 n - / i - . . . - / m - i + a - l . . n + am — 1 Qn(0,...,0) = * „ , m n / " a ~ 1 . Gn -1 (0,..., 0) n + a m — 1 Qn(n,...,0) =xn,\ n + ®— l—Qn-\{n- 1,...,0) n + a m — 1 <2„(0,...,n) = ^ „ , w - i 1 6 n - i(0 , . . . ,w-l) . TX I CCftT 1 As we can see, it is really helpful to use Q instead of C and X since we are sig-nificantly decreasing the time of computation. However, once the formulas for M assets written, it seems clear that the method becomes very time consuming due to the combinatorial explosion. We wi l l give an example of this in the numerical analysis part. 20 Chapter 5 Numerical Analysis. We have now all the tools to do an extensive numerical analysis. We will look first at some simulated data then we will study the universal portfolio's properties on some real data and we will discuss the results in a practical approach. 5.1 Simulated Data We study first the cases of 2 assets - 2 stocks and, 1 stock and 1 bond. The stocks and the bond will follow the processes defined in Chapter 3. In the case of 1 stock and 1 bond, we use the parameters u = 0.03, a = 0.06 and r = 0.03. In the case of 2 stocks, we use the parameters Ui = 0.04, G\ = 0.01 for stock 1 and u 2 = 0.05, 0"2 = 0.1. The sample size is 3000 intervals. We compute the Universal Portfolio using the recurrence method from T. Cover and Ordentlich's 96 paper and the Best Constant Rebalanced Portfolio using the formula given in part 3 in the case of two assets. 5.1.1 Exponent ia l G r o w t h Rate o f Wea l th . For both portfolio compositions, we would like to verify the main result of T. Cover's papers, ie the exponential growth rates of wealth are asymptotically con-verging. On Figure 1, one can observe the wealth obtained by the Universal Portfolio (blue) and the Best Constant Rebalanced Portfolio (red) in log scale in the case of 1 bond and 1 stock. One can see that in the first 1000 intervals, the Universal Portfolio's wealth oscillates around the BCRP's one. Then, the two wealth both increase but the BCRP's seems to get larger and larger than the Universal Portfolio's one, al-though still very close. 21 Chapter 5. Numerical Analysis. •_• '-Wealth'Implied by UP T BCRP (log scale) # Intervals Figure 5.1: Wealth obtained using the Universal Portfolio or the Best Constant Rebalanced Port-folio. Portfolio composed of 1 simulated stock and 1 simulated bond. On Figure 2, we show the difference between the two exponential growth rates of wealth. To obtain these, we simply take the log of the wealth obtained by each portfolio and we divide by n, the interval number. We can clearly see that Cover's results are verified. Indeed, the difference BCRP - UP is at first very large but con-verges to zero with n (the scale of the graph being 10-3). This result is even more observable in the case of two stocks. On Figure 3, we can again see the UP oscillating in the first 500 intervals but we can also clearly see that the two wealths are spreading linearly with the interval number for the rest of the sample. Figure 4, showing the difference of exponential growth rates of wealth again, illustrates the convergence discussed earlier. However, the difference doesn't seem to converge towards zero but towards 0.005 (increasing the sample size leads us to think that it converges very slowly toward zero). 5.1.2 Optimal Dirichlet Distribution We would like now to know if there is an optimality to find in the choice of the Dirichlet distribution. As said in the introduction, Cover and Ordentlich used the 22 Chapter 5. Numerical Analysis. Convergence of UP.andBCRP's Growth Rate of Wealth Figure 5.2: Convergence of the Universal Portfolio's exponential growth rate of wealth toward the best constant rebalanced portfolio's one. Portfolio composed of 1 simulated stock and 1 simulated bond. -Universal-Portfolio; v B e s t C o n s t a n t R e b a i a n c B d Portfolio Figure 5.3: Wealth obtained using the Universal Portfolio or the Best Constant Rebalanced Port-folio. Portfolio composed of 2 simulated stocks. 23 Chapter 5. Numerical Analysis. Convergence of UR and BCRP's Growth Rate of Wealth 0.035 r r— : 17 T. r 0.03 0':025 . 002 0 • -0.005: -" ° 0 1 0 500 10C0 1500 2000 2S00 3000 #Intervals Figure 5.4: Convergence of the Universal Portfolio's exponential growth rate of wealth toward the best constant rebalanced portfolio's one. Portfolio composed of 2 simulated stocks. Dirichlet uniform and the Dirichlet^,...,5). We are studying the same universal portfolio as in Chapter 4 (with a general Dirichlet(a a)) comparing the final wealth as a function of a . We choose to compute the portfolios for 10 values of a , starting with 0.1 and increasing by 0.1. Looking at Figure 5, we can obviously see that the wealth is different depend-ing of the choice of Dirichlet distribution. Even though the difference is small on the graph, on some large portfolios and looking at a large sample, the difference can be significant. It seems clear from the picture that the optimal Dirichlet distri-bution is not the one Cover and Ordentlich chose (Dirichlet(^,..., | in red) but the Dirichlet(0.1 0.1) in black. To be sure of this, we chose to do a Monte Carlo simulation with 10000 iterations and compute the average wealth for the 3 values of a that interest us the most: 0.1, 0.5 and 1. We obtained the following results: This is a stunning result, since by a 1 0.5 0.1 Average Final Wealth 0.4659 * 1 0 4 1 1.0487* 10 4 1 2.6208* 10 4 1 Table 5.1: Choice of the Optimal Dirichlet Distribution choosing a Dirichlet distribution (0.1,...,0.1), the final wealth is 250% larger than 24 Chapter 5. Numerical Analysis. 25 Chapter 5. Numerical Analysis. for the one considered in T. Cover's paper! 5.1.3 3 Assets Analys is . As we have already stated, the convergence rate gets better when we add more as-sets. However, to add more assets has a very big drawback since the computation time increases exponentially. In order to show this, we are now applying our algorithm to a portfolio with 3 simulated assets. We use the algorithm described in Chapter 4. However, we avoid any optional computations such as the Best Constant Rebalanced Portfolio or the proportions of wealth that one would invest in the Universal Portfolio. We are fo-cusing here on obtaining the wealth for every interval, and it will be enough to show our point. On a period of 2000 intervals, it took 0.2810 seconds to compute the wealth at each interval for the Universal Portfolio with 2 assets. However, in the same con-ditions, it took 292.039 seconds to compute the same thing in the case of 3 assets. One could question the efficiency of the code. This is why we would like to present the main part of the algorithm. First, we can observe the code for the case M=2 assets: % Matrix Q : N- l l ines (k), N- l columns (n) Q = zeros(N- l .N- l ) ; Q ( l , l ) = 1; for i = 2 : 1 : N- l Q ( l , i ) = y ( i , l ) * ( ( i - l ) - l / 2 ) / ( i - l ) * Q ( l , i - l ) ; end for i = 2 : 1: N- l Q ( i , i ) = x ( i , l ) * ( ( i - l ) - l / 2 ) / ( i - l ) * Q ( i - l , i - l ) ; end for n = 3 : 1 : N- l for k = 2 : 1 : n - l Q(k,n) = x ( n , l ) * ( ( ( k - l ) - l / 2 ) / ( n - l ) ) * Q ( k - l , n - l ) . . . + y ( n , l ) * ( ( ( n - l ) - ( k - l ) - l / 2 ) / ( n - l ) ) * Q ( k ) n - l ) ; end end */„ Wealth S 26 Chapter 5. Numerical Analysis. SUP = zeros(N-l ,1); for n = 1 : 1 : N- l for k = 1 : 1 : n SUP(n,l) = SUPGi.l) + Q(k,n); e n d e n d Some loops could actually be simplified, but the overall computation time looks good enough. However, in the case of 3 assets, the code is the following: Qtmp = zeros(N-l,N-1); QtmpU.l) =1; SUP = zeros(N-1,1); SUP(l . l ) = 1; Q = zeros(N-l ,N- l ) ; for i = 2 : 1 : N- l Q ( l , l ) = z ( i , l ) * ( ( i - l ) - l / 2 ) / ( ( i - l ) + l / 2 ) * Q t m p ( l , l ) ; Q ( i , l ) = x ( i , l ) * ( ( i - l ) - l / 2 ) / ( ( i - l ) + l / 2 ) * Q t m p ( i - l ) l ) ; Q ( l , i ) = y ( i , l ) * ( ( i - l ) - l / 2 ) / ( ( i - l ) + l / 2 ) * Q t m p ( l , i - l ) ; for 11 = 2 : 1 : i for 12 = 2 : 1 : i-11 Q(ll ,12) = x ( i > l ) * ( ( ( l l - l ) - l / 2 ) / ( ( i - l ) + l / 2 ) ) * Q t m p ( l l - l ) 1 2 ) + y ( i , l )* ( ( (12- l ) - l /2 ) / ( ( i - l )+ l /2 ) )*qtmp( l l ,12- l ) . . . + z ( i , l ) * ( ( ( i - l l - 1 2 - l ) - l / 2 ) / ( ( i - l ) + l / 2 ) ) * q t m p ( l l , 1 2 ) ; SUP(i . l ) = SUP(i , l ) + QUI,12); e n d e n d Qtmp = Q; e n d As one can see, there is not much to enhance. However, we can clearly give an answer to the computation times difference. It seems clear that in the case of 2 assets, just one loot is necessary for 11. But, in the case of 3 and more assets, we must execute, for every interval, M - l loops which explode our computation time... Hence, the performance of the algorithm is severely impacted by the time needed to compute the wealth and the proportions of wealth to invest on every asset. 27 Chapter 5. Numerical Analysis. 5.2 Real Data. We would like now to confirm the previous results on some real data. We selected several stocks and downloaded their daily prices from Yahoo Finance. The data set range is from January 1st 1997 to December 31st 2006, which represents approx-imately 2500 values. We computed the Universal Portfolio the exact same way as with simulated data. However, to compute the best constant rebalanced portfolio, we had to compute the portfolio for 0 < b < 1, increasing b by 0.1 at each step, and picked the value of b which leads to maximum final wealth. Obviously, this method is easy for 2 assets but becomes more time consuming for 3 assets and more. 5.2.1 Exponen t ia l G r o w t h Rate o f Wea l th . On Figure 6, one can see the wealth generated by the two portfolios composed of I B M and M S (Morgan Stanley). We start with an initial wealth of 100. On Figure 7, we show the exponential growth rate of wealth. As seen in the analysis of simulated data, we note that the growth rate of the universal portfolio oscillates around the best constant rebalanced portfolio's one. Then, after around 500 intervals, they seem to follow the same pattern. This is obviously also the case for the wealth. The difference of final wealth being only due to the different paths followed by the two portfolios in the first 500 intervals. We justify this by looking at figure 8 where we show the difference of the growth rates. One can clearly see that this difference is converging toward zero which confirms the main result of T. Cover's paper. Note : The results for other stocks being very similar, we don't show any fur-ther graphs. 5.2.2 O p t i m a l D i r i ch le t D i s t r i b u t i o n Let's look at the optimal Dirichlet distribution now. As we did for simulated data, we compute and plot the wealth for different Dirichlet distributions. As we have seen before, the difference on Figure 9 doesn't seem too big and we have to zoom in to observe any results. However, looking at the final wealth, we can now observe that the results are not the same. The wealth generated by the Universal Portfolio with a Dirichlet(0.1,...,0.1) for a composition I B M - M S is smaller than the one from the choices made by T. Cover. We show the results on Table 2. Looking at some other portfolio composition, we observe the same results. However, it seems 28 Chapter 5. Numerical Analysis. Wealth Impl.od by I P • BCRP -Universal Portfolio;. •Best Constant Rebalanced Portfolio 5500: tooo. 1500 # Intervals' Figure 5.6: Wealth obtained using the Universal Portfolio or the Best Constant Rebalanced Port-folio. Portfolio composed of 2 real stocks: IBM and MS (Morgan Stanley). 1^0 1 0 0 Wealth Implied by UP • BCRP (log-scale) -Universal Portfolio •/BestConstantRebalancedSbrtfoiio^  520 ,1000. 1500 .^Intervals :2000 .2500 3020 Figure 5.7: Exponential growth rate of wealth of Universal Portfolio and of best constant rebal-anced portfolio. Portfolio composed of 2 real stocks: IBM and MS (Morgan Stanley). 29 Chapter 5. Numerical Analysis. X IQ- 3 Convergence of UP and BCRP's Growth Rate of-Wealth 1500 # Intervals 2500 13000? Figure 5.8: Convergence of the Universal Portfolio's exponential growth rate of wealth toward the best constant rebalanced portfolio's one. Portfolio composed of 2 real stocks: I B M and M S (Morgan Stanley). a 1 0.5 0.1 Average Final Wealth 118.76 111.58 97:62 Table 5.2: Computation times for 2 and 3 assets. 30 Chapter 5. Numerical Analysis. to be a general result that in the case of real data, the Dirichlet(l,...,l) works the best. 10 f,, : Wealth Obtained by UP for Several Dirichlet Distributions vs. BCRP (log scale) io'L D(1, ,1) D(0.5 0 5) D(0 1, ,0 1) BCRP 20 40 60 80 Last 115 Intervals of Sample Data Set 100 120 Figure 5.9: Optimal Dirichlet distribution choice for portfolio composed of 2 real stocks: IBM and MS (Morgan Stanley). 5.2.3 Universa l Por t fo l io W i t h o u t Side I n f o r m a t i o n vs W i t h Side I n f o r m a t i o n The last problem we would like to study is the difference between the portfolio without side information and with side information. Intuitively, it would seem rea-sonable to think that if one has some side information, the results should be better. We would like to observe this. 31 Chapter 5. Numerical Analysis. First of all, we need to decide what side information we could have. In this pa-per, we wi l l look at the moving average of the returns over a certain number of intervals I (20 intervals for instance). We compute those for each price vector and choose to define the state 1 of side information as a day when the moving average on the I previous days of the first assets is superior than the one of the second as-set. State 2 wi l l be the opposite. Once we have this vector of side information, we cut the 2 relative prices vectors in 2 vectors depending on the state of information. Hence we obtain, 2 vectors for state 1 and 2 vectors for state 2. As we have seen earlier, we can compute the Universal Portfolio with Side In-formation as a multiplication of Universal Portfolio without Side Information at state i . Hence, we compute the Universal Portfolio for every couple of vectors and multiply the final wealth together to obtain the final wealth of the Universal Port-folio with Side Information. We compare the wealth of those 2 portfolios on Figure 10. The results are sur-Unneraal Portfolio IBM - MS with and withoul Sid* Information — with SI •without SI Figure 5.10: Wealth in log scale obtained by using a Universal Portfolio with or without side information. prising since the Universal Portfolio without Side Information seems to perform better than the one with side information. The graph shows the results for I B M and M S but the results are true for many other stocks (MSFT, W M T , etc). To be more 32 Chapter 5. Numerical Analysis. precise, we haven't found any combination of stocks that performs better with side information that without. One of the possible reason is certainly the type of infor-mation we gave. We have used different types of side information without being really successful. 5.3 Discussion. As we have explained in the introduction and shown in the numerical analysis, the Universal Portfolio needs some time before to have its exponential growth rate of wealth converging toward the best constant rebalanced portfolio's one. The order of time in our computations was 500 intervals, days in our case. However, as said in the introduction, the convergence rate increases slowly with the number of assets and of states of side information. Hence, we might be able to make the convergence rate reasonable (less than a year) by increasing significantly the size of the portfo-lio. Unfortunately, this has a very big drawback, since the computations become very important and time consuming. Another big issue can be also seen in the formula 5nC*n|y„) > 4(nl+ljS*n(Xn\yn)-Indeed, S* (xn |y„) being the wealth from the best constant rebalanced portfolio, for a large number of period, let say 1000, this wealth would be at worst 2000 larger than the one from the Universal Portfolio! This result means that even if the exponential growth rates of wealth converge, the final wealth from the Univer-sal Portfolio could be way lower than the one from the best constant rebalanced portfolio. This formula was an example in the introduction, but we can clearly see on the first figures of the numerical analysis that this result is true. This obviously makes sense since no one would expect a portfolio to beat another one in hindsight. Finally, the main issue is the trading costs. This problem has been quoted many times but we can add something. Looking at the figures, we can see that in the case of 2 stocks, the Universal Portfolio's wealth pattern is very similar to the pattern of the best stock in the portfolio. Hence, the difference of scale in the wealth between a simple buy and hold strategy and the Universal Portfolio is small. One would say that we don't know before the winning stock and that the Universal Portfolio does a good job picking the right one. But, in the case of the Universal Portfolio, we are rebalancing every day! Of course, ignoring the trading costs make us think this strategy could possibly be rewarding. However, as soon as the trading costs are 33 Chapter 5. Numerical Analysis. counted, the Universal Portfolio leads to bankruptcy in a few time intervals... We observe usually the exact same problem with rebalancing strategies, as Kelly bets, especially in low volatility markets. 34 Chapter 6 Conclusion. In this paper, we have analyzed the Universal Portfolio concept with and without side information. First, we wrote the full proof of the main theorem from T. Cover's 96 paper in order to be able to do the computations. We extended the model for a general Dirichlet distribution and wrote the full algorithm for the case of a portfolio with more than 2 assets. The idea to approach the BCRP's exponential growth rate of wealth asymptoti-cally is attractive but as we have seen in the numerical analysis, to approach the same growth rate doesn't mean to approach the same wealth at all. We have shown that the difference in final wealth between the UP and the BCRP can be huge. This result decreases strongly the viability of the strategy. Furthermore, the rate of con-vergence is quite slow, approximately 500 days for our data set. We also have shown that this gap of wealth can be decreased by choosing the right Dirichlet distribution. However, our results don't seem consistent when go-ing from a simulated data set to a real data set. Indeed, for all the stocks we tried, the Dirichlet(l,...,l) seems to be the optimal one, whereas the Dirichlet(0.1,...,0.1) seems to be optimal in the case of simulated data. Finally, the biggest problem is that the wealth obtained depends mainly on the size of the first investment. Indeed, the general pattern of the wealth when using the UP is very similar to the one when investing on the winning stock. If the stocks in the portfolio are winning then the UP will be winning, if they loose, the UP will loose too. However, in our analysis, we didn't even take account of the transaction costs which would clearly pull even downer the final wealth, especially in this type of rebalancing strategy. Nevertheless, the UP seems to be efficient in determining the winning stock but an experienced trader could probably see it without the use of the UP. 35 Bibliography [1] J. L . Kelly, Jr., " A new interpretation of information rate", Bel l Syst. Tech. J., vol. 35, pp. 917-926, 1956. [2] T. M . Cover, "Universal Portfolios", Math. Finance, vol. 1, no. 1, pp. 1-29, Jan. 1991. [3] T. M . Cover and E. Ordentlich, "Universal Portfolios with Side Information", IEEE Transactions on Information Theory, vol. 42, no. 2, pp. 348-363, March 1996. [4] A . Blum and A . Kalai, "Universal Portfolio with and without Transaction Costs", Annual Workshop on Computational Learning Theory, pp. 309-323, 1997. [5] P. H . Algoet and T. M . Cover, "Asymptotic optimality and asymptotic equipar-tition propoerties of log-optimum investment", Ann. Probab., vol. 16, no. 2, pp. 876-898, 1988. [6] R. Bel l and T. M . Cover, "Competitive optimality of logarithmic investment", Math. Oper. Res., vol. 5, pp. 161-166, 1980. 36 

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