The Universal Portfolio of many Portfolios

The Universal Portfolio of many Portfolios

Jakob J. Bosma

September 27, 2011

Supervisor: Prof. Theo K. Dijkstra

Abstract

The main aim of this investigation is to ascertain the impact of the number of assets on the performance of Cover’s (1991) universal portfolio. The strategy even-tually yields the same exponential growth rate of wealth as generated by the best constant rebalanced portfolio. However, for a finite number of investment periods the rates can differ substantially. We derive heuristically that the universal portfo-lio depends on the first two sample moments of the returns on asset prices, such that the strategy bears some resemblance to the optimal mean-variance strategies. In this light, the naive equal weighted portfolio is considered to be a competi-tive benchmark to assess the universal portfolio’s total return across five empiri-cal datasets for a finite timeframe. Results indicate that the universal portfolio’s performance, in terms of achieved total return, increases as the number of assets increase, and outperforms the other considered strategies. In the presence of trans-action costs these results continue to hold across the considered datasets.

Keywords: Universal portfolio, performance testing, exponential gradient; JEL: C58, G11.

∗_{Student nr: 1529951. Affiliation: University of Groningen, Faculty of Economics & Business, PO}

I.

Introduction

Since the introduction of Cover’s 1991 universal portfolio, the performance of this portfolio strategy received widespread academic attention on various aspects. The main aim of this investigation is to evaluate the universal portfolio’s performance when the investor can apply the strategy to a large number of assets. This strategy marks two key features in light of the number of assets available for investment that serve as the main motivation to conduct this study.

First, the guaranteed lower bound of the universal portfolio’s generated final wealth is the mean value line index.1 Notwithstanding this result, there exist more naive au-tomated strategies that may also achieve a higher total return for a large number of assets. For instance, DeMiguel, Garlappi, and Uppal (2009) find that the simple rebal-ancing strategy with equal weights assigned across assets is hard to be outperformed by more sophisticated mean-variance strategies when the number of assets is large. This result is primarily attributed to the fact that this naive strategy does not rely on any estimation approach, and has an inherent natural diversification of assets by as-signing equal weights. Hence, the excesses of returns have a smaller impact on total return as the number of assets become large.

Second, the universal portfolio is competitive with the best constant rebalanced portfolio as the number of trading periods for which the strategy is executed becomes large: The universal portfolio achieves eventually the same exponential growth rate as the best constant rebalanced portfolio. However, a caveat should be added since the rate of convergence is similar only for a "large" number of trading periods. There-fore, for a “small" number of trading periods it is not certain whether the universal portfolio will eventually achieve the same exponential rate of growth implied by the

1_{This constitutes the geometric mean of wealth generated by portfolios that only invest in one of the}

best constant rebalanced portfolio. Additionally, the rate of convergence between the universal portfolio’s growth rate and that of the best constant rebalanced strategy will be lower for a larger number of assets (Cover and Ordentlich, 1996). The intuition underlying this is that it takes more time for a larger number of assets to determine the optimal portfolio strategy. The uncertainty revolving around the speed of conver-gence provides scope for more naive strategies to outperform the universal portfolio in a fixed time frame.

In addition to the universal portfolio the automated strategy of Helmbold, Schapire, and Singer (1998) is included, the exponential gradient. Helmbold et al. (1998) show that the exponential gradient strategy’s performance also achieves the same exponen-tial growth rate as the best constant rebalanced portfolio and is therefore competi-tive with the universal portfolio. However, the rate of convergence to the exponential growth rate of the BCRP is lower than that of the universal portfolio. The authors show, for a finite number of trading days and a large number of assets, that the expo-nential gradient may outperform the universal portfolio in terms of the growth rate of generated wealth. This result is confirmed empirically by Helmbold et al. (1998) and constitutes the motivation to include the exponential gradient in this empirical investigation.

Real data are used to evaluate the performance of this strategy. In particular, daily total return index data are obtained for a set of portfolio indices that cover industry portfolios and world indices. This data is obtained from Kenneth R. French’s (2011) website2. The considered time frame ranges from July 1963 through December 2010.

The results indicate that the universal portfolio outperforms the equal weighted constantly rebalanced strategy across all considered datasets in terms of total return. Additionally, as the number of assets increases the universal portfolio generates a higher total return. Similarly, the universal portfolio also outperforms the exponential gradient. Overall, results show that the universal portfolio’s performance increases as the number of assets increase. This applies for both daily, weekly and monthly rebalancing.

This paper is outlined as follow. Section II provides a review of the portfolio strate-gies, followed by a description of the data in section III. The results of the portfolios’ performances are reported in IV. Concluding remarks are presented in V, and deriva-tions and two heuristic proofs are appended in section VI.

II.

Review of portfolio strategies

Consider m assets for investment over a period of n "days"3. The investor’s portfolio choice on a particular day is characterized by an array b that indicates the distribution of the investor’s wealth over all m assets. Furthermore, it is assumed that the investor can not go short on an asset and the elements of the array sum to one. This imposes the restriction e0_jb ≥ 0, ∀j ∈ {1, ..., m}, where ej denotes a columnvector for which

the jth element is equal to one and the remaining elements equal zero. In addition

ι0b=1, where ι denotes a columnvector of ones. Taken together these restriction form

2_{Website: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/index.html.}

the simplexB := {b|b∈ Rm_{, ι}0_{b=1, b}

j≥0 ∀j∈ {1, ..., m}}.

Let xi ∈ R+m denote a column vector where the elements are the assets’ ratio

of closing to opening price on day i. The elements of xi are price relatives or factor

changes in the assets’ prices. Additionally, let the m×n matrix xn = (x0, x1, ..., xn)

denote all price relatives up and until day n. Without loss of generality it is assumed that the price relatives of the m assets at the start of the first day, x0, are equal to one.

Hence, the price of asset j on the nth day is

Sn(ej, xn) = n

∏

i=1

e0_jxi.

Suppose that an investor pursues a strategy in which his wealth is rebalanced with respect to b on a daily basis across the m assets. The final wealth associated with this strategy is Sn(b, xn) = n

∏

If the investor is endowed with perfect foresight he could infer the optimal constant rebalancing of his wealth across the available assets. Let such an optimal portfolio be denoted by b∗, so b∗ =arg max b n

∏

and the realized wealth is defined by S∗n :=Sn(b∗, xn).

Cover (1991) reports the properties of the optimal constant rebalanced portfolio, and these motivate why this portfolio is of interest from a wealth maximizing perspec-tive. It is shown by Cover that wealth obtained by (2) exceeds the wealth obtained by investing only in the asset with the best overall performance.

gener-generated by the best constant rebalanced portfolio for large n (see for instance: Cover, 1991; Cover and Ordentlich, 1996; Ordentlich and Cover, 1998). Before this result is verified, the universal portfolio is introduced first.

A. The universal portfolio

The initial wealth allocation of the universal portfolio is uniform over the m assets. For subsequent trading days the portfolio is determined by the past performance weighted average of all possible portfolio strategies. Cover (1991) describes the intuition un-derlying the universal portfolio with an example in which a continuum of portfolio managers receive an equal amount of wealth and implement differing rebalancing portfolio strategies that exhaust all possible strategies. At the end of the day all gen-erated wealth is pooled and forms the wealth gengen-erated by the universal portfolio. Subsequently wealth is reallocated on the basis of past performance. Essentially, the universal portfolio constitutes a performance weighted average of all possible con-stant rebalancing strategies that are members of the earlier defined simplex. In this regard the concept performance is the total return achieved by the respective constant rebalancing strategy as defined by (1). A key element in this strategy is that it is free of any assumption with respect to the distribution of price relatives, which motivates the use of empirical data to assess the portfolio’s performance. The strategy is denoted by the adaptive rule (3), where k indexes the trading days between the first and the nth day: ˆb1 = 1 mι, ˆbk+1 = R b∈BbSk(b, xk)db R b∈BSk(b, xk)db ; (3)

where Sk(b, xk)is defined as (1) by substituting k for n. The update rule (3) generates

given by ˆ Sn(ˆb, xn) = n

∏

Cover and Ordentlich (1996) establish that the ratio of final wealth generated by the universal portfolio over the wealth generated by the best constant rebalanced strategy satisfies: ˆSn/Sn∗ ≥ 1/(n+1)m−1. This result is confirmed by Blum and Kalai (1999)

and is subsequently discussed.

The argument of Blum and Kalai (1999) revolves around the notion that portfolios that do not differ substantially are "near" to each other and are likely to perform also similarly in terms of generated final wealth. The key to their argument is that they show that the fraction of portfolios "near" to the optimal portfolio is considerably large. Blum and Kalai’s (1999) idea of a portfolio being near to the optimal portfolio, such as b∗, is best observed from the expression b = (1−ξ)b∗+ξz, where z ∈ B, and ξ a parameter for the degree of being near to b∗. Note that any portfolio with membership in the simplexB can be expressed as such. Furthermore, to show that near portfolios perform similar as the optimal portfolio it must hold that the final wealth generated by any constant rebalanced portfolio b is also at least (1−ξ)n times as large as that generated by the optimal constant rebalanced portfolio, since

Sn(b, xn) = n

∏

∏

∏

∏

by

Volm−1({(1−ξ)b∗+ξz : z∈ B}) = Volm−1({ξz : z∈ B})

=ξm−1Volm−1(B). (6)

Heuristically, the first equality is obtained by noting that (1−ξ)b∗ constitutes an Euclidean coordinate in the simplex space and does not contribute to the volume. The second equality is obtained from the volume of a regular simplex.4 From (6) it can be noted that a fraction ξm−1of portfolios with membership in the simplex Bperform at least (1−ξ)n as well as the BCRP. This property can be used to evaluate the perfor-mance of the universal portfolio.

Essentially, as Cover (1991) shows, the universal portfolio constitutes a perfor-mance weighted average of constant rebalanced portfolios with membership in B:

ˆ

Sn(ˆb, xn) = Eb∈B[S(b, xn)]. Combined with (5) it can then be stated that

ˆ Sn

S∗ n

≥Eb∈B[(1−ξ)n],

where ξ ∈ [0, 1]constitutes a random variable such that the expectation is to be evalu-ated over all possible constant rebalanced portfolios, for all deviations from the BCRP. The expectation can be transformed by means of an application of a moment identity for non-negative random variables:5

Eb∈B[(1−ξ)n] = Z 1 0 Probb∈B[(1−ξ) n _{≥v]dv =}Z 1 0 Probb∈B[ξ ≤ (1−v 1/n_)]dv.

4_{From a geometric perspective the volume of a regular (m−1)-dimensional simplex with edge}

length ξ, originating from the origin, can be denoted by ξm−1

(m−1)!

q

m

2m−1. The second equality is then

obtained by noting that{ξz: z∈ B}constitutes a simplex with edge length ξ.

5_{Let the random variable U ∈ R}+ _{have density f : R}+ _{→ [0, 1], and cumulative distribution}

function F : R+ _{→ [0, 1]. Then the first moment E[U] =} R∞

The density of ξ can be induced from (6), since for each ξ the proportion of portfolios with membership in the simplexB that perform at least (1−ξ)n as well as the BCRP is ξm−1. Hence, we have ˆ Sn S∗n ≥ Z 1 0 Probb∈B [ξ ≤ (1−v1/n)]dv = Z 1 0 (1−v1/n)m−1dv ≥ 1 (n+1)m−1, (7)

where the derivation of the last step is reserved for the appendix. Taking the logarithm of this expression and multiplying by 1/n allows for evaluating the asymptotic growth rate of wealth generated by the universal portfolio relative to that of the BCRP for a large number of trading periods:

lim n→∞inf n1 nln[Sˆn/S ∗ n] o = lim n→∞  −m−1 n ln[n+1]  =0.

Hence, the asymptotic growth rate of the universal portfolio’s generated wealth con-verges to that of the BCRP at a rate ofO(m−1

n ln[n+1]). Note that an increase in the

number of assets results in a decrease in the rate of convergence.

B. The exponential gradient strategy

rule that requires data only from the previous trading day.

Unlike the universal portfolio, the exponential gradient relies on the specification of a learning rate parameter. The parameter η is specified a nonnegative value and reflects the degree with which past price relatives determine the current exponential gradient portfolio. Larger values of η imply that past price relatives are taken rela-tively more into account in comparison to smaller values of η. In essence, the learning rate constitutes a momentum parameter, in the sense that a large price relative change of a particular asset results in a higher wealth allocation to that asset in the subsequent trading period.

The channel through which the learning rate determines the portfolio is best ob-served from the exponential gradient’s updating rule. Let the portfolio weights for the exponential gradient be ˜bk ∈ B, ∀k ∈ {1, ..., n}, and the final wealth on day

k ∈ {1, ..., n} be defined by ˜Sk := ∏ik=1˜bkxi. The portfolio weights of the

exponen-tial gradient follow the rule

˜bj 1= 1 m, ˜b j k+1 = ˜b_kjexp{ηe0_jxi/ ˜b 0 kxi} ∑m j=1˜b j kexp{ηe 0 jxi/ ˜b 0 kxi} ; (8)

where j∈ {1, ..., m}indexes the jthasset, and ej : m×1 denotes a zero vector for which

the jth element equals one. Note that for larger values of η the previous period’s price relatives receive larger weights in determining the next period’s portfolio. The equal weighted constant rebalanced portfolio strategy is obtained when η is set to zero.

Helmbold et al. (1998) establish that the ratio of the wealth generated by the expo-nential gradient strategy over the final wealth of the BCRP satisfies:

Similarly as for the universal portfolio, the asymptotic growth rate of the exponential gradient can also be evaluated by taking the logarithm of this expression and multi-plying by 1/n: lim n→∞inf n1 nln[S˜n/Sn] o = lim n→∞  − √ 2lnm 2√n  =0.

Hence, the asymptotic growth rate of wealth generated by the exponential gradient eventually converges to that of the BCRP as well.

However, the exponential growth rate of wealth associated with the exponential gradient converges at a rate of O(pln[m]/n). Whereas the rate of convergence for the universal portfolio is of O(m−1

n lnn). These rates indicate that the growth rate of

wealth generated by the universal portfolio converges at a faster rate to the best con-stant rebalanced portfolio, as the number of trading days grows large relative to the exponential gradient when there are two assets. However, for a "large" number of as-sets and a limited number of trading days the exponential gradient can outperform the universal portfolio in terms of generated wealth, since m impairs to a lesser ex-tend on the rate of convergence in the final wealth’s growth rate relative to that of the universal portfolio.

C. Commission fees

In this section the implications of commission fees on transactions are considered and the analysis of Blum and Kalai (1999) is closely followed. Typically a fixed percentage commission of cb, cs ∈ (0, 1) is charged. Respectively these denote the commission

fees associated with the values of buying and selling stocks. For notational purposes, suppose an investor decides on a particular day k to rebalance his portfolio from bold_k to

bnew_k+1, and his wealth equals Sk before rebalancing. Note that the difference in boldk and bnew_k+1 is that the latter denotes the optimal allocation of wealth for day k+1 and boldk

denotes the resulting allocation of wealth after the price relatives are realized on day k. Assuming that the investor finances the change in his distribution of wealth with funds obtained from his portfolio strategy, the current level of wealth decreases to αkSk, where the proportion of wealth left after paying the commission fees is denoted

by αk ∈ (0, 1), ∀k∈ {1, ..., n}. In order to maximize wealth after rebalancing the choice

of αk should be such that the following inequality is binding:

αkSk ≤Sk  1−cb

∑

∑

where j indexes the jthelement of bnew_k+1and bold, and k indexes the day. For reasonably

small commission fees the inequality is almost binding and wealth after rebalancing can be denoted by αkSk ≈Sk  1−cb

∑

∑

Note that the commission fees bear on the investor’s choice regarding the next day’s wealth allocation, due to their impact on the final return of constant rebalancing strategies. The implications for the universal portfolio are that commission fees are required to be incorporated in the update process. The universal portfolio is a per-formance weighted average of constant rebalancing strategies that are members of the simplex, and the performance of the strategies are influenced by the commission fees. In this regard, the final wealth associated with a constant rebalancing strategy, b, is denoted by: Sc_n(b, xn) = n

∏

Note that the index i is necessary for αi, since price relatives may proportionally differ

from each other across days. Furthermore, there is no need for the investor to rebalance on day n, and therefore αn is set to equal one throughout. In addition let the optimal

constant rebalancing portfolio with commission fees be denoted by bc∗, so

bc∗ =arg max b n

∏

and the realized wealth is defined by Scn∗ :=Snc(b∗, xn).

For any day k ∈ {1, ..., n−1} let final wealth be S_kc(b, xk) = _∏k_i=1αib0xi. Note

that the implications of a particular constant rebalancing strategy b are twofold: The strategy determines final wealth and the proportion of wealth that remains after com-mission fees are paid. Blum and Kalai (1999) therefore suggest to execute the universal portfolio when commission fees are included in the manner:

ˆbc₁ = 1 mι, ˆb c k+1 = R b∈BbSck(b, xk)db R b∈BSck(b, xk)db ; (10)

sell-ing assets, c = cs = cb, and for a "near" portfolio that is similarly denoted earlier by b= (1−ξ)b∗+ξz, and z ∈ Bthat: Sc_n Sc∗ n ≥ (1−ξ)(1+c)n.

The only difference with (5) is that n is replaced by(1+c)n. Since both n and(1+c)n comprise of constant terms, it is straightforward to evaluate the growth rate of wealth generated by the universal portfolio relative to that of the BCRP, where both strategies are subject to commission fees. In this regard, n can be substituted in (7) for(1+c)n, which yields Sc_n Sc∗ n ≥ 1 ((1+c)n+1)m−1

This allows for evaluating in a similar manner, as for the case without commis-sion fees, the asymptotic growth rate of wealth generated by the universal portfolio relative to that of the BCRP for a large number of trading periods. With proportional commission fees this yields:

lim n→∞inf n1 nln[Sˆ c n/Scn∗] o = lim n→∞  −m−1 n ln[(1+c)n+1]  =0.

Hence, also for the case of commission fees the asymptotic growth rate of the uni-versal portfolio’s generated wealth converges to that of the BCRP but at a rate of

O(m−_n1ln[(1+c)n+1]). It can be noted that transaction costs lower the rate of conver-gence.

D. Semi-constant rebalanced portfolios

against the implementation of such an approach, since it will not eventually achieve the same exponential growth rate of generated wealth as the BCRP. The reason to include a semi-constant rebalanced universal portfolio is to compare such a strategy with the exponential gradient, since for the latter there exists no approach similar as the method by Blum and Kalai (1999) for the universal portfolio in the wake of com-mission fees. ? contrastingly propose a semi-constant rebalanced portfolio to include commission fees.

In case current wealth should drop substantially due to the cost of rebalancing, the investor may refrain from pursuing (3). More specifically, if the costs of rebalancing6 would exceed a critical proportion, λ, of accumulated wealth used to finance the re-balancing the investor decides to adopt the end of last day’s distribution of wealth. Hence, the investor adopts the rule for the universal portfolio:

ˆbc₁= 1 mι, ˆb c k+1 =          R b∈BbSk(b, xk)db R b∈BSk(b, xk)db if λ≤1−αk; < ˆbc_k, xk > ι0x_k if λ>1−αk, (11)

where <., . >constitutes the inner product operator. The key difference between (3) and (11) is that the investor leaves his wealth distribution unaffected on day k+1 rela-tive to the end of the prior day’s end distribution,<bk, xk >/ι0xk, if the proportional

cost of rebalancing exceed the pre specified λ. Similarly, in the presence of commission fees the update rule of the exponential gradient adheres to:

˜bc 1 = 1 m, ˜b c,j k+1=              ˜bj kexp{ηe 0 jxi/ ˜b 0 xi} ∑m j=1˜b j kexp{ηe0jxi/ ˜b 0 xi} if λ≤1−αk; e0_j< ˜b c k, xk > ι0xk if λ>1−αk. (12)

6_{As a proportion of total wealth accumulated, these costs can be expressed as 1−}

E. Implementation of universal portfolio

Two main issues arise in evaluating (3) and (11). The first issue revolves around the numerical method employed to evaluate the integrals to determine the universal port-folio. Second, the issue of the impact of the number of assets on computing time. This latter part relates to the inclusion of a number of assets larger than two, which im-plies that the coordinates on the simplex for which the integral is evaluated becomes large. This may lead to the impracticality of time-consuming calculations to determine portfolio weights for a large number of assets.

A monte carlo method is adopted to evaluate the integrals in (3) and (11). The objective is to replicate the simplexB by a matrix∆ of column vectors that represent monte carlo allocations that satisfy the constraints in B, i.e. no shortselling and the individual allocations should sum up to one. To this end, let be ∆ := [b₁MC, ..., bMC_M ], an m× M matrix with random column vectors, where each of the column vectors

bMC_j are independently and identically uniformly distributed in accordance with the simplex. Furthermore, m denotes the number of assets, and M the number of monte carlo replications.

For each monte carlo replication bMC_j the sampling procedure consists of the fol-lowing steps:

1. Generate a(m−1)vector of independent variates, each uniformly distributed on[0, 1]; and sort the elements to obtain the vector with elements in ascending order.

2. Next, create the(m+1) ×1 vector z := [0, u0, 1]0.

3. The last step is to obtain bMC_j from the elements of z. Let the elements of z be indexed by i, then bMC_j := [z2−z1, ..., zi+1−zi, ..., zm+1−zm].

1986). Now we are in a position to evaluate (3): ˆb1 = 1 mι, ˆbk+1 = ∑_bMC j ∈∆b MC j Sk(bjMC, xk) ∑_bMC j ∈∆Sk(b MC j , xk) ,

and in a similar manner can (10) and (11) be evaluated.

The computing time of the universal portfolio strategies (3), (10) and (11) grows with each additional asset included in each trading period. Kalai and Vempala (2002) present an efficient implementation of the universal portfolio that is based on non-uniform random walks that are mixing over the simplex. Their approach is based on the work of Frieze and Kannan (1999) and implies that one does not need to evaluate the entire simplex in order to obtain reasonable approximations of (3) and (11). Rather than sampling the allocations uniformly over the simplex, Kalai and Vempala (2002) propose to sample the portfolios on the basis of their distribution of past performances. This procedure speeds up computing time, since portfolios with low performance are less likely to be sampled than those with a high past performance, such that smaller samples will suffice to calculate the universal portfolio relative to uniform sampling.

The method proposed by Kalai and Vempala (2002) is not adopted in this investi-gation, since the total computing time for the considered number of assets and time frame is deemed reasonable. Calculating the performance of the universal portfolio for the largest dataset, SBM100, took 12 hours for (3), (10) and (11) seperately on a Mac OS X 10.6.8. operating system.

III.

Data

book equity to market equity (book-to-market ratio). The second class of portfolios consists of aggregate industry indices.

TABLE I

List of considered datasets

Description of dataset # of assets: m Period Abbreviation 1 6 Portfolios Formed on Size and

Book-to-Market ratio.

6 July 1, 1963 - December 31, 2010 SBM6 2 25 Portfolios Formed on Size

and Book-to-Market ratio.

25 July 1, 1963 - December 31, 2010 SBM25 3 100 Portfolios Formed on Size

and Book-to-Market ratio.

100 July 1, 1963 - December 31, 2010 SBM100 4 10 Industry portfolios. 10 July 1, 1963 - December 31, 2010 IND10 5 49 Industry portfolios. 49 July 1, 1963 - December 31, 2010 IND49 Notes: The table lists the datasets that are analyzed; the number of assets included in each dataset; the considered timeframe. All datasets contain daily total returns (from Kenneth R. French’s website 2011: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/index.html). The last column con-tains abbreviations of the datasets that will be used throughout the text, figures and tables.

The stocks that are included in the formation of the portfolios formed on size and the book-to-market ratio encompass all NYSE, AMEX and NASDAQ stocks for which there exists market equity data for the month December in the lagged year until June of the contemporaneous year, and positive book equity data for the lagged year.

For the case of SBM6 the portfolios, which are constructed at the end of each June, are the intersections of two portfolios formed on size (market equity, ME) and 3 portfo-lios formed on the ratio of book equity to market equity (BE/ME). The size breakpoint for the contemporaneous year is the median NYSE market equity at the end of June of that year. BE/ME for June of the contemporaneous year is the book equity for the last fiscal year end in one year lagged divided by ME for December of the lagged year. The BE/ME breakpoints are the 30th and 70th NYSE percentiles. Hence, the two-by-three classification:

For instance, the stocks included in portfolio 5 have market equity values below the median of the NYSE, and book equity to market equity ratio between the 30th and 70th percentile. Note that for the case of SBM25 and SBM100 respectively there is a five-by-five and a ten-by-ten partitioning for portfolios, with equal distances between the breakpoint percentiles.

The portfolios in IND10 and IND49 consist of NYSE, AMEX and NASDAQ stocks and are portfolios based on the four digit SIC (Standard Industrial Classification) code at that time. One year lagged Compustat SIC codes for the fiscal year ending are used as classification. Table II reports the included industries.

TABLE II

Industries included in INDxx datasets

Dataset Industries included

IND10 Consumer non-durables, Durables, Manufacturing, Energy, High-technology, Telecommunication, Shops, Health, Utilities, Other (Mines, Construction, Transport, Hotels, Bus. Serv., Entertainment, Finance )

IND49 Agriculture, Food, Soda, Beer, Smoke, Toys, Entertain-ment, Books, Household goods, Clothing, Health, Med-ical Equipment, PharmaceutMed-icals, ChemMed-icals, Textiles, Construction, Building Material, Steel, Fabricated Prod-ucts, Machinery, Electrical Equipment, Autos, Aero, Ships, Guns, Gold, Mines, Coal, Oil, Utilities, Telecom-munication, Personal Services, Business Serv., Services, Management, Hardware, Software, Chips, Paper, Boxes, Transport, Wholesale, Retail, Banks, Insurance, Real Es-tate, Financial Trading, Other (Sanitary serv., Air condi-tioning, Irrigation)

IV.

Results

A. Empirical Results

However, since commission fees are also considered, daily rebalancing may lead to high transaction costs for the investor. To this end, the performances of the considered strategies are also evaluated for weekly and monthly rebalancing. Notwithstanding, in all cases the universal portfolio stands out as the best performing strategy in terms of realized total return. Additionally, for a larger number of assets the universal port-folio considerably outperforms the 1/m strategy in terms of achieved total return.

TABLE III

Portfolios’ total returns, daily rebalanced

Portfolio Strategy auxiliary commission fees Datasets

parameters (in basispoints) SBM6 SBM25 SBM100 IND10 IND49 m=6 m=25 m=100 m=10 m=49 1/m-strategy - 222.67 243.71 252.61 143.29 202.95 [0.12] [0.12] [0.12] [0.11] [0.12] „ 20 32.42 37.94 39.47 17.41 24.23 [0.08] [0.08] [0.08] [0.06] [0.07] universal portfolio - 293.25 410.64 627.03 185.31 405.03 [0.13] [0.14] [0.15] [0.12] [0.13] Blum and Kalai (1999) 20 162.47 267.65 350.53 98.37 226.32 method [0.11] [0.12] [0.13] [0.10] [0.12] semi const. 20 49.96 57.81 74.25 23.74 49.31 rebalancing [0.09] [0.09] [0.09] [0.07] [0.09] exponential gradient η=0.05 - 228.43 260.03 435.45 135.56 273.17 [0.12] [0.12] [0.14] [0.11] [0.13] „ „ 20 35.91 39.37 53.61 17.71 28.48 [0.08] [0.08] [0.09] [0.06] [0.07] „ η=0.10 - 235.51 265.71 473.81 132.02 292.43 [0.12] [0.12] [0.14] [0.11] [0.13] „ „ 20 37.87 45.16 59.67 17.73 32.67 [0.08] [0.08] [0.09] [0.06] [0.08] „ η=0.50 - 312.32 317.09 516.91 129.73 353.71 [0.13] [0.13] [0.14] [0.11] [0.13] „ „ 20 43.62 52.18 65.41 15.02 38.15 [0.08] [0.09] [0.09] [0.06] [0.08] Notes: The table contains total nominal returns achieved by the respective portfolio strategies over the period July1, 1963 through December 31, 2010. The strategies are rebalanced on a daily basis. The proportional commission fees are equal for both selling and buying assets. Yearly return rate equivalents are reported in brackets: yearly return rate= (total return)1/47.5_{−1. Futhermore, the "Blum and Kalai (1999) method" refers}

to the strategy (5). Apart from the Blum and Kalai (1999) method, all other strategies are implemented as described in section II.D. when commission fees are nonzero.

aver-TABLE IV

Portfolio total return, weekly rebalanced

Portfolio Strategy auxiliary commission fees Datasets

parameters (in basispoints) SBM6 SBM25 SBM100 IND10 IND49 m=6 m=25 m=100 m=10 m=49 1/m-strategy - 180.75 221.32 231.34 112.48 183.72 [0.12] [0.12] [0.12] [0.10] [0.12] „ 20 39.62 43.02 46.82 23.72 32.81 [0.08] [0.08] [0.08] [0.07] [0.08] universal portfolio - 253.32 383.74 591.73 162.89 367.24 [0.12] [0.13] [0.14] [0.11] [0.13] Blum and Kalai (1999) 20 170.31 255.48 374.10 105.26 259.62 method [0.11] [0.12] [0.13] [0.10] [0.12] semi const. 20 45.08 52.83 67.15 26.63 42.92 rebalancing [0.08] [0.09] [0.09] [0.06] [0.08] exponential gradient η=0.05 - 197.53 243.91 392.28 118.20 241.21 [0.12] [0.12] [0.13] [0.11] [0.12] „ „ 20 42.83 48.32 59.37 21.37 37.48 [0.08] [0.09] [0.09] [0.07] [0.08] „ η=0.10 - 203.92 244.73 412.91 121.48 272.41 [0.12] [0.12] [0.14] [0.11] [0.13] „ „ 20 41.37 49.47 67.48 18.57 37.14 [0.08] [0.09] [0.09] [0.06] [0.08] „ η=0.50 - 293.47 297.67 449.28 112.86 312.80 [0.13] [0.13] [0.14] [0.10] [0.13] „ „ 20 49.28 59.39 72.35 16.23 44.21 [0.09] [0.09] [0.09] [0.06] [0.08] Notes: The table contains total nominal returns achieved by the respective portfolio strategies over the period July1, 1963 through December 31, 2010. The strategies are rebalanced on a weekly basis. The proportional commission fees are equal for both selling and buying assets. Yearly return rate equivalents are reported in brackets: yearly return rate= (total return)1/47.5_{−1. Futhermore, the "Blum and Kalai (1999) method" refers}

to the strategy (5). Apart from the Blum and Kalai (1999) method, all other strategies are implemented as described in section II.D. when commission fees are nonzero.

age value line, respectively the geometric mean and arithmetic mean of the consid-ered asset prices on a given day.7 The motivation for the inclusion of these lines is based on the fact that the best constant rebalanced portfolio’s wealth exceeds these, and the mean value line defines the universal portfolio’s lower bound. Note that throughout the considered time period the universal portfolio’s performance exceeds the 1/m strategy in terms of total return. This result continues to hold for all con-sidered datasets and rebalancing periods. The exponential gradient portfolio strategy

7_{Let S(j)}

kbe the return obtained on the kthday of investing one unit of a currency in the jthasset of

the m possible assets. The mean value line at day k is then defined by∏m_j=1S(j)k

_m1

, which defines the lower bound of wealth generated by the universal portfolio (Cover, 1991). The average value line

TABLE V

Portfolio total return, monthly rebalanced

Portfolio Strategy auxiliary commission fees Datasets

parameters (in basispoints) SBM6 SBM25 SBM100 IND10 IND49 m=6 m=25 m=100 m=10 m=49 1/m-strategy - 161.38 192.25 207.46 96.32 154.37 [0.11] [0.12] [0.12] [0.10] [0.11] „ 20 46.89 49.38 54.29 38.19 54.69 [0.08] [0.09] [0.09] [0.08] [0.09] universal portfolio - 237.04 357.69 562.87 152.48 342.68 [0.12] [0.13] [0.14] [0.11] [0.13] Blum and Kalai (1999) 20 174.37 236.45 321.63 83.72 234.89 method [0.11] [0.12] [0.13] [0.10] [0.12] semi const. 20 67.36 68.89 81.39 35.95 57.30 rebalancing [0.09] [0.09] [0.10] [0.08] [0.09] exponential gradient η=0.05 - 183.28 226.05 376.11 92.43 196.87 [0.12] [0.12] [0.13] [0.10] [0.12] „ „ 20 49.49 52.68 63.27 30.65 46.98 [0.09] [0.09] [0.09] [0.07] [0.08] „ η=0.10 - 187.00 218.65 384.17 108.56 208.35 [0.12] [0.12] [0.13] [0.10] [0.12] „ „ 20 57.83 54.93 71.43 34.87 48.38 [0.09] [0.09] [0.09] [0.08] [0.09] „ η=0.50 - 228.32 245.54 410.73 106.93 295.49 [0.12] [0.12] [0.14] [0.10] [0.13] „ „ 20 63.59 67.74 77.32 28.43 57.38 [0.09] [0.09] [0.10] [0.07] [0.09] Notes: The table contains total nominal returns achieved by the respective portfolio strategies over the period July1, 1963 through December 31, 2010. The strategies are rebalanced on a monthly basis. The proportional commission fees are equal for both selling and buying assets. Yearly return rate equivalents are reported in brackets: yearly return rate= (total return)1/47.5_{−1. Futhermore, the "Blum and Kalai (1999) method" refers}

to the strategy (5). Apart from the Blum and Kalai (1999) method, all other strategies are implemented as described in section II.D. when commission fees are nonzero.

also outperforms the naive constant rebalancing strategy. Since this result continues to hold as the number of assets in the portfolio increases it contrasts the findings of DeMiguel et al. (2009). They conclude that only a small group of the investigated strategies outperforms the naive equal constant rebalanced portfolio. Part of the result could potentially be explained by the composition of the investigated datasets. Since, each considered asset constitutes a portfolio itself, and implies that some of the firm specific risks are already diversified away.

im-FIGURE 1

Portfolio performance for the SBM100 dataset

Nov−1943 Apr−1971 Sep−1998 Jan−2026 −1 0 1 2 3 4 5 6 7

Time (days), daily rebalancing

Natural log. of total return index

Universal Portfolio Naive (1/N) EG, η = 0.05 EG, η = 0.10 EG, η = 0.50 Mean Value Line Average value line

Aug−1957 Apr−1971 Dec−1984 Sep−1998 May−2012 −1 0 1 2 3 4 5 6 7

Time (days), weekly rebalancing

Natural log. of total return index

Universal Portfolio Naive (1/N) EG, η = 0.05 EG, η = 0.10 EG, η = 0.50 Mean Value Line Average value line

Nov−1943 Apr−1971 Sep−1998 Jan−2026 −1 0 1 2 3 4 5 6 7

Time (days), monthly rebalancing

Natural log. of total return index

Universal Portfolio Naive (1/N) EG, η = 0.05 EG, η = 0.10 EG, η = 0.50 Mean Value Line Average value line

Notes: Performances of the universal portfolio and the naive portfolio strategy are presented for the SBM100 dataset. Additionally, the exponential gradient has been included for various levels of the learning rate, η. Both figures display the graphs associated with the mean value line and the average value line, see footnote 7 for a definition. (Left) the universal portfolio’s is based on the asset allocation rule (3), and the naive strategy follows the rule where the investor rebalances every day his wealth allocation to 1/m. The exponential gradient function is based on (8). Dialy rebalancing is used. (Right) the universal portfolio’s is based on the asset allocation rule (3), and the naive strategy follows the rule where the investor rebalances every day his wealth allocation to 1/m. The exponential gradient function is based on (8). Weekly rebalancing is used.

pact through the updating rule of the exponential gradient on its overall performance. Helmbold et al. (1998) show empirically that the optimal learning rate is around 0.05. This is in contrast to the result obtained in this investigation, since the performance in terms of total return of the exponential gradient seems to improve as the learning rate increases. An investor lacks perfect foresight and is unable to set the learning rate appropriately before he implements the exponential gradient strategy. In this light these results, combined with those obtained by Helmbold et al. (1998), cast doubt on whether 0.05 constitutes the optimal learning rate for this strategy.

propor-tional commission fee of 20 basis points on both acquiring and selling assets reduces the yearly return rate equivalent of each strategy nearly by half. However, the uni-versal portfolio’s performance still stands out among the SBM datasets and achieves a yearly equivalent growth rate of nine percent for 100 assets included in the strat-egy. Commision fees prove to be less destructive on the achieved performance in the method proposed by Blum and Kalai (1999). Additionally, this method results in an approximate loss of twenty percent of total return relative to the case of no commis-sion fees. Figure 2 provides a visualization of the difference in terms of achieved total return between the method of Blum and Kalai (1999) and semi-constant rebalancing in the presence of commission fees for the IND10 dataset. An explanation for this result can be based on a distinction between the method proposed by Blum and Kalai (1999) and semi-constant rebalancing. In the former approach, commission fees are incorpo-rated in the performance of the portfolios, this implies that strategies that may have performed well but impose large expenses on the investor due to rebalancing efforts may have received less weight than strategies that require rebalancing to a lesser ex-tend but perform less well. Hence, the optimal portfolio is determined by taking into account the implications of commission fees.

FIGURE 2

Portfolio performance for the IND10 dataset

Nov−1943 Apr−1971 Sep−1998 Jan−2026 −1 0 1 2 3 4 5

Time (days), daily rebalancing

Natural log. of total return index

Blum and Kalai (1999) method Universal Portfolio Naive (1/N) EG, η = 0.05 EG, η = 0.10 EG, η = 0.50

Aug−1957 Apr−1971 Dec−1984 Sep−1998 May−2012 −1 0 1 2 3 4 5

Time (days), weekly rebalancing

Natural log. of total return index

Blum and Kalai (1999) method Universal Portfolio Naive (1/N) EG, η = 0.05 EG, η = 0.10 EG, η = 0.50

Nov−1943 Apr−1971 Sep−1998 Jan−2026 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time (days), monthly rebalancing

Natural log. of total return index

Blum and Kalai (1999) method Universal Port. Naive (1/N) EG, η = 0.05 EG, η = 0.10 EG, η = 0.50

Notes: Performances of the universal portfolio and the naive portfolio strategy are presented for the 10 portfolios in the IND10 dataset. Additionally, the exponential gradient has been included for various levels of the learning rate, η. (Left) the universal portfolio follows the rule (11) and (5), and the exponential gradient (12), where the fixed percentage commission cb =cs =20×

10−10,000_{, (20 basispoints) and the investor allows a maximum of ten percent of his wealth to be}

spend on rebalancing, λ=0.1. These costs also apply to the naive strategy. Daily rebalancing is used(Right) the universal portfolio follows the rule (11) and (5), and the exponential gradient (12), where the fixed percentage commission cb=cs=20×10−10,000, (20 basispoints) and the investor

allows a maximum of ten percent of his wealth to be spend on rebalancing, λ=0.1. These costs also apply to the naive strategy. Weekly rebalancing is used.

portfolio increases as the number of assets available for investment increases.

With respect to the rebalancing period, table III through V indicate that as the rebal-ancing period increases, the performance of all strategies decreases. However, when commission fees are incorporated overall performance increases. This result shows the impact commission fees have on the total return of a portfolio strategy, and that rebalancing less may prove to be more attractive as commission fees rise.

B. Analytical Results

other portfolio strategies. As it turns out the universal portfolio has in common with empirical mean-variance based strategies that it approximately depends on the first two sample moments of asset returns when the rebalancing periods are not large. For Result 1 we heuristically derive that the first two sample moments are determinants of the universal portfolio. The proof is presented in the appendix.

result 1 (universal portfolio’s dependence on stock return moments)

Let the m asset price returns for a day i ∈ {1, ..., k} be ri := xi−1, and the corresponding

first two sample moments up to day k be ˆµ_k := 1_k∑k_i=1riand ˆAk := 1_k∑ik=1rir0_i. Furthermore,

for b0µˆ_k the universal portfolio for day k+1 can then be solely expressed in terms of these

moments: ˆbk+1 = R b∈Bbφ(Aˆ −1 k µˆk, 1kAˆ −1 )db R b∈Bφ(Aˆ −1 k µˆk, 1kAˆ −1 )db ,

where φ(Aˆ−_k1µˆ_k,1_kAˆ−1)denotes a m-dimensional normal vector with mean vector ˆA−_k1µˆ_kand

covariance matrix 1_kAˆ−1.

Note that result 1 is derived under the condition that individual asset returns are close to zero. For longer rebalancing periods this result may not hold as could be the case for weekly or monthly rebalancing. Despite that result 1 does not provide a closed form solution it is apparent that if the shortselling constraint is alleviated the universal portfolio constitutes a normal vector conditional on the restriction that the indivdual asset weights sum to one. This result is heuristically derived and presented in Result 2.

result 2 (Universal portfolio and shortselling)

following constant vector when shortselling is allowed: E[ˆbk+1|ι0b=1] = Aˆ −1 k µˆk+ ˆ A−_k1ι(1−ι0Aˆ−_k1) ι0Aˆ−_k1ι

This result states that for small rebalancing periods the universal portfolio can be approximated by the above constant when shortselling is allowed. Although this re-sult does not necessarily hold for Cover’s (1991) universal portfolio, due to the restric-tion on short selling, it provides ground to benchmark the universal portfolio with the naive 1/m portfolio strategy. Since for a limited number of trading days optimal mean-variance based strategies are generally outperformed by the 1/m strategy.

Overall, for a large number of trading days, the universal portfolio weights con-verge to constant terms. Since for a large number of trading days, any additional observation contributes less and less to a change in values of the first two sample moments of asset price returns. However, this is only approximately true for small rebalancing periods in the strategy.

V.

Conclusion

A. Concluding remarks

the number of assets increases. Moreover, the results found by DeMiguel et al. (2009) provide further motivation to benchmark the universal portfolio’s performance with that of a more naive strategy. Since, the universal portfolio is found to be dependent on the first two sample moments of stock price returns it bears some resemblance to the optimal mean-variance portfolio strategies. DeMiguel et al. (2009) find that op-timal mean-variance strategies do generally not outperform equal weighted constant rebalanced strategies. Therefore, the equal weighted constant rebalanced portfolio is deemed to be an adequate benchmark to assess the universal portfolio’s performance. Moreover, an equal weighted constantly rebalanced portfolio may have diversification benefits as the number of assets increase.

Additionally, results suggest that the universal portfolio outperforms the exponen-tial gradient strategy designed by Helmbold et al. (1998). A noteworthy aspect of this result is that the exponential gradient performs optimally when the learning rate is set at 0.50, whereas Helmbold et al. (1998) find the optimal learning rate to be around 0.05. No apparent cause is found as to why this difference exists. However, this result casts doubt on whether 0.05 constitutes a general optimal learning rate.

B. Limitations and future research

Result 1 and 2 indicate that the universal portfolio converges to constant allocations of wealth across assets that are formed by the first two sample moments of asset price returns. This poses a caveat in the sense that new realizations of asset price returns contribute less and less to the formation of the first two sample moments. Hence, the universal portfolio can be slow in incorporating events that cause structural breaks in the asset price return series of the assets. Especially if the events occur after the universal portfolio has been evaluated over a considerable period. It should be noted that these results hold approximately for small rebalancing periods.

Possible extensions in this regard could revolve around the introduction of weights assigned to past performance realizations of the universal portfolio. These weights can typically be smaller for more historic observations, such that recent performances be-come more important to determine optimal portfolio weights. In particular, the nearest neighbor strategy of Györfi et al. (2008). Their approach revolves around a set of arti-ficial expert strategies for which the optimal portfolios are determined on a given day. Typically, these experts only take a subsample of past relative realizations into account. Based on their past performance the portfolio strategy is essential a weighted average of all the expert’s strategies, where the weights are based on their past performance. This bears close resemblance to the universal portfolio in the sense that strategies are based on their past performance, but goes one step further by also determining the optimal subset of historic price relative realizations.

References

Blum, A. and A. Kalai, 1999, Universal Portfolios With and Without Transaction costs, Machine Learning 35, 193–205.

Cover, T. M., 1991, Universal Portfolios, Mathematical Finance 1, 1–29.

Cover, T. M. and E. Ordentlich, 1996, Universal Portfolios with Side Information, IEEE Transactions on Information Theory 42, 348–363.

DeMiguel, V., L. Garlappi, and R. Uppal, 2009, Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy?, Review of Financial Studies 22, 1915– 1953.

Devroye, L., 1986, Non-Uniform Random Variate Generation (Springer-Verlag, New York).

French, K. R., 2011, Kenneth R. French’s website: http://mba.tuck.dartmouth.edu /pages/faculty/ken.french/index.html.

Frieze, A. and R. Kannan, 1999, Sobolev Inequalities and Sampling from Log-Concave Distributions, Annals of Applied Probability 9, 14–26.

Györfi, L., F. Udina, and H. Walk, 2008, Nonparametric Nearest Neighbor based Em-pirical Portfolio Selection Strategies, Statistics & Decisions 26, 145–157.

Helmbold, D. P., R. E. Schapire, and Y. Singer, 1998, On-Line Portfolio Selection Using Multiplicative Updates, Mathematical Finance 8, 325–347.

Kalai, A. and S. Vempala, 2002, Efficient Algorithms for Universal Portfolios, Journal of Machine Learning Research 3, 423–440.

VI.

Appendix

A. Derivation of (7)

This section presents a derivation for the equalityR₀1(1−v1/n)m−1dv≥ _(n+11_)m−1 in (7) that is taken from Blum and Kalai (1999). Note that change of variable y =v1/nis used combined with integration by parts, and yields

Z 1 0 (1 −v1/n)m−1dv =n Z 1 0 y n−1₍ 1−y)m−1dy =ny n_(1−y)m−1 n    1 0+ m−1 n Z 1 0 y n_(1−y)m−2_dy =nm−1 n Z 1 0 y n_(1−y)m−2_dy =nm−1 n m−2 n+1 Z 1 0 y n+1_(1−y)m−3_dy =... =n(m−1)!(n−1)! (n+m−2)! Z 1 0 y n+m−2 dy = 1 n+m−1 m−1  ≥ 1 (n+1)m−1. B. Proofs

Proof for result 1 (Heuristic)The final wealth at any given day k ∈ {1, ..., n}for a

con-stant rebalanced portfolio with weights b is denoted by Sk =∏ki=1b 0

xi. Price relatives

can be expressed in terms of asset price returns ri, as in xi = 1+ri. Substituting this

expression for total wealth on day k yields:

Sk(b, xk) = k

∏

∏

∑

Generally, b0ri ≈0 for daily total returns for stocks. Based on this observation, lets say

that the term ln(1+b0ri)can be expanded in the following Taylor series:

0 0

Hence, total wealth on day k can be expanded in a similar fashion and yields an ap-proximation to Sk. Additionally, the terms ˆµ_k := _k1∑k_i=1ri and ˆAk := 1_k∑ki=1rir0i are

introduced. Now we are in a position to state an approximation of Sk as:

Sk(b, xk) = exp{ k

∑

∑

This expression can be further rearranged by noting that (Abadir and Magnus, 2005):

b0µˆ_k−1 2b 0 _ˆ Akb= 1 2µˆ 0 kAˆkµˆ_k−1 2(b−Aˆ −1 k µˆk) 0 _ˆ Ak(b−Aˆ −1 k µˆk) (14)

Note that the universal portfolio without commission fees is constructed on the basis of the rule ˆb1 = 1 mι, ˆbk+1 = R b∈BbSk(b, xk)db R b∈BSk(b, xk)db . Substituting (14) then yields for ˆbk+1:

ˆbk+1 = R b∈Bbexp{−k2(b−Aˆ −1 k µˆk)0Aˆk(b−Aˆ −1 k µˆk)}db R b∈Bexp{−k2(b−Aˆ −1 k µˆk)0Aˆk(b−Aˆ −1 k µˆk)}db . (15)

Essentially (15) implies that ˆbk+1is the first moment of a m-dimensional normal vector

with mean parameter vector ˆA−_k1µˆ_k, covariance matrix Aˆ −1 k

k , conditional on

member-ship of the simplexB.

Proof for result 2 (Heuristic)In case shortselling is allowed, the set of potential

port-folio weights is {b ∈ Rm _{: ι}0_{b =}

1}. Based on (15) the universal portfolio can be expressed as: ˆbk+1|ι0bk+1 ∼ N ˆ A−_k1µˆ_k ι0Aˆ−_k1µˆ_k ! , 1_k Aˆ −1 k Aˆ −1 k ι ι0Aˆ−_k1 ι0Aˆ−_k1ι !! . (16)

This implies that the expected portfolio weights are: