QUANTITATIVE FINANCE RESEARCH CENTRE

Research Paper 184 September 2006

Approximating the Growth Optimal Portfolio with a Diversified World Stock Index

Truc Le and Eckhard Platen

ISSN 1441-8010 www.qfrc.uts.edu.au

QUANTITATIVE FINANCE RESEARCH CENTRE

Approximating the Growth Optimal

Portfolio

with a Diversified World Stock Index

Truc Le1 and Eckhard Platen1

September 14, 2006

Abstract. This paper constructs and compares various total return world

stock indices based on daily data. Due to diversification these indices

are noticeably similar. A diversification theorem identifies any diversified

portfolio as a proxy for the growth optimal portfolio. The paper constructs

a diversified world stock index that outperforms a number of other indices

and argues that it is a good proxy for the growth optimal portfolio. This has

applications to derivative pricing and investment management.

1991 Mathematics Subject Classification: primary 90A12; secondary 60G30, 62P20.JEL Classification: G10, G13Key words and phrases: world stock index, growth optimal portfolio, diversifica-tion, mean-variance portfolio selection, enhanced index fund.

1University of Technology Sydney, School of Finance & Economics and Department of

Mathematical Sciences, PO Box 123, Broadway, NSW, 2007, Australia

1 Introduction

In investment management there is a vital interest in identifying best perform-ing portfolios. Theoretically, it can be shown that the growth optimal portfolio(GOP), which maximizes expected logarithmic utility from terminal wealth, isthe portfolio that almost surely outperforms all other strictly positive portfoliosafter a sufficiently long time. This fascinating property of the GOP was discov-ered by Kelly (1956). It is also an optimal portfolio in a number of other sensesas discussed in Platen & Heath (2006). The long term outperformance of anystrictly positive portfolio by the GOP has been studied, for instance, in Latane(1959), Breiman (1961), Markowitz (1976) and Long (1990). In principle, theGOP is the portfolio that cannot be beaten in any reasonable systematic way.Reviews of this portfolio can be found in Hakansson & Ziemba (1995) and Platen(2005b).

Diversification is a classical concept in portfolio optimization, which has been ap-plied in practice for centuries. Some results on different notions of diversificationand diversified portfolios can be found, for instance, in Bjork & Naslund (1998),Hofmann & Platen (2000), Fernholz (2002), Platen (2004, 2005b) and Guan, Liu& Chong (2004). Under rather general assumptions it follows that diversifiedportfolios (DPs), where the fractions that are invested in different securities aresmall, behave similarly, see Platen & Heath (2006). If the market possesses aregularity property, under which the GOP is itself a DP, then in a large marketthe GOP can be asymptotically approximated by DPs. This result on diversi-fication has been derived for continuous markets in Platen (2004) and for thecase of jump diffusion markets in Platen (2005a). Since it does not assume anyparticular market dynamics, it endows the construction of a diversified proxy forthe GOP with a robustness property.

The aim of this paper is to construct diversified world stock market indices fromsector stock market index data. We shall argue that such diversified indices arereasonable proxies for the GOP and can be potentially used as enhanced indexfunds, see Scowcroft & Sefton (2003).

The paper is organized as follows: Section 2 introduces a jump diffusion financialmarket. Section 3 presents the GOP and points out that in the long run it ispathwise the best performing portfolio. Section 4 demonstrates that DPs ap-proximate the GOP. Section 5 constructs various diversified world stock indicesusing daily sector stock index data and suggests a good proxy of the GOP. Sec-tion 6 estimates GOP fractions and discusses related issues. Section 7 comparesindices and discusses their construction. Section 8 applies the constructed indexto log-return estimation.

2

2 Financial Market Model

This section introduces a jump diffusion financial market along the lines of thebenchmark approach as described in Platen (2002, 2006b) and Platen & Heath(2006). This approach presents a unified framework for financial modeling, in-vestment management, derivative pricing and risk measurement.

We model a financial market that evolves on an infinite time horizon R+ = [0,∞)and in which there are d ∈ N = 1, 2, . . . sources of trading uncertainty.These are defined on a filtered probability space (Ω,A,A, P ) where filtrationA = (At)t∈R+

satisfies the usual conditions and where A0 is the trivial σ-algebra.As usual, we regard At as modeling the information available at time t. Con-tinuous trading uncertainties are represented by m ∈ 1, 2, . . . , d independentstandard Wiener processes Bk = Bk

t , t ∈ R+, for k ∈ 1, 2, . . . ,m. Event-driven trading uncertainties are modeled by d−m counting processes of the formpk = pk

t , t ∈ R+, for k ∈ m + 1, . . . , d, whose intensities hk = hkt , t ∈ R+

are predictable positive processes satisfying

∫ t

0

hks ds < ∞ (2.1)

almost surely, for all t ∈ R+. The corresponding compensated normalized jumpmartingales qk = qk

t , t ∈ R+, for k ∈ m + 1, . . . , d, are represented by thestochastic differentials

dqkt = (hk

t )− 1

2 (dpkt − hk

t dt) (2.2)

for t ∈ R+. Thus, the trading uncertainties are modeled by the vector processW = Wt = (W 1

t , . . . ,Wmt ,Wm+1

t , . . . ,W dt )⊤, t ∈ R+ of independent (A, P )-

martingales with W 1t = B1

t , . . . ,Wmt = Bm

t and Wm+1t = qm+1

t , . . . ,W dt = qd

t

for all t ∈ R+. Note that the conditional variance of the kth source of tradinguncertainty is

E(

(

W kt+s − W k

t

)2∣

∣

∣At

)

= s (2.3)

for all k ∈ 1, 2, . . . , d and s, t ∈ R+.

We now specify d + 1 primary security accounts, one of which is a locally risklesssavings account S0 = S0

t , t ∈ R+, given by

S0t = exp

∫ t

0

rsds

< ∞ (2.4)

for t ∈ R+. Here r = rt, t ∈ R+ denotes the adapted short rate process. Thereare also d nonnegative risky primary security accounts, whose value processes aredenoted by Sj = Sj

t , t ∈ R+, for j ∈ 1, 2, . . . , d. These are typically stocks,with all dividends reinvested. In Section 5 we shall choose them to be sectorworld stock indices. Note that foreign savings accounts, bonds, corporate bonds

3

and other derivatives may potentially also form primary security accounts. Weassume that the dynamics of the risky assets satisfy the SDEs

dSjt = Sj

t−

(

ajtdt +

d∑

k=1

bj,kt dW k

t

)

, (2.5)

for t ∈ R+, Sj0 > 0, for all j ∈ 1, 2, . . . , d. Here we denote the appreciation

rate vector process by a = at = (a1t , . . . , a

dt )

⊤, t ∈ R+ and the volatility matrixprocess by b = bt = [bj,k

t ]dj,k=1, t ∈ R+. We assume that bt is an invertible

matrix for every t ∈ R+, with inverse b−1t . Furthermore, aj

t and bj,kt are almost

surely finite and predictable, ensuring that a unique strong solution of the systemof SDEs (2.5) exists. The inequality

bj,kt ≥ −

√

hkt (2.6)

for all t ∈ R+ and k ∈ m + 1, . . . , d, guarantees nonnegativity for each of therisky assets.

We can now define the market price of risk vector

θt = (θ1t , . . . , θ

dt )

⊤ = b−1t (at − rt 1), (2.7)

for t ∈ R+. Here 1 = (1, . . . , 1)⊤ denotes the vector of ones. To ensure that themarket is reasonable, in the sense that it is worth investing in risky assets, weassume that

θkt <

√

hkt (2.8)

for all t ∈ R+ and k ∈ m + 1, . . . , d. Furthermore, the total market price of riskis taken to be nonzero and finite, that is,

|θt| =

√

√

√

√

d∑

k=1

(θkt )

2 ∈ (0,∞). (2.9)

Finally, we assume thatd∑

k=1

d∑

j=0

θkt b−1j,k

t > 0 (2.10)

for all t ∈ R+. Using the expression (2.7) for the market prices of risk, we canrewrite the SDE (2.5) for Sδ

t in the form

dSjt = Sj

t−

(

rtdt +d∑

k=1

bj,kt

(

θkt dt + dW k

t

)

)

(2.11)

= Sjt−

(

rtdt + bjt (θtdt + dWt)

)

(2.12)

for t ∈ R+. Here we set bjt = (bj,1

t , . . . , bj,dt ), for all t ∈ R+ and all j ∈ 1, 2, . . . , d.

4

Let Sδ = Sδt , t ∈ R+ denote the value process of a portfolio, so that

Sδt =

d∑

j=0

δjt Sj

t , (2.13)

for t ∈ R+. Here δjt represents the number of units of the jth primary security

account held in the portfolio at time t. We call the predictable stochastic processδ = δt = (δ0

t , δ1t , . . . , δ

dt )

⊤, t ∈ R+ a strategy if the Ito integral

∫ t

0

δjs dSj

s (2.14)

exists for each j ∈ 0, 1, . . . , d and t ∈ R+. The portfolio Sδ is said to be self-financing if all changes in its value are due to gains or losses from trade in theprimary security accounts. This can be expressed as follows:

dSδt =

d∑

j=0

δjt dSj

t . (2.15)

In this paper we restrict our attention to self-financing portfolios.

Denote by ν+the set of strictly positive portfolio processes. For a given strategy

δ with strictly positive portfolio process Sδ ∈ ν+, denote by πδ,t = (π1

δ,t, . . . , πdδ,t)

⊤

the vector of fractions of wealth invested in the primary security accounts. For aportfolio Sδ, πj

δ,t is the fraction of wealth held in the jth primary security accountat time t:

πjδ,t = δj

t

Sjt

Sδt

(2.16)

for all t ∈ R+ and all j ∈ 0, 1, . . . , d. Note that these fractions can be negative,but they always sum to one, that is

∑dj=0π

jδ,t = 1. By equation (2.15) Sδ

t satisfiesthe following SDE, in terms of fractions:

dSδt = Sδ

t−

(

rt dt + π⊤

δ,tbt (θt dt + dWt))

. (2.17)

It follows by (2.6) that Sδ remains strictly positive if

d∑

j=1

πjδ,tb

j,kt > −

√

hkt (2.18)

almost surely, for all k ∈ m + 1, . . . , d and t ∈ R+. We assume that all primarysecurity accounts can jump to zero at any time. This is a realistic assumption,even though the intensities of such default events may be very small. This implies

that all fractions of wealth in a strictly positive portfolio Sδ ∈ ν+must be

nonnegative, that isπj

δ,t ≥ 0 (2.19)

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almost surely, for all j ∈ 0, 1, . . . , d and t ∈ R+.

Now, applying the Ito formula yields the following SDE for the logarithm of Sδt

d ln(Sδt ) = gδ

t dt+m∑

k=1

d∑

j=1

πjδ,tb

j,kt dW k

t +d∑

k=m+1

ln

(

1 +d∑

j=1

πjδ,t−

bj,kt√

hkt

)

dW kt , (2.20)

where the growth rate gδt is expressed as

gδt = rt +

m∑

k=1

d∑

j=1

πjδ,tb

j,kt θk

t −1

2

(

d∑

j=1

πjδ,tb

j,kt

)2

+d∑

k=m+1

(

d∑

j=1

πjδ,t−bj,k

t

(

θkt −

√

hkt

)

+ ln

(

1 +d∑

j=1

πjδ,t−

bj,kt√

hkt

)

hkt

)

(2.21)

for t ∈ R+.

3 Growth Optimal Portfolio

The growth optimal portfolio (GOP) is the central building block of the bench-mark approach, see Platen & Heath (2006). It is the portfolio that maximizesgrowth over all strictly positive portfolios.

Definition 3.1 A strictly positive portfolio process Sδ∗ ∈ ν+is called a GOP if

gδt ≤ gδ∗

t almost surely, for all t ∈ R+ and Sδ ∈ ν+.

By equation (2.17), see Platen (2006a) and Platen & Heath (2006), it follows thatthe GOP satisfies the SDE

dSδ∗t = Sδ∗

t−

(

rtdt +m∑

k=1

θkt (θ

kt dt + dW k

t ) +d∑

k=m+1

θkt

1 − θkt (h

kt )

−1/2(θk

t dt + dW kt )

)

(3.1)

for t ∈ R+, with Sδ∗0 > 0. Since a GOP is uniquely determined by (3.1), up to

its initial value, which we set to Sδ∗0 = 100, we call Sδ∗ the GOP. The optimal

growth rate of the GOP is given by

gδ∗t = rt +

1

2

m∑

k=1

(θkt )

2 −d∑

k=m+1

hkt

(

ln

(

1 +θk

t√

hkt − θk

t

)

+θk

t√

hkt

)

(3.2)

for all t ∈ R+. The corresponding vector of optimal fractions is

πδ∗,t = (π1δ,t, . . . , π

dδ,t)

⊤ = (b−1t )⊤ct, (3.3)

6

where the predictable vector process c = ct = (c1t , . . . , c

dt )

⊤, t ∈ R+ is given by

ckt =

θkt for k ∈ 1, 2, . . . ,m

θkt

1−θkt (hk

t )−1/2 for k ∈ m + 1, . . . , d (3.4)

for t ∈ R+. Since the GOP is a strictly positive portfolio, by (2.19) we know thatthe optimal fractions of the GOP are nonnegative:

πjδ∗,t ≥ 0 (3.5)

for all t ∈ R+ and all j ∈ 0, 1, . . . , d.It is shown in Platen & Heath (2006) that the GOP has remarkable propertieswhich single it out as the best performing portfolio according to various objec-tives. For instance, it has the maximum long term growth rate. That is, after asufficiently long time the trajectories of the GOP almost surely outperform thoseof every other portfolio. This fascinating property is summarized in the followingstatement, see Platen & Heath (2006) for the proof:

Theorem 3.2 The GOP Sδ∗ has the largest long term growth rate among all

strictly positive portfolios Sδ ∈ ν+in the sense that

lim supT→∞

1

Tln

(

SδT

Sδ0

)

≤ lim supT→∞

1

Tln

(

Sδ∗T

Sδ∗0

)

(3.6)

almost surely.

Due to the result of Theorem 3.2 any investor who has a sufficiently long timehorizon should invest according the GOP. Theoretically, if one has calibrated anappropriate model, then one can determine the optimal fractions according to(3.3). However, as pointed out in Frankfurter, Phillips & Seagle (1971), Merton(1980), Jorion (1986) and Broadie (1993) it is unrealistic to assume that one canestimate reliably expected returns from available data.

4 Diversified Portfolios and Approximate GOPs

Since the sufficiently accurate estimation of the fractions of the GOP seems inpractice not to be feasible we will describe a result that allows its approximation.More precisely, we describe a diversification theorem that will allow us to identifyproxies for the GOP. For each d ∈ N , let S(d) denote the market comprising S0

t

and the d risky assets S1t , . . . , S

dt . In the sequel, Sδ

(d)(t) denotes the value of aself-financing portfolio made up of assets from S(d).

7

Definition 4.1 A sequence of portfolios (Sδ(d))d∈N is said to be a sequence of

diversified portfolios (DPs), if there exist strictly positive constants K1, K2 andK3 ∈ (0,∞), independent of d, such that, for all d ∈ K3, K3 + 1, . . . one has

|πjδ,t| ≤

K2

d1/2+K1(4.1)

almost surely, for all j ∈ 0, 1, . . . , d and all t ∈ R+.

Intuitively, this means that in large markets a DP invests only small fractions ineach primary security account. Alternative definitions of diversification can befound in Litterman and the Quantitative Researches Group (2003), Luenberger(1997) or Fernholz (2002). We now define the so-called specific volatilities bysetting

σj,k(d)(t) =

θkt − bj,k

t if k ∈ 1, 2, . . . ,mθk

t − bj,kt

(

1 − θkt√hk

t

)

if k ∈ m + 1, . . . , d (4.2)

for all t ∈ R+ and all j, k ∈ 1, 2, . . . , d. We also set σ0,k(d)(t) = θk

t for all t ∈ R+

and k ∈ 1, 2, . . . , d. The specific volatilities are required to be finite. That is,for all d ∈ N , T ∈ R+ and j ∈ 0, 1, . . . , d, we suppose that

∫ T

0

d∑

k=1

(

σj,k(d)(t)

)2

dt ≤ CT < ∞ (4.3)

almost surely, where CT is some finite AT -measurable random variable, indepen-dent of d. Furthermore, it is assumed that

σj,k(d)(t) <

√

hkt (4.4)

holds almost surely, for all t ∈ R+, k ∈ m + 1, . . . , d and j ∈ 0, 1, . . . , d. Thetotal specific volatility with respect to the kth source of trading uncertainty is attime t defined by

σk(d)(t) =

d∑

j=0

|σj,k(d)(t)| (4.5)

for all t ∈ R+ and all k ∈ 1, 2, . . . , d.To establish the diversification theorem we require the following regularity prop-erty:

Definition 4.2 A sequence of markets (S(d))d∈N is called regular if there existsa constant K5 ∈ (0,∞), independent of d, such that

E(

(σk(d)(t))

2)

≤ K5 (4.6)

for all t ∈ R+, d ∈ N and k ∈ 1, 2, . . . , d.

8

The property above can be interpreted as saying that each source of tradinguncertainty influences only a restricted number of primary security accounts,when these are denominated in units of the GOP. From now on we assume that(S(d))d∈N is a regular sequence of markets.

For given d ∈ N and a portfolio Sδ(d) in the market S(d), the tracking rate Rδ

(d)(t)measures the distance from this portfolio to the GOP as follows:

Rδ(d)(t) =

d∑

k=1

(

d∑

j=0

πjδ,t σ

j,k(d)(t)

)2

(4.7)

for t ∈ R+. One can show by (2.17), (3.1) and (4.7) that Rδ∗(d)(t) = 0.

Definition 4.3 A sequence (Sδ(d))d∈N of strictly positive portfolios is called a se-

quence of approximate GOPs if the corresponding sequence of tracking rates van-ishes in probability. That is, for each ε > 0 we have

limd→∞

Rδ(d)(t)

P= 0 (4.8)

for all t ∈ R+.

The following diversification theorem is proved in Platen (2005a).

Theorem 4.4 For a regular sequence of markets (S(d))d∈N , each sequence (Sδ(d))d∈N

of DPs is a sequence of approximate GOPs.

Subject to the above regularity condition, this result states that any portfoliowith small fractions in all the primary security accounts is a reasonable proxy forthe GOP, if the market is large enough. It is highly significant that the validityof this statement is model independent.

5 Construction of Indices by Portfolio Generat-

ing Functions

Market indices are usually conceived as measures of general market performance.We aim to construct an index that measures the performance of the global worldstock market. Usually such an index would be constructed by using the ratiosof the market capitalization of each security in the market to the total marketcapitalization as the fractions for investment, see Scowcroft & Sefton (2003).Stocks with larger market capitalizations would then have larger fractions. Thistype of index is called a market capitalization weighted index (MCI) and is widely

9

used as a benchmark in investment management. However, since such an indextends to be dominated by a relatively small number of stocks, it is, in general,not a DP. Therefore, an MCI is unlikely to be an ideal proxy for the GOP

Our primary objective is to construct an investable, diversified global stock mar-ket index, which eventually could be useful as an enhanced index fund. Thisindex shall perform well in the long term and will be called world stock index(WSI). The construction of such a WSI shall be rule based and data driven. Itshould avoid subjective decisions.

In practice, the number of risky assets used to construct a portfolio is obviouslyfinite and often not even all that large. In this paper we shall construct proxiesfor the GOP comprising 104 sector stock market accumulation indices and riskyprimary security accounts.

Since the fractions in the GOP are nonnegative, we only consider portfolios with-out short sales, that is, all fractions must be in [0, 1]. A very simple DP withnonnegative fractions is the equally weighted index (EWI) with

πjδEWI ,t =

1

d(5.1)

for all j ∈ 1, 2, . . . , d.Let Sδ = Sδ

t , t ∈ R+ be the value process of a given portfolio and denote byπδ,t = (π1

δ,t, . . . , πdδ,t)

⊤ ∈ Rd the vector of its fractions at time t. This portfolio could

be, for example, the MCI or the portfolio corresponding to an asset allocationstrategy that permits short sales. Now, define a portfolio generating function(PGF) A : R

d → [0, 1]d that maps the vector of fractions, πδ,t ∈ Rd, into a vector

of nonnegative fractions

πδ,t = (π1δ,t

, . . . , πdδ,t

)⊤ = A (πδ,t) ∈ [0, 1]d (5.2)

such that∑d

j=1πj

δ,t= 1 for all t ∈ R+, see Fernholz (2002). Note that there is

considerable freedom for choosing a PGF. Many different classes of portfolios canbe generated in this way. In particular, we are interested in PGFs that make S δ

a DP in the sense of Definition 4.1. By virtue of Theorem 4.4 this leads to asequence of approximate GOPs .

We now describe a useful example of a PGF. Let p ∈ [0, 1] be an arbitraryconstant. The jth fraction of (5.2) is given by

πj

δ,t=

(πjδ,t + µt)

p

∑dl=1(π

lδ,t + µt)p

(5.3)

for all j ∈ 1, 2, . . . , d and t ∈ R+, where

µt ≥∣

∣

∣

∣

infj

πjδ,t

∣

∣

∣

∣

. (5.4)

10

Note that the fractions of S δ obtained in this way are nonnegative. If µt issufficiently large, then S δ will be a DP. It is also noted that as the portfolio sized gets larger, the fractions of S δ tend to 1/d which equals the fractions of theEWI, that is

limd→∞

πj

δ,t=

1

d= πj

δEWI ,t (5.5)

for all t ∈ R+.

The PGF in the above example has some similarity with a PGF introduced inFernholz (2002), where the fractions of the MCI are used to construct a PGF asfollows:

πj

δ,t=

(πjδMCI ,t)

p

∑dl=1(π

lδMCI ,t)

p(5.6)

for j ∈ 1, 2, . . . , d and t ∈ R+, where p ∈ [0, 1]. The resulting portfolio is calledthe diversity weighted index (DWI). For a given p ∈ (0, 1), this PGF magnifiesthe small fractions of the MCI and reduces its large fractions. Therefore, ittransforms the market portfolio into a more diversified portfolio. Still, (5.6) doesnot necessarily generate a DP in the sense of Definition 4.1. Note that in (5.6),if p = 0 the corresponding portfolio is the EWI and if p = 1 it is the MCI.

All PGFs that induce DPs with the same ranking of fractions generate portfolioswith rather similar behavior. For instance, a PGF of exponential form, given by

πj

δ,t=

expλπjδ,t

∑dl=1 expλπl

δ,t(5.7)

for all j ∈ 1, 2, . . . , d, t ∈ R+ and a value of λ slightly greater than zero, pro-duces almost the same dynamics as (5.3) with p = 1. In this case (5.7) is approx-imated by

πj

δ,t≃

1 + λπjδ,t

∑dl=1(1 + λπl

δ,t). (5.8)

We emphasize that diversification yields a certain robustness. We shall demon-strate that small changes in the fractions in the direction of those of the GOPimprove the long term performance of the resulting index.

6 Estimating GOP Fractions

The challenge is to identify potential changes in the estimated fractions of theGOP that may point in the direction of the true fractions of the GOP. Follow-ing Markowitz (1952, 1959), mean-variance portfolio optimization in our currentcontinuous time jump diffusion market was performed in Platen (2006b). It turnsout that the resulting efficient portfolios are always a combination of the savings

11

account and a mutual fund (MF) SδMF which, in general, is different from theGOP. Its value satisfies the SDE

dSδMFt = SδMF

t−

(

rt +d∑

k=1

θkt (θ

kt dt + dW k

t )

)

(6.1)

for all t ∈ R+. Only when the ratios θkt /√

hkt vanish for all k ∈ m + 1, . . . , d

and t ∈ R+, does the MF coincide with the GOP. For simplicity, we assume thatthis is the case for the real stock market, with sector stock market indices takingthe role of primary security accounts. It then follows from (3.3) and (3.4) thatthe vector of the fractions of the GOP satisfies

πδ∗,t = πδMF ,t = (b−1t )⊤θt = Σ−1

t (at − rt1) (6.2)

for t ∈ R+. In this formula Σt = btb⊤

t is the covariance matrix of log-returns andat is the vector of expected returns of the primary security accounts. Using thestandard estimation method the covariance matrix can be estimated from oneyear’s worth of observed daily data, assuming that it does not change too muchover the year. On each day we estimate in this manner the covariance matrixusing the most recent one year observation period. Any estimate of the vectorof expected returns is very unreliable however. Even under the simple Black-Scholes model it is impossible to obtain reasonable estimates for the expectedreturns based on the available data, see, for instance, Merton (1980). This is akey problem in the application of modern portfolio theory. Despite the fact thatexpected return estimates are unreliable, they may still contain some valuableinformation. Therefore, we will exploit potentially relevant information to dateabout expected returns by estimating their values from last year’s data, using thestandard estimation technique.

Relying directly on estimated daily changes of expected returns would cause in-vestors in the estimated GOP continually to make substantial adjustments totheir portions. Also these adjustments are, in general, inexplicable and resultin prohibitive transaction costs. Moreover, the GOP fractions estimated in sucha manner can be substantially negative or positive. A portfolio constructed inthis way, with such extreme fluctuations of its fractions, would not be pursuedby any reasonable investment manager, see Scowcroft & Sefton (2003). Withthis in mind, we propose a different strategy for constructing a well-performingdiversified WSI.

We use the fact that the GOP should be a DP with nonnegative fractions. Toapproximate it we first estimate the daily covariance matrix of log-returns and thedaily vector of expected returns from the most recent one year of data. By (6.2)this yields an estimate for the vector of fractions of the GOP. By applying the PGF(5.3) with a sufficiently large level µt ∈ R+ and an appropriate exponent p ∈ [0, 1],we obtain fractions of a DP. Importantly, these fractions are nonnegative and havethe same ranking as those estimated for the GOP.

12

The diversified index constructed, as described above, is then called the WSI. Itis rather similar to the EWI. Indeed, as we shall see, the fluctuations of both theWSI and the EWI are almost identical for the chosen parameters in our example.However, differences arise in their long term performance, as will be shown below.

7 Comparison of Indices

By using the methodology described above we now construct the WSI whosefractions are determined by the PGF (5.3), with p = 1 and µt = 1000. Theconstituents are 104 sector stock market total return indices, denominated in USdollars, as provided by DataStream Advance and their abbreviations are given atthe end of the paper. The daily data used cover the period from 01 January 1973to 31 August 2006. The data for the sector indices are displayed in Figure 1. The

0

20,000

40,000

60,000

80,000

100,000

120,000

1/1/73 28/9/76 25/6/80 22/3/84 18/12/87 15/9/91 12/6/95 9/3/99 4/12/02 31/8/06

Figure 1: Sector stock market indices

fractions of the WSI are comparable in magnitude. To illustrate this, we plotthese in Figure 2. For comparison, we have also constructed the equally weightedindex (EWI), the diversity weighted index (DWI) and the market capitalizationweighted index (MCI) using Formula (5.6) with p = 0, p = 0.5 and p = 1,respectively. For convenience, all indices have the same initial value of 100 USdollars at the starting date. They are shown separately in Figure 3. The WSI doesnot put an emphasis on the level of market capitalization, but consistently aimsto deduce the fractions of the GOP. In the long run it appears to outperform theDWI, the MCI and almost all sector indices. In our view this is not coincidental,but due to its proximity to the GOP. As shown in (5.5) for a large number of

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0.000

0.002

0.004

0.006

0.008

0.010

0.012

1/1/73 28/9/76 25/6/80 22/3/84 18/12/87 15/9/91 12/6/95 9/3/99 4/12/02 31/8/06

Figure 2: The fractions of the WSI.

EWI

MCI

DWI

WSI

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

1/1/73 28/9/76 25/6/80 22/3/84 18/12/87 15/9/91 12/6/95 9/3/99 4/12/02 31/8/06

$US

Figure 3: Constructed EWI, DWI, MCI and WSI with terminal values3469.92, 4402.71, 7960.45 and 8048.14, respectively.

constituents d, the fractions of the WSI are approximately the same as those ofthe EWI. This results in that the EWI is also in the proximity of the GOP.

We note from Figure 3 that the DWI performed better in the long term than theMCI, but its performance was not as good as the EWI. This is not surprisingsince the PGF (5.6) places the fractions of the DWI between those of the MCIand the EWI.

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We observe the performance of the WSI is slightly better than the EWI, seeFigure 3, this is due to the small fluctuations in the fractions of the WSI anddo not change drastically over time and are quite homogenous, see Figure 2.On the other hand, strict equal weighting of sector indices performed equallywell as the WSI. This suggests that for a DP with large number of constituentssmall fluctuations in the fractions do not have a major impact on the long termperformance of an index. However, the performance of the WSI can only bebetter but not worse than that of the EWI, which is due to the ranking of theestimated fractions of the WSI. It seems that the estimated weights of the GOPcontain some information about the ranking of the true fractions of the GOP.

It appears that the better performance of the WSI is partly due to the diversifiedinclusion of the stock market indices of emerging sectors, in particular softwaresand internet. Furthermore, the estimated weights of the GOP seem to containsome information about the ranking of the true fractions of the GOP.

8 Application of Constructed Index

As described in Platen & Heath (2006), there are many applications where theWSI can be used as proxy for the GOP. In particular, when the WSI dynamicsis properly modeled it allows the real world pricing of contingent claims in anincomplete market. In that case the WSI is taken as numeraire and the pricingmeasure is chosen to be the real world probability measure. Derivative pricesdenominated in the WSI can be interpreted as real world martingales and theexistence of an equivalent risk neutral probability measure is unnecessary. TheWSI has many other applications. As indicated previously, it is a good candidatefor an enhanced index fund with a short selling constraint.

Platen (2002) introduced, the so-called minimal market model, for the GOP andby extension, for the WSI. According to this model the daily log-returns of theGOP should be estimated to be Student t distributed with approximately fourdegrees of freedom. This prediction was investigated empirically in Fergusson &Platen (2006). We apply the same methodology to the indices considered here.We find, as in Fergusson & Platen (2006), that in the wide class of symmetricgeneralized hyperbolic distributions the Student t distribution offers the best fitfor daily log-returns for all the above indices when compared with the normal-inverse Gaussian, hyperbolic and variance gamma distributions.

We plot the empirical Student t log-return densities in log-scale in Figure 4. Theestimated Student t densities for the EWI, DWI, MCI and WSI have degrees offreedom 4.65, 4.62, 4.50, 4.65, respectively. Note that the degrees of freedom ofthe Student t densities are close to the value predicted by the stylized minimalmarket model, see Platen & Heath (2006). This is a rewarding result because wealready singled out the WSI as a good proxy for the GOP, due to its best long

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EWI

DWI

MCI

WSI-8

-6

-4

-2

0

2

4

-0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08

Figure 4: Estimated Student t densities of the log-returns of EWI, DWI, MCIand WSI.

term performance.

Conclusion

The proposed rule based method of constructing a proxy for the growth optimalportfolio has specific advantages over the methodologies of diversity weightingand market capitalization weighting. First, it relies entirely on observable andestimable information. Second, the approach is theoretically justified by searchingfor the diversified outperformance of the long-term growth rate. Lastly, it allowsthe investors to understand the index movements on the basis of expectations andcovariances of log-returns of the underlying assets in a classical sense and alsopresents a consistent platform for constructing diversified world stock indices.The proposed methodology is very tractable and easy to implement.

Acknowledgements

The authors like to thank Frank Howard for collecting data from DataStreamAdvance and for a preliminary study of sector and regional based indices involvingthe EWI and the MCI. They also thank Wolfgang Breymann for providing themaximum likelihood estimation algorithm used at the end of the paper. Alsothanks go to Hardy Hulley for valuable and stimulating comments on the paper.Finally, support by the ARC grant DP0559879 is acknowledged.

The WSI as shown in Figure 3 can be found at the webpage of the second author.

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Abbreviation

Sector stock indices are abbreviated by DataStream Advance: BMATS,GASDS,INSUP,

CHMSP,COMPH,FMFSH,HOMES,ELEQP,FORST, HVYCN,MEDAG,CNFIN,IMACH,DEFEN,HCPRO,FINAD,WASTE,

OILEP,OILSV,PIPEL,TYRES,NOFMS,RECSV,STEEL,ELETR,DURHP,FURNS,TOYSG,NDRHP,AUPRT,TRNSV,AUTOS,

APRET,BREWS,DISTV,CLTHG,CONPK,FDPRD,RESTS,CNELE,INVNK,PLTNM,TOBAC,HOTEL,RAILS,PAPER,PUBLS,

HIMPR,BUSUP,BDRET,FDRET,SPRET,MTUTL,CHEMS,ALUMN,TRAVL,PHRMC,OILIN,AEROS,MARIN,GAMNG,DIVIN,

BANKS,MEDSP,ASSET,OFFEQ,LFINS,PCINS,INSBR,ITINT,INVSV,RLDEV,SPFIN,SOFTD,BRDEN,COMMV,GOLDS,

PRSNL,COALM,DGRET,MINES,TELEQ,AIRLN,SEMIC,TRUCK,MEDEQ,REINS,BUSTE,ELECT,FLINS,TELFL,TELMB,

WATER,CMPSV,MORTF,REITS,RECPR,SPCSV,FOOTW,DELSV,BIOTC,SOFTW,DIAMD,INTNT.

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