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Munich Personal RePEc Archive

An alternative to portfolio selection
problem beyond Markowitz’s: Log
Optimal Growth Portfolio
Muteba Mwamba, John and Suteni, Mwambi
University of Johannesburg, University of Johannesburg

October 2010

Online at https://mpra.ub.uni-muenchen.de/50240/
MPRA Paper No. 50240, posted 02 Oct 2013 11:21 UTC

An alternative to portfolio selection problem beyond
Markowitz’s: Log Optimal Growth Portfolio.
Sutene Mwambi∗and Muteba Mwamba†
Department of Financial Economics/Econometrics
University of Johannesburg
March 10, 2010

Abstract
This paper constructs an alternative investment strategy to portfolio optimization model in the
framework of the Mean–Variance portfolio selection model. To differentiate it from the ubiquitously
applied Mean–Variance model, which is constructed on an assumption that returns are normally distributed, our model makes two assumptions: Firstly, that asset prices follow a Geometric Brownian
Motion and that secondly asset prices are Log-normally distributed meaning that continuously compounded returns are normally distributed. The traditional Mean–Variance optimization approach
has only one objective, which fails to capture the stochastic nature of asset returns and their correlations. This paper presents an alternative approach to the portfolio selection problem. The proposed
optimization model which is an optimal portfolio strategy is produced for investors of various risk
tolerance, taking into account the stochastic nature of the returns. Detailed analysis based on log–
optimal growth optimization and the application of the model are provided and compared to the
standard Mean–Variance approach.

1

Introduction to Portfolio Optimization

In this research paper, we construct the growth optimal portfolio (GOP) which is a strategic asset
allocation process more suited for those investors with a long term investment view and wish to maximize
their expected utility of terminal wealth. Growth optimal portfolio arise from the notion of computing
the investment internal rate of return which in essence is bent on constructing those portfolio that have
maximal growth. In principle we build a portfolio of risky asset that maximizes the geometric mean.
The paper has primarily been inspired and written in the framework of Modern Portfolio Theory
(MPT). Portfolio optimization in the context of portfolio theory is a classical problem in mathematical
finance which has spawned a great amount of important academic work. In particular it is one of the
well studied classical problems. Central to (MPT) is the Mean–Variance optimization theory(MVO), an
important model which was a major breakthrough developed in the 50s and 60s by Markowitz (1952,
1959), his paper opened a new era in the theory of portfolio selection. It plays a important and critical
role in determining passive portfolio investment strategies for rational investors and quantifies precisely
the relationship between risk and return. His theory set out a way of diversifying investment portfolios
so that for any degree of risk, the investor got the best return possible, or alternatively, for any risk, the
investor bore the lowest risk. Tobin (1958) built on Markowitz work with the formulation of the “Capital
market line” portfolio. Following Markowitz and Tobin, the theory generated a lot of interest and the
general equilibrium model capital asset pricing model “CAPM” was independently developed by Sharpe
(1964), Lintner (1965), Mossin (1966) .
Markowitz theories pervade the finance industry and are well-known to almost everyone vested in
portfolio management. Nonetheless, the Mean–Variance theory suffers from certain well-known drawbacks. The most notable one is that, it is a static optimization problem and is only concerned with
∗ Student
† Lecturer University of Johannesburg

1

solving single period investment strategy hence it can not optimally be applied to any rebalancing period. Unlike the GOP, it is a strategy that can be applied in multi–period framework and can also be
used for any rebalancing period such as a year, a month, a week or a day. However, in our case we
investigate the application in a continuous time period. Portfolio rebalancing is also what is commonly
known as “volatility pumping”. When volatility is high and a portfolio is rebalanced, an investor would
normally yield higher returns. In fact while volatility(risk) is considered bad in static portfolios, it is a
positive for growth optimal portfolio.
In order to understand growth portfolios approach, we begin with understanding assets dynamics and
how assets prices evolve over time. We characterize that asset behavior follow a stochastic process and
can adequately be modeled by stochastic differential equation the “Geometric Brownian motion(GBM)”.
Bachelier (1900) was the first to discover that stock prices changes are random and unpredictable. The
differential equation has been used in the field of asset pricing see Black & Scholes (1973) because of its
unique positivity property. This property of positivity is precisely the nature exhibited in a stock price
since stock prices can not be negative but returns can, hence the logarithmic or geometric growth.
The Mean–Variance model requires two model input parameters, thus, the first and second
 moments

and the model is built on the assumption that asset returns are normally distributed N µ, σ 2 and
therefore that the portfolio optimal solution is quantified on the premise that we know collectively the
mean vector and variance-covariance matrix. The growth model makes the assumption that continuously
compounded asset returns are normally distributed and that price relatives are log–normally distributed
a concept which was studied and introduced in financial economics in the late 1950s by Osborne (1959).
Campbell, Lo & MacKinlay (1997) outline two remarkable drawbacks of assets returns being independent and identically normally distributed. Firstly, most financial assets show limited liability meaning
that the maximum loss for an investment is equal to the total investment and no more. This implies
that the minimum return achievable is −100% 1 However, normal distribution is defined over the range
[−∞, ∞] hence, the assumption of normality clearly violets this lower bound of −100%.
Secondly, if single period returns are assumed to be normal, then multi-period returns cannot be
normal since they are simply products of single period returns. In probability theory, the sum of normal
single period returns are of course normal, but these sums have no economically meaningful interpretation. When analyzing optimal portfolios over longer time periods or on multiple time periods, the
normality hypothesis of returns leads to problems. This is because long-term returns are far from being
normally distributed. Undeniably, even over a single year, it can be shown that Log-normal distribution,
while still not perfect, is a much better approximation to the distribution of the observed historical
returns for common financial assets like stocks and bonds see Norstad (2005).
As discussed above, it is quite intuitive to see that asset prices cannot be negative but can only be
infinitely positive. It then follows logically that, the investment gross return at any point in time t,
t
) is bounded below. This fact is the basis for suggesting that asset price returns should
1 + Rt = ( SSt−1
t
be modelled as continuously compounded rt = ln( SSt−1
).2 . The Log-normal distribution often shows a
better fit to historical asset returns as compared to the traditional normal distribution when observed
over a longer time horizon. This view is also shared by Palczewski (2005), that asset returns are far
from being normal and deviate from the independent identically distribution (i.i.d) assumptions. This
formulation is extremely important given both the multitude of areas within economics where portfolio
models have found applications and the increasing acceptance of the Log-normality of price relatives in
the economic literature.
Any work such as this builds on the advances and ideas of so much knowledge already ascertained in
the field and that is this paper is based on the breakthroughs and developments of countless researchers
in the fields of finance and statistics and most of all mathematics. Although they are too numerous to
name in this space, I acknowledge the foundation that they have built in the field. The remainder of this
paper will be elucidated in the next four sections which are described in detail as follows:

1 It is easy to see that, since asset prices cannot be negative, the price relative

value that Rt can have, given that the gross return is 1 + Rt = SSt
2 If gross return is defined as 1 + R

St
t = (S

t−1

t−1

St
> 0. Then by definition the smallest
St−1

is -100%.

) then taking logarithms on both sides yields rt = ln(1 + Rt ) = ln( SSt ).
t−1

This implies that St = St−1 ert hence the expression ln( SSt ) is a representation of a continuously compounded rate of
t−1

return. Also, Rt = ert − 1, rt ∈ [−∞, ∞] unlike Rt , rt is not bounded

2

Section 2 and Section 3 : Introduces literature review and the theoretical background to Log–
optimal growth portfolios. The chapter begins the discussion on the central limit theorem and asset
dynamics paying particular attention to the Log–normality of asset prices and how the geometric mean
relates to simple average returns. It further presents the continuous time mathematics on Geometric
Brownian Motion, Ito’s Lemma and the quadratic form of the log–optimal optimization problems. Finally
the section deals with the Lagrange multipliers and how the Markowitz quadratic optimization problem
is solved using matrix algebra. We study optimization where an investor has more than one risky asset
in his portfolio and when a riskless asset is added to a portfolio of risk assets. Finally, the section finishes
with the capital market line and construction of tangency portfolio.
Section 4 and 5: The data requirements and test results are presented followed by the research conclusion. Five stocks randomly selected were used to illustrate how log optimal portfolio are constructed.
The data used are the monthly prices downloaded from I-Net bridge, covering the period from the 30
November 1999 up to 31 October 2009. That makes up nine years of monthly data, and in total 120
monthly observations. The assets under consideration were randomly selected from the Johannesburg
Securities Exchange namely, BHPBILL, WBHOVCO,NEDBANK,ABIL and SAB and these have been
chosen from different economic sectors.

2

Previous work and contribution to existing literature

Over the past 50 years a large number of papers have been dedicated to the Growth Optimal Portfolio
(GOP). In financial literature it has been applied in as diverse connections as portfolio theory and
gambling, utility theory, information theory, game theory, theoretical and applied asset pricing, insurance,
capital structure theory and event studies.
Theoretically, it can be shown that the GOP, which maximizes expected logarithmic utility from
terminal wealth, is the portfolio that almost indisputably outperforms all other strictly positive portfolios
after a sufficiently long time. Theory has its roots dating back to Benoulli (1954)with the introduction
of “expected utility theory” and using the geometric mean as a performance measure of risky portfolios.
Kelly (1956) is considered the father of the growth optimal portfolio where in his paper he applied it to
the gambling setting. He proposed maximizing the expected exponential growth rate of an investment
capital as an investment strategy in a gambling. It is comforting to know that there is a sound theoretical
basis for advocating a growth portfolio investment strategy. The Kelly view:that maximizing investment
growth of value is a self-evident superior strategy, probably resonates more with the investment sector.
However as Christensen (2005) notes that, the first origins of the GOP can be attributed to William
(1936). Following, William some remarkable papers on this theory can also be credited to Latane (1959),
Hakansson (1971), Elton & Gruber (1974a) and Fernholz & Shay (1982).
The most recent work found on the GOP is by Estrada (2010) with a comparative analysis between
the GOP and maximization of the Sharpe ratio. Hunt(2000, 2002, 2005b) attempts to test a simple,
practical investment strategy based on portfolios selected to have maximum expected growth rate. Le &
Platen (2006) on the other hand applies the theory studying well diversified world stock indices. Elton &
Gruber (1974b) argued the practicality of the geometric mean as another alternative portfolio selection
criterion in the GOP framework. The development of an algorithm to maximize the geometric mean
with assumption log-normally distributed asset prices was entailed by this paper. The methodology was
based on the proof that that the maximum geometric mean lies on the efficient frontier in the Mean–
Variance space. However, in the conclusion, they pointed out one aspect noting that the construction of
the portfolio has major implications for economic theory.
In Weide, Peterson & Maier (1977), it is noted that not much has been documented in terms of
finding the optimal solution to the wealth allocation problem of GOP. An investigation into the necessary
restrictions under which solutions exist for the case where the retuns distribution is discrete was done.
The restrictions were formulated based on the assumption that returns ri are discrete random variables
which can assume a finite number n of combinations of values.
On a different note, in Roll (1973), it is stated that if investors wish to maximize the probability of
achieving a given level of wealth within a fixed time, they should choose the GOP that is, the portfolio
with highest expected rate of increase in value. The paper investigated the repercussions for observed
common stock returns of all investors who choose, under utility maximization, such a portfolio. It is

3

further pointed out that the GOP model has caused a schism in academia. In some tests it performs
well while in other cases the results are rather perplexing. Again it has been shown to outperform the
Mean–Variance model but the results are so close due to the operational similarity of the two models.
The effects of long-term out-performance of any strictly positive portfolio by the GOP has been
studied, for instance, in Laten (1959). Latane is perceived to be the main scholar in financial economics
to have introduced the geometric mean as another approach to portfolio selection criteria and since then
the theory has recently received some attention in the academic circles.
Others like, Breiman (1961), Markowitz (1976) and Long (1990) also have made remarkable contributions to the theory. In principle, the GOP is the portfolio that cannot be beaten in any reasonable
systematic way. Reviews of this portfolio properties can be found in Hakansson & Ziemba (1995) where
in their investigation the GOP focused mainly on the relevance of the GOP for investment and gambling
setting.
The most far reaching study can be linked to other researchers like Luenberger (1998) and Hakansson
(1971) where the use of GOPs on the basis of investors expected utility portfolio maximization was
justified. However, the paper compared the Mean–Variance approach to portfolio selection with the
capital growth model. A comparison of the two models partly in terms of long-run results may accordingly
seem inequitable to the Mean–Variance approach. The reason being that former is a single period model
while the later is by definition a multi period model. However, both models are ultimately portfolio
models which claim to offer guidance to sensible portfolio choices at any given decision point.
In Platen (2005), the various roles that the GOP plays in finance are discussed and a conclusion is
made that the GOP can be interpreted as a fundamental building block in financial market modeling,
portfolio optimization and risk measurement and the various ways that the GOP is the best performing
portfolio are described. Elton, Gruber, Brown & Goetmann (2003) studied the GOP in the case of a
continuous market. No doubt, this property has fascinated many researchers and created a huge and
exciting literature on growth optimal investments, a field of study for financial economics.
In Samuelson (1971), evidence for the use of the geometric mean is given and argues, that the law
of large numbers or of the central limit theorem when applied to logs can show that a maximumgeometric-mean selection criterion does indeed make it ”virtually certain” that, in a ”long” sequence one
will end with a higher terminal wealth and utility, are given. The prescription to select a portfolio that
maximizes an investors expected utility is hardly new. Nor are applications in the area of asset allocation.
Particularly relevant in this respect is also very recent work by Cremers & Page (2005), and Tim &
Kritzman (2007) in which a full-scale optimization numerical search algorithm is used to find an asset
allocation that maximizes expected utility under a variety of assumptions about investor preferences. The
GOP, however, is unique in that it has dominating characteristics over all other investment strategies.
The determination of the true population mean and variance-covariance matrix is indeed a significant part to the entire optimization problem. In order to estimate the expected portfolio return, the
Mean–Variance methodology uses historical data on the assumption that the sample mean is a true representation of the population mean. The sample mean and sample variance are simply average values of
a finite data sample, and are not entirely true representation population parameters. As a consequence
of that, it is imperative that we revisit how we parameterize the Mean–Variance model. In addition, due
to the amount of volatility in asset returns, sample means may not provide accurate estimates of the true
population means. As a consequence, utilizing the sample version into a model expecting the population
equivalent can produce off-the-wall results. In particular, an investor could be inclined to invest large
amounts of money into securities and sectors that performed better than expected in the past.
Supporting this statement, Merton (1980) admits that there has been diminutive academic research
on estimating the expected return of assets in comparison to extensive research that has been conducted
since the pioneering work of Black & Scholes (1973) in option pricing where they used the second moment
as the input parameter in the model. Whether this argument still stands in the 21st century is yet to
determined. Perhaps, now the factor models or Bayes-Stein models address this shortcoming. Another
paper reinforcing this point is by Black (1993), who also agrees that estimates of expected return based
on past data are indeed inaccurate and we need theory to quantify returns.

4

3

Methodology: Log Optimal Portfolio Selection Criterion

As a preamble to understanding the growth optimal portfolio, we start by introducing the theory on the
geometric mean followed by a background to the mathematics of continuous time processes underlying
stochastic variables.

3.1

Geometric Mean and Asset Prices as Random Processes

In an investors wealth perspective, the growth of the asset over the entire period [0, T ] should be expressed
as a geometric mean. The use of geometric mean has far better properties in terms of the interpretation
of asset returns as compared to the arithmetic mean. In analyzing wealth over a longer period of time,
the geometric mean conveys what the average financial rate of return would have been over the whole
duration of the investment period.
t
−1 represents price return
Combining the results discussed in the previous section, where Rt = SSt−1
for a single period, between dates t − 1 and t. Then for n single periods returns, a sequence of asset
n
n
returns defined as {Rt }t=1 based on the sequence of asset prices {St }t=0 , the geometric mean return or
the so-called “time-weighted rate of return,” r(0,n) , can be expressed as follows:
r(0,n) =

n
Y

! n1

(1 + Rt )

t=1

− 1.

(1)

There is, however, a mathematical relationship between the geometric mean and the arithmetic mean of
logarithmic returns. The geometric mean can technically be viewed as an average of logarithm values of
asset returns. Thus:
! n1
“
”
n
Pn
St
Y
1
ln
(1 + Rt )
= e n t=1 St−1 .
(2)
t=1

1

Pn

ln

“

St

”

1

ln

“

Sn

Sn−1

···

S1

”

Sn−1 Sn−2
S0
and this represents the expresHowever, this also implies that, e n t=1 St−1 = e n
 
  n1
Q
n
. Hence from equation [2] we have ln SSn0 = ln ( t=1 (1 + Rt )) . This equation implies that
sion SSn0

the geometric mean is just a function of the mean of the log-returns 3 , if they are interpreted as statistics,
that is:
h “
”i
S
E ln S t

r(0,n) = e

t−1

− 1,

(3)

from the above equation, we can therefore infer that the objective of the Growth Optimal Portfolio (GOP)
rests in maximizing the mean (Gmax ) which is the same as maximizing the mean of the log-returns due
to the monoticity of the exponential function, thus:

  
St
,
(4)
Gmax = max E ln
St−1
where E is the expectation operator.
n
Now let {Rt }t=1 be a sequence of independent and identically distributed (i.i.d.) random variables
with finite mean
then so is the sequence:{Zt |Zt = ln (1 + Rt ) and t ∈ 1, 2, · · · , n} . Then it
 variance,
 and
Pn
is clear that,ln SSn0 = i=t Zt . Choosing a time horizon T and letting △t = T /n, we demonstrate the
 
distribution dynamics of ln SSn0 . First we look at the Central Limit Theorem (CLT) which says that:

Theorem 3.1. [Central Limit Theorem] Let random variables Y1 , Y2 · · · , be independent and identically
distributed with finite mean µ and variance σ 2 generated from any distribution. Then as the sample
sizes gets large, the sampling distribution of the mean approaches a normal distribution with mean µ and
variance σ 2 /N .
3 This is stated more formally in the textbook by Luenburger, D. (1998), page 420, were the rationale for the result is

as a result of the Strong Law of Large Numbers

5

 
Having the CLT in mind and that E(Zt ) = µ△t and V(Zt ) = σ 2 △t, then ln SSn0 /n is approximately
normally distributed:
 


ln SSn0
△t
.
(5)
∼ N µ△t, σ 2
n
n
 
 
√

Hence, ln SSn0 ∼ N µT, σ 2 T . Another way of expressing this is in the form ln SSn0 = µT + σ T ǫ.
where ǫ has a standard normal distribution. In particular, it is not an unreasonable assumption to
assume that the Zt above are normally distributed i.i.d. and then we have that (by definition),


p
St
ln
(6)
= Zt = µ△t + σ △tǫt ,
St−1

which is by definition just the discrete random walk, ǫt being just a sequence of standard normal variables.
Note that by the following theorem on random variable transformation;
Theorem 3.2. [Transformation of random variables] Let X be a continuous random variables having
probability density function fX . Suppose g (x) is strictly monotone, differentiable function of x. The
random variable Y defined by Y = g (X) has probability density function given by

 d −1

g (y)
if y = g (x) for some x,
fX g −1 (y) dy
fY (y) =
0
if y 6= g (x) for all x,


we have that, SSn0 ∼ Log-normal µT, σ 2 T which is an important result showing that price relatives
are log-normally distributed, given the assumptions above.

3.2

Geometric Wiener Process

We have so far given a thorough introduction to asset dynamics and specifically the stochastic process
of how asset prices change over time in discrete terms. Having said that, this section provides a more
rigorous approach to asset behavior in continuous time mathematics. We now formalize the stochastic
growth model for the optimal portfolio selection optimization problem. In continuous time mathematics,
the log normal random walk model in equation for non dividend paying asset is usually formulated in
terms of the following stochastic differential equation.
√
d ln S(t) = µdt + σ dtǫt .

(7)

Where the parameters µ and σ represent percentage drift rate or instantaneous rate of return and the
percentage rate of volatility respectively.
The two parameters could be constant or could depend on the
√
stock price and/or time. The term dtǫt is the standard wiener process where ǫt ∼ N (0, 1), the process
represents something like the infinitesimal stochastic change in ln S(t) over an infinitesimal instant of


2
time. It then follows that, µdt + σdw(t) ∼ N µdt, σ 2 dt , hence dS(t)
S(t) ∼ N µdt, σ dt . Equation [7]
√
cannot be interpreted as an ordinary differential equation, since the Brownian paths dtǫt are not
differentiable with respect to time. It was precisely for the purpose of dealing with differential equations
incorporating stochastic differentials that Itō developed what is now called the Itō calculus. The left hand
4
side represents the percentage change in asset value and it equivalent to stating it as d ln S(t) = dS(t)
S(t) ,
substituting the expression into equation [7], we obtain the following expression,
d ln S(t) =

dS(t)
S(t)

⇒ dS(t)

√
= µdt + σ dtǫt .

(8)

√
= µS(t)dt + σS(t) dtǫt .

(9)

4 If y = ln f (x), then the derivative of y is given by d ln S(t) = dS(t)
S(t)

6

√
which is the Itōs diffusion process, now let dtǫt = dw, Itōs lemma states that for the process above
and a function F (S, t) which is twice continuously differentiable in both S and t then, the change in
F (S(t), t), dF (S(t), t) is also an Itō process, subsequently,


∂F
∂F
1 ∂2F 2 2
∂F
dF (S, t) =
µS +
+
σ
S
σSdw.
(10)
dt +
2
∂S
∂t
2 ∂S
∂S
We know that S(t) follows the process defined by equation [9]. Then, the process followed by d ln S(t)
can easily be solved by defining a function F (S (t)) = ln (S (t)) and since
this is a function of only S

and not t, then the Itō lemma takes the form dF = µSF ′ + 12 σ 2 S 2 F ′′ dt + σSF ′ dw. Applying the Itō
lemma, we derive the governing process outlined by F (S(t)) = ln S(t). Then, the process followed by dF
is


√
1
dtǫt
dF = µ − σ 2 dt + σdwt where dF is representing d ln S and dwt is
2
(11)

Equation [11] concludes that d ln S follows the Wiener process. 5
Integrating and set the initial value condition F (0) = ln S (0) yields;

Z t
Z t
1
µ − σ 2 ds +
F (t) = F (0) +
σdw
2
0
0


1
= F (0) + µ − σ 2 t + σw(t).
2
We know that F (s) = ln S then,


1
ln S(t) − ln S(0) = µ − σ 2 t + σw(t)
2
 


1
S(t)
= µ − σ 2 t + σw(t)
ln
S(0)
2
1 2
S(t) = S(0)e(µ− 2 σ )t+σw(t) ,

(12)

discretized, this can be written as:
√

1 2
St = St−1 e(µ− 2 σ )△t+σǫ △t .

(13)

In essence we are stating that stock prices follow a Geometric Wiener process and that over any time
increment, △t,
with mean α△t, where
 the distribution of logarithmic returns is normally distributed
√
α = µ − 21 σ 2 , proportional to the time increment and the volatility, σ △t is proportional to the square
root of time increment.




S(t)
The probability distribution function of X = ln S(0)
∼ N µ − 12 σ 2 t, σ 2 t is;
f (X) = √

1
2πσ 2 t

− 12

e

"

(

)

X− µ− 1 σ 2 t
2
√
σ t

#2

(14)

S

ln S t
X
(µ− 12 σ2 )t+σt is lognormally distributed
and through Jacobian variable transformation
  Y= e =e 0 =e
  

with parameters µ − 21 σ 2 t and σ 2 t. E ln SS0t
= µ − 12 σ 2 t and the variance is, V ln SS0t
= σ2 t
This statement stated differently is saying that the the change in asset prices when modelled using the
Geometric Brownian motion are log-normally distributed. Applying theorem 3.2, the probability density
function of Y is as follows;
5 notice that the left hand side is a expression of the form a + bx and if x ∼ N (0, 1) then y = a + bx ∼ N

“`
√ ”
´
d ln S ∼ N µ − 12 σ 2 dt, σǫ dt

7

`

´
a, b2 . Hence

− 21

1

f (Y ) = √
e
y 2πσ 2 t

"

(

)

ln y− µ− 1 σ 2 t
2
√
σ t

#2

(15)

Putting all the pieces of information discussed so far together, we conclude that the Geometric
Brownian motion is lognomally distributed thus,



St
1
t, σ 2 t and
∼ LN
µ−
S0
2


 
1 2
1 2
The expected value E SS0t = E e(µ− 2 σ )t+ 2 σ t = eµt
If we put all the pieces of information discussed so far, then, by employing the GBM to estimate
the model input parameters, we significantly change the whole Markowitz’s setting and in particular we
construct a portfolio which maximizes the expected growth rate hence the name Log-Optimal growth
portfolio. The methodology of how this is achieved is the topic of our discussion the last chapter.

3.3

Log Optimal Portfolio Selection Criterion

We also focus on the key analytic tools employed in portfolio optimization methods where we introduce
the elements of stochastic calculus as an important tool in modeling of financial processes, see Wilmott
(2001) for a detailed introduction on the subject of SDE6 . We further review the standard analytic
approach based on mean-variance assumptions and then describe a more general procedure that assumes
investors seek to maximize expected utility. We will show, later, that mean-variance procedures are
special cases of the more general expected utility formulations. We term the more general approaches
Expected Utility Optimization and the traditional methods Mean-Variance optimization.
The problem dealt in this paper realistically deals also with the problem of rebalancing the portfolio
in any time horizon. In small time horizons, the portfolio rebalancing a log optimal growth portfolio is
constructed as a as continuous-time stochastic process.
Suppose an investor has a choice of investing in a finite number of correlated risky assets represented
by an n × 1 vector S = [S1 , S2 , · · · Sn ] where, without loss of generality, it can be assumed that each Si
n
represents the asset and its price. Then, for each asset {Si }i=1 , the price is assumed to be governed by
a Geometric Brownian Motion:
dSi
= µi dt + σi dwt , where i ∈ {1, 2, . . . , n} .
Si

(16)

√
By our earlier assumption, the assets prices are correlated through the Wiener process dwt = ǫi dt
x2
1
with a probability density function fdwt (x) = √2πdt
e− 2dt and the covariance, Cov [dwi , dwj ] = σij dt
where σij can be understood as the correlation between assets i and j. In the Markowitz framework the
Wiener vector, dw = [dw1 , dw2 , . . . , dwn ] has a multivariate Gaussian distribution hence,
1

′ −1
1
e[− 2 (x−µ) Σ (x−µ)] .

(17)
(2π) |Σ|
√
The vector µ is just the zero vector since ǫi dt ∼ N (0, dt), |Σ| is the determinant of the covariance
matrix Σ, and the n × n covariance matrix is defined having diagonal elements σi2 dt, for i = j and σij dt
for i 6= j. The symbol e is an exponential function representing the number approximately 2.7182818
In the multivariate framework we have the log price change for time (t − t0 ) from the univariate
solution in equation [12] as normally distributed,




σ2
(18)
ln Si,t ∼ N ln Si,t0 + µi − i (t − t0 ) , σi2 (t − t0 ) .
2
fdw (x1 , . . . , xn ) =

n
2

6 Stands for stochastic differential Equation

8

1
2

The asset prices at some future deterministic date (t), given that we know the asset price at time (t0 ) is
lognormally distributed and the standard deviation is proportional to the square root of the time interval
of how far ahead we are looking. From Equation [18] we have:
 
 


 
S(i,t)
S(i,t)
1 2
E ln
= µi − σi (t − t0 ) And the variance is V ln
= σi2 (t − t0 ).
S(i,t0 )
2
S(i,t0 )
Pn
n
Let the sequence of portfolio weights be denoted {ωi }i=1 , such that i=1 ωi = 1 represent the budget
constraint. The overall portfolio value P can be formulated in terms of the aforementioned stochastic
processes as follows:
n
X
dSi
dS2
dSn
dS1
dP
ωi
+ ω2
+ · · · + ωn
=
= ω1
P
S1
S2
Sn
Si
i=1

=

n
X

ωi (µi dt + σi dwi ) .

(19)

i=1

One thing to note is that the portfolio is a weighted sum of the assets.
In continuous time, the percentage change in the value of our portfolio, is normally distributed with
mean:
 
 X
n
n
Pt
1 X
E ln
ωi σi,j ωj (t − t0 );
(20)
ωi µi (t − t0 ) −
=
Pt0
2 i,j=1
i=1
and the portfolio variance is:

 
n
X
Pt
ωi σi,j ωj (t − t0 ).
=
V ln
Pt0
i,j=1

3.4

(21)

Deriving the Weights that Maximize Portfolio Growth Rates with Short
Selling Allowed

For n assets with linearly independent
growthireturns, a portfolio of risky assets modelled in continuous
h 
Pt
1
= ω ′ µ − 12 ω ′ Σω. Denote µ = ν + 21 ω ′ Σω as a column
E
ln
time has a growth rate ν = (t−t
Pt0
0)
vector representing the expected growth rates, let Σ denote the variance-covariance matrix of the growth
rates and ω represent a column vector of portfolio weights which are chosen to maximize the growth
rate G. Let θ = 1 be the risk aversion parameter this equal to a unity since it does not affect our finale
solution. In the Markowitz framework the portfolio optimization problem entails minimizing the portfolio
variance for some specified portfolio mean. However the duplex to the problem, is the maximization of
the portfolio growth for some specified portfolio mean- the GOP problem. Symbolically the the two
problems are formulated as follows;



θ ′
ω Σω |ω ′ µ = E (Rp ) ω ′ e = 1
(22)
2
ω


θ
argmax ω ′ µ − ω ′ Σω |ω ′ µ = E (Rp ) ω ′ e = 1 .
(23)
2
ω
Pn
where Rp in the portfolio return defined in the following way: E(Rp ) = i=1 ωi E(Ri ) and Ri is the i-th
asset return, and e is a column vector of ones. Notice that the risk aversion parameter θ
argmin

3.4.1

Minimizing the Portfolio variance (σp2 = ω ′ Σω) for a Specified Mean E(Rp )

The solution to the problem in [22] in the Mean–Variance framework can be easily be derived as follows:

9

Definition 3.1. A portfolio, P , is the minimum variance portfolio of all portfolios with mean return µp
if its portfolio vector of weights ω is the solution to the following unconstrained optimization problem,
n
o
argmin
ω ′ Σω |ω ′ µ = E (Rp ) ω ′ e = 1 .
(24)
2
ω
The Lagrangian function with λ and γ as multipliers is constructed for the optimization problem
above, hence:
ω ′ Σω
− λ [ω ′ µ − µp ] − γ [ω ′ e − 1] .
(25)
L=
2
We optimize the Langrangian by differentiating with respect to ω, λ and γ and set the derivative equal
to zero to yield these three sets of matrix equations:
dL
dw
dL
dλ
dL
dγ

=

Σw − λµ − γe = 0,

(26)

= w′ µ − µp = 0,

(27)

= w′ e − 1 = 0.

(28)

Solving for ω, notice that we can rewrite the first equation as follows,
ω = Σ−1 [λµ + γe] .

(29)

′
Recall from the equations in 26 that dL
dγ = ω e − 1 = 0, then by substitution, we have an equation of the
form,

λµ′ Σ−1 e + γe′ Σ−1 e = 1.

(30)

Similarly, we know that µp = ω ′ µ, hence
µp = λµ′ Σ−1 µ + γe′ Σ−1 µ.

(31)

In order to get the closed form solution for λ and γ we solve a system of two simultaneous equation(30
and 31),

   ′ −1
λµ Σ µ + γe′ Σ−1 µ
µp
.
(32)
=
λµ′ Σ−1 e + γe′ Σ−1 e
1
The system above can be represented in a more elaborate matrix form as follows:
 
   ′ −1
µ Σ µ e′ Σ−1 µ λ
µp
= ′ −1
.
1
µ Σ e e′ Σ−1 e γ
Notice in Equation 33 that e′ Σ−1 µ=µ′ Σ−1 e, then the expression can be simplified to:
 
  
A B λ
µp
=
,
1
B C γ

(33)

(34)

where we define:
A = µ′ Σ−1 µ, B = µ′ Σ−1 e and C = e′ Σ−1 e.
A closer inspection of all the entries in the matrix above reveals that they are scalars. Also notice that
the equation is a linear system of the form Λx = b, so solving for this system would be as x = Λ−1 b:
From the inverse of the matrix, thus solving for the two unknown multipliers γ and λ we get,

 
 
1
C −B µp
λ
,
(35)
=
1
γ
AC − BB −B A

10

Multiplying out the expression we get:
λ=

Cµp − B
AC − BB

and γ =

−Bµp + A
.
AC − BB

Finally putting all the pieces together we write down the solution for ω which will be a n × 1 vector of
portfolio weights as follows:
ω ∗ = Σ−1 [λµ + γe]
1
= Σ−1 [(Ae − Bµ) + µp (Cµ − Be)] .
D

(36)

The equation of the minimum variance set of the portfolio can then be shown to be
σp2 = ω ′ Σω = λµp + γ,
=

Cµ2p − 2Bµp + A
.
AC − B 2

(37)

The portfolio variance is a quadratic function of the mean portfolio return. In the (µp , σp ) space this
plots the parabola while in the (σp , µp ) space this plots the inverted parabola. The global minimum
variance portfolio (gmv) has the mean return which is found by differentiating the equation for the
d(σ 2 )
2Cµp −2B
variance and setting the derivative equal to zero dµpp = AC−BB
= 0. This yields µgmv
=B
p
C which
simplifies to the following,
Mean of Global Minimum Variance =

µ′ Σ−1 e
,
e′ Σ−1 e

(38)

2

2
It is easy to see that the variance of the GMV portfolio σgmv
=
manipulation this can be expressed as follows:

Minimum Variance =

CB
−2B B
C +A
C2
AC−BB

and further algebraic

1
.
e′ Σ−1 e

(39)

The portfolio weights for the global minimum variance portfolio
 be computed from
 (gmv) can also easily
B
B
µ
C
−B
+e
A−B
(
)
(
)
C
C
∗
−1
Equation [36] by substituting µp for B
and the weights are
C . This gives ω = Σ
AC−BB
therefore equal to the following,

Portfolio Weights of the GMV =
3.4.2

Σ−1 e
.
e′ Σ−1 e

Maximizing the Log-Optimal Growth portfolio for a Specified Mean E (Rp )

In the GOP framework, we demonstrate the similarity with the weights of both the GOP and the Mean–
Variance optimal weights. Solving for the optimal solution to this problem is similar to the problem
above and a complete derivation is outlined in appendix B. The Lagrangian is therefore,
L = ω′ µ −

ω ′ Σω
− λ1 [ω ′ µ − µp ] − γ1 [ω ′ e − 1]
2

(40)

It is easy to show that
ω = Σ−1 [µ − λ1 µ − γ1 e] .
Hence, the system above can be represented in a more elaborate matrix form as follows
 ′ −1
  ′ −1
 
µ Σ µ − µp
µ Σ µ e′ Σ−1 µ λ1
,
=
e′ Σ−1 µ − 1
µ′ Σ−1 e e′ Σ−1 e γ1
11

(41)

therefore the optimal weights are given as λ1 = CX−BY
andγ1 = AY −BX
, where D = AC − B 2 and
D
D
′ −1
′ −1
X = µ Σ µ − µp and Y = e Σ µ − 1. Hence, the weights are given as.
ω∗ = Σ−1 µ −

AY − BX −1
CX − BY −1
Σ µ−
Σ e.
D
D

(42)

The results display a remarkable similarity between the log optimal and mean variance optimization
problem as is evident from equation [42] which is identical to equation [36]. This can be shown as follows;
To show that the two equations are identical we need break the two expression into two parts, firstly,
equation [36] can be written as follows,
ω1∗ = λΣ−1 µ + γΣ−1 e and ω2∗ = Σ−1 µ − λ1 Σ−1 µ + γ1 Σ−1 e,
With simple algebraic manipulation, we can show that: γ = γ1 . Comparing the two weight vectors
above, we have that,
γΣ−1 e = γ1 Σ−1 e
(γ − γ1 ) Σ−1 e = 0.
The expression Σ−1 e is non zero vector, hence γ − γ1 = 0 or γ = γ1 . The second part we need to show
that λ = 1 − λ1 , we have that,
Σ−1 µ = Σ−1 µ − λ1 Σ−1 µ
(λ + λ1 )Σ−1 µ = Σ−1 µ
(λ + λ1 )µ = µ.
This implies that (λ + λ1 ) = 1, that is (λ = 1 − λ1 ) hence ω1∗ = ω2∗ .
3.4.3

Maximizing the portfolio growth regardless of the mean

Next we show that maximizing the portfolio growth rate without indicating the mean implies the optimal
mean growth for the portfolio, for an investor with risk aversion parameter θ. In matrix algebra notation,
the problem of maximizing the portfolio growth rate in continuous time is formulated as follows:

θ
argmax ω ′ µ − ω ′ Σω
2
ω


|ω ′ e = 1 .

(43)

Note that µ and Σ are time dependent and the subscripts denoting that are dropped for ease of analysis.
We now use the method of Lagrangian multipliers to solve the problem analytically. However we
arrive at the same solution using block matrix decomposition. Thus the Lagrangian expression is given
by:
θ
L(ω, λ) = ω ′ µ − ω ′ Σω − λ [ωe′ − 1] ,
2

(44)

To find the critical points of the Lagrangian, we determine the first order equations:
dL
= µ − θΣω − λe = 0
dω
dL
= 1 − ω ′ e = 0,
dλ

(45)
(46)

Notice that ω ′ e = e′ ω and that Σ−1 Σ exists which is just the identity matrix, then we have,
1 = e′ Σ−1 Σω.

(47)

Right multiplying e′ Σ−1 into equation [45] and solve for λ, we have,
λ

=

e′ Σ−1 µ − θ
.
e′ Σ−1 e
12

(48)

The optimal solution- the portfolio vector of weights (ω) that maximize the portfolio growth rate is:
1 −1
Σ [µ − λe]
θ
 

 ′ −1
1
eΣ µ−θ
= Σ−1 µ −
e
θ
e′ Σ−1 e



 ′ −1
eΣ µ−1
−1
Σ
e
= Σ−1 µ −
e′ Σ−1 e

ω∗ =

for θ = 1.

The portfolio mean E (Rp ) and variance V (Rp ) are calculated as follows:
 ′ −1

θ
eΣ µ
µ′ ω = θ−1 µ′ Σ−1 µ − θ−1 ′ −1 − ′ −1 µ′ Σ−1 e,
eΣ e
eΣ e

(49)

(50)

now let A = e′ Σ−1 µ, B = µ′ Σ−1 µ, C = e′ Σ−1 e and D = CB − A2 . Then the expression simplifies to
A
D
1
D
′
θC + C and the variance is simply V (Rp ) = ω Σω which is
θ2 C + C .
′

−1

µ
A
1
1
It is easy to see that limθ→∞ E (Rp ) → ee′Σ
Σ−1 e = C and limθ→∞ V (Rp ) → e′ Σ−1 e = C , so the
Mean–Variance portfolio mean and variance are asymptotes to the GOP ones hence the more risk averse
the investor is the more he/she is likely to miss out on the benefits of portfolio growth, namely that
the more inefficient the strategy is. This we know since the Markowitz strategy is less efficient than the
GOP/Log-optimal one. On the contrary, when limθ→0 , both E (Rp ) and V (Rp ) → +∞ showing that
risk-seeking investors benefit in the mean.

3.4.4

Growth Portfolio with a Riskless Asset

In the previous section, we discussed the construction of an efficient portfolio with risky assets. However
the same reasoning can be expanded with the inclusion of a riskless asset. The process followed by a
riskless asset is therefore dS
S = µf dt where µf representing the mean return of the riskless asset and that
the riskless asset is deterministic, since on assumption there is no risk, in other words the Wiener term
is zero. Now let P represent the vector of n risky assets, µf a unit vector of a riskless asset and e a
column vector of ones of dimension n+1. Then the budget constraint is therefore, noting the appropriate
notation for the weights:
ω ′ e + ωf = 1

⇔ ωf = 1 − ω ′ e so that this can be written as ω ′ e + (1 − ω ′ e) = 1.

The portfolio expected return is therefore stated as:
E [Rp ] = ω ′ µ + (1 − ω ′ e) µf ,
where µf represents a (n + 1) × 1 vector which is zero everywhere except for the position (n + 1, 1) and µ
the vector of means with the same dimension as the former but zero where the former is non-zero. The
log optimal maximization problem is formulated as follows;


1 ′
′
′
argmax (1 − ω ) µf + ω µ − ω Σω .
(51)
2
ω
Forming the Lagrange is: L = (1 − ω ′ ) µf + ω ′ µ − 12 ω ′ Σω, then the maximum (it can be shown that
dL
d2 L
dω 2 < 0) of this function occurs at dw = −µf + µ − Σω = 0. which happens when,
ω ∗ = Σ−1 (µ − µf ) .

(52)

However a more general formulation is to restate the problem as follows, where the mean is specified:


1 ′
′
′
argmin
ω Σω |E [Rp ] = ω µ + (1 − ω e) µf .
(53)
2
ω

13

The Lagrangian function associated to the problem above is:
L (ω, λ) =

1 ′
ω Σω + λ (E(Rp ) − ω ′ µ − (1 − ω ′ e)µf ) .
2

(54)

The first order conditions for ω to be a solution are therefore:
(
dL(ω,λ)
= Σω − λ (µ − eµf ) = 0
dω
dL(ω,λ)
= E(Rp ) − ω ′ µ − (1 − ω ′ e) µf = 0.
dλ
Solving for ω and noting that ω ′ µ = µ′ ω, we have,
E(Rp ) = ω ′ µ + (1 − ω ′ e) µf
= ω ′ [µ − eµf ] + µf
hence,
E(Rp ) − µf = ω ′ [µ − eµf ].
Also, we have that,
ω = Σ−1 λ [µ − eµf ]
′

ω ′ = λ [µ − eµf ] Σ−1
′

ω ′ [µ − eµf ] = λ [µ − eµf ] Σ−1 [µ − eµf ] .
Combining the two equations equal to ω ′ [µ − eµf ], we can solve for λ
′

E(Rp ) − µf = λ [µ − eµf ] Σ−1 [µ − eµf ]
λ=

E(Rp ) − µf
,
[µ − eµf ] Σ−1 [µ − eµf ]

Substituting λ in equation we have the general solution to the portfolio weights as follows:
E(Rp ) − µf
.
′
[µ − eµf ] Σ−1 [µ − eµf ]

ω = Σ−1 [µ − eµf ]
It is easy to show that the denominator,
′

[µ − eµf ] Σ−1 [µ − eµf ] = µ′ Σ−1 µ − µ′ Σ−1 eµf − µ′f e′ Σ−1 µ + µ′f e′ Σ−1 eµf
= µ′ Σ−1 µ − 2µ′ Σ−1 eµf + µ′f e′ Σ−1 eµf

= Cµ2f − 2Bµf + A
= H.

Hence, the portfolio variance V (Rp ) is therefore,
V (Rp ) = ω ′ Σω
′  −1

 −1
Σ (µ − eµf ) (E (Rp ) − µf )
Σ (µ − eµf ) (E (Rp ) − µf )
Σ
.
=
H
H
Notice that (E (Rp ) − µf ) and H are scalars hence, a simple algebraic manipulation we have,
2

V (Rp ) =

(E (Rp ) − µf )
H2

Which simplifies to,

′
Σ−1 (µ − eµf ) (µ − eµf ) .
2

=

(E (Rp ) − µf )
.
H

.
14

(55)

3.4.5

Conclusion

However the inclusion of the riskfree asset into a portfolio of risky assets maps out a straight line in
a (µp , σp ) two dimensional space. However, for a desired portfolio return E(Rp ) the growth optimal
portfolio is constructed by computing,
Gopt = ω ′ µ − ω ′ Σω,
where the portfolio weight ω is defined in equation [55].
ω = Σ−1 [µ − eµf ]

4

E(Rp ) − µf
.
′
[µ − eµf ] Σ−1 [µ − eµf ]

Data and Results

In this section I present a practical example demonstrating the application of log optimal maximization
algorithm. Five stocks from different economic sectors were randomly selected to illustrate how log
optimal growth portfolios are constructed. The data used are the daily prices downloaded from I-Net
bridge, covering the period from the 30 November 1999 up to 31 October 2009. That makes up nine
years of monthly data, and in total 120 monthly observations. The results we will derive are then the
portfolio weights the investor would allocate to each asset in order to achieve an optimal portfolio. The
assets under consideration are as follows BHPBILL, WBHOVCO,NEDBANK,ABIL and SAB.

4.1

Calculation 1: Markowitz portfolio optimization and Log-Optimal Growth
Portfolio When short selling is allowed

Given a target value E (Rp ) for the mean return of a portfolio, The efficient portfolio characterized by
Markowitz and the Log optimal growth portfolio for N = 5 is therefore derived as follows,

BIL
WBO
NED
ABL
SAB

BIL
0.008770
0.000569
0.000252
0.001357
0.003156

WBO
0.000569
0.006594
0.001717
0.002571
0.000467

NED
0.000252
0.001717
0.006045
0.003549
0.001697

ABL
0.001357
0.002571
0.003549
0.011630
0.000881

SAB
0.003156
0.000467
0.001697
0.000881
0.004876

Table 1: Variance Covariance Matrix
Having the variance covariance matrix and the individual asset means we can derive the portfolio
constants which in principle are functions of the market parameters.
A
e′ Σ−1 µ
5.3160

B
e′ Σ−1 µ
0.15451

C
e′ Σ−1 µ
394.7973

D
BC-AA
32.7419

Table 2: Constants
The weights for the efficient portfolio satisfy the the following equation,




1 
−1
−1 
−1
−1
ω∗ef f =
 AΣ e − BΣ µ +E (Rp ) AΣ µ − AΣ e 
D |
{z
}
{z
}
|
ω1

ω2

= ω1 + E (Rp ) ω2

15

(56)

Which yields,

8.77 0.57 0.25
0.57 6.59 1.72

ω1 = 5.316 ∗ 10−3 
0.25 1.72 6.04
1.36 2.57 3.55
3.16 0.47 1.70

1.36
2.57
3.55
11.63
0.88

and,


8.77
0.57

ω2 = E (Rp ) 5.316 ∗ 10−3 
0.25
1.36
3.16

0.57 0.25
6.59 1.72
1.72 6.04
2.57 3.55
0.47 1.70

−1 
 


3.16
1
1.64%

1
2.78%
0.47
 
 







1.70 5.316 × 1 − 0.15451 × 
0.05%





0.88
1
0.77%
4.88
1
0.96%
1.36
2.57
3.55
11.63
0.88

(57)


−1 

 
3.16
1.64%
1

1
2.78%
0.47


 
 





1.70 394.79 × 1 − 0.154 × 
0.05%





0.88
0.77%
1
1
4.88
0.96%
(58)

Hence the efficient frontier can be mapped as follows




0.08187
4.5709
(0.21544)
 36.2866 




∗




0.66051
ωef
=
+
E
(R
)
p (32.9226)
f


 0.10715 
 (4.2105) 
0.36592
(3.7244)

(59)

From the efficient set we derive the portfolio variance,
σp2 =

Portfolio1
Portfolio2
Portfolio3
Portfolio4
Portfolio5
Portfolio6
Portfolio7
Portfolio8
Portfolio9
Portfolio10


1
Cµ2p − 2Bµp + A = 394.7973 × µ2p − 2 × 5.3160 + 0.1545
32.7419
BIL
8.2
10.5
12.8
15.0
17.3
19.6
21.9
24.2
26.5
28.8

WBO
-21.5
-3.4
14.7
32.9
51.0
69.2
87.3
105.5
123.6
141.7

NED
66.1
49.6
33.1
16.7
0.2
-16.3
-32.7
-49.2
-65.6
-82.1

ABL
10.7
8.6
6.5
4.4
2.3
0.2
-1.9
-4.0
-6.1
-8.2

SAB
36.6
34.7
32.9
31.0
29.1
27.3
25.4
23.6
21.7
19.8

E[R]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5

V[R]
0.5
0.3
0.3
0.3
0.3
0.4
0.6
0.8
1.1
1.5

S[R]
6.9
5.8
5.2
5.1
5.5
6.4
7.6
9.0
10.5
12.1

(60)

Growth
-0.2
0.3
0.9
1.4
1.8
2.3
2.7
3.1
3.4
3.8

Table 3: A Combination of 20 Portfolios
By varying the portfolio desired return E(Rp ), we can trace out the efficient frontier and this is
depicted by the continuous line in figure 1. This demonstrates the diversification benefit of the growth
portfolio and the mean variance optimization.
The notable properties of the efficient set is that there exist a portfolio on the frontier which has the
minimum variance known as the global minimum variance portfolio and the mean, variance and standard
deviation are as follows;
µ′ Σ−1 e
5.316045
=
= 1.3465
e′ Σ−1 e
394.79733
1
1
2
σgmv
= ′ −1 =
= 0.2533
eΣ e
394.7973
σgmv = 5.033

µgmv =

16

(61)
(62)
(63)

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Figure 1: Feasible Region MVO and LOGP
The portfolio weights for the global minimum variance portfolio is therefore;

 

56.622
14.3%
 107.84  27.3%

 

Σ−1
85.74811
21.7%
ωgmv = ′ −1 = 394.797 × 
=




eΣ e
 19.9176   5.0% 
124.663
31.6%
PortGMV

BIL
14.342

WHO
27.317

NED
21.720

ABL
5.045

SAB
31.577

µgmv
1.3465

2
σgmv
0.2533

(64)

σ
5.0

G
1.220

Table 4: Global minimum Variance Portfolio

4.1.1

LOG-Optimal Portfolio

In case of the log optimal portfolio we can represent the the path followed by each asset as follows,
dS1
S1
dS2
S2
dS3
S3
dS4
S4
dS5
S5

= 0.016dt + 0.094dw1
= 0.028dt + 0.081dw2
= 0.0005dt + 0.078dw3
= 0.008dt + 0.108dw4
= 0.010dt + 0.07dw5

(65)

Assuming that the assets are correlated through the wiener term, then the solution to equation [36] is
exactly equal to the solution of equation [42]. The feasible region of the growth optimal portfolio and
mean variance portfolio

4.2

Calculations2:Efficient Portfolios with a Riskless Asset

With the inclusion of a risky asset and using µf = 2%



  
−0.004
1.64
1
2.78
1  0.008 



  
 − 2.00% 1 = −0.020
0.05
(µ − eµf ) = 



  
0.77
1 −0.012
1
−0.010
0.96

17

(66)


0.0088
0.0006

Σ−1 (µ − eµf ) = 
0.0003
0.0014
0.0032

0.0006
0.0066
0.0017
0.0026
0.0005

0.0003
0.0017
0.0060
0.0035
0.0017

0.0014
0.0026
0.0035
0.0116
0.0009

−1 
 

0.0032
−0.004
0.0091

 

0.0005
  0.008   2.3046 
−0.020 = −3.2907
0.0017
 
 

0.0009 −0.012 −0.4793
0.0049
−0.010
−1.1235

H = Cµ2f − 2 × µf + A = 394.7973 × 2.002 − 2 × 0.15451 × 2.00 + 5.3160

(67)

(68)

= 9.9792

(69)
(70)

The portfolio weights given a desired level of portfolio return E(Rp )


0.0091
 2.3046 

 E(Rp ) − 2.00

ω∗ = 
−3.2907 ×
H
−0.4793
−1.1235

(71)

The optimal portfolio mean,





0.0091
1.64
 2.3046  2.78



′
′


ω µ + (1 − ω e)µf = 
−3.2907 0.05 = 0.1198
−0.4793 0.77
−1.1235
0.96

(72)

and the optimal portfolio variance,

′ 
0.0091
0.0088 0.0006
 2.3046  0.0006 0.0066
 

 
ω ′ Σ−1 ω = 
−3.2907 0.0003 0.0017
−0.4793 0.0014 0.0026
0.0032 0.0005
−1.1235


−1 

0.0003 0.0014 0.0032
0.0091


0.0017 0.0026 0.0005
  2.3046 
−3.2907 = 9.98%
0.0060 0.0035 0.0017
 

0.0035 0.0116 0.0009 −0.4793
0.0017 0.0009 0.0049
−1.1235

(73)

A mapping of different portfolios is therefore tabulated below in table 5 and the graphical representation
is shown in
BIL
Portfolio1
Portfolio2
Portfolio3
Portfolio4
Portfolio5
Portfolio6
Portfolio7
Portfolio8
Portfolio9
Portfolio10

0.000
0.001
0.001
0.002
0.002
0.003
0.003
0.004
0.004
0.005

WBO
0.115
0.231
0.346
0.462
0.577
0.693
0.808
0.924
1.039
1.155

NED
-0.165
-0.330
-0.495
-0.660
-0.824
-0.989
-1.154
-1.319
-1.484
-1.649

ABL
-0.024
-0.048
-0.072
-0.096
-0.120
-0.144
-0.168
-0.192
-0.216
-0.240

SAB
-0.056
-0.113
-0.169
-0.225
-0.281
-0.338
-0.394
-0.450
-0.507
-0.563

Risk-Free
1.00
1.13
1.26
1.39
1.52
1.65
1.78
1.90
2.03
2.16
2.29

V[R]
0.000%
0.025%
0.100%
0.225%
0.401%
0.626%
0.902%
1.228%
1.603%
2.029%
2.505%

σ
0
0.01583
0.03166
0.04748
0.06331
0.07914
0.09497
0.11079
0.12662
0.14245
0.15828

Table 5: Portfolios with the inclusion of a Risk Free Asset

18

Growth
0.0200
0.0249
0.0295
0.0339
0.0380
0.0419
0.0455
0.0489
0.0520
0.0549
0.0575

E[Rp]
0.0200
0.0250
0.0300
0.0350
0.0400
0.0450
0.0500
0.0550
0.0600
0.0650
0.0700

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Figure 2: Log Optimal Growth Portfolio

5

Conclusions

In portfolio theory, optimization models play a critical role in determining portfolio investment strategies
for investors. Any investment strategy that maximizes portfolio growth has an intuitive appeal for
both the professional and non-professional investor. In Markowizt framework one works on diversifying
investment assets in order to minimize risk for a given level of return. In this paper we have looked at
another portfolio characteristic which is equally important in investment decision process that maximizes
the portfolio long term growth rate over a specified time period-the log optimal growth portfolio. We
applied our model to five randomly selected JSE stocks. While the Markowizt mean variance strategy
is static one period strategy(buy and hold) and a has a fixed time horizon, the log–optimal strategy is
dynamic and can be applied to any rebalancing period such as a year, a month a week or a day. The
model assumes that stock prices follow a geometric Brownian process and utilizes a stochastic differential
equation to quantify the distribution of asset returns. The stochastic process describe the probabilistic
evolution of the asset returns through the passage of time. Our model further assumes that asset prices
follow a log normal distribution. The solution to the differential equation leads us to conclude that
2
continuously compounded asset return r0,T = T1 ln( SST0 ) are normally distributed N ∼ (µ − 12 σ 2 , σT ).
This further implies that the future asset price ST is log normally distributed with E(ST ) = eµt . This is
a realistic approach to modeling asset prices since a variable that is normally distributed can on negative
values something that most financial prices can never do. The distribution of asset returns yields an
expected return of µ − 21 σ 2 rather than µ hence the log optimal aims at maximizing this geometric mean
representing the long term growth rate G = µ − 12 σ 2 . There is however a subtle difference between the
two expected value of assets returns7 . In our case the portfolio has been formulated in a continuous time
framework.If an investor strategy is to maximize long term capital growth then adopting a strategy that
maximizes the expected logarithm of returns is considered to be an optimal strategy.

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6

Appendix

A

Example



t−1
Lets define a simple asset returns as Rt = StS−S
where Rt ∼ N µ, σ 2 . Since asset prices are greater
t−1
than zero, we have St ≥ 0. By implication then:
1 + Rt =

St
≥ 0,
St−1

this implies that Rt ≥ −1

Then since Rt is assumed to be normally distributed, by symmetry, the probability of Rt ≤ −1 given
that µ = 0.5 and σ = 0.6 is;


Rt − µ
= P (−1.75) = 4.00%
P
σ
This implies that there is a 4% probability that prices can be negative which is contrary to our belief.

21

B

Derivation of Maximum Log Optimal Portfolio

θ
argmax ω ′ µ ω ′ Σω
2
ω

′

|ω µ = E (Rp )


ωe=1 .
′

The Lagrangian function with λ1 and γ1 as multipliers thus.
θ
L = ω ′ µ − ω ′ Σω − λ1 [ω ′ µ − µp ] − γ1 [ω ′ e − 1] .
2
Differentiating and setting the matrix equations to zero we have,
dL
dw
dL
dλ1
dL
dγ1

= µ − Σω − λ1 µ − γ1 e = 0,
= w′ µ − µp = 0,
= w′ e − 1 = 0.

Notice that we can rewrite the first equation with ω as a subject of the formula as follows,
ω = Σ−1 [µ − λ1 µ − γ1 e] .
The last two matrix equations are therefore written as,
′

ω ′ e = Σ−1 [µ − λ1 µ − γ1 e] e = 1

1 = µ′ Σ−1 e − λ1 µ′ Σ−1 e − γ1 e′ Σ−1 e

(74)

and similarly, we know that µp = ω ′ µ, then
µp = µ′ Σ−1 e − λ1 µ′ Σ−1 µ − γ1 e′ Σ−1 µ
The system above can be represented in a more elaborate matrix form as follows:
 −1
  ′ −1
 
µΣ µ − µp
µ Σ µ e′ Σ−1 µ λ1
= ′ −1
µΣ−1 e − 1
µ Σ e e′ Σ−1 e γ1

(75)

(76)

Which has a solution of the form,

Where we have defined:


1
A
AC − BB −B

−B
C


−1    ′ −1
µ Σ µ − µp
λ1
=
µ′ Σ−1 e − 1
γ1

(77)

A = µ′ Σ−1 µ, B = µ′ Σ−1 e and C = e′ Σ−1 e
Therefore the optimal weights
CX − BY
D
AY − BX
γ1 =
D

λ1 =

where D = AC − B 2 and X = µ′ Σ−1 µ − µp and Y = e′ Σ−1 µ − 1.
AY − BX −1
CX − BY −1
Σ µ−
Σ e
D
D
= Σ−1 µ − λ1 Σ−1 µ − γ1 Σ−1 e

ω∗ = Σ−1 µ −
.

22

(78)
</reference>

<statements>
1. Markowitz’s framework is consistent with either (a) quadratic utility over mean and variance or (b) joint normality of returns; in practice, it assumes the mean vector and covariance matrix of future returns are known or well‑estimated, which is rarely true.
2. Expected returns \(\mu\) are treated as exogenous inputs; in practice they are usually estimated via historical averages or simple forecasting models, under assumptions like i.i.d. returns and normality/log‑normality.
3. Mean–Variance: Return prediction / prior: Exogenous expected returns, typically from historical means or simple models; assumes known \(\mu,\Sigma\).
</statements>

Begin the assessment now. Output only the JSON list, without any conversational text or explanations.