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Statistical Analysis on the Advantages of Portfolio Diversification

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Statistical Analysis on the Advantages of Portfolio Diversification

Abstract

The classical mean-variance portfolio selection problem (PSP) pioneered by Markowitz is, undoubtedly, one of the most frequently studied areas in finance, and several financial analysts regard it as the foundation of modern portfolio theory (MPT). The model in its basic form deals with making a choice from a universe of assets to form a master asset known as portfolio of assets. The main aim of such a strategy is to achieve a reasonable trade-off given the conflicting objectives related to making a maximum possible return/profit at the most minimum risk possible, provided that the right choice of constituent assets is made and proper weights (fraction of investment funds) are correspondingly allotted. In this paper, we looked at the effects and advantages of constructing a reasonably diversified portfolio from a pool of assets while giving emphasis on the interrelationship existing among the portfolio's constituent assets.
Int. J. Pure Appl. Sci. Technol., 7(2) (2011), pp. 98-106
International Journal of Pure and Applied Sciences and Technology
ISSN 2229 - 6107
Available online at
www.ijopaasat.in
Research Paper,
Statistical Analysis on the Advantages of Portfolio
Diversification
Abubakar Yahaya
1,*
, Amina Hassan Abubakar
2
and Jamilu Garba
3
1, 2, 3
Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria.
* Corresponding author, e-mail: (ensiliyu2@yahoo.co.uk)
(Received: 2-8-11; Accepted: 14-10-11)
Abstract:
The classical mean-variance portfolio selection problem (PSP) pioneered
by Markowitz is, undoubtedly, one of the most frequently studied areas in finance, and
several financial analysts regard it as the foundation of modern portfolio theory
(MPT). The model in its basic form deals with making a choice from a universe of
assets to form a master asset known as portfolio of assets. The main aim of such a
strategy is to achieve a reasonable trade-off given the conflicting objectives related to
making a maximum possible return/profit at the most minimum risk possible, provided
that the right choice of constituent assets is made and proper weights (fraction of
investment funds) are correspondingly allotted. In this paper, we looked at the effects
and advantages of constructing a reasonably diversified portfolio from a pool of assets
while giving emphasis on the interrelationship existing among the portfolio’s
constituent assets.
Keywords:
Portfolio Selection, Diversification, Covariance, Correlation.
1. Introduction:
The main goal behind the concept of portfolio management is to combine various securities
and other assets into portfolios that address investor needs and then to manage those
portfolios so as to achieve the desired investments objectives. The investors’ needs are
mostly defined in terms of return and risk, and the portfolio manager makes a sound decision
aimed at maximizing return for investment risk undertaken [2].
The goal of investment decisions which is to maximize shareholders’ wealth and making
sound investment decisions that enhance shareholders’ wealth lies at the very heart of the
financial manager’s job. Wealth-enhancing investment decisions (corporate or personal)
cannot be made without understanding of the interplay between investment returns and
Int. J. Pure Appl. Sci. Technol., 7(2) (2011), 98-106. 99
investment risk. The risk-return relationship is central to investment decision making,
whether evaluating a single investment or choosing between alternative investments.
Potential investors, for instance, will assess the risk-return relationship or trade-off in
deciding whether to invest in company securities such as shares or bonds. Investors will
evaluate whether, in their view, the securities provide a return commensurate with their level
of risk.
The conventional mean-variance portfolio selection model introduced by Harry Markowitz is,
undoubtedly, the corner stone of MPT [7, 8]. The basic assumption underpinning this model
is that: a rational investor has a multivariate normally distributed asset returns characterized
by the first two moments of the distribution in which the first moment (portfolio expected
return) stands for the return/profit of the investment, while the second moment (Variance)
stands for the corresponding risk involved. Given this, among other assumptions; Markowitz
showed that an efficient portfolio lies on a parabola-like frontier of points, often regarded as
Efficient Frontier (EF). A portfolio of assets is considered efficient, if for any expected
return, there exists no other portfolio with lower risk (variance); similarly, for any given
value of risk, there exists no other portfolio with higher expected return. Hence, the EF is
made up of all efficient portfolios. Depending on how many efficient portfolios an
investor/fund manager wants to generate on the EF, there are several computationally
efficient algorithms that can compute a single point/portfolio or even all points/portfolios [1,
6].
The remaining parts of this paper are organized as follows: in the next section we show how
to derive the formula/expression for computing portfolio expected return and risk (variance)
from the returns of portfolio’s constituent assets. Furthermore, we introduce the classical
mean-variance portfolio selection model together with the mathematical formulation of the
problem after which we review some related literature on the model’s possible extensions. In
section 3 we present the main idea behind portfolio diversification and the effect of forming a
diversified portfolio from a set of correlated as well as uncorrelated assets. Finally, we
conclude and offer some recommendations in section 4.
2. Materials and Methods/Definitions and Preliminaries:
2.1 Portfolio Expected Return and Risk (Variance)
This section is further subdivided into two in which the first subsection deals with deriving
the formula for computing the portfolio expected return, while the other section focuses on
deriving the formula for computing the portfolio risk (variance).
2.1.1 Expected return of a portfolio
Suppose there are n assets with rates of return
12
, ,..., 1,2,...,
tt nt
rr r fort T
. The values for the
expected returns of these rates of returns are given by:
11 2 2
, , ...,
nn
rr r r r r
 .
Suppose further that, we construct a portfolio from these n assets each having a given fraction
of the investment fund, often regarded as weights
1,2,...,
i
wfori n . The corresponding
rate of return of the [constructed] portfolio in terms of the returns of individual assets is given
by:
11 2 2
1
...
Pnn
n
ii
i
rwrwrwr
wr


1
Int. J. Pure Appl. Sci. Technol., 7(2) (2011), 98-106. 100
By taking expectation on both sides of equation 1 above, we obtain:


1
1
1
n
Pii
i
n
ii
i
n
Pii
i
rwr
wr
rwr





2
From equation 2 above, it can easily be understood that, the portfolio’s expected return,
P
r
can be obtained by taking the weighted sum of the assets expected return.
2.1.2 Variance (Risk) of a portfolio
To determine the variance of portfolio return; let us now denote the variance of asset
i by
2
i
;
the covariance of assets
i and j by
ij
and let the portfolio’s variance be denoted by
2
. By
performing a straightforward computation, we now have:









2
2
2
11
11
11
11
,
PPP
nn
ii ii
ii
nn
ii i j j j
ij
nn
ij i i j j
ij
nn
ij i i j j
ij
ij i i j j
rr
wr wr
wr r wr r
ww r r r r
ww r r r r
recall that r r r r the












































11
nn
ijij
ij
n
ww



3
It is important to note at this point that, the variance (risk) of a portfolio’s return can be easily
computed from the covariances of the pairs of asset returns and the weights of the asset that
make up the portfolio.
2.2 The Classical Mean-Variance Model
The classical mean-variance model originally developed by Markowitz is aimed at finding a
portfolio of assets that seeks to minimize the (portfolio) risk subject to achieving a given
level of (portfolio) return. In this conventional formulation, the portfolio risk (objective
function) being minimized is quantified by the portfolio’s variance, which is the most
commonly used measure [2]. The model assumes a market composed of n different assets
Int. J. Pure Appl. Sci. Technol., 7(2) (2011), 98-106. 101
with corresponding expected returns
i
r
, and asset covariances
ij
. The aim is to find a set of
fractions (decision variables) w
i
of an investor’s investment fund to be assigned to each asset
i so as to minimize the risk (variance,
2
) of the entire portfolio’s expected return, while at
the same time ensuring that the portfolio’s expected return attains a specified target, say ξ.
The only (practical) restrictions that accompany this model are the portfolio’s return and
budget constraints.
The return constraint ensures allotting weights to assets that guarantee achieving a specified
target return, while the budget constraint ensures that these fractions or asset weights must be
nonnegative and their sum must also be unity. The classical Markowitz model can be
represented mathematically by:
2
11
1
1
1
01
nn
pijij
ij
n
ii
i
n
i
i
i
Minimize w w
Subject to
wr
w
w







4
5
6
7
Equation 4 represents the objective function (Portfolio Risk), while equations 5 and 6
respectively represent the return and budget constraints. Constraint 7 ensures that no asset’s
weight falls outside the interval [0, 1], which literally means no short sales are allowed. The
above is a convex quadratic programming problem, since the objective function is quadratic;
the covariance matrix
ij
is positive definite and the constraints composed of both linear
equalities and inequalities.
A quite number of researches have been done and many are still going on purposely to
produce a near-perfect model that tries as much as possible to address the needs of an
investor, which serves as an improvement or rather an extension to the Markowitz’s (basic)
model. Even a superficial glance at the model above reveals that, it comprises of three main
features namely, the objective, constraints and variables/bounds. It is also worthwhile to note
at this point that, most of the studies conducted on PSP focused on making some
modifications to one or more of these attributes. Researchers such as Estrada [9], Ballestero
[4], Konno and Yamazaki [5], as well as Feiring et al [3] substituted variance with other
alternative measures and went extra mile to argue that variance should not serve as a measure
of portfolio risk; because it penalizes both upward and downward deviations from the mean
return.
Other researchers’ interest on improving the basic model tilted towards resolving other issues
to do with the real world investment constraints. For instance, Chang et al [12] while
maintaining the classical objective (variance) introduced additional constraints involving
cardinality as well as floor & ceiling constraints, after which they applied Genetic Algorithms
(GA), Tabu Search (TS) and Simulated Annealing (SA) to solve the constrained PSP. Crama
and Schyns [13] used SA to solve the enriched Chang et al [12] model by further
incorporating turnover (purchase and sale) and trading constraints.
Int. J. Pure Appl. Sci. Technol., 7(2) (2011), 98-106. 102
3. Results and Discussion
3.1 Portfolio Diversification
It is worthwhile at this point to make it clear that, the total portfolio risk comprises of two
types of sub-risks namely: the market/systematic/nondiversifiable risk and the other known as
unique/firm-specific/non-systematic/diversifiable risk (i.e. Total Portfolio risk = Market risk
+ Non-market risk). Thus, it should be made known that, it is (theoretically) possible for
investors to construct portfolios and (possibly) eliminate (non-market/non-
systematic/diversifiable) part of the total portfolio risk through an intelligent investment
practice known as diversification.
Portfolio diversification is, essentially, a defensive technique to counter the problem of
investment risk. Imagine one wish to transport a basketful of eggs from one place to another;
there is, of course, different options open to him to accomplish that task. Some of these
options might include:
Carrying the whole basketful of eggs on foot at one go.
Rationing and taking the eggs on foot to the desired place on several journeys.
Carrying the whole basketful of eggs in a car to the desired place at one go.
Rationing and taking the eggs in a car to the desired place on several journeys.
Rationing the eggs among many people and each will be responsible for some.
Apparently, these are just some of the few possible options aimed at ensuring that the eggs
reach their destination safely. However, it is obvious that different risks are attached to the
different egg-carrying options. Therefore, if it makes sense to divide the eggs in some way to
reduce/spread the risk of disaster, it may also make sense to build an investment portfolio in
order to minimize investment risk; in the sense that, what is a good strategy for the carriage
of eggs may also be a good strategy for investment in marketable securities (stocks, shares,
e.t.c.).
An important way to reduce the risk of investment is to diversify one’s investments. For
instance, if a given investor’s portfolio consisted of [only] stocks from the banking sector,
there is every likelihood that such an investor would face a substantial loss in the value of
his/her portfolio if any major event adversely affected the banking industry – as was
witnessed in 2009 when there was a global economic meltdown that affected the banking
industry more than any other and whose impact is felt across the globe up till this time.
Portfolio diversification refers to mixing up your investments (i.e. making them more
diverse) and such a [good] practice guarantees an investor at least two main advantages as
follows:
Portfolio diversification helps an investor in minimizing the chances of losing all
his/her investment capital; if a given stock, sector or industry performs poorly in the
market.
It also helps an investor increases his chances of being in the right place at the right
time; if a given stock, sector or industry performs pretty well in the market.
Int. J. Pure Appl. Sci. Technol., 7(2) (2011), 98-106. 103
Let us now show how a (unique/firm-specific/non-systematic/diversifiable) risk can be
somehow eliminated using the strategy of portfolio diversification. Now notice that equation
1 above can be re-written as:

222
111,
[Sum of asset variances Sum of asset covariances]
nnn
Pii ijij
iijji
www




8
3.1.1 Correlated Assets
Two assets i and j are said to be correlated if
0.. 0
ij ij
ie

, and suppose that the
investment funds are shared equally among the portfolio’s constituent assets; that is, let
1 1,2,...,
i
wni n be a weight associated with asset i. Now the portfolio risk in equation
5 above can be re-written as:





2
2
22
111,
2
111,
1
1
1
1
1
nnn
ij
i
P
iijji
nnn
ij
i
iijji
nn
n
nn n nn
n
Average Variance Average Covariance
nn







9
Taking limits on both sides of equation 9 above leads to obtaining:



2
2
1
1
P
nn
P
n
n
lim lim Average Variance Average Covariance
nn
lim Average Covariance
 






10
The above relation can be graphically represented as in Figure 1 below:
Int. J. Pure Appl. Sci. Technol., 7(2) (2011), 98-106. 104
3.1.2 Uncorrelated Assets
Assets i & j are said to be uncorrelated if

0 . . 0, , 1,2,..., |
ij ij ij ij i j
ie i j n i j since


.
Again, if we let
1 1,2,...,
i
wni n
, the portfolio risk (equation 8 above) takes the form:

222
1
2
2
1
2
1
1
1
n
Pii
i
n
i
i
n
i
i
w
n
nn
Average Variance
n


11
Therefore, by taking limits:

2
2
1
0
P
nn
P
n
lim lim Average Variance
n
lim
 





12
Similarly, the above relation can be pictorially represented as shown in Figure 2 below:
Number of stocks
Diversifiable (non-Market) risk
Total Portfolio risk
0
Nondiversifiable (Market) risk
Figure 2
Int. J. Pure Appl. Sci. Technol., 7(2) (2011), 98-106. 105
Now concerning the decision to take on how many securities are needed to constitute a
portfolio that will be able to eliminate most or all of the non-systematic risk, opinion differs
among researchers. For instance, Statman [11] argue that: “a well diversified portfolio of
randomly chosen stocks must include at least 30 to 40 stocks.” However, this contradicts the
stand of Evans and Archer [10] in which they raise doubts concerning the economic
justification of increasing a portfolio size beyond 10 or so securities; as according to them:
“virtually all economic benefits of diversification are exhausted, provided the portfolio
contains ten or so stocks”.
4. Conclusions
From the above statistical analysis, it can easily be observed that, it is possible to diversify
away a portion of Total Portfolio risk. The other portion of the Total Portfolio risk that
remains even after such diversification is called the Market risk – a risk attributable to
market-wide sources.
It can also be observed that, the degree of relationship (correlation) existing among various
securities in the market plays a vital role in portfolio’s risk reduction; in the sense that if the
assets (that makes up a given portfolio) are correlated, the (diversifiable) risk cannot be
reduced lower than the Average Covariance. On the other hand, if portfolio’s constituent
assets are uncorrelated, the (diversifiable) risk can be completely eliminated, and the only
risk an investor or portfolio manager has to contend with, is the non-diversifiable, otherwise
known as the Market risk. In view of the foregoing analyses and discussions, we advice
investors and portfolio managers to consider uncorrelated securities in constructing their
portfolios as it is proven to be the most intelligent investment decision any investor/portfolio
manager can make.
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