ON THE PROTECTION OF INVESTMENT CAPITAL DURING … · can be found in the efficient frontier...
Transcript of ON THE PROTECTION OF INVESTMENT CAPITAL DURING … · can be found in the efficient frontier...
ECONOMIA INTERNAZIONALE / INTERNATIONAL ECONOMICS 2017- Volume 70, Issue 2 – May, 165-192
Authors::::
J.W. MUTEBA MWAMBA,
Department of Economics, University of Johannesburg, South Africa
L. MANTSHIMULI,
Department of Economics, University of Johannesburg, South Africa
ON THE PROTECTION OF INVESTMENT CAPITAL DURING FINANCIAL CRISIS IN SOUTH AFRICAN EQUITY MARKET: A
RISK-BASED ALLOCATION APPROACH
ABSTRACT
This paper constructs six portfolios using six risk-based asset allocation techniques and
compares the performance of these portfolios with that of the market portfolio proxied by the
Johannesburg All Share Index (JSE ALSI). We make use of the daily closing prices of eleven JSE
sector indices starting from August 2004 to September 2015. We divide this sample period into
three overlapping sub-samples representing the pre-crisis period, the crisis period, and the post-
crisis period. The performance analysis is based on the Sharpe and the Sortino ratios. The
covariance matrix, the most important input in the construction of these risk-based portfolios is
assumed to be constant, and time varying respectively. When it is assumed to be constant our
results show that during the pre-crisis period risk-based portfolios performed poorly than the
market portfolio. But during the crisis and post-crisis periods we find that risk-based portfolios
performed better than the market portfolio with the minimum correlation portfolio generating
the highest Sharpe and Sortino ratios. More investment capital during these two sample periods
is found to be mostly allocated to the property sector. However, when the covariance matrix is
assumed to be time varying the pre-crisis period is used as the in-sample space while the crisis
and post-crisis periods are used as the out-sample space. The forecasts of the time varying
covariances in the out-sample space are obtained with a multivariate GARCH model based on a
sixty rolling window forecast. Our results with forecasted covariances show that during the crisis
period all risk-based portfolios performed better than the market portfolio due to their ability to
protect investor’s capital during financial crisis. We find mixed results during the post-crisis
period: the equally weighted, the risk parity, and the minimum correlation portfolios performed
poorly while the rest of the risk –based portfolios performed better than the market portfolio
with the minimum variance portfolio generating the highest Sharpe and Sortino ratios. More
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investment capital is found to be allocated in the property, telecommunication, consumer
services, and health sectors when the forward looking approach is employed.
Keywords: Risk-Based Strategies, Markowitz Mean-Variance Framework, Financial Crises, Predictive Risk Measures, Asset
Allocation
JEL Classification: C44, C63, G01, G11, G12
RIASSUNTO
La protezione del capitale investito durante la crisi finanziaria
nel mercato azionario sudafricano: un approccio risk-based
Questo paper crea sei portafogli utilizzando tecniche di investimento risk-based e paragona
l’andamento di questi portafogli all’andamento del portafoglio di mercato rappresentato
dall’indice Johannesburg All Share Index (JSE ALSI). Si utilizzano i prezzi alla chiusura di
undici indici JSE nel periodo agosto 2004-settembre 2015. Poi si divide questo campione in tre
sottogruppi sovrapponibili che rappresentano il periodo pre-crisi, quello durante la crisi e il
post-crisi. L’analisi dell’andamento si basa sui rapporti Sharpe e Sortino. La matrice di
covarianza, l’elemento più importante nella costruzione di questi portafogli risk-based, si
presume rispettivamente costante e time-varying. Quando si considera costante i risultati
mostrano che durante il periodo pre-crisi i portafogli risk-based hanno rendimenti più bassi del
portafoglio di mercato. Invece durante la crisi e nel post-crisi i portafogli risk-based hanno un
andamento migliore del portafoglio di mercato e il portafoglio con correlazione minima genera i
livelli più alti di rapporti Sharpe e Sortino. Si riscontra che in questi due periodi campione vi è
stato un maggiore investimento di capitale prevalentemente nel settore immobiliare.
Comunque, quando la matrice di covarianza si considera time varying il periodo pre-crisi viene
utilizzato come lo spazio intra-campione mentre i periodi della crisi e post-crisi sono usati come
spazi extra-campione.
Le previsioni delle covarianze time varying nello spazio extra-campione sono state ottenute
tramite un modello GARCH multivariato. I risultati delle covarianze studiate mostrano che
durante la crisi tutti i portafogli risk-based hanno avuto un andamento migliore del portafoglio
di mercato a causa della loro capacità di proteggere il capitale investito durante la crisi
finanziaria. I risultati del periodo post-crisi sono invece eterogenei: i portafogli bilanciati, a
parità di rischio e a correlazione minima, hanno avuto un andamento peggiore mentre il resto
dei portafogli risk-based ha avuto una performance migliore del portafoglio di mercato e quello a
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varianza minima ha generato i più alti rapporti Sharpe e Sortino. Si riscontra che c’è stato
maggior investimento di capitale nel settore immobiliare, delle telecomunicazioni, dei servizi ai
consumatori e della sanità quando è stato adottato un approccio di lungo periodo.
1. INTRODUCTION Portfolio allocation as pioneered by Markowitz (1952) focuses mainly on the maximization of
the mean-variance utility function without paying attention to the protection of investment
capital during financial crises. Following the frequency and severity of portfolio losses during
the recent financial crises, there has been an increase in calls for improved portfolio allocation
methodologies. Portfolio allocation strategies should be able to protect investors’ capital and
result in higher relative returns in economic downturn. Hence the emergence of risk-based asset
allocation methods which focus on portfolio construction based on risk and diversification. This
paper focuses mainly on the comparison with the performance measures (Sharpe and Sortino
ratios) of risk-based asset allocation methods. Although there has been an extensive research
(see for example Allen, 2010 and Lee, 2011) on the performance of risk-based asset allocation
methods, no research has been done focusing on the performance of these methods in the
emerging markets especially in South African market.
The paper shows that the risk-based asset allocation techniques can be useful in tracking the
performance of the market portfolio during different economic business cycles. The study finds
that the minimum correlation portfolio performs better than all other risk-based asset
allocation portfolios during periods of economic downturn. This result is unique given that most
literatures suggest otherwise. For example Barber et. al. (2015) using data from developed
economies found that the minimum correlation portfolio does not consistently perform better
than other risk-based asset allocation portfolios.
Furthermore this paper finds that the risk-based asset allocation portfolios do not consistently
perform better than the naive equally Weighted Portfolio during different economic business
cycles. This finding is consistent with the literature (see DeMiguel et. al., 2009; Brown et. al.,
2013; Jorion, 1991 and Muteba Mwamba, 2012).
The Markowitz (1952) model remains the cornerstone of the modern portfolio theory. However
this model has faced a lot of criticisms for not having a forward approach, and for relying only on
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the past performance of assets which often leads to undiversified portfolios (see Roncalli, 2013
and Lee, 2011). This paper makes use of most recent techniques in portfolio theory namely risk-
based asset allocation techniques in order to have a forward looking approach and significantly
minimise the total risk of the portfolio. The following risk-based asset allocation techniques are
discussed in this paper: the equally weighted portfolio, the most diversified portfolio, the
minimum variance portfolio, the risk parity portfolio, the minimum correlation portfolio, and
the minimum conditional value at risk.
(CVaR) Portfolio. These risk-based allocation techniques aim at significantly minimising the
total portfolio risk and at protecting investors’ capital during period of economic downturn.
The equally weighted portfolio is a portfolio where all assets in the portfolio are given the same
weight. This type of asset allocation strategy is considered to be the simplest of risk-based
portfolios. DeMiguel et al. (2009) investigate the performance of this type of asset allocation
strategy and compare it to the mean-variance model. They find that the equally weighted
portfolio performs poorly in terms of Sharpe ratio. Similarly Kritzman et al. (2010) show that a
well optimized mean-variance portfolio usually outperforms better than the equally weighted
portfolio.
The most diversified portfolio is one of the risk-based asset allocation methods that has recently
gained increasing attention. Choueifaty and Coignard (2008) use the daily performances of the
Standard and Poor’s 500 and the Dow Jones Industrial average indices to compare the most
diversified portfolio with the market cap-weighted benchmark, the minimum variance portfolio
and the equal weighted portfolio. Their empirical results show that the most diversified portfolio
has higher Sharpe ratios than most market-cap weighted indices, and results in higher expected
returns in the long run.
The minimum variance portfolio is a portfolio of assets that has the lowest level of volatility, and
can be found in the efficient frontier without using expected returns as inputs. Empirical studies
(including Ledoit and Wolf, 2003) have shown that the minimum variance portfolio often yields
better out of sample results than the Markowitz mean-variance portfolio. Ledoit and Wolf
(2003) investigate the importance of using the minimum variance portfolio in asset allocation
problems by improving the estimation of the covariance matrices used as inputs in the
optimisation process. Using the data from the New York and the American Stock they find that
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the minimum variance portfolio does not outperform naively diversified portfolios such as the
Equally Weighted Portfolio in terms of the Sharpe ratio.
The method that receives the most attention of all the risk-based allocation methods is probably
the risk parity portfolio method. It has been studied extensively since the 2008 financial crises.
Asness et al. (2012) show that applying leverage in the risk-parity portfolio often results in higher
expected returns.
The minimum correlation portfolio is formed based on the minimum correlation algorithm
proposed by Varadi et al. (2012). They developed this technique with the aim of significantly
minimising the portfolio variance from the correlation function of the variance.
In the minimum conditional value at risk (CVaR) portfolio, the portfolio weights are found by
striking a balance between the return objectives of the portfolio manager and the allocation of
risk across the portfolio. The construction of the portfolio is based on work done by Boudt et al.
(2013) which makes use of the downside risk measure such as the CVaR rather than portfolio
variance as input to optimization problem.
The covariance matrix, the most important input in the construction of these risk-based asset
allocation portfolios is assumed to be constant and time varying respectively. When it is
assumed to be constant an empirical estimate of the covariance matrix is used. However when it
is assumed to be time varying, a multivariate GARCH model with normal distribution
innovations is used in what we refer to as a forward looking approach. This approach consists in
forecasting the covariance matrices using a sixty day rolling window forecasts. The remaining of
the paper proceeds as follows: in section 2 we describe the methodology for the construction of
risk-based asset allocation portfolios, section 3 discusses the empirical results and section 4
concludes the paper.
2. METHODOLOGY This section develops an understanding of how the six risk-based asset allocation portfolios are
constructed using constant and time varying covariance matrix respectively.
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Equally Equally Equally Equally WWWWeighted eighted eighted eighted PPPPortfolioortfolioortfolioortfolio
The equally weighted portfolio is considered to be the simplest portfolio of the risk-based
portfolios. All assets are assumed to have equal weights. The equally weighted portfolio is simply
given by:
��� = �� (1)
where � is the number of assets in the portfolio, � is the n x 1 vector of portfolio weights. The
weight of any asset j will be given by �� and will be the same as all other portfolios. The return
and variance of the portfolio depend only on the number of assets n and they can be given as
follows respectively:
�� = ��∑ ����� (2)
���� = ��� �∑ ������� + 2∑ ��,�������� � (3)
The equally weighted portfolio does not require any estimate of risk, hence the forecasted
covariance risk measures will not be used in this case.
Most DiversiMost DiversiMost DiversiMost Diversified fied fied fied PPPPortfolioortfolioortfolioortfolio
The Most Diversified Portfolio (MDP) is one of the risk-based asset allocation methods that has
recently gained increasing attention. The investment objective in this case is to achieve the
highest level of diversification in a portfolio. Choueifaty and Coignard (2008) introduce a
measure of portfolio diversification known as the diversification ratio (DR) which is used to
construct the most diversification portfolios. It is the ratio between the weighted average
volatility and the portfolio volatility and is given by:
����� = ∑ ������� ���� (4)
where ���� is the volatility of the portfolio, and �� and �� are the weight and the volatility of
asset j respectively.
Following work by Choueifaty et al. (2011) the diversification ratio can be decomposed into the
concentration ratio and the volatility-weighted average correlation as follows:
����� = [�����1 − $����� + $����]& � (5)
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where ���� is the volatility-weighted average correlation and $���� is the volatility-weighted
concentration ratio of the assets in the portfolio. The volatility-weighted average correlation is
given by:
���� = ∑ ��'�'�����(',�')�∑ ��'�'����')� � (6)
while the volatility-weighted concentration ratio by:
$���� = ∑ ��'�'���' ��∑ ��'�'��' ��� (7)
The concentration ratio measures the portfolio concentration and takes into account the
volatility of each asset, hence the ratio not only measures the concentration of weights, but also
the concentration of risks. When the correlation of assets in the portfolio are equal, it can be
deduced that the diversification ratio only changes with the concentration ratio, hence
maximising the diversification ratio will yield the same result as minimising the concentration
ratio.
The most diversified portfolio is defined as a portfolio which maximises the diversification ratio.
We can use the diversification ratio as given in Equation (5) or the original diversification ratio
in Equation (4) to get the most diversified portfolio:
�*+,=max ln ����� (8)
Subject to
- 1.� = 10 ≤ � ≤ 1 The Lagrange function for this optimisation problem can be written as follows:
ℒ��, 2∗, 2� =ln��.�� − �� ln��.∑�� + 2∗�67� − 1� + 2.� (9)
where 2∗ ∈ R and 2 ∈ ℝ�. The solution to the most diversified portfolio weights will be found by
satisfying the first order condition:
9ℒ��,:∗,:�9� = �
�;� − ∑��;∑� + 2∗6 + 2 = 0 (10)
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This equation is satisfied for the minimum values of ω and λ as per Karush–Kuhn–Tucker (KKT)
conditions. Roncalli (2013) analysed the core property1 of the MDP by looking at the correlation
between a certain asset < and the MDP. This analysis led to the MDP weights given as follows:
��*+, = ����*+,� �'���=>?��@'� A1 − (',=?
(BC D (11)
where �E�� is the idiosyncratic variance, ��,*, is the correlation between asset < and the market
portfolio given by :
��,*, = �=?�' (12)
��, the volatility off asset <, is given by FG���*� and �HI is the threshold correlation given by the
formula:
�HI = J�K∑ L',=?� ML',=?�N'� O
J∑ L',=?� ML',=?�N'� O
(13)
Since this paper takes a forward looking approach to risk-based asset allocation, forecasted
volatility of each asset �� and forecasted correlations � will be obtained by using a sixty day
rolling window of multivariate GARCH model.
Minimum Minimum Minimum Minimum Variance PortfolioVariance PortfolioVariance PortfolioVariance Portfolio
The Minimum Variance Portfolio is a portfolio of assets that has the lowest volatility and can be
found in the eKcient frontier without using expected returns as inputs. It is sometimes referred
to as the Global Minimum Variance Portfolio. Empirical studies have shown that the MVP often
yields better out-of sample results than the Markowitz based mean-variance portfolio (Ledoit
and Wolf, 2003 and Clark et al., 2006 ). Put simply, the minimum variance portfolio is found by
minimizing the portfolio variance and has the following optimisation problem:
�*P = min ���.Σw (14)
Subject to:
U67� = 1
1 The core property simply states that the MDP (with the long only constraint) is the only portfolio that has a
correlation between itself and other long only portfolios greater than or equal to their diversification ratios.
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where Σ is the N x N covariance matrix. Following work by Roncalli (2013), we can also find the
global minimum variance portfolio as follows: we first express the objective minimising the
variance for a given level of expected return as follows:
�*P = min�Vℝ A���.Σ − 2�.̅D (15)
Subject to:
U�.6 = 1,
where ̅ is a vector of asset returns of a portfolio, Σ is the N x N covariance matrix of the portfolio
and 2 is the risk aversion parameter. An investor can use the risk aversion parameter to scale the
level of risk that they prefer. The method of Lagrange multipliers is used for this optimisation
problem. The Lagrangian function for this system which is the same as for the maximise
expected return system can be defined as:
ℒ��, 2∗� = �.̅ − :��.∑� + 2∗�67� − 1� (16)
the first order diLerential equations for this equation are:
9ℒ��,:∗�9� = ̅ − 2Σw + 2∗6 (17)
9ℒ��,:∗�9:∗ = 6.� − 1 (18)
from Equation (17) we can solve for the minimum variance weight as:
�*P = ̅ − 2&�Σ&��̅ + 2∗6� (19)
Substituting this weight into Equation (18) we get:
2∗ = :&6;XM Y̅6;XM 6 (20)
We can further simplify Equation (19) into
�*P = ̅ − 2&�Σ&�̅ + ̅ −2&�Σ&�2∗6 (21)
and substituting λ∗ into this equation we get
�*P = XM 66;XM 6+ 2&�Σ&�̅ A1 − 6 67XM
6;XM 6D (22)
Equation (22) is the solution to the global minimum variance portfolio optimisation problem
and can be interpreted as follows: the first term determines the global minimum variance
portfolio and the second part of the equation shows the portfolios’ expected return relative to
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the return of the individual assets within the global minimum variance portfolio (Lee, 2011). In a
special case where all the assets expected excess returns are the same, the second part of
Equation (22) is zero and the investor will hold the global minimum variance portfolio. The
solution (optimal portfolio weights) to the optimisation problem is found using forecasted
covariance matrix Σ.
RiskRiskRiskRisk----parity parity parity parity PPPPortfolios: Inverse Volatility Portfolioortfolios: Inverse Volatility Portfolioortfolios: Inverse Volatility Portfolioortfolios: Inverse Volatility Portfolio
The risk parity method has been studied extensively since the 2008 financial crises (see Lee,
2011; Maillard et al., 2010 and Neukirch, 2008). It can be described as an asset allocation method
that aims to allocate market risk equally among diLerent asset classes. In multi-asset portfolios
where n > 2, the number of parameters can get very large and it is not possible to get analytical
solutions for the risk parity weights. Closed-form solutions can be found for some cases, e.g. one
can find solutions if it is assumed that correlation is the same for diLerent assets. If one assume a
constant correlation matrix with ��,�=�, the total risk contribution of asset < can be written as:
����� = ������� + �∑ ���[� �� ���������
= ��'��'�K(∑ �'� ���'������� (23)
which can also be written as:
����� = ��'��'�K(∑ �'')� ���'������� (24)
Using this equation and the ERC portfolio implies �����= �����∀<, ], the ERC portfolio satisfies
������1 − ������ + �∑ �P�P�P � = ���� A�1 − ������ + �∑ �P�P�P D (25)
It follows that ���� = ���� and we can use the fact that ∑ ������ = 1 to find the weights of the ERC
portfolio in this case:
�� = �'M ∑ ��M N�� (26)
This shows that the weight allocated to particular asset or sector index < is inversely proportional
to its volatility.
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Minimum Correlation PortfolioMinimum Correlation PortfolioMinimum Correlation PortfolioMinimum Correlation Portfolio
The Minimum Correlation Portfolio construction technique is not very popular and is a less
researched risk based asset allocation method. This might be as a result of the overreliance of
academics and practitioners on volatility in measuring risk. It is formed based on the minimum
correlation algorithm by Varadi et al. (2012). The algorithm stems from the formula of the
portfolio variance. The portfolio variance of a two asset portfolio is simply given by:
�,� = ������ + ������ + 2��������� (27)
we can extend this to a portfolio with n assets:
�,� = �.Σw
or
∑ ∑ ��,����� �������� (28)
where ��,�= ��,�����is the covariance between asset < and asset ], ��,� is the correlation
coeKcient and ��,�= ���. We can simply write the portfolio variance as:
�,� = ∑ ∑ ��,��������� �������� (29)
The correlation between two assets is simply given by:
��,� = �',��'�� (30)
In a minimum correlation portfolio the objective is to minimise correlations between assets in
the portfolio since this results in lower portfolio volatility. The minimum correlation algorithm
developed by Varadi et al. (2012) is used to find the minimum correlation portfolio weights. In
determining the weights, the weighted average correlation is used and this enables the
portfolio’s average correlation to be minimised by giving more weight to assets with low
correlation compared to the rest of portfolio and less weight to assets with high correlations
which ensures more diversification in the portfolio.
The concept of proportional weights is used because of its simplicity to calculate, as well as its
ability, to speed up the calculations. The minimum correlation algorithm uses the long only
constraint as this results in more stable portfolio weights solutions. The correlation matrix is
converted to a relative scale that is only positive to ensure the long only constraint holds. The
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rank weighted function2 is used in weighing of individual correlations to get the average
weighted correlation. Furthermore, each asset is normalised for volatility to ensure they have
the same risk. Portfolio weights are found by using the following steps:
I. We start by computing the correlation matrix,� the portfolio mean, ^,and portfolio
volatility, �,.
II. Create an adjusted correlation matrix, �_ by transforming each of the elements in the
correlation matrix the (-1; +1) space to the (1; 0) space using the normal inverse
transformation.
III. Calculate the average value for each row of the adjusted correlation matrix , �_, and use
this as initial portfolio weight estimates after the transformation, �..
IV. Compute the rank portfolio weight estimates:
�`a�b = �c�d��.�∑ �c�d��.��
V. Combine rank portfolio weights with adjusted correlation matrix to find new weights:
�� = �c�d��.�x�_∑ �c�d��.�� x�_
VI. Last but not least, find the minimum correlation portfolio weights by scaling portfolio
weights by the assets volatilities and normalise weights to sum to 1:
�*I, =�' �'f∑ �' �'f' (31)
Varadi et al. (2012) shows the algorithm for 2 cases, the case where the adjusted correlation
matrix is used, and the case where the raw correlation matrix is used to compute average
correlations. The first case is looked at because it makes much more sense to adjust the
correlation matrix into the (1; 0) space. The use of the average correlations also makes the
portfolio less sensitive to estimation errors, which is a very important aspect to always consider
in portfolio construction.
Minimum CVaR Minimum CVaR Minimum CVaR Minimum CVaR PPPPortfolioortfolioortfolioortfolio
Conditional Value-at-risk (CVaR) is an extension of Value at Risk that takes into account the
shape of the loss distribution in the tails. A portfolio based on the CVaR is analysed since it is an
improvement on the VaR in the sense that it adheres to the properties of coherent risk measures
and takes into consideration the shape of the loss distribution in the tails. Portfolio weights are
2 The average weighted correlation is ranked from highest to smallest e.g. (0.5 0.35 0.15) will be ranked (1 2 3).
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found by striking a balance between the return objectives of the portfolio manager and the
allocation of risk (CVaR) across the portfolio. The risk contribution of an asset, say asset <, using
a certain risk measure, RM, and given in terms of weights is given by:
�g� = ��9`h���9�'
(32)
hence the CVaR contribution of asset < can be written as:
$�$ic���j� = ��9kla`m�n�
9�' (33)
where the portfolio CVaR is defined as the expected portfolio return less than 0 when the return
is less than its j probability level quartile:
$ic���j� = −o!�|� ≤ −ic���j�% (34)
In other words, CVaR calculates the negative expected returns or losses of the portfolio for a
given probability level j exceeding the VaR. We define VaR as the negative value of the j quartile
of the portfolio returns. We can also define the CVaR contribution as the conditional expectation
of returns given that the portfolio loss is larger than the VaR threshold (Scaillet, 2004):
$�$ic���j� = −o!���|� ≤ −ic���j�% (35)
As with other risk measures the summation of CVaR risk contributions for all assets in the
portfolio, say �, will give a portfolio CVaR:
$ic���j� = ∑ �� 9kla`m�n�9�'�� (36)
hence we can also write the percentage CVaR risk contribution of asset < as:
%$�$ic���j� = k'kla`m�n�kla`m�n� (37)
To estimate CVaR risk contributions and use them in portfolio optimisation, we follow the
approach by Boudt et al. (2013). The actual risk contribution using CVaR can be found in two
ways. One of the ways is to find the risk contributions by replacing the expectation in Equation
35 simulated or historical data. In portfolio optimisation, there may be a large number of assets
in a portfolio which means risk contribution needs to be estimated for a large number of weights,
hence the need for a second way of estimating risk contributions. A simpler way in estimating
risk contributions will be to use analytical formulas. Boudt et al. (2013) showed that if returns
are conditionally normally distributed, CVaR is given by:
$ic���j� = −�.^ + √�.Σw ∅�tu�n (38)
where vn is the j quantile of the standard normal distribution and ∅ is the standard normal
density function. The CVaR contribution of asset < is given by:
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$�$ic���j� = �� w−^� + �Xx�'y�;Xx
∅�tu�n z (39)
These analytical formulas assume returns are normally distributed but it is known that financial
returns are not always normally distributed, hence we need a CVaR contribution estimator that
does not make the normal return assumption. Boudt et al. (2008) proposed a modified CVaR
contribution estimator based on the Cornish-Fisher expansions and it has shown accurate
estimates of CVaR contributions for assets with non-normal returns. It is important to note that
the Cornish-Fisher based estimates works well for tail probabilities j that are not too small,
because for smaller values of j it becomes less reliable, hence Boudt et al. (2008) set the
probability level at 5% in portfolio optimisation practices. Weights of the minimum CVaR
portfolio are found by minimising the portfolio CVaR:
�*kla`� = min�∈|$ic���j� (40)
}~�]�����∑ �� = 1��
where � is a set of feasible portfolio weights. The Lagrangian of the above optimisation problem
is given by:
ℒ��, 2� = $ic���j� + 2��.6 − 1� (41)
where 1 is the N x 1 vector with each element equal to 1. From the first order conditions results:
9ℒ��,:�9� = 9�kla`m�n�K:��;6&���
9� (42)
it is deduced that:
9kla`m�n�9� = −26 (43)
This means the partial derivatives will be equal and because we have the CVaR contributions
given by:
$�$ic���j� = �� 9kla`m�n�9�' ,
then there exists a unique constant k such that:
�*kla`� = d$�$ic��=�����j�, ∀< ∈ �1: �� (44)
This results in the weights for the minimum CVaR portfolio where d = 1 �*kla`f when we
have the full investment constraint, hence Equation 44 becomes:
�*kla`� = k'kla`m�n�kla`m�n� = %$�$ic���j� (45)
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This means the minimum CVaR portfolio weights coincide with the percentage CVaR allocation.
The variance-covariance matrix in Equations 38 and 39 are estimated using the multivariate
GARCH models.
3. EMPIRICAL ANALYSIS We make use of daily closing prices of the following Johannesburg Stock Exchange sector
indices: Financials (JFN) sector, Mining (JMN) sector, Industrials (JIN) sector, Technology
(JTC) sector, Properties (JPR) sector, Oil and Gas (JOG) sector, Basic Materials (JBM) sector,
Consumer Goods (JCG) sector, Health Care (JHC) sector, Telecommunications (JTE) sector,
and Consumer Services (JCS) sector indices. The dataset cover the period between August 2009
and September 2015. We divide this sample period into three overlapping sub-samples
representing the pre-crisis period (August 2004 to September 2007), the crisis period (October
2007 to September 2009), and the post-crisis period (October 2009 to September 2015). The
JSE All Share Index (ALSI) is used as a proxy of the market portfolio. Table 1 to Table 3 below
report the descriptive statistics of the data during the three sub-sample periods.
TABLE 1 - JSE Sectors Descriptive Statistics: Pre-Crisis
ALSIALSIALSIALSI JFNJFNJFNJFN JMNJMNJMNJMN JINJINJINJIN JTCJTCJTCJTC JPRJPRJPRJPR JOGJOGJOGJOG JBMJBMJBMJBM JCGJCGJCGJCG JHCJHCJHCJHC JTEJTEJTEJTE JCSJCSJCSJCS
Mean(%)Mean(%)Mean(%)Mean(%) 0.14% 0.11% 0.17% 0.13% 0.13% 0.12% 0.15% 0.14% 0.14% 0.11% 0.17% 0.13%
MedianMedianMedianMedian 0.22% 0.15% 0.22% 0.21% 0.05% 0.16% 0.16% 0.18% 0.09% 0.13% 0.18% 0.16%
Std. Dev(%)Std. Dev(%)Std. Dev(%)Std. Dev(%) 1.12% 1.23% 1.71% 1.02% 1.22% 0.75% 2.08% 1.49% 1.22% 1.30% 1.75% 1.01%
Variance(%)Variance(%)Variance(%)Variance(%) 0.01% 0.02% 0.03% 0.01% 0.01% 0.01% 0.04% 0.02% 0.01% 0.02% 0.03% 0.01%
KurtosisKurtosisKurtosisKurtosis 3.42 3.37 1.45 3.78 1.08 4.82 1.46 2.62 1.97 2.71 1.25 2.67
SkewnessSkewnessSkewnessSkewness -0.44 -0.19 -0.05 -0.56 0.24 -1.06 -0.13 -0.19 -0.12 0.06 0.12 -0.75
Minimum(%)Minimum(%)Minimum(%)Minimum(%) -6.48% -7.11% -6.64% -6.57% -3.94% -4.71% -8.38% -6.57% -7.08% -5.51% -5.67% -4.86%
Maximum(%)Maximum(%)Maximum(%)Maximum(%) 5.04% 7.06% 7.26% 4.80% 5.44% 2.53% 8.24% 7.05% 4.86% 6.48% 8.98% 0.13%
Table 1 reports the descriptive statistics of all market sectors during the pre-crisis period. It can
be seen that the mining sector and the telecommunication sector exhibit the highest rate of
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return followed by the market portfolio. Leading to the crisis period, Tables 1 shows that all
Johannesburg stock market sectors exhibit positive kurtosis suggesting the possibility of large
price drop for those sectors with negative skewness.
TABLE 2 - JSE Sectors Descriptive Statistics: Crisis Period
ALSIALSIALSIALSI JFNJFNJFNJFN JMNJMNJMNJMN JINJINJINJIN JTCJTCJTCJTC JPRJPRJPRJPR JOGJOGJOGJOG JBMJBMJBMJBM JCGJCGJCGJCG JHCJHCJHCJHC JTEJTEJTEJTE JCSJCSJCSJCS
Mean(%)Mean(%)Mean(%)Mean(%) -0.02% -0.02% -0.02% 0.01% -0.01% -0.02% 0.04% -0.02% 0.02% 0.06% 0.06% 0.02%
MedianMedianMedianMedian -0.05% -0.11% -0.19% -0.06% 0.02% -0.01% 0.01% -0.11% 0.00% 0.07% -0.13% -0.08%
Std. Dev(%)Std. Dev(%)Std. Dev(%)Std. Dev(%) 1.97% 2.09% 3.11% 1.70% 2.16% 1.08% 3.20% 3.01% 1.85% 1.64% 2.92% 1.59%
Variance(%)Variance(%)Variance(%)Variance(%) 0.04% 0.04% 0.10% 0.03% 0.05% 0.01% 0.10% 0.09% 0.03% 0.03% 0.09% 0.03%
KurtosisKurtosisKurtosisKurtosis 1.3 1.01 1.96 1.35 5.33 4.42 1.42 1.9 9.2 0.87 2.07 0.63
SkewnessSkewnessSkewnessSkewness 0.12 0.29 0.27 0.3 -0.15 -0.17 0.35 0.22 1.13 0.14 0.5 0.19
Minimum(%)Minimum(%)Minimum(%)Minimum(%) -7.30% -6.98% -11.28% -6.03% -12.74% -6.54% -10.10% -11.14% -7.17% -5.05% -10.40% -5.38%
Maximum(%)Maximum(%)Maximum(%)Maximum(%) 7.07% 8.43% 12.32% 7.44% 10.75% 4.16% 12.11% 11.81% 15.27% 6.22% 14.41% 0.02%
In contrast to Table 1 above, Table 2 shows that the oil and gas sector, the consumer goods
sector, the health sector, the telecommunication sector, and the consumer services sector
produce negative rate of return. The stock market during the financial crisis is characterised by
negative skewness and positive kurtosis respectively suggesting significant price drops during
this period. The few market sectors with positive rate of return exhibit moderately positive
return in the crisis period. These include the industrial sector, the oil and gas sector, the
consumer goods sector, the health care sector, the telecommunication sector, and the consumer
services sector.
However as it can be seen in Table 3, most market sectors are characterised by positive rate of
return in what we can refer to as the recovery period. Although characterised by positive
kurtosis in all market sectors, the pre-crisis period exhibit relatively insignificant negative
excess skewness indicating low probability of price drops.
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TABLE 3 - JSE Sectors Descriptive Statistics: Post-Crisis
ALSIALSIALSIALSI JFNJFNJFNJFN JMNJMNJMNJMN JINJINJINJIN JTCJTCJTCJTC JPRJPRJPRJPR JOGJOGJOGJOG JBMJBMJBMJBM JCGJCGJCGJCG JHCJHCJHCJHC JTEJTEJTEJTE JCSJCSJCSJCS
Mean(%)Mean(%)Mean(%)Mean(%) 0.05% 0.06% -0.01% 0.08% 0.12% 0.05% 0.13% 0.00% 0.09% 0.10% 0.04% 0.11%
MedianMedianMedianMedian 0.07% 0.08% -0.04% 0.13% 0.07% 0.07% 0.00% -0.01% 0.11% 0.09% 0.05% 0.14%
Std. Dev(%)Std. Dev(%)Std. Dev(%)Std. Dev(%) 0.96% 1.07% 1.52% 1.00% 1.11% 0.75% 2.50% 1.43% 1.07% 1.06% 1.52% 1.30%
Variance(%)Variance(%)Variance(%)Variance(%) 0.01% 0.01% 0.02% 0.01% 0.01% 0.01% 0.06% 0.02% 0.01% 0.01% 0.02% 0.02%
KurtosisKurtosisKurtosisKurtosis 1.48 1.68 0.97 1.82 33.96 4.1 48.65 0.87 1.52 1.28 1.05 1.99
SkewnessSkewnessSkewnessSkewness -0.15 -0.06 0.1 -0.17 2.21 -0.09 3.52 0.09 -0.18 -0.03 -0.03 0.16
Minimum(%)Minimum(%)Minimum(%)Minimum(%) -3.63% -5.32% -6.49% -4.24% -7.01% -4.42% -15.79% -5.97% -5.33% -4.91% -6.69% -4.77%
Maximum(%)Maximum(%)Maximum(%)Maximum(%) 4.32% 5.21% 6.12% 4.80% 16.32% 4.75% 36.45% 5.75% 4.73% 4.24% 6.94% 0.11%
Assuming constant covariance matrix, we construct six risk-based portfolios (equally weighted,
risk parity, most diversified, minimum correlation, minimum variance and the minimum
conditional value at risk portfolio) during each economic business cycles. The resulting capital
allocations (weights) are reported in Tables 4 to 6 below. Table 4 reports the constant
investment capital allocation to each market sector during the pre-crisis period while Table 5
and Table 6 report the constant investment capital allocation during the crisis and post-crisis
periods respectively.
During the pre-crisis period nearly all risk-based portfolios allocate more investment capital to
property sector with the exception of the minimum CVaR portfolio which allocates more
investment capital to oil and gas sector.
During the crisis period most risk-based portfolio continued to allocate more investment capital
to the property sector. The minimum variance portfolio together with the minimum CVaR risk-
based portfolios shifted large investment capital into the basic material sector.
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TABLE 4 - Risk-based Portfolio Estimated Weights: Pre-Crisis Back-test
Equally Equally Equally Equally WeightedWeightedWeightedWeighted
Maximum Maximum Maximum Maximum DiversifiedDiversifiedDiversifiedDiversified
Minimum Minimum Minimum Minimum VarianceVarianceVarianceVariance
Risk Risk Risk Risk ParityParityParityParity
Minimum Minimum Minimum Minimum CorrelationCorrelationCorrelationCorrelation
Minimum Minimum Minimum Minimum CVaRCVaRCVaRCVaR
JFNJFNJFNJFN 9.09% 9.01% 0.00% 9.22% 7.28% 10.04%
JMNJMNJMNJMN 9.09% 6.41% 0.00% 6.62% 6.19% 14.14%
JINJINJINJIN 9.09% 10.77% 2.71% 11.08% 6.81% 8.90%
JCTJCTJCTJCT 9.09% 9.59% 11.73% 9.31% 11.37% 4.83%
JPRJPRJPRJPR 9.09% 15.70% 52.96% 15.05% 22.00% 0.53%
JOGJOGJOGJOG 9.09% 5.44% 1.43% 5.44% 5.90% 14.96%
JBMJBMJBMJBM 9.09% 7.40% 4.17% 7.61% 6.66% 12.12%
JCGJCGJCGJCG 9.09% 9.19% 10.69% 9.26% 7.86% 8.22%
JHCJHCJHCJHC 9.09% 8.78% 5.03% 8.72% 8.86% 7.95%
JTEJTEJTEJTE 9.09% 6.55% 0.00% 6.48% 7.03% 11.79%
JCSJCSJCSJCS 9.09% 11.15% 11.28% 11.20% 10.03% 6.53%
TABLE 5 - Risk-based Portfolio Estimated Weights: Crisis Period Back-test
Equally Equally Equally Equally WeightedWeightedWeightedWeighted
Maximum Maximum Maximum Maximum DiversifiedDiversifiedDiversifiedDiversified
Minimum Minimum Minimum Minimum VarianceVarianceVarianceVariance
Risk Risk Risk Risk ParityParityParityParity
Minimum Minimum Minimum Minimum CorrelationCorrelationCorrelationCorrelation
Minimum Minimum Minimum Minimum CVaRCVaRCVaRCVaR
JFNJFNJFNJFN 9.09% 8.58% 0.00% 8.64% 7.66% 8.95%
JMNJMNJMNJMN 9.09% 5.65% 0.00% 5.81% 5.12% 14.49%
JINJINJINJIN 9.09% 10.43% 6.45% 10.62% 6.57% 8.44%
JCTJCTJCTJCT 9.09% 8.43% 2.00% 8.33% 9.21% 7.52%
JPRJPRJPRJPR 9.09% 17.02% 33.36% 16.66% 22.19% 2.94%
JOGJOGJOGJOG 9.09% 5.59% 0.00% 5.64% 5.59% 13.35%
JBMJBMJBMJBM 9.09% 5.82% 41.17% 5.98% 5.19% 14.19%
JCGJCGJCGJCG 9.09% 9.77% 4.54% 9.77% 9.56% 6.96%
JHCJHCJHCJHC 9.09% 11.19% 8.95% 11.02% 13.11% 5.21%
JTEJTEJTEJTE 9.09% 6.21% 0.00% 6.18% 5.78% 11.33%
JCSJCSJCSJCS 9.09% 11.32% 3.53% 11.35% 10.00% 6.61%
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TABLE 6 - Risk-based Portfolio Estimated Weights: Post-Crisis Period Back-test
Equally Equally Equally Equally WeightedWeightedWeightedWeighted
Maximum Maximum Maximum Maximum DiversifiedDiversifiedDiversifiedDiversified
Minimum Minimum Minimum Minimum VarianceVarianceVarianceVariance
Risk Risk Risk Risk ParityParityParityParity
Minimum Minimum Minimum Minimum CorrelationCorrelationCorrelationCorrelation
Minimum Minimum Minimum Minimum CVaRCVaRCVaRCVaR
JFNJFNJFNJFN 9.09% 9.84% 0.00% 10.09% 8.22% 9.21%
JMNJMNJMNJMN 9.09% 6.99% 0.00% 7.14% 7.15% 12.56%
JINJINJINJIN 9.09% 10.44% 0.00% 10.79% 7.44% 9.38%
JCTJCTJCTJCT 9.09% 10.10% 5.14% 9.75% 11.31% 3.80%
JPRJPRJPRJPR 9.09% 15.07% 16.28% 14.55% 17.76% 1.31%
JOGJOGJOGJOG 9.09% 4.47% 0.35% 4.34% 6.02% 16.95%
JBMJBMJBMJBM 9.09% 7.39% 18.16% 7.57% 7.27% 12.24%
JCGJCGJCGJCG 9.09% 9.98% 29.53% 10.13% 8.90% 7.52%
JHCJHCJHCJHC 9.09% 10.28% 6.69% 10.20% 10.48% 6.36%
JTEJTEJTEJTE 9.09% 7.18% 10.10% 7.14% 7.49% 10.81%
JCSJCSJCSJCS 9.09% 8.26% 13.74% 8.32% 7.98% 9.88%
After the crisis period, i.e. the post-crisis period, we saw a more diversified investment capital
allocation. Although the property sector remains the most preferred investment destination, the
amount of investment capital allocated in this sector was reduced and spread out in other
market sectors.
TABLE 7 - Portfolio Performance Measures: Pre-Crisis
EquallyEquallyEquallyEqually WeightedWeightedWeightedWeighted
MaximumMaximumMaximumMaximum DiversifiedDiversifiedDiversifiedDiversified
MinimumMinimumMinimumMinimum VarianceVarianceVarianceVariance
Risk Risk Risk Risk ParityParityParityParity
MinimumMinimumMinimumMinimum CorrelationCorrelationCorrelationCorrelation
MinimumMinimumMinimumMinimum CVaRCVaRCVaRCVaR
VolatilityVolatilityVolatilityVolatility 0.96 0.88 0.68 0.89 0.84 1.07
Cumulative Cumulative Cumulative Cumulative PerformancePerformancePerformancePerformance
191.09 183.60 167.58 183.91 182.46 200.43
Annualised Annualised Annualised Annualised ReturnsReturnsReturnsReturns
40.61 39.45 36.88 39.49 39.27 42.03
Sharpe RatioSharpe RatioSharpe RatioSharpe Ratio -1.80 -3.28 -8.04 -3.21 -3.65 -0.28
Sortino RatioSortino RatioSortino RatioSortino Ratio -2.88 -5.21 -12.46 -5.10 -5.77 -0.45
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Assuming constant covariance matrix and constant weights as reported in Table 4, 5, and 6, we
were able to analyse the performance of each risk-based portfolio during different economic
business cycles based on the portfolio Sharpe ratio, Sortino ratio, the annualised return,
volatility and cumulative performance. Table 7 to Table 9 report these performance criterions
during each economic business cycle. The volatility, the cumulative performance, and the
annualised returns are all reported in percentage change.
The Sharpe ratio is computed as the ratio of the difference between the market portfolio as
proxied by the Johannesburg All Share index and expected return of that specific risk-based
portfolio over the standard deviation of the risk-based portfolio, that is:
�ℎc���c�<� = `=����B&������
���<���c�<� = `=����B&������
where �*aYb�H, o���, ��, and G� represent the market portfolio return as proxied by the JSE
ALSI return, the risk-based portfolio expected return, the standard deviation of the risk-based
portfolio, and the beta of the risk-based portfolio respectively. A positive ratio indicates that the
market portfolio return is greater than the expected return of the risk-based portfolio, while a
negative ratio indicates the opposite.
TABLE 8 - Portfolio Performance Measures: Crisis Period
EquallyEquallyEquallyEqually WeightedWeightedWeightedWeighted
MaximumMaximumMaximumMaximum DiversifiedDiversifiedDiversifiedDiversified
MinimumMinimumMinimumMinimum VarianceVarianceVarianceVariance
Risk Risk Risk Risk ParityParityParityParity
MinimumMinimumMinimumMinimum CorrelationCorrelationCorrelationCorrelation
MinimumMinimumMinimumMinimum CVaRCVaRCVaRCVaR
VolatilityVolatilityVolatilityVolatility 1.65 1.45 1.65 1.46 1.38 1.92
Cumulative Cumulative Cumulative Cumulative PerformancePerformancePerformancePerformance
4.91 4.35 -4.87 4.31 4.01 4.35
Annualised Annualised Annualised Annualised ReturnsReturnsReturnsReturns
2.44 2.16 -2.48 2.15 2.00 2.17
Sharpe RatioSharpe RatioSharpe RatioSharpe Ratio1 4.09 4.47 1.10 4.43 4.56 3.37
Sortino RatioSortino RatioSortino RatioSortino Ratio 6.96 7.60 1.86 7.54 7.72 5.73
During the pre-crisis period both the Sharpe and Sortino ratios are found to be negative
indicating a poor performance of the market portfolio during this period. This economic period
was characterised by a cycle of interest rate hikes which helped risk-based portfolio strategies to
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perform better than naïve investment strategies proxied by the market portfolio. The equally
weighted portfolio generated the highest rate of return with relatively larger risk.
TABLE 9 - Portfolio Performance Measures: Post-Crisis Back-test
EquallyEquallyEquallyEqually
WeightedWeightedWeightedWeighted MaximumMaximumMaximumMaximum
DiversifiedDiversifiedDiversifiedDiversified MinimumMinimumMinimumMinimum VarianceVarianceVarianceVariance
Risk Risk Risk Risk ParityParityParityParity
MinimumMinimumMinimumMinimum CorrelationCorrelationCorrelationCorrelation
MinimumMinimumMinimumMinimum CVaRCVaRCVaRCVaR
VolatilityVolatilityVolatilityVolatility 0.89 0.82 0.83 0.82 0.80 1.04
Cumulative Cumulative Cumulative Cumulative PerformancePerformancePerformancePerformance
181.95 184.52 172.64 183.19 185.44 169.12
Annualised Annualised Annualised Annualised ReturnsReturnsReturnsReturns
19.25 19.43 18.57 19.33 19.50 18.31
Sharpe RatioSharpe RatioSharpe RatioSharpe Ratio 6.42 7.21 6.08 7.06 7.41 4.56
Sortino RatioSortino RatioSortino RatioSortino Ratio 10.81 12.06 10.13 11.80 12.44 7.77
However during the financial crisis as well as the post-crisis period the Sharpe ratio and the
Sortino ratio are found to be positive, an indication that during these two periods the market
portfolio performed better than the all risk-based investment strategies. We argue that this
market performance might have been due to the fact that risk behaviour is assumed to be
constant over time. To overcome this issue we model the covariance matrix by making use of the
Dynamic Conditional GARCH (DCC-GARCH) model with different marginal distributions. We
refer to this process as the forward looking approach since we estimate the covariance matrix
using a sixty day rolling window forecast. The resulting portfolio weights, i.e. investment capital
allocations, are time varying and therefore are not reported here due to space constraints.
We fit the covariance matrix to three multivariate GARCH models of order one namely the DCC-
GARCH(1,1) with multivariate normal distribution, the DCC-GARCH(1,1) with multivariate
student t distribution, and the DCC-GARCH(1,1) model with multivariate normal copula. The
estimated parameters are reported in Table 12a, 12b, and 12c in Appendix for the pre-crisis,
crisis, and post crisis period respectively. Based on the Akaike Information Criteria (AIC) and
the Bayes Information Criteria (BIC) it can be shown that the multivariate DCC-GARCH(1,1)
model with multivariate normal distribution best fits the covariance matrices during the pre-
crisis, the crisis, and the post-crisis periods respectively. Therefore we make use of the
multivariate DCC-GARCH(1,1) model with multivariate normal distribution to generate out-
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sample forecasts of (time varying) covariances used as inputs in the optimisation process of the
risk-based portfolios.
Given that the main purpose of this study is to compare the performance of risk-based asset
allocation techniques that is, the risk-based portfolios with that of the market portfolio, we
therefore focus the rest of this study on the performance of these risk-based portfolios during
the out-sample space considered here as the crisis and post-crisis periods. We make use of the
volatility, cumulative performance, monthly return, Sharpe ratio, and Sortino ratio as
performance criterions. These performance criterions are reported in Table 10 and Table 11
below.
TABLE 10 - Out-sample Portfolio Performance: The Crisis Period
EquallyEquallyEquallyEqually
WeightedWeightedWeightedWeighted MaximumMaximumMaximumMaximum
DiversifiedDiversifiedDiversifiedDiversified MinimumMinimumMinimumMinimum VarianceVarianceVarianceVariance
Risk Risk Risk Risk ParityParityParityParity
MinimumMinimumMinimumMinimum CorrelationCorrelationCorrelationCorrelation
MinimumMinimumMinimumMinimum CVaRCVaRCVaRCVaR
VolatilityVolatilityVolatilityVolatility 0.92% 0.98% 1.03% 0.88% 0.85% 0.98%
Cumulative Cumulative Cumulative Cumulative PerformancePerformancePerformancePerformance
2.46% 2.35% 1.76% 2.74% 2.69% 2.15%
Monthly Monthly Monthly Monthly ReturnsReturnsReturnsReturns
1.20% 1.15% 0.86% 1.34% 1.31% 1.05%
Sharpe RatioSharpe RatioSharpe RatioSharpe Ratio 1.91 1.74 1.38 2.15 2.21 1.63
Sortino RatioSortino RatioSortino RatioSortino Ratio 3.31 2.97 2.33 3.83 3.94 2.79
Table 10 reports the performance of the risk-based portfolios based on different performance
criterions during the first part of the out-sample space i.e. the crisis period. During this period
risk-based portfolios performed very well compared to the market portfolio. The differences
between the average returns of all risk-based portfolios and the market portfolio (i.e. the
numerator of the Sharpe and Sortino ratio as defined in this paper) are positive leading to
positive Sharpe and Sortino ratios. These results are consistent with findings obtained in
previous studies (see for example Allen, 2010; Rappoport and Nottebohm, 2012). The
performance of the risk-based portfolios in the second part of the out-sample space is reported
in Table 11 below.
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TABLE 11 - Out-sample Portfolio Performance: The Post-crisis Period
EquallyEquallyEquallyEqually WeightedWeightedWeightedWeighted
MaximumMaximumMaximumMaximum DiversifiedDiversifiedDiversifiedDiversified
MinimumMinimumMinimumMinimum VarianceVarianceVarianceVariance
Risk Risk Risk Risk ParityParityParityParity
MinimumMinimumMinimumMinimum CorrelationCorrelationCorrelationCorrelation
MinimumMinimumMinimumMinimum CVaRCVaRCVaRCVaR
VolatilityVolatilityVolatilityVolatility 0.76% 0.82% 0.85% 0.71% 0.69% 0.81%
Cumulative Cumulative Cumulative Cumulative performanceperformanceperformanceperformance
10.35% 11.33% 11.59% 9.17% 8.96% 11.06%
Annualised Annualised Annualised Annualised ReturnsReturnsReturnsReturns
4.97% 5.42% 5.54% 4.41% 4.31% 5.29%
Sharpe RatioSharpe RatioSharpe RatioSharpe Ratio -0.39 0.19 0.33 -1.21 -1.39 0.04
Sortino RatioSortino RatioSortino RatioSortino Ratio -0.70 0.35 0.61 -2.10 -2.41 0.07
Table 11 shows mixed performance results during the post-crisis period: the equally weighted
portfolio, the risk parity portfolio, and the minimum correlation portfolio performed poorly
during this period while the maximum diversified portfolio, minimum variance portfolio, and
the minimum CVaR portfolio performed very well during this sample period with the minimum
variance generating the highest Sharpe and Sortino ratios.
4. CONCLUSIONS This paper aimed at analysing the performance of risk-based portfolios (equally weighted, risk
parity, most diversified, minimum correlation, minimum variance and the minimum conditional
value at risk portfolios) using six daily closing prices of six JSE sector indices namely the
financials, mining, industrials, technology, properties, oil & gas, basic materials, consumer
goods, health care, telecommunications, and the consumer services indices starting from August
2004 to September 2015. To achieve this, the study assumed that the covariance matrix used in
the optimisation process of the six risk-based portfolios was constant, that is obtained using only
historical return series, and time varying meaning obtained using sixty days rolling window
forecast from a multivariate Dynamic Conditional Correlation GARCH (1,1) model. The data set
was divided into three different sub-sample periods representing different economic business
cycles observed on the global economic platform: the first sub-sample (August 2004 to
September 2007) representing the period that preceded the 2007-2008 global financial crisis,
the second sub-sample period representing the global financial crisis, and the last sub-sample
period representing the post-crisis or the recovery period.
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When the covariance matrix is firstly assumed to be constant, the study focuses mainly on the
period of each risk-based portfolio during each economic business cycle by making use of the
Sharpe ratio here defined as the difference between the market portfolio and risk-based
portfolio returns, divided by the standard deviation of the risk-based portfolio return and the
Sortino ratio. The results suggest that during the pre-crisis period risk-based portfolios
performed poorly than the market portfolio. This is due to the fact that the market performed
very well due to excellent market conditions coupled with higher interest rates. However during
the crisis and post-crisis periods it was found that risk-based portfolios performed better than
the market portfolio with the minimum correlation portfolio generating the highest Sharpe and
Sortino ratios. Large investment capital during the crisis and post-crisis periods was found to be
mostly allocated to the property sector.
However, when the covariance matrix is assumed to be time varying the pre-crisis period is used
as the in-sample space, while the crisis and post-crisis periods are used as the out-sample space.
The forecasts of the time varying covariances in the out-sample space were obtained by making
use of the multivariate DCC-GARCH(1,1) model based on a sixty rolling window forecast. The
study used three different multivariate DCC-GARCH(1,1) models: the first with multivariate
student t distribution, the second with a multivariate normal distribution, and the third one with
a multivariate normal copula. Based on the Akaike and Bayes Information criterion it was found
that the DCC-GARCH(1,1) with multivariate normal distribution best fitted the data. The DCC-
GARCH model with multivariate normal distribution was thereafter used to generate time
varying covariances using a sixty day rolling window forecast. It was found that during the crisis
period all risk-based portfolios performed better than the market portfolio due to their ability to
protect investor’s capital during financial crisis. However mixed results were found during the
post-crisis period: the equally weighted, the risk parity, and the minimum correlation portfolios
performed poorly in this out-sample space while the rest of the risk-based (most diversified,
minimum variance and the minimum conditional value at risk portfolio) portfolios performed
better than the market portfolio with the minimum variance portfolio generating the highest
Sharpe and Sortino ratios. Large investment capital were found to be allocated to the property,
telecommunication, consumer services, and health sectors when the forward looking approach
was employed.
On the protection of investment capital during financial crisis in the South African equity market 189
ECONOMIA INTERNAZIONALE / INTERNATIONAL ECONOMICS 2017- Volume 70, Issue 2 – May, 165-192
The findings in this study are very important not only for South African investors, but also for
investors in emerging economies characterised by relatively higher interest rates, moderate
political risk, and the risk of falling of commodity prices. The findings highlights the importance
of using risk-based asset allocation, especially the minimum correlation technique during period
of economic downturn in order to protect investor’s capital.
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On the protection of investment capital during financial crisis in the South African equity market 191
ECONOMIA INTERNAZIONALE / INTERNATIONAL ECONOMICS 2017- Volume 70, Issue 2 – May, 165-192
APPENDIX
TABLE 12A - DCC-GARCH Models Comparisons: Crisis Period
DCC GARCHDCC GARCHDCC GARCHDCC GARCH fitfitfitfit
DCC GARCHDCC GARCHDCC GARCHDCC GARCH fitfitfitfit
CopulaCopulaCopulaCopula GARCH FitGARCH FitGARCH FitGARCH Fit
ModelModelModelModel DCC(1,1) DCC(1,1) Copula GARCH with DCC order (1,1)
DistributionDistributionDistributionDistribution Multivariate t Multivariate Normal
Multivariate Normal
No. ParametersNo. ParametersNo. ParametersNo. Parameters 113 112 112
No. Series (Assets in Portfolio)No. Series (Assets in Portfolio)No. Series (Assets in Portfolio)No. Series (Assets in Portfolio) 11 11 11
No. ObservationsNo. ObservationsNo. ObservationsNo. Observations 789 789 789
LogLogLogLog----LikelihoodLikelihoodLikelihoodLikelihood 29387.39 29207.82 29214.32
Akaike Information Criteria (AIC)Akaike Information Criteria (AIC)Akaike Information Criteria (AIC)Akaike Information Criteria (AIC) -74.21 -73.75 -73.91
Bayes Information Criteria (BIC)Bayes Information Criteria (BIC)Bayes Information Criteria (BIC)Bayes Information Criteria (BIC) -73.54 -73.09 -73.51
TABLE 12B - DCC-GARCH Models Comparisons: Crisis Period
DCC GARCHDCC GARCHDCC GARCHDCC GARCH fitfitfitfit
DCC GARCHDCC GARCHDCC GARCHDCC GARCH fitfitfitfit
CopulaCopulaCopulaCopula GARCH FitGARCH FitGARCH FitGARCH Fit
ModelModelModelModel DCC(1,1) DCC(1,1) Copula GARCH with DCC order (1,1)
DistributionDistributionDistributionDistribution Multivariate t Multivariate Normal
Multivariate Normal
No. ParametersNo. ParametersNo. ParametersNo. Parameters 113 112 112
No. Series (Assets in Portfolio)No. Series (Assets in Portfolio)No. Series (Assets in Portfolio)No. Series (Assets in Portfolio) 11 11 11
No. ObservationsNo. ObservationsNo. ObservationsNo. Observations 500 500 500
LogLogLogLog----LikelihoodLikelihoodLikelihoodLikelihood 17502.29 17414.24 17430.93
Akaike Information Criteria (AIC)Akaike Information Criteria (AIC)Akaike Information Criteria (AIC)Akaike Information Criteria (AIC) -69.56 -69.21 -69.50
Bayes Bayes Bayes Bayes Information Criteria (BIC)Information Criteria (BIC)Information Criteria (BIC)Information Criteria (BIC) -69.21 -68.27 -69.01
192 J.W. Muteba Mwamba – L. Mantshimuli
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TABLE 12C - DCC-GARCH Models Comparisons: Post-Crisis
DCC GARCHDCC GARCHDCC GARCHDCC GARCH fitfitfitfit
DCC GARCHDCC GARCHDCC GARCHDCC GARCH fitfitfitfit
CopulaCopulaCopulaCopula GARCH FitGARCH FitGARCH FitGARCH Fit
ModelModelModelModel DCC(1,1) DCC(1,1) Copula GARCH with DCC order (1,1)
DistributionDistributionDistributionDistribution Multivariate t Multivariate Normal
Multivariate Normal
No. ParametersNo. ParametersNo. ParametersNo. Parameters 113 112 112
No. Series (Assets in Portfolio)No. Series (Assets in Portfolio)No. Series (Assets in Portfolio)No. Series (Assets in Portfolio) 11 11 11
No. ObservationsNo. ObservationsNo. ObservationsNo. Observations 1483 1483 1483
LogLogLogLog----LikelihoodLikelihoodLikelihoodLikelihood 59692.76 59355.23 59374.1
Akaike Information Criteria (AIC)Akaike Information Criteria (AIC)Akaike Information Criteria (AIC)Akaike Information Criteria (AIC) -80.35 -79.89 -79.99
Bayes Information CriteriaBayes Information CriteriaBayes Information CriteriaBayes Information Criteria (BIC)(BIC)(BIC)(BIC) -79.95 -79.49 -79.79