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

166 J.W. Muteba Mwamba – L. Mantshimuli

www.iei1946.it © 2017. Camera di Commercio di Genova

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

On the protection of investment capital during financial crisis in the South African equity market 167


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



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



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.



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)



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)



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.



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



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.



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



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).



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:



$�$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)



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



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


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%




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



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.



TABLE 3 - JSE Sectors Descriptive Statistics: Post-Crisis


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%




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



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.



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







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%



TABLE 6 - Risk-based Portfolio Estimated Weights: Post-Crisis Period Back-test







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


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



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









4.91 4.35 -4.87 4.31 4.01 4.35


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



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






181.95 184.52 172.64 183.19 185.44 169.12


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


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-



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




VolatilityVolatilityVolatilityVolatility 0.92% 0.98% 1.03% 0.88% 0.85% 0.98%


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


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.



TABLE 11 - Out-sample Portfolio Performance: The Post-crisis Period







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%


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.



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.



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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APPENDIX

TABLE 12A - DCC-GARCH Models Comparisons: Crisis Period

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






Multivariate Normal






Bayes Bayes Bayes Bayes Information Criteria (BIC)Information Criteria (BIC)Information Criteria (BIC)Information Criteria (BIC) -69.21 -68.27 -69.01



TABLE 12C - DCC-GARCH Models Comparisons: Post-Crisis






Multivariate Normal






Bayes Information CriteriaBayes Information CriteriaBayes Information CriteriaBayes Information Criteria (BIC)(BIC)(BIC)(BIC) -79.95 -79.49 -79.79

on the protection of investment capital during … · can be found in the efficient frontier...

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