A new approach to portfolio selection : mean-expected shortfall vs mean-variance portfolio optimization.

University of Amsterdam, Amsterdam Business School

Master of Science in International Finance

A New Approach to Portfolio Selection:

Mean-Expected Shortfall vs. Mean-Variance Portfolio

Optimization

September 2014

Supervisor : Simon Broda, Ph.D.

Student: Carla van Sintfiet

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Abstract

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Portfolio optimization and the risk-reward theory of finance are analyzed in this thesis by making a comparison between the classical mean-variance approach from Modern Portfolio Theory (MPT) and other quantile-based risk measures such as expected shortfall. We discuss expected shortfall properties and its advantages over other risk metrics for portfolio optimization as well as its role in risk management and financial regulation. For the purpose of this research study a portfolio containing five different asset classes (MSCI AC World IMI Index, Barclays Capital Aggregate Bond Index, Global Hedge Fund USD Index, DXY Currency Index and the three-month USD Libor rate) is built using daily returns data from January 1st 2000 until May 1st 2014. Weight and return constraints are imposed as well as a restriction on short selling. Under the assumption that 1,000 USD is invested into each portfolio and rebalancing occurs every 10 days the performance of four portfolios is tracked: the global minimum-variance portfolio (Rpminvar), the mean-variance efficient

portfolio (Rpeffvar), the global minimum-expected shortfall portfolio (RpminES) and the mean-expected

shortfall efficient portfolio (RpeffES). We used five different values for µp [0.00005, 0.0001, 0.0005,

0.001, 0.005], which is the minimum daily return demanded by the investor, and two different levels of β [0.99, 0.975] which is the specified probability level or quantile of the expected-shortfall function. This resulted in ten different scenarios. Subsequently, we tested for the significance of the excess returns for different combinations of portfolio pairs as well as a combination of each time series of portfolio returns and the returns of a global benchmark such as the S&P500. The evidence in this research suggests that on average for all values of µp, the global minimum ES portfolio

showed better results followed by the global minimum variance portfolio. A comparison between the expected shortfall results for the two β-quantiles used seems to indicate that on average performance is slightly better under β=0.975.

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Table of Contents

1. Introduction ... 4

2. Literature Review ... 9

3. Objectives, Data and Methodology ... 17

3.1. Objectives of this Research ... 17

3.2. Data and Descriptive Statistics ... 18

3.3. Method of Research ... 21

4. Empirical Research: Results & Discussion ... 26

5. Conclusion... 31

6. Appendix ... 34

7. Acknowledgements... 45

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1. Introduction

Modern Portfolio Theory (MPT) is the mathematical representation of the concept of diversification and one of the most important proposed theories in finance (Kondor and Varga-Haszonits, 2010). Modern Portfolio Theory seeks to maximize portfolio returns for a certain level of risk, or equivalently it seeks to minimize risk for a certain level of expected returns (Clarke, De Silva and Thorley, 2006). This theory suggests investing is a trade-off between risk and return (Baker, Bradley and Wurgler, 2011). Fundamental assumptions of this model are somewhat restrictive and have been widely challenged in recent years (Clarke, De Silva and Thorley, 2006). For example, Modern Portfolio Theory builds on many simplifying assumptions:

- Assets returns are modeled as a normally distributed function (or in general as an elliptically distributed random variable)

- Risk is measured by the standard deviation of returns - Investors are rational individuals and markets are efficient

- There are no market frictions (e.g. tax, transactions costs, limited divisibility of financial assets, market segmentation, etc.)

- There is no heterogeneity in investors (e.g. rich vs. poor, informed vs. uninformed, young vs. old, etc.)

- Static expected returns, variance and correlations -no forecastability in returns or volatility (e.g. financial analysis, accounting information, macroeconomic variables do not play any role in making an investment decision)

One of the contradictions to the Capital Asset Pricing Model (CAPM), Modern Portfolio Theory and the risk-return relationship it predicts is known as the low volatility anomaly (Dutt and Humphery-Jenner, 2013). This phenomenon is based on the observation that portfolios of low-volatility stocks have yielded higher risk-adjusted returns than portfolios composed of high-volatility stocks (Dutt and Humphery-Jenner, 2013).

The low volatility anomaly has been well documented especially in the U.S. and recently also in emerging markets (Dutt and Humphery-Jenner, 2013). For instance, when ranked in terms of volatility, it has been found that on average stocks in the bottom quantile earn higher returns than stocks found in other quantiles (Dutt and Humphery-Jenner, 2013; Baker, Bradley and Wurgler, 2011). This and other empirical evidence has rendered this phenomenon an important issue in portfolio management and furthermore, has triggered questions about the suitability of the capital

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asset pricing model in explaining asset returns (Baker, Bradley and Wurgler, 2011). Particular attention has been given to the choice of an adequate risk measure.

In this regard, the mean-variance portfolio optimization approach first proposed by Harry Markowitz in 1959 uses the standard deviation of portfolio returns as the predominant measure of risk in finance (Bertsimas et al. 2004). Due to its intuitiveness and simplicity of calculation standard deviation has been commonly used for practical financial decisions (De Giorgi, 2002). However, as a risk measure, it is inadequate and incapable of capturing true risk, especially when return distributions exhibit an asymmetric nature (Bertsimas et al., 2004; De Giorgi, 2002). This is mainly due to the fact that standard deviation is a symmetric measure and therefore it equally penalizes desirable positive returns and undesirable negative returns (Bertsimas et al., 2004; De Giorgi, 2002). The assumption of symmetric return distributions poses difficulties for model construction because in practice the distribution of asset returns is rather asymmetric, particularly when the portfolio includes financial derivatives such as options (Lemus Rodrigues, 1999; Bertsimas et al., 2004). For instance, previous research has shown that U.S. equity returns are distributed asymmetrically, i.e. with more returns found in the extreme tails (Lemus Rodrigues, 1999). Nonetheless, the assumption of normality is a good model approximation when the inter-trading period taken into consideration is short and continuously compounded returns are used (Lemus Rodriguez, 1999). This is the reason that the normal distribution has been widely used to model the dynamics of stock prices (Lemus Rodriguez, 1999). Despite these simplifications, the asymmetric distribution of portfolio returns represents yet a bigger challenge for mean-variance portfolio optimization beyond model construction (Bertsimas et al., 2004). This is particularly true during economic crises when spill-over volatility effects, higher correlations and greater contagion between assets and markets increase, thus magnifying risks (Bertsimas et al., 2004).

Furthermore, standard deviation is an inappropriate risk measure because it fails to describe the risk of low probability events (Bertsimas et al., 2004). For instance, standard deviation as a risk measure does not properly account for the existence of fat tails often observed in the distribution of financial returns (Bertsimas et al., 2004). In addition, standard deviation is not always consistent with expected utility maximization unless under the assumption of a multivariate normal or elliptically symmetric distribution (Bertsimas et al., 2004). Consistency with the utility maximization theorem is equivalent to say that for a given fixed expected return and utility function U, the investor would prefer the portfolio with the smallest standard deviation (Bertsimas et al., 2004). When working with more general distributions, however, standard deviation loses this property (Bertsimas et al., 2004). This confirms the notion that unless returns have a normal or elliptically symmetric

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distribution, the mean-variance portfolio optimization is not the most suitable approach for portfolio construction (Bertsimas et al., 2004).

The shortcomings of standard deviation as a measure of risk and the asymmetric nature of the distribution of returns commonly found in finance have encouraged academics to propose alternative downside-risk measures for portfolio optimization (Bertsimas et al., 2004). The financial industry, in recent years in particular, has made extensive use of quantile-based downside risk measures (Bertsimas et al., 2004). Among this category, the most common risk measure used by financial institutions is Value-at-Risk (VaR), which by definition is the worst loss that can be expected with a certain probability for a particular confidence level (Bertsimas et al., 2004). Therefore, the β%-VaR of a portfolio is the maximum amount α that can be lost with probability β for a pre-defined period of time t (Bertsimas et al., 2004; Rockafellar and Uryasev, 2000). This is often expressed as “β% confidence that a particular investment or portfolio will not lose more than α within a t period of

time”. In this manner, VaR is an answer to the natural and legitimate question on the risks run by a

portfolio (Acerbi and Tasche, 2001).

As it is customary in the practitioner’s world, risk metrics such as VaR are popular because of their convenience and usefulness; they summarize the risk associated with a financial position or portfolio into a single number (Rockafellar and Uryasev, 2000). This last feature makes VaR the most common and practical choice for risk managers in the day-to-day business. In addition, VaR is also useful for regulatory purposes in financial markets, for instance in the assessment of capital requirements in the banking industry. Despite the advantages of VaR as a risk metric, it is nevertheless unable to quantify the magnitude of the potential expected losses when a tail event outside of our confidence level occurs (Bertsimas et al., 2004). Furthermore, VaR is not a flawless risk measure because it does not comply with all the coherence axioms, so suitability and usage of VaR is somewhat limited in practical terms. With respect to coherence, VaR particularly fails to comply with the principle of sub-additivity (Acerbi and Tasche, 2001; Bertsimas et al., 2004).

Coherence axioms are key defining properties of any risk measure (Acerbi and Tasche, 2001) and compliance with these is critical. Then, if V is a set of random variables where V ϵ R, the function ρ is a risk measure if it meets the below criteria:

a) monotonicity: X ϵ V , X ≥ 0 ⇒ ρ (X) ≤ 0

b) sub-additivity: X, Y, X + Y ϵ V ⇒ ⇒ ρ (X + Y) ≤ ρ(X) + ρ(Y)

c) positive homogeneity: X ϵ V and h > 0, h X ϵ V ⇒ ρ (h X) = h ρ(X) d) translation invariance: X ϵ V and a ϵ R ⇒ ρ (X + a) = ρ(X) –a

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The property of sub-additivity, which VaR disregards, is based on the concept that a portfolio made of portfolios will have a maximum total risk figure equal to the sum of its individual sub-portfolios (Acerbi and Tasche, 2001). In most cases, the global risk of the portfolio will be less than the sum of its sub-portfolio’s partial risks due to risk diversification (Acerbi and Tasche, 2001). This is perhaps the most characteristic feature of a suitable risk measure and something that is pivotal to the whole concept of risk (Acerbi and Tasche, 2001). Sub-additivity is also a very important property for portfolio optimization as this is directly related to the convexity of the risk surface to be minimized in the space of portfolios (Acerbi and Tasche, 2001). Thus, for any measure of risk that meets the sub-additivity axiom, portfolio diversification leads to risk reduction (Acerbi and Tasche, 2001). On the other hand, for risk metrics that do not obey this principle diversification may produce an increase in risk even when partial risks are triggered by mutually exclusive events (Acerbi and Tasche, 2001). Since VaR does not conform to the principle of sub-additivity, it may actually discourage portfolio diversification (Acerbi and Tasche, 2001; Bertsimas et al., 2004).

A slightly different risk measure known as Expected Shortfall (ES), conditional-VaR or tail-VaR exists that addresses the deficiencies of VaR and it is defined as the conditional expectation of losses above a certain threshold amount α (Rockafellar and Uryasev, 2000). More simply explained, expected shortfall measures how large losses, below the expected return, can be if the return of the portfolio drops below its β-quantile (Rockafellar and Uryasev, 2000; Bertsimas et al., 2004). The β quantile of a random variable W is expressed as qβ(W)= inf {w|P(W ≤ w) ≥ β}, β ϵ [0,1]. Thus,

expected shortfall at level β is the average of the VaRs for all levels below β, i.e.:

ESβ (w) = 1/β ∫

Where is the Value-at-Risk and the above expression comes from E[W|W ≤ qβ (W)] = 1/β

∫ .

As a result, expected shortfall is greater than VaR at the same risk level, thus: ESβ (w) ≥ VaRβ (w)

where both ESβ (w) and VaRβ (w) are decreasing functions of β.

Although traditionally VaR has been a more popular risk metric than expected shortfall, the latter is becoming more appealing and it is certainly preferable over VaR because it is a coherent risk measure and has the ability to take large tail-event losses into account (Acerbi and Tasche, 2001; Kondor and Varga-Haszonits, 2010). Moreover, expected shortfall has recently attracted more interest from practitioners after a proposal from the Basel Committee on Banking Supervision to

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replace the 1% VaR as a measure of capital requirements for banks, for the 2.5% Expected Shortfall (Basel Committee on Banking Supervision 2013, p.3). This potential change aims to determine regulatory capital requirements with more precision and to better capture the possibility of tail events; thus more accurately depicting the risk associated with a financial position or a particular portfolio. It is for this reason that in the present research study, expected shortfall is selected as a downside-risk measure for portfolio optimization.

Thus, in light of the demerits and anomalies of traditional risk-return models as well as the developments in financial market regulation, a new approach to portfolio selection using two different risk metrics in the optimization process is tested and presented in this research study. The objective of this paper is to compare the performance of four different portfolios using two methods of portfolio selection: the expected shortfall optimization approach and the standard mean-variance optimization approach from Modern Portfolio Theory; which to this day continues to be widely used in the financial world.

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2. Literature Review

Low Volatility Anomaly

As introduced before, financial market inefficiencies such as the low volatility effect is a well- documented anomaly in today’s world (Soe, 2012). Of all anomalies in investment theory, none is as perplexing as the low volatility effect which challenges the notion that an asset’s expected return is proportional to its systematic risk (Soe, 2012). Contrary to the dictates of finance theory, the low volatility anomaly has resulted in high beta and high volatility stocks underperforming against low beta and low volatility stocks throughout long term horizons (Baker, Bradley and Wurgler, 2011). This has been observed not only in the U.S. but also in emerging and international markets (Baker, Bradley and Wurgler, 2011; Dutt and Dumphery-Jenner, 2013).

Previous studies by Pettengill et al. show that in up markets high beta equities earned a higher return than low beta equities and, on the contrary, during down markets the opposite occurred (Pettengill, Sundaram and Mathur, 1995). However, later it has been found that on a capital asset pricing model (CAPM) market-adjusted basis, the low volatility anomaly is present in both up and down markets (Pettengill, Sundaram and Mathur, 1995; Baker, Bradley and Wurgler, 2011). Similarly, Ang et al. found that risky stocks have yielded significantly lower average returns in samples taken across U.S. and international markets (Ang et al., 2006). In addition, it has been observed that low risk portfolios behave less volatile in general and therefore follow a much smoother trajectory to their higher values over time than high risk portfolios (Baker, Bradley and Wurgler, 2011).

In a study conducted by Clarke, De Silva and Thorley, minimum-variance portfolios in which security weights are independent from their expected returns are examined on a large scale from 1968 until 2005 (Clarke, De Silva and Thorley, 2006). It was found that minimum variance portfolios have approximately three-fourths the realized risk of the market, represented by standard deviation, and only two-thirds the realized risk represented by beta (Clarke, De Silva, and Thorley, 2006). Also, it was observed that this lower risk does not come at the expense of lower returns as most of the low volatility portfolios showed a high realized Sharpe ratio (Clarke, De Silva, and Thorley, 2006). Furthermore, although minimum variance portfolios appear to have a value and small size bias, the imposition of factor neutrality constraints using stock characteristics or factor return sensitivities had little effect on the minimum variance portfolios as these showed a high Sharpe ratio regardless of the constraints (Clarke, De Silva, and Thorley, 2006). Additionally, long-short minimum variance portfolios delivered greater value added than long-only minimum variance portfolios.

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Thus, historical evidence from these and other studies is consistent with a low beta-high alpha pattern and, in this way, it suggests a further deterioration of the risk-return trade-off proposed by the CAPM (Baker, Bradley and Wurgler, 2011). Moreover, the difference between the returns of high-beta and low-beta equities has widened since 1983, despite the sophistication of quantitative techniques and the proliferation of institutional investment managers (Baker, Bradley and Wurgler, 2011).

The findings here discussed are nothing new; however, to this day they have not been fully investigated or exploited (Baker, Bradley and Wurgler, 2011). This is mainly caused by the difficulty of explaining the low volatility phenomenon with traditional financial models (Baker, Bradley and Wurgler, 2011). Since the CAPM is only one equilibrium model of risk-rewards with simplifying and in some cases unrealistic assumptions, it is plausible that beta is simply not an adequate measure of risk (Baker, Bradley and Wurgler, 2011). It is for this reason that considerable efforts have been devoted to the development of more sophisticated models and the search for a more adequate risk metric (Baker, Bradley and Wurgler, 2011).

Due to the financial crises experienced in recent years and the rise in market volatility, the minimum volatility phenomenon and the potential use of low-volatility investment strategies has gained importance in the last decade (Soe, 2012). This is particularly relevant for risk management as well as for portfolio management, which in the past has mostly relied on traditional modeling and optimization techniques (Soe, 2012).

The Search for an Adequate Risk Measure

In this regard, portfolio optimization techniques seek to minimize the risk parameter in the portfolio given a minimum expected return demanded by the investor (De Giorgi, 2002). Thus, portfolio optimization in its most general form can be expressed mathematically as the following problem: Find w that solves: minimize risk (Rw) s.t. E[Rp]≥ µp

Where R is a vector of risky returns, w represents the weights of the optimal portfolio that satisfies the return constraint, i.e. the expected return µp , and hence Rw represents the return of the

portfolio.

Moreover, the function risk (.) may vary with the risk metric being used –variance, VaR or expected shortfall – hence it can take the following forms:

11 i) Risk (Rw) = σw

ii) Risk (Rw) = ESβ (Rw) = C-VaRβ (Rw) for β ϵ [0,1]

iii) Risk (Rw) = VaRβ (Rw) for β ϵ [0,1]

In the last two cases and under the particular assumption of multivariate normally distributed returns, VaR and ES can be fully characterized by the mean and variance of the portfolio (De Giorgi, 2002). This is easy to illustrate if we assume R, the vector of returns, is multivariate Gaussian distributed with mean µ and variance-covariance matrix V:

R ~ N(µ, V) and let β ϵ [0,1] then we have the following results for ES and VaR ESβ (R) = ρ(z(β))/β * σ - µ

VaRβ (R) = z(β) * σ - µ

Where ρ(.) is the density of the standard normal distribution ρ(z) = _√

And z(β) = Φ-1 (1-β) while Φ(.) is the cumulative distribution function of the standard normal

distribution Φ(x) = _√ ∫ dz

The corresponding proofs to these results can be found in the Appendix section of this paper. In this manner, it is easy to see that under multivariate normally distributed returns, our risk function is reduced to the below linear expression (De Giorgi, 2002):

Risk (Rw) = ϒ(β) * σ - µ

Where a fixed parameter ϒ(β) is a function of the level of β that takes the value z(β) for VaR and ρ (z(β))/β for expected shortfall (De Giorgi, 2002). If we assume that β is some value ϵ (0, 0.5) then the expression ϒ(β) is positive for both VaR and expected shortfall (De Giorgi, 2002). Moreover, it has been shown in the literature that when the risk function has a linear form such as Risk (Rw) = ϒ(β) * σ - µ , just as depicted above, the solution to the optimization problem using VaR and ES as measure of risk is analogous to the solution to portfolio selection using the classical mean-variance approach (De Giorgi, 2002).

Under a multivariate normal distribution, a similar result is presented by Rockafellar and Uryasev (2000). In their study, Rockafellar and Uryasev compared the minimum expected shortfall or C-VaR approach with the mean-variance approach to portfolio selection. For the sake of consistency, when comparing the mean C-VaR portfolio optimization approach with the mean variance approach,

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losses and gains are expressed in percentage terms (Rockafellar and Uryasev, 2000). In this research β-ES or β C-VaR is defined as the conditional expectation of losses above an amount α with respect to a specified probability level β, where common values of β are 0.90, 0.95 and 0.99.

In order to illustrate these results, we consider the case where vector w represents the weights of a portfolio of financial instruments:

w = (w1, w2, w3 … wn) with wj being the position in instrument j and

the following constraints are imposed: wj ≥ 0 for j=1,… n and with ∑ j = 1

If yi is the return on instrument j, the random vector of returns is represented by:

y= (y1, …, yn).

Where the vector y is independent of w. In addition, vector y has a joint distribution of the various asset returns and it has density p(y) (Rockafellar and Uryasev, 2000). In this way, the return on portfolio w is the sum of the returns of the individual asset classes in the portfolio multiplied by their corresponding weight wj; thus we have portfolio return as a function of w and y. Analogously, the

loss function is the negative of this, which is expressed as: f(w,y) = -[w1y1, w2y2, … wnyn] = -wTy

As long as the density of y, i.e. p(y), is continuous with respect to y the cumulative distribution function of the loss of the portfolio represented by f(w,y) will itself be continuous (Rockafellar and Uryasev, 2000). As the main objective is to minimize expected shortfall or C-VaR, this can be expressed as:

Minimize ESβ (w) = min Fβ (w, α)

Where ESβ (w) is the expected shortfall at level β, which it is the average of the VaRs for all levels

below β just as defined above. Thus, the β-ES or β C-VaR of the loss associated with any w ϵ W can be determined from the above formula. Then, Fβ (w, α) which is a convex function of w and α is the

performance function with respect to β- ES or CVaR and β-VaR. This is defined as:

Fβ(w, α) = α + (1-β)^-1 ∫ Ty - α ]+ p(y)dy

Thus, minimizing β-ES or β C-VaR is equivalent to minimizing the performance function Fβ (w,α)

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associated with portfolio w be expressed in terms of the mean m and variance V of the vector of returns y we have:

µ(w) = -wTm and σ2(w)= wT V w

Assuming that the weights and portfolio losses are normally distributed with β=0.5 and that the three below conditions are imposed:

1) A linear constraint of a target return µ(w) ≤ -r and

2) No short selling, i.e. each weight is greater than zero wj ≥ 0 for j=1,…, n

3) All weights add up to 1, i.e. ∑ j = 1

Then, the set of portfolios W that satisfy these constraints is convex and is the same for the mean-CVaR, mean-VaR and mean-variance portfolio selection process (Rockafellar and Uryasev, 2000). This is expressed as:

W = {w : w that satisfies µ(w) ≤ -r and wj ≥ 0 for j=1,… n and with ∑ j = 1 }

Hence, the evidence suggests that given normally distributed returns the solution to the problem of minimizing C-VaR, VaR and variance is the same (Rockafellar and Uryasev, 2000). This result implies that a common vector of portfolio weights w* (W ϵ R) is optimal for all criteria and portfolio selection methods (Rockafellar and Uryasev, 2000).

To illustrate this point further, when the target return constraint imposed at optimality µ(w) ≤ -r is replaced by a more restrictive one such as µ(w) = -r, a smaller set of portfolios W’ is obtained that satisfies this new restriction (Rockafellar and Uryasev, 2000). These portfolios form part of a new frontier in the optimization process (Rockafellar and Uryasev, 2000). Thus, for a vector w ϵ W’ we have that CVaR/ES and VaR are expressed as:

ESβ (w) = ϒ(β) * σ(w) - µ(w) and

VaRβ (w) = δ(β) * σ(w) - µ(w)

where the parameters ϒ(β)> 0 and δ(β)>0.

Minimizing these expressions for C-VaR and VaR is equivalent to minimizing the variance over w ϵ W’, in other words this is equivalent to mean-variance optimization (Rockafellar and Uryasev, 2000). Therefore, under a target return constraint applied to both minimization problems, any portfolio w* that minimizes the variance σ(w) over w ϵ W’ is also optimal for minimizing C-VaR and VaR (Rockafellar and Uryasev, 2000). The same conclusion is also reached by Hurlimann (2002) for

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elliptical distributions in general. In his study it is shown that when we deal with elliptical distributions, the expected shortfall approach to portfolio selection is analogous to the mean-variance Markowitz approach (Hurlimann, 2002). Naturally, this conclusion for elliptical distributions is a generalization of the corresponding result for multivariate normal distributions presented by Rockafellar and Uryasev.

In addition, research studies on portfolio selection using different risk measures suggest that for multivariate Gaussian distributions or at least elliptically symmetric return distributions, the impact of VaR or expected-shortfall limits on the mean-variance portfolio framework reduces the set of efficient portfolio allocations (De Giorgi, 2002). For instance, there are examples in the literature that show that under mean-ES portfolio optimization with ESβ=5%, the set of efficient portfolios is

reduced with respect to the mean-variance approach (De Giorgi, 2002). This supports the evidence that efficient frontiers constructed using VaR or ES are subsets of the mean-variance efficient frontier (De Giorgi, 2002). In fact, under the assumption of multivariate distribution of asset’s returns, for every mean-variance efficient portfolio (µ,σ) which differs from the global minimum-variance portfolio, there is a β-quantile such that this portfolio corresponds to the global minimum ESβ portfolio (De Giorgi, 2002). The same rationale applies to VaRβ for a particular β-quantile (De

Giorgi, 2002). Then graphically, the efficient frontier under the various risk metrics, are a subset of the boundary above their corresponding global minimum risk portfolios (De Giorgi, 2002).

Consistent with the point above, it has been observed that mean-variance efficient portfolios can be inefficient under Value-at-Risk or Expected Shortfall; however, the opposite does not seem to occur (De Giorgi, 2002). In fact, empirical research suggests that every mean-VaR and mean-ES efficient portfolio is also mean-variance efficient; however every mean-variance efficient portfolio is not necessarily mean-VaR or mean-ES efficient (De Giorgi, 2002). A portfolio that is mean-variance efficient will be (µ,risk) efficient, being this measure of risk VaR or ES, only under special circumstances. The global minimum variance portfolio, in particular, is never mean-risk efficient. In light of this evidence, some authors have found that the mean-risk (µ,risk) approach to portfolio selection using VaR or ES does not really represent an improvement with respect to the classical mean-variance portfolio approach (De Giorgi, 2002). This conclusion has been drawn even when no assumptions about the returns were made (De Giorgi, 2002). For instance, there are studies on the Swiss Market Index that exemplify this point where the mean-expected shortfall portfolio allocation is very similar to the mean-variance asset allocation (De Giorgi, 2002).

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Until now, we have considered the particular case when returns are normally distributed. This greatly simplified the analytical formulation of the mean-variance framework in which the risk function can be replaced by variance, expected shortfall or VaR. However, when returns are not normally distributed the mean-risk optimization problem gives as a result a completely different set of optimal weights which varies with the risk metrics used (Lemus Rodriguez, 1999). For example, when the financial assets have asymmetric returns the efficient frontier could still be convex but the optimal or efficient weights computed using one risk metric can become inefficient under a different risk metric (Lemus Rodriguez, 1999). The same result is observed when the risk metric in use is not convex and hence the efficient frontier is also not convex (Lemus Rodriguez, 1999). Thus, under a more general asymmetric return distribution the asset allocations between the two approaches may differ significantly (Bertsimas et al., 2004).

In addition, asset allocations depend heavily on the β-quantile associated with the risk parameter (Bertsimas et al., 2004). In this regard, the level of beta needs to be carefully chosen as it has been observed that mean-variance and mean-shortfall efficient frontiers could be empty for a β-quantile higher than a certain level (De Giorgi, 2002). Thus, the risk sensitivity of any given portfolio is directly related and may significantly be affected by β (Bertsimas et al., 2004). Despite this conclusion, it is worth noting that in some cases non-normality of returns may provide degenerated solutions as well (Lemus Rodriguez, 1999).

Furthermore, expected shortfall is a convex function of portfolio weights as opposed to VaR which in general is not a convex function (Bertsimas et al., 2004). This particular feature of expected shortfall gives it an advantage over VaR and makes the mean-ESβ approach to portfolio selection more

attractive than the mean-VaRβ criterion (De Giorgi, 2002). Thus, previous studies by Bertsimas et al.,

suggest that mean-ES portfolio optimization may be preferable to the classical mean-variance optimization (Bertsimas et al., 2004). This is particularly observed when we are working with asymmetric distributions (Hurlimann, 2002). However, this is also applicable to a normal or elliptic distribution of returns because mean-ES optimization leads to an efficient and stable computation of the same optimal weights but does not require the cumbersome estimation of large covariance matrices (Bertsimas et al., 2004). Moreover, in some cases computational results have shown that the efficient frontier constructed via mean- expected shortfall optimization outperforms the efficient frontier constructed via mean-variance optimization (Bertsimas et al., 2004).

Contrary to what has been previously discussed, there is evidence that shows that optimization of some portfolios- especially large portfolios- is not always feasible under the expected shortfall method (Kondor and Varga-Haszonitz, 2010). It seems that expected shortfall can become

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unbounded from below in some cases, which appears to be a probabilistic event and this depends directly on the sample (Kondor and Varga-Haszonitz, 2010). Thus, for a finite sample size there will always be cases where portfolio optimization cannot be carried out under a particular coherent risk metric (Kondor and Varga-Haszonitz, 2010).

Moreover, there is research indicating that coherent risk measures are sensitive to sample fluctuations and they break down if one of the assets in the portfolio dominates the others (Kondor and Varga-Haszonitz, 2010). Dominance in this context refers to the situation in which the returns of a portfolio are greater than the returns of another similar portfolio at all times (Kondor and Varga-Haszonitz, 2010). In fact, the dominance of one portfolio is sufficient for the instability of any coherent risk measure. Nevertheless, the necessary conditions for the instability of expected shortfall have not been fully determined to this day (Kondor and Varga-Haszonitz, 2010). Additionally, it is suggested that instability is a common property of every coherent measure but it is not restricted to them since it has been found that VaR can also suffer from instability (Kondor and Varga-Haszonitz, 2010). Furthermore, it is implied that the phenomenon of instability in other coherent measures behaves in a similar way as for the case of expected shortfall (Kondor and Varga-Haszonitz, 2010).

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3. Objectives, Data and Methodology

3.1. Objectives of this Research

As introduced before, financial market anomalies coupled to the inadequacy of a proper risk measure that accounts for an asymmetric distribution of returns have motivated scholars and practitioners to bring forward different approaches to portfolio selection. In the present research study, the classical mean-variance approach is compared to the alternative mean-expected shortfall approach.

The objective of this study is to compare the performance of four different portfolio strategies –the global minimum variance portfolio, the efficient mean-variance portfolio which is the one regarded as the optimal portfolio proposed by Markowitz in the Capital Asset Pricing Model (Modern Portfolio Theory), the global minimum mean-ES portfolio and the efficient mean-ES portfolio. These portfolios are interesting to compare because of what these strategies can tell us about the suitability of classical risk-return models v.s. alternative models. For instance, the global minimum variance portfolio is interesting to analyze because in recent years there is plenty of evidence- conflicting with basic finance principles - that suggests that high-beta, high-volatility stocks have consistently underperformed low-beta, low-volatility stocks particularly in the U.S. market (Baker, M.; Bradley, B.; Wurgler, J., 2011). Comparing the performance of the global minimum variance portfolio to the mean-variance efficient portfolio, therefore, would be of value in shedding some light on the performance of the two portfolios in terms of volatility.

Likewise and as previously discussed, the mean-variance approach to portfolio selection proposed by Markowitz has failed to depict an accurate picture of the distribution of financial returns and as a consequence new risk metrics are examined for further improvement of this model. For this reason, it is interesting to analyze the performance of the portfolios on the efficient frontier subject to a different measure of risk, in this case, expected shortfall. Accordingly, the global minimum mean-ES portfolio and the efficient mean-ES portfolio are singled out from the mean-expected shortfall frontier and compared each to its equivalent portfolio under the classical mean-variance approach. Thus, the main question we want to answer in this exercise is: if we had invested $1,000 USD in these 4 different strategies from January 1st 2000 until May 1st 2014, which one would have been the most profitable (highest value) at the end of the experiment? Are the excess returns between

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different strategies significantly different from zero? How do these results compare with the returns of a global benchmark?

3.2. Data and Descriptive Statistics

Data was gathered to construct a multivariate asset-class portfolio using historical financial data from data providers Factset and Bloomberg. The assets used for the construction of the portfolio are:

-The MSCI AC World IMI equity index (Factset ticker symbol: MS664204) which is a broad global equity benchmark that includes a diverse portfolio of micro to large cap equities in developed countries and small to large cap equities in Emerging market countries.

-The Barclays Capital Aggregate Bond Index (Bloomberg lookup symbol: LEGATRUU) which is a market –capitalization weighted index based on wide variety of investment grade bonds traded in the US. The index includes Treasury securities, Government agency bonds, Mortgage-backed bonds, corporate bonds, and a small amount of foreign bonds traded in the U.S.

-The USD Global Hedge Fund index (Factset ticker symbol: HFRX) which is a daily NAV (Net Asset Value) index and a global industry standard for performance across the hedge fund industry.

-The DXY Currency index (Bloomberg lookup symbol: USDX) which is an index based on the value of the U.S. dollar relative to a basket of other (hard) foreign currencies.

-A money-market index such as the returns on the 3-month USD LIBOR.

In addition to this, the S&P 500 was used as a global benchmark. The daily returns were obtained from Yahoo Finance for the period under research, from January 1st 2000 until May 1st 2014.

Whenever indices are used for the construction of the portfolios, the total return index is used as opposed to the price return index. A total return index tracks the capital gains of the index and assumes all dividends, coupons or any other cash distribution is reinvested back into the index. Thus, a total return index is a more accurate representation of the performance of an index and it is for this reason that it was the preferred choice for portfolio construction.

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The daily returns on the different asset classes are expressed in percentages. MATLAB is used as a computational tool for the optimization and back testing exercise. Eviews is used for the hypothesis testing exercise.

In table 1, the summary statistics for the different asset classes in the global market portfolio and the S&P500 are presented. There are 3740 observations across all asset classes. As depicted in the first column of table 1, the daily returns of the MSCI AC World IMI index varies between -7.3% (min) and 8.62% (max). The average daily return is 0.016% and as we can see there is some variation in the returns as the standard deviation reported for this equity index is 1.06%. In the second column, we have Barclays Capital Aggregate Bond index with a minimum daily return of 1.95% and a maximum of 2.82%. The average daily return is very low at 0.02% and so is standard deviation at 0.35%. Kurtosis is 3.27, which suggests a very symmetric distribution that approximates normality.

In the third column, the Global Hedge Fund index has a minimum daily return of -3.93% and a maximum return of 5.8%. The mean daily return is 0.015% and standard deviation is the lowest of all assets in the market portfolio at 0.29%. This can also be implied from the high value for kurtosis 124.7, which suggests this is an asymmetric distribution with heavy tails. In the fourth column, we have the DXY currency index with a minimum daily return of -2.73% and a maximum daily return of 2.52%. The average daily return for the DXY currency index is very close to zero at exactly -0.006%. Standard deviation is low at 0.52% and kurtosis is also very low at 1.42. In the fifth column we have the returns of the 3-month USD Libor, with a minimum and maximum daily return of 0.22% and 6.87% respectively. The mean daily return is 2.37% and standard deviation is 2.1%.

Lastly, in the sixth column we have the daily returns of the S&P 500. The minimum and maximum daily returns are -9.47% and 10.96% respectively. These values are greater in absolute terms than the minimum and maximum daily returns of the MSCI AC World IMI index shown in column 1. The same is observed in terms of standard deviation where at 1.28% the standard deviation of the S&P 500 is greater than the standard deviation of the MSCI AC World IMI Index. The average daily return is 0.007% -lower than the MSCI index- and kurtosis is greater at 8.21.

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Equity

Fixed Income

Hedge Funds

Currency

LIBOR

Benchmark

MSCI AC World IMI

Index

Barclays Capital

Agg Bond Index

Global Hedge

Fund Index USD

DXY Currency

Index

3-Month USD

Libor

S&P 500

Number of Observations

3740

3740

3740

3740

3740

3740

Minimum

-7.309%

-1.955%

-3.926%

-2.726%

0.223%

-9.470%

Maximum

8.682%

2.817%

5.780%

2.520%

6.869%

10.957%

Mean

0.016%

0.021%

0.015%

-0.006%

2.376%

0.007%

Median

0.070%

0.013%

0.000%

0.000%

1.718%

0.021%

Variance

0.011%

0.001%

0.001%

0.003%

0.044%

0.016%

Standard Deviation

1.057%

0.348%

0.289%

0.516%

2.104%

1.281%

Skewness

-0.364

0.156

5.169

-0.021

0.663

-0.182

Kurtosis

7.099

3.273

124.680

1.417

-0.971

8.208

Table 1. Descriptive Statistics

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3.3. Method of Research

Portfolio returns are a linear combination of n random variables. It is important to look at the multivariate distribution of assets returns and how the univariate distribution of the portfolio returns is changed by the different weights of the assets composing the portfolio (De Giorgi, 2002). This is not always an easy task to undertake.

Given a collection of n assets and a vector of random returns R= (R1, R2, R3,… Rn) the main goal of

portfolio selection is the determination of the optimal portfolio weights with respect to some meaningful criterion. If vector w= (w1, w2 … wn) represent the fractions of the portfolio held in each

asset, then Rp = R * w describes the portfolio return, which should be optimized in some way. For the

purpose of the research, we assume that the joint distribution of Rp is continuous and investors are

interested in maximizing their utility.

Thus, we first constructed the efficient frontier for the mean-variance and the mean-ES approach. This is essentially the construction of a set of portfolios that achieve the lowest possible risk- being this variance or expected shortfall- for any given target rate of return. The graphs depicting these two frontiers are plotted below in this section of the paper. From the set of portfolios on the variance efficient frontier we single out the global minimum-variance portfolio and the mean-variance efficient portfolio. From the mean-expected shortfall efficient frontier, we select the second pair of portfolios; the global minimum-ES portfolio and the mean-ES efficient portfolio. For this purpose, we define the mean-variance portfolio optimization problem as:

min variance= min w’∑w s.t. Rp= w’ * R ≥ µp and wj >0

where µp is the daily return demanded by the investor.

The Markowitz optimal portfolio has minimal variance σ2p for a given mean μp. For the construction

of the global minimum variance portfolio, the constraint on returns is lifted and therefore the portfolio is defined as:

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Figure 1. An illustration of the mean-variance portfolio selection problem

Analogously, we consider the mean- expected shortfall portfolio optimization problem, which we define as;

min ESβ (w) = min ES (w’ R) s.t. Rp= w’ * R ≥ µp,

where µp is the daily return demanded by the investor.

Just as it was described for the global minimum-variance portfolio, the global minimum-expected shortfall portfolio is defined as:

min ESβ (w) = min ES (w’ R) s.t. Rp= w’ * R and wj > 0

3 4 5 6 7 8 9 10 11 12 13 x 10-6 0.8 1 1.2 1.4 1.6 1.8 2 2.2x 10

-4 _{Mean-Variance Portfolio Optimization}

Variance E x p e c te d P o rt fo lio R a te o f R e tu rn

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Figure 2. An illustration of the mean-variance portfolio selection problem

Through a back testing exercise we assess the performance of these four portfolios. For a meaningful analysis a thirteen-and-a-half year period is used in this exercise; the total period comprises from January 1st 2000 until May 1st 2014, which accounts for 3740 days in total.

Next, we do a re-optimization of portfolio weights using a moving window time frame. For this purpose, we compute the vector of optimal weights for each portfolio using the observations from the immediate last 1000 days in a rolling-window fashion starting from day 1001 until day 3740. For instance, on day 1001, all asset returns in the global portfolio from day 1 until day 1000 are used to calculate the optimal asset allocation of the portfolio for day 1001. Similarly, on day 1002, all asset returns from day 2 until day 1001 are used to calculate the optimal asset allocation of the portfolio for day 1002, and so forth. This calculation continues until day 3740, so the result is a matrix of optimal weights w for every day from day 1001 until day 3740. This is done for all four portfolios, so this exercise results in four matrices of optimal weights.

After this we proceed to calculate portfolio returns Rp every 10 business days (or 2 calendar weeks).

Therefore, the portfolio return Rp from t to t+10 following a 10-day interval computation is defined

as: 6 6.5 7 7.5 8 8.5 9 9.5 x 10-3 1.5 2 2.5 3 3.5 4 4.5 5x 10

-4 _{Mean-Expected Shortfall Portfolio Optimization}

Expected Shortfall E x p e c te d P o rt fo lio R a te o f R e tu rn

24 Rpt+10 = Returns t+10 * wt

Based on these returns we define the next period’s portfolio value as:

Vt+10 = Vt * (Rpt+10 + 1)

The underlying assumption in the backtesting exercise is that we invest 1,000 USD in every strategy on day 1001 so we fixed Vt1001 at 1,000. Then, we calculate the value of the portfolio following a

10-day interval from 10-day 1001 until 10-day 3740. We assume that past losses remain and that any gains are reinvested back into the particular portfolio so that nothing is withdrawn from these portfolios for the entire duration of the thirteen-and-a-half-year experiment. Next, the portfolio values are calculated at the specified points and the results are plotted in a graph. The final value of the portfolio at the end of the experiment following a 10-day rebalancing frequency is defined as Vt3740.

We repeat this exercise using five different values for µp [0.00005, 0.0001, 0.0005, 0.001, 0.005],

which was previously defined as the minimum daily return demanded by the investor, and two different levels of β [0.99, 0.975], which is the quantile or specified probability level of the expected shortfall function. In this manner, we generate ten different scenarios.

Subsequently, the time series of returns for the four portfolios are compared for further analysis. In order to investigate the meaningfulness of our results, we proceed to test the significance of the excess return of each pair of portfolios. For this purpose, we regress Rp1 – Rp2 on a constant c,

where Rp1 and Rp2 are any two time series of portfolio returns we want to compare and ε is an error

vector, therefore E(εi) = 0. In this manner, a regression model without predictors is established in

order to assess the significance of the constant; this equation in general form is expressed as:

Rp1 - Rp2 = c + εt

We check the t-ratio [tα=0.05 = ĉi/SE(ĉi)] using HAC (heteroskedasticity-consistent) standard errors and

perform a two-sided t-test at the α=5% significance level in order to investigate whether the constant is significantly different from zero. The hypothesis is stated as: H0 : c= 0 ; H1: c ≠ 0.

In this particular case, the variables we want to test are: Rpminvar , the returns on the global

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the returns on the global minimum-ES portfolio; and finally RpeffES , the returns on the mean-ES

efficient portfolio. Consequently, we establish linear regression equations 1 until 6 in order to test the significance of the excess returns of each pair of portfolios. The equations are stated below:

Rpminvar – Rpeffvar = c + εt (1) RpminES – RpeffES = c + εt (2) RpeffES – Rpeffvar = c + εt (3) RpminES – Rpminvar = c + εt (4) RpeffES – Rpminvar = c + εt (5) RpminES – Rpeffvar = c + εt (6)

In addition, regressions 7 until 10 are established in order to compare the returns of each of the portfolios to the returns of a global benchmark such as the S&P500 index. The equations are stated below: Rpminvar – RpS&P500 = c + εt (7) Rpeffvar – RpS&P500 = c + εt (8) RpminES – RpS&P500 = c + εt (9) RpeffES – RpS&P500 = c + εt (10)

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4. Empirical Research: Results & Discussion

As previously mentioned, the question we want to answer is; what strategy is more profitable? Hypothetically speaking if we had invested 1,000 USD on January 1st 2000 until May 1st 2014, which strategy would have earned the highest return? Which portfolio would have ended up with the highest value? When comparing two different strategies; is the difference in returns from two portfolios significantly different from zero? How does the performance of these portfolios compare to a global benchmark such as the S&P 500?

The results of this experiment are presented in the tables below. These tables show the final value of the four portfolios after rebalancing every 10 days for the period under research (from January 1st 2000 until May 1st 2014) using five different values for µp [0.00005, 0.0001, 0.0005, 0.001, 0.005],

which is the daily rate of return demanded by the investor, and two different β-quantiles for expected shortfall [0.99, 0.975].

Table 2. Mean –variance efficient portfolio for different values of µp

0.00005 1,222.90 0.0001 1,265.50 0.0005 1,046.50 0.001 817.40 0.005 635.47 In USD µp

0.99 0.975

1,123.10

1,165.50 Beta In USD

Table 4. Mean–ES efficient portfolio for different values of µp and levels of β

0.99 0.975 0.00005 1,205.90 1,221.20 0.0001 1,262.00 1,252.50 µp 0.0005 1,098.80 1,071.70 0.001 869.64 828.54 0.005 637.31 634.89 In USD Beta

As expected, the global minimum variance portfolio remains constant at 1,107.30 USD and is not affected by different values of µp or levels of β (value not shown). In table 2, the value of the

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value, 1,222.90 USD, was observed at µp=0.00005, and finally the lowest value, 635.47 USD, was

observed at µp=0.005. As it is expected, the level of β did not have any effect on the performance of

the mean-variance efficient portfolio (table 2).

Contrary to the above but in line with our expectations, the global-minimum ES portfolio in table 3 was only affected by the level of β and not however by the different values of µp, for which the value

of the portfolio remained constant. As depicted in table 3, the value of the portfolio reached 1,123.10 USD and 1,165.50 USD for β=0.99 and β=0.975 respectively (table 3). Finally, in table 4 the final value of mean-expected shortfall efficient portfolio was the highest at µp=0.0001 with 1,262.00

USD under β=0.99 and 1,252.50 USD under β=0.975. The second best performance was observed at µp=0.00005 with 1,205.90 USD under β=0.99 and 1,221.20 USD under β=0.975, while the lowest

performance was observed at µp=0.005 with 637.31 USD for β=0.99 and 634.89 USD for β=0.975. In

accordance with our expectations, the value of the mean-ES efficient portfolio was affected by the value of µp as well as the level of β (table 4).

Table 5. Comparison between portfolios for different levels of µp and β

Final Value of Portfolios in USD µp= 0.00005 µp= 0.0001 µp= 0.0005 µp= 0.001 µp= 0.005 µp= 0.00005 µp= 0.0001 µp= 0.0005 µp= 0.001 µp= 0.005 Minimum Variance 1107.30 1107.30 1107.30 1107.30 1107.30 1107.30 1107.30 1107.30 1107.30 1107.30 Efficient Variance 1222.90 1265.50 1046.50 817.40 635.47 1222.90 1265.50 1046.50 817.40 635.47 Minimum Expected Shortfall 1123.10 1123.10 1123.10 1123.10 1123.10 1165.50 1165.50 1165.50 1165.50 1165.50 Efficient Expected Shortfall 1205.90 1262.00 1098.80 869.64 637.31 1221.20 1252.50 1071.70 828.54 634.89 β=0.99 β=0.975

A comparison across portfolios in table 5 shows the best performance at high values of µp [0.0005,

0.001, 0.005] was achieved by the global minimum-ES portfolio with 1,123.10 USD under β=0.99 and 1,165.50 USD under β=0.975 (table 5, values shown in green). At low values of µp [0.00005 and

0.0001] the best performer was the mean-variance efficient portfolio with 1,222.90 USD and 1,265.50 USD respectively (table 5, values shown in green). This was observed for both levels of β (table 5, values shown in green).

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In the same manner, the worst performer was the global minimum-variance portfolio at low values of µp [0.0005, 0.001] and the mean-variance efficient portfolio at higher values of µp [0.0005, 0.001].

This is observed for both levels of β (table 5, values shown in red). At µp=0.005, the mean-variance

efficient portfolio was the worst performer under β=0.99 with 635.47 USD (table 5, value shown in red), whereas under β=0.975 the mean-ES efficient portfolio was the worst performer yielding a final value of 634.89 USD (table 5, value shown in red).

The results in this research study suggest that on average across all values of µp, the global

minimum-ES portfolio reached its highest final values for both levels of β; 1,123.10 USD for β=0.99 and 1,165.50 USD for β=0.975 (table 6). This is also observed at the total level where the global minimum-ES portfolio averages 1,144.30 USD (table 6, value shown in blue). Furthermore, across all portfolios and all values of µp, the results seem to indicate that performance is slightly better at

β=0.975 where portfolio value averages 1,068.03 USD (table 6, value shown in blue).

Table 6. Average performance of portfolios

Portfolios Average for all µp

β= 0.99

Average for all µp

β= 0.975 Total Average

Minimum Variance 1107.30 1107.30 1107.30

Efficient Variance 997.55 997.55 997.55

Minimum Expected Shortfall 1123.10 1165.50 1144.30

Efficient Expected Shortfall 1014.73 1001.76 1008.25

Total Average 1060.67 1068.03

After regressing equations 1 until 6 in order to test the significance of the excess returns of each combination of portfolio pairs, and equations 7 until 10 in order to test the significance of the performance of each portfolio with the performance of the S&P500 we observe that in general there are very few occurrences in which the null hypothesis is rejected. These results imply that in the majority of the cases the excess returns of the different combination of portfolio pairs are not significantly different from zero.

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For regression equations 1 until 6 there were three occurrences under β=0.99 and µp=0.00005

(equations 1, 2 and 6) in which the null hypothesis was rejected suggesting that the excess returns of Rpminvar over Rpeffvar, RpminES, over RpeffES and RpminES, over Rpeffvar were significantly different from zero

(table 7, values shown in blue). Likewise, for linear regressions 1 until 6, under β=0.99 and µp=0.0001, there was only one occurrence in which the null hypothesis was rejected. This is the case

for equation 2 where the excess returns of RpminES, over RpeffES had a p-value of 0.0383 and therefore

was significantly different from zero (table 7, values shown in blue). A similar result is observed under β=0.99 and µp=0.001 (table 7, values shown in blue) where equation 3 had a p-value of

0.0336, which suggests that the excess returns of RpeffES, over Rpeffvar were significantly different

from zero. Contrary to the above, all tests under β=0.99 for µp=0.0005 and µp=0.005 (table 7) failed

to reject the null hypothesis suggesting that the excess returns were not significantly different from zero.

Similarly, for equations 1 until 6 under β=0.975 and µp=0.00005, there were two occurrences

(equations 2 and 5) in which the null hypothesis was rejected suggesting that the excess returns of RpminES, over RpeffES and RpeffES, over Rpminvar were significantly different from zero (table 7, values

shown in blue). On the other hand, the same equations at µp=0.0001, µp=0.0005, µp=0.001 and

µp=0.005 (tables 7) showed no occurrences where the null hypothesis was rejected, implying that

excess returns were not significantly different from zero.

In the same manner, in regression equations 7 until 10 where we assess the significance of the excess returns of each portfolio against the returns of a global index such as the S&P500, we observe that for all values of µp and levels of β the test fails to reject the null hypothesis. This suggests that

the excess returns of Rpminvar, Rpeffvar, RpminES, and RpeffES portfolios over the returns of the S&P500

were not significantly different from zero.

As we can observe from the results, the excess returns between the global minimum-ES portfolio and its efficient counterpart are significant and different from zero at µp=0.00005 for both levels of

β. A similar result is observed for µp=0.0001, although only for the case of expected shortfall under

β=0.99. The excess returns of the global minimum variance portfolio over the mean-variance efficient portfolio were significant at β=0.99 and µp= 0.00005, this is in line with our expectations.

The excess returns between the two efficient portfolios is only significant and different from zero at β=0.99 and µp=0.001. The results from the hypothesis tests for all ten scenarios are detailed below

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Eq Regressions µp = 0.00005 µp = 0.0001 µp= 0.0005 µp= 0.001 µp= 0.005 µp = 0.00005 µp = 0.0001 µp= 0.0005 µp= 0.001 µp= 0.005

1 Rpminvar – Rpeffvar = c + εt -2.4829 -1.6779 0.0264 0.432 0.809 n/a n/a n/a n/a n/a

0.0132 0.0934 0.9789 0.6661 0.4186 n/a n/a n/a n/a n/a

2 RpminES – RpeffES = c + εt -2.6417 -2.0669 -0.0448 0.3485 0.8235 -2.2769 -1.3235 0.0826 0.4904 0.8834

0.0084 0.0383 0.9642 0.7281 0.4099 0.0227 0.1855 0.9342 0.6242 0.3774

3 RpeffES – Rpeffvar = c + εt -0.4681 -0.066 1.0229 2.1226 0.6558 -0.0514 0.2493 0.6098 0.8099 -0.2459

0.6386 0.9475 0.3051 0.0336 0.5133 0.959 0.8026 0.5413 0.4178 0.8058

4 RpminES – Rpminvar = c + εt 0.44 n/a n/a n/a n/a 1.3623 n/a n/a n/a n/a

0.6609 n/a n/a n/a n/a 0.1747 n/a n/a n/a n/a

5 RpeffES – Rpminvar = c + εt 1.7486 1.7509 0.075 -0.3228 -0.8045 1.9725 1.6438 0.0225 -0.4077 -0.8136

0.0804 0.08 0.9405 0.7464 0.4212 0.049 0.0997 0.982 0.6841 0.4174 6 RpminES – Rpeffvar = c + εt -2.2127 -1.6679 0.0565 0.4555 0.828 -1.2553 -1.1123 0.1307 0.5142 0.8794

0.0267 0.0958 0.955 0.6488 0.4075 0.2086 0.2665 0.8961 0.6072 0.3785

7 Rpminvar – RpS&P500 = c + εt -0.91 n/a n/a n/a n/a n/a n/a n/a n/a n/a

0.3629 n/a n/a n/a n/a n/a n/a n/a n/a n/a

8 Rpeffvar – RpS&P500 = c + εt -0.7105 -0.6421 -0.6679 -0.919 -1.1864 n/a n/a n/a n/a n/a

0.4753 0.5199 0.5049 0.3583 0.2362 n/a n/a n/a n/a n/a

9 RpminES – RpS&P500 = c + εt -0.8697 n/a n/a n/a n/a -0.801 n/a n/a n/a n/a

0.3847 n/a n/a n/a n/a 0.4232 n/a n/a n/a n/a

10 RpeffES – RpS&P500 = c + εt -0.7291 -0.6406 -0.6045 -0.8409 -1.183 -0.7083 -0.6596 -0.6363 -0.9014 -1.1864

0.4639 0.5197 0.547 0.3991 0.2375 0.4784 0.5097 0.5254 0.3684 0.2356

β=0.99 β=0.975

Table 7. Hypothesis testing results for different values of µp and levels of β

This table looks at the results of the hypothesis testing for all linear regression equations at the α= 5% significance level (critical value = 1.96). The regressions use daily data from day 1001 (November 3rd 2003) until day 3740 (April 30th 2014). In this table, HAC (heteroskedasticity consistent) standard errors are used and therefore robust t-statistics are reported. P-values are reported right below t-statistics. T-test and p-values for

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5. Conclusion

The advent of modern portfolio theory and mean-variance portfolio optimization methods provided a rigid mathematical framework for modeling the risk-return relationship of investment portfolios. Although this model has been the cornerstone of finance for many years due to its simplicity, it is far from flawless as it lacks flexibility and precision. In this regard, the simplifying assumptions underlying the classical risk-return relationship and the true nature of the distribution of asset returns have played a significant role in the inability of this model to fully explain the risk-return characteristic of portfolios. This has resulted not only in financial market anomalies but most importantly in the general realization that developing a more robust model which uses the “right” measure of risk is essential (Baker, Bradley and Wurgler, 2011). As a consequence, the risk metric used in the classical risk-reward framework proposed by Markowitz has been called into question (Soe, 2012).

Thus, empirical research in the recent years has shed light on phenomena such as the low volatility anomaly and other financial market inefficiencies which have contributed further to the debate of the adequacy of the mean-variance framework for portfolio optimization (Clarke, De Silva and Thorley, 2006). For instance, numerous academic studies suggest that low-volatility strategies consistently outperform benchmark portfolios on a risk-adjusted basis over a long investment horizon (Soe, 2012). Moreover, the financial crises experienced in the last two decades and the subsequent developments in financial market regulation have emphasized on the importance of having a sound risk metric for portfolio optimization and the search for low-volatility investment strategies in highly volatile markets (Soe, 2012; Clarke, De Silva and Thorley, 2006). With these findings in mind, the present research study has compared the performance of four portfolios: the global minimum-variance portfolio, the mean-variance efficient portfolio, the global minimum-ES portfolio and the mean-ES efficient portfolio using two different frameworks for portfolio optimization: the classical mean-variance approach and the expected-shortfall approach.

The results showed that on average for all values of µp and levels of β the global minimum-expected

shortfall portfolio had the highest value at the end of the thirteen-and-a-half year experiment. The global minimum-variance portfolio performed second to the global minimum-ES portfolio on average and the mean-variance efficient portfolio seemed to have performed the worst out of all strategies on average. Furthermore, the evidence suggests that on average the performance of the

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portfolios was slightly better under the β=0.975 quantile than under the commonly used β=0.99 quantile.

When analyzing the significance of the excess returns for different combinations of portfolio pairs, it was observed that the majority of the hypothesis tests failed to reject the null hypothesis. These results imply that the excess returns were not significantly different from zero. An exception was particularly noted for µp=0.00005 under both β=0.99 and β=0.975 where excess returns were

significant and different from zero for some equations (equations 1,2,6 and 2,5 respectively). The regressions for which the excess returns were significant and different from zero, such as equations 1 and 2, compare the returns between the global minimum portfolios and the mean-efficient portfolios for both variance and expected shortfall. The results suggest that for low values of µp, the

global minimum portfolios performed significantly different from the mean efficient portfolios. In a similar fashion, equation 6 is significant for the excess returns of RminES over Reffvar under β=0.99 and

µp=0.00005, while equation 5 is significant for the excess returns of ReffES over Rminvar under β=0.975

and µp=0.00005.

Contrary to the above results, when we analyzed the significance of the excess returns of each of the four portfolios against the returns of the S&P500, all the tests failed to reject the null hypothesis. This suggests that the excess returns for all pairs of portfolios were not significantly different from zero. The implication of these findings is that the four portfolios considered in this research did not perform better or worse than the S&P500, which was not in line with our expectations.

To summarize, a comparison across different combination of portfolio pairs seems to indicate that the global-minimum expected shortfall strategy is the most profitable on average. The significance of our results from the hypothesis tests become more evident for low levels of µp under both β=0.99

and β=0.975, where again the difference between the global minimum strategies and their efficient counterparts is significant. This is in line with our expectations and it is supported by some research studies in the literature that show expected shortfall strategies outperform mean-variance portfolios on the efficient frontier.

Although the evidence in this study shows that the global minimum-ES portfolio had a better performance on average at the end of the experiment, the results are not conclusive as there are various limitations to this research. For instance, in this experiment we do not take into account investor’s individual preferences, investor’s profile and risk aversion, tax considerations for