Novel approaches for portfolio construction using second order stochastic dominance
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Abstract
In the last decade, a few models of portfolio construction have been proposed which apply second order stochastic dominance (SSD) as a choice criterion. SSD approach requires the use of a reference distribution which acts as a benchmark. The return distribution of the computed portfolio dominates the benchmark by the SSD criterion. The benchmark distribution naturally plays an important role since different benchmarks lead to very different portfolio solutions. In this paper we describe a novel concept of reshaping the benchmark distribution with a view to obtaining portfolio solutions which have enhanced return distributions. The return distribution of the constructed portfolio is considered enhanced if the left tail is improved, the downside risk is reduced and the standard deviation remains within a specified range. We extend this approach from long only to longshort strategies which are used by many hedge fund and quant fund practitioners. We present computational results which illustrate (1) how this approach leads to superior portfolio performance (2) how significantly better performance is achieved for portfolios that include shorting of assets.
Keywords
Portfolio optimisation Stochastic dominance Reference distribution Left tail Downside risk1 Introduction
Second order stochastic dominance (SSD) has been long recognised as a rational criterion of choice between wealth distributions (Hadar and Russell 1969; Bawa 1975; Levy 1992). Empirical tests for SSD portfolio efficiency have been proposed in Post (2003), Kuosmanen (2004). In recent times SSD choice criterion has been proposed (Dentcheva and Ruszczynski 2003, 2006; Roman et al. 2006) for portfolio construction by researchers working in this domain. The approach described in Dentcheva and Ruszczynski (2003, 2006) first considers a reference (or benchmark) distribution and then computes a portfolio which dominates the benchmark distribution by the SSD criterion. In Roman et al. (2006) a multiobjective optimisation model is introduced in order to achieve SSD dominance. This model is both novel and usable since, when the benchmark solution itself is SSD efficient or its dominance is unattainable, it finds an SSD efficient portfolio whose return distribution comes close to the benchmark in a satisficing sense. The topic continues to be researched (Dentcheva and Ruszczynski; Fábián et al. 2011a, b; Post and Kopa 2013; Kopa and Post 2015; Post et al. 2015; Hodder et al. 2015; Javanmardi and Lawryshy 2016) from the perspective of modelling as well as that of computational solution.
These models start from the assumption that a reference (benchmark) distribution is available. It was shown in Roman et al. (2006) that the reference distribution plays a crucial role in the selection process: there are many SSD efficient portfolios and the choice of a specific one depends on the benchmark distribution used. SSD efficiency does not necessarily make a return distribution desirable, as demonstrated by the optimal portfolio with regards to maximum expected return. It was shown in Roman et al. (2006) that this portfolio is SSD efficient  however, it is undesirable to a large class of decisionmakers.
In the last two decades, quantitative analysts in the fund management industry have actively debated about the benefit of active fund management in contrast to passive investment. Passive investment equates to holding a portfolio determined by the constituents of a chosen market index. Active fund managers are engaged in finding portfolios which provide better return than that of a passive index portfolio. Set against this background the index is a natural benchmark (reference) distribution which an active fund manager would like to dominate. There have been several papers under the topic of “enhanced indexation” (di Bartolomeo 2000) which discuss alternative ways of doing better than the passive index portfolio. It has been shown empirically that return distributions of financial indices are SSD dominated (Post 2003; Kuosmanen 2004; Post and Kopa 2013; Kopa and Post 2015; Post et al. 2015).
In Roman et al. (2013) we introduced SSDbased models for enhanced indexation and reported encouraging practical results. An essential aspect of our approach to portfolio construction can be articulated by the qualitative statement “reduction of the downside risk and improvement of the upside potential”. This can be translated as finding return distributions with high expected value and skewness, meaning a left tail that is closer to the mean. An index does not necessarily (indeed very rarely) possess these properties. Thus the SSD dominant portfolio solutions, when we choose an index as the benchmark, do not necessarily have return distributions with a short left tail, high skewness and controlled standard deviation. Research effort in this direction include SSD based models in which, by appropriately selecting model parameters, the left tail of the resulting distribution can be partially controlled, in the sense that more “weight” can be given to tails at specified levels of confidence (Kopa and Post 2015; Hodder et al. 2015), see also Javanmardi and Lawryshy (2016).
In this paper, we propose a different approach that stems from a natural question to ask: how should we choose the reference distribution in SSD models such that the resulting portfolio has a return distribution that, in addition to being SSD efficient, has specific desirable properties, in the form of (1) high skewness and (2) standard deviation within a range?
 (a)
we propose a method of reshaping, or enhancing, a given (reference) distribution, namely, that of a financial index, in order to use it as a benchmark in SSD optimisation models;
 (b)
we formulate and solve SSD models that include longshort strategies which are established financial practice to cope with changing financial regimes (bull and bear markets);
 (c)
we investigate empirically the insample and outof sample performance of portfolios obtained using enhanced benchmarks and longshort strategies.
2 Portfolio optimisation using SSD
We consider a portfolio selection problem with one investment period. Let n denote the number of assets into which we may invest. A portfolio \( x=(x_1, \ldots x_n) \in \mathbb {R}^n \) represents the proportions of the portfolio value invested in the available assets. Let the ndimensional random vector \( R = (R_1, \ldots , R_n)\) denote the returns of the different assets at the end of the investment period.
It is usual to consider the distribution of R as discrete, described by the realisations under a finite number of scenarios S; scenario j occurs with probability \(p_j\) where \(p_j>0\) and \(p_1+ \cdots +p_S=1\). Let us denote by \(r_{ij}\) the return of asset i under scenario j. The random return of portfolio x is denoted by \( R_{x}\), with \( R_{x} := x_1R_1+ \cdots x_nR_n\).
 (a)
\( \text{ E } ( U(R) ) \ge \text{ E } ( U( R^{\prime } ) ) \) holds for any nondecreasing and concave utility function U for which these expected values exist and are finite.
 (b)
\( \text{ E } ( [ t  R ]_+ ) \;\le \; \text{ E } ( [ t  R^{\prime } ]_+ ) \) holds for each \( t \in \mathbb {R} \).
 (c)
\( \text{ Tail }_{\alpha }( R ) \;\ge \; \text{ Tail }_{\alpha }( R^{\prime } ) \) holds for each \( 0 < \alpha \le 1 \), where \( \text{ Tail }_{\alpha }( R ) \) denotes the unconditional expectation of the least \( \alpha *100\% \) of the outcomes of R.
The equivalence of the above relations is well known since the works of Whimore and Findlay (1978) and Ogryczak and Ruszczynski (2002). From the first relation, the importance of SSD in portfolio selection can be clearly seen: it expresses the preference of rational and riskaverse decision makers.
Remark 1
The definition of \(\text{ Tail }_{\alpha }( R ) \) is an informal definition. For a formal definition, quantile functions can be used. Denote by \(F_R\) the cumulative distribution function of a random variable R. If there exists t such that \(F_R(t) = \alpha \) then \(\text{ Tail }_{\alpha }( R ) = \alpha E(R  R \le t)\)—which justifies the informal definition. For the general case, let us define the generalised inverse of \(F_R\) as \(F_R^{1}(\alpha ) := \text{ inf } \{t  F_R(t) \ge \alpha \}\) and the second quantile function as \(F_R^{2}(\alpha ) := \int _{0}^{\alpha } F_R^{1}(\beta )d\beta \) and \(F_R^{2}(0):= 0\). With these notations, \(\text{ Tail }_{\alpha }( R ): = F_R^{2}(\alpha ).\)
Let \( X \subset \mathbb {R}^n \) denote the set of the feasible portfolios, we assume that X is a bounded convex polyhedron. A portfolio \( {x}^{\star } \) is said to be SSDefficient if there is no feasible portfolio \( {x} \in X \) such that \( R_{x} \succ _{_{SSD}} R_{{x}^{\star }} \).
Recently proposed portfolio optimisation models based on the concept of SSD assume that a reference (benchmark) distribution \(R^{\text {ref}}\) is available. Let \(\hat{\tau }\) be the tails of the benchmark distribution at confidence levels \( \frac{1}{S}, \ldots , \frac{S}{S}\); that is, \(\hat{\tau } = (\hat{\tau }_1, \ldots , \hat{\tau }_S) = \big ( \text {Tail}_{\frac{1}{S}}R^{\text {ref}}, \ldots , \text {Tail}_{\frac{S}{S}}R^{\text {ref}} \big )\).
Remark 2
The cutting plane formulation above has a huge number of constraints (3), referred to in Fábián et al. (2011a) as “cuts”. The specialised solution method (Fábián et al. 2011a) adds cuts at each iteration until optimality is reached; it is shown that in practice, only a few cuts are needed. For example, all models with 10,000 scenarios of assets returns were solved with less than 30 iterations. For more details, the reader is referred to Fábián et al. (2011a).
Remark 3
In case that optimisation of the worst partial achievement has multiple optimal solutions, all of them improve on the benchmark (if the optimum is positive) but not all of them are guaranteed to improve until SSD efficiency is attained. The model proposed in Roman et al. (2006) has a slightly different objective function that included a regularisation term, in order to guarantee SSD efficiency for the case multiple optimal solutions. This term was dropped in the cutting plane formulation, the advantage of this being huge decrease in computational difficulty and solution time; just as an example, models with tens of thousands of scenarios were solved within seconds, while the original model formulation in Roman et al. (2006) could only deal with a number of scenarios in the order of hundreds. In Fábián et al. (2011a), extensive computational results are reported. For relatively small datasets, SSD models were solved with both the cutting plane formulation and the original formulation including the regularisation term; in all instances, both formulations led to the same optimal solutions. For more details, the reader is referred to Fábián et al. (2011a).
The tails of the benchmark distribution \((\hat{\tau }_1, \ldots , \hat{\tau }_S)\) are the decisionmaker’s input. When the benchmark is not SSD efficient, the solution portfolio has a return distribution that improves on the benchmark until SSD efficiency is achieved. In case the benchmark is SSD efficient, the model finds the portfolio whose return distribution is the benchmark. For instance, if the benchmark is the return distribution of the asset with the highest expected return, the solution portfolio is that where all capital is invested in this asset.
“Unattainable” reference distributions are discussed in Roman et al. (2006), where the (SSD efficient) solution portfolio has a return distribution that comes as close as possible to dominating the benchmark; this is obtained by minimising the largest difference between the tails of these two distributions. However, simply setting “high targets” (i.e. a possibly unrealistic benchmark) does not solve the problem of finding a portfolio with a “good” return distribution, e.g. one having a short left tail/high skewness and controlled standard deviation.
In recent research, the most common approach is to set the benchmark as the return distribution of a financial index. This is natural since discrete approximations for this choice can be directly obtained from publicly available historical data, and also due to the meaningfulness of interpretation  it is common practice to compare / make reference to an index performance. The financial index distribution is “achievable” since there exists a feasible portfolio that replicates the index and empirical evidence (Roman et al. 2006, 2013; Post and Kopa 2013) suggests that this distribution is in most cases not SSD efficient. While it is safe to say that generally a portfolio that dominates an index with relation to SSD can be found, there is no guarantee that this portfolio will have desirable properties.
In this work, we use the distribution of a financial index as a starting point; we enhance it in the sense of increasing skewness by a decisionmaker’s specified amount while keeping standard deviation within a (decisionmaker specified) range.
3 Reshaping the reference distribution
We propose a method of reshaping an original reference distribution and achieving a synthetic (improved) reference distribution.
The pink area in the figure represents the density curve of what we consider to be an improved reference distribution. Desirable properties include a shorter left tail (reduced probability of large losses), which translates into higher skewness, and a higher expected return, which is equivalent to a higher mean. A smaller standard deviation is not necessarily desirable, as it might limit the upside potential of high returns. Instead, we require the standard deviation of the new distribution to be within a specified range from the standard deviation of the original distribution.
We hereby introduce a method for transforming the original reference distribution into a synthetic reference distribution given target values for the first three statistical moments (mean, standard deviation and skewness).
Let the original reference distribution be represented by a sample \(Y = (y_1, \ldots , y_S)\) with mean \(\mu _Y\), standard deviation \(\sigma _Y\) and skewness \(\gamma _Y\).
Given target values \(\mu _T\), \(\sigma _T\) and \(\gamma _T\), our goal is to find a distribution \(Y'\) with values \(y'_s, s = 1, \ldots , S\), such that \(\mu _Y' = \mu _{T}\), \(\sigma _{Y'} = \sigma _T\) and \(\gamma _{Y'} = \gamma _T\).
In statistics, this problem is related to test equating. It is commonly found in standardized testing, where multiple test forms are needed because examinees must take the test at different occasions and one test form can only be administered once to ensure test security. However, test scores derived from different forms must be equivalent. Let us consider two test forms, say, Form V and Form W. It is generally assumed that the examinee groups that take test forms V and W are sampled from the same population, and differences in score distributions are attributed to form differences (e.g. more difficult questions in V than in W). Equating forms V and W involves modifying V scores so that the transformed V scores have the same statistical properties as W.
 1.
Step 1: Arbitrarily define \(\mu _T\), \(\gamma _T\) and \(\sigma _T\).
 2.Step 2: Using the singlevariable Newton–Raphson iterative method (Press et al. (1988), pp. 362–367), find d so that \(Y + d Y^2\) has skewness \(\gamma _{(Y + d Y^2)} = \gamma _T\). Using the standard skewness formula for a discrete sample, the skewness of the original reference distribution \(\gamma _Y\) is given by:In order to find d we need to solve the equation:$$\begin{aligned} \gamma _Y = \frac{ \frac{1}{n} \sum _{i = 1}^{n} (y_i  \mu _{Y})^3 }{ \Big [ \frac{1}{n1} \sum _{i = 1}^{n} (y_i  \mu _Y)^2 \Big ]^ {3/2} } \end{aligned}$$$$\begin{aligned} \frac{ \frac{1}{n} \sum _{i = 1}^{n} \big ( y_i + d y_i^2  (\frac{1}{n} \sum _{j=1}^{n} y_j + d y_j^ 2) \big )^3 }{ \Big [ \frac{1}{n1} \sum _{i = 1}^{n} \big ( y_i + d y_i^2  (\frac{1}{n} \sum _{j=1}^{n} y_j + d y_j^ 2) \big )^2 \Big ]^ {3/2} }  \gamma _T = 0 \end{aligned}$$
 3.
Step 3: Let \(g = \frac{\sigma _T}{\sigma _{Y + d Y^2}}\). Then \(g (Y + d Y^2) \) will have \(\gamma _{g (Y + d Y^2)} = \gamma _T\) and \(\sigma _{g (Y + d Y^2)} = \sigma _T\) since a linear transformation (multiplication of constant g) does not change the skewness of a distribution.
 4.
Step 4: Let \(h = \mu _T  \mu _{g(Y + dY^2)}\). Then \(Y' = h + g(X + dX^2)\) will have \(\mu _{Y'} = \mu _T\), \(\gamma _{Y'} = \gamma _T\) and \(\sigma _{Y'} = \sigma _T\) since adding a constant does not change the skewness or standard deviation of a distribution.
 5.Step 5: We then find \(Y'\) by computing:$$\begin{aligned} y'_s = gdy_s^2 + gy_s + h, \;\;\;\;\;\; \forall s = 1, \ldots , S \end{aligned}$$
4 Longshort modelling
When shortselling is allowed, the amount available for purchases of stocks in long positions is increased. Suppose we borrow from an intermediary a specified number of units of asset i (\(i = 1, \ldots , n\)), corresponding to a proportion \(x_i^\) of capital. We sell them immediately in the market and hence have a cash sum of \((1 + \sum _{i=1}^{n} x_i^)C\) to invest in long positions; where C is the initial capital available.
In longshort practice, it is common to fix the total amount of shortselling to a prespecified proportion \(\alpha \) of the initial capital. In this case, the amount available to invest in long positions is \((1 + \alpha )C\). A fund that limits their exposure with a proportion \(\alpha = 0.2\) is usually referred to as a 120/20 fund. For modelling this situation, to each asset \(i \in {1, \ldots ,n}\) we assign two continuous nonnegative decision variables \(x_i^+, x_i^\), representing the proportions invested in long and short positions in asset i, and two binary variables \(z_i^+, z_i^\) that indicate whether there is investment in long or short positions in asset i. For example, if 10% of the capital is shorted in asset i, we write this as \(x_i^+=0\), \(x_i^=0.1\), \(z_i^+=0\), \(z_i^=1\).
5 Computational results
5.1 Motivation, dataset and methodology
 1.
To investigate the effect of using a benchmark distribution, reshaped by modifying skewness and/or standard deviation, in SSDbased portfolio optimisation models; the return distributions of the resulting portfolios, as well as their outofsample performance, are compared to those of portfolios obtained using the original benchmark.
 2.
To investigate the performance of various longshort strategies in comparison with the long only strategy, as used in SSDbased models with original and reshaped benchmarks.
The original benchmark distribution is obtained by considering the historical daily rates of return of FTSE100 during the same time period. We implement models (2)–(4) and (5)–(12) for different values of \(\alpha , \varDelta _{\gamma }\) and \(\varDelta _{\sigma }.\)
We used an Intel(R) Core(TM) i53337U CPU @ 1.80 GHz with 6GB of RAM and Linux as operating system. The Branchandcut algorithm was written in C++ and the backtesting framework was written in R (R Core Team 2015); we used CPLEX 12.6 (IBM 2015) as mixedinteger programming solver.
The methodology we adopt is successive rebalancing over time with recent historical data as scenarios. We start from the beginning of our data set. Given insample duration of S days, we decide a portfolio using data taken from an insample period corresponding to the first \(S+1\) days (yielding S daily returns for each asset). The portfolio is then held unchanged for an outofsample period of 5 days. We then rebalance (change) our portfolio, but now using the most recent S returns as insample data. The decided portfolio is then held unchanged for an outofsample period of 5 days, and the process repeats until we have exhausted all of the data. We set \(S = 564\) (approximately the number of trading days in 2.5 years) ; the total outofsample period spans almost 5 years (January 2010–October 2014).
Once the data has been exhausted we have a time series of 1201 portfolio return values for outofsample performance, here from period 565 (the first outofsample return value, corresponding to 04/01/2010) until the end of the data.
5.2 Longshort and longonly comparison
We test \(\alpha = 0\), \(\alpha = 0.2\), \(\alpha = 0.5\) and \(\alpha = 1.0\), thus we consider 100/0 (longonly), 120/20, 150/50 and 200/100 portfolios. The benchmark distribution is that of FTSE100. Given the portfolio holding period of 5 days, during the outofsample evaluation period there are a total of 240 rebalances. For each rebalance, we assign a time limit of 60 s.
5.2.1 Insample analysis
Table 1 shows insample statistics regarding optimal solution values. Under Optimal object value, three columns are reported: (1) Mean, showing the average of the optimal objective values in each rebalance; (2) Min and (3) Max, showing respectively the minimum and maximum optimal objective values in each rebalance.
Long only and longshort, insample statistics
Long/short  Optimal objective value  

Mean  Min  Max  
100/0  0.001234  0.000760  0.001868 
120/20  0.001697  0.001019  0.002359 
150/50  0.002114  0.001177  0.002899 
200/100  0.002434  \(\)0.000055  0.003471 
Long/short, number of stocks in the composition of optimal portfolios
Long/short  Long positions  Short positions  Total  

Mean  Min  Max  Mean  Min  Max  Mean  Min  Max  
100/0  10.7  5  21  –  –  –  10.7  5  21 
120/20  13.3  6  23  4.1  1  9  17.4  8  30 
150/50  18.3  10  29  8.5  2  18  26.8  13  44 
200/100  25.9  17  37  16.8  9  25  42.7  27  59 
Table 2 shows how many assets on average were in the composition of optimal portfolios—also reported are minimum and maximum numbers, for each value of \(\alpha \). Statistics are shown for assets held long, short and also for the complete set. From the table we can see that the addition of shorting tends to increase the number of stocks picked. This is expected, since the higher \(\alpha \) is, the higher is the exposure. For instance, if \(\alpha = 0.5\) we have 0.5C in repayment obligations and 1.5C in long positions, having a total exposure of 2C in different assets.
However, even in long/short models, the cardinality of the optimal portfolios is not high (26.8 for 150/50, about a quarter of the total of 100 companies from FTSE100, and 42.7 for 200/100), thus we consider the introduction of cardinality constraints to be unnecessary.
5.2.2 Outofsample performance

Final value: Normalised final value of the portfolio at the end of the outofsample period.

Excess over RFR (%): Annualised excess return over the risk free rate. For FTSE100 we used a flat yearly risk free rate of 2%.

Sharpe ratio: Annualised Sharpe ratio (Sharpe 1966) of returns.

Sortino ratio: Annualised Sortino ratio (Sortino and Price 1994) of returns.

Max drawdown (%): Maximum peaktotrough decline (as percentage of the peak value) during the entire outofsample period.

Max recovery days: Maximum number of days for the portfolio to recover to the value of a former peak.

Daily returns—Mean: Mean of outofsample daily returns.

Daily returns—SD: Standard deviation of outofsample daily returns.
Long/short, outofsample performance statistics
Long/short  Final  Excess over  Sharpe  Sortino  Max draw  Max reco  Daily returns  

value  RFR (%)  ratio  ratio  down (%)  very days  Mean  SD  
FTSE100  1.18  1.57  0.097  0.134  18.83  481  0.00019  0.00985 
100/0  2.19  15.97  0.855  1.242  14.55  150  0.00071  0.01071 
120/20  2.60  20.24  0.998  1.446  12.33  151  0.00086  0.01141 
150/50  2.78  21.93  0.983  1.410  14.15  183  0.00093  0.01246 
200/100  2.53  19.58  0.824  1.170  17.65  244  0.00086  0.01340 
As we increase \(\alpha \) up to 0.5, both the mean and the standard deviation of the daily returns increase. As a consequence, although 150/50 achieved better returns than 120/20, the latter obtained higher Sharpe and Sortino ratios, as well as a lower maximum drawdown. Adding shorting seems to bring better performance at the expense of greater risk.
The overall performance of 200/100 portfolios was better than the longonly portfolio, but worse than 120/20 and 150/50. The 200/100 portfolio is more volatile (both insample and outofsample), which may be the reason for its lower final value when compared to the other cases. Moreover, some rebalances were not solved within 60 s, yielding suboptimal solutions. In the next sections, where we analyse the effects of reshaping the reference distribution, we ommit 200/100 results as they show similar behaviour to the ones presented in this section.
5.3 Reshaping the reference distribution: increased skewness
We now test how reshaping the reference distribution impacts both insample and outofsample results. For \(\alpha = 0\), \(\alpha = 0.2\), \(\alpha = 0.5\), we test the effects of different values of \(\varDelta _{\gamma }\), more specifically, we test \(\varDelta _{\gamma } = [0, 1, 2, 3, 4, 5]\). In all tests in this section, \(\varDelta _{\sigma } = 0\), that is, the standard deviation is unchanged.
Setting \(\varDelta _{\gamma } = 0\) is equivalent to optimising with the original reference distribution. In the rest of the cases, the skewness is increased to \(\gamma _T = \gamma _Y + (\gamma _Y \varDelta _{\gamma })\). For example, if \(\varDelta _{\gamma } = 1\) and \(\gamma _Y < 0\) then \(\gamma _T = 0\). If \(\varDelta _{\gamma } = 1\) and \(\gamma _Y > 0 \) then \(\gamma _T = 2 \gamma _Y\).
5.3.1 Insample results

(iv) \(\textit{ScTail}_{\alpha }\): the \(\alpha \) “scaled” tail, defined as in Sect. 2 as the conditional expectation of the least \(\alpha *100\%\) of the outcomes.

(v) \(\textit{EP}_{\rho }\): Expected conditional profit at \(\rho \%\) confidence level, equivalent to CVaR\(_{\rho }\) but calculated from the right tail of the distribution. Let \(S^{\rho } = \lceil S (\rho /100) \rceil \). EP\(_{\rho }\) is defined as \(\frac{1}{S  S^{\rho }1} \sum _{s=S^{\rho }}^{S} r^p_s\).
We also compare insample solutions in terms of their SSD relation. Let Y be the optimal portfolio (with worst partial achievement \(\overline{\mathcal {V}}\)) that solves the enhanced indexation model and dominates the original reference distribution with respect to SSD. Accordingly, let \(Y'\) be the optimal porfolio (with worst partial achievement \(\overline{\mathcal {V}'}\)) that dominates the synthetic (improved) reference distribution. Let also \(R^Y = [r^Y_1 \le \cdots \le r^Y_S]\) and \(R^{Y'} = [r^{Y'}_1 \le \cdots \le r^{Y'}_S]\) be the ordered set of insample returns for Y and \(Y'\).
It is clear that if \(\overline{\mathcal {V}} \ne \overline{\mathcal {V}'}\), \(Y \not \succeq _{\text {SSD}} Y'\) and \(Y' \not \succeq _{\text {SSD}} Y\). If, for instance, \(Y' \succeq _{\text {SSD}} Y\), then \(Y'\) would have been chosen instead of Y as the optimal solution of the enhanced indexation model with the original reference distribution since \(\overline{\mathcal {V}'} > \overline{\mathcal {V}}\). If \(\overline{\mathcal {V}} = \overline{\mathcal {V}'}\), it is possible, albeit unlikely, that \(Y' \succeq _{\text {SSD}} Y\) or \(Y \succeq _{\text {SSD}} Y'\)  that could be the case where the mixedinteger programming model had multiple optima.
Assuming that most likely neither \(Y \succeq _{\text {SSD}} Y'\) nor \(Y' \succeq _{\text {SSD}} Y\), we can measure which solution is “closer” to dominating the other. For \(Y'\), we compute, for each rebalance, \(\mathcal {S}_{Y'} \subset \{1, ..., S\}\) where \(s \in \mathcal {S}_{Y'}\) if \(\text {Tail}_{\frac{s}{S}} R^{Y'} > \text {Tail}_{\frac{s}{S}} R^Y\), i.e. \(\mathcal {S}_{Y'}\) represents the number of times that the unconditional expectation of the least s scenarios of \(Y'\) is greater than the equivalent for Y. We also compute, for each rebalance, \(\mathcal {S}_{Y} \subset \{1, ..., S\}\) where \(s \in \mathcal {S}_{Y}\) if \(\text {Tail}_{\frac{s}{S}} R^{Y} > \text {Tail}_{\frac{s}{S}} R^{Y'}\).

(ix) \(\overline{\mathcal {S}_{Y'}}\): Mean value of \(\mathcal {S}_{Y'}\) over all rebalances.

(x) \(\overline{\mathcal {S}_{Y}}\): Mean value of \(\mathcal {S}_{Y}\) over all rebalances.
Changing \(\varDelta _{\gamma }\): average values of insample statistics for return distributions of optimal portfolios and of reshaped benchmarks; the average benchmark mean and standard deviation are 0.00028 and 0.01247 respectively
Long/short  \(\varDelta _\gamma \)  Insample portfolio performance  Insample benchmark performance  Dominance  

Mean  SD  Skewness  \(\hbox {ScTail}_{05}\)  \(\hbox {EP}_{95}\)  Skewness  \(\hbox {ScTail}_{05}\)  \(\hbox {EP}_{95}\)  \(\overline{\mathcal {S}_{Y'}}\)  \(\overline{\mathcal {S}_{Y}}\)  
100/0  0  0.00152  0.01201  \(\)0.04113  \(\)0.02597  0.02719  \(\)0.03861  \(\)0.02898  0.02790  \(\)  – 
100/0  1  0.00151  0.01200  0.00006  \(\)0.02573  0.02731  0.10116  \(\)0.02833  0.02849  256.80  307.20 
100/0  2  0.00149  0.01195  0.03633  \(\)0.02542  0.02728  0.24094  \(\)0.02766  0.02906  263.38  300.62 
100/0  3  0.00145  0.01183  0.06341  \(\)0.02498  0.02705  0.38071  \(\)0.02698  0.02961  300.58  263.42 
100/0  4  0.00139  0.01165  0.08358  \(\)0.02448  0.02664  0.52049  \(\)0.02628  0.03015  331.70  232.30 
100/0  5  0.00129  0.01136  0.08703  \(\)0.02386  0.02588  0.66026  \(\)0.02556  0.03067  362.88  201.12 
120/20  0  0.00198  0.01195  \(\)0.03403  \(\)0.02508  0.02756  \(\)0.03861  \(\)0.02898  0.02790  \(\)  – 
120/20  1  0.00198  0.01196  0.00364  \(\)0.02492  0.02771  0.10116  \(\)0.02833  0.02849  240.76  323.24 
120/20  2  0.00197  0.01192  0.04218  \(\)0.02464  0.02773  0.24094  \(\)0.02766  0.02906  272.75  291.25 
120/20  3  0.00194  0.01184  0.07870  \(\)0.02424  0.02763  0.38071  \(\)0.02698  0.02961  295.34  268.66 
120/20  4  0.00191  0.01174  0.11291  \(\)0.02378  0.02743  0.52049  \(\)0.02628  0.03015  313.20  250.80 
120/20  5  0.00186  0.01160  0.14315  \(\)0.02327  0.02711  0.66026  \(\)0.02556  0.03067  328.19  235.81 
150/50  0  0.00240  0.01194  \(\)0.03456  \(\)0.02445  0.02806  \(\)0.03861  \(\)0.02898  0.02790  \(\)  – 
150/50  1  0.00240  0.01194  0.00500  \(\)0.02428  0.02824  0.10116  \(\)0.02833  0.02849  270.30  293.70 
150/50  2  0.00239  0.01191  0.04945  \(\)0.02396  0.02832  0.24094  \(\)0.02766  0.02906  281.27  282.73 
150/50  3  0.00237  0.01186  0.09315  \(\)0.02355  0.02831  0.38071  \(\)0.02698  0.02961  292.37  271.63 
150/50  4  0.00233  0.01178  0.13320  \(\)0.02309  0.02820  0.52049  \(\)0.02628  0.03015  298.49  265.51 
150/50  5  0.00229  0.01167  0.16910  \(\)0.02260  0.02798  0.66026  \(\)0.02556  0.03067  305.06  258.94 
From Table 4, we observe that increasing \(\varDelta _{\gamma }\) increases the insample skewness and also ScTail\(_{05}\) of the optimal distributions—which was expected, since increasing skewness reduces the left tail. We can also observe that the mean and standard deviation tend to decrease as we increase the skewness in the synthetic distribution. This decrease is very small or nonexistent in the case of smaller deviations from the original skewness e.g. \(\gamma =1\) but more pronounced for larger values of \(\gamma \).
The skewness of the solution portfolios, although clearly increasing in line with increasing the skewness of the benchmark, is considerably below the skewness values set by the improved benchmark. The solution portfolios have on average much higher expected value and less variance than the improved benchmark.
The statistics for \(\hbox {EP}_{95}\) reach a peak somewhere between \(\varDelta _{\gamma } = 1\) and \(\varDelta _{\gamma } = 2\), and their values decrease for higher \(\varDelta _{\gamma }\). This might be due to the synthetic distribution having “unrealistic” properties if its third moment differs too much from that of the original distribution. We do observe, nevertheless, that increasing skewness also makes insample portfolios based on synthetic distributions “closer” to dominating those based on the original distribution, since \(\overline{\mathcal {S}_{Y'}}\) increases and \(\overline{\mathcal {S}_{Y}}\) decreases as \(\varDelta _{\gamma }\) grows.
We do not report the cardinality of the optimal portfolios as we have observed very little change in the number of stocks held due to changes in \(\varDelta _{\gamma }\).
5.3.2 Outofsample performance
Changing \(\varDelta _{\gamma }\), outofsample performance statistics
Long/short  \(\varDelta _{\gamma }\)  Final  Excess over  Sharpe  Sortino  Max draw  Max reco  Daily returns  

value  RFR (%)  ratio  ratio  down (%)  very days  Mean  SD  
FTSE100  1.18  1.57  0.097  0.134  18.83  481  0.00019  0.00985  
100/0  0  2.19  15.97  0.855  1.242  14.55  150  0.00071  0.01071 
100/0  1  2.21  16.16  0.870  1.267  13.44  150  0.00072  0.01064 
100/0  2  2.17  15.67  0.856  1.245  13.69  150  0.00070  0.01051 
100/0  3  2.09  14.73  0.818  1.190  13.54  150  0.00067  0.01039 
100/0  4  2.00  13.69  0.778  1.131  13.07  150  0.00063  0.01018 
100/0  5  1.95  13.10  0.768  1.112  12.66  150  0.00061  0.00990 
120/20  0  2.60  20.24  0.998  1.446  12.33  151  0.00086  0.01141 
120/20  1  2.71  21.27  1.047  1.521  12.32  151  0.00089  0.01139 
120/20  2  2.72  21.39  1.062  1.545  11.82  160  0.00090  0.01129 
120/20  3  2.69  21.09  1.053  1.532  12.05  162  0.00089  0.01124 
120/20  4  2.62  20.44  1.033  1.501  12.05  178  0.00087  0.01113 
120/20  5  2.57  19.91  1.028  1.497  12.22  183  0.00085  0.01092 
150/50  0  2.78  21.93  0.983  1.410  14.15  183  0.00093  0.01246 
150/50  1  2.77  21.86  0.980  1.405  14.48  183  0.00093  0.01247 
150/50  2  2.78  21.96  0.988  1.420  14.64  184  0.00093  0.01241 
150/50  3  2.73  21.53  0.978  1.406  14.70  185  0.00091  0.01232 
150/50  4  2.71  21.33  0.978  1.407  14.35  196  0.00091  0.01221 
150/50  5  2.73  21.46  0.999  1.437  13.97  184  0.00091  0.01203 
We observe that the volatility of outofsample returns tend to reduce as we optimise over synthetic distributions with higher skewness values. The mean returns and excess returns over the risk free rate tend to increase slightly (up to somewhere between \(\varDelta _{\gamma } = 1\) and \(\varDelta _{\gamma } = 2\)) and then start to drop. These results are consistent with observed insample statistics. Changing skewness had little impact in terms of drawdown and recovery from drops. While returns do tend to decrease for higher value of \(\varDelta _{\gamma }\), its reduced standard deviation might appeal to riskaverse investors. The best values of Sharpe and Sortino ratios are generally obtained when \(\varDelta _{\gamma } = 1\), 2 or 3. In particular, the 120/20 strategy with \(\varDelta _{\gamma }\)=1 or 2 seems to have the best riskreturn characteristics.
5.4 Reshaping the reference distribution: modified standard deviation
In this section, we test how portfolios behave for different values of \(\varDelta _{\sigma }\). Since increasing skewness slightly seems to be the best choice, according to our results in the previous section, for the next tests we set \(\varDelta _{\gamma } = 1\).
We report results for \(\varDelta _{\sigma } = [0.1, 0, 0.1, 0.3, 0.5]\); for example, if \(\varDelta _{\sigma } = 0.1\), the synthetic reference distribution will have a 10% increase in its standard deviation when compared to the original reference distribution.
5.4.1 Insample results
Changing \(\varDelta _{\sigma }\): Average insample statistics for return distributions of optimal portfolios and of improved benchmarks; the average benchmark mean is 0.00028; the average benchmark skewness is \(0.03861\) for \(\varDelta _\gamma = 0\) and 0.10116 for \(\varDelta _\gamma = 1\)
Long/short  \(\varDelta _\gamma \)  \(\varDelta _\sigma \)  Insample portfolio performance  Insample benchmark performance  Dominance  

Mean  SD  Skewness  \(\hbox {ScTail}_{05}\)  \(\hbox {EP}_{95}\)  SD  \(\hbox {ScTail}_{05}\)  \(\hbox {EP}_{95}\)  \(\overline{\mathcal {S}_{Y'}}\)  \(\overline{\mathcal {S}_{Y}}\)  
100/0  0  0  0.00152  0.01201  \(\)0.04113  \(\)0.02597  0.02719  0.01247  \(\)0.02898  0.02790  –  – 
100/0  1  \(\)0.1  0.00129  0.01083  \(\)0.02138  \(\)0.02343  0.02451  0.01123  \(\)0.02547  0.02567  497.25  66.75 
100/0  1  0  0.00151  0.01200  0.00006  \(\)0.02573  0.02731  0.01247  \(\)0.02833  0.02849  256.80  307.20 
100/0  1  0.1  0.00169  0.01316  0.00791  \(\)0.02809  0.03003  0.01372  \(\)0.03119  0.03131  40.74  523.26 
100/0  1  0.3  0.00195  0.01537  0.01004  \(\)0.03251  0.03520  0.01622  \(\)0.03691  0.03695  35.23  528.77 
100/0  1  0.5  0.00214  0.01756  0.04954  \(\)0.03696  0.04062  0.01871  \(\)0.04263  0.04259  28.97  535.03 
120/20  0  0  0.00198  0.01195  \(\)0.03403  \(\)0.02508  0.02756  0.01247  \(\)0.02898  0.02790  –  – 
120/20  1  \(\)0.1  0.00175  0.01076  \(\)0.00967  \(\)0.02248  0.02475  0.01123  \(\)0.02547  0.02567  499.05  64.95 
120/20  1  0  0.00198  0.01196  0.00364  \(\)0.02492  0.02771  0.01247  \(\)0.02833  0.02849  240.75  323.25 
120/20  1  0.1  0.00217  0.01314  0.01127  \(\)0.02733  0.03056  0.01372  \(\)0.03119  0.03131  43.48  520.52 
120/20  1  0.3  0.00246  0.01543  0.03121  \(\)0.03190  0.03611  0.01622  \(\)0.03691  0.03695  37.52  526.48 
120/20  1  0.5  0.00267  0.01768  0.05574  \(\)0.03644  0.04146  0.01871  \(\)0.04263  0.04259  31.83  532.17 
150/50  0  0  0.00240  0.01194  \(\)0.03456  \(\)0.02445  0.02806  0.01247  \(\)0.02898  0.02790  \(\)  – 
150/50  1  \(\)0.1  0.00212  0.01076  \(\)0.00258  \(\)0.02194  0.02540  0.01123  \(\)0.02547  0.02567  480.19  83.81 
150/50  1  0  0.00240  0.01194  0.00500  \(\)0.02428  0.02824  0.01247  \(\)0.02833  0.02849  270.30  293.70 
150/50  1  0.1  0.00263  0.01313  0.01193  \(\)0.02667  0.03110  0.01372  \(\)0.03119  0.03131  56.66  507.34 
150/50  1  0.3  0.00300  0.01545  0.02802  \(\)0.03130  0.03664  0.01622  \(\)0.03691  0.03695  50.20  513.80 
150/50  1  0.5  0.00329  0.01775  0.04156  \(\)0.03590  0.04199  0.01871  \(\)0.04263  0.04259  43.97  520.03 
We can see that increasing \(\varDelta _{\sigma }\) increases the standard deviation of the return distributions of optimal portfolios, but also increases their mean and skewness at the same time.
The standard deviation of the return distribution of optimal portfolios follows the same pattern as set by the improved benchmarks, in the sense that it increases (or decreases) with increased (or decreased) standard deviation of the benchmark. It is, on average, slightly lower compared to the standard deviation of the benchmark. Also, the optimal portfolios have consuderably better mean and CVaR values than their corresponding benchmarks.
For \(\varDelta _{\sigma } = 0.1\), we obtain portfolios whose return distributions have better risk characteristics in the form of lower standard deviation and higher tail value. On the other hand, their mean returns, EP\(_{95}\) and even skewness are lower. As we increase \(\varDelta _{\sigma }\), ScTail\(_{05}\) gets lower, but EP\(_{95}\) gets higher.
We also observe that, when increasing \(\varDelta _{\sigma }\), the portfolios obtained based on the original reference distribution are “closer” to dominating the portfolio obtained via the synthetic benchmark.
Once again, we do not report the cardinality of the optimal portfolios as very little alteration was observed due to changes in \(\varDelta _{\sigma }\).
Similarly to Figs. 3 and 5 shows the difference between optimising over the original and synthetic benchmarks, this time for varying standard deviation in addition to skewness. We compare, for the 150/50 case, the 180^{th} rebalance of the original benchmark and the equivalent synthetic benchmark where \(\varDelta _\gamma = 1\) and \(\varDelta _\sigma = 0.5\). Again, the left panel compares the different benchmark distributions and the right panel compares the corresponding portfolios obtained when solving the model against each benchmark.
5.4.2 Outofsample performance
Table 7 reports outofsample performance statistics. In accordance to our insample results, a higher value for \(\varDelta _{\sigma }\) implies higher returns but also higher risk. As an example, the portfolio obtained when \(\alpha = 0.5\) (150/50) and \(\varDelta _{\sigma } = 0.5\) had the highest final value (4.88) and the highest yearly excess return (37.52%), but also the highest standard deviation of returns (0.01782). As further measure of risk, the maximum drawdown also increased as we increase \(\varDelta _{\sigma }\).
Changing \(\varDelta _{\sigma }\), outofsample performance statistics
Long/short  \(\varDelta _{\gamma }\)  \(\varDelta _{\sigma }\)  Final  Excess over  Sharpe  Sortino  Max draw  Max reco  Daily returns  

value  RFR (%)  ratio  ratio  down (%)  very days  Mean  SD  
FTSE100  1.18  1.57  0.097  0.134  18.83  481  0.00019  0.00985  
100/0  0  0  2.19  15.97  0.855  1.242  14.55  150  0.00071  0.01071 
100/0  1  \(0.1\)  1.92  12.75  0.772  1.122  12.53  150  0.00059  0.00960 
100/0  1  0  2.21  16.16  0.870  1.267  13.44  150  0.00072  0.01064 
100/0  1  0.1  2.43  18.54  0.907  1.317  14.21  150  0.00081  0.01159 
100/0  1  0.3  2.82  22.35  0.931  1.352  17.54  337  0.00095  0.01339 
100/0  1  0.5  3.11  24.92  0.930  1.342  21.96  321  0.00106  0.01479 
120/20  0  0  2.60  20.24  0.998  1.446  12.33  151  0.00086  0.01141 
120/20  1  \(0.1\)  2.31  17.24  0.948  1.373  10.95  150  0.00075  0.01037 
120/20  1  0  2.71  21.27  1.047  1.521  12.32  151  0.00089  0.01139 
120/20  1  0.1  3.23  25.93  1.151  1.679  13.60  159  0.00105  0.01239 
120/20  1  0.3  3.88  30.96  1.170  1.703  17.75  182  0.00123  0.01426 
120/20  1  0.5  4.32  33.99  1.128  1.638  23.43  308  0.00135  0.01606 
150/50  0  0  2.78  21.93  0.983  1.410  14.15  183  0.00093  0.01246 
150/50  1  \(0.1\)  2.45  18.75  0.934  1.345  13.18  193  0.00081  0.01136 
150/50  1  0  2.77  21.86  0.980  1.405  14.48  183  0.00093  0.01247 
150/50  1  0.1  3.20  25.66  1.038  1.481  15.91  150  0.00106  0.01360 
150/50  1  0.3  4.20  33.17  1.122  1.603  17.99  153  0.00132  0.01578 
150/50  1  0.5  4.88  37.52  1.105  1.572  21.35  161  0.00148  0.01782 
We run the models for higher values of \(\varDelta _{\sigma }\), i.e. \(\varDelta _{\sigma }>0.5\) but due to space constraints we do not report the results here. It was observed that, after a certain level (\(\varDelta _{\sigma } > 0.6\)), outofsample returns tend to drop while standard deviation continues to increase. Parameters \(\varDelta _{\sigma }\) and \(\varDelta _{\gamma }\) should be adjusted according to investor constraints and aversion to risk.
In summary, using a benchmark distribution with a slightly lower standard deviation tends to provide lower returns on average but at the same time is a “safer” choice, having better risk characteristics. Increasing standard deviation of the benchmark is a somewhat riskier choice, but it can provide significantly higher returns.
6 Summary and conclusions
This paper is a natural sequel to our earlier work on the topic of enhanced indexation based on SSD criterion and reported in Roman et al. (2013). In that approach we computed SSD efficient portfolios that dominate (if possible) an index which is chosen as the benchmark, that is, the reference distribution. In this paper we introduce a modified / reshaped benchmark distribution with the purpose of obtaining improved SSD efficient portfolios, whose return distributions possess superior (desirable) properties.
The first step in reshaping the original benchmark is to increase its skewness. The amount by which skewness should be increased is specified by the decision maker and is stated as a proportion of the original skewness. Our numerical results show that, by using a reshaped benchmark with higher skewness, we indeed obtain SSD efficient portfolios with better, that is, shorter left tails, as measured insample by CVaR and skewness. On the other hand, if the skewness of the benchmark is excessively increased, the improvement in the left tail of the solution portfolio has adverse effects on the rest of the distribution, with lower returns on average and reduced right tail. We observed that as long as skewness is not increased excessively, the improvement in the left tail comes at no or little marginal cost for the rest of the distribution. We observed that, for an increase in the benchmark skewness of up to 100%, the improvement in the left tail of the solution portfolio is substantial and comes at virtually no cost to the rest of the return distribution. The outofsample results in this case are also considerably better than in the case of using the original benchmark.
We also experimented with a reshaped benchmark by changing the standard deviation by various amounts. We observed in Tables 6, 7 and Fig. 6 that reducing the benchmark standard deviation reflects in portfolios that have lower standard deviations and better tail characteristics, but also worse overall returns, both insample and outofsample (lower mean, skewness, Sharpe and Sortino ratios). Moreover, by increasing the benchmark standard deviation up to 30% of its original value, we obtained more volatile portfolios with better overall return characteristics, both insample and outofsample (higher mean, standard deviation, skewness, Sharpe and Sortino ratios). These results are consistent as they are reported for longonly as well as for longshort combinations (120/20 and 150/50).
We also observed consistent better performance, both insample and outofsample, of longshort portfolios as compared to longonly portfolios.
Overall, our numerical experiments have shown that, by reshaping the return distribution of a financial index and using it as a benchmark in longshort SSD models, it is possible to obtain superior portfolios with better insample and outofsample performance.
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