Novel approaches for portfolio construction using … · Comput Manag Sci (2017) 14:257–280 DOI...

Comput Manag Sci (2017) 14:257–280DOI 10.1007/s10287-017-0274-9

ORIGINAL PAPER

Novel approaches for portfolio construction usingsecond order stochastic dominance

Cristiano Arbex Valle1 · Diana Roman2 ·Gautam Mitra1,3

Received: 23 March 2016 / Accepted: 16 January 2017 / Published online: 2 February 2017© The Author(s) 2017. This article is published with open access at Springerlink.com

Abstract In the last decade, a few models of portfolio construction have been pro-posed 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 theSSD criterion. The benchmark distribution naturally plays an important role since dif-ferent benchmarks lead to very different portfolio solutions. In this paper we describea novel concept of reshaping the benchmark distribution with a view to obtaining port-folio solutions which have enhanced return distributions. The return distribution of theconstructed portfolio is considered enhanced if the left tail is improved, the downsiderisk is reduced and the standard deviation remains within a specified range. We extendthis approach from long only to long-short strategies which are used by many hedgefund and quant fund practitioners. We present computational results which illustrate(1) how this approach leads to superior portfolio performance (2) how significantlybetter performance is achieved for portfolios that include shorting of assets.

Keywords Portfolio optimisation · Stochastic dominance · Reference distribution ·Left tail · Downside risk

B Diana [email protected]

Cristiano Arbex [email protected]

Gautam [email protected]

1 Optirisk Systems, One Oxford Road, Uxbridge UB9 4DA, UK

2 Brunel University, Kingston Lane, Uxbridge UB8 3PH, UK

3 University College London (UCL), London, UK

123

http://crossmark.crossref.org/dialog/?doi=10.1007/s10287-017-0274-9&domain=pdf

http://orcid.org/0000-0002-6716-593X

258 C. A. Valle et al.

1 Introduction

Second order stochastic dominance (SSD) has been long recognised as a rationalcriterion 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 andRuszczynski 2003, 2006;Roman et al. 2006) for portfolio constructionby researchers working in this domain. The approach described in Dentcheva andRuszczynski (2003, 2006) first considers a reference (or benchmark) distribution andthen computes a portfolio which dominates the benchmark distribution by the SSDcriterion. In Roman et al. (2006) a multi-objective optimisation model is introduced inorder to achieve SSD dominance. This model is both novel and usable since, when thebenchmark solution itself is SSD efficient or its dominance is unattainable, it finds anSSD efficient portfolio whose return distribution comes close to the benchmark in asatisficing 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 ofmodellingas well as that of computational solution.

These models start from the assumption that a reference (benchmark) distributionis available. It was shown in Roman et al. (2006) that the reference distribution playsa crucial role in the selection process: there are many SSD efficient portfolios and thechoice of a specific one depends on the benchmark distribution used. SSD efficiencydoes not necessarily make a return distribution desirable, as demonstrated by theoptimal portfolio with regards to maximum expected return. It was shown in Romanet al. (2006) that this portfolio is SSD efficient - however, it is undesirable to a largeclass of decision-makers.

In the last two decades, quantitative analysts in the fund management industryhave actively debated about the benefit of active fund management in contrast topassive investment. Passive investment equates to holding a portfolio determined bythe constituents of a chosenmarket index. Active fundmanagers are engaged in findingportfolios which provide better return than that of a passive index portfolio. Set againstthis background the index is a natural benchmark (reference) distribution which anactive fund manager would like to dominate. There have been several papers under thetopic of “enhanced indexation” (di Bartolomeo 2000) which discuss alternative waysof doing better than the passive index portfolio. It has been shown empirically thatreturn distributions of financial indices are SSD dominated (Post 2003; Kuosmanen2004; Post and Kopa 2013; Kopa and Post 2015; Post et al. 2015).

In Roman et al. (2013) we introduced SSD-based models for enhanced indexationand reported encouraging practical results. An essential aspect of our approach toportfolio construction can be articulated by the qualitative statement “reduction of thedownside risk and improvement of the upside potential”. This can be translated asfinding return distributions with high expected value and skewness, meaning a lefttail 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 choosean index as the benchmark, do not necessarily have return distributions with a short lefttail, high skewness and controlled standard deviation. Research effort in this direction

123

Novel approaches for portfolio construction using second… 259

include SSD based models in which, by appropriately selecting model parameters, theleft 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 questionto ask: how should we choose the reference distribution in SSD models such that theresulting 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) standarddeviation within a range?

The contributions of this paper are summarised as follows:

(a) we propose a method of reshaping, or enhancing, a given (reference) distribu-tion, namely, that of a financial index, in order to use it as a benchmark in SSDoptimisation models;

(b) we formulate and solve SSD models that include long-short strategies which areestablished financial practice to cope with changing financial regimes (bull andbear markets);

(c) we investigate empirically the in-sample and out-of sample performance of port-folios obtained using enhanced benchmarks and long-short strategies.

The rest of the paper is organised in the following way. In Sect. 2 we presentportfolio optimisationmodels based on the SSD concept and the role of the benchmark/ reference distribution. Themethod of reshaping a benchmark distribution is presentedin Sect. 3. In Sect. 4we extend the long-only formulation presented in Sect. 2 to includelong-short strategies as discussed in (b) above. Section 5 contains the results of ournumerical experiments. We compare the in-sample and out-of sample performanceof portfolios obtained in SSD models, using an original benchmark and a reshapedbenchmark. The comparison is made in a long-only setting as well as in the context ofvarious long-short strategies. A summary and our conclusions are presented in Sect. 6.

2 Portfolio optimisation using SSD

We consider a portfolio selection problem with one investment period. Let n denotethe number of assets into which we may invest. A portfolio x = (x1, . . . xn) ∈ R

n

represents the proportions of the portfolio value invested in the available assets. Letthe n-dimensional random vector R = (R1, . . . , Rn) denote the returns of the differentassets at the end of the investment period.

It is usual to consider the distribution of R as discrete, described by the realisationsunder a finite number of scenarios S; scenario j occurs with probability p j wherep j > 0 and p1+· · ·+ pS = 1. Let us denote by ri j the return of asset i under scenarioj . The random return of portfolio x is denoted by Rx , with Rx := x1R1 + · · · xn Rn .We remind that second-order stochastic dominance (SSD) is a preference relation

among random variables (representing portfolio returns) defined by the followingequivalent conditions:

(a) E(U (R)) ≥ E(U (R′)) holds for any nondecreasing and concave utility functionU for which these expected values exist and are finite.

123


(b) E([t − R]+) ≤ E([t − R′]+) holds for each t ∈ R.(c) Tailα(R) ≥ Tailα(R′) holds for each 0 < α ≤ 1, where Tailα(R) denotes the

unconditional expectation of the least α ∗ 100% of the outcomes of R.

If the relations above hold, the random variable R is said to dominate the randomvariable R′ with respect to second-order stochastic dominance (SSD); we denote thisby R �SSD R′. The strict relation R �SSD R′ is similarly defined, if, for example, inaddition to (c), there exists α in (0,1) such that Tailα(R) > Tailα(R′).

The equivalence of the above relations is well known since the works of Whimoreand Findlay (1978) and Ogryczak and Ruszczynski (2002). From the first relation,the importance of SSD in portfolio selection can be clearly seen: it expresses thepreference of rational and risk-averse decision makers.

Remark 1 The definition of Tailα(R) is an informal definition. For a formal definition,quantile functions can be used. Denote by FR the cumulative distribution functionof a random variable R. If there exists t such that FR(t) = α then Tailα(R) =αE(R|R ≤ t)—which justifies the informal definition. For the general case, let usdefine the generalised inverse of FR as F−1

R (α) := inf{t |FR(t) ≥ α} and the secondquantile function as F−2

R (α) := ∫ α

0 F−1R (β)dβ and F−2

R (0) := 0.With these notations,Tailα(R) := F−2

R (α).

Let X ⊂ Rn denote the set of the feasible portfolios, we assume that X is a bounded

convex polyhedron. A portfolio x� is said to be SSD-efficient if there is no feasibleportfolio x ∈ X such that Rx �SSD Rx� .

Recently proposed portfolio optimisation models based on the concept of SSDassume that a reference (benchmark) distribution Rref is available. Let τ̂ be the tails ofthe benchmark distribution at confidence levels 1

S , . . . , SS ; that is, τ̂ = (τ̂1, . . . , τ̂S) =(

Tail 1SRref, . . . ,Tail S

SRref

).

Assuming equiprobable scenarios as in Roman et al. (2006, 2013), Fábián et al.(2011a, b), the model in Fábián et al. (2011b) optimises the worst difference betweenthe “scaled” tails of the benchmark and of the return distribution of the solutionportfolio; the α- “scaled” tail is defined as 1

αTailα(R) . The decision variables are the

portfolio weights x1, . . . , xn and V = min1≤s≤S1s

(Tail s

S(Rx ) − τ̂s

), representing the

worst partial achievement of the differences between the scaled tails of the portfolioreturn and the scaled tails of the benchmark. The scaled tails of the benchmark are( S1 τ̂1,

S2 τ̂2 . . . , S

S τ̂S). Using a cutting plane representation (Fábián et al. 2011a) themodel can be written as:

maxV (1)

subject to:

n∑

i=1

xi = 1 (2)

V ≤ 1

s

∑

j∈Js

n∑

i=1

ri j xi − S

sτ̂s ∀Js ⊂ {1, . . . , S}, |Js | = s, s = {1, . . . , S} (3)

123


V ∈ R, xi ∈ R+, ∀i ∈ {1 . . . n} (4)

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 (Fábiánet al. 2011a) adds cuts at each iteration until optimality is reached; it is shown that inpractice, only a few cuts are needed. For example, all models with 10,000 scenariosof assets returns were solved with less than 30 iterations. For more details, the readeris referred to Fábián et al. (2011a).

Remark 3 In case that optimisation of the worst partial achievement has multipleoptimal 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. Themodel proposed in Roman et al. (2006) has a slightly different objective functionthat included a regularisation term, in order to guarantee SSD efficiency for the casemultiple optimal solutions. This termwas dropped in the cutting plane formulation, theadvantage of this being huge decrease in computational difficulty and solution time;just as an example, models with tens of thousands of scenarios were solved withinseconds, while the original model formulation in Roman et al. (2006) could only dealwith a number of scenarios in the order of hundreds. In Fábián et al. (2011a), extensivecomputational results are reported. For relatively small datasets, SSD models weresolved with both the cutting plane formulation and the original formulation includingthe regularisation term; in all instances, both formulations led to the same optimalsolutions. For more details, the reader is referred to Fábián et al. (2011a).

The tails of the benchmark distribution (τ̂1, . . . , τ̂S) are the decision-maker’s input.When the benchmark is not SSD efficient, the solution portfolio has a return distri-bution that improves on the benchmark until SSD efficiency is achieved. In case thebenchmark is SSD efficient, the model finds the portfolio whose return distribution isthe benchmark. For instance, if the benchmark is the return distribution of the assetwith the highest expected return, the solution portfolio is that where all capital isinvested in this asset.

“Unattainable” reference distributions are discussed in Roman et al. (2006), wherethe (SSD efficient) solution portfolio has a return distribution that comes as closeas possible to dominating the benchmark; this is obtained by minimising the largestdifference between the tails of these two distributions. However, simply setting “hightargets” (i.e. a possibly unrealistic benchmark) does not solve the problem of findinga portfolio with a “good” return distribution, e.g. one having a short left tail/highskewness and controlled standard deviation.

In recent research, the most common approach is to set the benchmark as the returndistribution of a financial index. This is natural since discrete approximations for thischoice can be directly obtained from publicly available historical data, and also dueto the meaningfulness of interpretation - it is common practice to compare / makereference 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 inmost cases not SSD efficient. While it is safe to say that generally a portfolio that

123


dominates an index with relation to SSD can be found, there is no guarantee that thisportfolio will have desirable properties.

In this work, we use the distribution of a financial index as a starting point; weenhance it in the sense of increasing skewness by a decision-maker’s specified amountwhile keeping standard deviation within a (decision-maker specified) range.

3 Reshaping the reference distribution

We propose a method of reshaping an original reference distribution and achieving asynthetic (improved) reference distribution.

To clarify what we mean by improved reference distribution, let us consider theblue area in Fig. 1 to be the density curve of the original reference distribution, in thisexample closely symmetrical and with a considerably long left-tail.

The pink area in the figure represents the density curve of what we consider tobe an improved reference distribution. Desirable properties include a shorter left tail(reduced probability of large losses), which translates into higher skewness, and ahigher expected return, which is equivalent to a higher mean. A smaller standarddeviation is not necessarily desirable, as it might limit the upside potential of highreturns. Instead, we require the standard deviation of the new distribution to be withina specified range from the standard deviation of the original distribution.

We hereby introduce a method for transforming the original reference distributioninto a synthetic reference distribution given target values for the first three statisticalmoments (mean, standard deviation and skewness).

Let the original reference distribution be represented by a sample Y = (y1, . . . , yS)with mean μY , standard deviation σY and skewness γY .

Fig. 1 Density curves for the original and improved reference distributions

123


Given target values μT , σT and γT , our goal is to find a distribution Y ′ with valuesy′s, s = 1, . . . , S, such that μ′

Y = μT , σY ′ = σT and γY ′ = γT .In statistics, this problem is related to test equating. It is commonly found in stan-

dardized testing, where multiple test forms are needed because examinees must takethe test at different occasions and one test form can only be administered once to ensuretest security. However, test scores derived from different forms must be equivalent.Let us consider two test forms, say, Form V and FormW . It is generally assumed thatthe examinee groups that take test forms V andW are sampled from the same popula-tion, and differences in score distributions are attributed to form differences (e.g. moredifficult questions in V than in W ). Equating forms V and W involves modifying Vscores so that the transformed V scores have the same statistical properties as W .

If our intention was solely to preserve the first two moments, a linear equatingmethod would be appropriate. Linear equating (Kolen and Brennan 1995) takes theform:

Y ′ = σT

[Y − μY

σY

]

+ μT

In order, however, to preserve the first three moments, we make use of a quadraticcurve equating method proposed by Wang and Kolen (1996). In selecting a nonlinearequating function, the authors aimed for a method that was more flexible than linearequating and would still preserve its beneficial properties such as using statistics withsmall random errors that are computationally simple. The method works as follows:

1. Step 1: Arbitrarily define μT , γT and σT .2. Step 2: Using the single-variable Newton–Raphson iterative method (Press et al.

(1988), pp. 362–367), find d so that Y + dY 2 has skewness γ(Y+dY 2) = γT . Usingthe standard skewness formula for a discrete sample, the skewness of the originalreference distribution γY is given by:

γY =1n

∑ni=1(yi − μY )3

[1

n−1

∑ni=1(yi − μY )2

]3/2

In order to find d we need to solve the equation:

1n

∑ni=1

(yi + dy2i − ( 1n

∑nj=1 y j + dy2j )

)3

[1

n−1

∑ni=1

(yi + dy2i − ( 1n

∑nj=1 y j + dy2j )

)2]3/2− γT = 0

3. Step 3: Let g = σTσY+dY2

. Then g(Y + dY 2) will have γg(Y+dY 2) = γT and

σg(Y+dY 2) = σT since a linear transformation (multiplication of constant g) doesnot change the skewness of a distribution.

4. Step 4: Let h = μT −μg(Y+dY 2). Then Y′ = h+g(X+dX2)will haveμY ′ = μT ,

γY ′ = γT and σY ′ = σT since adding a constant does not change the skewness orstandard deviation of a distribution.

123


5. Step 5: We then find Y ′ by computing:

y′s = gdy2s + gys + h, ∀s = 1, . . . , S

To apply this method we need to define values for γT , σT and μT . In this specificcontext, these values should not be independent from the original distribution. Forinstance, if the skewness of the return distribution of a financial index is very low,simply setting a very high value for the target skewness might render it impossibleto find a combination of assets (which are to some extent correlated to the originalreference distribution) that dominates the synthetic distribution.

In preliminary tests, we noticed that large differences in values for μT have littleimpact in out-of-sample results. Therefore, in our computation experiments, we setμT = μY and we introduce two parameters to define the amount by which the targetfor the other moments differ from the original values: Δγ and Δσ , both defined in R.Given these parameters, γT and σT are defined as:

γT = γY + |γY |Δγ

σT = σY + σYΔσ

4 Long-short modelling

When short-selling is allowed, the amount available for purchases of stocks in longpositions is increased. Suppose we borrow from an intermediary a specified numberof units of asset i (i = 1, . . . , n), corresponding to a proportion x−

i of capital. We sellthem immediately in the market and hence have a cash sum of (1 + ∑n

i=1 x−i )C to

invest in long positions; where C is the initial capital available.In long-short practice, it is common to fix the total amount of short-selling to a

pre-specified proportion α of the initial capital. In this case, the amount available toinvest in long positions is (1+α)C . A fund that limits their exposure with a proportionα = 0.2 is usually referred to as a 120/20 fund. For modelling this situation, to eachasset i ∈ 1, . . . , 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 twobinary variables z+i , z−i that indicate whether there is investment in long or shortpositions in asset i . For example, if 10% of the capital is shorted in asset i , we writethis as x+

i = 0, x−i = 0.1, z+i = 0, z−i = 1.

As in Roman et al. (2013), V = min1≤s≤S1s

(Tail s

S(RT (x+ − x−)) − τ̂s

)denotes

the worst partial achievement; the scaled long/short reformulation of the achievement-maximisation problem is written as:

maxV (5)

subject to:

n∑

i=1

x+i = 1 + α (6)

123


n∑

i=1

x−i = α (7)

x+i ≤ (1 + α)z+i ∀i ∈ N (8)

x−i ≤ αz−i ∀i ∈ N (9)

z+i + z−i ≤ 1 ∀i ∈ N (10)

s

SV + τ̂s ≤ 1

S

∑

j∈Js

n∑

i=1

ri j (x+i − x−

i ) ∀Js ⊂ {1, . . . , S}, |Js | = s, s = {1, . . . , S}

(11)

V ∈ R, x+i , x−

i ∈ R+, z+i , z−i ∈ {0, 1}, ∀i ∈ N (12)

To solve this formulation, we implemented a branch-and-cut algorithm (Padbergand Rinaldi 1991), which modifies the basic branch-and-bound strategy by attemptingto identify new inequalities before branching a partial solution. Since the branch-and-bound algorithm begins with a relaxed formulation of the problem, a solution cannotbe accepted as candidate for branching unless it violates no constraints of type (11).In order to identify violated constraints we employ the separation algorithm proposedby Fábián et al. (2011a).

5 Computational results

5.1 Motivation, dataset and methodology

The aims of this computational study are:

1. To investigate the effect of using a benchmark distribution, reshaped by modi-fying skewness and/or standard deviation, in SSD-based portfolio optimisationmodels; the return distributions of the resulting portfolios, as well as their out-of-sample performance, are compared to those of portfolios obtained using theoriginal benchmark.

2. To investigate the performance of various long-short strategies in comparison withthe long only strategy, as used in SSD-based models with original and reshapedbenchmarks.

We use real-world historical daily data (closing prices) taken from the universe ofassets defined by the Financial Times Stock Exchange 100 (FTSE100) index over theperiod 09/10/2007 to 07/10/2014 (1765 trading days). The data was collected fromThomson Reuters Datastream (2014) and adjusted to account for changes in indexcomposition. This means that our models use no more data than was available atthe time, removing susceptibility to the influence of survivor bias. For each asset wecompute the corresponding daily rates of return.

The original benchmark distribution is obtained by considering the historical dailyrates of return of FTSE100 during the same time period.We implement models (2)–(4)and (5)–(12) for different values of α,Δγ and Δσ .

123


We used an Intel(R) Core(TM) i5-3337U CPU@ 1.80 GHz with 6GB of RAM andLinux as operating system. The Branch-and-cut algorithm was written in C++ and thebacktesting framework was written in R (R Core Team 2015); we used CPLEX 12.6(IBM 2015) as mixed-integer programming solver.

The methodology we adopt is successive rebalancing over time with recent histor-ical data as scenarios. We start from the beginning of our data set. Given in-sampleduration of S days, we decide a portfolio using data taken from an in-sample periodcorresponding to the first S+1 days (yielding S daily returns for each asset). The port-folio is then held unchanged for an out-of-sample period of 5days. We then rebalance(change) our portfolio, but now using the most recent S returns as in-sample data. Thedecided portfolio is then held unchanged for an out-of-sample period of 5days, andthe process repeats until we have exhausted all of the data. We set S = 564 (approxi-mately the number of trading days in 2.5years) ; the total out-of-sample period spansalmost 5years (January 2010–October 2014).

Once the data has been exhausted we have a time series of 1201 portfolio returnvalues for out-of-sample performance, here from period 565 (the first out-of-samplereturn value, corresponding to 04/01/2010) until the end of the data.

5.2 Long-short and long-only comparison

We test α = 0, α = 0.2, α = 0.5 and α = 1.0, thus we consider 100/0 (long-only), 120/20, 150/50 and 200/100 portfolios. The benchmark distribution is that ofFTSE100. Given the portfolio holding period of 5 days, during the out-of-sampleevaluation period there are a total of 240 rebalances. For each rebalance, we assign atime limit of 60 s.

5.2.1 In-sample analysis

Table 1 shows in-sample statistics regarding optimal solution values. Under Optimalobject value, three columns are reported: (1)Mean, showing the average of the optimalobjective values in each rebalance; (2) Min and (3) Max, showing respectively theminimum and maximum optimal objective values in each rebalance.

The first thing we note is that, with the exception of 200/100 portfolios, in allrebalances a positive optimal value was obtained, which means that the solver founda portfolio that dominates the index return distribution with respect to SSD. Up to150/50, all rebalances were solved to optimality within 60 s. This is not the case,however, for the 200/100 portfolios. Optimality was not proven within the time limitfor about 15% (35 out of 240) of all 200/100 rebalances. In some cases we were notable to find a portfolio that dominates its corresponding benchmark: in Table 1, theminimum solution value found for 200/100 was −0.000055.

Despite that, overall, we observe that adding shorting improves the quality of in-sample solutions, with the average, minimum and maximum of optima being higherwhen α is higher (with the exception of the minimum solution value for 200/100 asstated above).

123


Table 1 Long only andlong-short, in-sample 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

Table 2 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 optimalportfolios—also reported are minimum and maximum numbers, for each value of α.Statistics are shown for assets held long, short and also for the complete set. From thetable we can see that the addition of shorting tends to increase the number of stockspicked. This is expected, since the higher α is, the higher is the exposure. For instance,if α = 0.5 we have 0.5C in repayment obligations and 1.5C in long positions, havinga total exposure of 2C in different assets.

However, even in long/short models, the cardinality of the optimal portfolios is nothigh (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 tobe unnecessary.

5.2.2 Out-of-sample performance

Figure 2 shows graphically the performance of each of the four strategies (100/0,120/20, 150/50 and 200/100) as well as that of the financial index (FTSE100), asrepresented by their actual returns over the out-of sample period January 2010 toOctober 2014.

From the figure it is clear that portfolios with shorting (α > 0) achieved better per-formance than the long-only portfolio, although it is also apparent that the variabilityof returns was larger. This can be confirmed by analysing Table 3, which presentsseveral out-of-sample statistics. The meaning of each column is outlined below:

– Final value: Normalised final value of the portfolio at the end of the out-of-sampleperiod.

– Excess over RFR (%): Annualised excess return over the risk free rate. ForFTSE100 we used a flat yearly risk free rate of 2%.

123


Fig. 2 Long/short out-of-sample performance, 2010–2014

Table 3 Long/short, out-of-sample 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

– Sharpe ratio: Annualised Sharpe ratio (Sharpe 1966) of returns.– Sortino ratio: Annualised Sortino ratio (Sortino and Price 1994) of returns.– Max drawdown (%): Maximum peak-to-trough decline (as percentage of the peakvalue) during the entire out-of-sample period.

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

– Daily returns—Mean: Mean of out-of-sample daily returns.– Daily returns—SD: Standard deviation of out-of-sample daily returns.

As we increase α up to 0.5, both the mean and the standard deviation of the dailyreturns increase. As a consequence, although 150/50 achieved better returns than120/20, the latter obtained higher Sharpe and Sortino ratios, as well as a lower max-imum drawdown. Adding shorting seems to bring better performance at the expenseof greater risk.

The overall performance of 200/100 portfolios was better than the long-only port-folio, but worse than 120/20 and 150/50. The 200/100 portfolio is more volatile (bothin-sample and out-of-sample), which may be the reason for its lower final value when

123


compared to the other cases. Moreover, some rebalances were not solved within 60s,yielding suboptimal solutions. In the next sections, where we analyse the effects ofreshaping the reference distribution, we ommit 200/100 results as they show similarbehaviour 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 in-sample and out-of-sample results. For α = 0, α = 0.2, α = 0.5, we test the effects of different valuesof Δγ , more specifically, we test Δγ = [0, 1, 2, 3, 4, 5]. In all tests in this section,Δσ = 0, that is, the standard deviation is unchanged.

Setting Δγ = 0 is equivalent to optimising with the original reference distribution.In the rest of the cases, the skewness is increased to γT = γY +(|γY |Δγ ). For example,if Δγ = 1 and γY < 0 then γT = 0. If Δγ = 1 and γY > 0 then γT = 2γY .

5.3.1 In-sample results

Table 4 presents in-sample results for different values of Δγ . Results are reporteddifferently when compared to those in Sect. 5.2.1. A direct comparison of optimalsolution values is no longer valid since the models are being optimised over different(synthetic) reference distributions. We therefore report other in-sample statistics, suchas (i) Mean, (ii) SD and (iii) Skewness: average in-sample mean, standard deviationand skewness of optimal return distributions over all 240 rebalances. Other reportedin-sample statistics are:

– (iv) ScTailα: the α- “scaled” tail, defined as in Sect. 2 as the conditional expectationof the least α ∗ 100% of the outcomes.

– (v) EPρ : Expected conditional profit at ρ% confidence level, equivalent to CVaRρ

but calculated from the right tail of the distribution. Let Sρ = �S(ρ/100)�. EPρ

is defined as 1S−Sρ−1

∑Ss=Sρ r

ps .

Also reported are the average in-sample (vi) Skewness, (vii) ScTailα and (viii) EPρ

for the synthetic benchmark. As only Δγ was changed, we do not report the averagein-sample mean and standard deviation for the benchmark as these values match theoriginal in all cases.

We also compare in-sample solutions in terms of their SSD relation. Let Y bethe optimal portfolio (with worst partial achievement V) that solves the enhancedindexation model and dominates the original reference distribution with respect toSSD. Accordingly, let Y ′ be the optimal porfolio (with worst partial achievement V ′)that dominates the synthetic (improved) reference distribution. Let also RY = [rY1 ≤· · · ≤ rYS ] and RY ′ = [rY ′

1 ≤ · · · ≤ rY′

S ] be the ordered set of in-sample returns for Yand Y ′.

It is clear that if V = V ′, Y �SSD Y ′ and Y ′�SSD Y . If, for instance, Y ′ �SSD Y ,

then Y ′ would have been chosen instead of Y as the optimal solution of the enhancedindexation model with the original reference distribution since V ′ > V . If V = V ′,

123


it is possible, albeit unlikely, that Y ′ �SSD Y or Y �SSD Y ′ - that could be the casewhere the mixed-integer programming model had multiple optima.

Assuming that most likely neither Y �SSD Y ′ nor Y ′ �SSD Y , we can measurewhich solution is “closer” to dominating the other. For Y ′, we compute, for eachrebalance,SY ′ ⊂ {1, ..., S}where s ∈ SY ′ if Tail s

SRY ′

> Tail sSRY , i.e. |SY ′ | represents

the number of times that the unconditional expectation of the least s scenarios of Y ′ isgreater than the equivalent for Y .We also compute, for each rebalance,SY ⊂ {1, ..., S}where s ∈ SY if Tail s

SRY > Tail s

SRY ′

.The following columns are included in Table 4:

– (ix) |SY ′ |: Mean value of |SY ′ | over all rebalances.– (x) |SY |: Mean value of |SY | over all rebalances.

From Table 4, we observe that increasing Δγ increases the in-sample skewnessand also ScTail05 of the optimal distributions—which was expected, since increas-ing skewness reduces the left tail. We can also observe that the mean and standarddeviation tend to decrease as we increase the skewness in the synthetic distribution.This decrease is very small or non-existent in the case of smaller deviations from theoriginal skewness e.g. γ = 1 but more pronounced for larger values of γ .

The skewness of the solution portfolios, although clearly increasing in line withincreasing the skewness of the benchmark, is considerably below the skewness valuesset by the improved benchmark. The solution portfolios have on average much higherexpected value and less variance than the improved benchmark.

The statistics for EP95 reach a peak somewhere between Δγ = 1 and Δγ = 2, andtheir values decrease for higher Δγ . This might be due to the synthetic distributionhaving “unrealistic” properties if its third moment differs too much from that of theoriginal distribution.We do observe, nevertheless, that increasing skewness alsomakesin-sample portfolios based on synthetic distributions “closer” to dominating thosebased on the original distribution, since |SY ′ | increases and |SY | decreases as Δγ

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 Δγ .Figure 3 highlights the differences between optimising over the original benchmark

and over the synthetic benchmark. We compare, for the 150/50 case, histograms forthe original benchmark distribution from the 180th rebalance (out of 240) and theequivalent synthetic benchmark distribution, where Δγ = 5 and Δσ = 0. We chosethe 180th rebalance as the corresponding figures approximate the observed averageproperties. The histograms are in line with the average properties observed in Table4. The left panel shows the difference in the benchmark distributions; the syntheticbenchmark distribution has a reduced left tail, especially concerning the worst casescenarios, at the expense of having the “peak” slightly towards the left, as compared tothe original benchmark.The right panel compares the returndistributions of the optimalportfolios, obtained using the original and the synthetic benchmark. The portfoliobased on the synthetic benchmark has a reduced left tail, at the expense of a marginallylower mean.

123


Table4

Chang

ing

Δγ:average

values

ofin-sam

plestatistic

sforreturn

distributio

nsof

optim

alpo

rtfolio

sandof

reshaped

benchm

arks;the

averagebenchm

arkmeanand

standard

deviationare0.00

028and0.01

247respectiv

ely

Lon

g/short

Δγ

In-sam

plepo

rtfolio

performance

In-sam

plebenchm

arkperformance

Dom

inance

Mean

SDSk

ewness

ScTail 0

5EP 9

5Sk

ewness

ScTail 0

5EP 9

5|S

Y′|

|SY|

100/0

00.00

152

0.01

201

−0.041

13−0

.025

970.02

719

−0.038

61−0

.028

980.02

790

−–

100/0

10.00

151

0.01

200

0.00

006

−0.025

730.02

731

0.10

116

−0.028

330.02

849

256.80

307.20

100/0

20.00

149

0.01

195

0.03

633

−0.025

420.02

728

0.24

094

−0.027

660.02

906

263.38

300.62

100/0

30.00

145

0.01

183

0.06

341

−0.024

980.02

705

0.38

071

−0.026

980.02

961

300.58

263.42

100/0

40.00

139

0.01

165

0.08

358

−0.024

480.02

664

0.52

049

−0.026

280.03

015

331.70

232.30

100/0

50.00

129

0.01

136

0.08

703

−0.023

860.02

588

0.66

026

−0.025

560.03

067

362.88

201.12

120/20

00.00

198

0.01

195

−0.034

03−0

.025

080.02

756

−0.038

61−0

.028

980.02

790

−–

120/20

10.00

198

0.01

196

0.00

364

−0.024

920.02

771

0.10

116

−0.028

330.02

849

240.76

323.24

120/20

20.00

197

0.01

192

0.04

218

−0.024

640.02

773

0.24

094

−0.027

660.02

906

272.75

291.25

120/20

30.00

194

0.01

184

0.07

870

−0.024

240.02

763

0.38

071

− 0.026

980.02

961

295.34

268.66

120/20

40.00

191

0.01

174

0.11

291

−0.023

780.02

743

0.52

049

−0.026

280.03

015

313.20

250.80

120/20

50.00

186

0.01

160

0.14

315

−0.023

270.02

711

0.66

026

−0.025

560.03

067

328.19

235.81

150/50

00.00

240

0.01

194

−0.034

56−0

.024

450.02

806

−0.038

61−0

.028

980.02

790

−–

150/50

10.00

240

0.01

194

0.00

500

−0.024

280.02

824

0.10

116

−0.028

330.02

849

270.30

293.70

150/50

20.00

239

0.01

191

0.04

945

−0.023

960.02

832

0.24

094

−0.027

660.02

906

281.27

282.73

150/50

30.00

237

0.01

186

0.09

315

−0.023

550.02

831

0.38

071

−0.026

980.02

961

292.37

271.63

150/50

40.00

233

0.01

178

0.13

320

−0.023

090.02

820

0.52

049

−0.026

280.03

015

298.49

265.51

150/50

50.00

229

0.01

167

0.16

910

−0.022

600.02

798

0.66

026

−0.025

560.03

067

305.06

258.94

123


Fig. 3 Left panel: original benchmark (Δγ = 0, Δσ = 0) and a synthetic benchmark with added skew-ness (Δγ = 5,Δσ = 0). Right panel: the distribution of 150/50 optimal portfolios obtained using thebenchmarks on the left

Fig. 4 Changing Δγ , out-of-sample performance, 2010–2014


Out-of-sample performance statistics for variations in Δγ are presented in both Table5 and Fig. 4. Regarding the figure, we choose 120/20 portfolios and show the perfor-mance graphic for the index (FTSE100), the portfolio based on the original referencedistribution and portfolios based on synthetic distributions where Δγ = {1, 5}. Theout-of-sample performance of the other 120/20 tested parameters (Δγ = {2, 3, 4})was relatively similar to the ones displayed, hence for readability we did not includethem in the graphic. The table, however, includes the full results.

We observe that the volatility of out-of-sample returns tend to reduce aswe optimiseover synthetic distributions with higher skewness values. The mean returns and excessreturns over the risk free rate tend to increase slightly (up to somewhere betweenΔγ = 1 andΔγ = 2) and then start to drop. These results are consistent with observed

123


Table5

Chang

ing

Δγ,o

ut-of-sampleperformance

statistics

Lon

g/short

Δγ

Final

Excessover

Sharpe

Sortino

Max

draw

-Max

reco-

Daily

returns

value

RFR

(%)

ratio

ratio

down(%

)very

days

Mean

SD

FTSE

100

1.18

1.57

0.09

70.13

418

.83

481

0.00

019

0.00

985

100/0

02.19

15.97

0.85

51.24

214

.55

150

0.00

071

0.01

071

100/0

12.21

16.16

0.87

01.26

713

.44

150

0.00

072

0.01

064

100/0

22.17

15.67

0.85

61.24

513

.69

150

0.00

070

0.01

051

100/0

32.09

14.73

0.81

81.19

013

.54

150

0.00

067

0.01

039

100/0

42.00

13.69

0.77

81.13

113

.07

150

0.00

063

0.01

018

100/0

51.95

13.10

0.76

81.11

212

.66

150

0.00

061

0.00

990

120/20

02.60

20.24

0.99

81.44

612

.33

151

0.00

086

0.01

141

120/20

12.71

21.27

1.04

71.52

112

.32

151

0.00

089

0.01

139

120/20

22.72

21.39

1.06

21.54

511

.82

160

0.00

090

0.01

129

120/20

32.69

21.09

1.05

31.53

212

.05

162

0.00

089

0.01

124

120/20

42.62

20.44

1.03

31.50

112

.05

178

0.00

087

0.01

113

120/20

52.57

19.91

1.02

81.49

712

.22

183

0.00

085

0.01

092

150/50

02.78

21.93

0.98

31.41

014

.15

183

0.00

093

0.01

246

150/50

12.77

21.86

0.98

01.40

514

.48

183

0.00

093

0.01

247

150/50

22.78

21.96

0.98

81.42

014

.64

184

0.00

093

0.01

241

150/50

32.73

21.53

0.97

81.40

614

.70

185

0.00

091

0.01

232

150/50

42.71

21.33

0.97

81.40

714

.35

196

0.00

091

0.01

221

150/50

52.73

21.46

0.99

91.43

713

.97

184

0.00

091

0.01

203

123


in-sample statistics. Changing skewness had little impact in terms of drawdown andrecovery from drops. While returns do tend to decrease for higher value of Δγ , itsreduced standard deviation might appeal to risk-averse investors. The best values ofSharpe and Sortino ratios are generally obtained when Δγ = 1, 2 or 3. In particular,the 120/20 strategy with Δγ =1 or 2 seems to have the best risk-return characteristics.

5.4 Reshaping the reference distribution: modified standard deviation

In this section, we test how portfolios behave for different values ofΔσ . Since increas-ing skewness slightly seems to be the best choice, according to our results in theprevious section, for the next tests we set Δγ = 1.

We report results for Δσ = [−0.1, 0, 0.1, 0.3, 0.5]; for example, if Δσ = 0.1,the synthetic reference distribution will have a 10% increase in its standard deviationwhen compared to the original reference distribution.

5.4.1 In-sample results

Table 6 shows in-sample results for different values of Δσ . We also display resultsfor the original reference distribution (when Δσ = Δγ = 0). Apart from this case,onlyΔσ is changed. Due to this, when compared to Table 4, we replaced the Skewnesscolumn under In-sample benchmark performance by SD, that is, the average in-samplebenchmark standard deviation.

We can see that increasing Δσ increases the standard deviation of the return distri-butions of optimal portfolios, but also increases their mean and skewness at the sametime.

The standard deviation of the return distribution of optimal portfolios follows thesame pattern as set by the improved benchmarks, in the sense that it increases (ordecreases) 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 theircorresponding benchmarks.

For Δσ = −0.1, we obtain portfolios whose return distributions have better riskcharacteristics in the form of lower standard deviation and higher tail value. On theother hand, their mean returns, EP95 and even skewness are lower. As we increaseΔσ ,ScTail05 gets lower, but EP95 gets higher.

We also observe that, when increasing Δσ , the portfolios obtained based on theoriginal reference distribution are “closer” to dominating the portfolio obtained viathe synthetic benchmark.

Once again, we do not report the cardinality of the optimal portfolios as very littlealteration was observed due to changes in Δσ .

Similarly to Figs. 3 and 5 shows the difference between optimising over the originaland synthetic benchmarks, this time for varying standard deviation in addition to skew-ness. We compare, for the 150/50 case, the 180th rebalance of the original benchmarkand the equivalent synthetic benchmark where Δγ = 1 and Δσ = 0.5. Again, the left

123


Table6

Chang

ing

Δσ:A

verage

in-sam

plestatistic

sforreturn

distributio

nsof

optim

alpo

rtfolio

sandof

improved

benchm

arks;the

averagebenchm

arkmeanis0.00

028;

the

averagebenchm

arkskew

ness

is−0

.038

61forΔ

γ=

0and0.10

116forΔ

γ=

1

Lon

g/short

Δγ

Δσ

In-sam

plepo

rtfolio

performance

In-sam

plebenchm

arkperformance

Dom

inance

Mean

SDSk

ewness

ScTail05

EP 9

5SD

ScTail 0

5EP 9

5|S

Y′|

|SY|

100/0

00

0.00

152

0.01

201

−0.041

13−0

.025

970.02

719

0.01

247

−0.028

980.02

790

––

100/0

1−0

.10.00

129

0.01

083

−0.021

38−0

.023

430.02

451

0.01

123

−0.025

470.02

567

497.25

66.75

100/0

10

0.00

151

0.01

200

0.00

006

−0.025

730.02

731

0.01

247

−0.028

330.02

849

256.80

307.20

100/0

10.1

0.00

169

0.01

316

0.00

791

−0.028

090.03

003

0.01

372

−0.031

190.03

131

40.74

523.26

100/0

10.3

0.00

195

0.01

537

0.01

004

−0.032

510.03

520

0.01

622

−0.036

910.03

695

35.23

528.77

100/0

10.5

0.00

214

0.01

756

0.04

954

−0.036

960.04

062

0.01

871

−0.042

630.04

259

28.97

535.03

120/20

00

0.00

198

0.01

195

−0.034

03−0

.025

080.02

756

0.01

247

−0.028

980.02

790

––

120/20

1−0

.10.00

175

0.01

076

−0.009

67−0

.022

480.02

475

0.01

123

−0.025

470.02

567

499.05

64.95

120/20

10

0.00

198

0.01

196

0.00

364

−0.024

920.02

771

0.01

247

−0.028

330.02

849

240.75

323.25

120/20

10.1

0.00

217

0.01

314

0.01

127

−0.027

330.03

056

0.01

372

−0.031

190.03

131

43.48

520.52

120/20

10.3

0.00

246

0.01

543

0.03

121

−0.031

900.03

611

0.01

622

−0.036

910.03

695

37.52

526.48

120/20

10.5

0.00

267

0.01

768

0.05

574

−0.036

440.04

146

0.01

871

−0.042

630.04

259

31.83

532.17

150/50

00

0.00

240

0.01

194

−0.034

56−0

.024

450.02

806

0.01

247

−0.028

980.02

790

−–

150/50

1−0

.10.00

212

0.01

076

−0.002

58−0

.021

940.02

540

0.01

123

−0.025

470.02

567

480.19

83.81

150/50

10

0.00

240

0.01

194

0.00

500

−0.024

280.02

824

0.01

247

−0.028

330.02

849

270.30

293.70

150/50

10.1

0.00

263

0.01

313

0.01

193

−0.026

670.03

110

0.01

372

−0.031

190.03

131

56.66

507.34

150/50

10.3

0.00

300

0.01

545

0.02

802

−0.031

300.03

664

0.01

622

−0.036

910.03

695

50.20

513.80

150/50

10.5

0.00

329

0.01

775

0.04

156

−0.035

900.04

199

0.01

871

−0.042

630.04

259

43.97

520.03

123


Fig. 5 Left panel: original benchmark (Δγ = 0, Δσ = 0) and a synthetic benchmark with added standarddeviation and skewness (Δγ = 1,Δσ = 0.5). Right panel: distribution of optimal portfolios obtained usingthe benchmarks on the left

panel compares the different benchmark distributions and the right panel compares thecorresponding portfolios obtained when solving the model against each benchmark.

Increasing standard deviation leads to fatter left and right tails in both the syntheticbenchmark (as compared to the original benchmark) and the distribution of the opti-mal portfolio (as compared to the portfolio optimised with the original benchmark).However, especially with the optimised portfolios, the tail “fattening” is much morepronounced on the right side. This is clearly a matter of choice: if an investor has a lessrisk-averse profile, than increasing the standard deviation of the benchmark will yieldriskier but potentially more rewarding portfolios. Once again, the properties observedin the histograms are in line with the results in Table 6: as we increaseΔσ , we observea slightly worse tail, but better mean and EP. The risk/return profile of in-sample port-folios could be adjusted by increasing standard deviation to increase potential returnand, at the same time, increasing the skewness to reduce the probability of extremelosses.


Figure 6 shows the performance graphic for the index (FTSE100), a 120/20 portfoliobased on the original reference distribution and 120/20 portfolios based on syntheticdistributions where Δγ = 1 and Δσ ∈ {−0.1, 0, 0.1, 0.3, 0.5}. From the graphicwe can see that as we increase volatility and returns of in-sample portfolios, we alsoincrease both for out-of-sample portfolios. The highest returns are obtained whenΔσ = 0.5, although this seems to be the portfolio with the highest variability. Per-formance graphics for other values of α are similar, but not reported due to spaceconstraints.

Table 7 reports out-of-sample performance statistics. In accordance to our in-sampleresults, a higher value forΔσ implies higher returns but also higher risk.As an example,the portfolio obtained when α = 0.5 (150/50) and Δσ = 0.5 had the highest finalvalue (4.88) and the highest yearly excess return (37.52%), but also the highest standarddeviation of returns (0.01782). As further measure of risk, the maximum drawdownalso increased as we increase Δσ .

123


Fig. 6 Changing Δσ , out-of-sample performance, 2010–2014

The index itself, being equivalent to a highly-diversified portfolio, generally haslower volatility (0.00985) than portfolios composed of a smaller amount of assets .However, the portfolio obtained when Δσ < 0 and α = 0 had a smaller standarddeviation regardless of being composed of less assets. Furthermore, although its finalvalue is not as high as those when Δσ ≥ 0, it is still much higher than that of theindex, making it a potentially safer choice for investors accostumed to index-trackingfunds.

We run the models for higher values of Δσ , i.e. Δσ > 0.5 but due to space con-straints we do not report the results here. It was observed that, after a certain level(Δσ > 0.6), out-of-sample returns tend to drop while standard deviation continues toincrease. Parameters Δσ and Δγ should be adjusted according to investor constraintsand aversion to risk.

In summary, using a benchmark distributionwith a slightly lower standard deviationtends 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 asomewhat 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 indexationbased on SSD criterion and reported in Roman et al. (2013). In that approach we com-puted SSD efficient portfolios that dominate (if possible) an index which is chosen asthe benchmark, that is, the reference distribution. In this paper we introduce a modi-fied / reshaped benchmark distribution with the purpose of obtaining improved SSDefficient portfolios, whose return distributions possess superior (desirable) properties.

The first step in reshaping the original benchmark is to increase its skewness. Theamount by which skewness should be increased is specified by the decision maker andis stated as a proportion of the original skewness. Our numerical results show that, by

123


Table7

Chang

ing

Δσ,o

ut-of-sampleperformance

statistics

Lon

g/short

Δγ

Δσ

Final

Excessover

Sharpe

Sortino

Max

draw

-Max

reco-

Daily

returns

value

RFR

(%)

ratio

ratio

down(%

)very

days

Mean

SD

FTSE

100

1.18

1.57

0.09

70.13

418

.83

481

0.00

019

0.00

985

100/0

00

2.19

15.97

0.85

51.24

214

.55

150

0.00

071

0.01

071

100/0

1−0

.11.92

12.75

0.77

21.12

212

.53

150

0.00

059

0.00

960

100/0

10

2.21

16.16

0.87

01.26

713

.44

150

0.00

072

0.01

064

100/0

10.1

2.43

18.54

0.90

71.31

714

.21

150

0.00

081

0.01

159

100/0

10.3

2.82

22.35

0.93

11.35

217

.54

337

0.00

095

0.01

339

100/0

10.5

3.11

24.92

0.93

01.34

221

.96

321

0.00

106

0.01

479

120/20

00

2.60

20.24

0.99

81.44

612

.33

151

0.00

086

0.01

141

120/20

1−0

.12.31

17.24

0.94

81.37

310

.95

150

0.00

075

0.01

037

120/20

10

2.71

21.27

1.04

71.52

112

.32

151

0.00

089

0.01

139

120/20

10.1

3.23

25.93

1.15

11.67

913

.60

159

0.00

105

0.01

239

120/20

10.3

3.88

30.96

1.17

01.70

317

.75

182

0.00

123

0.01

426

120/20

10.5

4.32

33.99

1.12

81.63

823

.43

308

0.00

135

0.01

606

150/50

00

2.78

21.93

0.98

31.41

014

.15

183

0.00

093

0.01

246

150/50

1−0

.12.45

18.75

0.93

41.34

513

.18

193

0.00

081

0.01

136

150/50

10

2.77

21.86

0.98

01.40

514

.48

183

0.00

093

0.01

247

150/50

10.1

3.20

25.66

1.03

81.48

115

.91

150

0.00

106

0.01

360

150/50

10.3

4.20

33.17

1.12

21.60

317

.99

153

0.00

132

0.01

578

150/50

10.5

4.88

37.52

1.10

51.57

221

.35

161

0.00

148

0.01

782

123


using a reshaped benchmark with higher skewness, we indeed obtain SSD efficientportfolios with better, that is, shorter left tails, as measured in-sample by CVaR andskewness.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 restof the distribution, with lower returns on average and reduced right tail. We observedthat as long as skewness is not increased excessively, the improvement in the left tailcomes 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 lefttail of the solution portfolio is substantial and comes at virtually no cost to the restof the return distribution. The out-of-sample results in this case are also considerablybetter than in the case of using the original benchmark.

We also experimented with a reshaped benchmark by changing the standard devi-ation by various amounts. We observed in Tables 6, 7 and Fig. 6 that reducing thebenchmark standard deviation reflects in portfolios that have lower standard deviationsand better tail characteristics, but also worse overall returns, both in-sample and out-of-sample (lowermean, skewness, Sharpe and Sortino ratios).Moreover, by increasingthe benchmark standard deviation up to 30% of its original value, we obtained morevolatile portfolios with better overall return characteristics, both in-sample and out-of-sample (higher mean, standard deviation, skewness, Sharpe and Sortino ratios).These results are consistent as they are reported for long-only as well as for long-shortcombinations (120/20 and 150/50).

We also observed consistent better performance, both in-sample and out-of-sample,of long-short portfolios as compared to long-only portfolios.

Overall, our numerical experiments have shown that, by reshaping the return dis-tribution of a financial index and using it as a benchmark in long-short SSD models,it is possible to obtain superior portfolios with better in-sample and out-of-sampleperformance.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons license, and indicate if changes were made.

References

Bawa VS (1975) Optimal rules for ordering uncertain prospects. J Financ Econ 2(1):95–121Dentcheva D, Ruszczynski A (2003) Optimization with Stochastic dominance constraints. SIAM J Optim

14(2):548–566Dentcheva D, Ruszczynski A (2006) Portfolio optimization with Stochastic dominance constraints. J Bank

Finance 30(2):433–451Dentcheva D, Ruszczynski A (2010) Inverse cutting plane methods for optimization problems with second-

order stochastic dominance constraints. Optim J Math Progr Oper Res 59(3):323–338diBartolomeoD (2000)The enhanced index fund as an alternative to indexed equitymanagement.Northfield

Information Services, BostonFábián C, Mitra G, Roman D (2011a) Processing second-order stochastic dominance models using cutting-

plane representations. Math Progr 130(1):33–57Fábián C, Mitra G, Roman D, Zverovich V (2011b) An enhanced model for portfolio choice with SSD

criteria: a constructive approach. Quant Finance 11(10):1525–1534

123

http://creativecommons.org/licenses/by/4.0/


Hadar J, Russell W (1969) Rules for ordering uncertain prospects. Am Econ Rev 59(1):25–34Hodder JE, Jackwerth JC, Kolokolova O (2015) Improved portfolio choice using second order stochastic

dominance. Rev Finance 19(1):1623–1647IBM ILOG CPLEX Optimizer (2015). https://www-01.ibm.com/software/commerce/optimization/

cplex-optimizer/. Accessed 3 Sept 2015Javanmardi L, Lawryshy Y (2016) A new rank dependent utility approach to model risk averse preferences

in portfolio optimization. Ann Oper Res 237(1):161–176Kolen MJ, Brennan RL (1995) Test equating: methods and practices. Springer, New YorkKopa M, Post T (2015) A general test for SSD portfolio efficiency. OR Spectr 37(1):703–734Kuosmanen T (2004) Efficient diversification according to stochastic dominance criteria. Manag Sci

50(10):1390–1406Levy H (1992) Stochastic dominance and expected utility: survey and analysis. Manag Sci 38(4):555–593Ogryczak W, Ruszczynski A (2002) Dual stochastic dominance and related mean-risk models. SIAM J

Optim 13(1):60–78PadbergM,RinaldiG (1991)Abranch-and-cut algorithm for resolution of large scale of symmetric traveling

salesman problem. SIAM Rev 33:60–100Post T (2003) Empirical tests for stochastic dominance efficiency. J Finance 58(5):1905–1931Post T, Fang Y, Kopa M (2015) Linear tests for DARA stochastic dominance. Manag Sci 61(1):1615–1629Post T, Kopa M (2013) General linear formulations of stochastic dominance criteria. Eur J Oper Res

230(2):321–332Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1988) Numerical recipes in C: the art of scientific

computing. Cambridge University Press, New York, NYR Core Team (2015) R: A language and environment for statistical computing. R Foundation for Statistical

Computing, Vienna. https://www.R-project.org/Roman D, Darby-Dowman K, Mitra G (2006) Portfolio construction based on stochastic dominance and

target return distributions. Math Progr 108(2):541–569Roman D, Mitra G, Zverovich V (2013) Enhanced indexation based on second-order stochastic dominance.

Eur J Oper Res 228(1):273–281Sharpe WF (1966) Mutual fund performance. J Bus 39(1):119–138Sortino FA, Price LN (1994) Performance measurement in a downside risk framework. J Invest 3(3):59–64ThomsonReutersDatastream (2014). http://online.thomsonreuters.com/Datastream/.Accessed 3Sept 2015Wang T, Kolen MJ (1996) A quadratic curve equating method to equate the first three moments in equiper-

centile equating. Appl Psychol Meas 20(1):27–43Whimore GA, Findlay MCE (1978) Stochastic dominance: an approach to decision-making under risk.

Lexington Books, Lanham

123

https://www-01.ibm.com/software/commerce/optimization/cplex-optimizer/

https://www-01.ibm.com/software/commerce/optimization/cplex-optimizer/

https://www.R-project.org/

http://online.thomsonreuters.com/ Datastream/

Novel approaches for portfolio construction using … · Comput Manag Sci (2017) 14:257–280 DOI...

Documents

Transcript of Novel approaches for portfolio construction using … · Comput Manag Sci (2017) 14:257–280 DOI...