A nonlinear optimisation model for constructing minimal drawdown portfolios
AA nonlinear optimisation model for constructing minimaldrawdown portfolios
C.A. Valle and J.E. Beasley Departamento de Ciˆencia da Computa¸c˜ao,Universidade Federal de Minas Gerais,Belo Horizonte, MG 31270-010, [email protected] Mathematical Sciences, Brunel University, Uxbridge UB8 3PH, [email protected] 2019
Abstract
In this paper we consider the problem of minimising drawdown in a portfolio of financialassets. Here drawdown represents the relative opportunity cost of the single best missedtrading opportunity over a specified time period. We formulate the problem (minimisingaverage drawdown, maximum drawdown, or a weighted combination of the two) as a non-linear program and show how it can be partially linearised by replacing one of the nonlinearconstraints by equivalent linear constraints.Computational results are presented (generated using the nonlinear solver SCIP) for threetest instances drawn from the EURO STOXX 50, the FTSE 100 and the S&P 500 with dailyprice data over the period 2010-2016. We present results for long-only drawdown portfoliosas well as results for portfolios with both long and short positions. These indicate that (onaverage) our minimal drawdown portfolios dominate the market indices in terms of return,Sharpe ratio, maximum drawdown and average drawdown over the (approximately 1800trading day) out-of-sample period.
Keywords:
Index out-performance; Nonlinear optimisation; Portfolio construction; Portfo-lio drawdown; Portfolio optimisation
Given a portfolio of financial assets then, as time passes, the price of each asset changes andso by extension the value of the portfolio changes. Portfolio drawdown is a measure of currentportfolio value when compared to the maximum value achieved by the same portfolio of assetsin the recent past. It gives insight into how much the portfolio has fallen in value by comparingits value now with the best (maximum) value it had in the recent past.In this paper we adopt an optimisation approach to the problem of deciding a portfoliothat minimises portfolio drawdown. The structure of this paper is as follows. In Section 2 wegive an example to illustrate the concept of drawdown and then review the relevant literaturerelating to deciding portfolios that minimise drawdown. We also discuss the context of our work,Operational Research applied to a financial portfolio optimisation problem. In Section 3 we givea nonlinear formulation of the problem of deciding a portfolio that minimises drawdown. Our1 a r X i v : . [ q -f i n . R M ] A ug ormulation incorporates cash inflow/outflow, and can be used either to create an initial portfoliofrom cash or to rebalance an existing portfolio. Transaction costs associated with buying orselling an asset are included. We indicate how we can partially linearise our formulation. Wealso discuss some computational issues associated with our nonlinear formulation and presentthe amendments necessary to deal with shorting. In Section 4 we give computational results forconstructing minimum drawdown portfolios for three different problem instances derived fromuniverses defined by major equity markets, involving up to 500 assets. We present results forlong-only drawdown portfolios as well as results for portfolios with both long and short positions.Finally in Section 5 we present our conclusions. In this section we first give an example to illustrate the concept of drawdown. We then presentour literature review relating to deciding portfolios that minimise drawdown. We also discussthe context of our work to show that it follows the general pattern seen in the literature forOperational Research applied to financial portfolio optimisation problems.
To illustrate drawdown consider the solid line in Figure 1 where we show the value of a portfolioover time (starting from a value of 50 at time one). At time 6 the portfolio has value 60 andwe can see that the maximum value of the portfolio was 90 (at time 4). At time 6 therefore thedrawdown associated with this portfolio can be defined as 100(90 − /
90 = 33 . − /
90 = 55 . − /
70 = 14 . Solid portfolio Dotted portfolioTime Value Return (%) Drawdown (%) Value Return (%) Drawdown (%)1 50 0 50 02 70 33.65 0 77.45 43.76 03 60 -15.42 14.29 55.97 -32.48 27.734 90 40.55 0 29.76 -63.17 61.585 40 -81.09 55.56 57.11 65.18 26.266 60 40.55 33.33 60 4.94 22.53Mean 3.65 17.20 3.65 23.02Standard deviation 52.84 52.84
The value of drawdown at any particular time t is (in relative percentage terms) the oppor-2igure 1: Drawdown illustrated l l l l l l Time P o r tf o li o v a l ue tunity cost of the single best missed trading opportunity associated with selling the portfolio atsome point before t and then repurchasing it at t . It represents the (percentage) value forgoneby not having sold the portfolio at its previous maximum value point (the single best point atwhich to sell the portfolio) and then repurchasing it at time t .So in the example considered above we start at time 1 with a portfolio of assets. At time 6had we sold the portfolio at time 4 (when its value was at a maximum) and then repurchasedit at time 6 we would have banked 90 in cash from the sale of the portfolio at time 4, andwould need to spend 60 at time 6 to repurchase the portfolio. So at time 6 we would be in thesame situation as we started at time 1, holding the portfolio of assets, but now also with 30 incash. Since we did not avail ourselves of this trading opportunity our percentage missed profitis 100(30) /
90 = 33 . sequence of values achieved, in other words it is a path-dependent measure. This contrasts with path/sequence independent summary statistics suchas mean and variance/standard deviation calculated from an entire set of values. For this reasonany permutation of the set of returns associated with either of the two portfolios shown in Table 1will result in different portfolios with different drawdown values (but all such portfolios will havevalue 50 at time 1, value 60 at time 6, the same set of returns, the same mean return and thesame standard deviation in return). To structure our literature review we first discuss papers that consider drawdown within adiscrete time setting. In such papers financial asset values are assumed to be known, e.g. fromhistoric data, at discrete points in time. This is a standard setting for portfolio optimisationsuch as adopted in mean-variance optimisation [42].We then discuss papers that use a continuous time setting. Such papers typically start fromthe premise that the process underlying asset price dynamics is some form of Brownian motion(e.g. geometric Brownian motion with drift) and involve stochastic differential equations.We exclude from explicit consideration here a number of papers which, whilst using draw-down as a statistic with which to evaluate portfolio performance, have as their primary focusother matters.
Alexander and Baptista [2] introduced a drawdown constraint into the standard mean-varianceportfolio approach due to Markowitz [42]. Their approach was a scenario based one in whichasset returns in each of a number of different scenarios are known. They defined drawdown asthe worst (minimum) portfolio return seen over all the scenarios considered. With drawdowndefined in this way limiting drawdown merely involves adding linear constraints (one for eachscenario considered) to the standard quadratic program associated with mean-variance portfolio4ptimisation. Using one data set containing ten assets they illustrated how the standard un-constrained mean-variance efficient frontier compared with the drawdown constrained efficientfrontier.Yao et al [66] extended the work of Alexander and Baptista [2] from a theoretical standpoint.They investigated the composition and geometric features of the mean-variance efficient frontierwith a drawdown constraint. One numeric example involving eight assets was presented.Baghdadabad et al [7] linked drawdown to the Capital Asset Pricing Model (CAPM) [35,56, 60]. In the CAPM the setting is that different assets achieve different returns, against acommon background of a risk-free rate and a market return. The hypothesis is that the returnon any asset in excess of the risk-free rate is a linear function, namely a constant (alpha) plus aconstant (beta) multiplied by the difference between the return on the market and the risk-freerate. Values for alpha and beta, which are typically different for different assets, can be obtainedfrom data using standard linear regression. Drawdown is defined in [7] as the loss in portfoliovalue when compared with a previous maximum value. They presented two beta values basedupon maximum drawdown and average drawdown. These were computed using the covarianceof a function that involves the sum of asset drawdown and asset return above/below its mean.Extensive results were presented for their beta values when computed for 11737 mutual fundsover the period 2000-2011.Zabarankin et al [68] also linked drawdown to the CAPM. Their approach is fundamentallydifferent from that of Baghdadabad et al [7]. They considered Conditional Drawdown-at-Risk(CDaR), which is defined as the average of a specified percentage of the largest drawdowns overan investment horizon. Their approach was a scenario (sample-path) based approach in whichasset returns in each of a number of different scenarios were known (as were the probabilitiesof each scenario). Drawdown was defined using cumulative portfolio return since the initialinvestment and was equal to the difference between the maximum cumulative portfolio returnover a specified number of preceding periods and the current portfolio cumulative return. Theyproposed either minimising portfolio CDaR subject to a constraint on portfolio expected returnat the end of the investment horizon; or maximising portfolio expected return at the end ofthe investment horizon subject to a constraint on portfolio CDaR. Underlying their work isthe earlier work presented in the literature [17, 18] relating to the definition of conditionaldrawdown. The main focus of their paper was on the CAPM, with CDaR alpha and CDaR betabeing defined, these being analogous to alpha and beta in the CAPM. Values for CDaR alphaand CDaR beta were computed and presented for 80 hedge fund indices.Mohr and Dochow [45] considered how expert judgment can be incorporated into portfolioselection so as to minimise maximum drawdown. In their work drawdown was defined as thedifference between the maximum portfolio value to date and current portfolio value. Computa-tional results were presented for portfolios containing just two assets formed from a NYSE dataset containing daily price data for 36 assets over 5651 days until the end of 1985.
Grossman and Zhou [26] considered the case of an investor who wants to lose no more than afixed percentage of the maximum value their portfolio has achieved to date. This constraint isequivalent to saying that drawdown (defined in percentage terms, as in the example considered5n Table 1 above) in any period is limited. For a portfolio comprising just one risky asset and arisk-free asset they derive an optimal investment policy.Magdon-Ismail and Atiya [38] presented results relating maximum drawdown to the meanreturn and Sharpe ratio [57, 58, 59]. Related work is presented in Magdon-Ismail et al [39] whoconsidered the distribution of drawdown and its expected value.Pospisil and Vecer [48] considered a financial instrument whose value depends not only onthe price of an underlying asset, but also on other factors such as running maximum drawdown.They examined the sensitivity of the value of this instrument with respect to drawdown. Theirwork draws on earlier work presented by the same authors in [47]. In their work drawdown wasdefined as the difference between the maximum portfolio value to date and current portfoliovalue.Yu et al [67] build on the work of Grossman and Zhou [26] and considered the case ofderiving the optimal investment policy for a portfolio comprising of a number of risky assetsand a risk-free asset. Numeric results for a single risky asset and for two risky assets werepresented.Chen et al [20] considered two cases: a portfolio with two risky assets and a portfolio with onerisky asset and a risk-free asset. They focused on minimising the probability that a significantdrawdown occurs over portfolio lifetime. In their work drawdown is defined in percentage terms,as in the example considered in Table 1 above, and a significant drawdown is one that exceedsa specified percentage. Related work can be seen in Angoshtari et al [5] for a portfolio with apayout rate, one risky asset and a risk-free asset and in Angoshtari et al [6] for a portfolio witha constant payout rate, one risky asset and a risk-free asset.Goldberg and Mahmoud [25] considered Conditional Expected Drawdown (CED), which isthe expected value of maximum drawdown given that a specified maximum drawdown thresholdhas been exceeded. CED is the tail mean of the distribution of maximum drawdowns. Draw-down was defined as the difference between the maximum portfolio value to date and currentportfolio value. Although set within a continuous time framework their work does not dependon restrictive assumptions as to the underlying stochastic process. They presented CED valuesfor a US equity index, a US bond index and three fixed-mix portfolios using daily data over theperiod 1982-2013.Mahmoud [40] considered drawdown as a temporal path-dependent risk measure within astochastic continuous time framework. In their work drawdown was defined as the differencebetween the maximum portfolio value to date and current portfolio value. Although this paperis primarily theoretical in nature drawdown associated with two US equity and bond indicesover the period 1978-2013 is presented for illustration.
The basic context of our work is financial portfolio optimisation from an Operational Research(OR) perspective. Here the financial problem is to decide a portfolio of financial assets (i.e. decidehow much to invest in each asset) so as to optimise some objective subject to constraints upon theinvestments made. The constraints applied might limit the investment made in any particularasset as well as specify the total investment that is to be made.In dealing with financial portfolio optimisation from an OR perspective portfolio choice is6nderpinned by a formal explicit optimisation model, which is solved either optimally or heuris-tically (e.g. using some metaheuristic). Typically a heuristic is adopted when the underlyingoptimisation model is (computationally) hard to solve, for example when it involves integervariables and/or nonlinearities.For any particular financial portfolio optimisation problem the general pattern is that it firstappears in the academic literature in journals related to the financial sphere (as would seemnatural). A number of years after its first appearance in the financial literature it captures theattention of one or more OR workers and the problem then appears in journals more commonlyassociated with OR. This general pattern is illustrated and evidenced below with reference tothree example financial portfolio optimisation problems.
Markowitz mean-variance portfolio optimisation with cardinality constraints
Here the problem is to find a portfolio that best balances risk against return, where risk ismeasured using variance in portfolio return, but where in addition there are constraints on thenumber of assets that can be in the portfolio chosen. This is the classic mean-variance portfoliooptimisation approach due to Markowitz [42], but enhanced with cardinality constraints.One early work in the finance literature formalising the problem of restricting the num-ber of assets in a mean-variance portfolio is that of Jacob [30] which appeared in a financejournal in 1974. Early work in the literature from an OR perspective in OR journals canbe found in [9, 16, 41]. That work dates from 1996 some 22 years after appearance of theproblem in the financial literature. Recent work in OR journals relating to the problem in-cludes [4, 15, 22, 28, 29, 31, 33, 34, 36, 65, 70].
Equity index tracking
Here the problem is to find a portfolio that best matches the return on a specified equitymarket index, but without holding all of the assets that are present in the index (so excludingfull replication of the index).Early work in the finance literature dealing with equity index tracking can be found inRudd [51], which appeared in a finance journal in 1980. Early work in the literature from anOR perspective in OR journals can be found in [1, 8]. That work dates from 1994 some 14 yearsafter appearance of the problem in the financial literature. Recent work in OR journals relatingto the problem includes [4, 19, 23, 44, 52, 55, 64].
Enhanced indexation (enhanced index tracking)
Here the problem is to find a portfolio that exceeds the return on a specified equity marketindex.Early published work in the finance field dating from from the late 1990’s dealing withenhanced indexation can be found in [24, 49, 54]. Early work in the literature from an ORperspective in an OR journal can be found in [14]. That work dates from 2009 some 11 yearsafter appearance of the problem in the financial literature. Recent work in OR journals relatingto the problem includes [11, 21, 27, 44, 50, 69].Note here that for the three financial portfolio optimisation problems considered above it took7ver a decade in each case before the problem first captured the attention of OR workers.The problem considered in this paper of finding a minimal drawdown portfolio, followsexactly the same pattern as evidenced above . This problem, as seen in the detailedliterature survey presented, has first appeared in the financial literature. The work presentedin this paper is one of the first to consider the problem from an OR perspective. We believethat the financial portfolio optimisation problem relating to minimal drawdown considered inthis paper will increasingly attract the attention of OR workers.
In this section we formulate the problem of deciding a portfolio that minimises drawdown.Our formulation incorporates cash inflow/outflow, and can be used either to create an initialportfolio from cash or to rebalance an existing portfolio. Transaction costs associated withbuying or selling an asset are included.Our formulation involves a nonlinear definition of drawdown. We indicate how we canpartially linearise our formulation by replacing one of the nonlinear constraints by equivalentlinear constraints. We also discuss some computational issues associated with our nonlinearformulation. We present the amendments to the formulation necessary to deal with shorting.The approach adopted in formulating the problem below is a backward-looking future-blind approach. Here we assume that we are at time T , with no knowledge of future asset prices,but with information as to historic asset prices over the time period [1 , T ]. This contrasts witha forward-looking future-assumed-known simulation style approach of generating one ormore sample paths (scenarios) for future asset prices and using that sample path informationto decide a portfolio, e.g. [2, 66, 68]. Suppose we observe over time 1 , , . . . , T the value (price) of N assets. Given this informationthe decision problem we face is how can we best invest at time T in a portfolio which, hadwe held it over [1 , T ], would best minimise an appropriate objective involving drawdown. Tointroduce our notation, let D be the number of time periods associated with calculating drawdown P t be the value ( ≥
0) of the portfolio at time td t be the portfolio drawdown value ( ≥
0) at time tM t be the maximum portfolio value ( ≥
0) over the period [ t, t − , . . . , max [1 , t − D ]]Suppose the current time is t . Then the portfolio currently has value P t . In the immediate D time periods preceding time t the portfolio had values P τ , τ = t − , . . . , max [1 , t − D ]. Attime t therefore the drawdown associated with the portfolio is given by: M t = max [ P τ | τ = t, t − , . . . , max [1 , t − D ]] t = 1 , . . . , T (1) d t = 100( M t − P t ) /M t t = 1 , . . . , T (2)8quation (1) defines the maximum portfolio value seen over the time period[ t, t − , . . . , max [1 , t − D ]]. Equation (2) defines drawdown as the reduction in portfolio valuefrom the best portfolio value, expressed as a percentage of M t . Defining drawdown in relativepercentage terms as in equation (2) is common in the literature, especially when reporting onthe performance of a portfolio, e.g. see [3, 10, 12, 32, 37].The use of τ = t in the maximization term involving P τ in equation (1) is deliberate andensures that if the current value P t is superior to the values achieved in the preceding D timeperiods then drawdown d t takes the value zero (since in that case M t = P t ).The role of D , drawdown lookback , is that it gives us flexibility not to be forced to lookinto the entire past when calculating drawdown, but instead simply look D periods into thepast. So drawdown at time t is derived by comparing the portfolio value at time t with thebest (maximum) portfolio value over the preceding D time periods. Using a time window fordrawdown lookback has been seen previously in the literature, e.g. in [68]. If we consider all possible time periods t = 1 , . . . , T then we would like drawdown to be small.There are a number of possibilities here, for example: • minimise average drawdown • minimise maximum drawdown • minimise a weighted sum of maximum drawdown and average drawdownTo proceed we shall first concentrate on minimising average drawdown. Let: V it be the value (price) of asset i at time tA i be the number of units of asset i held in the current portfolio δ i be the maximum proportion of the portfolio value at time T that can be placed inasset iC be the total value ( ≥
0) of the current portfolio [ A i ] at time T , (cid:80) Ni =1 A i V iT , pluscash change (either new cash to be invested or cash to be taken out) f bi be the fractional cost of buying one unit of asset i at time T , so that the costincurred in buying one unit of asset i at time T is f bi V iT f si be the fractional cost of selling one unit of asset i at time T , so that the costincurred in selling one unit of asset i at time T is f si V iT γ be the limit (0 ≤ γ ≤
1) on the proportion of C that can be consumed by transac-tion costWith respect to decision variables, let: x i be the number of units ( ≥
0, so we exclude shorting) of asset i to be held in theportfolio G i be the transaction cost ( ≥
0) associated with buying or selling asset i at time T Then our nonlinear formulation of the problem of minimising average drawdown is:9in T (cid:88) t =1 d t /T (3)subject to equations (1),(2) and: P t = N (cid:88) i =1 V it x i t = 1 , . . . , T (4) V iT x i ≤ δ i P T i = 1 , . . . , N (5) G i ≥ f si ( A i − x i ) V iT i = 1 , . . . , N (6) G i ≥ f bi ( x i − A i ) V iT i = 1 , . . . , N (7) N (cid:88) i =1 G i ≤ γC (8) P T = C − N (cid:88) i =1 G i (9) x i , G i ≥ i = 1 , . . . , N (10) d t , P t , M t ≥ t = 1 , . . . , T (11)Equation (3) minimises average drawdown. Equation (4) defines the portfolio value variables( P t ). Clearly these variables can be eliminated by algebraic substitution, but we have left themin the formulation seen above for clarity of exposition. Equation (5) limits the proportion of theportfolio value at time T placed in any asset appropriately. Equation (6) defines the transactioncost associated with selling asset i , where we have sold the asset if the current holding A i isgreater than the new holding x i . Equation (7) defines the transaction cost associated withbuying asset i , where we have bought the asset if the new holding x i is greater than the currentholding A i . Equation (8) limits the total transaction cost. Equation (9) is a balance constraintwhich ensures that the value of the portfolio after trading at time T is equal to its value beforetrading (after accounting for the cash change, so C ) minus the total transaction cost incurred.Equations (10),(11) are the non-negativity constraints.The above formulation: optimise equation (3) subject to equations (1),(2),(4)-(11), minimisesaverage drawdown. Essentially in that formulation we are drawing on historic asset price data[ V it ] over the period t = 1 , . . . , T to decide how we can best improve our existing portfolio ofassets [ A i ], whilst taking account of any cash inflow/outflow, so as to minimise average drawdownover the period considered.In the computational results reported later we used asset sets [ i, i = 1 , . . . , N ] drawn fromequity (stock) indices. Optionally, if so desired, we can include in the asset set a risk-free assetto represent investment in an interest-bearing “cash” account.10ote that there is one technical subtlety here - strictly the equations defining G i do not di-rectly constrain G i to be equal to the correct transaction cost (which is max[ f si ( A i − x i ) V iT , f bi ( x i − A i ) V iT ]). Rather G i is bounded below by the correct transaction cost (as the inequalities (equa-tions (6),(7)) above indicate). We would expect that the numeric value for G i which we getfrom the optimisation to be equal to the correct transaction cost found by direct calculationusing the numeric x i value given by the optimisation, since otherwise we would have unallocatedwealth (namely G i − max[ f si ( A i − x i ) V iT , f bi ( x i − A i ) V iT ]) which is left untouched, so making nocontribution to reducing portfolio drawdown. Computationally if we wish to ensure that thissituation never occurs then we simply amend the objective function (equation (3)) to minimiseΓ (cid:80) Tt =1 d t /T + (cid:80) Ni =1 G i where Γ is a large positive constant, thus ensuring that we minimiseaverage drawdown as well the transaction cost variables G i . In our formulation equations (1) and (2) are nonlinear, because of the presence of the maximisa-tion term in equation (1) and because of the division in equation (2). A partial linearisation of this formulation can be made since the maximisation term, equation (1), can be removed andreplaced by the linear inequalities: M t ≥ P τ τ = t, t − , . . . , max [1 , t − D ] t = 1 , . . . , T (12)The formulation given above to minimise average drawdown then becomes: optimise equa-tion (3) subject to equations (2),(4)-(12).To show that replacing nonlinear equation (1) by linear equation (12) is valid is simple using aproof by contradiction. Suppose that when we optimise equation (3) subject to equations (2),(4)-(12) we obtain, for some time period t , a value for M t than satisfies equation (12) but doesnot satisfy equation (1). This can only occur if we have strict inequality, i.e. M t > P τ τ = t, t − , . . . , max [1 , t − D ]. Now equation (2) can be written as d t = 100(1 − P t /M t ) and weare minimising a function that involves d t . So we would like M t as small as possible, consistentwith the constraints upon it, so as to make the negative contribution of the P t /M t term in d t = 100(1 − P t /M t ) as great as possible. The strict inequality for M t in relation to P τ impliesthat we can reduce the numeric value for M t , thereby reducing the value for d t , and hencecontradicting the assumption that we already had the optimal solution which involved optimisingequation (3). Hence of the inequalities in equation (12) for time period t one (or more) will besatisfied with equality at optimality, ensuring that M t will indeed satisfy equation (1) andcorrespond to the appropriate maximum value over the period [ t, t − , . . . , max [1 , t − D ]].Note here that whether replacing nonlinear equation (1) by linear equation (12) is valid ornot depends upon the particular optimisation problem involved. In other words equation (12)is not a general linearisation of equation (1) that applies in all circumstances. For example ifwe were to maximise (cid:80) Tt =1 d t /T then this would be achieved by letting M t → ∞ , which giventhe finite nature of the investment in each asset implied by equation (9) would mean that whilstequation (12) was satisfied equation (1) would not be satisfied.11 .4 Other objectives To minimise maximum drawdown we introduce an artificial variable d max ( ≥
0) to representmaximum drawdown defined by: d max = max [ d t | t = 1 , . . . , T ] (13)The formulation then is: min d max (14)subject to equations (2),(4)-(12) and: d max ≥ d t t = 1 , . . . , T (15) d max ≥ λ and λ are the weights ( >
0) that we place on maximum drawdown and averagedrawdown respectively than to minimise the weighted sum of maximum drawdown and averagedrawdown we: min λ d max + λ T (cid:88) t =1 d t /T (17)subject to equations (2),(4)-(12),(15),(16).In relation to a minor technical issue here then, purely in the case when the objective isequation (14), it is possible that when using equation (12) the numeric values assigned by theoptimiser to the M t variables could be strictly greater than every element in the right-hand sideof equation (1). However, since equation (12) is a ≥ constraint, alternative optimal solutionsexist where the M t values are artificially decreased to satisfy equation (1).Note here that we have defined drawdown as the reduction in portfolio value from the bestportfolio value, expressed as a percentage of the best value ( M t ) achieved, i.e. d t = 100( M t − P t ) /M t , equation (2). The approach given in this paper, including the partial linearisation givenabove, also applies (with only minor modifications) if (e.g. as in [18]) we define drawdown using d t = 100( M t − P t ) /P t , i.e. as the reduction in portfolio value from the best portfolio value,expressed as a percentage of the portfolio value P t . Readers familiar with the numeric solution of nonlinear programs will be aware that they arecomputationally much more challenging than solving linear programs. Computational benefithowever can sometimes be gained by imposing (non-trivial) bounds on decision variables. Lim-ited computational experience indicated that for the solver (SCIP [53]) we used this was indeedthe case and so here we indicate the bounds we used.A valid upper bound for x i is given by δ i C/V iT , from equations (5),(9). A valid upperbound for G i is γC , from equation (8). Valid lower and upper bounds for P t ( P mint and P maxt respectively) can be found by optimising P t subject to equations (4)-(11). This is a simple linear12rogram to solve, where we minimise P t to find the value for P mint , maximise P t to find the valuefor P maxt .Valid lower and upper bounds for M t ( M mint and M maxt respectively) can be found byusing M mint = max [ P minτ | τ = t, t − , . . . , max [1 , t − D ]] and M maxt = max [ P maxτ | τ = t, t − , . . . , max [1 , t − D ]]. Valid lower and upper bounds for d t are given by max [0 , − P maxt /M mint )] and min [100 , − P mint /M maxt )].The bounds that we used therefore were: x i ≤ δ i C/V iT i = 1 , . . . , N (18) G i ≤ γC i = 1 , . . . , N (19) P mint ≤ P t ≤ P maxt t = 1 , . . . , T (20) M mint ≤ M t ≤ M maxt t = 1 , . . . , T (21) max [0 , − P maxt /M mint )] ≤ d t ≤ min [100 , − P mint /M maxt )] t = 1 , . . . , T (22)Limited computational experience also indicated that for the nonlinear solver (SCIP) we usedbenefit could be gained if we replaced equality equation (2) by an inequality, as in equation (23)below: d t ≥ M t − P t ) /M t t = 1 , . . . , T (23)Replacing the equality definition of d t by this inequality definition can be shown to be validusing a very similar argument as was used above in terms of replacing equality equation (1) byinequality equation (12). In the formulation given above we have excluded shorting. The amendments necessary to includeshorting are as outlined below. For simplicity of exposition here we shall henceforth assume thatall transaction costs are zero. Amending the formulation given previously above to incorporatethe notational changes seen below relating to x i is relatively straightforward (and is not givenhere for space reasons). Incorporating the transaction costs associated with shorting can be donein a very similar manner as in the formulation above where we had transaction costs associatedwith buying/selling an asset (equations (6)-(10)).Introduce x Li , x Si as the number of units ( ≥
0) of asset i that we choose to hold in long/shortpositions respectively. Let δ Li and δ Si be the maximum proportions of the portfolio value at time T that can be placed in asset i in long/short positions respectively. Let ∆ L and ∆ S be themaximum overall proportions of the portfolio value at time T that can be placed in long/shortpositions respectively.Then the decision variables x i are no longer non-negative, but are instead unrestricted insign, and we have: 13 i = x Li − x Si i = 1 , . . . , N (24) V iT x Li ≤ δ Li P T i = 1 , . . . , N (25) V iT x Si ≤ δ Si P T i = 1 , . . . , N (26) N (cid:88) i =1 V iT x Li ≤ ∆ L P T (27) N (cid:88) i =1 V iT x Si ≤ ∆ S P T (28)Equation (24) relates the previously defined variable for the investment in asset i to the newvariables for investment in long/short positions. Equations (25),(26) limit the proportion of theportfolio value placed in any asset at time T appropriately, for both long and short positions.Equations (27),(28) limit the portfolio investment at time T in long and short positions.In the presence of shorting then (although dependent on the values given to ∆ L and ∆ S ) itis possible that the in-sample portfolio value P t could potentially be negative in any time period t . Allowing the in-sample portfolio value to be in a loss position, even if by so doing we minimisesome function of drawdown, would not in our view be desirable. To avoid this when shortingwe simply retain equation (11) which ensures that P t is never negative. Hence, although we canshort one or more assets, we never let the overall portfolio value go negative. In this section we first discuss the data we used and the methodology adopted. We then presentresults for long-only drawdown portfolios, followed by result for portfolios with both long andshort positions.
We used three test instances drawn from major equity markets, the EURO STOXX 50, theFTSE 100 and the S&P 500. For these markets we used daily price data over the period 2010-2016 (inclusive). The test instances we used have been manually curated to ensure that we knowon any day the exact composition of the index. This means that when we come to rebalancethe drawdown portfolio we only consider for inclusion in the portfolio assets that are in theindex at the moment of rebalance. This means that our results use no more information thanwas available at the time, removing susceptibility to the influence of survivor bias.As noted in [43] most published analysis as to the size of transaction costs refers to USmarkets, and different estimates apply to equities with different capitalisations (companies ofdifferent sizes). As our test instances involve a number of different equity markets (two ofwhich are non-US) over a long time period then, given the difficulty in accurately estimating14ppropriate transaction costs, we assume in the results given below that transaction costs arezero.The formulations discussed above that we investigated computationally were: • formulation MINAVG: minimise average drawdown, optimise equation (3) subject to equa-tions (4)-(12),(18)-(23) • formulation MINMAX: minimise maximum drawdown, optimise equation (14) subject toequations (4)-(12),(15),(16),(18)-(23)We also examined these formulations, but with the addition of shorting (so these formu-lations, but modified as discussed above using equations (24)-(28)). We denote the shortingformulations as MINAVG-S and MINMAX-S respectively.Note here that although MINAVG, MINMAX, MINAVG-S and MINMAX-S are nonlinearprograms the solver we used, SCIP [53], is capable of solving them to proven global optimal-ity [13, 61, 62, 63]. This is because SCIP restricts the type of nonlinear expression allowed. If,within the computational time limit allowed, SCIP cannot terminate with the proven optimalsolution then it terminates with the best feasible solution found, but as well provides a guaran-teed percentage deviation from optimal for that solution. For a technical explanation as to howSCIP can achieve proven global optimality see Vigerske and Gleixner [62, 63].We used an Intel Xeon @ 2.40GHz with 32GB of RAM and Linux as the operating system.The code was written in C++.The methodology we adopted is successive periodic rebalancing over time. We start fromthe beginning of our data set. We decide a portfolio using data taken from an in-sample periodcorresponding to the first T days. This portfolio is then held unchanged for a specified out-of-sample period. We then rebalance (change) our portfolio, but now using the most recent T daysas in-sample data. The decided portfolio is then held unchanged for the specified out-of-sampleperiod, and the process repeats until we have exhausted all of the data.These results given below are for T = 30 and D = 20, so an in-sample time period of 30(trading) days with a drawdown lookback of 20 (trading) days. We rebalanced the drawdownportfolio every 10 (trading) days, so an out-of-sample period of 10 days. The motivation for usinglow values for these parameters was that we were adopting a strategy of frequent rebalancingwith a limited time horizon in terms of the immediate past. Such values seemed appropriatefor the context of our work (funds that seek to produce profit, or at least avoid loss, in theshort-term; as discussed above).In terms of computation time we set a time limit for each rebalance of max(500, 7 N ) seconds.With some seven years of data the results shown below involve approximately 180 rebalances foreach instance/case solved. We initialised the solution process using C = 1000 and A i = 0 i =1 , . . . , N . At each rebalance we set [ A i ] equal to the portfolio [ x i ] decided at the last rebalance.This corresponds to a self-financing strategy, rebalancing with no cash inflow/outflow. For long-only portfolios, so using formulations MINAVG and MINMAX, we examined two casesfor the maximum proportion limit ( δ i , equation (5)), one where we limited the proportion to 0.1,15o at most 10% of portfolio value could be invested in any asset, and one where the proportionlimit was 1, so potentially all of the investment could be in just a single asset.The results obtained are shown in Table 2. In this table we (for space reasons) also showresults associated with shorting, and these will discussed later below.To produce the first three in-sample columns in Table 2 we first compute using the drawdownportfolio as decided by the optimiser, over the in-sample period associated with each rebalance:the average daily (logarithmic) return for that portfolio over the in-sample period; the maximum(%) drawdown for that portfolio over the in-sample period; and the average (%) drawdown forthat portfolio over the in-sample period. Here drawdown is calculated as in equations (1) and(2), so involving drawdown lookback. The values shown in Table 2 are then the averages forthese three factors, as averaged over all rebalances. Note here that as we rebalance every 10days the in-sample periods overlap.The fourth in-sample column in Table 2 gives the average computation time (per rebalance,in seconds) and the fifth and final in-sample column the percentage of optimal solutions found.As mentioned above although we are solving a nonlinear problem SCIP [53] is able of solvingproblems such as MINAVG and MINMAX to proven global optimality and so this final in-sample column gives the percentage of rebalances for which the solution derived was proved tobe optimal within the computational time limit imposed.The first three out-of-sample columns in Table 2 have the same headings as the first threein-sample columns, but are computed using a single time series of out-of-sample portfoliovalues . This single time series of out-of-sample portfolio values is produced by amalgamatingtogether successive (non-overlapping) 10 day out-of-sample periods (one for each rebalance). Forthe instances shown in Table 2 this single out-of-sample time series contained approximately 1800daily portfolio values.The fourth out-of-sample column in Table 2 shows the Sharpe ratio [57, 58, 59] annualisedas in Pope and Yadav [46] using 252 trading days in a year. Here, because we have used threedifferent indices over a significant time period, we for ease of comparison use a risk-free rate ofzero.The final out-of-sample column in Table 2 shows the percentage of out-of-sample days inwhich the cumulative drawdown portfolio value exceeded the index (when both were normalisedto one at the start of out-of-sample period). With a normalisation to one at the start of out-of-sample period we are comparing cumulative return over time as achieved by the drawdownportfolio and the index. This statistic gives insight into the probability that the drawdownportfolio exceeds the index on any arbitrarily chosen day in the entire out-of-sample time period.For comparison purposes we also show in Table 2 the values associated with the index, i.e. thevalues for the factors tabulated that would be achieved were we to simply rebalance to the indexportfolio every 10 days.For the EURO STOXX 50 we can see from Table 2 that in-sample all four drawdown port-folios have superior return performance, as well as superior maximum and average drawdownperformance when compared to the index itself. It should be emphasised here that althoughwe restrict the set of assets that can be in the drawdown portfolio to the assets that are in theindex (for consistency of comparison) the MINAVG and MINMAX approaches that we haveformulated above make no use of index values nor of any knowledge as to how the index value16s computed (in terms of a weighted sum of index asset prices).Out-of-sample all four drawdown portfolios effectively dominate the index in terms of (cu-mulative) return achieved. Numerically for over 99% of the out-of-sample days the drawdownportfolios have a cumulative return that exceeds that of the index. With respect to drawdownout-of-sample three of the four drawdown portfolios dominate the index with respect to bothmaximum and average drawdown. Sharpe ratios for all four of the drawdown portfolios arebetter than the Sharpe ratio for the index.Figure 2 shows graphically, for the EURO STOXX 50, the out-of-sample performance foreach of the MINAVG and MINMAX portfolios summarised in Table 2, as well as the performanceof the index (all values normalised to one at the start of the out-of-sample period).Figure 2: Out-of-sample performance: EURO STOXX 50 V a l ue EURO STOXX 50MINAVG 0.1MINAVG 1MINMAX 0.1MINMAX 1
One item of note for the EURO STOXX 50 is the significant difference between the max-imum drawdown values in-sample and out-of-sample (e.g. 3.41% and 19.51% respectively forMINAVG 0.1). This arises because the in-sample maximum drawdown is an average over ap-proximately 180 rebalances of 180 maximum drawdowns, each computed from a time seriescontaining 30 values; whereas the maximum drawdown out-of-sample is a single value computedfor a time series containing approximately 1800 values. As such these maximum drawdownvalues are not directly comparable.For the FTSE 100 we can see from Table 2 that in-sample all four drawdown portfolioshave superior return performance, as well as superior maximum and average drawdown perfor-mance when compared to the index itself. Out-of-sample Table 2 shows that three of the fourdrawdown portfolios effectively dominate the index in terms of (cumulative) return achieved.With respect to drawdown out-of-sample two of the four drawdown portfolios (MINAVG 0.1and MINMAX 0.1) dominate the index with respect to both drawdown measures. Sharpe ratios17or all four of the drawdown portfolios are better than the Sharpe ratio for the index.For the S&P 500 we can see from Table 2 that in-sample all four drawdown portfolios havesuperior return performance, as well as superior maximum and average drawdown performancewhen compared to the index itself.Of particular note for the S&P 500 is the high percentage of optimal solutions obtained(higher than the corresponding cases for either the EURO STOXX 50 or the FTSE 100). Forthe S&P 500 we have the number of assets N equal to 500, much larger that the values of N for the EURO STOXX 50 ( N = 50) or FTSE 100 ( N = 100). Although a larger value for N increases the number of decision variables and linear constraints the number of nonlinearconstraints depends only upon T (see equation (2)), and all the cases examined in Table 2 have T = 30. The reason why we have a much higher percentage of optimal solutions found for theS&P 500 is, we believe, due to the fact that for this index the growth over the period is muchlarger than for either the EURO STOXX 50 or the FTSE 100 (compare the average returnon the index given in Table 2 for the S&P 500 with that for the EURO STOXX 50 or theFTSE 100). If an index is growing then one might reasonably suppose that it is easier to find aportfolio with drawdown zero, or close to zero (i.e. a portfolio that effectively grows for all/mostof the in-sample period). The very low average in-sample drawdown values seen in Table 2 forthe S&P 500 support this argument that because of the growth in the index optimal drawdownportfolios will be more easily found.For the S&P 500 we have from Table 2 that out-of-sample all four drawdown portfoliosdominate the index in terms of maximum and average drawdown, as well as in terms of theSharpe ratio. Over the (approximately 1800 day) out-of-sample period all of the drawdownportfolios exceed the index in terms of cumulative return for over 89% of days.The average values in Table 2 show that all of the drawdown portfolio cases examineddominate the index in terms of return, maximum drawdown and average drawdown both in-sample and out-of-sample. Sharpe ratios for all four of the drawdown portfolios are better thanthe corresponding Sharpe ratios for the indices. Of particular note here is the performance ofthe MINMAX 0.1 portfolio which out-of-sample has the highest return and the lowest maximumdrawdown with the associated drawdown portfolios exceeding the index for 99.6% of days in theout-of-sample period. For drawdown portfolios involving shorting, so using formulations MINAVG-S and MINMAX-S, we set the maximum proportion limits ( δ Li and δ Si equations (25),(26)) to 0.1, so at most10% of portfolio value could be invested (long or short) in any asset. We used ∆ S = 0 . T that couldbe placed in short positions (across all assets) was at most 10%. Due to the nature of shorting(where short positions generate funds to purchase assets) we typically have ∆ L = 1 + ∆ S , sohere we used ∆ L = 1 . Date V a l ue EURO STOXX 50MINAVG−S 0.1MINMAX−S 0.1
For the FTSE 100 only maximum drawdown is better out-of-sample than the equivalentcases without shorting. For the S&P 500 it also seems clear that (on balance) allowing shortingdoes not pay off in terms of improved out-of-sample performance.Comparing the results for MINAVG-S 0.1 and MINMAX-S 0.1 with shorting to the associatedindex values we have that for all three instances the out-of-sample shorting results correspond toportfolios that are superior to the index in terms of maximum drawdown, average drawdown andSharpe ratio. For the EURO STOXX 50 and the FTSE 100 the drawdown portfolios provide anaverage daily return that exceeds that of the index. For the S&P 500 the returns provided falljust below that of the index, although note that, as occurred with the non-shorting portfolios,we have that over the (approximately 1800 day) out-of-sample period the drawdown portfoliosexceed the index in terms of cumulative return for over 89% of days.Although obviously dependent on the instances examined, as well as the various parametervalues adopted (such as for T , D , δ i , δ Li , δ Si , ∆ L and ∆ S ), these results would indicate thatlong-only drawdown portfolios would be preferable to drawdown portfolios that allow shorting.However we would note that the drawdown portfolios with shorting that we examined still19ffectively dominate the index. In this paper we have considered the problem of minimising drawdown in a portfolio of financialassets. We discussed how the work presented in this paper follows the general pattern of dealingwith a financial portfolio optimisation problem that first appeared in the financial literature nowbeing considered from an Operational Research perspective.We formulated the problem (minimising either average drawdown, maximum drawdown, ora weighted combination of the two) as a nonlinear program and showed how it can be partiallylinearised by replacing one of the nonlinear constraints by equivalent linear constraints. Ourformulation incorporated cash inflow/outflow, and can be used either to create an initial portfoliofrom cash or to rebalance an existing portfolio. Transaction costs associated with buying orselling an asset were included.Computational results were presented (generated using the nonlinear solver SCIP) for threetest instances drawn from the EURO STOXX 50, the FTSE 100 and the S&P 500 with dailyprice data over the period 2010-2016. We considered long-only drawdown portfolios as well asportfolios with both long and short positions.Our results showed that (within the computational time limits imposed) we were able to findglobally optimal minimal drawdown portfolios for a significant percentage of the cases examined.Our results indicated that (on average) our minimal drawdown portfolios dominated the marketindices in terms of return, Sharpe ratio, maximum drawdown and average drawdown over the(approximately 1800 trading day) out-of-sample period.Finally we would note here that we believe that the work presented in this paper indicatesthe value of adopting an Operational Research perspective on a financial portfolio optimisationproblem. 20 a b l e : C o m pu t a t i o n a l r e s u l t s I n s t a n ce C a s e P r o p o r t i o n I n - s a m p l e O u t - o f - s a m p l e li m i t A v e r ag e M a x i m u m A v e r ag e A v e r ag e O p t i m a l A v e r ag e M a x i m u m A v e r ag e A nnu a li s e d P e r ce n t ag e o f d a y s d a il y d r a w d o w nd r a w d o w n c o m pu t a t i o n s o l u t i o n s d a il y d r a w d o w nd r a w d o w nSh a r p ec u m u l a t i v e p o r t f o li o r e t u r n ( % )( % )t i m e ( s ec s ) f o und ( % ) r e t u r n ( % )( % ) r a t i o v a l u ee x cee d s i nd e x E U R O S T O XX I nd e x . . . . . . . M I NAV G . . . . . . . . . . . M I NAV G . . . . . . . . . . M I N M AX . . . . . . . . . . . M I N M AX . . . . . . . . . . M I NAV G - S . . . . . . . . . . . M I N M AX - S . . . . . . . . . . . F T S E I nd e x . . . . . . . M I NAV G . . . . . . . . . . . M I NAV G . . . . . . . . . . M I N M AX . . . . . . . . . . . M I N M AX . . . . . . . . . . M I NAV G - S . . . . . . . . . . . M I N M AX - S . . . . . . . . . . . S & P I nd e x . . . . . . . M I NAV G . . . . . . . . . . . M I NAV G . . . . . . . . . . M I N M AX . . . . . . . . . . M I N M AX . . . . . . . . . . M I NAV G - S . . . . . . . . . . M I N M AX - S . . . . . . . . . . . A v e r ag e I nd e x . . . . . . . M I NAV G . . . . . . . . . . . M I NAV G . . . . . . . . . . M I N M AX . . . . . . . . . . . M I N M AX . . . . . . . . . . M I NAV G - S . . . . . . . . . . . M I N M AX - S . . . . . . . . . . . eferences ∼ stefan/minlpsoft.pdf Last accessed August13 2019.[14] Canakgoz NA, Beasley JE. Mixed-integer programming approaches for index tracking andenhanced indexation. European Journal of Operational Research 2009;196(1):384–399.[15] Cesarone F, Scozzari A, Tardella F. A new method for mean-variance portfolio optimizationwith cardinality constraints. Annals of Operations Research 2013;205(1):213–234.[16] Chang TJ, Meade N, Beasley JE, Sharaiha YM. Heuristics for cardinality constrained port-folio optimisation. Computers & Operations Research 2000;27(13):1271-1302.[17] Chekhlov A, Uryasev S, Zabarankin M. Portfolio optimization with drawdown constraints.In: Asset and Liability Management Tools, ed. B. Scherer (Risk Books, London, 2003)pp.263–278.[18] Chekhlov A, Uryasev S, Zabarankin M. Drawdown measure in portfolio optimization. In-ternational Journal of Theoretical and Applied Finance 2005;8(1):13–58.[19] Chen C, Kwon RH. Robust portfolio selection for index tracking. Computers & OperationsResearch 2012;39(4):829-837.20] Chen XF, Landriault D, Li B, Li DC. On minimizing drawdown risks of lifetime investments.Insurance: Mathematics and Economics 2015;65(November):46–54.[21] Filippi C, Guastaroba G, Speranza MG. A heuristic framework for the bi-objective en-hanced index tracking problem. OMEGA - International Journal of Management Science2016;65:122–137.[22] Gao JJ, Li D. Optimal cardinality constrained portfolio selection. Operations Research2013;61(3):745–761.[23] Giuzio M, Ferrari D, Paterlini S. Sparse and robust normal and t-portfolios by penalized Lq-likelihood minimization. European Journal of Operational Research 2016;250(1):251-261.[24] Gold M, Ali P. Portable alpha strategies offer greater scope. JASSA 2001;4(Summer):23–26.[25] Goldberg LR, Mahmoud O. Drawdown: from practice to theory and back again. Mathe-matics and Financial Economics 2017;11(3):275–297.[26] Grossman SJ, Zhou ZQ. Optimal investment strategies for controlling drawdowns. Mathe-matical Finance 1993;3(3):241–276.[27] Guastaroba G, Mansini R, Ogryczak W, Speranza MG. Linear programming models basedon Omega ratio for the enhanced index tracking problem. European Journal of OperationalResearch 2016;251(3):938–956.[28] Guijarro F. A similarity measure for the cardinality constrained frontier in the mean-variance optimization model. Journal of the Operational Research Society 2018;69(6):928–945.[29] Hardoroudi ND, Keshvari A, Kallio M, Korhonen P. Solving cardinality constrained mean-variance portfolio problems via MILP. Annals of Operations Research 2017;254(1-2):47–59.[30] Jacob NL. A limited-diversification portfolio selection model for the small investor. TheJournal of Finance 1974;29(3):847–856.[31] Lee Y, Kim MJ, Kim JH, Jang JR, Kim WC. Sparse and robust portfolio selection viasemi-definite relaxations. 2019, to appear in Journal of the Operational Research Society.[32] Lhabitant F. Hedge funds: Quantitative insights. Wiley, London 2009.[33] Liagkouras K, Metaxiotis K. A new efficiently encoded multiobjective algorithm for thesolution of the cardinality constrained portfolio optimization problem. Annals of OperationsResearch 2018;267(1-2):281–319.[34] Liagkouras K, Metaxiotis K. Handling the complexities of the multi-constrained portfo-lio optimization problem with the support of a novel MOEA. Journal of the OperationalResearch Society 2018;69(10):1609–1627.[35] Lintner J. The valuation of risk assets and the selection of risky investments in stockportfolios and capital budgets. The Review of Economics and Statistics 1965;47(1):13–37.[36] Liu YJ, Zhang WG, Wang JB. Multi-period cardinality constrained portfolio selection mod-els with interval coefficients. Annals of Operations Research 2016;244(2):545–569.[37] Madhogarhia PK, Lam M. Dynamic asset allocation. Journal of Asset Management2015;16(5):293–302.[38] Magdon-Ismail M, Atiya AF. Maximum drawdown. Risk 2004;17(10):99–102.[39] Magdon-Ismail M, Atiya AF, Pratap A, Abu-Mostafa YS. On the maximum drawdown ofa Brownian motion. Journal of Applied Probability 2004;41(1):147–161.[40] Mahmoud O. The temporal dimension of risk. Journal of Risk 2017;19(3):57–83.[41] Maringer D, Kellerer H. Optimization of cardinality constrained portfolios with a hybridlocal search algorithm. OR Spectrum 2003;25(4):481-495.[42] Markowitz H. Portfolio selection. The Journal of Finance 1952;7(1):77–91.[43] Meade N, Beasley JE. Detection of momentum effects using an index out-performancestrategy. Quantitative Finance 2011;11(2):313–326.[44] Mezali H, Beasley JE. Quantile regression for index tracking and enhanced indexation.Journal of the Operational Research Society 2013;64(11):1676–1692.45] Mohr E, Dochow R. Risk management strategies for finding universal portfolios. Annals ofOperations Research 2017;256(1):129–147.[46] Pope PF, Yadav PK. Discovering errors in tracking error. Journal of Portfolio Management1994;20(2):27–32.[47] Pospisil L, Vecer J. Partial differential equation methods for the maximum drawdown. TheJournal of Computational Finance 2008;12(2):59–76.[48] Pospisil L, Vecer J. Portfolio sensitivity to changes in the maximum and the maximumdrawdown. Quantitative Finance 2010;10(6):617–627.[49] Riepe M, Werner M. Are enhanced index mutual funds worthy of their name? Journal ofInvesting 1998;7(2):6–15.[50] Roman D, Mitra G, Zverovich V. Enhanced indexation based on second-order stochasticdominance. European Journal of Operational Research 2013;228(1):273–281.[51] Rudd A. Optimal selection of passive portfolios. Financial Management 1980;9(1):57–66.[52] Sant’Anna LR, Filomena TP, Guedes PC, Borenstein D. Index tracking with controllednumber of assets using a hybrid heuristic combining genetic algorithm and non-linear pro-gramming. Annals of Operations Research 2017;258(2):849–867.[53] SCIP: Solving constraint integer programs. Available from http://scip.zib.de/ Last accessedAugust 13 2019.[54] Scowcroft S, Sefton J. Enhanced indexation. In: Advances in portfolio construction andimplementation, Satchell SE and Scowcroft A (eds) 2003, pages 95–124.[55] Scozzari A, Tardella F, Paterlini S, Krink T. Exact and heuristic approaches for the indextracking problem with UCITS constraints. Annals of Operations Research 2013;205(1):235-250.[56] Sharpe WF. Capital asset prices: A theory of market equilibrium under conditions of risk.The Journal of Finance 1964;19(3):425–442.[57] Sharpe WF. Mutual fund performance. The Journal of Business 1966;39(1):119–138.[58] Sharpe WF. Adjusting for risk in portfolio performance measurement. The Journal of Port-folio Management 1975;1(2),Winter:29–34.[59] Sharpe WF. The Sharpe ratio. The Journal of Portfolio Management 1994;21(1),Fall:49–58.[60] Treynor J. Towards a theory of market value of risky assets. 1961, Unpublished manuscript.Edited version available fromhttp://dx.doi.org/10.2139/ssrn.628187 Last accessed August 13 2019.[61] Vigerske S. Private communication, April 2017.[62] Vigerske S, Gleixner A. SCIP: Global optimization of mixed-integer nonlinear programs ina branch-and-cut framework. ZIB Report 16-24 (May 2017). Available fromhttps://opus4.kobv.de/opus4-zib/frontdoor/index/index/docId/5937 Last accessed August13 2019.[63] Vigerske S, Gleixner A. SCIP: Global optimization of mixed-integer nonlinear programs ina branch-and-cut framework. Optimization Methods and Software 2018:33(3):563–593.[64] Wu LC, Tsai IC. Three fuzzy goal programming models for index portfolios. Journal of theOperational Research Society 2014;65(8):1155–1169.[65] Xidonas P, Hassapis C, Mavrotas G, Staikouras C, Zopounidis C. Multiobjective portfoliooptimization: bridging mathematical theory with asset management practice. Annals ofOperations Research 2018;267(1–2):585–606.[66] Yao HX, Lai YZ, Maa QH, Zheng HB. Characterization of efficient frontier for mean–variance model with a drawdown constraint. Applied Mathematics and Computation2013;220:770–782.[67] Yu XJ, Xie SY, Xu WJ. Optimal portfolio strategy under rolling economic maximum draw-down constraints. Mathematical Problems in Engineering 2014:Article ID 787943, availablefrom http://dx.doi.org/10.1155/2014/787943 Last accessed August 13 2019.68] Zabarankin M, Pavlikov K, Uryasev S. Capital Asset Pricing Model (CAPM) with draw-down measure. European Journal of Operational Research 2014;234(2):508–517.[69] Zhao ZH, Xu FM, Wang MH, Zhang CY. A sparse enhanced indexation model with (cid:96) /2