An Empirical Analysis of Alternative Portfolio Selection Criteria

## Keywords

portfolio optimisation

## Review Status

Unreviewed

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# Abstract

In modern portfolio theory, financial portfolios are characterised by a desired property, the ‘reward’, and something undesirable, the ‘risk’. While these properties are commonly identified with mean and variance of returns, respectively, we test alternative specifications like partial and conditional moments, quantiles, and drawdowns. More specifically, we analyse the empirical performance of portfolios selected by optimising risk-reward ratios constructed from these alternative functions. We find that these portfolios in many cases outperform our benchmark (minimum-variance), in particular when long-run returns are concerned. However, we also find that all the strategies tested seem quite sensitive to relatively small changes in the data. The main theme throughout our results is that minimising risk, as opposed to maximising reward, often leads to good out-of-sample performance. In contrast, adding a reward-function to the selection criterion improves a given strategy often only marginally.

# Introduction

An alleged weakness of mean-variance optimisation (Markowitz [9]) is that selecting portfolios only on the basis of the first two moments of portfolio returns should not be appropriate, given the considerable body of evidence of the non-Gaussian nature of financial time series. To investigate this criticism, we empirically evaluate portfolio selection criteria that have been proposed as alternatives to the mean-variance rule, thus we replace the mean and variance by alternative measures of ‘reward’ and ‘risk’. These alternative functions explicitly take into account certain empirical regularities (‘stylised facts’) of financial prices like fat tails or asymmetric return distributions. We provide robustness checks for our empirical results by trying to capture the uncertainty around the point estimates that are usually presented in empirical studies on portfolio optimisation. To solve the portfolio problems we use a heuristic optimisation technique, Threshold Accepting, which is capable of optimising portfolios under all the different selection criteria discussed. Threshold Accepting works directly on the empirical distribution function of portfolio returns, it does not require approximating the data by a parametric distribution.

Our study links to two strands of financial literature. Firstly, there is the large number of studies investigating the empirical performance of mean-variance optimisation; the main finding here is that a straightforward estimation of the required parameters, that is the assets' means and their variance-covariance matrix, and their ‘plugging-into’ the objective function, very often leads to undiversified portfolios that perform poorly out-of-sample. These estimation difficulties are by now well documented; Brandt [3] gives a very good overview.

Secondly, there are several theoretical studies on desirable properties of risk and performance measures (notably Artzner et al. [1]), in particular such measures that capture non-Gaussian properties of the data. (The Sharpe ratio [11], though probably the most widely used mapping of a portfolio's desirability into a single real number, inherits the alleged weakness of mean-variance optimisation.) Thus, in recent years, a large number of alternative risk and performance measures have been proposed. For an overview, see for example Bacon [2].

In practice, these new performance measures are mainly used for ex-post comparison of different funds or strategies, but rarely for ex-ante optimisation. The main reason is the difficulty to optimise portfolios with such objective functions, in particular in conjunction with constraints and real-world data, since the resulting optimisation problems are often not convex and cannot be solved with standard techniques like linear or quadratic programmes.

To judge their effectiveness, we decompose alternative performance measures into their building blocks, like partial moments or quantiles, and then test whole classes of performance measures (risk-reward ratios) based on these building blocks. As an example, we do not just investigate the ratio of lower to upper partial moment with exponent one (the Omega function, see Keating and Shadwick [8]), but test such ratios of partial moments for many different exponents. This indicates whether, generally, partial moments are an effective element to be included in portfolio selection criteria.

Our overall results indicate that incorporating alternative reward and risk measures into the portfolio optimisation process does result in improvements over our benchmark, the minimum-variance (MV) portfolio. This improvement is clearest when long-run returns are concerned, where for instance portfolios selected by minimising lower partial moments or functions of the drawdown often perform very well. When risk-adjusted returns are considered, the results become less clear, but mainly because it is less clear how to adjust returns for risk in these cases. Since a protracted discussion is not the purpose of this paper, we settled on computing the Sharpe ratio for all strategies that we tested. We find that many alternative risk measures have higher Sharpe ratios, albeit often only marginally so. For a true ‘horse race’ between different objective functions, a more careful investigation of the risk-return characteristics of selected portfolios will be necessary, though.

Regarding the estimation problems, our study suggests that changing the objective function, in particular using selection criteria that only consider functions of the lower tail of the return distribution as risk, considerably exacerbates the estimation problem. Instead on relying solely on historical data, we thus use a resampling method to approximate the distribution function of returns.

# The investor's problem

## Reward and risk

An objective function is a mapping of portfolio returns into a real number. We will always assume a given return sample $r=[r_{1}\; r_{2}\; \ldots \; r_{{n_S}}]'$, where $n_S$ is the number of observations or, more generally, the number of scenarios. We will not assume a parametric distribution for the returns and always work with the empirical cumulative distribution function (CDF) of our sample.

Markowitz's selection rule states to choose portfolios that are mean-variance efficient. A corresponding objective function, to be minimised, can be written as

(1)
\begin{align} \displaystyle \mathcal{M}_2(r) - \lambda\,\mathcal{M}_1(r)\, , \end{align}

where $\mathcal{M}_1(r)$ and $\mathcal{M}_2(r)$ are the mean and variance of returns, respectively, and $\lambda$ is a measure of risk-aversion. This function includes only mean and variance, without regard to the overall shape of the return distribution. Furthermore, it only cares for final wealth, not for the path that wealth takes between time $0$ and $T$.

More generally, we may argue that a portfolio of risky assets has a desirable property (here, mean return) and an undesirable one (here, variance of returns), where we refer to these properties as ‘reward’ and ‘risk’, respectively. Building blocks for such functions may be the following quantities.

### Partial moments

For a given sample of portfolio returns, the identity

(2)
\begin{align} r = \!\! \underbrace{\phantom{x}r_d\phantom{x}}_{\textrm{desired return}} \!\!\! + \, \underbrace{\max(r - r_d,0)}_{\textrm{upside}} - \underbrace{\max(r_d - r,0)}_{\textrm{downside}} \end{align}

always holds for a chosen ‘desired return’ threshold $r_d$ (Scherer [10]). Partial moments are a convenient way to distinguish between returns above and below $r_d$, that is the ‘upside’ and ‘downside’ terms in Equation (2); they can be estimated as

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\begin{split} \mathcal{P}^{+}_{\gamma}(r_d) &= \frac{1}{{n_S}}\sum_{\phantom{x}r>r_d} \left(r - r_d \right)^\gamma\, , \\ \mathcal{P}^{-}_{\gamma}(r_d) &= \frac{1}{{n_S}}\sum_{\phantom{x}r<r_d} \left(r_d - r \right)^\gamma\, . \end{split}

The superscripts $+$ and $-$ indicate the tail (ie, upside and downside). Partial moments take two more parameters: an exponent $\gamma$, and the threshold $r_d$. The expression ‘$r > r_d$’ indicates to sum only over those returns that are greater than $r_d$.

A well-known partial moment is the semivariance, given by $\mathcal{P}^{-}_{2}(\mathcal{M}_1(r))$. The square root of this expression, sometimes called ‘downside deviation’, is used as the risk function in several performance measures like the Sortino and the Upside Potential ratio (Sortino [12]), or Kappa$_2$ (Kaplan and Knowles [7]).

### Conditional moments

Conditional moments can be estimated by

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\begin{split} \mathcal{C}^{+}_{\gamma}(r_d) &= \frac{1}{\#\{r>r_d\}}\sum_{\phantom{x}r>r_d} \left(r - r_d \right)^\gamma\, ,\\ \mathcal{C}^{-}_{\gamma}(r_d) &= \frac{1}{\#\{r<r_d\}}\sum_{\phantom{x}r<r_d} \left(r_d - r \right)^\gamma\, , \end{split}

where again $+$ and $-$ indicate the tail, and ‘$\#\{r>r_d\}$’ is a counter for the number of return observations higher than $r_d$.

For a threshold $r_d$, the lower partial moment of order $\gamma$ equals the lower tail's conditional moment of the same order, times the lower partial moment of order 0. That is,

(5)
\begin{split} \mathcal{P}^{+}_{\gamma}(r_d) &= \mathcal{C}^{+}_{\gamma}(r_d) \mathcal{P}^{+}_{0}(r_d)\, ,\\ \mathcal{P}^{-}_{\gamma}(r_d) &= \mathcal{C}^{-}_{\gamma}(r_d) \mathcal{P}^{-}_{0}(r_d)\, . \end{split}

The partial moment of order 0 is simply the probability of obtaining a return beyond $r_d$. Still, for a given $r_d$, both conditional and partial moments convey different information, since both the probability and the conditional moment need to be estimated from the data to obtain a partial moment. In other words, the conditional moment measures the magnitude of returns around $r_d$, while the partial moment also takes into account the probability of such returns.

(If $r_d$ is chosen equal to some quantile of the return distribution, as is the convention for conditional moments like Expected Shortfall, we do not centre around $r_d$, but replace $r$ by $\max(r,0)$ in moments for the upper tail, and by $\min(r,0)$ for the lower tail. See the paper for the rationale of this approach.)

### Quantiles

A quantile of the CDF of a sample $r$ is defined as

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\begin{align} \mathcal{Q}_{q} &= \operatorname{CDF}^{-1}(q) = \min\{r\,|\,\operatorname{CDF}(r) \geq q\}\, , \end{align}

where $q$ may range from 0% to 100% (we drop the %-sign in subscripts). In words, the $q$th quantile is a number $\mathcal{Q}_{q}$ such that $q$ of the observations are smaller, and $(100\%-q)$ larger than $\mathcal{Q}_{q}$.

Value-at-Risk (VaR) is the loss only to be exceeded with a given, usually small, probability at the end of a defined horizon. Thus, VaR is a quantile of the return distribution; in our notation, VaR for a probability of 1% can be written as $\mathcal{Q}_{1}$. Quantiles may also be used as reward measures; we could for example maximise a higher quantile (eg, the 90th).

### Drawdowns

The functions described so far are usually applied to the distribution of final wealth. We may, however, as well observe the evolution of a portfolio over time, and compute the drawdown $\mathcal{D}$ of our portfolio. In our study, we look at three drawdown functions: the drawdown's mean, its maximum and its standard deviation. (For a tutorial on how to compute drawdown, go here.)

## The optimisation problem

Our objective functions $\Phi$ will be ratios of risk and reward to be minimised. Ratios have the advantage of being easy to communicate and interpret. Even though numerically, linear combinations are often more stable and thus preferable, working with ratios practically never caused problems in our experiments. We generally ‘safeguarded’ our objective function, though, for cases where numerator or denominator could switch signs while moving through the search space. Ratios that use the mean return for reward, for instance, are not directly interpretable anymore if mean returns are negative.

With $x = [x_1\; x_2\; \ldots\; x_{\na}]'$ the holdings of the individual assets and $\mathcal{J}$ the set of assets in the portfolio, the problem, including constraints, can be written as

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\begin{array} {l} \displaystyle \min_{x} \; \Phi(r) \\ x_j^{\inf} \leq x_j \leq x_j^{\sup} \qquad j \in \mathcal{J}\, , \\ K_{\inf} \leq \#\{\mathcal{J}\} \leq K_{\sup} \, . \end{array}

$x_j^{\inf}$ and $x_j^{\sup}$ are minimum and maximum holding sizes, respectively, for those assets included in the portfolio (ie, those in $\mathcal{J}$). $K_{\inf}$ and $K_{\sup}$ are cardinality constraints which set a minimum and maximum number of assets in $\mathcal{J}$. We do not include minimum return constraints.

# Data and methodology

## Data and Software

We use a heuristic method, Threshold Accepting, to optimise our portfolio holdings. The great advantage of Threshold Accepting is its flexibility, as it allows to optimise portfolio holdings under all the discussed objective functions (also, in fact, if the risk and reward measures were added as constraints). The optimisation algorithms are written for Matlab R2008a and can be downloaded from http://comisef.eu. A single portfolio optimisation over several thousand scenarios takes less than 10 seconds on a PC with an Intel T9300 2.5 GHz (this includes the generation of the scenarios). Still, the large number strategies tested and in particular the robustness checks would have amounted to a serial computing time of more than two years. Thus, many of the computations were distributed with Matlab's Parallel Computing Toolbox on the Myrinet Cluster of the University of Geneva. Myrinet is a Linux Cluster with 32 nodes, each a Sun V60x dual Intel Xeon 2.8GHz with 2 GB of RAM. For more details see http://spc.unige.ch.

The data set consists of more than 500 price series of European companies, all denominated in euro, spanning the period from January 1998 to March 2008. For each company, we also have a market capitalisation series; for a given period, we keep only companies with a reasonable minimum market value (more than EUR4 billion) as a proxy for sufficient market liquidity.

We set $x^{\inf}=1\%$ and $x^{\sup}=5\%$; an upper cardinality is 50; there is no minimum return constraint. We do not include a riskless asset. Since our algorithm works with actual position sizes, that is integer numbers, a small fraction of less than 1% of the portfolio is usually left uninvested.

## The distribution of portfolio returns

The optimisation procedure used for objective functions that rely on the CDF of end-of-period portfolio returns (ie, not on the path of portfolio wealth) is scenario-based and can be divided into two stages: constructing scenarios, and then finding the portfolio that optimises the selection criterion for these scenarios. This is equivalent to working with the empirical CDF (the step function). In the simplest case, every historical return constitutes one scenario, hence explicitly modelling the data is not necessary for our algorithm. Our test suggest, however, that the method of scenario creation strongly influences the out-of-sample performance of selected portfolios. Thus we use a resampling-based procedure to create scenarios; details are described in the paper.

For objective functions that need a path of portfolio wealth (drawdowns), we work with historical data, since our scenario generation method does not capture serial dependencies.

## Moving-window backtest, rebalancing and transaction costs}

We conduct rolling-window backtests for the different strategies with a historical window of length $H$, and an out-of-sample holding period of length $F$. $H$ was set to around 250 days, that is one year; $F$ was set to around 90 days (three months). Thus we optimise at point in time $t_1$ on data from $t_1-H$ to $t_1-1$, the resulting portfolio is held until $t_2 = t_1+F$. At this point, a new optimal portfolio is computed, using data from $t_2-H$ until $t_2-1$, and the existing portfolio is rebalanced. This new portfolio is then held until $t_3=t_2+F$, and so on.

Transaction costs are set to 10 basis points which should approximately reflect the actual cost of an institutional investor. We also ran backtests with higher transaction costs, and, alternatively, with turnover constraints (not reported here). We found that transaction costs did not qualitatively influence the results, whereas turnover constraints led to a markedly worse performance. This result is of course conditional on our relatively low frequency of rebalancing.

## Uncertainty

Assume a few return observations are randomly selected from the original sample that we use to find an optimal portfolio, ie, the ‘in-sample’ sample, and deleted. Thus the historical time series have changed, the scenarios will be created differently, and the composition of the optimal portfolio will change. If the portfolio selection method is robust, we would expect the portfolio optimised on the perturbed data to be similar to the original optimal portfolio, and thus to exhibit a similar performance. After all, the change in the historical data is only small. Repeating this procedure many times, we obtain a collection of out-of-sample wealth paths. The distribution of these paths gives an indication of the sensitivity of a particular strategy to a particular data set. This procedure is analogous to repeatedly re-estimating a regression equation from jackknifed data to obtain a sampling distribution of the coefficients. For our problem, however, it appears difficult to judge what a given norm of the difference between two weight-vectors practically means. Thus we do not compare the differences in the obtained portfolio weights (which would be the counterparts to the regression coefficients), but we compare the differences in resulting out-of-sample results (to stay with the regression model analogy, we rather look at a distribution of forecast errors). This confounds the different sources of uncertainty described above, but still should give a a rough idea of robustness. The result of this procedure is not just a point estimate for the performance of a specific strategy, but rather a collection of realisations.

# Results

We discuss some of our findings here; a complete list of the strategies' results can be found in the paper.

## Partial moments and conditional moments

### Partial moments

For a long-term investor, who is mainly interested in final wealth, objective functions based on partial moments seem to offer a superior alternative to minimum-variance. From our tests, a robust suggestion on how to construct the objective function is not to rely on upper partial moments of a higher order (higher than 1.5, say). Portfolios based on objective functions with $\mathcal{P}^{+}$ of order 3 or higher perform always poorly, sometimes even resulting in losses over the whole test period. In any case, the curvature of the objective function should be more pronounced for losses than for gains. That is, the exponent $\gamma$ of the moments should be chosen higher for losses than for gains.

All tables include quartile plots, that is ‘reduced-form’ boxplots (Tufte [13]): they only print the median (the dot in the middle) and the whiskers of the boxplot. The following figure illustrates their construction.

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The limits of the whiskers are, just like in the standard box plot, given by $\max\left[\mathcal{Q}_{25} - \operatorname{IQR}, \min(r)\right]$ and $\min\left[\mathcal{Q}_{75}+ \operatorname{IQR}, \max(r)\right]$.

A further result is that using partial moments of order 0 (ie, frequencies of losses or gains) works well in many instances. The ratio of the frequencies of losses to frequencies of gains, for instance, is a successful objective function; combining the frequencies of gains as the reward with a lower partial moment of higher order gives consistently good results.

The following table gives the median returns over the out-of-sample paths of selected strategies. The symbol ‘$c$’ stands for ‘constant’, so for instance reward equal to $c$ and risk equal to $\mathcal{M}_2$ gives the MV-portfolio. We include quartile plots, a variant of box plots, to illustrate the distribution of the returns. The examples at the bottom of the table demonstrate the effect of relying on upper partial moments of higher order.

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\begin{align} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % selected PM, I %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \setlength{\unitlength}{1.0pt} \begingroup \everymath{\scriptstyle} \begin{tabular}{p{1.1cm}|p{1.1cm}|p{1.1cm}ll} reward&risk & median & \begin{picture}(100,1) \put(0,2){\line(1,0){200}} \put(0,2){\line(0,-1){4}} \put(10,2){\line(0,-1){2}} \put(20,2){\line(0,-1){2}} \put(30,2){\line(0,-1){2}} \put(40,2){\line(0,-1){2}} \put(50,2){\line(0,-1){4}} \put(60,2){\line(0,-1){2}} \put(70,2){\line(0,-1){2}} \put(80,2){\line(0,-1){2}} \put(90,2){\line(0,-1){2}} \put(100,2){\line(0,-1){4}} \put(110,2){\line(0,-1){2}} \put(120,2){\line(0,-1){2}} \put(130,2){\line(0,-1){2}} \put(140,2){\line(0,-1){2}} \put(150,2){\line(0,-1){4}} \put(160,2){\line(0,-1){2}} \put(170,2){\line(0,-1){2}} \put(180,2){\line(0,-1){2}} \put(190,2){\line(0,-1){2}} \put(200,2){\line(0,-1){4}} \put(195,10){{\scriptsize 20\%}} \put(145,10){{\scriptsize 15\%}} \put(95,10){{\scriptsize 10\%}} \put(45,10){{\scriptsize 5\%}} \put(-5,10){{\scriptsize 0\%}} \end{picture}\phantom{XXXXXXXXXXXXXX}&\\ %\usecolor{black} $c$&$\mathcal{M}_2$&$13.76$&\begin{picture}(100,1)\put(110.4,2){\line(1,0){19.5}}\put(137.6,2){\circle*{2}}\put(142.9,2){\line(1,0){18.7}}\end{picture} &{\tiny MV-portfolio} \\ [0.5ex] $\mathcal{P}^{+}_{0}$& $\mathcal{P}^{-}_{0}$& $15.56$&\begin{picture}(100,1)\put(131.8,2){\line(1,0){18.1}}\put(155.6,2){\circle*{2}}\put(162.6,2){\line(1,0){16.1}}\end{picture}\\ $\mathcal{P}^{+}_{1}$& $\mathcal{P}^{-}_{1}$& $15.83$&\begin{picture}(100,1)\put(132.5,2){\line(1,0){17.3}}\put(158.3,2){\circle*{2}}\put(164.7,2){\line(1,0){15}}\end{picture}&{\tiny Omega} \\ $\mathcal{P}^{+}_{1}$& $\mathcal{P}^{-}_{2}$& $15.56$&\begin{picture}(100,1)\put(138.3,2){\line(1,0){11.9000000000000}}\put(155.6,2){\circle*{2}}\put(160.1,2){\line(1,0){14.3000000000000}}\end{picture}&{\tiny Upside Potential ratio}\\ $\mathcal{P}^{+}_{1.5}$& $\mathcal{P}^{-}_{1.5}$& $16.23$&\begin{picture}(100,1)\put(137.2,2){\line(1,0){18.2000000000000}}\put(162.3,2){\circle*{2}}\put(169.5,2){\line(1,0){19.8}}\end{picture}\\ $\mathcal{P}^{+}_{1.5}$& $\mathcal{P}^{-}_{2}$& $15.95$&\begin{picture}(100,1)\put(136.4,2){\line(1,0){17.8}}\put(159.5,2){\circle*{2}}\put(166.1,2){\line(1,0){14.7}}\end{picture}\\ $\mathcal{P}^{+}_{2}$& $\mathcal{P}^{-}_{2}$& $15.32$&\begin{picture}(100,1)\put(125.7,2){\line(1,0){21.4}}\put(153.2,2){\circle*{2}}\put(163.3,2){\line(1,0){24.1}}\end{picture}&{\tiny Volatility skewness}\\[0.5ex] $\mathcal{P}^{+}_{3}$& $\mathcal{P}^{-}_{2}$& $6.35$&\begin{picture}(100,1)\put(32.5,2){\line(1,0){20.3}}\put(63.5,2){\circle*{2}}\put(73,2){\line(1,0){30.2}}\end{picture}\\ $\mathcal{P}^{+}_{4}$& $\mathcal{P}^{-}_{2}$& $2.38$&\begin{picture}(100,1)\put(-8,2){\line(1,0){22.3}}\put(23.8,2){\circle*{2}}\put(31.2,2){\line(1,0){25.4}}\end{picture}\\ $\mathcal{P}^{+}_{4}$& $\mathcal{P}^{-}_{3}$& $7.76$&\begin{picture}(100,1)\put(48.2,2){\line(1,0){20.1}}\put(77.6,2){\circle*{2}}\put(91,2){\line(1,0){34.2}}\end{picture}\\ &&& \begin{picture}(100,1) \put(0,2){\line(1,0){200}} \put(0,2){\line(0,1){4}} \put(10,2){\line(0,1){2}} \put(20,2){\line(0,1){2}} \put(30,2){\line(0,1){2}} \put(40,2){\line(0,1){2}} \put(50,2){\line(0,1){4}} \put(60,2){\line(0,1){2}} \put(70,2){\line(0,1){2}} \put(80,2){\line(0,1){2}} \put(90,2){\line(0,1){2}} \put(100,2){\line(0,1){4}} \put(110,2){\line(0,1){2}} \put(120,2){\line(0,1){2}} \put(130,2){\line(0,1){2}} \put(140,2){\line(0,1){2}} \put(150,2){\line(0,1){4}} \put(160,2){\line(0,1){2}} \put(170,2){\line(0,1){2}} \put(180,2){\line(0,1){2}} \put(190,2){\line(0,1){2}} \put(200,2){\line(0,1){4}} \put(195,-10){{\scriptsize 20\%}} \put(145,-10){{\scriptsize 15\%}} \put(95,-10){{\scriptsize 10\%}} \put(45,-10){{\scriptsize 5\%}} \put(-5,-10){{\scriptsize 0\%}} \end{picture} \end{tabular} \endgroup \end{align}

A main result, which is a recurring theme also for other objective functions, is that solely minimising a risk function, and disregarding reward altogether, leads to the selection of well-performing portfolios. Hence minimising a lower partial moment already outperforms MV, even though using upper partial moments of order 0 to 1.5 helps to improve the performance further. The following table gives some results for lower partial moments. There we also added the mean return as the reward function since this conforms with specific objective functions discussed in the literature, for instance the ‘Kappa’. As can be seen, including the mean return generally improves the average result, but at the price of increased data sensitivity, that is wider distributions.

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\begin{align} \setlength{\unitlength}{1.0pt} \begingroup \everymath{\scriptstyle} \begin{footnotesize} \begin{tabular}{p{1.1cm}|p{1.1cm}|p{1.1cm}ll} reward&risk & median & \begin{picture}(100,1) \put(0,2){\line(1,0){200}} \put(0,2){\line(0,-1){4}} \put(10,2){\line(0,-1){2}} \put(20,2){\line(0,-1){2}} \put(30,2){\line(0,-1){2}} \put(40,2){\line(0,-1){2}} \put(50,2){\line(0,-1){4}} \put(60,2){\line(0,-1){2}} \put(70,2){\line(0,-1){2}} \put(80,2){\line(0,-1){2}} \put(90,2){\line(0,-1){2}} \put(100,2){\line(0,-1){4}} \put(110,2){\line(0,-1){2}} \put(120,2){\line(0,-1){2}} \put(130,2){\line(0,-1){2}} \put(140,2){\line(0,-1){2}} \put(150,2){\line(0,-1){4}} \put(160,2){\line(0,-1){2}} \put(170,2){\line(0,-1){2}} \put(180,2){\line(0,-1){2}} \put(190,2){\line(0,-1){2}} \put(200,2){\line(0,-1){4}} \put(195,10){{\scriptsize 20\%}} \put(145,10){{\scriptsize 15\%}} \put(95,10){{\scriptsize 10\%}} \put(45,10){{\scriptsize 5\%}} \put(-5,10){{\scriptsize 0\%}} \end{picture}\phantom{XXXXXXXXXXXXXX}&\label{tab:pureLPM}\\ %\usecolor{black} $c$&$\mathcal{M}_2$&$13.76$&\begin{picture}(100,1)\put(110.4,2){\line(1,0){19.5}}\put(137.6,2){\circle*{2}}\put(142.9,2){\line(1,0){18.7}}\end{picture} &{\tiny MV-portfolio} \\ [0.5ex] $c$& $\mathcal{P}^{-}_{0}$& $15.00$&\begin{picture}(100,1)\put(132.8,2){\line(1,0){12.7000000000000}}\put(150,2){\circle*{2}}\put(154,2){\line(1,0){8}}\end{picture}\\ $\mathcal{M}_1$& $\mathcal{P}^{-}_{0}$& $16.23$&\begin{picture}(100,1)\put(133.1,2){\line(1,0){20.7}}\put(162.3,2){\circle*{2}}\put(167.6,2){\line(1,0){14.8000000000000}}\end{picture}\\[0.5ex] $c$& $\mathcal{P}^{-}_{1}$& $14.90$&\begin{picture}(100,1)\put(135.9,2){\line(1,0){10.3000000000000}}\put(149,2){\circle*{2}}\put(153,2){\line(1,0){5.9}}\end{picture}\\ $\mathcal{M}_1$& $\mathcal{P}^{-}_{1}$& $16.18$&\begin{picture}(100,1)\put(136.9,2){\line(1,0){18.8}}\put(161.8,2){\circle*{2}}\put(168.3,2){\line(1,0){16.6}}\end{picture}&{\tiny Kappa$_1$}\\[0.5ex] $c$& $\mathcal{P}^{-}_{1.5}$& $14.80$&\begin{picture}(100,1)\put(134.6,2){\line(1,0){8.69999999999999}}\put(148,2){\circle*{2}}\put(150.6,2){\line(1,0){10.9}}\end{picture}\\ $\mathcal{M}_1$& $\mathcal{P}^{-}_{1.5}$& $16.11$&\begin{picture}(100,1)\put(134.9,2){\line(1,0){19.8}}\put(161.1,2){\circle*{2}}\put(167.9,2){\line(1,0){12.4000000000000}}\end{picture}\\[0.5ex] $c$& $\mathcal{P}^{-}_{2}$& $14.61$&\begin{picture}(100,1)\put(133,2){\line(1,0){9.6}}\put(146.1,2){\circle*{2}}\put(149.3,2){\line(1,0){10}}\end{picture}\\ $\mathcal{M}_1$& $\mathcal{P}^{-}_{2}$& $16.18$&\begin{picture}(100,1)\put(142.3,2){\line(1,0){14.7}}\put(161.8,2){\circle*{2}}\put(166.8,2){\line(1,0){8.79999999999998}}\end{picture}&{\tiny Sortino ratio, Kappa$_2$}\\[0.5ex] $c$& $\mathcal{P}^{-}_{3}$& $14.83$&\begin{picture}(100,1)\put(132.7,2){\line(1,0){11.300}}\put(148.3,2){\circle*{2}}\put(151.5,2){\line(1,0){11.1000000000000}}\end{picture}\\ $\mathcal{M}_1$& $\mathcal{P}^{-}_{3}$& $15.67$&\begin{picture}(100,1)\put(136,2){\line(1,0){15.7}}\put(156.7,2){\circle*{2}}\put(162.2,2){\line(1,0){8.70000000000002}}\end{picture}&{\tiny Kappa$_3$}\\ &&& \begin{picture}(100,1) \put(0,2){\line(1,0){200}} \put(0,2){\line(0,1){4}} \put(10,2){\line(0,1){2}} \put(20,2){\line(0,1){2}} \put(30,2){\line(0,1){2}} \put(40,2){\line(0,1){2}} \put(50,2){\line(0,1){4}} \put(60,2){\line(0,1){2}} \put(70,2){\line(0,1){2}} \put(80,2){\line(0,1){2}} \put(90,2){\line(0,1){2}} \put(100,2){\line(0,1){4}} \put(110,2){\line(0,1){2}} \put(120,2){\line(0,1){2}} \put(130,2){\line(0,1){2}} \put(140,2){\line(0,1){2}} \put(150,2){\line(0,1){4}} \put(160,2){\line(0,1){2}} \put(170,2){\line(0,1){2}} \put(180,2){\line(0,1){2}} \put(190,2){\line(0,1){2}} \put(200,2){\line(0,1){4}} \put(195,-10){{\scriptsize 20\%}} \put(145,-10){{\scriptsize 15\%}} \put(95,-10){{\scriptsize 10\%}} \put(45,-10){{\scriptsize 5\%}} \put(-5,-10){{\scriptsize 0\%}} \end{picture} \end{tabular} \end{footnotesize} \endgroup \end{align}

As seen for the MV-portfolio, simply minimising the historic ‘variability’ of returns often is an advisable strategy; lower partial moments may be just an alternative way to measure this variability. Upper partial moments, though meant to capture reward, confound return and risk, since maximising an upper partial moment inevitably also increases variability. In fact, a strategy of solely maximising an upper partial moment (of any order) leads to portfolios that perform very poorly, with a final wealth often barely breaking even over the whole period. In contrast, the rather counterintuitive strategy of minimising an upper partial moment (ie, reward) gives at least positive returns, albeit low ones when compared with other strategies (see table below).

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\begin{align} \begingroup \setlength{\unitlength}{1.0pt} \everymath{\scriptstyle} \begin{footnotesize} \begin{tabular}{p{1.1cm}|p{1.1cm}|p{1.1cm}ll}reward&risk & median & \begin{picture}(100,1) \put(0,2){\line(1,0){200}} \put(0,2){\line(0,-1){4}} \put(10,2){\line(0,-1){2}} \put(20,2){\line(0,-1){2}} \put(30,2){\line(0,-1){2}} \put(40,2){\line(0,-1){2}} \put(50,2){\line(0,-1){4}} \put(60,2){\line(0,-1){2}} \put(70,2){\line(0,-1){2}} \put(80,2){\line(0,-1){2}} \put(90,2){\line(0,-1){2}} \put(100,2){\line(0,-1){4}} \put(110,2){\line(0,-1){2}} \put(120,2){\line(0,-1){2}} \put(130,2){\line(0,-1){2}} \put(140,2){\line(0,-1){2}} \put(150,2){\line(0,-1){4}} \put(160,2){\line(0,-1){2}} \put(170,2){\line(0,-1){2}} \put(180,2){\line(0,-1){2}} \put(190,2){\line(0,-1){2}} \put(200,2){\line(0,-1){4}} \put(195,10){{\scriptsize 20\%}} \put(145,10){{\scriptsize 15\%}} \put(95,10){{\scriptsize 10\%}} \put(45,10){{\scriptsize 5\%}} \put(-5,10){{\scriptsize 0\%}} \end{picture}\phantom{XXXXXXXXXXXXXX}&\label{tab:pureUPM}\\ %\usecolor{black} $c$&$\mathcal{M}_2$&$13.76$&\begin{picture}(100,1)\put(110.4,2){\line(1,0){19.5}}\put(137.6,2){\circle*{2}}\put(142.9,2){\line(1,0){18.7}}\end{picture} &{\tiny MV-portfolio} \\[0.5ex] $\mathcal{P}^{+}_{2}$& $c$& \makebox[0.2\width][r]{$-0.70$} &\begin{picture}(100,1)\put(-24,2){\line(1,0){11.2}}\put(-7,2){\circle*{2}}\put(-2.2,2){\line(1,0){9.9}}\end{picture}\\ $\mathcal{P}^{+}_{3}$& $c$& \makebox[0.2\width][r]{$-0.96$}&\begin{picture}(100,1)\put(-26.9,2){\line(1,0){12.9}}\put(-9.6,2){\circle*{2}}\put(-4.9,2){\line(1,0){11.1}}\end{picture}\\[0.5ex] $c$& $\mathcal{P}^{+}_{2}$& $12.56$&\begin{picture}(100,1)\put(97.2,2){\line(1,0){20}}\put(125.6,2){\circle*{2}}\put(132,2){\line(1,0){11.7}}\end{picture}\\ $c$& $\mathcal{P}^{+}_{3}$& $13.16$&\begin{picture}(100,1)\put(115.9,2){\line(1,0){11.4}}\put(131.6,2){\circle*{2}}\put(135.7,2){\line(1,0){9.00000000000003}}\end{picture}\\ &&& \begin{picture}(100,1) \put(0,2){\line(1,0){200}} \put(0,2){\line(0,1){4}} \put(10,2){\line(0,1){2}} \put(20,2){\line(0,1){2}} \put(30,2){\line(0,1){2}} \put(40,2){\line(0,1){2}} \put(50,2){\line(0,1){4}} \put(60,2){\line(0,1){2}} \put(70,2){\line(0,1){2}} \put(80,2){\line(0,1){2}} \put(90,2){\line(0,1){2}} \put(100,2){\line(0,1){4}} \put(110,2){\line(0,1){2}} \put(120,2){\line(0,1){2}} \put(130,2){\line(0,1){2}} \put(140,2){\line(0,1){2}} \put(150,2){\line(0,1){4}} \put(160,2){\line(0,1){2}} \put(170,2){\line(0,1){2}} \put(180,2){\line(0,1){2}} \put(190,2){\line(0,1){2}} \put(200,2){\line(0,1){4}} \put(195,-10){{\scriptsize 20\%}} \put(145,-10){{\scriptsize 15\%}} \put(95,-10){{\scriptsize 10\%}} \put(45,-10){{\scriptsize 5\%}} \put(-5,-10){{\scriptsize 0\%}} \end{picture} \end{tabular} \end{footnotesize} \endgroup \end{align}

### Conditional moments

The following table gives the results (in terms of annualised returns) for minimising lower conditional moments of order one, where the threshold $r_d$ is chosen as a certain quantile. This specification corresponds to minimising Expected Shortfall.

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\begin{align} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % ES, I %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \setlength{\unitlength}{1.0pt} \begingroup \everymath{\scriptstyle} \begin{footnotesize} \begin{tabular}{p{1.1cm}|p{1.1cm}|p{1.1cm}ll} reward&risk & median & \begin{picture}(100,1) \put(0,2){\line(1,0){200}} \put(0,2){\line(0,-1){4}} \put(10,2){\line(0,-1){2}} \put(20,2){\line(0,-1){2}} \put(30,2){\line(0,-1){2}} \put(40,2){\line(0,-1){2}} \put(50,2){\line(0,-1){4}} \put(60,2){\line(0,-1){2}} \put(70,2){\line(0,-1){2}} \put(80,2){\line(0,-1){2}} \put(90,2){\line(0,-1){2}} \put(100,2){\line(0,-1){4}} \put(110,2){\line(0,-1){2}} \put(120,2){\line(0,-1){2}} \put(130,2){\line(0,-1){2}} \put(140,2){\line(0,-1){2}} \put(150,2){\line(0,-1){4}} \put(160,2){\line(0,-1){2}} \put(170,2){\line(0,-1){2}} \put(180,2){\line(0,-1){2}} \put(190,2){\line(0,-1){2}} \put(200,2){\line(0,-1){4}} \put(195,10){{\scriptsize 20\%}} \put(145,10){{\scriptsize 15\%}} \put(95,10){{\scriptsize 10\%}} \put(45,10){{\scriptsize 5\%}} \put(-5,10){{\scriptsize 0\%}} \end{picture}\phantom{XXXXXXXXXXXXXX}&\label{tab:ES}\\ %\usecolor{black} $c$&$\mathcal{M}_2$&$13.76$&\begin{picture}(100,1)\put(110.4,2){\line(1,0){19.5}}\put(137.6,2){\circle*{2}}\put(142.9,2){\line(1,0){18.7}}\end{picture} &{\tiny MV-portfolio} \\[0.5ex] $c$&$\mathcal{C}^-_1(\mathcal{Q}_1)$&$14.88$&\begin{picture}(100,1)\put(127.471,2){\line(1,0){16.105}}\put(148.806,2){\circle*{2}}\put(154.763,2){\line(1,0){14.1}}\end{picture}\\ $c$&$\mathcal{C}^-_1(\mathcal{Q}_5)$&$14.98$&\begin{picture}(100,1)\put(130.263,2){\line(1,0){13.361}}\put(149.824,2){\circle*{2}}\put(153.51,2){\line(1,0){14.8290000000000}}\end{picture}\\ $c$&$\mathcal{C}^-_1(\mathcal{Q}_{10})$&$14.58$&\begin{picture}(100,1)\put(130.321,2){\line(1,0){10.792}}\put(145.797,2){\circle*{2}}\put(150.07,2){\line(1,0){13.435}}\end{picture}\\ $c$&$\mathcal{C}^-_1(\mathcal{Q}_{20})$&$14.80$&\begin{picture}(100,1)\put(130.846,2){\line(1,0){12.924}}\put(147.98,2){\circle*{2}}\put(152.5,2){\line(1,0){9.82500000000002}}\end{picture}\\ &&& \begin{picture}(100,1) \put(0,2){\line(1,0){200}} \put(0,2){\line(0,1){4}} \put(10,2){\line(0,1){2}} \put(20,2){\line(0,1){2}} \put(30,2){\line(0,1){2}} \put(40,2){\line(0,1){2}} \put(50,2){\line(0,1){4}} \put(60,2){\line(0,1){2}} \put(70,2){\line(0,1){2}} \put(80,2){\line(0,1){2}} \put(90,2){\line(0,1){2}} \put(100,2){\line(0,1){4}} \put(110,2){\line(0,1){2}} \put(120,2){\line(0,1){2}} \put(130,2){\line(0,1){2}} \put(140,2){\line(0,1){2}} \put(150,2){\line(0,1){4}} \put(160,2){\line(0,1){2}} \put(170,2){\line(0,1){2}} \put(180,2){\line(0,1){2}} \put(190,2){\line(0,1){2}} \put(200,2){\line(0,1){4}} \put(195,-10){{\scriptsize 20\%}} \put(145,-10){{\scriptsize 15\%}} \put(95,-10){{\scriptsize 10\%}} \put(45,-10){{\scriptsize 5\%}} \put(-5,-10){{\scriptsize 0\%}} \end{picture} \end{tabular} \end{footnotesize} \endgroup \end{align}

We see that again we improve on MV, in particular do we obtain denser distributions. That is, the results appear slightly less sensitive to particular data sets.

Conditional moments offer more possibilities than Expected Shortfall, however. The following table gives results for ratios of upper to lower conditional moments of different orders. In the literature, this ratio has also been called the ‘Generalised Rachev ratio’.

(13)
\begin{align} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Gen. Rachev %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \setlength{\unitlength}{1.0pt} \begingroup \everymath{\scriptstyle} \begin{footnotesize} \begin{tabular}{p{1.1cm}|p{1.1cm}|p{1.1cm}ll} reward&risk & median & \begin{picture}(100,1) \put(0,2){\line(1,0){200}} \put(0,2){\line(0,-1){4}} \put(10,2){\line(0,-1){2}} \put(20,2){\line(0,-1){2}} \put(30,2){\line(0,-1){2}} \put(40,2){\line(0,-1){2}} \put(50,2){\line(0,-1){4}} \put(60,2){\line(0,-1){2}} \put(70,2){\line(0,-1){2}} \put(80,2){\line(0,-1){2}} \put(90,2){\line(0,-1){2}} \put(100,2){\line(0,-1){4}} \put(110,2){\line(0,-1){2}} \put(120,2){\line(0,-1){2}} \put(130,2){\line(0,-1){2}} \put(140,2){\line(0,-1){2}} \put(150,2){\line(0,-1){4}} \put(160,2){\line(0,-1){2}} \put(170,2){\line(0,-1){2}} \put(180,2){\line(0,-1){2}} \put(190,2){\line(0,-1){2}} \put(200,2){\line(0,-1){4}} \put(195,10){{\scriptsize 20\%}} \put(145,10){{\scriptsize 15\%}} \put(95,10){{\scriptsize 10\%}} \put(45,10){{\scriptsize 5\%}} \put(-5,10){{\scriptsize 0\%}} \end{picture}\phantom{XXXXXXXXXXXXXX}&\label{tab:Rachev}\\ %\usecolor{black} $c$&$\mathcal{M}_2$&$13.76$&\begin{picture}(100,1)\put(110.4,2){\line(1,0){19.5}}\put(137.6,2){\circle*{2}}\put(142.9,2){\line(1,0){18.7}}\end{picture} &{\tiny MV-portfolio} \\[0.5ex] $\mathcal{C}^{+}_{1}(\mathcal{Q}_{80})$& $\mathcal{C}^{-}_{1}(\mathcal{Q}_{20})$& $15.98$&\begin{picture}(100,1)\put(135,2){\line(1,0){17.3}}\put(159.8,2){\circle*{2}}\put(165.8,2){\line(1,0){20.1000000000000}}\end{picture}\\ $\mathcal{C}^{+}_{1}(\mathcal{Q}_{80})$& $\mathcal{C}^{-}_{2}(\mathcal{Q}_{20})$& $15.54$&\begin{picture}(100,1)\put(139,2){\line(1,0){12.2}}\put(155.4,2){\circle*{2}}\put(159.4,2){\line(1,0){6.89999999999998}}\end{picture}\\ $\mathcal{C}^{+}_{1.5}(\mathcal{Q}_{80})$& $\mathcal{C}^{-}_{1.5}(\mathcal{Q}_{20})$& $15.84$&\begin{picture}(100,1)\put(139,2){\line(1,0){13.9000000000000}}\put(158.4,2){\circle*{2}}\put(164.8,2){\line(1,0){17.7}}\end{picture}\\ $\mathcal{C}^{+}_{2}(\mathcal{Q}_{80})$& $\mathcal{C}^{-}_{2}(\mathcal{Q}_{20})$& $15.21$&\begin{picture}(100,1)\put(132.4,2){\line(1,0){13.9}}\put(152.1,2){\circle*{2}}\put(158.2,2){\line(1,0){17.2}}\end{picture}\\[0.5ex] $\mathcal{C}^{+}_{1}(\mathcal{Q}_{50})$& $\mathcal{C}^{-}_{1}(\mathcal{Q}_{50})$& $16.36$&\begin{picture}(100,1)\put(146.1,2){\line(1,0){13.8}}\put(163.6,2){\circle*{2}}\put(169,2){\line(1,0){13.7}}\end{picture}\\ $\mathcal{C}^{+}_{1}(\mathcal{Q}_{50})$& $\mathcal{C}^{-}_{2}(\mathcal{Q}_{50})$& $15.50$&\begin{picture}(100,1)\put(142.3,2){\line(1,0){9.19999999999999}}\put(155,2){\circle*{2}}\put(159,2){\line(1,0){10.1}}\end{picture}\\ $\mathcal{C}^{+}_{1.5}(\mathcal{Q}_{50})$& $\mathcal{C}^{-}_{1.5}(\mathcal{Q}_{50})$& $16.95$&\begin{picture}(100,1)\put(153.9,2){\line(1,0){10.7000000000000}}\put(169.5,2){\circle*{2}}\put(175.6,2){\line(1,0){14.9}}\end{picture}\\ $\mathcal{C}^{+}_{2}(\mathcal{Q}_{50})$& $\mathcal{C}^{-}_{2}(\mathcal{Q}_{50})$& $16.57$&\begin{picture}(100,1)\put(142.9,2){\line(1,0){16.2000000000000}}\put(165.7,2){\circle*{2}}\put(171.3,2){\line(1,0){17.2}}\end{picture}\\ &&& \begin{picture}(100,1) \put(0,2){\line(1,0){200}} \put(0,2){\line(0,1){4}} \put(10,2){\line(0,1){2}} \put(20,2){\line(0,1){2}} \put(30,2){\line(0,1){2}} \put(40,2){\line(0,1){2}} \put(50,2){\line(0,1){4}} \put(60,2){\line(0,1){2}} \put(70,2){\line(0,1){2}} \put(80,2){\line(0,1){2}} \put(90,2){\line(0,1){2}} \put(100,2){\line(0,1){4}} \put(110,2){\line(0,1){2}} \put(120,2){\line(0,1){2}} \put(130,2){\line(0,1){2}} \put(140,2){\line(0,1){2}} \put(150,2){\line(0,1){4}} \put(160,2){\line(0,1){2}} \put(170,2){\line(0,1){2}} \put(180,2){\line(0,1){2}} \put(190,2){\line(0,1){2}} \put(200,2){\line(0,1){4}} \put(195,-10){{\scriptsize 20\%}} \put(145,-10){{\scriptsize 15\%}} \put(95,-10){{\scriptsize 10\%}} \put(45,-10){{\scriptsize 5\%}} \put(-5,-10){{\scriptsize 0\%}} \end{picture} \end{tabular} \end{footnotesize} \endgroup \end{align}

## Quantiles

The following table gives results for VaR. The quantiles have been chosen to correspond to the tests for Expected Shortfall, given in the table above. Average returns are roughly the same as for MV, but the results seem more unstable (ie, wider distributions).

(14)
\begin{align} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % VaR, I %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \setlength{\unitlength}{1.0pt} \begingroup \everymath{\scriptstyle} \begin{footnotesize} \begin{tabular}{p{1.1cm}|p{1.1cm}|p{1.1cm}ll} reward&risk & median & \begin{picture}(100,1) \put(0,2){\line(1,0){200}} \put(0,2){\line(0,-1){4}} \put(10,2){\line(0,-1){2}} \put(20,2){\line(0,-1){2}} \put(30,2){\line(0,-1){2}} \put(40,2){\line(0,-1){2}} \put(50,2){\line(0,-1){4}} \put(60,2){\line(0,-1){2}} \put(70,2){\line(0,-1){2}} \put(80,2){\line(0,-1){2}} \put(90,2){\line(0,-1){2}} \put(100,2){\line(0,-1){4}} \put(110,2){\line(0,-1){2}} \put(120,2){\line(0,-1){2}} \put(130,2){\line(0,-1){2}} \put(140,2){\line(0,-1){2}} \put(150,2){\line(0,-1){4}} \put(160,2){\line(0,-1){2}} \put(170,2){\line(0,-1){2}} \put(180,2){\line(0,-1){2}} \put(190,2){\line(0,-1){2}} \put(200,2){\line(0,-1){4}} \put(195,10){{\scriptsize 20\%}} \put(145,10){{\scriptsize 15\%}} \put(95,10){{\scriptsize 10\%}} \put(45,10){{\scriptsize 5\%}} \put(-5,10){{\scriptsize 0\%}} \end{picture}\phantom{XXXXXXXXXXXXXX}&\label{tab:Q}\\ %\usecolor{black} $c$&$\mathcal{M}_2$&$13.76$&\begin{picture}(100,1)\put(110.4,2){\line(1,0){19.5}}\put(137.6,2){\circle*{2}}\put(142.9,2){\line(1,0){18.7}}\end{picture} &{\tiny MV-portfolio} \\[0.5ex] $c$&$\mathcal{Q}_{1}$&$14.44$&\begin{picture}(100,1)\put(118.796,2){\line(1,0){16.686}}\put(144.386,2){\circle*{2}}\put(151.235,2){\line(1,0){17.951}}\end{picture}\\ $c$&$\mathcal{Q}_{5}$&$14.17$&\begin{picture}(100,1)\put(119.318,2){\line(1,0){13.8460000000000}}\put(141.728,2){\circle*{2}}\put(148.141,2){\line(1,0){12.136}}\end{picture}\\ $c$&$\mathcal{Q}_{10}$&$14.19$&\begin{picture}(100,1)\put(111.489,2){\line(1,0){22.166}}\put(141.907,2){\circle*{2}}\put(148.432,2){\line(1,0){19.1250000000000}}\end{picture}\\ $c$&$\mathcal{Q}_{20}$&$13.70$&\begin{picture}(100,1)\put(106.488,2){\line(1,0){21.955}}\put(136.999,2){\circle*{2}}\put(143.08,2){\line(1,0){19.804}}\end{picture}\\ &&& \begin{picture}(100,1) \put(0,2){\line(1,0){200}} \put(0,2){\line(0,1){4}} \put(10,2){\line(0,1){2}} \put(20,2){\line(0,1){2}} \put(30,2){\line(0,1){2}} \put(40,2){\line(0,1){2}} \put(50,2){\line(0,1){4}} \put(60,2){\line(0,1){2}} \put(70,2){\line(0,1){2}} \put(80,2){\line(0,1){2}} \put(90,2){\line(0,1){2}} \put(100,2){\line(0,1){4}} \put(110,2){\line(0,1){2}} \put(120,2){\line(0,1){2}} \put(130,2){\line(0,1){2}} \put(140,2){\line(0,1){2}} \put(150,2){\line(0,1){4}} \put(160,2){\line(0,1){2}} \put(170,2){\line(0,1){2}} \put(180,2){\line(0,1){2}} \put(190,2){\line(0,1){2}} \put(200,2){\line(0,1){4}} \put(195,-10){{\scriptsize 20\%}} \put(145,-10){{\scriptsize 15\%}} \put(95,-10){{\scriptsize 10\%}} \put(45,-10){{\scriptsize 5\%}} \put(-5,-10){{\scriptsize 0\%}} \end{picture} \end{tabular} \end{footnotesize} \endgroup \end{align}

## Drawdowns

The following table gives the results for final wealth. Since many ex post performance measures used in the industry (like the Calmar or Sterling ratio) use mean portfolio return for reward, we also include such ratios here.

(15)
\begin{align} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % drawdowns, I %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \setlength{\unitlength}{1.0pt} \begingroup \everymath{\scriptstyle} \begin{footnotesize} \begin{tabular}{p{1.1cm}|p{1.1cm}|p{1.1cm}ll} reward&risk & median & \begin{picture}(100,1) \put(0,2){\line(1,0){200}} \put(0,2){\line(0,-1){4}} \put(10,2){\line(0,-1){2}} \put(20,2){\line(0,-1){2}} \put(30,2){\line(0,-1){2}} \put(40,2){\line(0,-1){2}} \put(50,2){\line(0,-1){4}} \put(60,2){\line(0,-1){2}} \put(70,2){\line(0,-1){2}} \put(80,2){\line(0,-1){2}} \put(90,2){\line(0,-1){2}} \put(100,2){\line(0,-1){4}} \put(110,2){\line(0,-1){2}} \put(120,2){\line(0,-1){2}} \put(130,2){\line(0,-1){2}} \put(140,2){\line(0,-1){2}} \put(150,2){\line(0,-1){4}} \put(160,2){\line(0,-1){2}} \put(170,2){\line(0,-1){2}} \put(180,2){\line(0,-1){2}} \put(190,2){\line(0,-1){2}} \put(200,2){\line(0,-1){4}} \put(195,10){{\scriptsize 20\%}} \put(145,10){{\scriptsize 15\%}} \put(95,10){{\scriptsize 10\%}} \put(45,10){{\scriptsize 5\%}} \put(-5,10){{\scriptsize 0\%}} \end{picture}\phantom{XXXXXXXXXXXXXX}&\label{tab:DD}\\ %\usecolor{black} $c$&$\mathcal{M}_2$&$13.76$&\begin{picture}(100,1)\put(110.4,2){\line(1,0){19.5}}\put(137.6,2){\circle*{2}}\put(142.9,2){\line(1,0){18.7}}\end{picture} &{\tiny MV-portfolio} \\[0.5ex] $c$&$\mathcal{D}_{\mathrm{mean}}$&$15.34$&\begin{picture}(100,1)\put(129.368,2){\line(1,0){15.992}}\put(153.397,2){\circle*{2}}\put(159.585,2){\line(1,0){21.339}}\end{picture}\\ $\mathcal{M}_1$&$\mathcal{D}_{\mathrm{mean}}$&$15.56$&\begin{picture}(100,1)\put(129.656,2){\line(1,0){19.844}}\put(155.572,2){\circle*{2}}\put(164.368,2){\line(1,0){14.1920000000000}}\end{picture}\\ $c$&$\mathcal{D}_{\max}$&$14.53$&\begin{picture}(100,1)\put(114.286,2){\line(1,0){23.239}}\put(145.32,2){\circle*{2}}\put(153.38,2){\line(1,0){21.881}}\end{picture}\\ $\mathcal{M}_1$&$\mathcal{D}_{\max}$&$14.93$&\begin{picture}(100,1)\put(115.764,2){\line(1,0){24.122}}\put(149.321,2){\circle*{2}}\put(157.359,2){\line(1,0){22.579}}\end{picture}&{\tiny Calmar ratio}\\ $c$&$\mathcal{D}_{\mathrm{std}}$&$15.31$&\begin{picture}(100,1)\put(123.487,2){\line(1,0){21.668}}\put(153.061,2){\circle*{2}}\put(159.6,2){\line(1,0){19.8730000000000}}\end{picture}\\ $\mathcal{M}_1$&$\mathcal{D}_{\mathrm{std}}$&$15.37$&\begin{picture}(100,1)\put(124.544,2){\line(1,0){21.303}}\put(153.689,2){\circle*{2}}\put(160.214,2){\line(1,0){18.106}}\end{picture}\\ &&& \begin{picture}(100,1) \put(0,2){\line(1,0){200}} \put(0,2){\line(0,1){4}} \put(10,2){\line(0,1){2}} \put(20,2){\line(0,1){2}} \put(30,2){\line(0,1){2}} \put(40,2){\line(0,1){2}} \put(50,2){\line(0,1){4}} \put(60,2){\line(0,1){2}} \put(70,2){\line(0,1){2}} \put(80,2){\line(0,1){2}} \put(90,2){\line(0,1){2}} \put(100,2){\line(0,1){4}} \put(110,2){\line(0,1){2}} \put(120,2){\line(0,1){2}} \put(130,2){\line(0,1){2}} \put(140,2){\line(0,1){2}} \put(150,2){\line(0,1){4}} \put(160,2){\line(0,1){2}} \put(170,2){\line(0,1){2}} \put(180,2){\line(0,1){2}} \put(190,2){\line(0,1){2}} \put(200,2){\line(0,1){4}} \put(195,-10){{\scriptsize 20\%}} \put(145,-10){{\scriptsize 15\%}} \put(95,-10){{\scriptsize 10\%}} \put(45,-10){{\scriptsize 5\%}} \put(-5,-10){{\scriptsize 0\%}} \end{picture} \end{tabular} \end{footnotesize} \endgroup \end{align}

# Conclusion

In this study we investigated the empirical performance of alternative selection criteria in portfolio optimisation problems. Our main findings are that alternative risk and performance measures in many cases improve on the MV-benchmark but that the estimation problems for these alternative functions also become more severe. Furthermore, all the strategies tested (including MV) are quite sensitive to relatively small changes in the data.

The recurring theme throughout our study was that minimising risk, as opposed to maximising reward, often lead to good out-of-sample performance; stated differently, low historical variability of portfolio returns was a predictor of good future performance. Our suggestion for constructing objective functions is thus to spend most effort here, as there seem better ways to measure this variability than variance. In particular, selection criteria based on partial and conditional moments, and drawdown performed well, with functions based on quantiles being less satisfying. In contrast, a careful design of a reward function may improve the strategy in terms of returns, but in many of our tests it also lead to a higher sensitivity to the data sample.