# Journal of Investment Strategies

**ISSN:**

2047-1238 (print)

2047-1246 (online)

**Editor-in-chief:** Ali Hirsa

####
Abstract

We formulate the portfolio allocation problem from a trading point of view, allowing both long and short positions and taking trading and interest rate costs into account. Expressions are derived for the portfolio profit or loss (PL) that may result over a holding period. The expected profit (EP) and the expected loss (EL) are taken as measures of reward and risk. Optimal portfolios are considered to be allocations that maximize EP subject to EL being below a specified fraction of the EP. Simple expressions are shown for the reward and risk contributions of individual stocks to the portfolio expected PL. This optimal portfolio approach is referred to as the EP–EL method, and it is compared with a method based on maximal expected PL subject to controlled volatility measured by expected absolute PL deviation. The calculations required for these optimal portfolios are formulated as linear programming problems. Extensive results based on the market trading data of 12 stocks are provided to illustrate the properties of the EP–EL method. Among others, this leads to allocations that simultaneously maximize EP and EL while keeping the latter below an acceptable fraction of the former.

####
Introduction

## 1 Introduction

Investment markets are often said to be driven by greed and fear; in less controversial terms, these two factors are called “reward” and “risk”. Decisions on portfolio allocations must balance these two opposing factors. For this purpose, formal measures of reward and risk are needed. In this paper, “portfolio profit or loss” (PL) will refer to the net monetary result experienced over a given period due to investment in a portfolio with a specific allocation of a number of financial instruments, typically traded stocks. The PL may be positive, in which case it amounts to a profit, or negative, in which case it amounts to a loss. The notion of “return” rather than PL is commonly used in portfolio theory, but we will use the term “return” specifically for stock price returns, and it may be less confusing to the reader if we do not also use it for portfolio PL in the specific trading context discussed here.

Markowitz (1968) introduced the mean–variance (MV) portfolio model, in which reward and risk are measured by the expectation and the variance of the portfolio PL, respectively. In this paper, we take the view that trading requires taking calculated risks to achieve the highest reward. More precisely, when using the MV model in our context, we have to find the portfolio that maximizes the expected PL subject to limiting the variance to a specified level acceptable to the trader. Such portfolios are said to be on the MV efficient frontier. An enormous literature has grown up over time as researchers and asset managers have developed and implemented the MV model in diverse directions. New contributions to this literature appear regularly (see, for example, Wenzelburger 2020), but it is not feasible to try to survey these developments here. It suffices to mention that there are still some aspects of the MV model that are unsatisfactory both in principle and in practice. From a principle point of view, it is still debatable whether variance is actually a good measure of risk, since variance is affected by both upside and downside PL fluctuations. Thus, variance measures PL volatility, and it is debatable whether volatility and risk are the same. Saft (2014) argues eloquently that measures of risk should be concerned about prospective losses rather than volatility. Indeed, there is a growing literature on loss-based risk measures (see, for example, Cont et al 2013; Chen et al 2018). Righi and Borenstein (2018) compare 11 risk measures, some of which are loss based. Among other conclusions, they state that “despite this lack of dominance, loss-deviation risk measures consistently exhibit advantage regarding performance, having good practical results in our context”. Another issue relates to the notion of the MV efficient frontiers when we wish to allow for short selling, as occurs in a trading context (see, for example, Brennan and Lo 2010).

On the matter of balancing reward and risk in the MV model, Sharpe (1966) introduced what came to be known as the Sharpe ratio, namely the ratio of PL expectation to its variance. This is a single index that may be taken as a measure to rank portfolios in terms of their qualities. A similar index was introduced by Sortino and Price (1994), replacing the variance by the lower half second moment of the PL probability distribution. These ratios too may be criticised on various aspects, including their dependence only on the first two moments of the probability distribution of portfolio PL. This led Keating and Shadwick (2002) to introduce the Omega measure, which involves the probability distribution of PL more fully rather than just by way of its first two moments. This measure continues to attract attention (see, for example, Metel et al (2017), Balder and Schweizer (2017) and the references therein).

To define the Omega measure, let $F$ denote the distribution function of the portfolio PL. Then Omega is defined as

$$\mathrm{\Omega}(D)=\frac{{\int}_{D}^{\mathrm{\infty}}(1-F(x))dx}{{\int}_{-\mathrm{\infty}}^{D}F(x)dx},$$ | (1.1) |

where $D$ is a threshold parameter that may be chosen by the investor as a demarcation point between the loss and profit of the portfolio PL. The two factors in the Omega ratio may also be interpreted as reward and risk measures. To see this, denote the portfolio PL by the random variable $X$ with distribution function $F$. Then the numerator in (1.1) may be written as

$${\int}_{D}^{\mathrm{\infty}}(1-F(x))\mathrm{d}x=?[X-D]I(X\ge D)=?{[X-D]}^{+}$$ | (1.2) |

with $I(A)$ denoting the indicator function of the event $A$ and with ${x}^{+}=\mathrm{max}(x,0)$ denoting the positive part of $x$. Denoting the negative part of $x$ by ${x}^{-}=-\mathrm{min}(x,0)$, the denominator in (1.1) may be written as

$$ | (1.3) |

If we take $D=0$ for simplicity of interpretation, then ${X}^{+}$ may be thought of as the profit part of the PL while ${X}^{-}$ is the loss part of $X$. Thus, the numerator and denominator of (1.1) may be interpreted as the expected profit (EP) and expected loss (EL) of the portfolio. With $D>0$ we are more conservative on what counts as profit and what counts as loss, but in this paper we prefer to subsume such matters into the expression of portfolio PL and therefore fix $D=0$. Rather than simply ranking portfolios, as mentioned above, we pursue the aim of “maximizing reward subject to controlling risk”. For this purpose, we use the two factors in the Omega ratio as separate measures of reward and risk. By analogy with the MV approach, this suggests finding the portfolio that maximizes the EP subject to limiting the EL level. An attractive feature of EP and EL as reward and risk measures is that they are both expressed in monetary terms. This means that we can directly compare reward and risk. For example, if a given portfolio has an EL that is only one-fifth (or some other small factor) of its EP, then it is intuitively clear that this would be a relatively risk-averse portfolio. Contrast this to the MV approach, in which it is not so clear what the interpretation is of a portfolio having its PL variance equal to one-fifth (or some other factor) of its expected PL.

In Section 2, we detail the investment context and the notation to be used throughout this paper. Our context is mainly motivated from a trading point of view. The trader has to take a combination of long and/or short positions in a number of financial instruments (usually stocks). At the start of the holding period the trader may borrow and/or receive capital on which interest will be paid and/or received. When the trader closes the positions at the end of the holding period, a PL is made. The issue is to find a good combination of instruments to use.

We formulate the optimal EP–EL portfolio allocation model that is the main focus of this paper. If we follow the MV model but replace the PL variance by the mean deviation (MD) of the PL as the risk measure, then many of the relevant technical expressions are closely analogous to those of the EP–EL model. This variation of the MV model will be referred to as the M-MD model and it serves as a useful comparison to the EP–EL model in our context. We discuss the M-MD model briefly in Section 3. Section 4 details the actual calculation of EP–EL and the M-MD allocations, focusing on an approach that combines Monte Carlo simulation with linear programming optimization. Using some theoretical models and an empirical data set from stocks traded on the Johannesburg Stock Exchange, Section 5 explores the properties of the EP–EL model and contrasts them with those of the M-MD model. Section 6 concludes with a brief summary and an outline of future research issues.

## 2 The long and short EP–EL portfolio allocation

To formulate the portfolio context dealt with in this paper, consider $N$ stocks in which we wish to invest by taking long and/or short positions at the beginning of a period. Suppose we are considering either buying shares (“going long”) or selling shares (“going short”) in the $n$th stock at a price ${p}_{n}$. If we buy, we denote the number of shares bought by ${k}_{n}^{+}$, and if we sell, we denote the number by ${k}_{n}^{-}$. Hence, the net position in the stock is ${k}_{n}={k}_{n}^{+}-{k}_{n}^{-}$. Suppose further that we are prepared to invest up to a total amount ${B}^{+}$ (our long budget) on the long side of our portfolio and have a total exposure of up to ${B}^{-}$ (our short budget) in the short side of our portfolio of stocks. Then the first two requirements on the portfolio allocation are that

$$\sum _{n=1}^{N}{k}_{n}^{+}{p}_{n}\le {B}^{+}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}\sum _{n=1}^{N}{k}_{n}^{-}{p}_{n}\le {B}^{-}.$$ | (2.1) |

Assume that trading cost is proportional at the rate ${c}_{\mathrm{b}}$ when buying and ${c}_{\mathrm{s}}$ when selling. Then the total amount paid out to get the long part of the portfolio position is $(1+{c}_{\mathrm{b}}){\sum}_{n=1}^{N}{k}_{n}^{+}{p}_{n}$. Assume that this amount must be borrowed from a bank that will charge interest at a rate of ${i}_{\mathrm{b}}$ over the holding period. Then the amount $(1+{i}_{\mathrm{b}})(1+{c}_{\mathrm{b}}){\sum}_{n=1}^{N}{k}_{n}^{+}{p}_{n}$ will be owed to the bank at the end of the holding period due to the long positions.

Assume next that we will exit the long positions at the end of the holding period and denote the exit price of the $n$th stock by ${q}_{n}$. We will get paid back an amount $(1-{c}_{\mathrm{s}}){\sum}_{n=1}^{N}{k}_{n}^{+}{q}_{n}$ from exiting the long part of the portfolio. Hence, our total PL from all the transactions on the long side will be

$${X}_{\mathrm{L}}=(1-{c}_{\mathrm{s}})\sum _{n=1}^{N}{k}_{n}^{+}{q}_{n}-(1+{i}_{\mathrm{b}})(1+{c}_{\mathrm{b}})\sum _{n=1}^{N}{k}_{n}^{+}{p}_{n}.$$ | (2.2) |

As to the short side of the portfolio, the total amount paid to us on taking the short positions is $(1-{c}_{\mathrm{s}}){\sum}_{n=1}^{N}{k}_{n}^{-}{p}_{n}$. Assume that this amount can be deposited at a bank that will pay us interest at the same rate of ${i}_{\mathrm{s}}$ over the holding period. Hence, at the end of the holding period the credit at the bank due to our short positions will be $(1+{i}_{\mathrm{s}})(1-{c}_{\mathrm{s}}){\sum}_{n=1}^{N}{k}_{n}^{-}{p}_{n}$. When we exit the short positions we must buy back those shares, and this will cost us $(1+{c}_{\mathrm{b}}){\sum}_{n=1}^{N}{k}_{n}^{-}{q}_{n}$. Hence, our total PL from the short side of the portfolio will be

$${X}_{\mathrm{S}}=(1+{i}_{\mathrm{s}})(1-{c}_{\mathrm{s}})\sum _{n=1}^{N}{k}_{n}^{-}{p}_{n}-(1+{c}_{\mathrm{b}})\sum _{n=1}^{N}{k}_{n}^{-}{q}_{n}.$$ | (2.3) |

Let ${r}_{n}=({q}_{n}-{p}_{n})/{p}_{n}$ denote the arithmetic return of the $n$th share so that ${q}_{n}={p}_{n}(1+{r}_{n})$. In addition, let ${w}_{n}^{+}={k}_{n}^{+}{p}_{n}$ and ${w}_{n}^{-}={k}_{n}^{-}{p}_{n}$ denote the amounts relevant to the long or short positions in the $n$th share before costs. Substituting into (2.2) and (2.3) and rearranging terms, these become

$${X}_{\mathrm{L}}=(1-{c}_{\mathrm{s}})\sum _{n=1}^{N}{w}_{n}^{+}({r}_{n}-{c}_{\mathrm{f}}^{+})\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{X}_{\mathrm{S}}=(1+{c}_{\mathrm{b}})\sum _{n=1}^{N}{w}_{n}^{-}({c}_{\mathrm{f}}^{-}-{r}_{n}).$$ | (2.4) |

Here,

$$\begin{array}{cc}\hfill {c}_{\mathrm{f}}^{+}& =\frac{(1+{i}_{\mathrm{b}})(1+{c}_{\mathrm{b}})}{(1-{c}_{\mathrm{s}})}-1\approx {i}_{\mathrm{b}}+{c}_{\mathrm{b}}+{c}_{\mathrm{s}},\hfill \\ \hfill {c}_{\mathrm{f}}^{-}& =\frac{(1+{i}_{\mathrm{s}})(1-{c}_{\mathrm{s}})}{(1+{c}_{\mathrm{b}})}-1\approx {i}_{\mathrm{s}}-{c}_{\mathrm{b}}-{c}_{\mathrm{s}}\hfill \end{array}\}$$ | (2.5) |

are factors depending on the trading costs and interest rates. The approximations in these expressions are valid when the cost factors are numerically small. From (2.4), it is clear that these factors act as thresholds on the stock returns to make positive contributions to the portfolio PL. On the long side, we need ${r}_{n}>{c}_{\mathrm{f}}^{+}$, and on the short side, we need $$. On the long side, increasing interest rate and trade costs all make this trading threshold larger, thus acting against the trader. By contrast, on the short side, the interest rate acts in favor of the trader but both trading costs are against the trader.

The total PL from the long and short parts together will be

$$X={X}_{\mathrm{L}}+{X}_{\mathrm{S}}.$$ | (2.6) |

We need to decide on the number of shares ${k}_{n}^{+}$ or ${k}_{n}^{-}$, but since ${p}_{n}$ is known we can determine them from ${k}_{n}^{+}$ $={w}_{n}^{+}/{p}_{n}$ and ${k}_{n}^{-}$ $={w}_{n}^{-}/{p}_{n}$ once we have decided on the corresponding amounts ${w}_{n}^{+}$ or ${w}_{n}^{-}$. Therefore, we can regard the choice of $\{({w}_{n}^{+},{w}_{n}^{-}),n=1,\mathrm{\dots},N\}$ as our main concern, and we will refer to them as the portfolio stock allocations.

The stock return ${r}_{n}$ depends on the end-of-period price ${q}_{n}$, and this is unknown at the time of deciding on the amounts ${w}_{n}^{+}$ or ${w}_{n}^{-}$ to trade in the $n$th stock. We regard the returns ${r}_{n}$ as random variables. We measure reward and risk by the portfolio EP and EL, respectively. These are now simply given by

$$?{X}^{+}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}?{X}^{-},$$ | (2.7) |

and we will choose the portfolio allocation to

$$\text{maximize}?{X}^{+}\text{subject to the constraint that}?{X}^{-}\le \lambda ?{X}^{+},$$ | (2.8) |

where $\lambda $ is a risk tolerance parameter chosen to balance reward and risk in terms of these measures. So this portfolio allocation maximizes the EP subject to the ratio of EL to EP being below the level $\lambda $. The choice of a smaller $\lambda $ implies less risk tolerance, while a larger $\lambda $ implies more risk tolerance. In addition, we have the budget constraints (2.1), which can now be written as

$$\sum _{n=1}^{N}{w}_{n}^{+}\le {B}^{+}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}\sum _{n=1}^{N}{w}_{n}^{-}\le {B}^{-}.$$ | (2.9) |

Further constraints may be introduced to force diversification or other properties into the portfolio allocation. For example, if we wish to have no more than a given fraction ${a}^{+}$ of our long budget ${B}^{+}$ invested in any one stock and no more than a given fraction ${a}^{-}$ of our short budget ${B}^{-}$ invested in any one stock, then we may require that

$${w}_{n}^{+}\le {a}^{+}{B}^{+}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{w}_{n}^{-}\le {a}^{-}{B}^{-},n=1,\mathrm{\dots},N.$$ | (2.10) |

We refer to the solution of the optimization problem (2.8)–(2.10) as the EP–EL portfolio allocation model.

An attractive feature of the EL risk measure is that it has simple expressions for the contribution each stock makes to the portfolio risk. This contrasts strongly with the equivalent problem for the MV approach (see, for example, Griveau-Billion et al 2013; Maillard et al 2010). Looking only at the long part, its EL is

$$ | (2.11) |

This expresses the long portfolio EL risk as a sum of terms due to the individual stocks, and the corresponding risk contribution of the investment in the $n$th stock is given by

$$ | (2.12) |

Similarly, for the short part of the portfolio we have

$$ | (2.13) |

which expresses the short portfolio EL risk as a sum of terms due to the individual stocks, and the corresponding risk contribution of the investment in the $n$th stock is given by

$$ | (2.14) |

The total portfolio EL is

$$ | (2.15) |

which gives the contribution of the long part in total to the overall portfolio EL as $$ and the contribution of the short part in total to the overall portfolio EL as $$. Moreover, the contribution of the $n$th stock on its own to the overall portfolio risk is given by expressions similar to (2.12) and (2.14), namely

$$ | (2.16) |

All these contributions can be expressed as fractions by dividing them by their totals. It is straightforward to find similar expressions for the profit contributions of individual stocks to the portfolio total EP.

In addition to EP and EL, other items of interest from a trading point of view are the probability of a profit, the probability of a loss and the expected size of a profit, given there is a profit, and the expected size of a loss, given there is a loss. Formally, these items are $P(X>0)$, $$, $?(X\mid X>0)=?{X}^{+}/P(X>0)$ and $$. The calculation of these items is discussed in Section 4.

## 3 The long and short M-MD portfolio allocation

Here we modify the MV model by replacing the variance of PL by its mean deviation (MD), which yields the M-MD model. MD is a well-known alternative to variance as a measure of volatility and we now use it as a measure of risk analogous to that in the MV model. Many of the expressions involved in the M-MD model are closely analogous to those of the EP–EL model, and this makes the M-MD model a useful comparator to the EP–EL model.

From (2.4) and (2.6), the reward measured by expected PL is given by

$$?X=?{X}_{\mathrm{L}}+?{X}_{\mathrm{S}}=(1-{c}_{\mathrm{s}})\sum _{n=1}^{N}{w}_{n}^{+}({\mu}_{n}-{c}_{\mathrm{f}}^{+})+(1+{c}_{\mathrm{b}})\sum _{n=1}^{N}{w}_{n}^{-}({c}_{\mathrm{f}}^{-}-{\mu}_{n})$$ | (3.1) |

with ${\mu}_{n}=?{r}_{n}$. Let $Y=X-?X$. Then, from (2.4), (2.6) and (3.1), we get

$$Y=(1-{c}_{\mathrm{s}})\sum _{n=1}^{N}{w}_{n}^{+}({r}_{n}-{\mu}_{n})+(1+{c}_{\mathrm{b}})\sum _{n=1}^{N}{w}_{n}^{-}({\mu}_{n}-{r}_{n}).$$ | (3.2) |

Hence, $$. Substituting from (3.2) yields

$?YI(Y>0)$ | $=(1-{c}_{\mathrm{s}}){\displaystyle \sum _{n=1}^{N}}{w}_{n}^{+}?({r}_{n}-{\mu}_{n})I(Y>0)$ | |||

$\mathrm{\hspace{1em}\hspace{1em}}+(1+{c}_{\mathrm{b}}){\displaystyle \sum _{n=1}^{N}}{w}_{n}^{-}?({\mu}_{n}-{r}_{n})I(Y>0)$ | (3.3) | |||

and | ||||

$$ | $$ | |||

$$ | (3.4) |

Subtracting these two expressions gives us $\mathrm{MD}(X)$. The M-MD model would choose the allocation to

$$\text{maximize}?X\text{subject to the constraint that}\mathrm{MD}(X)\le \kappa ,$$ | (3.5) |

where $\kappa $ is a risk tolerance parameter chosen to balance reward and risk in terms of these measures. Further budget and diversification constraints may be added as for the EP–EL model. Note that typically the zero allocation ${w}_{n}^{+}=0$, ${w}_{n}^{-}=0$ for $n=1,\mathrm{\dots},N$ will be a feasible solution and this has $X=0$ so that the optimal solution will automatically satisfy $?X\ge 0$. The next section explains the required calculations in more detail. It is also possible to get expressions for the $\mathrm{MD}$ risk contributions due to the individual stocks by splitting up the sums in (3.3) and (3.4) according to the stock index $n$.

## 4 Calculation of optimal allocations and contributions

In principle, given a probability model for the stock returns $\{{r}_{n},n=1,\mathrm{\dots},N\}$, the values of $?{X}^{+}$ and $?{X}^{-}$ in the EP–EL model at any allocation can be calculated (maybe by numerical integration) and the optimal allocation can be found by applying some nonlinear programming method. We propose a Monte Carlo solution that uses generated stock returns from such a given model and then estimates the optimal EP–EL (or M-MD) allocation by solving a linear programming (LP) problem. The same method can be used when we start out with relevant observational stock returns data.

To formulate the method, suppose we have a model specifying the distribution of the return vector $\{{r}_{n},n=1,\mathrm{\dots},N\}$ to be experienced over the holding period. Then we generate a matrix of returns $R=\{{r}_{jn},j=1,\mathrm{\dots},J;n=1,\mathrm{\dots},N\}$ with the rows independent and identically distributed according to the assumed model. Given an allocation $\{({w}_{n}^{+},{w}_{n}^{-}),n=1,\mathrm{\dots},N\}$, set

$${x}_{j}=(1-{c}_{\mathrm{s}})\sum _{n=1}^{N}{w}_{n}^{+}[{r}_{jn}-{c}_{\mathrm{f}}^{+}]+(1+{c}_{\mathrm{b}})\sum _{n=1}^{N}{w}_{n}^{-}[{c}_{\mathrm{f}}^{-}-{r}_{jn}],$$ | (4.1) |

which is the PL we would get over the holding period should the return vector $\{{r}_{jn},n=1,\mathrm{\dots},N\}$ actually realize over that period. The corresponding PL would be ${x}_{j}^{+}$ and ${x}_{j}^{-}$, respectively. Taking averages over the rows of the generated return matrix $R$, we get the following estimates of the EP and EL, respectively:

$$\widehat{?{X}^{+}}=\frac{1}{J}\sum _{j=1}^{J}{x}_{j}^{+}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}\widehat{?{X}^{-}}=\frac{1}{J}\sum _{j=1}^{J}{x}_{j}^{-}.$$ | (4.2) |

These estimates can be plugged into the expressions in (2.8) so that the estimated version of the EP–EL portfolio allocation problem becomes

$$\text{maximize}\frac{1}{J}\sum _{j=1}^{J}{x}_{j}^{+}\text{subject to the constraint that}\frac{1}{J}\sum _{j=1}^{J}{x}_{j}^{-}\le \lambda \frac{1}{J}\sum _{j=1}^{J}{x}_{j}^{+}.$$ | (4.3) |

This problem can be expressed as a linear programming (LP) problem in terms of the variables $\{({w}_{n}^{+},{w}_{n}^{-}),n=1,\mathrm{\dots},N\}$ and $\{({x}_{j}^{+},{x}_{j}^{-}),j=1,\mathrm{\dots},J\}$ with the objective function

$$\frac{1}{J}\sum _{j=1}^{J}{x}_{j}^{+}.$$ |

In addition to the constraint in (4.3), the variable constraints (4.1) can be written in the linear form

$$(1-{c}_{\mathrm{s}})\sum _{n=1}^{N}{w}_{n}^{+}({r}_{jn}-{c}_{\mathrm{f}}^{+})+(1+{c}_{\mathrm{b}})\sum _{n=1}^{N}{w}_{n}^{-}({c}_{\mathrm{f}}^{-}-{r}_{jn})-{x}_{j}^{+}+{x}_{j}^{-}=0$$ | (4.4) |

for $j=1,\mathrm{\dots},J$. The budget constraints are

$$\sum _{n=1}^{N}{w}_{n}^{+}\le {B}^{+}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}\sum _{n=1}^{N}{w}_{n}^{-}\le {B}^{-}.$$ | (4.5) |

Further constraints such as (2.10) may be introduced to force diversification or other properties into the portfolio allocation. Standard LP solvers can be used to solve this problem, yielding estimates of the EP–EL optimal allocation. Modern LP solvers can handle large data sets. We used sets of $J=5000$ generated returns in the illustrations in Section 5, where more details are given.

Once we have solved for the optimal EP–EL portfolio, we can estimate the contributions of the individual stocks to the portfolio EP and EL as given above. For example, the estimate of the contribution (2.16) is

$$ | (4.6) |

Further, the estimates of probability of a profit and expected size of a profit conditionally given there is a profit are given by

$$\frac{1}{J}\sum _{j=1}^{J}I({x}_{j}>0)\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}\frac{{\sum}_{j=1}^{J}{x}_{j}^{+}}{{\sum}_{j=1}^{J}I({x}_{j}>0)},$$ |

respectively. Similar expressions hold for the probability of a loss and the expected size of a loss given there is a loss.

Regarding the M-MD model, if we know ${?}_{n}={\mu}_{n}$ explicitly from the assumed probability model, then $?X$, the portfolio expected PL, follows from (3.1). Alternatively we can estimate ${\mu}_{n}$ from the generated data by using

$${\widehat{\mu}}_{n}=\frac{1}{J}\sum _{j=1}^{J}{r}_{jn},$$ |

and then substituting this into (3.1). In this case, our estimate of $?X$ is given by (3.1) with $?{r}_{n}$ replaced by ${\widehat{\mu}}_{n}$, ie,

$$\widehat{?X}=(1-{c}_{\mathrm{s}})\sum _{n=1}^{N}{w}_{n}^{+}({\widehat{\mu}}_{n}-{c}_{\mathrm{f}}^{+})+(1+{c}_{\mathrm{b}})\sum _{n=1}^{N}{w}_{n}^{-}({c}_{\mathrm{f}}^{-}-{\widehat{\mu}}_{n}).$$ | (4.7) |

To estimate $\mathrm{MD}(X)$, set

$${y}_{j}=(1-{c}_{\mathrm{s}})\sum _{n=1}^{N}{w}_{n}^{+}[{r}_{jn}-{\widehat{\mu}}_{n}]+(1+{c}_{\mathrm{b}})\sum _{n=1}^{N}{w}_{n}^{-}[{\widehat{\mu}}_{n}-{r}_{jn}],j=1,\mathrm{\dots},J.$$ | (4.8) |

Then $?YI(Y>0)$ in (3.3) is estimated by

$$\frac{1}{J}\sum _{j=1}^{J}{y}_{j}I({y}_{j}>0)=\frac{1}{J}\sum _{j=1}^{J}{y}_{j}^{+}.$$ |

Similarly, $$ is estimated by $(1/J){\sum}_{j=1}^{J}{y}_{j}^{-}$ so that

$$\frac{1}{J}\sum _{j=1}^{J}[{y}_{j}^{+}+{y}_{j}^{-}]$$ |

estimates $\mathrm{MD}(X)$.

The estimated version of the M-MD optimal allocation problem becomes

$$\text{maximize}\widehat{?X}\text{subject to the constraint}\frac{1}{J}\sum _{j=1}^{J}[{y}_{j}^{+}+{y}_{j}^{-}]\le \kappa .$$ | (4.9) |

This problem can also be expressed as an LP problem in terms of the variables $\{({w}_{n}^{+},{w}_{n}^{-}),n=1,\mathrm{\dots},N\}$ and $\{({y}_{j}^{+},{y}_{j}^{-}),j=1,\mathrm{\dots},J\}$, with the objective function in (4.7) and variable constraints (4.8) written in the linear form

$$(1-{c}_{\mathrm{s}})\sum _{n=1}^{N}{w}_{n}^{+}[{r}_{jn}-{\widehat{\mu}}_{n}]+(1+{c}_{\mathrm{b}})\sum _{n=1}^{N}{w}_{n}^{-}[{\widehat{\mu}}_{n}-{r}_{jn}]-{y}_{j}^{+}+{y}_{j}^{-}=0.$$ | (4.10) |

Budget and diversification constraints as above can be added. After optimization, further items of interest such as profit and risk contributions with the M-MD model can be calculated.

## 5 Illustrative exploration of the EP–EL and M-MD models

To illustrate and explore the properties of the methodologies, a number of parameters need to be specified and we have to choose suitable data on which to base practically relevant models. The specifications used here are the following.

- (1)
The risk tolerance parameters $\lambda $ for the EP–EL method and $\kappa $ for the M-MD method are varied over their ranges. For the EP–EL portfolio, $\lambda $ has the natural range $(0,1)$ as it does not seem reasonable to allow EL to be larger than EP. For the M-MD portfolio, we first find the portfolio with maximal risk, say ${\mathrm{MD}}_{\mathrm{max}}$, by taking a large initial value for $\kappa $. Subsequently, we then range $\kappa $ over the interval from $0$ to ${\mathrm{MD}}_{\mathrm{max}}$.

- (2)
The long and short budget levels ${B}^{+}$ and ${B}^{-}$ are each fixed at the nominal level of 100.

- (3)
For the long and short diversification parameters ${a}^{+}$ and ${a}^{-}$ three values will be used: 1, 0.5 and 0.25. These allow the full budget to be placed in one stock, or no more than one-half or just one-quarter of the budget may be placed in one stock, respectively.

- (4)
For the buying and selling proportional trading costs ${c}_{\mathrm{b}}$ and ${c}_{\mathrm{s}}$, we take 20 basis points (bps) (ie, 0.002) as the base case. These values are currently achievable via contract for difference trading. We will also vary them to study their effects.

- (5)
The borrowing and lending interest rates depend on the length of the holding period. We will use a monthly holding period and then take both ${i}_{\mathrm{b}}$ and ${i}_{\mathrm{s}}$ at 0.005 (ie, an annual rate of 6%), but we will also vary them when assessing their effects.

- (6)
The most common distributional model of stock returns data is to assume that they follow a multivariate normal distribution, ie, that the return vector $\{{r}_{n},n=1,\mathrm{\dots},N\}$ is $N(?,?)$-distributed with $\{{\mu}_{n},n=1,\mathrm{\dots},N\}$ the expected returns and $?$ their covariance matrix. We will use this model and also replace the normal distribution by heavier tailed distributions and use observational returns data directly.

In order to make the choice of parameters for the models in item (6) practically relevant, we use data for 12 large cap stocks on the Johannesburg Stock Exchange (JSE): NPN, BTI, BHP, CFR, AGL, FSR, SBK, AMS, VOD, MTN, SOL and SLM. The sheet entitled “stock details” in the“Table JSE data” spreadsheet in the supplementary material^{1}^{1} 1 All supplementary material is available from the authors upon request. shows their full names and market capitalization at June 2019, when this study was started. The “closing prices” spreadsheet shows the monthly closing prices of the stocks over the period January 2012 to August 2019 and the “Rel Log Prices” spreadsheet shows the logarithms of the prices relative to their January 2012 values. Figure 1 graphs these relative log prices over time. We use three market scenarios for illustration purposes: a bullish market, a bearish market and a nontrending market. As far as the 12 JSE stocks are concerned, the graph shows that the period January 2012 to August 2014 was largely bullish, September 2014 to January 2016 was largely bearish and September 2017 to August 2019 was mostly nontrending. We will also use the full data set for a fourth illustration scenario.

For the illustrative results to follow, we take the buying prices as ${p}_{n}=1$ for $n=1,\mathrm{\dots},N=12$, which entails that the number of shares to buy and the amount invested in each stock are the same, ie, ${k}_{n}={w}_{n}$. With the long and short budgets at 100 each, the share allocations may also be interpreted as percentages of the budgets invested in each stock.

### 5.1 Bullish market

Parameter | Stock | NPN | BTI | BHP | CFR | AGL | FSR | SBK | AMS | VOD | MTN | SOL | SLM |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Expectation | 0.042 | 0.015 | 0.011 | 0.029 | $-$0.003 | 0.023 | 0.010 | $-$0.006 | 0.011 | 0.016 | 0.015 | 0.026 | |

SD | 0.074 | 0.046 | 0.070 | 0.069 | 0.082 | 0.051 | 0.049 | 0.079 | 0.064 | 0.049 | 0.046 | 0.048 | |

Correlations | NPN | 1.000 | 0.287 | 0.450 | 0.387 | 0.312 | $-$0.029 | $-$0.006 | $-$0.008 | 0.386 | 0.146 | 0.316 | 0.066 |

BTI | 0.287 | 1.000 | 0.230 | 0.511 | 0.022 | 0.207 | 0.000 | $-$0.259 | 0.214 | 0.226 | 0.359 | 0.317 | |

BHP | 0.450 | 0.230 | 1.000 | 0.394 | 0.800 | 0.203 | 0.145 | 0.428 | 0.282 | 0.048 | 0.504 | 0.106 | |

CFR | 0.387 | 0.511 | 0.394 | 1.000 | 0.080 | $-$0.070 | $-$0.071 | $-$0.068 | 0.546 | 0.099 | 0.469 | 0.069 | |

AGL | 0.312 | 0.022 | 0.800 | 0.080 | 1.000 | 0.142 | 0.092 | 0.728 | 0.116 | $-$0.127 | 0.379 | $-$0.089 | |

FSR | $-$0.029 | 0.207 | 0.203 | $-$0.070 | 0.142 | 1.000 | 0.720 | 0.082 | 0.275 | 0.130 | $-$0.013 | 0.562 | |

SBK | $-$0.006 | 0.000 | 0.145 | $-$0.071 | 0.092 | 0.720 | 1.000 | 0.137 | 0.389 | 0.299 | 0.155 | 0.569 | |

AMS | $-$0.008 | $-$0.259 | 0.428 | $-$0.068 | 0.728 | 0.082 | 0.137 | 1.000 | 0.166 | $-$0.067 | 0.222 | $-$0.119 | |

VOD | 0.386 | 0.214 | 0.282 | 0.546 | 0.116 | 0.275 | 0.389 | 0.166 | 1.000 | 0.452 | 0.215 | 0.312 | |

MTN | 0.146 | 0.226 | 0.048 | 0.099 | $-$0.127 | 0.130 | 0.299 | $-$0.067 | 0.452 | 1.000 | 0.142 | 0.414 | |

SOL | 0.316 | 0.359 | 0.504 | 0.469 | 0.379 | $-$0.013 | 0.155 | 0.222 | 0.215 | 0.142 | 1.000 | 0.140 | |

SLM | 0.066 | 0.317 | 0.106 | 0.069 | $-$0.089 | 0.562 | 0.569 | $-$0.119 | 0.312 | 0.414 | 0.140 | 1.000 |

Table 1 shows the estimated expected monthly returns of the stock prices, their standard deviations and their correlation matrix. In line with the bullish market of this period, the expected returns are mostly positive, ranging from 0.042 for NPN to $-$0.006 for AMS. The standard deviations are quite large, ranging from 0.046 to 0.082. Most correlations are positive and some are large, in line with what we would expect, eg, 0.800 for the pair AGL and BHP (both commodity stocks), 0.728 for AGL and AMS (the latter being an affiliate of the former) and 0.720 for SBK and FSR (both banks).

We now explore the properties of an EP–EL allocation assuming that the returns to be experienced over the holding period follow a normal distribution with the bullish market parameters of Table 1. To start with, both risk diversification factors are set at $1$, ie, ${a}^{+}={a}^{-}=1$, which allows single stock long and short portfolios. Trading costs are taken at 20bps and interest rates at 0.5%. When optimal allocations are calculated using a set of $J=5000$ simulation-generated returns, some simulation error may be involved in the results. In order to estimate the size of such errors, we carried out 100 independent such sets of $J=5000$ simulation runs, each of the overall process, then averaged all the relevant items over these 100 independent sets and computed their standard errors as well. The “Table Bullish market” spreadsheet in the supplementary material provides full numerical and graphical details from which the results discussed below are extracted.

NPN | BTI | BHP | CFR | AGL | FSR | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$?$ | Allo | SE | Allo | SE | Allo | SE | Allo | SE | Allo | SE | Allo | SE |

0.125 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.150 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.160 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.165 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.170 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.175 | 3 | 3 | 0 | 0 | 0 | 0 | 2 | 2 | $-$1 | 1 | 2 | 2 |

0.180 | 9 | 5 | 0 | 0 | 0 | 0 | 5 | 2 | $-$4 | 2 | 8 | 4 |

0.190 | 32 | 5 | 0 | 0 | 0 | 0 | 12 | 2 | $-$12 | 2 | 22 | 3 |

0.200 | 44 | 3 | 0 | 0 | 0 | 0 | 11 | 1 | $-$17 | 1 | 25 | 2 |

0.210 | 51 | 2 | 0 | 0 | 0 | 0 | 8 | 1 | $-$20 | 1 | 24 | 1 |

0.220 | 57 | 2 | 0 | 0 | 0 | 0 | 5 | 1 | $-$23 | 1 | 22 | 1 |

0.230 | 63 | 2 | 0 | 0 | 0 | 0 | 3 | 1 | $-$25 | 1 | 20 | 2 |

0.240 | 69 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | $-$28 | 1 | 17 | 2 |

0.250 | 75 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | $-$31 | 1 | 14 | 2 |

0.275 | 89 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | $-$37 | 2 | 5 | 2 |

0.300 | 99 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | $-$42 | 3 | 0 | 0 |

0.350 | 100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $-$32 | 5 | 0 | 0 |

0.400 | 100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $-$5 | 3 | 0 | 0 |

0.450 | 100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.500 | 100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

SBK | AMS | VOD | MTN | SOL | SLM | |||||||

$?$ | Allo | SE | Allo | SE | Allo | SE | Allo | SE | Allo | SE | Allo | SE |

0.125 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.150 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.160 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.165 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.170 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.175 | 0 | 0 | 0 | 0 | $-$2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 |

0.180 | 0 | 0 | 0 | 0 | $-$6 | 3 | 0 | 0 | 0 | 0 | 5 | 2 |

0.190 | 0 | 0 | 0 | 0 | $-$15 | 2 | 0 | 0 | 0 | 0 | 14 | 2 |

0.200 | 0 | 0 | 0 | 0 | $-$12 | 1 | 0 | 0 | 0 | 0 | 16 | 1 |

0.210 | 0 | 0 | 0 | 0 | $-$8 | 1 | 0 | 0 | 0 | 0 | 15 | 1 |

0.220 | 0 | 0 | 0 | 0 | $-$4 | 1 | 0 | 0 | 0 | 0 | 15 | 2 |

0.230 | 0 | 0 | 0 | 0 | $-$2 | 1 | 0 | 0 | 0 | 0 | 13 | 2 |

0.240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 | 2 |

0.250 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 10 | 2 |

0.275 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 1 |

0.300 | 0 | 0 | $-$8 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.350 | 0 | 0 | $-$50 | 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.400 | 0 | 0 | $-$93 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.450 | 0 | 0 | $-$100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.500 | 0 | 0 | $-$100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Table 2 shows the averaged allocations (“Allo”) and their standard errors (“SE”) obtained in this way as functions of the risk tolerance $\lambda $ (“Lam”). The first column gives the grid of values used for $\lambda $. The remaining 12 double columns show the corresponding average optimal stock allocations and their simulation standard errors. For ease of reading, all allocations and standard errors were rounded to the nearest integer.

EL | EP | Ratio | ||||
---|---|---|---|---|---|---|

$?$ | EL_est | SE | EP_est | SE | est | SE |

0.125 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

0.150 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

0.160 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

0.165 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

0.170 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

0.175 | 0.034 | 0.034 | 0.192 | 0.193 | 0.016 | 0.016 |

0.180 | 0.119 | 0.060 | 0.657 | 0.335 | 0.051 | 0.026 |

0.190 | 0.407 | 0.060 | 2.139 | 0.314 | 0.158 | 0.023 |

0.200 | 0.559 | 0.029 | 2.786 | 0.146 | 0.197 | 0.009 |

0.210 | 0.650 | 0.014 | 3.083 | 0.068 | 0.211 | 0.000 |

0.220 | 0.731 | 0.015 | 3.307 | 0.068 | 0.221 | 0.000 |

0.230 | 0.815 | 0.016 | 3.528 | 0.071 | 0.231 | 0.000 |

0.240 | 0.901 | 0.018 | 3.740 | 0.073 | 0.241 | 0.000 |

0.250 | 0.991 | 0.019 | 3.950 | 0.076 | 0.251 | 0.000 |

0.275 | 1.237 | 0.024 | 4.485 | 0.087 | 0.276 | 0.000 |

0.300 | 1.499 | 0.025 | 5.003 | 0.086 | 0.300 | 0.000 |

0.350 | 2.019 | 0.032 | 5.786 | 0.093 | 0.349 | 0.000 |

0.400 | 2.557 | 0.023 | 6.500 | 0.074 | 0.394 | 0.003 |

0.450 | 2.639 | 0.020 | 6.602 | 0.037 | 0.400 | 0.005 |

0.500 | 2.639 | 0.020 | 6.602 | 0.037 | 0.400 | 0.005 |

Table 3 shows the average estimated optimal EL, EP and their ratio together with their respective simulation standard errors.

In practice, we would need to specify a value to use for the risk tolerance parameter $\lambda $. For the values in the last two rows of Table 2, the optimal allocation was long 100 shares of NPN and short 100 of AMS both with zero standard errors. This means that all 100 repeated simulation sets produced this allocation, so we can be quite certain this allocation is optimal even when based on a finite ($J=5000$) sample size. The corresponding bottom two rows of Table 3 show that the average EL and EP values of this allocation were 2.639 (SE 0.020) and 6.602 (SE 0.037), respectively, while the risk/reward ratio was about 0.4. Even if we take the total exposure to stocks as 200 (the sum of the long and short budgets allocated), the monthly EP reward as a percentage of this total exposure would be about 3.3% at the EL risk of 1.3%. This seems quite acceptable from a monthly trading point of view and would imply that taking $\lambda $ just above 0.4 would be reasonable. If we take $\lambda $ even larger, then the constraint $\mathrm{EL}\le \lambda \mathrm{EP}$ is no longer binding, the optimal portfolio allocation stays fixed at long 100 of NPN and short 100 of AMS, while EL and EP stays at the levels noted above. Looking at the other allocation entries in Table 3, we see that their EL and EP values are all smaller, so we may also describe this limiting portfolio as the EP–EL portfolio with simultaneous maximal EL and maximal EP and can refer to it simply as the “maximal portfolio”.

If the trader wished to be more conservative and specified a smaller value, such as $\lambda =0.25$, then the EP–EL method would produce not a unique allocation but suggested long positions in NPN, FSR and SLM coupled with a short position in AGL, and there is some leeway in terms of the actual split among these stocks as indicated by the corresponding averages and standard errors. Use of sample sizes larger than 5000 in the simulation-based LP optimization would be needed if more definite allocations were required. Going on to even more conservative specifications such as $$, the optimal allocations over the 100 repeated sets were all found to not invest at all. This shows that the trading hurdles caused by the costs and interest rates make it impossible to attain a very low risk portfolio that is actually invested in stocks under the circumstances reflected by the case under consideration here.

Returning to the maximal portfolio, the third and fourth rows of Table 4 show the estimated percentage risk and profit contributions to the maximal portfolio due to NPN and AMS. As to EL, about 27% is contributed by NPN and 73% by AMS. The EP contributions are about 61% and 39%, respectively, from these two stocks. In all cases, the simulation standard errors are less than 1% (if 0% is indicated, the SE is actually below 0.5% due to rounding). So NPN is simultaneously the better stock from a loss and a profit contributor point of view.

NPN | AMS | |||
---|---|---|---|---|

Con | SE | Con | SE | |

EL | 27 | 1 | 73 | 1 |

EP | 61 | 0 | 39 | 0 |

MD | 46 | 0 | 54 | 0 |

$?X$ | 83 | 1 | 17 | 1 |

Next we turn to the M-MD model. Table 5 shows the M-MD allocation results. Here, the first column gives the grid of values used for the risk tolerance parameter $\kappa $ and the contents of the other columns are analogous to those of Table 2. Note that the risk measure $\mathrm{MD}$ of the M-MD model behaves quite differently from EL of EP–EL especially at small risks. There is some similarity between the two sets of allocations, and as $\kappa $ increases the M-MD model also eventually settles on long 100 NPN and short 100 AMS shares, which is the same as the maximal EP–EL portfolio. Table 6 shows the estimated $\mathrm{MD}$ and $?X$ values and their ratios (together with standard errors) as functions of $\kappa $. The final portfolio also has simultaneous maximal risk and expected PL as measured by $\mathrm{MD}$ and $?X$, respectively. So it may also be referred to as the maximal M-MD portfolio.

Note that the estimated expected PL of this maximal portfolio M-MD is given as 3.962. Recall the relation $?X=?{X}^{+}-?{X}^{-}$. From Table 3, we get $\mathrm{EP}=6.602$ and $\mathrm{EL}=2.639$ so that $\mathrm{EP}-\mathrm{EL}=6.602-2.639=3.963$, agreeing with the M-MD entry for $?X$. The numerical values of the risk measure $\mathrm{MD}$ are very different from those of EL, which reflects that $\mathrm{MD}$ measures risk quite differently and does not have the monetary loss interpretation of EL. Although the maximal portfolio has expected PL of 3.962 and $\mathrm{MD}$ risk of 8.668, it is not immediately clear what these numbers imply about the acceptability of this portfolio from a practical trading point of view.

The last two rows of Table 4 show the expected $\mathrm{MD}$ and PL contributions by these risk and reward measures for the maximal portfolio. When comparing them with those of EP–EL, the stock NPN appears even stronger in terms of reward contributions, while the two stocks are more evenly matched in terms of the risk contributions.

NPN | BTI | BHP | CFR | AGL | FSR | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$?$ | Allo | SE | Allo | SE | Allo | SE | Allo | SE | Allo | SE | Allo | SE |

0.087 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |

0.173 | 2 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | $-$1 | 0 | 2 | 0 |

0.260 | 3 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | $-$1 | 0 | 3 | 0 |

0.347 | 5 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | $-$2 | 0 | 4 | 0 |

0.433 | 6 | 0 | $-$1 | 0 | 0 | 0 | 4 | 0 | $-$2 | 0 | 6 | 0 |

0.867 | 13 | 0 | $-$2 | 0 | $-$1 | 0 | 9 | 0 | $-$5 | 0 | 12 | 0 |

1.300 | 19 | 0 | $-$3 | 0 | $-$2 | 0 | 13 | 0 | $-$7 | 0 | 18 | 0 |

1.734 | 26 | 0 | $-$4 | 1 | $-$3 | 1 | 18 | 0 | $-$10 | 1 | 24 | 0 |

2.167 | 32 | 0 | $-$2 | 0 | $-$1 | 0 | 18 | 0 | $-$13 | 1 | 27 | 1 |

2.600 | 38 | 0 | 0 | 0 | 0 | 0 | 15 | 0 | $-$14 | 0 | 27 | 1 |

3.034 | 48 | 1 | 0 | 0 | 0 | 0 | 10 | 1 | $-$19 | 1 | 25 | 1 |

3.467 | 57 | 1 | 0 | 0 | 0 | 0 | 5 | 1 | $-$23 | 1 | 22 | 1 |

3.901 | 66 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | $-$27 | 1 | 19 | 2 |

4.334 | 75 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $-$31 | 1 | 14 | 2 |

4.767 | 83 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $-$34 | 1 | 9 | 2 |

5.201 | 91 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $-$38 | 1 | 4 | 1 |

5.634 | 99 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $-$42 | 1 | 0 | 0 |

6.068 | 100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $-$42 | 4 | 0 | 0 |

6.501 | 100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $-$38 | 4 | 0 | 0 |

6.934 | 100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $-$34 | 5 | 0 | 0 |

7.368 | 100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $-$31 | 6 | 0 | 0 |

7.801 | 100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $-$25 | 4 | 0 | 0 |

8.235 | 100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $-$13 | 2 | 0 | 0 |

8.668 | 100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

SBK | AMS | VOD | MTN | SOL | SLM | |||||||

$?$ | Allo | SE | Allo | SE | Allo | SE | Allo | SE | Allo | SE | Allo | SE |

0.087 | 0 | 0 | 0 | 0 | $-$1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.173 | 0 | 0 | 0 | 0 | $-$2 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |

0.260 | 0 | 0 | 0 | 0 | $-$3 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |

0.347 | 0 | 0 | 1 | 0 | $-$4 | 0 | 1 | 0 | 0 | 0 | 2 | 0 |

0.433 | 0 | 0 | 1 | 0 | $-$5 | 0 | 1 | 0 | 0 | 0 | 3 | 0 |

0.867 | 0 | 0 | 2 | 0 | $-$10 | 0 | 2 | 0 | 0 | 0 | 6 | 0 |

1.300 | 0 | 0 | 3 | 0 | $-$16 | 0 | 4 | 0 | 0 | 0 | 9 | 0 |

1.734 | 0 | 0 | 5 | 0 | $-$21 | 0 | 5 | 1 | 0 | 0 | 12 | 1 |

2.167 | 0 | 0 | 3 | 0 | $-$22 | 0 | 2 | 1 | 0 | 0 | 15 | 1 |

2.600 | 0 | 0 | 0 | 0 | $-$18 | 1 | 0 | 0 | 0 | 0 | 17 | 1 |

3.034 | 0 | 0 | 0 | 0 | $-$11 | 1 | 0 | 0 | 0 | 0 | 16 | 1 |

3.467 | 0 | 0 | 0 | 0 | $-$4 | 1 | 0 | 0 | 0 | 0 | 15 | 1 |

3.901 | 0 | 0 | 0 | 0 | $-$1 | 0 | 0 | 0 | 0 | 0 | 13 | 2 |

4.334 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 10 | 2 |

4.767 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 7 | 2 |

5.201 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 1 |

5.634 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

6.068 | 0 | 0 | $-$14 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

6.501 | 0 | 0 | $-$29 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

6.934 | 0 | 0 | $-$42 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

7.368 | 0 | 0 | $-$54 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

7.801 | 0 | 0 | $-$67 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

8.235 | 0 | 0 | $-$83 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

8.668 | 0 | 0 | $-$100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

MD | $\mathrm{E}?$ | Ratio | ||||
---|---|---|---|---|---|---|

$?$ | ${\text{??}}_{\text{???}}$ | SE | $\mathrm{E}{?}_{\text{???}}$ | SE | est | SE |

0.087 | 0.087 | 0.000 | 0.041 | 0.002 | 2.193 | 0.100 |

0.173 | 0.173 | 0.001 | 0.115 | 0.003 | 1.514 | 0.033 |

0.260 | 0.260 | 0.001 | 0.187 | 0.005 | 1.404 | 0.038 |

0.347 | 0.347 | 0.001 | 0.275 | 0.005 | 1.265 | 0.024 |

0.433 | 0.433 | 0.001 | 0.340 | 0.004 | 1.275 | 0.016 |

0.867 | 0.867 | 0.003 | 0.705 | 0.007 | 1.230 | 0.012 |

1.300 | 1.300 | 0.004 | 1.069 | 0.009 | 1.217 | 0.009 |

1.734 | 1.734 | 0.005 | 1.433 | 0.011 | 1.211 | 0.009 |

2.167 | 2.167 | 0.007 | 1.782 | 0.014 | 1.216 | 0.009 |

2.600 | 2.600 | 0.008 | 2.091 | 0.015 | 1.244 | 0.008 |

3.034 | 3.034 | 0.009 | 2.348 | 0.018 | 1.293 | 0.009 |

3.467 | 3.467 | 0.011 | 2.570 | 0.020 | 1.350 | 0.010 |

3.901 | 3.901 | 0.012 | 2.770 | 0.024 | 1.409 | 0.012 |

4.334 | 4.334 | 0.013 | 2.953 | 0.026 | 1.468 | 0.012 |

4.767 | 4.767 | 0.015 | 3.118 | 0.028 | 1.530 | 0.013 |

5.201 | 5.201 | 0.016 | 3.278 | 0.030 | 1.588 | 0.014 |

5.634 | 5.634 | 0.017 | 3.436 | 0.033 | 1.641 | 0.015 |

6.068 | 6.068 | 0.019 | 3.557 | 0.035 | 1.707 | 0.016 |

6.501 | 6.501 | 0.020 | 3.643 | 0.038 | 1.786 | 0.018 |

6.934 | 6.934 | 0.021 | 3.717 | 0.040 | 1.868 | 0.020 |

7.368 | 7.368 | 0.023 | 3.787 | 0.043 | 1.948 | 0.022 |

7.801 | 7.801 | 0.024 | 3.851 | 0.045 | 2.029 | 0.023 |

8.235 | 8.235 | 0.025 | 3.907 | 0.048 | 2.110 | 0.025 |

8.668 | 8.668 | 0.027 | 3.962 | 0.051 | 2.191 | 0.027 |

The graphs in Figure 2 show the efficient frontiers of the EP–EL and M-MD allocations, plotting reward against risk at varying values of the risk tolerances. These graphs have the familiar form of such frontiers. However, they start at $(0,0)$, which corresponds to not investing at all and therefore carries both zero risk and zero reward, thus forming the “minimal portfolio” here. The frontiers increase monotonically with their risk tolerance parameters and end at the points corresponding to the maximal portfolio.

Next we tighten the risk diversification requirement by allowing at most one-half of the portfolio budgets to be invested in one stock.^{2}^{2} 2 In order to save space, we drop the columns showing the respective simulation standard errors for this case. The reader can find these in the “Table Bullish market” spreadsheet EP-EL-2 in the supplementary material. This allowed us to combine the equivalents of Tables 2–4 into one table. The results in Table 7 are quite similar to those in Tables 2–4 except that, if the risk tolerance parameter increases sufficiently, the optimal allocations are split over two stocks on each of the long and short sides. Here, the maximal portfolio is long 50 shares in each of NPN and CFR and short 50 shares in each of AGL and AMS. These four stocks are now the reward and risk contributors. Regarding the EL contributions, NPN now contributes almost nothing, CFR about 14%, AGL about 39% and AMS about 47%. With the EP reward measure NPN contributes about 30%, CFR about 24%, AGL about 20% and AMS about 26%. Note that NPN contributed somewhat more than CFR, in line with the previous finding that NPN was the first choice when a full budget single stock long position was allowed. When this is not allowed, CFR is the second best long choice. A similar remark applies to the two short stocks. Clearly, NPN is also the best stock on the EL contribution score, and the two short positions contribute most of the EL risk. With a total exposure of 200, the EP is about 2.7% of total exposure at an EL risk of 1.1%, which again is acceptable from a trading point of view.

$?$ | NPN | BTI | BHP | CFR | AGL | FSR | SBK | AMS | VOD | MTN | SOL | SLM | EL | EP | Ratio |

0.125 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.000 | 0.000 | 0.000 |

0.150 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.000 | 0.000 | 0.000 |

0.160 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.000 | 0.000 | 0.000 |

0.165 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.000 | 0.000 | 0.000 |

0.170 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.000 | 0.000 | 0.000 |

0.175 | 3 | 0 | 0 | 1 | $-$1 | 2 | 0 | 0 | $-$2 | 0 | 0 | 1 | 0.033 | 0.191 | 0.014 |

0.180 | 9 | $-$1 | 0 | 5 | $-$4 | 7 | 0 | 1 | $-$6 | 0 | 0 | 4 | 0.111 | 0.616 | 0.047 |

0.190 | 30 | 0 | 0 | 12 | $-$12 | 21 | 0 | 1 | $-$14 | 0 | 0 | 14 | 0.390 | 2.053 | 0.148 |

0.200 | 43 | 0 | 0 | 12 | $-$18 | 24 | 0 | 0 | $-$13 | 0 | 0 | 17 | 0.552 | 2.759 | 0.192 |

0.210 | 49 | 0 | 0 | 10 | $-$21 | 22 | 0 | 0 | $-$7 | 0 | 0 | 19 | 0.649 | 3.089 | 0.210 |

0.220 | 50 | 0 | 0 | 11 | $-$23 | 18 | 0 | $-$1 | $-$2 | 0 | 0 | 22 | 0.716 | 3.256 | 0.220 |

0.230 | 50 | 0 | 0 | 13 | $-$24 | 12 | 0 | $-$2 | 0 | 0 | 0 | 25 | 0.778 | 3.382 | 0.230 |

0.240 | 50 | 0 | 0 | 15 | $-$25 | 8 | 0 | $-$5 | 0 | 0 | 0 | 27 | 0.836 | 3.485 | 0.240 |

0.250 | 50 | 0 | 0 | 17 | $-$24 | 5 | 0 | $-$9 | 0 | 0 | 0 | 28 | 0.895 | 3.582 | 0.250 |

0.275 | 50 | 0 | 0 | 20 | $-$22 | 2 | 0 | $-$19 | 0 | 0 | 0 | 28 | 1.050 | 3.817 | 0.275 |

0.300 | 50 | 0 | 0 | 23 | $-$20 | 1 | 0 | $-$30 | 0 | 0 | 0 | 26 | 1.218 | 4.060 | 0.300 |

0.350 | 50 | 0 | 0 | 33 | $-$24 | 0 | 0 | $-$45 | 0 | 0 | 0 | 17 | 1.606 | 4.589 | 0.350 |

0.400 | 50 | 0 | 0 | 44 | $-$43 | 0 | 0 | $-$49 | 0 | 0 | 0 | 6 | 2.065 | 5.169 | 0.399 |

0.450 | 50 | 0 | 0 | 49 | $-$50 | 0 | 0 | $-$50 | 0 | 0 | 0 | 1 | 2.245 | 5.382 | 0.417 |

0.500 | 50 | 0 | 0 | 50 | $-$50 | 0 | 0 | $-$50 | 0 | 0 | 0 | 0 | 2.246 | 5.383 | 0.417 |

NPN | CFR | AGL | AMS | ||||||||||||

EL | $-$1 | 14 | 39 | 47 | |||||||||||

EP | 30 | 24 | 20 | 26 |

Table 8 shows the equivalent results for the M-MD model. Note that, though risk and reward are measured differently here, the maximal portfolio is again the same as that of EP–EL. The M-MD reward and risk contributions from the stocks are numerically different from those of EP–EL, but this is to be expected in view of the different measures used there. Graphs of the efficient frontiers for the two methods under the present specifications are available in the supplementary material. They are qualitatively similar to those in Figure 2.

$?$ | NPN | BTI | BHP | CFR | AGL | FSR | SBK | AMS | VOD | MTN | SOL | SLM | MD | $\mathrm{E}?$ | Ratio |

0.072 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | $-$1 | 0 | 0 | 1 | 0.072 | 0.060 | 1.196 |

0.144 | 2 | 0 | 0 | 2 | $-$1 | 2 | 0 | 0 | $-$2 | 0 | 0 | 1 | 0.144 | 0.120 | 1.196 |

0.216 | 3 | $-$1 | 0 | 2 | $-$1 | 3 | 0 | 1 | $-$3 | 1 | 0 | 2 | 0.216 | 0.181 | 1.196 |

0.288 | 4 | $-$1 | $-$1 | 3 | $-$2 | 4 | 0 | 1 | $-$4 | 1 | 0 | 2 | 0.288 | 0.241 | 1.196 |

0.360 | 5 | $-$1 | $-$1 | 4 | $-$2 | 5 | 0 | 1 | $-$5 | 1 | 0 | 3 | 0.360 | 0.301 | 1.196 |

0.720 | 11 | $-$2 | $-$1 | 8 | $-$4 | 10 | 0 | 2 | $-$9 | 2 | 0 | 5 | 0.720 | 0.602 | 1.196 |

1.080 | 16 | $-$3 | $-$2 | 12 | $-$7 | 15 | 0 | 3 | $-$14 | 3 | 0 | 8 | 1.080 | 0.904 | 1.196 |

1.440 | 22 | $-$4 | $-$3 | 15 | $-$9 | 20 | 0 | 4 | $-$18 | 5 | 0 | 11 | 1.440 | 1.205 | 1.196 |

1.800 | 27 | $-$5 | $-$4 | 19 | $-$11 | 25 | 0 | 5 | $-$23 | 6 | 0 | 14 | 1.800 | 1.506 | 1.196 |

2.161 | 32 | $-$2 | $-$2 | 19 | $-$13 | 27 | 0 | 4 | $-$23 | 3 | 0 | 16 | 2.161 | 1.798 | 1.202 |

2.521 | 37 | 0 | 0 | 17 | $-$14 | 28 | 0 | 0 | $-$20 | 0 | 0 | 18 | 2.521 | 2.069 | 1.219 |

2.881 | 45 | 0 | 0 | 12 | $-$18 | 25 | 0 | 0 | $-$14 | 0 | 0 | 17 | 2.881 | 2.297 | 1.255 |

3.241 | 50 | 0 | 0 | 9 | $-$22 | 22 | 0 | 0 | $-$5 | 0 | 0 | 18 | 3.241 | 2.484 | 1.305 |

3.601 | 50 | 0 | 0 | 13 | $-$25 | 11 | 0 | $-$3 | 0 | 0 | 0 | 26 | 3.601 | 2.613 | 1.379 |

3.961 | 50 | 0 | 0 | 17 | $-$24 | 4 | 0 | $-$10 | 0 | 0 | 0 | 29 | 3.961 | 2.697 | 1.470 |

4.321 | 50 | 0 | 0 | 20 | $-$22 | 2 | 0 | $-$19 | 0 | 0 | 0 | 28 | 4.321 | 2.767 | 1.563 |

4.681 | 50 | 0 | 0 | 23 | $-$21 | 1 | 0 | $-$28 | 0 | 0 | 0 | 26 | 4.681 | 2.829 | 1.656 |

5.041 | 50 | 0 | 0 | 26 | $-$20 | 0 | 0 | $-$36 | 0 | 0 | 0 | 24 | 5.041 | 2.886 | 1.748 |

5.401 | 50 | 0 | 0 | 29 | $-$21 | 0 | 0 | $-$42 | 0 | 0 | 0 | 21 | 5.401 | 2.939 | 1.840 |

5.761 | 50 | 0 | 0 | 33 | $-$24 | 0 | 0 | $-$45 | 0 | 0 | 0 | 17 | 5.761 | 2.987 | 1.931 |

6.121 | 50 | 0 | 0 | 37 | $-$30 | 0 | 0 | $-$48 | 0 | 0 | 0 | 13 | 6.121 | 3.031 | 2.022 |

6.482 | 50 | 0 | 0 | 40 | $-$37 | 0 | 0 | $-$49 | 0 | 0 | 0 | 10 | 6.482 | 3.070 | 2.114 |

6.842 | 50 | 0 | 0 | 43 | $-$44 | 0 | 0 | $-$50 | 0 | 0 | 0 | 7 | 6.842 | 3.107 | 2.205 |

7.202 | 50 | 0 | 0 | 50 | $-$50 | 0 | 0 | $-$50 | 0 | 0 | 0 | 0 | 7.202 | 3.138 | 2.299 |

NPN | CFR | AGL | AMS | ||||||||||||

MD | 17 | 20 | 28 | 35 | |||||||||||

$?X$ | 52 | 31 | 6 | 11 |

Next we tightened the risk diversification further by allowing at most a quarter of the budgets to go into one stock. Tables 9 and 10 show the results in the same manner as for the previous diversification cases. The outcomes are quite similar except that, as the risk tolerance parameter increases, the optimal allocations are now split over four stocks on the long side and only two on the short side, so that the maximal portfolio is long 25 shares in each of NPN, CFR, FSR and SLM and short 25 shares in each of AGL and AMS. Evidently, whereas there were enough stocks with returns large enough to go long in, only these two were available to go short in during this bullish market, implying that only half the short budget was used.

$?$ | NPN | BTI | BHP | CFR | AGL | FSR | SBK | AMS | VOD | MTN | SOL | SLM | EL | EP | Ratio |

0.125 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.000 | 0.000 | 0.000 |

0.150 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.000 | 0.000 | 0.000 |

0.160 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.000 | 0.000 | 0.000 |

0.165 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.000 | 0.000 | 0.000 |

0.170 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.003 | 0.018 | 0.002 |

0.175 | 3 | 0 | 0 | 3 | $-$1 | 3 | 0 | 0 | $-$3 | 1 | 0 | 2 | 0.046 | 0.260 | 0.023 |

0.180 | 8 | $-$1 | $-$1 | 7 | $-$3 | 7 | 0 | 0 | $-$7 | 1 | 0 | 6 | 0.114 | 0.633 | 0.054 |

0.190 | 18 | $-$1 | $-$1 | 16 | $-$8 | 18 | 0 | 1 | $-$14 | 2 | 0 | 15 | 0.301 | 1.583 | 0.135 |

0.200 | 23 | 0 | 0 | 22 | $-$11 | 23 | 0 | 0 | $-$15 | 1 | 0 | 22 | 0.449 | 2.246 | 0.186 |

0.210 | 25 | 0 | 0 | 25 | $-$14 | 25 | 0 | 0 | $-$11 | 0 | 0 | 25 | 0.536 | 2.553 | 0.210 |

0.220 | 25 | 0 | 0 | 25 | $-$15 | 25 | 0 | $-$1 | $-$5 | 0 | 0 | 25 | 0.590 | 2.684 | 0.220 |

0.230 | 25 | 0 | 0 | 25 | $-$16 | 25 | 0 | $-$3 | $-$2 | 0 | 0 | 25 | 0.642 | 2.790 | 0.230 |

0.240 | 25 | 0 | 0 | 25 | $-$16 | 25 | 0 | $-$7 | $-$1 | 0 | 0 | 25 | 0.692 | 2.882 | 0.240 |

0.250 | 25 | 0 | 0 | 25 | $-$15 | 25 | 0 | $-$12 | 0 | 0 | 0 | 25 | 0.741 | 2.965 | 0.250 |

0.275 | 25 | 0 | 0 | 25 | $-$15 | 25 | 0 | $-$21 | 0 | 0 | 0 | 25 | 0.868 | 3.156 | 0.275 |

0.300 | 25 | 0 | 0 | 25 | $-$21 | 25 | 0 | $-$25 | 0 | 0 | 0 | 25 | 0.991 | 3.323 | 0.298 |

0.350 | 25 | 0 | 0 | 25 | $-$25 | 25 | 0 | $-$25 | 0 | 0 | 0 | 25 | 1.045 | 3.391 | 0.308 |

0.400 | 25 | 0 | 0 | 25 | $-$25 | 25 | 0 | $-$25 | 0 | 0 | 0 | 25 | 1.045 | 3.391 | 0.308 |

0.450 | 25 | 0 | 0 | 25 | $-$25 | 25 | 0 | $-$25 | 0 | 0 | 0 | 25 | 1.045 | 3.391 | 0.308 |

0.500 | 25 | 0 | 0 | 25 | $-$25 | 25 | 0 | $-$25 | 0 | 0 | 0 | 25 | 1.045 | 3.391 | 0.308 |

NPN | CFR | AGL | FSR | AMS | SLM | ||||||||||

EL | $-$1 | 13 | 35 | 1 | 44 | 8 | |||||||||

EP | 24 | 18 | 14 | 10 | 19 | 15 |

The EL and EP percentage contributions to the maximal portfolio are also shown, and these can be interpreted as before. NPN is still the preferred best stock to invest in from an EL and EP contributions point of view, and the two short stocks contribute most of the EL. This seems entirely natural, in that this market period experienced an overall bullish trend and it would be risky to take short positions during such a period. With a total exposure of 200 the EP of the maximal portfolio is about 1.7% of total exposure at an EL risk of 0.5%, which is still acceptable from a trading point of view.

$?$ | NPN | BTI | BHP | CFR | AGL | FSR | SBK | AMS | VOD | MTN | SOL | SLM | MD | $\mathrm{E}?$ | Ratio |

0.040 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | $-$1 | 0 | 0 | 0 | 0.040 | 0.033 | 1.198 |

0.080 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | $-$1 | 0 | 0 | 1 | 0.080 | 0.067 | 1.198 |

0.120 | 2 | 0 | 0 | 1 | $-$1 | 2 | 0 | 0 | $-$2 | 0 | 0 | 1 | 0.120 | 0.100 | 1.198 |

0.160 | 2 | 0 | 0 | 2 | $-$1 | 2 | 0 | 0 | $-$2 | 1 | 0 | 1 | 0.160 | 0.134 | 1.198 |

0.200 | 3 | $-$1 | 0 | 2 | $-$1 | 3 | 0 | 1 | $-$3 | 1 | 0 | 2 | 0.200 | 0.167 | 1.198 |

0.401 | 6 | $-$1 | $-$1 | 4 | $-$2 | 6 | 0 | 1 | $-$5 | 1 | 0 | 3 | 0.401 | 0.335 | 1.198 |

0.601 | 9 | $-$2 | $-$1 | 6 | $-$4 | 8 | 0 | 2 | $-$8 | 2 | 0 | 5 | 0.601 | 0.502 | 1.198 |

0.801 | 12 | $-$2 | $-$2 | 8 | $-$5 | 11 | 0 | 2 | $-$10 | 3 | 0 | 6 | 0.801 | 0.669 | 1.198 |

1.002 | 15 | $-$3 | $-$2 | 11 | $-$6 | 14 | 0 | 3 | $-$13 | 3 | 0 | 8 | 1.002 | 0.836 | 1.198 |

1.202 | 18 | $-$3 | $-$3 | 13 | $-$7 | 17 | 0 | 3 | $-$15 | 4 | 0 | 9 | 1.202 | 1.004 | 1.198 |

1.402 | 21 | $-$4 | $-$3 | 15 | $-$8 | 19 | 0 | 4 | $-$18 | 4 | 0 | 11 | 1.402 | 1.171 | 1.198 |

1.603 | 24 | $-$4 | $-$3 | 17 | $-$9 | 22 | 0 | 4 | $-$20 | 5 | 0 | 12 | 1.603 | 1.338 | 1.198 |

1.803 | 25 | $-$4 | $-$3 | 20 | $-$10 | 24 | 0 | 4 | $-$22 | 6 | 0 | 15 | 1.803 | 1.501 | 1.202 |

2.004 | 25 | $-$3 | $-$2 | 22 | $-$10 | 25 | 0 | 2 | $-$23 | 6 | 0 | 19 | 2.004 | 1.654 | 1.212 |

2.204 | 25 | $-$1 | $-$1 | 23 | $-$10 | 25 | 0 | 0 | $-$22 | 4 | 0 | 22 | 2.204 | 1.796 | 1.228 |

2.404 | 25 | 0 | 0 | 24 | $-$12 | 25 | 0 | 0 | $-$18 | 1 | 0 | 25 | 2.404 | 1.924 | 1.250 |

2.605 | 25 | 0 | 0 | 25 | $-$14 | 25 | 0 | 0 | $-$11 | 0 | 0 | 25 | 2.605 | 2.022 | 1.289 |

2.805 | 25 | 0 | 0 | 25 | $-$16 | 25 | 0 | 0 | $-$4 | 0 | 0 | 25 | 2.805 | 2.101 | 1.336 |

3.005 | 25 | 0 | 0 | 25 | $-$17 | 25 | 0 | $-$3 | 0 | 0 | 0 | 25 | 3.005 | 2.168 | 1.387 |

3.206 | 25 | 0 | 0 | 25 | $-$15 | 25 | 0 | $-$12 | 0 | 0 | 0 | 25 | 3.206 | 2.218 | 1.446 |

3.406 | 25 | 0 | 0 | 25 | $-$13 | 25 | 0 | $-$18 | 0 | 0 | 0 | 25 | 3.406 | 2.259 | 1.509 |

3.606 | 25 | 0 | 0 | 25 | $-$14 | 25 | 0 | $-$23 | 0 | 0 | 0 | 25 | 3.606 | 2.294 | 1.573 |

3.807 | 25 | 0 | 0 | 25 | $-$20 | 25 | 0 | $-$25 | 0 | 0 | 0 | 25 | 3.807 | 2.323 | 1.640 |

4.007 | 25 | 0 | 0 | 25 | $-$25 | 25 | 0 | $-$25 | 0 | 0 | 0 | 25 | 4.007 | 2.347 | 1.709 |

NPN | CFR | AGL | FSR | AMS | SLM | ||||||||||

MD | 14 | 16 | 22 | 7 | 29 | 12 | |||||||||

$?X$ | 35 | 21 | 4 | 15 | 7 | 18 |

Finally, Figure 3 displays the EP and EL curves as functions of the risk tolerance parameter for all three diversification cases on a single graph to aid comparison of the results. These show that the tighter diversification requirement is effective in lowering risk (bottom three curves) but at the price of also lowering reward (top three curves), as we would anticipate. However, the EL/EP ratio for the maximal portfolios drops from about 0.40 to 0.31 in the process, so the diversification appears to be worthwhile from this point of view.

### 5.2 Bearish market

Next we turn to the bearish market scenario. The “Table Bearish market” spreadsheets in the supplementary material provide full details for this scenario, including the relevant parameter estimates. Graphs and tables of EP–EL and M-MD allocations similar to those above can be extracted from that spreadsheet. Keeping space limitations in mind, we simply summarize the main features below and leave it to the reader to consult the supplementary material.

In line with a bearish market, more expected returns are now negative, ranging to as low as $-0.085$ for AGL, but there are also some positive ones, such as for NPN. Standard deviations and correlations are of a similar order to those during the preceding bullish market. Compared with trading over the bullish market, the bearish market turns out to have been better, reflecting the usefulness of being able to go short as well as long. The nonzero allocations started at lower risk tolerance and the EP rose more quickly and to higher levels, reaching its maximal level below $\lambda =0.15$. This was due to long allocations focusing on the favorable price-increasing stocks (especially NPN) and short allocations focusing correctly on declining stocks (eg, AGL). It is also evident that a tightened diversification is not always beneficial, since here the EL/EP ratio increased from 0.13 to 0.15 for the maximal portfolios when moving the diversification parameter from 1 to 0.25. This is due to NPN being an outstanding performer over this time, so that it is most beneficial to concentrate the long part of the allocation on NPN. It is notable that the EL and EP contributions largely came from the short allocations, again in line with what we would anticipate when short positions are allowed during bearish markets. It is again true that the maximal portfolios for the EP–EL model are the same as those for the M-MD model.

### 5.3 Nontrending market

The “Table Nontrending market” spreadsheets in the supplementary material provide details regarding the nontrending market scenario. We summarize the main features below.

During this period, the expected stock price returns are mostly closer to zero than in the other markets, with NPN, which now has a value of 0.0067 after being positive during the other markets, being especially notable. Both AGL and AMS now have positive expected returns, which contrasts strongly with their negative values in the other markets. Standard deviations and correlations are of similar order. The optimal allocation results for this market are comparable with those of the bullish market in terms of EP, but the EL values are notably smaller, implying somewhat better trading opportunities. Compared with the other markets, the maximal allocations are now to different stocks, with AGL and AMS tending to be included on the long side and stocks such as BTI, MTN, VOD and SOL included on the short side, while the favorite NPN of the bullish and bearish markets is largely avoided. The reward and risk ratios of the maximal portfolios are again acceptable from a trading point of view.

### 5.4 Full data set

Rather than dividing the full data set into bullish, bearish and nontrending subsets, the full data set was also used as a possible scenario. The “Table Full data set” spreadsheets in the supplementary material provide the results. We summarize the main features below.

Over this time NPN still has a relatively strong expected stock price return, which suggests that it should figure well on the long allocations. Indeed, this happened for all three diversification cases. Only MTN and SOL had slightly negative expected returns, suggesting they may figure on short allocations, as did happen. Standard deviations and correlations are of a similar order as before, but it is hard to make allocation predictions on these statistics. The most visible feature of the results for this case is that the maximal portfolios require higher risk/reward ratios (ie, more risk tolerance) than in the previous cases, where sustained market trends were more present and exploitable. The reward and risk ratios of the maximal portfolios are rather unattractive from a trading point of view, being in the region of 0.6 here compared with below 0.4 in the other cases.

### 5.5 Trading cost and interest rate effects

All our results so far were for the base case regarding trading costs and interest rates, ie, ${c}_{\mathrm{b}}={c}_{\mathrm{s}}=0.002$ and ${i}_{\mathrm{b}}={i}_{\mathrm{s}}=0.005$. Here we briefly look at the effects of varying these costs and rates. There are many possible variations in the settings of these parameters, and it is not feasible to report on their effects if we consider all possibilities, especially when we also combine them with variations in the settings of the other parameters and the choice of market types. Generally, we anticipate smaller costs and rates will yield more favorable trading results. We found that this indeed happens and will illustrate it in terms of the following three cases. In addition to the base case, we also use two other cases: half base values (ie, ${c}_{\mathrm{b}}={c}_{\mathrm{s}}=0.001$ and ${i}_{\mathrm{b}}={i}_{\mathrm{s}}=0.0025$), and zero costs and rates (ie, ${c}_{\mathrm{b}}={c}_{\mathrm{s}}=0.000$ and ${i}_{\mathrm{b}}={i}_{\mathrm{s}}=0.000$). Using the data of the bearish market, the multivariate normal assumption for the stock returns and with the diversification factors at 0.5 each, Figure 4 shows some results restricted to the comparison of the EP and EL curves as functions of the risk tolerance parameter $\lambda $.

We see that both the EL and EP curves start to rise at lower risk tolerance values as the costs and rates decrease, ie, market entry becomes more attractive at lower risk tolerance or the trading hurdles become smaller. Moreover, the EP reaches higher levels, and the EL lower levels, in the maximal portfolios as costs and rates decrease. This confirms our prediction above. The “Table Trading Cost effects” spreadsheets in the supplementary material provide further details.

### 5.6 Correlation effects

Here we look into the effects of simply assuming the returns to be independently normally distributed rather than correlated, as we have done until now. Again, it is not feasible to report on the effects of the correlation assumptions combined with all the possible variations of parameter settings used earlier. We therefore show only limited results that we found to be representative of the greater generality. Figure 5 compares the EL and EP curves between the correlated and uncorrelated assumptions using the data of the bearish market, with the diversification factors both at 0.5 each and using the base case for the cost and rates setting.

The graph shows that the optimal portfolio allocations behave fairly similarly to the functions of the risk tolerance $\lambda $. The “Table Correlation effects” spreadsheet in the supplementary material provides further details. Among these are that the maximal portfolios use exactly the same allocations. The estimated EP value of the maximal portfolio when assuming zero correlations is less than 4% lower than the EP value when using the sample correlations. However, the estimated EL value is about 30% lower with the zero correlation assumption than with the sample correlations. Thus, it appears that the correlations do influence the risk side of the maximal portfolio allocation. The contributions are also shown, and here it is again apparent that the main impact is on the risk side.

### 5.7 Heavy-tailed distributional effects

It is well known that stock returns often follow heavier tailed distributions than the normal distribution. Here we look at the effects of replacing the joint normality assumption with the joint ${t}_{12}$ distribution (moderately heavy) and ${t}_{6}$ distribution (heavy). Figure 6 reports some results, again using the data of the bearish market, both the diversification factors at 0.5 and the base case for the trading and interest costs. The curves identified as “norm”, “t12” and “t6” refer to using the multivariate normal, ${t}_{12}$ and ${t}_{6}$ distributions, respectively. The effect of assuming a heavier tailed distribution is that the optimal allocation curves start rising at higher risk tolerance levels, in effect avoiding the greater risk levels associated with heavier tails. The “Table Distribution effects” spreadsheet in the supplementary material reports more details. Among these are that the maximal portfolios use exactly the same allocations but the corresponding EP and EL values are both estimated to be higher under the heavier tailed assumption than under the normal assumption. This seems natural since the heavier tailed assumption will produce occasional large stock returns to both the upside and the downside, which implies a possible higher profit but at a greater risk of loss.

### 5.8 Portfolio based directly on observational returns

As mentioned earlier, we can use observational stock returns directly rather than making parametric distributional assumptions, estimating their parameters and then generating the matrix of stock returns $R$ to use in the simulation-based procedures of Section 4. Here we illustrate this approach by using the returns data directly to calculate the corresponding optimal allocations and its related items and then to compare the results with those found above using distributional assumptions. We use the returns data of the bullish market with the base case settings as discussed above and allow single stock allocations. Table 11 shows the optimal allocations at varying risk tolerance levels and can be compared with Table 2, which does the same except that the normality assumption is included in the process. Note that simulation standard errors are not relevant here.

EL- | EP- | Ratio- | |||||||||||||

$?$ | NPN | BTI | BHP | CFR | AGL | FSR | SBK | AMS | VOD | MTN | SOL | SLM | obs | obs | obs |

0.125 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.150 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.160 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.165 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.170 | 32 | $-$2 | 0 | 23 | $-$1 | 2 | 0 | 0 | $-$30 | 24 | 4 | 11 | 0.335 | 1.903 | 0.176 |

0.175 | 40 | $-$1 | 0 | 22 | $-$5 | 0 | 0 | 2 | $-$27 | 14 | 0 | 19 | 0.399 | 2.256 | 0.177 |

0.180 | 46 | 0 | 0 | 17 | $-$8 | 0 | 0 | 4 | $-$25 | 9 | 0 | 22 | 0.444 | 2.434 | 0.182 |

0.190 | 51 | 0 | 0 | 14 | $-$10 | 0 | 0 | 3 | $-$23 | 0 | 0 | 30 | 0.529 | 2.739 | 0.193 |

0.200 | 54 | 0 | 0 | 11 | $-$9 | 0 | 0 | 0 | $-$20 | 0 | 0 | 33 | 0.595 | 2.968 | 0.201 |

0.210 | 62 | 0 | 0 | 6 | $-$11 | 0 | 0 | 0 | $-$14 | 0 | 0 | 31 | 0.688 | 3.264 | 0.211 |

0.220 | 66 | 0 | 0 | 3 | $-$12 | 0 | 0 | 0 | $-$8 | 0 | 0 | 29 | 0.768 | 3.452 | 0.223 |

0.230 | 71 | 0 | 0 | 1 | $-$14 | 0 | 0 | 0 | $-$2 | 0 | 0 | 26 | 0.861 | 3.690 | 0.233 |

0.240 | 76 | 0 | 0 | 2 | $-$16 | 0 | 0 | 0 | 0 | 0 | 0 | 21 | 0.939 | 3.898 | 0.241 |

0.250 | 81 | 0 | 0 | 3 | $-$19 | 0 | 0 | 0 | 0 | 0 | 0 | 15 | 1.021 | 4.075 | 0.251 |

0.275 | 96 | 0 | 0 | 3 | $-$32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1.276 | 4.624 | 0.276 |

0.300 | 100 | 0 | 0 | 0 | $-$43 | 0 | 0 | $-$6 | 0 | 0 | 0 | 0 | 1.498 | 5.004 | 0.299 |

0.350 | 100 | 0 | 0 | 0 | $-$24 | 0 | 0 | $-$51 | 0 | 0 | 0 | 0 | 2.004 | 5.736 | 0.349 |

0.400 | 100 | 0 | 0 | 0 | $-$8 | 0 | 0 | $-$89 | 0 | 0 | 0 | 0 | 2.593 | 6.516 | 0.398 |

0.450 | 100 | 0 | 0 | 0 | 0 | 0 | 0 | $-$100 | 0 | 0 | 0 | 0 | 2.745 | 6.710 | 0.409 |

0.500 | 100 | 0 | 0 | 0 | 0 | 0 | 0 | $-$100 | 0 | 0 | 0 | 0 | 2.745 | 6.710 | 0.409 |

NPN | AMS | ||||||||||||||

EL-obs | 19 | 81 | |||||||||||||

EL-norm | 27 | 73 | |||||||||||||

EP-obs | 57 | 39 | |||||||||||||

EP-norm | 61 | 39 |

Comparing them, we find that while the allocations differ somewhat as the risk tolerance increases, there is also similarity between the stocks selected, and both eventually converge to the same maximal allocation, namely long 100 shares of NPN and short 100 shares of AMS. Figure 7 shows the results graphically. The EP and EL contributions calculated from using the normal assumption and the observations directly are also quite similar.

The “Table Bullish observed returns” spreadsheet in the supplementary material reports more details.

### 5.9 Comparison of maximal portfolios

Table 12 compares the maximal portfolios under the normality assumption, the direct use of the observed returns, the zero correlation assumption and the ${t}_{6}$ distribution assumption. Their effects are shown on the corresponding EL and EP values and the probability of a profit (ProbP) as well as the EL given that there is a loss (ELGL) and the EP given that there is a profit (EPGP). All the scenarios and the three diversification settings are used. An expanded version of this table can be found in the “Table maximal portfolios” spreadsheet in the supplementary material, where the allocations among the stocks are also shown.

These allocations are the same in most cases, with some differences for the nontrending market and the full data set. This implies that the allocations themselves are quite robust against variations in how the returns data is modeled, ie, how to trade does not depend sensitively on the data used. However, the estimates of the EL and EP values and their ratios do vary more substantially, and this is probably to be anticipated since the modeling assumptions feed directly into the EL and EP numerical values. Aside from the full data set case, the smallest value of the probability of a profit is 0.625 but their average value is over 0.7. Also, the EPGP is mostly substantially larger than ELGL, which enhances our confidence in the quality of those portfolios. As noted above, the bearish market again stands out as particularly favorable, which may seem contrary to anticipation but is the overall result of being able to go short and selecting the right stocks to do so, for which the EP–EL method is effective.

(a) Bullish | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Div. fact. | Distr | EL | SEEL | EP | SEEP | Ratio | SERat | ProbP | SEProbP | ELGL | SEELGL | EPGP | SEEPGP |

1.00 | Normal | 2.639 | 0.020 | 6.602 | 0.037 | 0.400 | 0.005 | 0.643 | 0.002 | 7.350 | 0.044 | 10.246 | 0.042 |

Obs | 2.745 | — | 6.710 | — | 0.409 | — | 0.719 | — | 9.760 | — | 9.336 | — | |

Zero corr | 2.619 | 0.025 | 6.569 | 0.032 | 0.399 | 0.005 | 0.643 | 0.002 | 7.328 | 0.050 | 10.222 | 0.048 | |

t6 | 3.365 | 0.030 | 7.342 | 0.040 | 0.458 | 0.007 | 0.629 | 0.002 | 9.077 | 0.055 | 11.666 | 0.051 | |

0.50 | Normal | 2.246 | 0.019 | 5.383 | 0.029 | 0.417 | 0.005 | 0.637 | 0.002 | 6.174 | 0.045 | 8.478 | 0.035 |

Obs | 2.550 | — | 5.703 | — | 0.447 | — | 0.719 | — | 9.065 | — | 7.935 | — | |

Zero corr | 1.727 | 0.016 | 4.865 | 0.024 | 0.355 | 0.006 | 0.659 | 0.002 | 5.069 | 0.033 | 7.378 | 0.029 | |

t6 | 2.270 | 0.025 | 5.435 | 0.037 | 0.418 | 0.006 | 0.641 | 0.002 | 6.325 | 0.052 | 8.477 | 0.055 | |

0.25 | Normal | 1.045 | 0.010 | 3.391 | 0.016 | 0.308 | 0.004 | 0.680 | 0.002 | 3.260 | 0.020 | 4.993 | 0.018 |

Obs | 1.221 | — | 3.566 | — | 0.342 | — | 0.656 | — | 3.552 | — | 5.434 | — | |

Zero corr | 0.757 | 0.007 | 3.090 | 0.014 | 0.245 | 0.004 | 0.711 | 0.002 | 2.622 | 0.017 | 4.343 | 0.016 | |

t6 | 1.061 | 0.011 | 3.406 | 0.020 | 0.311 | 0.006 | 0.684 | 0.002 | 3.360 | 0.025 | 4.976 | 0.022 | |

(b) Bearish | |||||||||||||

Div. fact. | Distr | EL | SEEL | EP | SEEP | Ratio | SERat | ProbP | SEProbP | ELGL | SEELGL | EPGP | SEEPGP |

1.00 | Normal | 1.447 | 0.020 | 11.478 | 0.048 | 0.127 | 0.002 | 0.791 | 0.002 | 6.959 | 0.056 | 14.518 | 0.045 |

Obs | 1.425 | — | 11.448 | — | 0.124 | — | 0.765 | — | 6.055 | — | 14.971 | — | |

Zero corr | 1.448 | 0.018 | 11.456 | 0.051 | 0.126 | 0.002 | 0.792 | 0.002 | 6.954 | 0.051 | 14.469 | 0.051 | |

t6 | 2.160 | 0.028 | 12.203 | 0.058 | 0.177 | 0.003 | 0.767 | 0.002 | 9.272 | 0.080 | 15.910 | 0.075 | |

0.50 | Normal | 1.006 | 0.012 | 8.901 | 0.031 | 0.113 | 0.002 | 0.804 | 0.002 | 5.135 | 0.043 | 11.065 | 0.030 |

Obs | 1.063 | — | 8.971 | — | 0.119 | — | 0.882 | — | 9.037 | — | 10.167 | — | |

Zero corr | 0.701 | 0.020 | 8.617 | 0.034 | 0.081 | 0.002 | 0.836 | 0.002 | 4.272 | 0.065 | 10.308 | 0.047 | |

t6 | 1.171 | 0.016 | 9.051 | 0.033 | 0.129 | 0.002 | 0.800 | 0.002 | 5.841 | 0.052 | 11.320 | 0.034 | |

0.25 | Normal | 1.032 | 0.012 | 6.822 | 0.030 | 0.151 | 0.002 | 0.770 | 0.002 | 4.535 | 0.032 | 8.820 | 0.026 |

Obs | 1.183 | — | 6.949 | — | 0.170 | — | 0.765 | — | 5.028 | — | 9.088 | — | |

Zero corr | 0.489 | 0.007 | 6.261 | 0.025 | 0.078 | 0.001 | 0.840 | 0.002 | 3.062 | 0.028 | 7.451 | 0.023 | |

t6 | 0.825 | 0.010 | 6.574 | 0.026 | 0.125 | 0.002 | 0.801 | 0.002 | 4.140 | 0.038 | 8.209 | 0.031 | |

(c) Nontrending | |||||||||||||

Div. fact. | Distr | EL | SEEL | EP | SEEP | Ratio | SERat | ProbP | SEProbP | ELGL | SEELGL | EPGP | SEEPGP |

1.00 | Normal | 1.967 | 0.018 | 7.091 | 0.031 | 0.278 | 0.031 | 0.697 | 0.002 | 6.395 | 0.072 | 10.170 | 0.075 |

Obs | 1.948 | — | 7.080 | — | 0.275 | — | 0.625 | — | 5.195 | — | 11.329 | — | |

Zero corr | 2.211 | 0.058 | 7.352 | 0.066 | 0.301 | 0.005 | 0.683 | 0.003 | 6.977 | 0.129 | 10.760 | 0.123 | |

t6 | 2.936 | 0.067 | 8.091 | 0.076 | 0.363 | 0.005 | 0.666 | 0.003 | 8.779 | 0.146 | 12.155 | 0.138 | |

0.50 | Normal | 1.234 | 0.011 | 4.955 | 0.023 | 0.249 | 0.003 | 0.708 | 0.002 | 4.250 | 0.030 | 7.008 | 0.025 |

Obs | 1.315 | — | 5.029 | — | 0.262 | — | 0.708 | — | 4.509 | — | 7.100 | — | |

Zero corr | 1.593 | 0.016 | 5.312 | 0.027 | 0.300 | 0.005 | 0.683 | 0.002 | 5.027 | 0.037 | 7.775 | 0.031 | |

t6 | 2.165 | 0.019 | 5.870 | 0.034 | 0.369 | 0.006 | 0.661 | 0.002 | 6.379 | 0.044 | 8.885 | 0.044 | |

0.25 | Normal | 0.934 | 0.008 | 3.487 | 0.016 | 0.268 | 0.003 | 0.698 | 0.002 | 3.111 | 0.024 | 4.995 | 0.021 |

Obs | 1.156 | — | 3.692 | — | 0.313 | — | 0.708 | — | 3.963 | — | 5.213 | — | |

Zero corr | 1.042 | 0.010 | 3.583 | 0.017 | 0.291 | 0.005 | 0.687 | 0.002 | 3.334 | 0.022 | 5.213 | 0.017 | |

t6 | 1.435 | 0.014 | 3.978 | 0.021 | 0.361 | 0.006 | 0.661 | 0.002 | 4.234 | 0.028 | 6.018 | 0.027 | |

(d) Full data set | |||||||||||||

Div. fact. | Distr | EL | SEEL | EP | SEEP | Ratio | SERat | ProbP | SEProbP | ELGL | SEELGL | EPGP | SEEPGP |

1.00 | Normal | 2.990 | 0.119 | 4.990 | 0.133 | 0.597 | 0.011 | 0.585 | 0.003 | 7.038 | 0.234 | 8.540 | 0.240 |

Obs | 3.006 | — | 5.045 | — | 0.596 | — | 0.598 | — | 7.475 | — | 8.439 | — | |

Zero corr | 3.207 | 0.028 | 5.279 | 0.033 | 0.607 | 0.008 | 0.578 | 0.002 | 7.605 | 0.054 | 9.127 | 0.054 | |

t6 | 3.952 | 0.035 | 6.032 | 0.039 | 0.655 | 0.010 | 0.571 | 0.002 | 9.221 | 0.066 | 10.555 | 0.060 | |

0.50 | Normal | 1.830 | 0.042 | 3.209 | 0.049 | 0.570 | 0.007 | 0.591 | 0.002 | 4.423 | 0.092 | 5.431 | 0.095 |

Obs | 1.974 | — | 3.330 | — | 0.593 | — | 0.565 | — | 4.541 | — | 5.891 | — | |

Zero corr | 2.243 | 0.030 | 3.618 | 0.033 | 0.620 | 0.007 | 0.577 | 0.002 | 5.289 | 0.055 | 6.284 | 0.064 | |

t6 | 2.810 | 0.050 | 4.191 | 0.049 | 0.671 | 0.010 | 0.565 | 0.002 | 6.465 | 0.094 | 7.413 | 0.094 | |

0.25 | Normal | 1.306 | 0.069 | 2.088 | 0.076 | 0.622 | 0.011 | 0.574 | 0.003 | 3.142 | 0.142 | 3.714 | 0.144 |

Obs | 1.133 | — | 1.899 | — | 0.597 | — | 0.609 | — | 2.896 | — | 3.120 | — | |

Zero corr | 1.500 | 0.054 | 2.275 | 0.058 | 0.659 | 0.007 | 0.567 | 0.003 | 3.459 | 0.110 | 4.016 | 0.113 | |

t6 | 1.893 | 0.064 | 2.672 | 0.072 | 0.709 | 0.000 | 0.556 | 0.002 | 4.264 | 0.132 | 4.805 | 0.138 |

## 6 Remarks

There are many further aspects that can be explored, of which we list just two.

- •
The estimates of the features of interest in Section 4 used simple sample averages. It is well known that sample averages do not yield robust estimates and that we can usually improve on the results by weighting the data in order to limit the effects of outliers. Doing this should make the eventual portfolio allocations and their features more robust. This may be important since the results of the optimization steps in particular can be quite sensitive to outlier effects.

- •
The observational stock return data is actually 12-dimensional multivariate time series vectors with time lengths of 32, 17 and 24 months for the bullish, bearish and nontrending markets, respectively. Although we take their correlations into account explicitly when using the normal and $t$-distributional assumptions, we ignored their possible time dependence when we used them to calculate the optimal allocations and other features of the EP–EL and M-MD models. To address this lack of dynamic modeling, we could fit a vector autoregressive time series model and then generate the return data set to use with the optimal LP allocation method from such a dynamic model. This should improve the results, especially in a predictive context.

Pursuit of these matters is beyond the scope of this paper.

## 7 Conclusion

In this paper, we formulated the portfolio allocation of a trader in terms of the optimization of EP as a reward measure subject to controlling the EL as a risk measure. We argued that this formulation allows for an intuitively appealing assessment and interpretation of risk and return in trading contexts. Moreover, these measures are convenient in that important information such as the contributions of the individual stocks to portfolio risk and reward are readily calculated. We illustrated the approach extensively in terms of stock price histories of a number of large cap stocks traded on the JSE and found that the results are attractive from a trading point of view.

Some issues requiring further attention are listed above. A further important matter is that this paper was written for a single-period trading decision problem. However, trading is a repetitive process and it is a challenge to extend the formulation and results to take this into account. Above, we assumed that the trader will close their positions at the end of the holding period. However, if a subsequent period requires the trader to take a position in the same stock in the portfolio of the prior period, unnecessary closing and reopening costs may occur. Hence, the basic expressions for the PL should be changed to take such issues into account. We leave these questions to future research.

## Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper. The opinions, findings and conclusions or recommendations expressed are those of the authors; the Department of Science and Innovation (DSI) accepts no liability whatsoever in this regard.

## Acknowledgements

The suggestions and input from the anonymous referee led to improvements in this paper, which are gratefully acknowledged. This work is based on research supported in part by the DSI of South Africa.

## References

- Balder, S., and Schweizer, N. (2017). Risk aversion vs the Omega ratio: consistency results. Finance Research Letters 21, 78–84 (https://doi.org/10.1016/j.frl.2016.12.012).
- Brennan, T. J., and Lo, A. N. (2010). Impossible frontiers. Management Science 56(6), 905–923 (https://doi.org/10.1287/mnsc.1100.1157).
- Chen, Y., Sun, F., and Hu, Y. (2018). Coherent and convex loss-based risk measures for portfolio vectors Positivity 22, 399–414 (https://doi.org/10.1007/s11117-017-0517-6).
- Cont, R., Deguest, R., and He, X. D. (2013). Loss-based risk measures. Statistics and Risk Modelling 30(2), 133–167 (https://doi.org/10.1524/strm.2013.1132).
- Griveau-Billion, T., Richard, J.-C., and Ronelli, T. (2013). A fast algorithm for computing high-dimensional risk parity portfolios. Working Paper, Social Science Research Network (https://doi.org/10.2139/ssrn.2325255)
- Keating, C., and Shadwick, W. F. (2002). A universal performance measure. Journal of Performance Measurement 6, 59–84.
- Maillard, S., Roncalli, T., and Teiletche, J. (2010). The properties of equally weighted risk contribution portfolios. Journal of Portfolio Management 36(4), 60–70 (https://doi.org/10.3905/jpm.2010.36.4.060).
- Markowitz, H. M. (1968). Portfolio Selection: Efficient Diversification of Investments. Yale University Press, New Haven, CT.
- Metel, M. R., Pirvu, T. A., and Wong, J. (2017). Risk management under omega measure. Risks 5, 27 (https://doi.org/10.3390/risks5020027).
- Righi, M. B., and Borenstein, D. (2018). A simulation comparison of risk measures for portfolio optimization. Finance Research Letters 24, 105–112 (https://doi.org/10.1016/j.frl.2017.07.013).
- Saft, J. (2014). Never confuse risk and volatility. Reuters, September 10. URL: https://reut.rs/2Qj9omm.
- Sharpe, W. F. (1966). Mutual fund performance. Journal of Business January, 119–138.
- Sortino, F. A., and Price, L. N. (1994). Performance measurement in a downside risk framework. Journal of Investing 3(3), 59–64 (https://doi.org/10.3905/joi.3.3.59).
- Wenzelburger, J. (2020). Mean–variance analysis and the modified market portfolio. Journal of Economic Dynamics and Control 111, 103821 (https://doi.org/10.1016/j.jedc.2019.103821).

Only users who have a paid subscription or are part of a corporate subscription are able to print or copy content.

To access these options, along with all other subscription benefits, please contact info@risk.net or view our subscription options here: http://subscriptions.risk.net/subscribe

You are currently unable to print this content. Please contact info@risk.net to find out more.

You are currently unable to copy this content. Please contact info@risk.net to find out more.

Copyright Infopro Digital Limited. All rights reserved.

As outlined in our terms and conditions, https://www.infopro-digital.com/terms-and-conditions/subscriptions/ (point 2.4), printing is limited to a single copy.

If you would like to purchase additional rights please email info@risk.net

Copyright Infopro Digital Limited. All rights reserved.

You may share this content using our article tools. As outlined in our terms and conditions, https://www.infopro-digital.com/terms-and-conditions/subscriptions/ (clause 2.4), an Authorised User may only make one copy of the materials for their own personal use. You must also comply with the restrictions in clause 2.5.

If you would like to purchase additional rights please email info@risk.net