#
Delivering Left-Skewed Portfolio Payoff Distributions in the Presence of Transaction Costs^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

^{1}, is unacceptable even with a chance of big gains.

^{2}In fact, the literature tends to ignore the distribution of actual outcomes that eventuate from the optimal investment strategy.

^{3}

^{4}. Thus, our solutions will be numerical and parameter-specific. We will study a hypothetical base-case scenario

^{5}that involves an initial outlay invested in a pension fund to grow and be collected as a lump sum after a given optimisation horizon.

^{6}, it is proposed in this paper that by optimising a utility measure, which has a reference-payoff level and which is concave on each side to this level, new “cautious-relaxed" strategies may be obtained. A cautious-relaxed strategy will be adopted by agents who are cautious in ensuring that payoffs do not fall much below a reference value, but relaxed about exceeding that reference value. In brief, it will generate a left-skewed payoff distribution, as opposed to the right-skewed ones that result from maximisation of a usual risk-averse (concave) utility function. A cautious-relaxed strategy satisfies the needs of loss-avoiding investors.

## 2. Wealth dynamics

^{7}(shares) and the other risk free (cash). Let the price $p\left(t\right)$ per share of the risky asset change according to the equation

^{8}

^{9}In this paper, we suppose a fee with force $cx\left(t\right)$ will be charged, where $c>0$ is a constant.

^{10}If the investor does not draw from the fund before T and the above fees are charged, wealth will evolve according to the following equation

## 3. Performance Measures

#### 3.1. Aggregate Reward

^{11}. Once an objective function is proposed, the investor’s strategy can be computed as a solution to a stochastic optimal control problem determined by the objective, where the system’s dynamics are given by (5), (4) and (3). The solution provides an optimal investment strategy $\mu \left(x\right(t),t)$ that generates control $u\left(t\right)=\mu \left(x\right(t),t)$, also referred to as a strategy realisation. As argued in the introduction, the solution should also include practical information about the resulting payoff distribution. This will allow the investor to decide what they can reasonably expect for their pension, and find the objective function that satisfactorily represents their preferences. In particular, knowledge of the distribution of $x\left(T\right)$ is useful to the investor, as it helps describe the risks associated with obtaining a particular realisation of the objective.

^{12}will be used in this paper as a model of the pension fund problem. Furthermore, we will assume that the fraction of the fund invested in the risky asset at $t=0$ is zero and drop ${u}_{0}$ from the notation.

^{13}

#### 3.2. Discussion

^{14}Overall, using (7) as a utility measure may appear non-standard.

**Figure 1.**Risk-averse ${h}_{M}$, prospect theoretic ${h}_{P}$ and the “double" concave ${h}_{C}$ performance measures for calibrated models.

## 4. Investment Strategies and Pension Distributions without Transaction Costs

#### 4.1. Optimal strategies

^{15}by the specialised software SOCSol from [12] (also, see [28]).

^{16}This implies that a cautious-relaxed investor will have to rebalance their portfolio very frequently, and expensively if b is large. The shares’ exposure adjustments of the Merton investor are evidently much smaller.

#### 4.2. Pension distributions

Statitic | Caut.-Relax. | Merton |
---|---|---|

Mode of $x\left(10\right)$ | $93,790 | $55,300 |

Mean of $x\left(10\right)$ | $74,922 | $86,596 |

Median of $x\left(10\right)$ | $83,373 | $73,082 |

Std. dev. of $x\left(10\right)$ | $21,723 | $55,042 |

Coeff. of skew. | –1.017 | 2.164 |

P$\left(x\right(10)>\$105,000)$ | 0 | 0.273 |

P$\left(x\right(10)>\$80,000)$ | 0.562 | 0.438 |

P$\left(x\right(10)<\$40,000)$ | 0.1077 | 0.15 |

^{17}indicates a large difference between the Merton’s and cautious-relaxed distributions.

## 5. Cautious-relaxed strategies and pension distributions with transaction costs

#### 5.1. Optimal strategies

**Figure 5.**Control rules for $t=2$ and $t=7$ in $x\left(t\right)$ (left panel) and $u\left(t\right)$ (right panel) for $b=0.005$.

**Figure 6.**Control rules for $t=2$ and $t=7$ in $x\left(t\right)$ and $u\left(t\right)$ for $b=0.05$.

#### 5.2. Pension Distributions

Statistic | $\mathbf{b}\mathbf{=}\mathbf{0}$ | $\mathbf{b}\mathbf{=}\mathbf{0.005}$ | $\mathbf{b}\mathbf{=}\mathbf{0.01}$ | $\mathbf{b}\mathbf{=}\mathbf{0.05}$ | $\mathbf{b}\mathbf{=}\mathbf{0.1}$ | Merton |
---|---|---|---|---|---|---|

Mean of $x\left(10\right)$ | $74,922 | $74,000 | $73,657 | $70,651 | $68,493 | $86,596 |

Median of $x\left(10\right)$ | $83,373 | $80,133 | $78,815 | $71,297 | $67,966 | $73,082 |

Std. dev. of $x\left(10\right)$ | $21,723 | $20,741 | $20,464 | $18,108 | $15,640 | $55,042 |

Coeff. of skew. of $x\left(10\right)$ | –1.017 | –0.8499 | –0.78 | –0.331 | 0.0158 | 2.164 |

P$\left(x\right(10)>\$80,000)$ | 0.562 | 0.503 | 0.475 | 0.333 | 0.233 | 0.438 |

P$\left(x\right(10)<\$40,000)$ | 0.1077 | 0.0938 | 0.0875 | 0.0561 | 0.0317 | 0.15 |

#### 5.3. Advice

## 6. Conclusion

## Acknowledgments

## Conflicts of Interest

## A. Parameters

r | α | σ | c |
---|---|---|---|

0.05 | 0.085 | 0.2 | .005 |

## B. SOCSol and Yield Distributions

## C. The Merton Investor

#### C.1. The Classic Utility Measure

#### C.2. The Distribution

^{18}that after fitting the optimal control (22) to the state equation (5), wealth $x\left(t\right)$ is a Geometric Brownian Motion that follows

## D. Why Zero-Investment Can Be Profitable in This Model

^{19}will be allocated to the risk-free asset. We can say that wealth at time 6 has reached the secure-investment level ${x}^{S}\left(6\right)$. Using $u\left(\tau \right)=0$ for $\tau \ge 6$ from ${x}^{S}\left(6\right)$ causes every $x\left(\tau \right)$ to the right of the x-intercept of the strategy graph in Figure 2, to be reached at the right time for the investor to continue the strategy $u\left({x}^{S}\left(\tau \right)\right)=0$.

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^{1}(1) When the mass of the distribution is concentrated on the left and the right tail is longer, the distribution is said right- or positively skewed; (2) when the mass of the distribution is concentrated on the right the left tail is longer, the distribution is said left- or negatively skewed.^{2}Interestingly, in the context of static portfolio management, [8] also question the wisdom of “traditional optimisation" for some investors, leverage-averse in their case.^{3}Exceptions to this include [9], [10]. The work done on the distribution builder in the first, enables subjects to build their desired pension distribution subject to a budget constraint. However, the results lead to distributions that are right skewed, which could be due to the setting of a (low) reference point at the amount guaranteed by the risk-free asset. In [10], quantiles are proposed as an effective way to evaluate the success and failings of a portfolio.^{4}Finding analytic solutions to the resulting PDEs would be a substantive research project which may not yield any results as the study subject are nonlinear PDEs. A “semi" analytic solution could be obtained by a functional expansion. We pursue numerical solutions in this paper, which are reliable and easy to interpret for parameter-specific problems.^{5}Notwithstanding the obtained solution’s parameter-specificity, our analysis can be extended to other cases through the use of specialised software (see [12]).^{7}This will be a synthetic aggregate good if there are many risky assets.^{8}Constraint (3) means no short selling or borrowing. This restriction has been weakened in the literature; however, it may be reasonable to keep it in a situation of a pension fund investor.^{9}A study of the impact of management incentives on investment strategies performed in [17] reports that maximising management revenue from fees changes little the investment strategies.^{10}An argument for using $z\left(t\right)$ as control instead of $u\left(t\right)$ can be found in [21]. If $u(\xb7)$ – the “fast" control – is Lebesgue measurable, Proposition 3.2 in that publication establishes that a solution $x\left(t\right)$ to a differential equation which contains $u\left(t\right)$, but not $z\left(t\right)$, can be approximated by a solution obtained from an equation where $z\left(t\right)$ is introduced as in (4).^{11}This publication deals with portfolio choice models for both pension funds and life assurance companies on a macro scale i.e., where many investors contribute to the fund. In that sense, our one-pension management problem is micro.^{12}Other constraints could be added, e.g., $x\left(t\right)\u2a7e0$.^{13}This measure (7) was also used in, among others, [16], [13] and [17]. A loss-averse utility function that is concave on each side of the reference point (so, “similar" to (7)) was proposed in [22]. However, for the original parameters adopted by [22], that function is only “lightly” concave and did not generate left-skewed distributions, see [15].^{14}These authors solved the problem by splitting it into subproblems and found that the optimal strategy is one in which the investor takes on aggressive gambling strategies. The strategies computed in [23] and [24] still generate right skewed distributions, which we deem not preferable by pension fund investors.^{15}The problem with an analytical solution is that the as long as α in the utility measure (7) is just any number greater than 1, little can be said about a closed-form strategies and value functions. This is because t and x in $V(t,x)$ (as in (18)) for the boundary problem with $V(T,x\left(T\right))=-{({x}_{T}-x\left(T\right))}^{a},\phantom{\rule{0.166667em}{0ex}}a>1$ appear non-separable. Nevertheless, even if $V(x,t)$ were obtained in an analytical form, a closed-form for the payoff-density function would still be an open problem. We also note that closed-form solutions in [22] were obtained for a similar but non-identical, utility function. More importantly their independent variable is not the current (observable) wealth but state price density.^{17}The skewness coefficient is calculated as $E\left[{\left(\frac{X-\mathrm{mean}}{\mathrm{stand}.\phantom{\rule{0.166667em}{0ex}}\mathrm{dev}.}\right)}^{3}\right]$. It provides a measure of asymmetry in the distribution.^{19}We can see in [11] that $u\left(t\right)$ is never zero for less volatile risky assets.

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**MDPI and ACS Style**

Krawczyk, J.B.
Delivering Left-Skewed Portfolio Payoff Distributions in the Presence of Transaction Costs. *Risks* **2015**, *3*, 318-337.
https://doi.org/10.3390/risks3030318

**AMA Style**

Krawczyk JB.
Delivering Left-Skewed Portfolio Payoff Distributions in the Presence of Transaction Costs. *Risks*. 2015; 3(3):318-337.
https://doi.org/10.3390/risks3030318

**Chicago/Turabian Style**

Krawczyk, Jacek B.
2015. "Delivering Left-Skewed Portfolio Payoff Distributions in the Presence of Transaction Costs" *Risks* 3, no. 3: 318-337.
https://doi.org/10.3390/risks3030318