Delivering Left-Skewed Portfolio Payoff Distributions in the Presence of Transaction Costs †
2. Wealth dynamics
3. Performance Measures
3.1. Aggregate Reward
4. Investment Strategies and Pension Distributions without Transaction Costs
4.1. Optimal strategies
4.2. Pension distributions
|Std. dev. of||$21,723||$55,042|
|Coeff. of skew.||–1.017||2.164|
5. Cautious-relaxed strategies and pension distributions with transaction costs
5.1. Optimal strategies
5.2. Pension Distributions
|Std. dev. of||$21,723||$20,741||$20,464||$18,108||$15,640||$55,042|
|Coeff. of skew. of||–1.017||–0.8499||–0.78||–0.331||0.0158||2.164|
Conflicts of Interest
B. SOCSol and Yield Distributions
C. The Merton Investor
C.1. The Classic Utility Measure
C.2. The Distribution
D. Why Zero-Investment Can Be Profitable in This Model
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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.
- 2Interestingly, in the context of static portfolio management,  also question the wisdom of “traditional optimisation" for some investors, leverage-averse in their case.
- 3Exceptions to this include , . 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 , quantiles are proposed as an effective way to evaluate the success and failings of a portfolio.
- 4Finding 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.
- 5Notwithstanding the obtained solution’s parameter-specificity, our analysis can be extended to other cases through the use of specialised software (see ).
- 7This will be a synthetic aggregate good if there are many risky assets.
- 8Constraint (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.
- 9A study of the impact of management incentives on investment strategies performed in  reports that maximising management revenue from fees changes little the investment strategies.
- 10An argument for using as control instead of can be found in . If – the “fast" control – is Lebesgue measurable, Proposition 3.2 in that publication establishes that a solution to a differential equation which contains , but not , can be approximated by a solution obtained from an equation where is introduced as in (4).
- 11This 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.
- 12Other constraints could be added, e.g., .
- 13This measure (7) was also used in, among others, ,  and . A loss-averse utility function that is concave on each side of the reference point (so, “similar" to (7)) was proposed in . However, for the original parameters adopted by , that function is only “lightly” concave and did not generate left-skewed distributions, see .
- 14These 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  and  still generate right skewed distributions, which we deem not preferable by pension fund investors.
- 15The 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 (as in (18)) for the boundary problem with appear non-separable. Nevertheless, even if 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  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.
- 17The skewness coefficient is calculated as . It provides a measure of asymmetry in the distribution.
- 19We can see in  that is never zero for less volatile risky assets.
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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
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/risks3030318Chicago/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