Stochastic dominance analysis views the data through a discriminating lens that usually exposes three levels of preference orderings: first, second, and third degree dominance. There is a stronger ordering process with each level of dominance, thereby enabling an improved decision making process. Rational investors perspicaciously construct portfolios exclusively from this subset of the population of feasible investment opportunities.
The prospect theory of Kahneman and Tversky (1979) and the cumulative prospect theory of Tversky and Kahneman (1992) are descriptive models for decision making that summarize violations of the expected utility theory (Giorgi and Hens, 2006). In this paper, we showed that the resulting optimal portfolios are efficient in the sense of the CPT-TSD. To the best of our knowledge this is the first attempt to apply CPT-TSD to portfolio management study.
Interestingly, our CPT-TSD iterations of 50 ETF components that operated continuously from 1998 through 2008 revealed that this unmanageable inventory of investment selections could be reduced to single-digit efficient sets. Portfolios assembled from the efficient sets were evaluated against stock market benchmarks in examining buy-and-hold and portfolio adjustment strategies. Portfolio strategy-weighted tactics testing disclosed a functional tendency to ascertain CPT-TSD portfolios--but no uncommon facility to collectively assemble funds into portfolios-capable of producing market-competitive performance.
The analysis reported in this research study advocates the financial reconnaissance methodology that characterizes CPT-TSD as a correlative and utilitarian competitor of the mean-variance model. It provides financial advisors and investment professionals with a collateral and feasible strategy to reveal relative investment preferences by discriminating among and parsing the universe of portfolio management.
One implication of the lack of CPSD is that many of the existing models may be based on either inappropriate utility functions, or incorrect returns distributions, or both. The results point to the need to search over probability distributions that capture higher-order moments in preferences, such as skewness and kurtosis. In addition to that, the lack of CPSD results suggest that research the has been devoted to formulating models that depart from the assumptions of complete and frictionless markets may be useful in so far as they are informative about the nature of preferences and about higher-order moments in the probability distributions of the assets.(See Grant and Quiggin, 2001). That will absolutely violate the investors” original investment motivation and utility requirement if we omitted CPSD assumption. It will also guide to the wrong conclusion and bias portfolio.
This paper has provided a non-parametric approach based on SD method and CPT utility to examine the ETF based portfolio without the need to specify the underlying utility and probability functionality. The empirical results was not wholly interpreted to imply that active constructed portfolio might in fact be excellent and performance improvement to passive investment strategy. But, there was some evidence that CPT-TSD portfolio dominated lower degree CPTSD criteria and MV rule. This study
mainly indicates that, to determine the efficient set of passive portfolios, portfolio management needs to consider the active manipulation inside.
In Taiwan, ETF related products were launched just a short history (started in 2003). This study scrutiny the ETFs performance pattern and further application in academic viewpoint and practical operation as well. It also shed a light on passive investment management integrated in active manipulation skill with higher order cumulative prospect stochastic dominance method. Truly, this research suggests a better and safer investment instrument philosophy under solid theoretical foundation.
The suggested methodology in this paper can be a useful filter for stock construction for DARA investors to maximize their expected utilities, and not sacrificing their wealth enlargement along with the skewed and non-normal characteristics of stock returns. This finding leads us to conjecture the CPT-TSD approach can exploit more information to decide on portfolio management than traditional TSD rule. Then the study shows the fact CPT-TSD effectively dominated the other portfolio strategies as we expected. Actually, the big 50 market price weighted stocks do not mean the best 50 of the market. It is worth noted that it only stand for the shadow of the stock exchange index. More useful information emerges than traditional SD comparisons because of the completed consideration of utility function: one stock dominates another stock on the von Newman utility assumption while the reverse dominance relationship can be found on the S shape utility condition.
Shortly, CPT-TSD analysis did not disclose any atypical capacity to generate portfolios that outpace the market. This finding supports and sustains the efficient market hypothesis of modern portfolio theory. This doctrine asserts that investment
research and trading in a relatively fruitless search for incorrectly valued securities.
Such carnivorous consumption of resources is regarded as futile by the efficient market theory which argues that securities prices contain all publicly available and relevant information and thus are correctly priced.
Inferences gleaned from modern portfolio theory are that investors should not engage in market-timing or commit funds to the top-ranked mutual funds as proclaimed by investment publications. Rather, investors are advised to accumulate shares in index funds and hold them over a long period of time. Interestingly, most investors and financial professionals do not adhere to this advice. Only a small portion of mutual fund assets is invested in index funds. Furthermore, the number of
"managed" equity funds has increased dramatically and there has been ineluctable growth in their assets under management.
Investors and financial professionals often seek to "beat" the market by implementing strategies at variance with the canon of modern portfolio theory.
Markets are not perfectly efficient. This inefficiency coupled with the inclination and appealing opportunity to compete against the market motivates many investors to reject indexing and embrace an alternative tactics. It is the issue about behavioral finance and we could not totally understand the investor’s intention and real desire if we overlook the CPT based theory. In sum, CPT-TSD iterations reduced a large number of stocks (50) into a manageable handful (nearly 4 to 5), greatly simplifying portfolio selection. Having eliminated roughly 90% of the component stocks in ETF 50 Taiwan, a logical issue to investigate is whether an investor or financial advisor could endow a competitive asset accumulation by constructing a portfolio in accordance with
probability weighting function transformation in this paper realizes the empirical study and portfolio application by CPT foundation.
This paper also found some evidence of stock picking ability among various portfolio strategies in the ranking priority test. Certainly, the recommended weighted method was investigated and four weighted tactics were ranked pair-wised. CPT-TSD strategy with price weighted or equal weighted depends on monthly return sorting basis or weekly return sorting basis are examined in the study for pursuing the best strategy-tactics fitness.
Trading is costly. Although most academic researchers and many financial planners are persuaded that index funds should be the core component of a portfolio, the results of this study imply that CPT-TSD strategy may be moderately successful in detecting few stocks which have the potential of transcending market rates of return.
Last but not least, portfolios components are always re-selected in both MV and CPT-TSD context; but there existed obvious difference components between portfolios of MV and CPT-TSD. The possible and reasonable explanation is that the quadratic utility function of MV assumption is fundamentally different to the S shape utility function of CPT. In the near future, it would surely be interesting to follow if CPT-TSD relations among some stocks would disappear over a considerable length of time. Additional years of data will need to be harvested and scrutinized and the sensibility analysis of weighting function of CPT before resolving the question of whether CPT-TSD application is an essential or supplemental investment analysis metric. Certainly, we suggest exploring the optimal data threshold window and portfolio revaluation point in the near future.
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Appendix A. The Proof of PSD Rule
Stochastic dominance for Cumulative Prospect Theory can be written as: F dominate G iff:
We integrate equation (1) by parts:
That leaves us following:
∫
0 '( )[ −( ( ))− −( ( ))] +∫
0 '( )[ +( ( ))− +( ( ))] ≥0:We integrate by parts again:
∫
We move the RHS to LHS:
Then we decompose the second term and transform as shown below:
We can further separate the two integrals in the second term of Equation (6):
The first element on the RHS of equation (7) cancels with the first element on the RHS of equation (6). We can write inequality (5) as follows:
That concludes:
∀x
∫
0x[ω
+(G(u))−ω
+(F(u))]du≥0 and
∫
x0[ω
−(G(u))−ω
−(F(u))]du≥0 (9)Appendix B. Complete MV Criteria Metric
2379 2454 2880 2301 2887 2317 2353 2352 2401 2388 2379 FALSE FALSE FALSE FALSE FALSE 2454 FALSE FALSE
2880 FALSE
2301 FAL FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE 2887
2317 TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE 2353 FALSE FALSE FALSE FALSE FALSE FALSE 2352 TRUE TRUE FALSE FALSE TRUE FALSE TRUE
2401 TRUE TRUE FALSE
2388 FALSE FALSE FALSE FALSE
2324 TRUE TRUE TRUE FALSE TRUE TRUE FALSE TRUE 2330 TRUE TRUE TRUE FALSE TRUE TRUE FALSE TRUE 2408 FALSE FALSE FALSE
1326 TRUE TRUE TRUE TRUE FALSE FALSE TRUE TRUE FALSE TRUE 2204 TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE 2344 FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE 2323 FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE 2349 FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE 2105 TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE 1301 TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE 2201 TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE 2912
2002 TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE 2303 TRUE TRUE FALSE FALSE FALSE TRUE TRUE FALSE TRUE
2603 TRUE FALSE
1303 TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE 2308 TRUE TRUE TRUE FALSE FALSE TRUE TRUE FALSE TRUE 2311 TRUE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE 9904
2409 FALSE FALSE FALSE
2357 TRUE TRUE TRUE FALSE FALSE TRUE
2886 TRUE
Complete MV Criteria Metric (Continued)
2379 2454 2880 2301 2887 2317 2353 2352 2401 2388
2610 TRUE FALSE
1402 TRUE TRUE TRUE TRUE FALSE FALSE TRUE TRUE FALSE TRUE 2475 FALSE FALSE
2382 TRUE TRUE FALSE FALSE TRUE 2890
2356 TRUE TRUE TRUE FALSE FALSE TRUE
2881 TRUE
1216 TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE 3045
2891
2412 TRUE TRUE TRUE
2801 TRUE FALSE
2883 TRUE
2882 TRUE
3009
numbers 19 21 22 9 7 7 21 14 8 19
5
Complete MV Criteria Metric (Continued)
2324 2330 2408 1326 2204 2344 2323 2349 2105 1301 2379 TRUE
2454- 2880
2301 FALSE FALSE TRUE TRUE FALSE FALSE 2887
2317 TRUE TRUE TRUE TRUE TRUE TRUE 2353 TRUE
2352 TRUE 2401 TRUE 2388 TRUE
2324 FALSE TRUE TRUE TRUE
2330- TRUE TRUE TRUE
2408
1326 TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE 2204 TRUE TRUE TRUE TRUE TRUE TRUE
2344 TRUE FALSE
2323 FALSE FALSE TRUE TRUE TRUE 2349 TRUE
2105 TRUE TRUE TRUE FALSE TRUE TRUE TRUE
1301 TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE 2201 TRUE TRUE TRUE FALSE TRUE TRUE TRUE
2912
2002 TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE 2303 FALSE FALSE TRUE TRUE TRUE TRUE
2603
1303 TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE 2308 TRUE TRUE TRUE TRUE TRUE TRUE
2311 FALSE FALSE TRUE TRUE TRUE TRUE 9904
2409
2357 TRUE 2886
2325 FALSE TRUE TRUE TRUE
Complete MV Criteria Metric (Continued)
2324 2330 2408 1326 2204 2344 2323 2349 2105 1301 2610
1402 TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE 2475
2382 TRUE 2890
2356 TRUE 2881
1216 TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE 3045
2891 2412 2801 2883 2882 3009
numbers 11 11 28 3 2 18 13 17 2 1 5
Complete MV Criteria Metric (Continued)
2201 2912 2002 2303 2603 1303 2308- 2311 9904 2409 2379 FALSE FALSE FALSE TRUE
2454 FALSE FALSE FALSE
2880 FALSE FALSE
2301 FALSE FALSE FALSE FALSE FALSE FALSE TRUE
2887 TRUE TRUE
2317 FALSE TRUE TRUE TRUE
2353 FALSE FALSE FALSE TRUE 2352 FALSE FALSE FALSE TRUE
2401 FALSE TRUE TRUE TRUE
2388 FALSE FALSE FALSE TRUE 2324- FALSE TRUE FALSE TRUE
2330 FALSE TRUE FALSE TRUE
2408 FALSE FALSE FALSE FALSE 1326 FALSE FALSE FALSE TRUE TRUE TRUE TRUE FALSE TRUE 2204 TRUE TRUE TRUE TRUE TRUE TRUE TRUE 2344 FALSE FALSE FALSE TRUE 2323 FALSE FALSE FALSE TRUE 2349 FALSE FALSE FALSE TRUE 2105 TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE 1301 FALSE FALSE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE 2201 FALSE TRUE TRUE TRUE TRUE TRUE TRUE
291 TRUE
2002 TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE 2303 FALSE FALSE FALSE TRUE FALSE TRUE
2603 FALSE FALSE
1303 FALSE FALSE FALSE TRUE TRUE TRUE TRUE TRUE TRUE 2308 FALSE TRUE TRUE FALSE TRUE 2311 FALSE FALSE FALSE TRUE 9904
2409 FALSE FALSE FALSE
2357 FALSE TRUE FALSE TRUE
2886 TRUE TRUE
2325 FALSE FALSE FALSE TRUE
Complete MV Criteria Metric (Continued)
2201 2912 2002 2303 2603 1303 2308- 2311 9904 2409
2610 FALSE TRUE
1402 FALSE FALSE FALSE TRUE TRUE FALSE TRUE FALSE TRUE
2475 FALSE FALSE FALSE
2382 FALSE TRUE FALSE TRUE
2890 TRUE TRUE
2356 FALSE TRUE FALSE TRUE
2881 TRUE TRUE
1216 TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
3045 TRUE
2891 TRUE TRUE
2412 TRUE TRUE TRUE
2801 FALSE FALSE
2883 TRUE TRUE
2882 TRUE TRUE
3009 TRUE
numbers 3 11 0 9 18 2 8 11 22 28
5 Eff
Complete MV Criteria Metric (Continued)
2357 2886 2325 2609 2610 1402 2475 2382 2890 2356 2379 FALSE FALSE FALSE TRUE FALSE FALSE 2454 FALSE FALSE FALSE TRUE FALSE
2880 FALSE FALSE
2301 FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE
2887 FALSE
2317 TRUE FALSE TRUE FALSE FALSE TRUE TRUE FALSE TRUE 2353 FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE 2352 FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE 2401 FALSE FALSE FALSE TRUE FALSE 2388 FALSE FALSE FALSE TRUE FALSE 2324 FALSE FALSE TRUE FALSE FALSE TRUE FALSE FALSE FALSE 2330 FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE 2408 FALSE FALSE FALSE FALSE FALSE 1326 TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE FALSE FALSE 2204 TRUE FALSE TRUE TRUE TRUE TRUE TRUE FALSE TRUE 2344 FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE 2323 FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE 2349 FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE 2105 TRUE FALSE TRUE TRUE TRUE TRUE TRUE FALSE TRUE 1301 TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE FALSE TRUE 2201 TRUE FALSE TRUE TRUE TRUE TRUE TRUE FALSE TRUE 2912
2002 TRUE FALSE TRUE TRUE TRUE TRUE TRUE FALSE TRUE 2303 FALSE FALSE TRUE FALSE FALSE TRUE FALSE FALSE FALSE
2603 FALSE FALSE FALSE FALSE
1303 TRUE FALSE TRUE TRUE FALSE TRUE TRUE TRUE FALSE TRUE 2308 TRUE FALSE TRUE FALSE FALSE TRUE TRUE FALSE FALSE 2311 FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE 9904
2409 FALSE FALSE FALSE TRUE FALSE 2357 FALSE FALSE FALSE TRUE TRUE FALSE
2886 FALSE
2325 FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE
Complete MV Criteria Metric (Continued)
2357 2886 2325 2609 2610 1402 2475 2382 2890 2356
2610 FALSE FALSE
1402 FALSE FALSE TRUE FALSE FALSE TRUE TRUE FALSE FALSE
2475 FALSE FALSE FALSE FALSE
2382 FALSE FALSE FALSE TRUE FALSE 2890
2356 TRUE FALSE FALSE FALSE TRUE TRUE FALSE
2881 TRUE FALSE
1216 TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE FALSE TRUE 3045
2891
2412 TRUE TRUE TRUE TRUE FALSE
2801 FALSE FALSE
2883 FALSE FALSE
2882 TRUE TRUE
3009
numbers 11 3 13 8 7 4 31 13 1 8 5
Complete MV Criteria Metric (Continued)
2881 1216 3045 2891 2412 2801 2883 2882 3009 2379 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 2454 FALSE FALSE FALSE FALSE FALSE FALSE FALSE 2880 FALSE FALSE FALSE FALSE FALSE FALSE 2301 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
2887 FALSE FALSE FALSE
2317 FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE 2353 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 2352 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 2401 FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE 2388 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 2324 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 2330 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 2408 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 1326 FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE
2317 FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE 2353 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 2352 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 2401 FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE 2388 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 2324 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 2330 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 2408 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 1326 FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE