The equivalence between partial moments and stochastic dominance dates back to Bawa [1] and Fishburn [2]. We present a test for first, second, and third degree stochastic dominance between two variables using Lower Partial Moments. The results uphold Hadar and Russell’s [3] original conclusions about the odd moments of preferred prospects. We recall Nawrocki’s [4] research comparing Mean/Variance portfolios against the continuum of risk-averse investors using Lower Partial Moments. The excess skewness of the LPM portfolios clearly demonstrates the preference of positive skewness for risk-averse investors. Finally, we provide an algorithm for efficiently determining stochastic dominance efficient sets among large numbers of variables.
Stochastic dominance (SD) is a very powerful risk analysis tool. It converts the probability distribution of an investment into a cumulative probability curve. Next, math analysis of the cumulative probability curve is used to determine if one investment is superior to another investment. Stochastic dominance has two major advantages: It works for all probability distributions and it includes all possible risk averse utility assumptions. Its major disadvantage is that an optimization algorithm for selecting stochastically dominant efficient portfolios has never been developed limiting it to a two-step process where 1) the SD analysis is run and 2) the resulting securities are run through a portfolio optimization process. This actually is not too great of an issue because in practice, stochastic dominance can be run on thousands of securities and portfolio optimization algorithms are limited to about 150 securities due to singularity errors.1 In this paper, we propose using a stochastic dominance algorithm using lower partial moments for step (1) and then using mean-semivariance or UPM/LPM algorithm for step (2).2
The use of semivariance or lower partial moment analysis to approximate stochastic dominance has already been suggested. Bey [
Since stochastic dominance is an analysis of cumulative distribution functions; only below target deviations are considered over the interval [−∞, target] with the target encompassing all values of X. This “for all X” target condition is directly responsible for the generalization to all possible risk averse utility assumptions.
This paper proposes an integration of stochastic dominance analysis and lower partial moment analysis by defining a stochastic dominance (SD) test via the Lower Partial Moments (LPM) of the investment’s probability distribution. From this SD-LPM test, we can quickly and efficiently determine the SD efficient set and generate optimal portfolios from that reduced universe of securities for any objective function (Mean/Variance, UPM/ LPM, etc.). Equation (1) represents the LPM of an investment,
where
Bawa [
Fishburn [
Therefore, Equation (3) will be the statistic used for FSD testing, Equation (4) will be the statistic used for SSD testing, and Equation (5) will be the statistic used for TSD testing. We note that Equation (3) is a probability whereby the target deviation is taken to the 0th degree. The LPM degree 0 is simply the empirical cumulative distribution function (CDF) of the distribution. When considering how far a deviation is from its target, increasing the degree per Equation (4) will give a linear weighting. This is the area of the distribution to the left of the target, h. Finally, if an investor is more sensitive to below target observations, increasing the degree further will compensate this behavior. The LPM Degree 2 used in Equation (5) is equivalent to the semivariance statistic. These equations must be used from every target in the return distribution to generalize for all risk-averse investor types.
In a bid to infer SD from aggregate statistics (and skip the every target “for all h” inconvenience), others have suggested various CDF tests to approximate SD results. For example, Klecan et al. [
“The set A is first-degree [resp., second-degree] maximal if no prospect in A is weakly stochastically dominated by another prospect in A. First-degree dominance implies second-degree dominance, and second- degree maximality implies first degree maximality.” Klecan et al. [
Developing an efficient routine to determine SD was originally hampered by computing power coupled with arcane techniques. Computing power has increased dramatically since the 1970s, thus enabling portfolio sizes unattainable back then. Furthermore, the use of partial moments are a computationally efficient method of determining CDFs flexible enough for the entire distribution and extending themselves to area analysis required for the higher SD degrees.
“Finally, in order to conduct tests of SSD efficiency, all the calculations required for the FSD test must be made regardless of whether one is interested in FSD results; for TSD experiments, all the calculations required for FSD and SSD tests must be made.” Porter, Wart and Ferguson [
The use of LPMs in the SD test eliminates the calculation redundancy Porter et al. [
A complete presentation of the R code and commentary on the modules is presented in the Appendix A and Appendix C of this paper. Here, we are providing a general overview of how each SD routine performs its task on individual securities or aggregate portfolios.
To determine FSD between two variables we combine the observations into a single vector. Next we rank the observations in ascending order. This is the target vector. Using the combined sorted observations we compare CDFs from each target using Equation (3).
If the
In an added bid of efficiency we incorporate an additional output vector for Y such that it can be checked simultaneously for “Y FSD X”.
To determine SSD the same initial combining and ranking procedure is performed on the variables. However, we are no longer comparing CDFs rather areas of the functions up to that specific target point. These areas are compared using Equation (4) which is the degree 1 LPM.
If the
Again, using the identical combined sorted observation vector (yet another efficiency since it does not have to be generated for each degree tested) the two variables of interest are compared with their squared areas below a target, or simply their semivariance. Equation (5) represents this consideration.
If the
We incorporate these SD routines into a “SD Efficient Set” algorithm. The specific R code is also provided in the Appendix C. A full discussion of the algorithm is in Appendix A using a data frame (a list of vectors of equal lengths in R) of security or portfolio returns, we are able to generate the SD efficient set for the desired degree and return a reduced data frame. A visualization of the problem is presented by borrowing a reference to Braid Theory, which is an abstract geometric theory studying the everyday braid concept.
Our first step is to rank the securities or portfolios in ascending order by their LPM from the maximum observation across all variables. Using the terms “Base” and “Challenger” from Porter et al. [
The resulting SD Efficient data frame can then be easily implemented (within the same command line) into an optimization routine. We verify the output from our method versus the DOMIN1 routine originally presented in Porter et al. [
One potential question is whether a simple reward to semivariability (R/SV) ratio ranking would be equivalent to the LPM SD algorithm.
While stochastic dominance implies mean/Semivariance or LPM dominance, LPM dominance does not imply stochastic dominance because of the nature of aggregate statistics versus individual observations. This is analogous to the common phrase: “While dependence implies correlation, correlation does not imply dependence.” The stringent criterion of stochastic dominance defies implication from aggregate distributional statistics.
Given LPM’s ability to consider multiple investor preferences, from a practical standpoint, it seems the best candidate to proxy stochastic dominance. We demonstrated how skewness is evident under SSD and TSD. This is not surprising given Nawrocki’s [
We have also provided an algorithm for efficiently determining SD efficient portfolios. The use of lower partial moments is consistent with the procedure originally proposed by Porter et al. [
R/SV Asset | Mean | Variance | Skewness | Kurtosis | Semi Deviation | R/SV | Rank |
---|---|---|---|---|---|---|---|
N P S Pharmaceuticals | 1.05186 | 0.02573 | 0.53738 | 2.89342 | 7.36910 | 0.5372 | 15 |
Under Armour | 1.04309 | 0.00858 | 0.47474 | 2.76997 | 3.64820 | 1.0591 | 2 |
Hain Celestial | 1.03491 | 0.00481 | 0.24761 | 2.92033 | 2.95020 | 1.0909 | 1 |
Gartner Inc | 1.02787 | 0.00382 | 0.07640 | 2.39715 | 2.83310 | 0.9035 | 4 |
Ross Stores | 1.02764 | 0.00343 | 0.05922 | 2.26037 | 2.61250 | 0.9780 | 3 |
P P G Industries | 1.02707 | 0.00356 | 0.28773 | 3.56232 | 2.79310 | 0.8925 | 5 |
Visa Inc. | 1.02064 | 0.00303 | -0.75621 | 5.66774 | 3.28380 | 0.5692 | 14 |
Lorillard Inc | 1.02045 | 0.00396 | 1.23399 | 5.63855 | 2.65120 | 0.6856 | 8 |
Amgen Inc | 1.01988 | 0.00279 | 0.05548 | 3.07874 | 2.76420 | 0.6545 | 9 |
O G E Energy | 1.01469 | 0.00268 | 0.61313 | 5.26623 | 2.70910 | 0.4791 | 18 |
Waste Connection | 1.01305 | 0.00168 | -0.11832 | 2.11724 | 2.26070 | 0.5220 | 16 |
Procter & Gamble | 1.01010 | 0.00131 | 0.45848 | 2.82344 | 1.81910 | 0.4971 | 17 |
R/SV Asset | Mean | Variance | Skewness | Kurtosis | Semi Deviation | R/SV | Rank |
---|---|---|---|---|---|---|---|
N P S Pharmaceuticals | 1.05186 | 0.02573 | 0.53738 | 2.89342 | 7.36910 | 0.5372 | 15 |
Under Armour | 1.04309 | 0.00858 | 0.47474 | 2.76997 | 3.64820 | 1.0591 | 2 |
Hain Celestial | 1.03491 | 0.00481 | 0.24761 | 2.92033 | 2.95020 | 1.0909 | 1 |
Gartner Inc | 1.02787 | 0.00382 | 0.07640 | 2.39715 | 2.83310 | 0.9035 | 4 |
Ross Stores | 1.02764 | 0.00343 | 0.05922 | 2.26037 | 2.61250 | 0.9780 | 3 |
P P G Industries | 1.02707 | 0.00356 | 0.28773 | 3.56232 | 2.79310 | 0.8925 | 5 |
Lorillard Inc | 1.02045 | 0.00396 | 1.23399 | 5.63855 | 2.65120 | 0.6856 | 14 |
Waste Connection | 1.01305 | 0.00168 | -0.11832 | 2.11724 | 2.26070 | 0.5220 | 16 |
Procter & Gamble | 1.01010 | 0.00131 | 0.45848 | 2.82344 | 1.81910 | 0.4971 | 17 |
The Bey-Kearns-Burgess [
variances. This is verified with a 67 security universe example providing identical outputs for SSD and TSD.
One potential weakness of any empirical risk analysis approach is estimation error. Both SD and LPM are nonparametric and do not require knowledge of the underlying probability function. In simulation tests that we have conducted, the LPM measures are less sensitive to estimation error than either the mean or variance no matter which distribution is assumed.
Again, one consideration that needs to be reiterated is these risk analysis tools are only for below target deviations. When considering the reflection effect from Prospect Theory, above target deviations will have their own investor preferences (risk seeking for gains and risk aversion with gains). These positive observations are only considered when the target approaches the maximum observation under the stochastic dominance test. Stochastic dominance therefore cannot reward the right tail in a manner commensurate with the means it penalizes the left tail observations. But, this was never its intended purpose given the underlying utility assumptions of Hadar and Russel [
In order to generalize further, one would have to expand the analysis into an Upper Partial Moment/Lower Partial Moment (UPM/LPM) framework, capable of incorporating the often observed four-fold pattern of risk behavior identified in prospect theory and expected utility theory such as the UPM/LPM optimization model described by Viole and Nawrocki [
The close relationship between lower partial moments and stochastic dominance has been known since Porter [
Future research should extend the analysis to the use of UPM/LPM models which are superior to SD and LPM models for incorporating the full range of utility functions available with expected utility theory and prospect theory.
FredViole,DavidNawrocki, (2016) LPM Density Functions for the Computation of the SD Efficient Set. Journal of Mathematical Finance,06,105-126. doi: 10.4236/jmf.2016.61012
This section of the paper will provide the R code commentary for the stochastic dominance test using lower partial moments. The code can be found in Appendix C.
1) Module 1: Lower Partial Moment Function (LPM)
The LPM function is a fairly straightforward interpretation of Equation (1) whereby only below target observations are summed and raised to the loss aversion degree (n), and then divided by the number of observations.
2) Module 2: First Degree Stochastic Dominance (FSD)
Section A: Sort the variables in ascending order. Combine the vectors and sort the combined vector. Create output vectors for areas used in plots.
B: Create an output vector to store the instances of CDF inequality.
C: Uses the sorted X and Y variables as the LPM target under n = 0. Thus all observations are used in the stringent “for all h” qualification. If
D: Uses the same logic as (C) above, only tests the inverse CDF relationship
E: Plots the CDFs of each variable.
F: Reads the output vectors. If the output vector has 0 instances of
G: Test case.
We can see in
For three h values (−0.99, −0.98, and −0.97)
3) Module 3: Second Degree Stochastic Dominance (SSD)
Section A: Sort the variables in ascending order. Combine the vectors and sort the combined vector. Create output vectors for areas used in plots.
B: Create an output vector to store the instances of area inequality.
C: Uses the sorted X and Y variables as the LPM target under n = 1. Thus all observations are used in the
stringent “for all h” qualification. If
instance is recorded into the output vector (output_x[i] < −0). Conversely, if
D: Uses the same logic as (C) above, only tests the inverse area relationships
E: Plots the cumulative areas of each variable.
F: Reads the output vectors. If the output vector has 0 instances of
then X SSD Y.
G: Test case.
positive values (a good thing), while simultaneously
values (also a good thing). This is evident when examining the skewness between both variables.
Skew(X) Skew(Y)
0.06529391 0.04430945
Nawrocki [
“For instance, we have indicated in the paper that FSD implies a certain relationship between the odd moments (and sometimes also between the even moments) of the prospects under consideration. Consequently, given that P is preferred to P’ for all monotonic utility functions, we can immediately say that all the odd moments around zero of P are larger than the respective moments of P’.”
Even when dominance interruptions occur in the histogram, the cumulative area to that point is not enough to negate the dominance of X over Y. Revisiting the LPMs from the FSD interval of question tells a different story when areas are compared:
The areas in
Target | LPM (0,target,x) | LPM (0,target,y) | LPM (1,target,x) | LPM (1,target,y) |
---|---|---|---|---|
[−1.0] | 0.165 | 0.166 | 0.0793658 | 0.0867652 |
[−0.99] | 0.170 | 0.167 | 0.0810362 | 0.0884257 |
[−0.98] | 0.170 | 0.169 | 0.0827362 | 0.0901005 |
[−0.97] | 0.172 | 0.170 | 0.0844422 | 0.0917979 |
[−0.96] | 0.173 | 0.174 | 0.0861641 | 0.0935241 |
[−0.95] | 0.174 | 0.176 | 0.0878957 | 0.0952721 |
Target | LPM (0,target,x) | LPM (0,target,y) |
---|---|---|
[−1.0] | 0.165 | 0.166 |
[−0.99] | 0.17 | 0.167 |
[−0.98] | 0.17 | 0.169 |
[−0.97] | 0.172 | 0.17 |
[−0.96] | 0.173 | 0.174 |
[−0.95] | 0.174 | 0.176 |
used in FSD.
4) Module 4: Third Degree Stochastic Dominance (TSD)
Section A: Sort the variables in ascending order. Combine the vectors and sort the combined vector. Create output vectors for areas used in plots.
B: Create an output vector to store the instances of (area of the cumulative distribution)2 inequality.
C: Uses the sorted X and Y variables as the LPM target under n = 2. Thus all observations are used in the stringent “for all h” qualification. If
instance is recorded into the output vector (output_x[i] < −0). Conversely, if
D: Uses the same logic as (C) above, only tests the inverse area relationships
E: Plots the cumulative areas squared for each variable.
F: Reads the output vectors. If the output vector has 0 instances of
then X TSD Y.
G: Test case.
quite clear. Alternatively viewed as a histogram,
To extend the stochastic dominance tests and examine multiple portfolios, we use inspiration from Braid Theory. Braid Theory is an abstract geometric theory studying the everyday braid concept and we envision the CDFs as strings in these braids. Braids will nullify SD and avoid placing the CDFs in the “Dominated Set”.
By testing for SD using the final ranks for that SD degree, we can derive the SD efficient sets. If a portfolio is dominated by a higher final ranked one, it is out of the efficient set. Example in Row 5 Column 4:
・ The highest final ranked portfolio is the “Base”. Test the “Base” portfolio against the next highest ranked, the “Challenger”. 4 v 2. No SD exists. 2 joins the “Current Base” vector to run unidirectional5 SD tests from.
・ Test the new “Base” against the next “Challenger”. 2 v 3.2 SD 3. 3 is placed in the “Dominated Set”.
・ Test the new “Base” against the next “Challenger”. 2 is the last entry in the “Current Base”. 2 v 1. No SD
exists. Test the remaining “Current Base” vector against 1. 4 v 1. No SD exists, 1 joins the “Current Base”.
・ There are no more “Challengers”. Stop the procedure. The SD efficient set is the final ranked set less the “Dominated Set” {4,2,1}.
・ #1 (the largest minimum value) can never be dominated, thus it is in every efficient set.
This procedure is different than that proposed by Porter, Wart and Ferguson [
This section of the paper will provide the R code for the stochastic dominance test using lower partial moments.
1) Module 1: Lower Partial Moment Function (LPM)
2) Module 2: First Degree Stochastic Dominance (FSD) [Bi-directional]
3) Module 3: Second Degree Stochastic Dominance (SSD) [Bi-directional]
4) Module 4: Third Degree Stochastic Dominance (TSD) [Bi-directional]
5) Module 5: Stochastic Dominance Efficient Set (SDES) & Uni-directional SD Tests