OPTIMAL ALGORITHMS AND LOWER PARTIAL MOMENT: EX-POST RESULTS
by David N. Nawrocki College of Commerce and Finance Villanova University Villanova, PA 19085
[email protected]
[email protected]
Acknowledgements: The author wishes to thank Ghassem Homaifar for his valuable contributions to this paper.
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OPTIMAL ALGORITHMS AND LOWER PARTIAL MOMENT: EX POST RESULTS ABSTRACT Portfolio management in the finance literature has typically used optimization algorithms to determine security allocations within a portfolio in order to obtain the best tradeoff between risk and return. These algorithms, despite some improvements, are restrictive in terms of an investor's risk aversion (utility function). Since individual investors have different levels of risk aversion, this paper proposes two portfolio optimization algorithms that can be tailored to the specific level of risk aversion of the individual investor and performs ex-post evaluation tests of the algorithm performance.
I.
INTRODUCTION
Optimization algorithms in portfolio management either require restrictive investor utility functions or the utility analysis is eliminated in a CAPM or APT equilibrium environment. While the utility-free methodology is useful, it is based on the separation theorem and, therefore, on an environment with zero transactions costs and unlimited borrowing. Since financial markets are characterized by both transaction costs and limited borrowing (margin requirements), a utility-dependent approach to portfolio analysis may prove useful since it can be empirically implemented. Utility-based portfolio analysis has continued the traditional mean-variance approach with some attempts at broadening the meanvariance utility function.1 Another alternative is to select another risk measure characterized by a general set of utility functions. One such measure proposed by Fishburn(1977) is the n-degree lower partial moment. Specifically, the higher the degree of the LPM measure, the greater the risk aversion of the investor. The purpose of this paper is to propose two optimization algorithms using the n-degree LPM and to test the ex-post performance of the LPM algorithms. Testing different degrees of the LPM is important for two reasons: First, there is a strong relationship between n-degree lower partial moment and stochastic dominance. This relationship is developed through the work of Bawa(1975,1978), Bawa and Lindenberg(1977), and Fishburn(1977). These authors provide mathematic proofs that stochastic dominance is equivalent to all degrees of n-degree LPM. Bey(1979) provides empirical evidence
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that semivariance (ES) efficient portfolios are part of the second degree stochastic dominance (SSD) (60-65%) and the third degree stochastic dominance (TSD) (90%) efficient sets. This relationship of the n-degree LPM to stochastic dominance is important since stochastic dominance does not make any distributional assumptions and assumes a very general set of utility functions. Since stochastic dominance cannot be used as an algorithm to efficiently select optimal security allocations, a lower partial moment algorithm that closely approximates the stochastic dominance efficient set may prove useful for portfolio selection. Second, Fishburn(1977) shows that by varying the degree of the n-degree lower partial moment, the risk measure can accurately reflect (when compared to semivariance-cosemivariance and variance-covariance analysis) investor utility towards risk and below target returns, i.e. the utility functions defined by n-degree LPM are as general as the utility functions defined by stochastic dominance. While Levy and Markowitz(1979) and Kroll, Levy and Markowitz(1984) have found that a limited number of utility functions can be approximated by a judiciously chosen utility function defined by mean-variance, the n-degree LPM defines a general class of utility functions. Because of the potential benefits of an n-degree LPM algorithm, two LPM algorithms are proposed and tested. Using stochastic dominance, the performance of the two algorithms is compared to the traditional variance-covariance portfolio analysis. The structure of this paper is as follows: First, a discussion of some issues concerning the LPM is presented. Second, the two LPM critical line algorithms are described along with the methodology of the empirical test. Finally, the empirical results are presented along with concluding comments. II.
SOME RELEVANT ISSUES
A. The Computational Formula for N-Degree LPM and CLPM Bawa(1975,1978) and Fishburn(1977), in developing the relationship between LPM and stochastic dominance, define n-degree LPM as follows:
h LPMn (h,F) = ∫ (h - R) df(R) -ì
(1)
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This yields the following computational formulas for LPM and CLPM (co-lower partial moment). 1 m n LPM = - ∑ [Max(0,(h – R )] in m t=1 it
(2)
1 m n-1 CLPM = - ∑ [Max(0,(h – R )] (h – R ) ij,n-1 m t=1 ij jt
(3)
where h is the target return, m is the number of observations, n is the LPM degree, and Rit is the periodic return for security i during period t. This form of the calculation means that LPM is nonnegative for all values of n. N-degree LPM is not comparable to the standard statistical moments of the distribution where investors exhibit a preference for higher values of odd moments (skewness) and a dislike for higher values of even moments (variance, kurtosis).2 LPM is based on the investor's risk attitude towards below target (h) returns. Whenever n2) algorithms earn the highest returns. During the 48-24 month simulation, the SLPM algorithms enjoy the best performance with the terminal wealth increasing with the degree of the LPM measure.
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During the 24-12 simulation period, the SLPM algorithm enjoys the highest terminal wealths and again, the terminal wealth increases as n increased. In all cases (72-60, 48-24 and 24-12), the higher order (n>2) LPM measures provide the best investment performance. In all cases, the covariance algorithm had lower final wealths than one or both of the LPM algorithms. TABLE 3 Average Terminal Wealth Values for Optimal Portfolio Algorithms Hist 72 Hold 60 Covariance Symmetric LPM Degree = 1.0 2.0 3.0 4.0 5.0 Asymmetric LPM Degree = 1.0 2.0 3.0 4.0 5.0
9.09 8.77 8.49 8.14 7.93 7.62 6.26 6.50 15.21 13.34 15.01
Hist 48 Hold 24 | | | | | | | | | | | | | | |
5.62 7.05 8.89 8.94 10.27 11.30 9.95 3.68 10.95 4.12 2.42
| | | | | | | | | | | | | | |
Hist 24 Hold 12 | 7.34 | | 8.72 | 10.15 | 10.59 | 10.07 | 13.60 | | 3.04 | 4.54 | 3.90 | 3.70 | 7.62 | |
C. Reward to Semivariability Results In order to measure the risk/return performance of the algorithms, reward to semivariability (R/SV) ratios are computed and presented in Table 4. The Sharpe and Treynor measures that are typically used are statistically biased. Ang and Chua(1979) demonstrate this bias and show the R/SV ratios to be unbiased performance measures. With 240 monthly observations, the results are not expected to be affected by estimation error.
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TABLE 4 Average Reward to Semivariability Ratios for Optimal Algorithms Hist 72 Hold 60 | Hist 48 Hold 24 | Hist 24 Hold 12| | | | Covar. .1708 | .1129 | .1454 | | | | Symmetric | | | LPM 1.0 .1705 | .1458 | .1658 | 2.0 .1644 | .1763 | .1904 | 3.0 .1575 | .1683 | .1971 | 4.0 .1550 | .1940 | .1880 | 5.0 .1471 | .2053 | .2240 | | | | Asymmetric | | | LPM 1.0 .1326 | .1805 | .0274 | 2.0 .1330 | .0591 | .0823 | 3.0 .2096 | .1980 | .0610 | 4.0 .2009 | .0743 | .0589 | 5.0 .1952 | .0086 | .1319 |
The results in Table 4 are identical to the terminal wealth results in Table 3. The ALPM algorithm has the best results with the 72-60 simulation (degrees of 3, 4 and 5). The SLPM provides the best performance with the 48-24 and with the 24-12 simulations.
D. Second Degree Stochastic Dominance Results Tables 3 and 4 cannot be subjected to statistical analysis because of the criticisms made by Fishburn(1977) concerning the distribution or utility assumption behind the statistical test. In fact, both the terminal wealth and the R/SV techniques assume restrictive utility functions. In order to draw any conclusions, a performance measure that satisfies Fishburn's objections needs to be used. The second degree stochastic dominance (SSD) meets this need since it does not make any distributional assumption and utilizes a very general utility function (for all nò1, which describes all risk averse behavior when risk is measured with below target returns).8 The portfolios that are second degree stochastic dominant are determined using the Porter, Wart and Ferguson(1973) algorithm corrected for the Vickers and Altman(1979) bias.
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TABLE 5
Second Degree Stochastic Dominance Results for Optimal Portfolio Algorithms using Covariance and Co-Lower Partial Moment Analysis for 125 Securities from 1958 to 1977. Hist 72 Hold 60 | Hist 48 Hold 24 | Hist 24 Hold 12| 5 10 15 | 5 10 15 | 5 10 15 | | | | X X | | | | | | Symmetric LPM | | | Degree = 1.0 X | X | X | 3.0 X | | X X | 4.0 X | X | X X | 5.0 | X | X X X | | | | Asymmetric LPM | | | Degree = 1.0 X | | | 2.0 X | | | 3.0 X | | | 4.0 X X | X | | 5.0 X | | | | | | | | | X - Undominated by SSD | | | Num. Stocks Algorithms Covariance
All of the portfolios denoted by X in Table 5 are SSD efficient. For the 72-60 simulation, none of the algorithms exhibit a clear dominance over the other algorithms. It should be noted that EV efficient portfolios are also part of the SSD efficient set.9 Because SSD includes a broad range of utility functions, both the ALPM and SLPM algorithms provide a valid alternative to mean-variance analysis by simply being included in the SSD efficient set. With the 48-24 month simulation periods and the 24-12 month periods, both the ALPM portfolios and the covariance portfolios are dominated by the SLPM algorithm. Here it is clear that the SLPM model is providing performance that is superior to mean-variance analysis.
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The results are sensitive to the length of the historic and holding period. With the shorter historic-holding periods (48-12 and 24-12), the SLPM algorithms dominate the variance-covariance algorithm. The shorter subperiods may place a premium on forecasting ability and the SLPM may simply be the better forecast model as per Elton et al.(1978). The ALPM model and the covariance model provide their best performance during the longer 72-60 simulation. Both of the LPM algorithms exhibit very good performance from using higher degrees of LPM (n>2). The main conclusion from these results is that LPM analysis provides investment performance that is very competitive with traditional covariance analysis, thus broadening the range of utility choices available to investors. In addition, these results indicate that future empirical tests of the lower partial moment should include the higher LPM moments.
VI.
SUMMARY AND CONCLUSIONS
This study tests two algorithms that employ higher degrees of the n-degree lower partial moment risk measure in order to provide investors with a broader range of utility choices. The first algorithm, the asymmetric matrix algorithm (ALPM), derives from the optimal Hogan and Warren[9] semivariance algorithm. The second algorithm, the symmetric algorithm (SLPM), is an ad hoc heuristic that simplifies the CLPM matrix. The two algorithms are tested along with the traditional variance-covariance algorithm. Portfolio performance is evaluated using terminal wealth, reward to semivariability and stochastic dominance. For historic estimation periods of 72 months and subsequent holding periods of 60 months, the stochastic dominance results indicate that no portfolio algorithm consistently dominates the others. All of the algorithms are providing portfolios that are members of the second degree stochastic dominance efficient set. Even though the LPM portfolios are not superior to covariance analysis, they are providing investors with different utility choices because they members of the SSD efficient set. Support for this statement also comes from the terminal wealth and reward to semivariability results that indicate that the SLPM algorithms are either comparable or superior to covariance analysis. For shorter periods such as 48-24 month and 24-12 month historic/holding periods, the SLPM algorithm utilizing higher LPM
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moments clearly dominates the other two algorithms. Overall, the SLPM portfolios provide the most consistent performance. The optimal ALPM algorithm performs well only during the longer 72-60 month subperiods. Because of its added complexity, the ALPM algorithm does not have the empirical support to recommend it. Since the SLPM algorithms provide portfolios that are members of the SSD efficient set when compared to the traditional covariance algorithm, the SLPM algorithms have to be considered a viable alternative to covariance analysis. The fact that SLPM analysis handles a broader range of investor utility functions increases the attractiveness of the technique.
FOOTNOTES 1. See Aivazian et al.(1984), Levy and Markowitz(1979) and Kroll et al.(1984) for a discussion of this area. 2. See Scott and Horvath(1980). 3. Remember that n=4.6 was the maximum value in Fishburn's (1977) survey of investor utility functions. 4. Survival bias has not been found to have a significant effect on empirical results. See Ball and Watts(1979) and Sunder(1980). 5. See Kroll and Levy(1980) for a study on estimation error. 6. A second data set containing more recent data (1972-1987) was also tested. Because of corporate restructuring that occured in the 1980s, only 100 of the original 125 securities were available for study. The empirical results from this data support the same conclusions that are derived from the original study. Therefore, they are not presented. 7. Duvall and Quinn(1981), Kane(1982), Simkowitz and Beedles(1978). 8. See Saunders et al.(1980) for a discussion of using stochastic dominance for portfolio evaluation. 9. See Bey(1979).
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