Computer Handholders Investment Software Research Paper Series TAILORING ASSET ALLOCATION TO THE INDIVIDUAL INVESTOR David N. Nawrocki -- Villanova University ABSTRACT Asset allocation has typically used optimization algorithms to determine security allocations within a portfolio in order to obtain the best tradeoff between risk and return. These techniques, by using the variance as a measure of risk restrict the investor to one level of risk aversion (utility function) which has to fit all investors. Since individual investors have different levels of risk aversion, this paper proposes a heuristic portfolio selection technique that can match the risk measure to the specific level of risk aversion of the investor. The technique is tested with 34 years of monthly data to demonstrate its use. INTRODUCTION Imagine the human race as a set of wooden pegs. Each peg is a different geometric shape. Along comes an asset allocation system with a hammer and woodcarving knife. It proceeds to whittle and pound until each peg fits into the round hole that is the quadratic utility function assumed by covariance or beta analysis. Does asset allocation have to do this? Not really. There have been two techniques that alleviate this problem: stochastic dominance and lower partial moment (LPM). While stochastic dominance is theoretically superior to variance analysis, a lack of a usable asset allocation algorithm has rendered it less than useful. The lower partial moment, on the other hand, has both usable algorithms and the ability to be tailored to the individual investor. This paper proposes a simple asset allocation technique that uses the lower partial moment as a measure of risk. The technique is tested with 34 years of monthly data and its portfolio performance results are compared to optimal covariance and semivariance techniques. The purpose of this paper is to show that an investor can control the skewness of a portfolio through the choice of a risk measure and that the appropriate level of risk aversion depends on the frequency of revising the portfolio.
LOWER PARTIAL MOMENT The lower partial moment (LPM) traces back to Markowitz(1959), who suggests that semivariance analysis be used to handle skewed security return distributions and investors who did not have quadratic utility functions. Bawa(1975) and Fishburn(1977) demonstrate that the semivariance is a special case of lower partial moment analysis. They derive the n-degree LPM where the semivariance is a special case (n=2) of LPM. The variable n is the degree to which deviations below the target return are raised. In the case of the semivariance, the below-target deviations are squared. If n=3, then the belowtarget deviation is cubed. LPM itself is equivalent to stochastic dominance analysis. Both Bawa and Fishburn show that third degree stochastic dominance (TSD) efficient sets contain all LPM efficient sets where n>2 and second degree stochastic dominance (SSD) contains all LPM efficient sets where n>1. This is important since it shows that the lower partial moment provides both of the benefits provided by stochastic dominance: a general utility model that fits most investors and no restrictive assumptions governing the probability distribution of security returns. Fishburn(1977) continues by demonstrating that the lower partial moment can match the utility functions of a number of classic investors who have been described in utility theory literature. He states that investors have a target rate of return that they wish to achieve and that they wish to avoid below-target returns. The lower partial moment ignores above-target returns and penalizes below target returns. As the degree n increases, the investor is increasingly unhappy with below target returns. As the degree n decreases, the investor will be more willing to accept below target returns. The n-degree lower partial moment is mathematically defined by the following equation: 1 m n LPMn(h) = - Max[0,(h - Rt)] (1) m t=1 where n is the degree of the lower partial moment (n>0), h is the target return that the investor does not wish to go below, Rt is the return for a security for period t, and m is the number of periods used to calculate the LPM. Note that the above-target returns (Rt > h) provide negative numbers. Given the choice of a zero or a negative number, the maximization (Max) function will select the zero. Only below-target returns (Rt < h) will provide a positive deviation that is raised to the n power and added into the LPM calculation. LPM will only provide nonnegative values. In the n-degree LPM model, n=1.0 is the boundary line between risk averse behavior and risk seeking behavior. When n>1.0, the investor is averse to risk and attempts to minimize it. When n<1.0, the investor seeks to add additional risk to a portfolio. A DEMONSTRATION OF THE N-DEGREE LOWER PARTIAL MOMENT
Consider the following example of two investments, Company A and Company B. Both have the same expected return and variance. However, they have different skewness values and below target returns. If the target return (h) is 15%, the LPM can be calculated for different values of n. Example of N-Degree Lower Partial Moment Compared to Variance Measure Company A Company B Return Prob. Return Prob. -5.00 0.20 10.00 0.80 20.00 0.80 35.00 0.20 Investment A B E(R) 15.00 15.00 Variance 100.00 100.00 Skewness -1.50 1.50 LPM n=0.5 h=15.0 0.89 1.79 LPM n=1.5 h=15.0 17.89 8.94 LPM n=2.0 h=15.0 80.00 20.00 LPM n=3.0 h=15.0 1600.00 100.00 Notice that when n<1, investment A is superior to investment B, i.e. LPMA is less than LPMB. When n>1, investment A is now considered (by LPM) to be riskier than investment B. As n increases, A receives a heavier utility penalty from the LPM measure. The utility choice clearly becomes more risk averse as n increases. Meanwhile, the variance measure does not differentiate between the two investments. Fishburn provides a methodology that allows the risk aversion of an investor to be measured in terms of the n-degree LPM. Laughhunn, Payne and Crum (1980) use Fishburn's methodology to test a sample of corporate middle managers for their attitude towards risk. They find that only 9% of the managers have a utility function that is in the same general area as the semivariance (n=2). This lends support to the notion that a general utility model such as the n-degree lower partial moment is needed. ASSET ALLOCATION USING LOWER PARTIAL MOMENT Once the risk measure is chosen, the asset allocation algorithm is next. Typically, an optimal algorithm is used that requires a large number of computations. A heuristic algorithm that provides an approximate solution as opposed to an optimal solution offers an attractive alternative to optimal number crunchers. There are a number of advantages to a heuristic algorithm. First, a heuristic offers a solution that is in the neighborhood of an optimal solution. Second, it requires very few calculations and minimizes computer turnaround time. Third, a simple model may be a better forecaster of future conditions than a more
complex model. Elton, Gruber and Urich (1978) provide support to this last statement. They find that an average correlation value will forecast a future covariance matrix better than the current covariance matrix. Lastly, maximum investment constraints (e.g. a mutual fund may not have more than 5% of its portfolio in one security) will move a solution so far from the optimal solution that a heuristic can be very effective in this case. For example, Sharpe (1967) uses a simple ranking heuristic for his mutual fund algorithm. Heuristic algorithms by Sharpe (1967) and Elton, Gruber and Padberg (1976) provide the basis for the heuristic used for this study. In this heuristic, the securities are ranked by their reward-to-risk ratio, or, Zi = (E(Ri) - Rf)/ LPMn (2) Where Zi is the reward-to-risk ratio for security i, E(Ri) is the expected return for security i, Rf is the Treasury Bill rate of return, and LPMn is the n-degree lower partial moment. The allocation for each security in the portfolio is calculated by ranking the securities by their Zi values and dividing each Zi by the sum of the positive Zi values, or k Xi = Zi / Zj for all Zj>0 (3) j=1 where Xi is the allocation for security i and k is the number of securities with positive Zi values. Alternatively, k could be the desired number of securities in the portfolio, e.g. k=5 would select a 5 security portfolio. A DESCRIPTION OF THE EMPIRICAL TEST Monthly data for 135 securities from January 1954 to December 1987 (408 observations) are used to simulate the performance of the LPM heuristic technique. The data is from the University of Chicago, Center for Research in Security Prices (CRSP) data tape. The estimation period for this study is 48 months which is long enough to minimize estimation error (Kroll and Levy, 1980). Holding periods of 3, 6, 12, 24, 36 and 48 months are simulated. The portfolios are revised to new portfolio allocations at the end of each holding period using a 1% transaction cost. The first four years 1954-57 are used to compute the initial portfolios selected in the simulation. The holding periods commence in January 1958 and end in December 1987 encompassing 360 monthly observations. Portfolio returns for portfolio sizes of 5, 10 and 15 securities are computed for the 1958-87 period and are evaluated using reward to semivariability ratios and second degree stochastic dominance. An optimal covariance algorithm and an optimal semivariance algorithm are utilized to provide a benchmark for comparison. A 135 security, equal allocation, buy-and-hold (no rebalancing back to original weights) portfolio was also
computed for benchmark purposes. The target value (h) for the LPM calculation was set to a zero percent return (0%) and the Rf for the Zi values was set to an annual rate of 3.6%. The degree of the LPM was varied from 0 to 15. EMPIRICAL RESULTS The most important empirical results in terms of the heuristic algorithm are presented in Table 1. By penalizing below target returns, the lower partial moment places a emphasis on investors preferring positive skewness instead of negative skewness. As n increases, the skewness of the portfolio should also increase. In Table 1, the average skewness for the 5, 10, and 15 security portfolios is presented. The skewness is in the ex-post returns of the portfolios selected by the optimal algorithms and the heuristic LPM algorithm. With 360 monthly observations, the sample skewness should be fairly stable. While Lau and Wingender (1989) point out that the sampling error of skewness has not been fully investigated in the finance literature, Kroll, Levy and Markowitz(1984) use the following formula to compute significant skewness: Std. Dev. = square root of (6/n) where n is equal to the number of observations.
Table 1 - Skewness Values for LPM Heuristics, Covariance, Semivariance Algorithms for the Period 1958-1987. Historic Periods of 48 Months and Holding Periods of 48,36,24,12,6, and 3 Months. 48-48 48-36 48-24 48-12 48-6 48-3 Market -.3044* -.3044* -.3044* -.3044* -.3044* -.3044* Covariance -.0422.0899 -.0546 -.0505 -.1127 -.1738 Semivariance -.1296.1356 -.1258 -.1600.1793 -.1712 LPM Heuristics 0.0.1114.1467.1122.1071.0818 -.0902 1.0.0553.1234.0756 -.0660 -.0156 -.1461 1.2.0345.1221.0719 -.0669 -.0104 -.1343 1.4.0182.1311.0713 -.1371 -.0226 -.1264 1.6 -.0119.1346.0833 -.1143.0555 -.0652 2.0 -.0628.1692.1110 -.0394.0797.0349 2.8 -.0694.1579.1934 -.0183.1390.0470 3.0 -.0860.1633.2115 -.0184.1807.0697 4.0 -.1325.2322.2771*.0333.1885.1622 4.6 -.1401.2353.3093*.0792.2083.1695 5.0 -.0828.2535.3446*.0999.2353.2582* 6.0 -.0354.3007*.4287*.1602.3079*.3391* 7.0.0069.3573*.4975*.1969.3506*.4410* 8.0.0477.3883*.5413*.2312.3819*.5057* 9.0.0799.4274*.5855*.2708*.4080*.5714* 10.0.1096.4457*.6129*.2944*.4254*.6166* 11.0.1332.4552*.6281*.3131*.4367*.6556* 12.0.1517.4587*.6376*.3282*.4524*.6928* 13.0.1755.4596*.6503*.3465*.4654*.7216* 14.0.1881.4596*.6505*.3626*.4806*.7474* 15.0.1994.4540*.6551*.3791*.4956*.7685* * - significant at 2 standard deviations (.2582) std. dev. = square root of (6/360) =.1291 In every case, the LPM heuristic portfolios show an increase in skewness compared to the buy-and-hold portfolio with equal weights. For holding periods of 36 months or less, the skewness of the heuristic LPM portfolios increase for n>1.4. For the 48 month holding period, the skewness of the portfolios decreases until n=4.6 and then increases as n is increased to 15. In every case, the higher degree LPM heuristics (n>5) exhibit higher skewness values than the buy-and-hold portfolio, the optimal covariance portfolios, and the optimal semivariance portfolios. By increasing the degree of the LPM risk measure, an investor can control (increase) the amount of skewness in the resulting portfolio. This is counter to the results of Singleton
and Wingender (1986) who state that skewness does not persist and cannot be forecasted. This conclusion is not as strong for the 48 month holding period, but it may indicate that the ability of the LPM heuristic to forecast portfolio skewness drops off after 36 months. It is also clear that the n-degree LPM is providing the investor with a wealth of utility choices as far as skewness of portfolio returns is concerned. Table 2 provides insight into the risk-return performance of the various portfolios using the reward to semivariability ratio. Studies such as Ang and Chua (1979) have shown the traditional measures of investment performance such as the Sharpe reward to variability ratio or the Treynor reward to volatility ratio to be statistically biased. The reward to semivariability on the other hand is not biased. The main drawback of the R/SV ratio is that it includes only the LPM n=2 utility function.
Table 2 - Reward to Semivariability Ratios for LPM Heuristics, Covariance, Semivariance Algorithms for the Period 1958-1987. Historic Periods of 48 Months and Holding Periods of 48,36,24,12,6, and 3 Months. 48-48 48-36 48-24 48-12 48-6 48-3 Market.1658.1658.1658.1658.1658.1658 Covariance.0842.1230.1383.1494.1636.1536 Semivariance.1490.1744.1636.1541.1736.1505 LPM Heuristics 0.0.1130.1345.1712.1757.2017.1839 1.0.1335.1705.1984.2288.2430*.2274 1.2.1366.1737.2117.2299.2390.2287* 1.4.1412.1835.2221*.2346.2365.2204 1.6.1544.2039.2110.2363*.2310.2062 2.0.2001.1972.2089.2096.2129.2018 2.8.2242.2201*.2186.1946.1927.1951 3.0.2348.2127.2155.1839.1850.1875 4.0.2442.2095.2098.1711.1633.1707 4.6.2467*.2120.2044.1741.1631.1698 5.0.2416.2002.2005.1667.1579.1585 6.0.2383.1938.1905.1544.1501.1491 7.0.2327.1893.1848.1482.1477.1463 8.0.2374.1912.1854.1456.1484.1431 9.0.2378.1909.1825.1427.1504.1421 10.0.2352.1898.1794.1405.1504.1409 11.0.2318.1905.1767.1386.1513.1427 12.0.2301.1915.1743.1362.1521.1428 13.0.2278.1924.1732.1350.1549.1431 14.0.2261.1916.1718.1354.1577.1451 15.0.2241.1920.1707.1350.1612.1476 For each holding period, there is a range of LPM portfolios that outperform the optimal algorithms and the sample buy-and-hold portfolio. For 48 months, the best LPM performance shows at n>2, while the best performance sits at n=4.6. For 36 months, the best performance is n=2.8. For 24 months or less, the best performance is at n<2. For holding periods of 48, 36, and 24 months, most of the LPM heuristics outperform the optimal algorithms. However, for 12 months or less, the best performance is limited to the range of n=0 to n=3. For longer holding periods, the best performance is with the more risk averse LPM degrees n>2. For shorter periods the best performance is with the more aggressive and less risk averse LPM degrees (n<2). Looking back at Table 1, the best performance for all holding periods is for portfolios with low or negative skewness. Increasing the level of skewness in a portfolio does not bring about better risk-return performance as measured by the reward to semivariability ratio.
Since the reward to semivariability does represent a very restrictive utility function, an evaluation technique with a more general utility function is needed. The best technique in this case is probably the second degree stochastic dominance which contains all LPM utility functions for n>1.
Table 3 - Second Degree Stochastic Dominance Results for LPM Heuristics, Covariance, Semivariance Algorithms for the Period 1958-1987 for Portfolio Sizes of 5, 10, and 15 Securities. Historic Periods of 48 Months and Holding Periods of 48, 36, 24, 12, 6, and 3 Months. Portfolios From Each Period Are Evaluated Separately. 48-48 48-36 48-24 48-12 48-6 48-3 51015 51015 51015 51015 51015 51015 Market Covariance X Semivariance X X X X X LPM Heuristics 0.0 1.0 X X X X X X X X X 1.2 X X X X X X X X X 1.4 X X X X X X X X X X X 1.6 X X X X X X X X X X 2.0 X X X X X X X X X X X X X 2.8 X X X X X X X X 3.0 X X X X X X X X X X X X X X 4.0 X X X X X X X X X X X X 4.6 X X X X X X X 5.0 X X X X X X X X X X X X X 6.0 X X X X X X X X X 7.0 X X X X X X 8.0 X X X X X 9.0 X X X X X X X 10.0 X X X X X X X 11.0 X X X X X X X X X 12.0 X X X X X X X 13.0 X X X X X X 14.0 X X X X X X 15.0 X X X X X
Table 3 presents the portfolios that are undominated by second degree stochastic dominance. Portfolios from each holding period are evaluated only with other portfolios from the same holding period. The optimal semivariance algorithm has some undominated portfolios for the 6, 12, 24, and 36 month holding periods. For the most part, the LPM heuristic dominates the results. With the LPM results, the same result as in Table 2 is evident. As the holding period is reduced the best performance is obtained from lower degree LPM measures. The range of LPM portfolios with all three portfolio sizes dominant tells the story. Holding LPM Period Range 48 11 < n < 15 36 11 < n < 14 (also n=3) 24 3 < n < 7 12 1.4 < n < 6 6 1 < n < 2 3 1 < n < 2 Finally, Table 4 provides the results of second degree stochastic dominance applied to all of the portfolios. The best results are for the LPM heuristic for 24 and 48 month holding (revision) periods. The 36 month and 3 month holding periods are not competitive. The 6 and 12 month portfolios are undominated only for n=1.0.
Table 4 - Second Degree Stochastic Dominance Results for LPM Heuristics, Covariance, Semivariance, Algorithms for the Period 1958-1987 for Portfolio Sizes of 5, 10 and 15 securities. Historic Perios of 48 Months and Holding Perios of 48, 36, 24, 12, 6, and 3 Months. Portfolios From Each Period Are Evaluated Together. 48-48 48-36 48-24 48-12 48-6 48-3 51015 51015 51015 51015 51015 51015 Market Covariance Semivariance X LPM Heuristics 0.0 1.0 X X X X X X X 2.0 X 3.0 X X 4.0 X X X X 5.0 X X X X X 6.0 X X X X X 7.0 X X X X 8.0 X X X X 9.0 X X X X 10.0 X X X X 11.0 X X X X X 12.0 X X X 13.0 X X X 14.0 X X 15.0 X X X
Table 4 shows the same trend as in the other tables, i.e. as the holding period decreases, the degree of the best LPM portfolios also decreases. This is evident by looking at the range of LPM portfolios that have two of the three portfolios dominant. Holding LPM Period Range 48 4 < n < 15 36 None 24 4 < n < 12 12 n = 1 6 n = 1 3 n = 1 SUMMARY AND CONCLUSIONS The major issue of this paper is whether the risk measure can be changed to match the risk aversion of an investor. The n-degree lower partial moment does have the capability of utilizing a wide range of utility functions and can be applied simply by the investor. The empirical test came to the following conclusions: 1. With shorter holding periods (36 months or less), the skewness of a portfolio can be managed using the n-degree lower partial moment, i.e. increases in the level of relative skewness can be achieved by using higher degrees of the LPM risk measure. 2. For longer holding periods, the best portfolio performance is achieved by higher degrees of LPM. These higher degrees represent higher risk aversion on the part of the investor. If the investor wishes to manage a portfolio to a unique utility function, then longer holding periods (greater than 24 months) is recommended. 3. For short holding (revision) periods, investor preference for skewness is irrelevant and the most aggressive utility functions n<2 provide the best performance. With holding periods less than 12 months, there seems to be only one relevant utility function, n=1.0. 4. Stochastic dominance results show that for long holding (revision) periods a wide range of LPM degrees (4.0 < n < 15.0) are members of the SSD efficient set. For short holding periods, the range of SSD dominant LPM portfolios narrows to 1.0 < n < 2.0.
REFERENCES Ang, J.S. and J.H. Chua, "Composite Measures for the Evaluation of Investment Performance," Journal of Financial and Quantitative Analysis, June 1979, pp. 361-384. Bawa, V.S.,"Optimal Rules for Ordering Uncertain Prospects," Journal of Financial Economics, March 1975, pp. 95-121. Elton,E.J., Gruber,M.J., and M.W. Padberg, "Simple Criteria for Optimal Portfolio Selection," Journal of Finance, December 1976, pp. 1341-1357. Elton,E.J., Gruber,M.J. and T. Urich, "Are Betas Best?," Journal of Finance, December 1978, pp. 1376-1384. Fishburn, P.C., "Mean-Risk Analysis with Risk Associated with Below-Target Returns," American Economic Review, March 1977, pp. 116-126. Kroll,Y. and H. Levy, "Sampling Errors and Portfolio Efficient Analysis," Journal of Financial and Quantitative Analysis, September 1980, pp. 655-688. Kroll,Y., Levy,H. and H.M. Markowitz, "Mean-Variance Versus Direct Utility Maximization", Journal of Finance, March 1984, pp. 47-62. Lau, H. and J.R. Wingender, "The Analytics of the Intervaling Effect on Skewness and Kurtosis of Stock Returns," The Financial Review, May 1989, pp. 215-234. Laughhunn, D.J., Payne, J.W. and R. Crum, "Managerial Risk Preferences for Below- Target Returns," Management Science, December 1980, pp. 1238-1249. Markowitz, H.M., Portfolio Selection: Efficient Diversification of Investments, New York: John Wiley and Sons, 1959. Sharpe, W., "A Linear Programming Algorithm for Mutual Fund Portfolio Selection," Management Science, March 1967, pp. 499-510. Singleton, J.C. and J.R. Wingender, "Skewness Persistence in Common Stock Returns," Journal of Financial and Quantitative Analysis, September 1986, pp. 335-341.