Parameter Estimation Techniques, Optimization Frequency, and Equity Portfolio Return Enhancement*

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1 Parameter Estimation Techniques, Optimization Frequency, and Equity Portfolio Return Enhancement* By Glen A. Larsen, Jr. Kelley School of Business, Indiana University, Indianapolis, IN 46202, USA, Bruce G. Resnick Babcock Graduate School of Management, Wake Forest University, Winston-Salem, NC 27109, USA, Significant assistance provided by Guozhong Lin, Kelley School of Business, Indiana University.

2 Abstract Various ex ante portfolio parameter estimation techniques and optimization/holding period frequency intervals are tested for their ability to enhance managed portfolio returns relative to a benchmark. The potential for return enhancement is accomplished by optimizing over highly correlated size-based constituent portfolios of the benchmark. Overall, the results suggest that it is possible to consistently achieve enhanced returns at much the same level of return per unit of risk as the benchmark portfolio. Keywords: Parameter Estimation, Optimization Frequency, Enhanced Returns

3 1. Introduction Equity portfolio return enhancement strategies focus on generating higher returns than a benchmark portfolio, without taking on excessive risk. John Loftus (1997) of Pacific Investment Management Company, a firm that sells an enhanced indexing product, notes that risk-controlled active management would be a more appropriate description of enhanced indexing. One popular strategy of return enhancement involves using mispriced stock index futures to create a synthetic index portfolio that offers enhanced returns at the same level of risk as the cash index. (1) Alternatively, active managers may attempt to enhance returns by modifying the composition of the securities held in the portfolio relative to the benchmark portfolio. The issue of return enhancement through portfolio modification is one of capital market theory. According to theory, in an efficient market, the theoretical market portfolio offers the highest level of return per unit of risk as measured by Sharpe measure of performance. The theoretical market portfolio is a capitalization-weighted portfolio of all risky assets. As a proxy for the theoretical market portfolio, an index that is representative of the market is often used as a benchmark portfolio. If the market is efficient and capital market theory holds, then it will be difficult to outperform a well-diversified, valueweighted benchmark portfolio according to standard risk-adjusted measures of portfolio performance. In a classic study, Jensen (1968) reviews the performance of 115 mutual fund managers over the period 1955 to On a risk-adjusted basis, the average mutual fund return before expenses was 0.1 percent less than that which could have been achieved by an investment of comparable risk in T-bills and the market index. Jensen s finds no evidence that any portfolio manager is able to

4 consistently outperform the popular market indexes. If these findings are accepted, then the costs associated with active equity portfolio management may not purchase an enhanced return on a portfolio. These costs consist of the research costs associated with uncovering mispriced stocks, the transaction costs of buying and selling stocks to take advantage of mispricing, and the transaction costs incurred in trying to time the market. Elton, Gruber, Das, and Hlavka (1993) find that active managers with low turnover outperform managers with high turnover. They indicate that the difference in performance is likely due to the added transaction costs associated with higher turnover. The same results seem to prevail with individual investors. Barber and Odeans (1998) obtain six years' worth of data on 78,000 customers from a large discount brokerage firm. They divide the customers into three categories: active traders (those who made more than 48 trades per year, comprising 7.7 percent of the total), affluents (those with more than $100,000 in equity but with fewer than 48 trades), and generals (everyone else). Their results indicate that the investors who trade the most have both the lowest gross return and the lowest net return. Carhart (1997) reexamines the issue of consistency in mutual fund performance in a manner similar to Elton, Gruber, Das, and Hlavka. He finds that the evidence of persistence in relative performance is concentrated at two extremes across active managers. However, much of that persistence is due to expenses and transaction costs rather than gross investment return. Carhart s findings suggest that there may be a small group of active managers who can with some consistency outperform a passive portfolio, and they do so by having fewer expenses and transactions costs. While a benchmark portfolio may be ex post inefficient relative to the optimal ex post tangency portfolio constructed from its constituent securities, it is not certain that the optimal ex ante tangency

5 portfolio will provide a higher Sharpe measure of performance than the benchmark. Much depends on the ex ante estimation of the portfolio input parameters under uncertainty. Capital market theory suggests that a well-diversified, value-weighted benchmark portfolio and the ex ante optimal tangency portfolio should exhibit a similar Sharpe measure of portfolio performance. Theory also suggests that an active manager will only be able to construct an ex ante portfolio with a higher expected return than the benchmark through the use of leverage. Leverage allows the active manager to extend the expected return on the optimal ex ante investment portfolio, while maintaining a constant Sharpe measure. If leverage is a constraint for the active manager, which is often the case, then strict return enhancement will be difficult to achieve. Modern portfolio theory (MPT) dictates that the lower the degree of pairwise correlations that exists between pairs of securities, the greater is the potential for Sharpe measure enhancement from ex ante optimization. Thus, when securities exhibit a high degree of pairwise correlation, ex ante optimization should provide little in the way of an enhanced Sharpe performance measure. It is, however, an empirical question whether ex ante return estimation and optimization techniques can provide strict return enhancement in risk-return space. Strict return enhancement, without the use of leverage, may be possible if a return estimation technique provides for securities with higher realized returns to carry larger weights in the ex ante optimal portfolio than in the benchmark portfolio. The purpose of this study to empirically investigate the potential for various ex ante portfolio parameter estimation techniques and optimization/holding period frequency intervals to enhance managed portfolio returns relative to a benchmark, while maintaining the benchmark s Sharpe measure of performance. Section 2 reviews portfolio optimization and describes the various parameter estimation techniques used in the study. Section 3 outlines the data and methodology. Section 4

6 outlines the results. Section 5 provides a summary and conclusion. 2. Portfolio Optimization and Parameter Estimation Techniques Portfolio optimization over the N size stock portfolios results in a (N x 1) investment weight vector X, where the elements X i, i=1,...,n, represent the optimal investment weights for the N portfolios (i= N denotes the number of portfolios). The optimization is a quadratic programming problem that can be stated as: Max X T µ/x T VX, (1) subject to: ΣX i = 1.0 and X i 0 for i = 1,...,N where µ is the (N x 1) vector of expected excess returns on the N portfolios. In this study, the ex ante portfolio input parameter estimation techniques developed by Jobson and Korkie (1980, 1981) and Jorion (1985, 1986) are employed. This literature has established that the expected return vector is the critical input for successfully implementing modern portfolio theory, i.e., identifying the ex ante optimal investment weights. Conventional estimation of the variance-covariance matrix works well. Let us examine the optimization strategy using the expected excess return equation µ = (1 - w^ )Y + w^ 1 Y o, (2) where Y is the (N x 1) ex post (historical) sample mean-return vector of the N portfolios that make up the benchmark portfolio, 1 is a vector of ones, Y o denotes the mean return from the ex post minimum-

7 variance portfolio, and w^ represents the estimated shrinkage factor for shrinking the elements of Y toward Y o. Equation (2) is a Bayes-Stein expression derived by Jorion for estimating the ex ante expected excess return vector to use in solving the portfolio problem. It is, however, general enough to encompass other models. If w^ = 0, the resulting vector of estimated expected returns contains the ex post classical sample means. These estimates result in identifying the weights of the ex post (or historical) tangency portfolio as the ex ante optimal investment weights. This method implicitly assumes there is no estimation risk in the classical sample estimates and it is labeled the certainty-equivalencetangency (CET) portfolio technique. A second method, which is due to the simulation results of Jobson and Korkie (1980, 1981), is to arbitrarily set w^ = 1. This technique identifies the optimal ex ante investment weights as those of the ex post minimum-variance portfolio (MVP). The MVP technique implicitly assumes that there is no useful asset-specific information in Y because it is not required as input to solve the portfolio problem. A third technique is the Bayes-Stein (BST) method developed by Jorion (1985, 1986), which uniquely estimates the shrinkage factor according to the equation: (N +2)(L - 1) w= ˆ, (3) -1 (N +2)(L -1)+(Y -Y 01) LV (L - N - 2)(Y - Y 01) where L represents the length of the time series of the sample observations and V is the usual (N x N) sample variance-covariance matrix of portfolio returns. Using Jorion s w^ in equation (2), the BST optimal ex ante tangency portfolio can be determined. When using a uniquely determined w^, equation (2) can potentially result in a uniform improvement on Y or Y o as estimates of the expected

8 return because it relies on a more general model that includes them both as special cases. To implement either the CET, MVP or the BST parameter estimation technique requires obtaining a historical time series sample of returns, R t+1 (i=1,..., N) to calculate the Y, Y 0, V, and w^ necessary to calculate the expected excess return vector specified by equation (2). In the next section, we test three portfolio parameter estimation techniques: i) CET ii) iii) MVP, and BST. Reinganum (1983) and Banz (1981) have shown that portfolios of small market capitalization firms have outperformed the stock indices consisting of both large and small capitalization firms. The CET and BST ex ante return estimation techniques both use information in the historical returns. Given this fact, it is possible that the CET and BST techniques may permit the ex ante optimal portfolio to place more weight on the smaller "size based" portfolios relative to the benchmark thereby enhancing overall return performance. Because the MVP return estimation technique estimates each security s expected return as a common value, small capitalization firms are not distinguishable from other size firms in the optimization on the basis of expected return, and thus the MVP estimation procedure is likely to fail to enhance portfolio return performance. We also test three optimization/holding periods: i) 3-month ii) iii) 9-month, and 12-month.

9 The optimization/holding period frequency necessary to promote enhanced returns is an important issue because it is directly related to management costs. Larsen and Resnick (1998) empirically demonstrate that when managing a portfolio designed to track a value-weighted index, frequent optimization may not be necessary. The performance results are compared with one another and with the Center for Research in Security Prices (CRSP) Value-Weighted (VW) index as the benchmark portfolio. 3. Data and Methodology We use the CRSP monthly return data 10 value-weighted, size-based market capitalization portfolios, P1 (smallest) to P10 (largest), and the CRSP Value-Weighted (VW) Index. These portfolios conform to the value-weight constraint of capital market theory. As previously mentioned, the potential for return enhancement and a stable Sharpe measure is examined using size-based constituent portfolios. This study uses a time series of 588 months of return data for each size portfolio and index that spans the time period through The first 60 months of returns are used in the initial portfolio parameter estimation period, leaving 528 out-of-sample months over which performance measures are compared. Table I provides a summary of the CRSP return data over the entire 528 months out-of-sample period. The table shows the pairwise correlation coefficients between size portfolio pairs and the mean return and standard deviation of the CRSP VW Index. The table indicates that the size portfolio pairs have relatively high positive pairwise correlation with one another. The lowest pairwise correlation of.62 is between the smallest size portfolio, P1, and the largest size portfolio, P10. The portfolio returns

10 appear to satisfy our requirement for relative high positive pairwise correlations between the smaller portfolios that make up the benchmark. For each out-of-sample performance test, the previous 60 corresponding monthly returns for each of the 10 sizes stock portfolios are used to estimate the input parameters to solve the ex ante optimal investment weights for each optimization technique. Various out-of-sample holding periods are then used for performance testing. A particular ex ante parameter estimation technique and optimization/holding period combination is referred to as a strategy. For example, if the holding period is 12 months, for the first holding period covering months 61 through 72, the estimation period covers months 1 through 60. For the second holding period covering months 73 through 84, the estimation period covers months 13 through 72. Each subsequent pair of estimation and holding periods is shifted forward in time by 12 months in this case. The performance of each strategy is measured by the Sharpe (1966) reward-to-variability measure of portfolio performance and the average return. (2) Additionally, dominance analysis is performed on both the Sharpe and average return measures. The dominance analysis shows the number in times the out-of-sample holding periods that one strategy has a larger average Sharpe or average return measure than each other strategy. A strategy is said to dominate another strategy if the former strategy has a larger Sharpe measure or return value than the latter in at least 50 percent of the out-of-sample holding periods. (3) Jobson and Korkie (1981b) have developed a z-statistic for determining a statistical difference in Sharpe measures is not very powerful in small samples. We therefore rely on the dominance test to determine the relative performance of each technique. 4. Average Performance Results

11 Table II presents the average performance results of the ex ante enhancement strategies. Both CET and BST techniques provide higher returns at much the same return/risk levels. For the 3-month holding period, the average VW return is 1.04% and the average Sharpe measure is By comparison, over the same 3-month optimization/holding period, the CET (BST) technique produces an average return and Sharpe measure of 1.24 (1.22) and 0.62 (0.61). For the 6-month optimization/holding period, the average VW return is 1.04% and its average Sharpe measure is The corresponding 6-month results for the CET (BST) technique are an average return and Sharpe measure of 1.31 (1.26) and 0.42 (0.44). The 12-month holding period produces an average VW return of 1.04% and average Sharpe measure of The corresponding 12-month results for the CET (BST) technique are an average return of 1.33 (1.22) and average Sharpe measure of 0.35 (0.34). In summary, the average returns of the CET and BST techniques are substantially higher than for the VW Index while their average Sharpe measures are very similar in magnitude to those of VW benchmark. Table III presents a dominance analysis comparing all investment strategies under each estimation technique to the CRSP VW benchmark. A number in the table denotes the number of times in the outof-sample holding periods that the row strategy has a larger Sharpe value than the strategy at the top of the table. A row strategy is said to dominate the strategy at the top if it has a larger Sharpe value in more than 50% of the out-of-sample holding periods. The Sharpe dominance results from the different techniques show little variability. For example, when using a 3-month optimization/holding period interval, the CET, MVP, and BST strategies dominate the VW index 81, 83 and 87 times, respectively, out of 176 out-of-sample tests, all close to but not exceeding 50 percent. Similar results hold when using a 12-month optimization/holding period. However, for the 88 6-month out-of-sample holding

12 periods, the CET and BST strategies dominate the VW index 45 and 47 times, respectively. The high positive pairwise correlations between the constituent size-based portfolios is reason why the ex ante strategies do not produce a marked improvement in Sharpe measures relative to the benchmark. The critical issue, however, is still the potential for return enhancement. Table IV presents a dominance analysis of average returns under each strategy relative to the VW benchmark. The format of the table is analogous to Table III. Table IV indicates that the BST technique dominates or equals the return performance of the VW benchmark for all optimization/holding period frequencies: 90 out of 176 times in the 3-month tests, 44 out of 88 times in the 6-month, and 23 out of 44 in the 12-month period. The CET technique produces excellent performance with both the 6- months and 12-month optimization/holding periods, beating the VW benchmark in 52 out of 88 holding periods and 25 out of 44 holding periods, respectively. The MVP strategy does not dominate the VW benchmark in any of the optimization/holding periods. Overall, the results suggest that using an optimization/holding period of less than 6 months is not helpful in enhancing returns in risk-return space relative to the VW benchmark. Table V presents the average portfolio investment weights and their standard deviations for the constituent portfolios for the out-of-sample tests from employing each of the three parameter estimation techniques. The table indicates that sized-based portfolio P10 has the largest average investment weight under all techniques and optimization/holding periods. For example, when using the BST estimation technique and a 6-month optimization/holding period, the average investment weight for P10 is.388 and.143 for P1. More importantly, it should be noted that the CET and BST techniques tend to weight the smallest size portfolio, P1, more heavily in the ex ante optimal portfolios than does the MVP technique. The larger average weight assigned to P1 by the CET and BST techniques is the reason for the higher

13 average portfolio returns produced by these strategies and implies that historical returns contain useful information when securities are stratified by market capitalization. This useful information is lost by the MVP technique which estimates each expected return the same; consequently, in the ex ante optimization less investment will be made in the high return, sized-based portfolio P1. 5. Summary and Conclusion This study empirically examines whether, when using parameter inputs estimated from historical data, if attempts to identify the ex ante optimal portfolio weights result in enhanced returns in comparison to the CRSP VW benchmark portfolio. Three techniques for controlling estimation risk are used: a CET technique, a MVP technique due to Jobson and Korkie (1980, 1981), and a Bayes-Stein (BST) technique derived by Jorion (1985, 1986). Three optimization/holding period lengths are tested: 3- month, 6-month, and 12-month. Combinations of techniques and optimization/holding periods are referred to as strategies. Optimization is performed over size-based portfolios created from all securities contained in the benchmark. Overall, the results suggest that enhanced returns can be consistently achieved at much the same Sharpe measure of performance as the benchmark portfolio by optimizing across constituent portfolios that are stratified by some measure that provides return persistence (in this case size, but "sector" might also work) and by using return estimation techniques that can pick up on this return persistence. The most promising results obtain using the CET strategy. This strategy provides average returns that are 20 to 30 percent higher than the VW benchmark while matching or exceeding the benchmark s Sharpe measure. Consistent enhanced return results also obtain when using the BST strategy. In particular,

14 using a 6-month optimization/holding period interval dominates using 3-month and 12-month intervals.

15 NOTES (1) In creating a synthetic index fund, it is assumed that the futures contract is fairly priced. Suppose, instead, that the stock index futures price is less than the theoretical futures price. If that situation occurs, the index fund manager can enhance the indexed portfolio s return by buying the futures and buying Treasury-bills. That is, the return on the futures/treasury-bill portfolio will be greater than that on the underlying index when the position is held to the settlement date. Alternatively, if the futures contract is expensive based on its theoretical price, an index fund manager who owns stock index futures and Treasury-bills will swap that portfolio for the stocks in the index. (2) In calculating the ex ante optimal investment weights for each strategy and the resulting Sharpe measures, the monthly risk-free rate is assumed to be zero. A positive risk-free locates the tangency portfolio higher up on the efficient frontier where fewer assets are likely to make up the optimal portfolio. Thus, as Jorion (1985) notes, a positive risk-free rate would accentuate any undesirable characteristics of the tangency portfolio. Using an assumed zero rate leads to a conservative measure of the effect of estimation risk on all assets. Eun and Resnick (1988, 1994) and Larsen and Resnick (1999) also assume a zero risk-free rate. (3) Professional money managers are often concerned about the consistency of a strategy as well as magnitude. A strategy that provides statistically superior performance, but due to only a few high value periods, may not be preferred to a strategy that provides enhanced performance of lesser magnitude, but over a majority of periods.

16 REFERENCES Banz, R., The Relationship between Return and Market Value of Common Stock, Journal of Financial Economics, March, Barber, Brad M., and Terrance Odean, "The Common Stock Investment Performance of Individual Investors," working paper, Graduate School of Management, University of California, Davis, Carhart, M., 1997, On Persistence in Mutual Fund Performance, Journal of Finance 52, Dumas, B. and B. Jacquillat, 1990, Performance of Currency Portfolios Chosen by a Bayesian Technique: , Journal of Banking and Finance 14, Elton, E. J., M. J. Gruber, S. Das, and M. Hlavka, 1993, Efficiency with Costly Information: A Reinterpretation of Evidence from Managed Portfolios, Review of Financial Studies 1, Jensen, Michael C., 1968, The Performance of Mutual Funds in the Period , Journal of Finance 23, Jobson, J. and B. Korkie, 1980, Improved Estimation and Selection Rules for Markowitz Portfolios, Paper presented at the Western Finance Association meeting. Jobson, J. and B. Korkie, 1981, Putting Markowitz Theory to Work, Journal of Portfolio Management 7, Jobson, J. and B. Korkie, 1981, Performance Hypothesis Testing with the Sharpe and Treynor Measures, Journal of Finance 36, Jorion, P., 1985, International Portfolio Diversification with Estimation Risk, Journal of Business 58, Jorion, P., 1986, Bayes-Stein Estimation for Portfolio Analysis, Journal of Financial and Quantitative Analysis 21, Jorion, P. and S. Khoury, 1996, Financial Risk Management (Cambridge, MA: Blackwell). Larsen, G. and B. Resnick, 1998, Empirical Insights on Indexing, The Journal of Portfolio Management 25, Loftus, J., 1997, Enhanced Equity Indexing, Chapter 4 in Professional Perspectives on Indexing, Frank J. Fabozzi, ed. (New Hope, PA: Frank J. Fabozzi Associates), 34. Markowitz, H., 1952, Portfolio Selection, Journal of Finance 7,

17 Reinganum, M., 1983, Portfolio Strategies for Small Caps vs. Large, Journal of Portfolio Management, Winter, Sharpe, W., 1966, Mutual Fund Performance, Journal of Business, A Supplement, No. 1, Part 2, Tobin, J., 1958, Liquidity Preference as Behavior Towards Risk, The Review of Economic Studies 26,

18 Table I Summary Statistics of the Monthly Returns: (a) VW EW P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 MN SD VW % 4.15% EW % 4.91% P % 6.48% P % 5.68% P % 5.43% P % 5.18% P % 5.02% P % 4.92% P % 4.75% P % 4.66% P % 4.43% P % 4.06% (a) The upper-right triangle provides the correlation matrix. MN and SD, respectively, denote the mean return and standard deviation of returns over the 528 month period. While the entire data set covers the 588 month period , the first 60 months of data are used to start the parameter estimation procedures.

19 Table II Average Performance Results of the Ex Ante Investment Strategies(a) Optimization (Holding) Period Ex Ante Strategy 3 month 6 month 12 month CET MN(%) SD(%) SHPb MVP MN(%) SD(%) SHPb BST MN(%) SD(%) SHPb VW MN(%) SD(%) SHPb (a)in each cell, the three numbers represent the average of out-of-sample values. MN, SD, and SHP respectively, denote the mean portfolio return, portfolio standard deviation and Sharpe reward-to-variability ratio. (b)the SHP ratios are calculated over the 3, 6, and 12 month optimization periods. Compare only within periods and not across periods.

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21 Table III Sharp Ratio Dominance Analysis of the Out-of-Sample Performance(a) Optimization period 3 month (176)(b) 6 month (88)(b) 12 month (44)(b) Technique Technique Technique VW CET MVP BST VW CET MVP BST VW CET MVP BST VW VW VW CET CET CET MVP MVP MVP BST BST BST (a) A number in the table represents the number of times, out of ex ante test periods, that the left-hand-side strategy had a larger out-of-sample reward-to-variability ratio than the strategy at the top. For example, the 6 month BST strategy had a larger reward-to-variability ratio than did the 6 month MVP strategy in 47 out of 88 out-of-sample tests. A left-hand-side strategy is said to dominate the strategy at the top if it has a larger Sharpe value in more than 50% of the out-of-sample holding periods. (b) The number in parentheses indicates the number of ex ante test periods in each optimization (holding) period. VW - The benchmark CRSP Value Weight Index.

22 Table IV Return Dominance Analysis of the Out-of-Sample Performance (a) Optimization period 3 month (176) (b) 6 month (88) (b) 12 month (44) (b) Technique Technique Technique VW CET MVP BST VW CET MVP BST VW CET MVP BST VW VW VW CET CET CET MVP MVP MVP BST BST BST (a) A number in the table represents the number of times, out of ex ante test periods, that the left-hand-side strategy had a larger out-of-sample average monthly return than the strategy at the top. For example, the 12 month BST strategy had a larger average monthly return than did the 12 month CET strategy in 26 out of 44 out-of-sample tests. A left-hand-side strategy is said to dominate the strategy at the top if it has a larger average monthly return in more than 50% of the out-of-sample holding periods. (b) The number in parentheses indicates the number of ex ante test periods in each optimization (holding) period. (c) The benchmark CRSP Value Weight Index.

23 Optimization (Holding) period Table V Average Stock Investment Weights and Standard Deviations Estimation Technique P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 3 month CET MEAN STDEV MVP MEAN STDEV BST MEAN STDEV month CET MEAN STDEV MVP MEAN STDEV BST MEAN STDEV month CET MEAN STDEV MVP MEAN STDEV BST MEAN STDEV (a) The standard deviations of the investment weights for the out-of-sample tests appear in shaded areas.

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