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Further Analysis of Efficient Portfolios with the USER Data

is a quantitative research analyst at McKinley Capital Management, LLC, in Anchorage, Alaska. [email protected]

M anish Kumar

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TaR optimization technique emphasizes systematic risk rather than total risk in portfolio optimization. We test two portfolio construction models—MV and EAW—and two portfolio optimization techniques—MV and TaR. To address these issues, we construct efficient portfolios with the USER data, solving for security weights using MV and EAW port­ folio construction models for the 1997–2009 period. The MV portfolio construction models with fixed security the upper bound perform very well compared with the EAW portfolio construction models. MV portfolios with a 4% security upper bound outperform the EAW 1%, 2%, and 3% strategies. One must use the EAW 4% and EAW 5% strategies to outperform the MV portfolio construction model with a 4% upper bound. We find that EIT portfolio construction models are extremely useful if a manager is more concerned with underperforming an index; however, the portfolio manager must be aggressive with the EAW strategy to outperform a traditional MV portfolio construction strategy. We employ MV and TaR optimization techniques to test whether the EAW strategies of 1%, 2%, 3%, 4%, and 5% (weight deviations from the index, or benchmark, weights) outperform MV strategies using 4% and 7% maximum security weights. We will show that MV portfolios using the TaR optimization technique outperform the MV optimization

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is a quantitative research analyst at McKinley Capital Management, LLC, in Anchorage, Alaska. [email protected]

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Eli K rauklis

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n this study, we address two issues in portfolio construction and management with the Guerard, Gultekin, and Xu [2012] U.S. Expected Return (USER) data. First, we test the issue of whether a Markowitz mean–variance (MV) portfolio construction model [1956, 1959, 1987] with a fixed upper bound on security weights dominates the Markowitz enhanced indextracking (EIT) portfolio construction model [1987] in which security weights are an absolute deviation from the security weight in the index. We shall refer to the absolute deviation from the benchmark weight, the enhanced index portfolio construction weight, as the equal active weighting (EAW) portfolio construction model. The Markowitz EIT model is similar to the Mean-Absolute portfolio optimization model by Konno and Yamazaki [1991]. The security weights are the primary decision variables to be solved in efficient portfolios. Second, using portfolio variance as the relevant risk measure and the Blin– Bender Advanced Portfolio Technologies (APT) tracking error at risk (TaR) optimization technique that emphasizes systematic, or market risk, we test whether a (traditional) mean–variance optimization technique dominates the risk–return trade-off curve. The APT measure of portfolio tracking error at risk, TaR, estimates the magnitude that the portfolio return may deviate from the benchmark return over one year. Specifically, the

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is the director of quantitative research at McKinley Capital Management, LLC, in Anchorage, Alaska. [email protected]

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John B. Guerard, Jr.,

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John B. Guerard, Jr., Eli Krauklis, and Manish Kumar

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where Σ iN=1w i = 1. The security weights summing to one indicates that the portfolios are fully invested. The Markowitz framework measured risk as the portfolio standard deviation, its measure of dispersion, or total risk. One seeks to minimize risk, as measured by the covariance matrix in the Markowitz framework, and hold constant expected returns. Elton et al. [2007] write a more modern version of the traditional Markowitz mean–variance problem as a maximization problem:

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Constructing Efficient Portfolios

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The Markowitz portfolio construction approach seeks to identify the efficient frontier, the point at which returns are maximized for a given level of risk or risk is minimized for a given level of return. For the seminal discussion of portfolio construction and management, see Markowitz [1959]. The portfolio expected return, E(Rp ), is calculated by taking the sum of the security weights, w, multiplied by their respective expected returns. The portfolio standard deviation is the sum of weighted security covariances. N



E(Rp ) = ∑ w i E(Ri ) i =1

N





(1)

N

σ 2p = ∑ ∑ w i w j σ ij i =1 j =1



(2)

E( R p ) − R F σ 2p

(3)

where Σ iN=1w i = 1, and N

N

N

i =1

i =1 j =1

σ 2p = ∑ w i2 σ i2 + ∑ ∑ w i w j σ ij i≠ j

and R F is the risk-free rate (90-day T-bill yield). The optimal portfolio weights are given by:

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technique during the 1997–2009 period. Both optimization techniques produce statistically significant asset selection. An alternative approach to the TaR optimization technique is the systematic tracking error optimization technique reported by Wormald and van der Merwe [2012]. We apply data mining corrections model testing to examine whether the USER model is statistically significantly different from the average of a set of frequently used quantitative models. We show that the data mining corrections test must be used with an appropriate lambda. Lambda is a measure of the trade-off between expected return and risk, as measured by the portfolio standard deviation. Generally, the higher the lambda, the higher the expected ratio of expected return to standard deviation. That is, creating portfolios with less than optimal lambdas produces portfolio excess returns that are not statistically different from zero, whereas appropriate lambdas create portfolios that are statistically significant. In the King’s English, benchmark-hugging portfolio construction techniques can significantly destroy asset selection. We assume that the portfolio manager seeks to maximize the combination of portfolio geometric mean (GM), Sharpe ratio (ShR), information ratio (IR), and asset selection in the Barra attribution analysis. If a portfolio manager has models that produce a slightly different ordering of these criteria, we maximize the GM (Latane [1959]) as the ultimate criteria, since it is well known that risk is implicit in the GM (Markowitz, Chap. 9 [1959]).

∂θ =0 ∂w i

As in the initial Markowitz analysis, one minimizes risk by setting the partial derivative of the portfolio risk with respect to the security weights—the portfolio decision variables—to zero. Thus, we may write the Elton–Gruber version of the efficient frontier as: ∂θ = −( λw1σ 1 j + λw 2 σ 2i + λw 3 σ 3i + λw i σ i2 + λw N σ Ni ) ∂w i + R i − RF = 0

(4)

Equation 4 has become the standard, modern MBA Markowitz formulation. Modern portfolio theory evolved with the introduction of the capital asset pricing model (CAPM). Implicit in the development of the CAPM by Sharpe [1964], Lintner [1965], and Mossin [1966] is that investors are compensated for bearing systematic (market) risk, not total risk. Systematic risk is measured by the stock

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beta. The beta is the slope of the market model in which the stock return is regressed as a function of the market return.1 An investor is not compensated for bearing risk that may be diversified away from the portfolio. The CAPM holds that the return to a security is a function of the security’s beta.

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Extensions to the Traditional Mean–Variance Model

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A second extension to the mean–variance approach involves the minimization of the tracking error against an index. Markowitz [1987, 2000] rewrites the general portfolio construction model variance, V, to be minimized as:

where

Rjt = expected security return at time t; E(R Mt ) = expected return on the market at time t; R F = risk-free rate; βj = security beta; and ej = randomly distributed error term.

Var(RM )

(6)

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where W T = (W1, …, Wn ) = the weights of an index, X are the portfolio weights, and r T = (r1, …, rn ) = security returns. One can create portfolios by allowing portfolio weights to differ from index weights by plus or minus 1%, EAW1, 2%, EAW2, 3%, EAW3, 4%, EAW4, or 5%, EAW5. Obviously, one can use an infinite set of EAW variations. We restrict this analysis to EAW1, EAW2, EAW3, EAW4, and EAW5 for simplicity.

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The difficulty of measuring beta and its corresponding security market line (SML) gave rise to extramarket measures of risk found in the work of Rosenberg [1974], Rosenberg and Marathe [1979], Ross [1976], and Ross and Roll [1980].2 The fundamentally based domestic Barra risk model was developed in the series of studies by Rosenberg and thoroughly discussed in Rudd and Clasing [1982] and Grinhold and Kahn [1999]. The total excess return for a multiple-factor model (MFM) in the Rosenberg methodology for security j, at time t, dropping the subscript t for time, may be written like this:

(8)

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Cov(R j , RM )

V = (X – W )T C(X – W )



An examination of the CAPM beta, its measure of systematic risk, from the capital market line equilibrium condition, follows. βj =

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Rjt = R F + βj [E(R Mt ) – R F ] + ej



based on statistically estimated (orthogonal) principal components, as described in the APT model of Blin, Bender, and Guerard [1997].

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E(R j ) = ∑ β jk fk + e j



(7)

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The nonfactor, or asset-specific return on security j, is the residual risk of the security after removing the estimated impacts of the K factors. The term f is the rate of return on factor “k.” A single-factor model, in which the market return is the only estimated factor, is obviously the basis of the CAPM. Accurate characterization of portfolio risk requires an accurate estimate of the covariance matrix of security returns. An alternative to the fundamentally based Barra risk model is a risk model

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Portfolio Construction, Management, and Analysis

The USER simulation conditions are identical to those described in Guerard, Gultekin, and Xu [2012]. We use monthly optimization with 8% turnover, and 125 basis points, each way, of transaction costs. We use the APT risk model and optimizer described in Blin, Bender, and Guerard [1997] to create portfolios during the 1997–2009 period by varying the portfolio lambda. One seeks to maximize the GMs, ShRs, and IRs of portfolios. However, what if one wants to be considered a “concentrated portfolio manager” who does not hold 300–500 stocks? How many securities should one employ in portfolios using MV and EAW construction models with a monthly set of 3,000 expected return and covariance data? Can a manager construct efficient portfolios of 3,000 stock universes with fewer than 100 securities in the portfolios? The answer is shown in Exhibit 1 below. It is well known that as one raises the portfolio lambda, the expected return of the portfolio rises and the number of securities in the optimal portfolio falls (see

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Exhibit 1

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USER Mean–Variance and EAW Portfolio Construction Models

in portfolios than the MV optimization technique? As shown in Exhibit 2, a lambda of 200 implies optimal portfolios of 99.7 (100) stocks with MV, whereas MVTaR requires only 77.8 (78) stocks. The Blin–Bender TaR optimization procedure allows a manager to use fewer stocks in his or her portfolios than would a traditional MV optimization technique manager for a given lambda. We produce an efficient frontier in Exhibit 3 by varying the portfolio lambdas. Note that EAW1, EAW2, and EAW3 TaR portfolios require more stocks than MVTaR and are statistically dominated in the risk–return trade-off. One must move to EAW4 and EAW5 strategies to outperform MVTaR models. Moreover, the USER EAW1 curve in Exhibit 3 shows no risk–return trade-off. An investor would be hardpressed to outperform if he or she used an EAW1 strategy (unless he or she managed an index-­enhanced product). The GMs, ShRs, and IRs for MV and MVTaR support the use of lambda 200 and the MVTaR approach, as shown in Exhibit 4. The Zephyr style analysis shown in Exhibit 5 illustrates the robust regression-weighted expected return model plots on the growth (right) side of the Zephyr box. USER can be used to manage portfolios seeking to outperform a growth benchmark (or the S&P 500 Index, as stated in Guerard, Gultekin, and Xu [2012]). As one increases the USER portfolio construction lambda, the portfolios contain more securities with smaller market capitalizations.

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Blin, Bender, and Guerard [1997]). Lambda—a measure of risk acceptance, the inverse of the risk-aversion acceptance level of the Barra system—is a decision variable to determine the optimal number of securities in a portfolio. Guerard, Chettiappan, and Xu [2010] report a lambda of 200 that maximized the GM in non-U.S. growth portfolios. We find that a lambda of 200 is necessary with MV, EAW3, EAW4, and EAW5 portfolio construction models for the USER data to create portfolios with fewer than 100 securities. Does the use of the TaR optimization technique produce a higher or lower number of average securities

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Source: Sungard APT output files, June 2011

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Exhibit 2

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USER Model Average Number of Securities in Optimal Portfolios

Source: Sungard APT output files, June 2011.

Data Mining Corrections

Markowitz and Xu [1994] developed a methodology that sought to determine the level of expected future outperformance of models. The Markowitz and Xu data mining corrections (DMC) test assumes that the T period historic returns are identically and independently distributed (i.i.d.). The DMC assumes that future returns are drawn from the same population (also i.i.d.).

gb = loge (1 + GM b )

(9)

Since portfolio managers test many models, not just the best model, the GM is no longer the best unbiased estimate of the true, underlying population portfolio return.

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Exhibit 3

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USER TaR MV, EAW Strategies, 1/1997–12/2009

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Source: Sungard APT output files, June 2010

Exhibit 4

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Optimal Portfolio Risk–Return Summary Statistics

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Source: Sungard APT output files

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Markowitz and Xu [1994] set yit as the logarithm of one plus the return for the ith portfolio selection model in period t; yit is of the form yt = µi + zt + εit



In Markowitz and Xu [1994] Model I, it is assumed that the portfolio return is observable and assumed to be a function of the return of a market index. Moreover, Model I returns are assumed to be independent of time and other error terms of other models. Markowitz and Xu proposed an alternative model, Model III, in which model covariances need not be zero. That is, portfolio model returns may be correlated. In Model III,

(10)

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(11)

Markowitz and Xu did not require the models to be independent of one another. Thus, Model III is not only the general case (Model I being a special case of Model III), but Model III is consistent with testing in business-world portfolio construction and testing. Finally, the appropriate estimate of µit in Model I is not the average return

where µi is a model effect, zt is a period effect, and εit is a random deviation.

yit = µi + εit ,



∑ r= i

T

y

t −1 it

T



(12)

but rather The Journal of Investing    5

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Exhibit 5

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Zephyr Style Analysis

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r = ∑ ri /n i =1

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Source: Date: June 2011.

(13)



µˆ = r + β ( ri − r )

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The estimate of µi is regressed back to the average return (the grand average), (14)



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where 0 < β < 1. The best linear estimate of the unknown µi, is µˆ i = E µ + β ( ri − E µ ) β=



cov( ri , µ ) Var( ri )



(15) (16)



Thus, β is the regression coefficient of µi as a function of ri. Does the return of the best model deviate (sta-

tistically significantly) from the average model return? The best linear unbiased estimate of the expected model return vector, µ, is our estimate of the percentage of expected out performance that should be continued with future use of the model. The Markowitz-Xu DMC test did not use a “holdout period,” as they can be routinely data mined as well.3 Guerard, Gultekin, and Xu [2012] report a USER variable DMC coefficient of 0.74, which is highly statistically significant, having an F-value of 3.791. Thus, one could expect 74% of the excess returns of the USER model relative to the average return to be continued. More importantly, the USER model produces a higher GM than the average model GM (of the 16 models) that could have been used to manage a U.S. equity portfolio during the January 1998–December 2009 period. A lambda of 200 produces a DMC coefficient

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of 0.70 (F = 3.50) for the January 1998–December 2009, whereas a lambda of 5 produces a DMC coefficient of 0.22 (F = 0.60) for the same period. The use of a lower lambda leads the asset manager to destroy statistically significant asset selection.

Bloch, M., J. Guerard, H. Markowitz, P. Todd, and G. Xu. “A Comparison of Some Aspects of the U.S. and Japanese Equity Markets.” Japan and the World Economy, 5 (1993), pp. 3-26. Elton, E.J., M.J. Gruber, and M.N. Gültekin. “Expectations and Share Prices.” Management Science, 27 (1981), pp. 975-987.

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Summary and Conclusions

Elton, E.J., M.J. Gruber, S.J. Brown, and W.N. Goetzman. Modern Portfolio Theory and Investment Analysis. John Wiley & Sons, Inc., 7th ed., 2007.

The USER data produces statistically significant asset selection in the most widely used portfolio construction and management systems during the 1998– 2009 period. The Markowitz MV portfolio construction model dominates the EAW model at weighting differences of less than 4%. One maximizes the GM by using TaR optimization techniques. The Markowitz–Xu DMC test allows one to reject data mining at appropriate levels of lambda.

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Guerard, J.B., S. Chettiappan, and G. Xu. “Stock-Selection Modeling and Data Mining Corrections: Long-Only Versus 130/30 Models.” In Handbook of Portfolio Construction: Contemporary Applications of Markowitz Techniques, edited by J.B. Guerard, Springer, 2010.

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The authors appreciate the comments and suggestions of Professor James Vander Weide. Gunlin Xu verified the DMC calculation. Any errors remaining are the responsibility of the authors. 1 Harry Markowitz often (always) reminds his audiences and readers that he discussed the possibility of looking at security returns relative to index returns in Portfolio Selection: Efficient Diversification of Investment (Chapter 4, footnote 1, p. 100 [1959]). 2 The reader is referred to Chapter 2 of Guerard [2010] for a history of multiindex and multifactor risk models. 3 Markowitz and Xu [1994] tested the DPOS strategies in Bloch et al. [1993], and the best model produced a Model III β of 0.59, which was statistically significant. The MQ variable passes the DMC test criteria for both U.S. and non-U.S. markets, indicating that the stock selection and portfolio construction methodologies produce superior returns that are not due to chance. The MQ variable, when compared to the average of most models shown in Table 4 of Markowitz and Xu [1994], has a DMC coefficient of 0.47 and is highly statistically significant, having an F-value of 1.872. Thus, one could expect 47% of the excess returns of the MQ model relative to the average return to be continued.

Guerard, J.B., Jr., ed. Handbook of Portfolio Construction: Contemporary Applications of Markowitz Techniques, New York: Springer, 2010.

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Endnotes

Grinhold, R. and R. Kahn. Active Portfolio Management New York: McGraw-Hill/Irwin, 1999.

References

Blin, J.M., S. Bender, and J.B. Guerard, Jr. “Earnings Forecasts, Revisions and Momentum in the Estimation of Efficient Market-Neutral Japanese and U.S. Portfolios.” In Research in Finance 15, edited by A. Chen, 1997.

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Guerard, J.B., Jr., M.N. Gultekin, and G. Xu. “Investing with Momentum: The Past, Present, and Future.” Journal of Investing, Spring 2012. Konno, H. and H. Yamazaki. “Mean-Absolute Deviation Portfolio Optimization Model and its Application to the Tokyo Stock Exchange,” Management Science, 37(1991), pp. 519-531. Latane, H.A. “Criteria for Choice Among Risky Ventures.” Journal of Political Economy, 67 (1959), pp. 144-155. ——. Tuttle, and C.P. Jones. Security Analysis and Portfolio Management, New York: The Roland Press. 2nd Edition, 1975. Lintner, J. “The Valuation of Risk Assets on the Selection of Risky Investments in Stock Portfolios and Capital Investments.” The Review of Economics and Statistics, Vol. 47, No. 1 (1965), pp. 13-37. Markowitz, H.M. “Portfolio Selection.” Journal of Finance, 7 (1952), pp. 77-91. ——. “The Optimization of a Quadratic Function Subject to Linear Constraints.” Naval Research Logistics Quarterly, 3 (1956), pp. 111-133. The Journal of Investing    7

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——. Portfolio Selection: Efficient Diversification of Investment. Cowles Foundation Monograph No. 16, New York: John Wiley & Sons, 1959.

Wormald, L., and E. van der Merwe. “Constrained Optimization for Portfolio Construction,” The Journal of Investing, Spring 2012.

——. Mean–Variance Analysis in Portfolio Choice and Capital Markets. Oxford: Basil Blackwell, 1987; New Hope, PA: Frank J. Fabozzi Associates, 2000.

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Markowitz, H.M., and G. Xu. “Data Mining Corrections: Simple and Plausible.” The Journal of Portfolio Management, Vol. 21 (Fall 1994), pp. 60-69.

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Rosenberg, B. “Extra-Market Components of Covariance in Security Returns.” Journal of Financial and Quantitative Analysis, 9 (1974), pp. 263-274.

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Mossin, J. “Equilibrium in a Capital Asset Market.” Econometrica, 34 (1966), pp. 768-783. Ramnath, S., S. Rock, and P. Shane. 2008. “The Financial Forecasting Literature: A Taxonomy with Suggestions for Further Research.” International Journal of Forecasting, 24, pp. 34-75.

General Disclaimer Statement McKinley Capital Management, LLC (“McKinley Capital”) is a registered investment adviser under the Securities and Exchange Commission Investment Advisers Act of 1940. One or multiple authors may be currently employed by McKinley Capital. The firm may not be held liable for comments made outside the scope of this presentation. The material provided herein has been prepared for institutional clients and financially knowledgeable and sophisticated high net-worth individuals. This material may contain confidential and/ or proprietary information and may only be relied upon for this report. The data is unaudited and may not correspond to similar calculations provided in other reports. This is not an offer to purchase or sell any security or service, is not ref lective of composite or individual portfolio ownership and may not be relied upon for investment purposes. Investments and commentary were based on information available at the time and are subject to change without notice. Any positive comments regarding specific securities may no longer be applicable and should not be relied up for investment purposes. No one security is profitable all of the time and there is always the possibility of selling it at a loss. No one formula or set of data will be 100% reliable or profitable at any given period. Past performance is not indicative of future returns. Investments are subject to immediate change without notice. Because McKinley Capital’s investment process is proprietary, composite returns and individual client returns may at various times materially differ from stated benchmarks. The author(s) may utilize a combination of firm specific and general publicly available sources and resources to prepare material or prove content. Charts, graphs and other visual presentations and text information are derived from internal, proprietary, and/or service vendor technology sources and/or may have been extracted from other firm data bases. As a result, the tabulation of certain reports may not precisely match other published data. Data may have originated from various sources including but not limid to Bloomberg, ClariFi, MSCI/Barra, Russell Indices, and/ or other systems and programs. Please refer to the MSCI/Barra, Russell Investments, and FTSE web sites for complete details on all related indices. Future investments may be made under different economic conditions, in different securities and using different investment strategies. Global investing also carries additional risks and/or costs including but not limited to, political, economic, financial market, currency exchange, liquidity, accounting, and trading capability risks. Shorting and derivatives may materially increase overall risk and negatively affect returns in the portfolio. Investors must consider total costs including management fees, expenses, brokerage commissions, custodial services, and individual tax considerations when estimating a potential return. Actual investment advisory fees incurred by institutional and high net-worth clients may vary. A fee schedule for McKinley Capital is described in Form ADV Part 2A, which can be obtained from: McKinley Capital Management, LLC, 3301 C Street, Suite 500, Anchorage, Alaska 99503, 1.907.563.4488 or via the firm’s website at: www.mckinleycapital.com. All information is believed to be correct but accuracy cannot be guaranteed. Please review the references section of this report for further details on locating complete source information.

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——. “Investment in the Long Run: New Evidence for an Old Rule.” Journal of Finance, 31 (1976), pp. 1273-1286.

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Rosenberg, B., and V. Marathe. “Tests of Capital Asset Pricing Hypotheses.” In Research in Finance 1, edited by H. Levy, 1979.

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Ross, S.A. “The Arbitrage Theory of Capital Asset Pricing.” Journal of Economic Theory, 13 (1976), pp. 341-360.

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Ross, S.A., and R. Roll. “An Empirical Investigation of the Arbitrage Pricing Theory.” Journal of Finance, 35 (1980), pp. 1071-1103.

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Rudd, A., and H.K. Clasing. Modern Portfolio Theory: The Principles of Investment Management. Homewood, IL: DowJones Irwin, 1982.

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Sharpe, W.F. “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk.” Journal of Finance, 19 (1964), pp. 425-442. Vander Weide, J.H. “Principles for Lifetime Portfolio Selection: Lessons from Portfolio Theory.” In Handbook of Portfolio Construction: Contemporary Applications of Markowitz Techniques. Edited by J.B. Guerard, Jr., New York: Springer, 2010.

To order reprints of this article, please contact Dewey Palmieri at dpalmieri@ iijournals.com or 212-224-3675.

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