Asset Classes and Portfolio Diversification: Evidence from a Stochastic Spanning Approach

Nguyen, Duc Khuong and Topaloglou, Nikolas and Walther, Thomas (2020): Asset Classes and Portfolio Diversification: Evidence from a Stochastic Spanning Approach.

Abstract

We propose a stochastic spanning approach to assess whether a traditional portfolio of stocks and bonds spans augmented portfolios including commodities, foreign exchange, and real estate. We empirically show that in all seven portfolio combinations, the augmented portfolio is not spanned by the traditional one. Our results are further confirmed by both parametric and non-parametric tests in an out-of-sample setting. Therefore, traditional investors can generally benefit in terms of higher Sharpe ratios from augmenting their portfolio with alternative asset classes. Additional analysis demonstrates that diversification benefits can be explained by the current state of the U.S. economy and stock markets.

• Item Type: MPRA Paper
• Language: English
• Keywords: Stochastic Dominance, Stochastic Spanning, Commodities, FX, Real Estate, Diversification
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General; C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling; E - Macroeconomics and Monetary Economics > E3 - Prices, Business Fluctuations, and Cycles > E32 - Business Fluctuations ; Cycles; G - Financial Economics > G1 - General Financial Markets > G10 - General; G - Financial Economics > G1 - General Financial Markets > G11 - Portfolio Choice ; Investment Decisions; G - Financial Economics > G1 - General Financial Markets > G12 - Asset Pricing ; Trading Volume ; Bond Interest Rates; G - Financial Economics > G1 - General Financial Markets > G15 - International Financial Markets

References

Abadie, A.: “Bootstrap Tests for Distributional Treatment Effects in Instrumental Variable Models.” Journal of the American Statistical Association, 97, (2002), 284–292.

Abanomey, W. S. and I. Mathur: “The Hedging Benefits of Commodity Futures in International Portfolio Diversification.” The Journal of Alternative Investments, 2, (1999), 51–62.

Adams, Z.; R. F¨uss; and D. G. Kaiser: “Macroeconomic Determinants of Commodity Futures Returns.” “The Handbook of Commodity Investing,” , F. J. Fabozzi; R. Füss; and D. G. Kaiser, eds., chapter 4, 87–112. Hoboken, NJ, USA: John Wiley & Sons, Inc. (2008).

Adams, Z. and T. Glück: “Financialization in commodity markets: A passing trend or the new normal?” Journal of Banking & Finance, 60, (2015), 93–111.

Ankrim, E. M. and C. R. Hensel: “Commodities in Asset Allocation: A Real-Asset Alternative to Real Estate?” Financial Analysts Journal, 49, (1993), 20–29.

Anson, M. J.: “Spot Returns, Roll Yield, and Diversification with Commodity Futures.” The Journal of Alternative Investments, 1, (1998), 16–32.

Anson, M. J.: “Maximizing Utility with Commodity Futures Diversification.” The Journal of Portfolio Management, 25, (1999), 86–94.

Arouri, M. E. H.; D. K. Nguyen; and K. Pukthuanthong: “Diversification benefits and strategic portfolio allocation across asset classes: the case of the US markets.” (2014). URL https://ideas.repec.org/p/ipg/wpaper/2014-294.html.

Arvanitis, S.; M. Hallam; T. Post; and N. Topaloglou: “Stochastic Spanning.” Journal of Business & Economic Statistics, 37, (2019), 573–585.

Baker, S. R.; N. Bloom; and S. J. Davis: “Measuring Economic Policy Uncertainty.” The Quarterly Journal of Economics, 131, (2016), 1593–1636.

Barrett, G. F. and S. G. Donald: “Consistent Tests for Stochastic Dominance.” Econometrica, 71, (2003), 71–104.

Basak, S. and A. Pavlova: “A Model of Financialization of Commodities.” Journal of Finance, 71, (2016), 1511–1556.

Bawa, V. S.: “Optimal, rules for ordering.” Journal of Financial Economics, 2, (1975), 95–121.

Bawa, V. S.; J. N. Bodurtha Jr.; M. R. Rao; and H. L. Suri: “On Determination of Stochastic Dominance Optimal Sets.” The Journal of Finance, 40, (1985), 417.

Belousova, J. and G. Dorfleitner: “On the diversification benefits of commodities from the perspective of euro investors.” Journal of Banking and Finance, 36, (2012), 2455–2472.

Berger, D.; K. Pukthuanthong; and J. J. Yang: “International diversification with frontier markets.” Journal of Financial Economics, 101, (2011), 227–242.

Berk, J. B.: “Necessary Conditions for the CAPM.” Journal of Economic Theory, 73, (1997), 245–257.

Bessler, W. and D. Wolff: “Do commodities add value in multi-asset portfolios? An out-of-sample analysis for different investment strategies.” Journal of Banking and Finance, 60, (2015), 1–20.

Bhardwaj, G.; G. Gorton; and G. Rouwenhorst: “Facts and Fantasies about Commodity Futures Ten Years Later.” (2015). URL http://www.nber.org/papers/w21243.pdf.

Bjornson, B. and C. A. Carter: “New Evidence on Agricultural Commodity Return Performance under Time-Varying Risk.” American Journal of Agricultural Economics, 79, (1997), 918–930.

Bodie, Z.: “Commodity futures as a hedge against inflation.” The Journal of Portfolio Management, 9, (1983), 12–17.

Bodie, Z. and V. I. Rosansky: “Risk and Return in Commodity Futures.” Financial Analysts Journal, 36, (1980), 27–39.

Boubaker, S.; D. Gounopoulos; D. K. Nguyen; and N. Paltalidis: “Reprint of: Assessing the effects of unconventional monetary policy and low interest rates on pension fund risk incentives.” Journal of Banking and Finance, 92, (2018), 340–357.

Brueggeman, W. B.; A. H. Chen; and T. G. Thibodeau: “Real Estate Investment Funds: Performance and Portfolio Considerations.” Real Estate Economics, 12, (1984), 333–354.

Burns, W. L. and D. R. Epley: “The performance of portfolios of REITs + stocks.” The Journal of Portfolio Management, 8, (1982), 37–42.

B¨uy¨uksahin, B.; M. S. Haigh; and M. A. Robe: “Commodities and Equities: Ever a Market of One?” The Journal of Alternative Investments, 12, (2009), 76–95.

Campbell, J. Y.; K. Serfaty-de Medeiros; and L. M. Viceira: “Global Currency Hedging.” The Journal of Finance, 65, (2010), 87–121.

Carrieri, F.; V. Errunza; and K. Hogan: “Characterizing World Market Integration through Time.” Journal of Financial and Quantitative Analysis, 42, (2007), 915–940.

Carroll, R.; T. Conlon; J. Cotter; and E. Salvador: “Asset allocation with correlation: A composite tradeoff.” European Journal of Operational Research, 262, (2017), 1164–1180.

Chamberlain, G.: “A characterization of the distributions that imply mean-Variance utility functions.” Journal of Economic Theory, 29, (1983), 185–201.

Chan, K. F.; S. Treepongkaruna; R. Brooks; and S. Gray: “Asset market linkages: Evidence from financial, commodity and real estate assets.” Journal of Banking and Finance, 35, (2011), 1415–1426.

Chandrashekaran, V.: “Time-Series Properties and Diversification Benefits of REIT Returns.” Journal of Real Estate Research, 17, (1999), 91–112.

Chen, H.-C.; K.-Y. Ho; C. Lu; and C.-H. Wu: “Real Estate Investment Trusts.” The Journal of Portfolio Management, 31, (2005), 46–54.

Cheung, C. S. and P. Miu: “Diversification benefits of commodity futures.” Journal of International Financial Markets, Institutions and Money, 20, (2010), 451–474.

Christoffersen, P.; V. Errunza; K. Jacobs; and H. Langlois: “Is the Potential for International Diversification Disappearing? A Dynamic Copula Approach.” Review of Financial Studies, 25, (2012), 3711–3751.

Cotter, J.; E. Eyiah-Donkor; and V. Pot`ı: “Predictability and diversification benefits of investing in commodity and currency futures.” International Review of Financial Analysis, 50, (2017), 52–66.

Daigler, R. T.; B. Dupoyet; and L. You: “Spicing Up a Portfolio with Commodity Futures: Still a Good Recipe?” The Journal of Alternative Investments, 19, (2017), 8–23.

Daskalaki, C. and G. Skiadopoulos: “Should investors include commodities in their portfolios after all? New evidence.” Journal of Banking and Finance, 35, (2011), 2606–2626.

Daskalaki, C.; G. Skiadopoulos; and N. Topaloglou: “Diversification benefits of commodities: A stochastic dominance efficiency approach.” Journal of Empirical Finance, 44, (2017), 250–269.

Davidson, R.: “Testing for Restricted Stochastic Dominance: Some Further Results.” Review of Economic Analysis, 1, (2009), 34–59.

Davidson, R. and J.-Y. Duclos: “Statistical Inference for Stochastic Dominance and for the Measurement of Poverty and Inequality.” Econometrica, 68, (2000), 1435–1464.

de Roon, F. A.; T. E. Nijman; and B. J. Werker: “Currency hedging for international stock portfolios: The usefulness of mean-variance analysis.” Journal of Banking & Finance, 27, (2003), 327–349.

DeMiguel, V.; L. Garlappi; and R. Uppal: “Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy?” Review of Financial Studies, 22, (2009), 1915–1953.

Demiralay, S.; S. Bayraci; and H. Gaye Gencer: “Time-varying diversification benefits of commodity futures.” Empirical Economics, 56, (2019), 1823–1853.

Edwards, F. R. and J. M. Park: “Do managed futures make good investments?” Journal of Futures Markets, 16, (1996), 475–517.

Elton, E. J.; M. J. Gruber; and J. C. Rentzler: “Professionally Managed, Publicly Traded Commodity Funds.” The Journal of Business, 60, (1987), 175.

Erb, C. B. and C. R. Harvey: “The Strategic and Tactical Value of Commodity Futures.” Financial Analysts Journal, 62, (2006), 69–97.

Eun, C. S. and B. G. Resnick: “Exchange Rate Uncertainty, Forward Contracts, and International Portfolio Selection.” The Journal of Finance, 43, (1988), 197–215.

Fischmar, D. and C. Peters: “Portfolio analysis of stocks, bonds, and managed futures using compromise stochastic dominance.” Journal of Futures Markets, 11, (1991), 259–270.

Fogler, H. R.: “20% in Real Estate: Can Theory Justify it?” The Journal of Portfolio Management, 10, (1984), 6–13.

Fortenbery, T. R. and R. J. Hauser: “Investment Potential of Agricultural Futures Contracts.” American Journal of Agricultural Economics, 72, (1990), 721.

Friedman, H. C.: “Real Estate Investment and Portfolio Theory.” The Journal of Financial and Quantitative Analysis, 6, (1971), 861– 874.

Froot, K. A.: “Hedging Portfolios with Real Assets.” The Journal of Portfolio Management, 21, (1995), 60–77.

Galvani, V. and A. Plourde: “Portfolio diversification in energy markets.” Energy Economics, 32, (2010), 257–268.

Gao, X. and F. Nardari: “Do Commodities Add Economic Value in Asset Allocation? New Evidence from Time-Varying Moments.” Journal of Financial and Quantitative Analysis, 53, (2018), 365–393.

Geman, H. and C. Kharoubi: “WTI crude oil Futures in portfolio diversification: The time-to-maturity effect.” Journal of Banking & Finance, 32, (2008), 2553–2559.

Georgiev, G.: “Benefits of Commodity Investment.” The Journal of Alternative Investments, 4, (2001), 40–48.

Georgiev, G.; B. Gupta; and T. Kunkel: “Benefits of Real Estate Investment: Some Diversification Benefit in particular Allocations.” The Journal of Portfolio Management, 29, (2003), 28–33.

Gibbons, M. R.; S. A. Ross; and J. Shanken: “A Test of the Efficiency of a Given Portfolio.” Econometrica, 57, (1989), 1121.

Gibson, R. C.: “The rewards of multiple-asset-class investing.” Journal of Financial Planning, 12, (1999), 50–59. Glen, J. and P. Jorion: “Currency Hedging for International Portfolios.” The Journal of Finance, 48, (1993), 1865–1886.

Goldstein, M. A. and E. F. Nelling: “REIT Return Behavior in Advancing and Declining Stock Markets. ”Real Estate Finance, 15, (1999), 68–77.

Gorton, G. and K. G. Rouwenhorst: “Facts and Fantasies about Commodity Futures.” Financial Analysts Journal, 62, (2006), 47–68.

Graham, M.; J. Kiviaho; and J. Nikkinen: “Short-term and long-term dependencies of the S&P 500 index and commodity prices.” Quantitative Finance, 13, (2013), 583–592.

Grauer, R. R. and N. H. Hakansson: “Gains From Diversifying Into Real Estate: Three Decades of Portfolio Returns Based on the Dynamic Investment Model.” Real Estate Economics, 23, (1995), 117–159.

Greer, R. J.: “Conservative Commodities: A Key Inflation Hedge.” The Journal of Portfolio Management, 4, (1978), 26–29.

Greer, R. J.: “Methods for Institutional Investment in Commodity Futures.” The Journal of Derivatives, 2, (1994), 28–36.

Greer, R. J.: “The Nature of Commodity Index Returns.” The Journal of Alternative Investments, 3, (2000), 45–52.

Halpern, P. and R. Warsager: “The Performance of Energy and Non-Energy Based Commodity Investment Vehicles in Periods of Inflation.” The Journal of Alternative Investments, 1, (1998), 75–81.

Harvey, C. R. and A. Siddique: “Conditional Skewness in Asset Pricing Tests.” The Journal of Finance, 55, (2000), 1263–1295.

Henriksen, T. E. S.: “Properties of Long/Short Commodity Indices in Stock and Bond Portfolios.” The Journal of Alternative Investments, 20, (2018), 51–68.

Henriksen, T. E. S.; A. Pichler; S. Westgaard; and S. Frydenberg: “Can commodities dominate stock and bond portfolios?” Annals of Operations Research, 282, (2019), 155–177.

Herbst, A. F.: “Gold versus U.S. Common Stocks: Some Evidence on Inflation Hedge Performance and Cyclical Behavior.” Financial Analysts Journal, 39, (1983), 66–74.

Huang, J. z. and Z. K. Zhong: “Time Variation in Diversification Benefits of Commodity, REITs, and TIPS.” Journal of Real Estate Finance and Economics, 46, (2013), 152–192.

Huberman, G. and S. Kandel: “Mean-Variance Spanning.” The Journal of Finance, 42, (1987), 873.

Hudson-Wilson, S.; F. J. Fabozzi; and J. N. Gordon: “Why Real Estate?” The Journal of Portfolio Management, 29, (2003), 12–25.

Hudson-Wilson, S.; J. N. Gordon; F. J. Fabozzi; M. J. P. Anson; and S. M. Giliberto: “Why real estate? And how? where? and when?” Journal of Portfolio Management, 20, (2005), 12–22.

Hung, J.-C.; M.-C. Lee; and H.-C. Liu: “Estimation of value-at-risk for energy commodities via fat-tailed GARCH models.” Energy Economics, 30, (2008a), 1173–1191.

Hung, K.; Z. Onayev; and C. C. Tu: “Time-Varying Diversification Effect of Real Estate in Institutional Portfolios: When Alternative Assets Are Considered.” Journal of Real Estate Portfolio Management, 14, (2008b), 241–261.

Ibbotson, R. G. and L. B. Siegel: “Real Estate Returns: A Comparison with Other Investments.” Real Estate Economics, 12, (1984), 219–242.

Idzorek, T. M.: “Strategic Asset Allocation and Commodities.” “Intelligent commodity investing: New strategies and practical insights for informed decision making,” , H. Till and J. Eagleeye, eds., 113–177. London: Risk Books (2007).

Irwin, S. H.; T. R. Krukemyer; and C. R. Zulauf: “Investment performance of public commodity pools: 1979-1990.” Journal of Futures Markets, 13, (1993), 799–820.

Irwin, S. H. and D. Landa: “Real estate, futures, and gold as portfolio assets.” The Journal of Portfolio Management, 14, (1987), 29–34.

Jensen, G. R.; R. R. Johnson; and J. M. Mercer: “Efficient use of commodity futures in diversified portfolios.” Journal of Futures Markets, 20, (2000), 489–506.

Jensen, G. R.; R. R. Johnson; and J. M. Mercer: “Tactical Asset Allocation and Commodity Futures.” The Journal of Portfolio Management, 28, (2002), 100–111.

Johnson, R. R. and G. R. Jensen: “The Diversification Benefits of Commodities and Real Estate in Alternative Monetary Conditions.” The Journal of Alternative Investments, 3, (2001), 53–61.

Kallberg, J. G.; C. H. Liu; and D. W. Greig: “The Role of Real Estate in the Portfolio Allocation Process.” Real Estate Economics, 24, (1996), 359–377.

Kaplan, P. D. and S. L. Lummer: “Update: GSCI Collateralized Futures as a Hedging and Diversification Tool for Institutional Portfolios.” The Journal of Investing, 7, (1998), 11–17.

Kilian, L.: “Not All Oil Price Shocks Are Alike: Disentangling Demand and Supply Shocks in the Crude Oil Market.” American Economic Review, 99, (2009), 1053–1069.

Kilian, L.: “Measuring global real economic activity: Do recent critiques hold up to scrutiny?” Economics Letters, 178, (2019), 106–110.

Kremer, P. j.: “Comparing three generations of commodity indices: New evidence for portfolio diversification.” Alternative Investment Analyst Review, 3, (2015), 30–43.

Kroencke, T. A.; F. Schindler; and A. Schrimpf: “International diversification benefits with foreign exchange investment styles.” Review of Finance, 18, (2014), 1847–1883.

Kuhle, J.: “Portfolio Diversification and Return Benefits: Common Stock vs. Real Estate Investment Trusts (REITs).” Journal of Real Estate Research, 2, (1987), 1–9.

Kuosmanen, T.: “Efficient Diversification According to Stochastic Dominance Criteria.” Management Science, 50, (2004), 1390–1406.

Lee, S. L.: “The return due to diversification of real estate to the US mixed-asset portfolio.” Journal of Real Estate Portfolio Management, 11, (2005), 19–28. 37 Linton, O.; E. Maasoumi; and Y.-J. Whang: “Consistent Testing for Stochastic Dominance under General Sampling Schemes.” Review of Economic Studies, 72, (2005), 735–765.

Linton, O.; T. Post; and Y. J. Whang: “Testing for the stochastic dominance efficiency of a given portfolio.” Econometrics Journal, 17.

Liu, E. X.: “Portfolio Diversification and International Corporate Bonds.” Journal of Financial and Quantitative Analysis, 51, (2016), 959–983.

Lizieri, C.: “After the Fall: Real Estate in the Mixed-Asset Portfolio in the Aftermath of the Global Financial Crisis.” The Journal of Portfolio Management, 39, (2013), 43–59.

Lombardi, M. J. and F. Ravazzolo: “On the correlation between commodity and equity returns: Implications for portfolio allocation.” Journal of Commodity Markets, 2, (2016), 45–57.

Lummer, S. L. and L. B. Siegel: “GSCI Collateralized Futures.” The Journal of Investing, 2, (1993), 75–82.

McDonald, J. G. and B. H. Solnick: “Valuation and Strategy for Gold Stocks.” The Journal of Portfolio Management, 3, (1977), 29–33.

Miles, M. and J. Mahoney: “Is commercial real estate an inflation hedge?” Real Estate Issues, 13, (1997), 31–45.

Mull, S. R. and L. A. Soenen: “U.S. REITs as an Asset Class in International Investment Portfolios.” Financial Analysts Journal, 53, (1997), 55–61.

Nijman, T. E. and L. A. P. Swinkels: “Strategic and Tactical Allocation to Commodities for Retirement Savings Schemes.” “The Handbook of Commodity Investing,” , F. J. Fabozzi; R. F¨uss; and D. G. Kaiser, eds., chapter 22, 522–546. Hoboken, NJ, USA: John Wiley & Sons, Inc. (2008).

Owen, J. and R. Rabinovitch: “On the Class of Elliptical Distributions and their Applications to the Theory of Portfolio Choice.” The Journal of Finance, 38, (1983), 745–752.

Platanakis, E.; A. Sakkas; and C. Sutcliffe: “Harmful diversification: Evidence from alternative investments.” The British Accounting Review, 51, (2019), 1–23.

Pojarliev, M. and R. M. Levich: A New Look at Currency Investing. CFA Institute (2012). ISBN 9781934667545.

Post, T.: “Empirical Tests for Stochastic Dominance Efficiency.” The Journal of Finance, 58, (2003), 1905–1931.

Post, T. and V. Pot`ı: “Portfolio Analysis Using Stochastic Dominance, Relative Entropy, and Empirical Likelihood.” Management Science, 63, (2017), 153–165.

Post, T. and P. Versijp: “Multivariate Tests for Stochastic Dominance Efficiency of a Given Portfolio.” Journal of Financial and Quantitative Analysis, 42, (2007), 489–515.

Pukthuanthong, K.; R. Roll; and A. Subrahmanyam: “A Protocol for Factor Identification.” The Review of Financial Studies, 32, (2019), 1573–1607.

Robichek, A. A.; R. A. Cohn; and J. J. Pringle: “Returns on Alternative Investment Media and Implications for Portfolio Construction.” The Journal of Business, 45, (1972), 427–443.

Roman, D.; K. Darby-Dowman; and G. Mitra: “Portfolio construction based on stochastic dominance and target return distributions.” Mathematical Programming, 108, (2006), 541–569.

Russell, W. R. and T. K. Seo: “Representative Sets for Stochastic Dominance Rules.” “Studies in the Economics of Uncertainty,” , T. B. Fomby and T. K. Seo, eds., 59–76. New York, NY: Springer New York (1989).

Sa-Aadu, J.; J. Shilling; and A. Tiwari: “On the Portfolio Properties of Real Estate in Good Times and Bad Times.” Real Estate Economics, 38, (2010), 529–565.

Satyanarayan, S. and P. Varangis: “Diversification Benefits of Commodity Assets in Global Portfolios.” The Journal of Investing, 5, (1996), 69–78.

Scaillet, O. and N. Topaloglou: “Testing for stochastic dominance efficiency.” Journal of Business and Economic Statistics, 28, (2010), 169–180.

Scherer, B. and L. He: “The Diversification Benefits of Commodity Futures Indexes: A Mean-Variance Spanning Test.” “The Handbook of Commodity Investing,” , F. J. Fabozzi; R. Füss; and D. G. Kaiser, eds., chapter 10, 241–265. Hoboken, NJ, USA: Wiley. ISBN 9780470117644 (2008).

Schneeweis, T. and R. B. Spurgin: “Comparisons of Commodity and Managed Futures Benchmark Indexes.” The Journal of Derivatives, 4, (1997), 33–50.

Seiler, M. J.; J. R. Webb; and F. C. N. Myer: “Diversification Issues in Real Estate Investment.” Journal of Real Estate Literature, 7, (1999), 163–179.

Shalit, H. and S. Yitzhaki: “Marginal Conditional Stochastic Dominance.” Management Science, 40, (1994), 670–684.

Shanken, J.: “Multivariate tests of the zero-beta CAPM.” Journal of Financial Economics, 14, (1985), 327–348.

Shanken, J.: “Testing Portfolio Efficiency when the Zero-Beta Rate is Unknown: A Note.” The Journal of Finance, 41, (1986), 269.

Sherman, E. J.: “Gold: A conservative, prudent diversifier.” The Journal of Portfolio Management, 8, (1982), 21–27.

Silvennoinen, A. and S. Thorp: “Financialization, crisis and commodity correlation dynamics.” Journal of International Financial Markets, Institutions and Money, 24, (2013), 42–65.

Simaan, Y.: “Portfolio Selection and Asset Pricing—Three-Parameter Framework.” Management Science, 39, (1993), 568–577.

Simon, D. P.: “A conditional assessment of the relationships between commodity and equity indexes.” Journal of Alternative Investments, 16, (2013), 30–51.

Solnik, B. H.: “Why Not Diversify Internationally Rather Than Domestically?” Financial Analysts Journal, 30, (1974), 48–54.

Sortino, F. A.; R. v. D. Meer; A. Plantinga; and H. Forsey: “The Upside Potential Ratio: What Are We Trying to Measure?” Senior Consultant, 6, (2003), 28–29.

Tang, K. and W. Xiong: “Index Investment and the Financialization of Commodities.” Financial Analysts Journal, 68, (2012), 54–74.

Webb, J. R.; R. J. Curcio; and J. H. Rubens: “Diversification Gains from including Real Estate in Mixed Asset Portfolios.” Decision Sciences, 19, (1988), 434–452.

Webb, J. R. and J. H. Rubens: “Portfolio Considerations in the Valuation of Real Estate.” Journal of the American Real Estate and Urban Economics Association, 14, (1986), 1986.

Webb, J. R. and J. H. Rubens: “How Much in Real Estate? A Surprising Answer.” The Journal of Portfolio Management, 13, (1987), 10–14.

Yan, L. and P. Garcia: “Portfolio investment: Are commodities useful?” Journal of Commodity Markets, 8, (2017), 43–55.

You, L. and R. T. Daigler: “Using Four-Moment Tail Risk to Examine Financial and Commodity Instrument Diversification.” Financial Review, 45, (2010), 1101–1123.

You, L. and R. T. Daigler: “A Markowitz Optimization of Commodity Futures Portfolios.” Journal of Futures Markets, 33, (2013), 343–368.

Zerbst, R. H. and B. R. Cambon: “Real Estate: Historical Returns and Risks.” The Journal of Portfolio Management, 10, (1984), 5–20.

Ziemba, W. T.: “The Symmetric Downside-Risk Sharpe Ratio.” The Journal of Portfolio Management, 32, (2005), 108–122.

Ziobrowski, B. J. and A. J. Ziobrowski: “Higher Real Estate Risk and Mixed-Asset Portfolio Performance.” Journal of Real Estate Portfolio Management, 3, (1997), 107–115.