# Risk minimization in multi-factor portfolios: What is the best strategy?

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## Abstract

Exposures to risk factors, as opposed to individual securities or bonds, can lead to an ex-ante improved risk management and a more transparent and cheaper way of developing active asset allocation strategies. This paper provides an extensive analysis of eight state-of-the-art risk-minimization schemes and compares risk factor performance in a conditional performance analysis, contrasting good and bad states of the economy. The investment universe spans a total of 25 risk factors, including size, momentum, value, high profitability and low investments, from five non-overlapping regions (i.e., USA, UK, Japan, Developed Europe ex. UK and, Asia ex. Japan). Considering as investment period the interval from May 2004 to June 2015, our results show that each single factor yields positive premia in exchange for risk, which can lead to considerable underperformance and extensive recovery periods during times of crisis. The best factor investments can be found in Asia ex. Japan and the US. However, risk factor based portfolio construction across the various regions enables the investor to exploit low correlation structures, reducing the overall volatility, as well as tail- and extreme risk measures. Finally, the empirical results point towards the long-only global minimum variance portfolio, as the best risk minimization strategy.

## Keywords

Risk factors Minimum risk portfolio Regularization Portfolio optimization Transaction cost## References

- Amenc, N., Goltz, F., Lodh, A., & Martellini, L. (2015).
*Scientific beta multi-strategy factor indices: Combining factor tilts and improved diversification*. ERI Scientific Beta Publication.Google Scholar - Asness, C., Frazzini, A., Israel, R., & Moskowitz, T. (2015). Fact, fiction, and value investing.
*Journal of Portfolio Management*,*42*(1), 34–52.CrossRefGoogle Scholar - Bai, J., & Ng, S. (2002). Determining the number of factors in approximate factor models.
*Econometrica*,*70*(1), 191–221.CrossRefGoogle Scholar - Becker, F., Guertler, M., & Hibbeln, M. (2015). Markowitz versus michaud: Portfolio optimization strategies reconsidered.
*The European Journal of Finance*,*21*(4), 269–291.CrossRefGoogle Scholar - Benartzi, S., & Thaler, R. (2001). Naive diversification ststrategies defined contribution plans.
*American Economic Review*,*91*(1), 79–98.CrossRefGoogle Scholar - Bessler, W., & Wolff, D. (2015). Do commodities add value in multi-asset-portfolios? An out-of-sample analysis for different investment strategies.
*Journal of Banking and Finance*,*60*, 1–20.CrossRefGoogle Scholar - Bessler, W., Opfer, H., & Wolff, D. (2017). Multi-asset portfolio optimization and out-of-sample performance: An evaluation of Black–Litterman, mean-variance, and naive diversification approaches.
*The European Journal of Finance, 23*(1), 1–30.Google Scholar - Best, M., & Grauer, J. (1991). On the sensitivity of mean-variance-efficient portfolios to changes in asset means: Some analytical and computational results.
*The Review of Financial Studies*,*4*(2), 315–342.CrossRefGoogle Scholar - Broadie, M. (1993). Computing efficient frontiers using estimated parameters.
*Annals of Operations Research*,*45*(1), 2158.CrossRefGoogle Scholar - Brodie, J., Daubechies, I., DeMol, C., Giannone, D., & Loris, D. (2009). Sparse and stable markowitz portfolios.
*Proceedings of the National Academy of Science*,*106*(30), 12,267–12,272.CrossRefGoogle Scholar - Bruder, B., & Roncalli. T. (2012).
*Managing risk exposures using the risk budgeting approach*. SSRN www.ssrn.com/abstract=2009778. - Chopra, V., & Ziemba, W. (1993). The effect of errors in means, variances, and covariances on optimal portfolio choice.
*Journal of Portfolio Management*,*19*, 6–12.CrossRefGoogle Scholar - Chordia, T., & Shivakumar, L. (2002). Momentum, business cycle, and time-varying expected returns.
*Journal of Finance*,*57*(2), 985–1019.CrossRefGoogle Scholar - Choueifaty, Y., & Coignard, Y. (2008). Toward maximum diversification.
*Journal of Portfolio Management*,*34*(4), 40–51.CrossRefGoogle Scholar - Choueifaty, Y., Froidure, T., & Reynier, J. (2011). Properties of the most diversified portfolio.
*Journal of Investment Strategies*,*2*(2), 49–70.CrossRefGoogle Scholar - Christoffersen, P., Errunza, V. R., Jacobs, K., & Jin, X. (2010)
*Is the potential for international diversification disappearing?*. sSRN: http://ssrn.com/abstract=1573345. - Clarke, R., de Silva, H., & Thorley, S. (2011). Minimum variance portfolio composition.
*Journal of Portfolio Management*,*37*(2), 31–45.CrossRefGoogle Scholar - Cooper, I. (2006). Asset pricing implications of non-convex adjustment costs and irreversibility of investment.
*The Journal of Finance*,*61*(1), 139–170.CrossRefGoogle Scholar - Cooper, M., Gulen, H., & Schill, M. (2008). Asset growth and the cross section of stock returns.
*Journal of Finance*,*63*(4), 1609–1651.CrossRefGoogle Scholar - Coqueret, G., & Milhau, V. (2014).
*Estimating covariance matrices for portfolio optimization*. ERI Scientific Beta White Paper.Google Scholar - Daly, J., Crane, M., & Ruskin, H. (2008). Random matrix theory filters in portfolio optimisation: A stability approach.
*Physica A: Statistical Mechanics and Its Applications*,*387*(16–17), 4248–4260.CrossRefGoogle Scholar - De Souza Oliveira, T. (2014).
*Discount rates, market frictions, and the mystery of the size premium*. Ph.D. thesis, University of Southern Denmark.Google Scholar - Deguest, R., Martellini, L., & Meucci, A. (2013).
*Risk parity and beyond—From asset allocation to risk allocation decisions*. SSRN: http://ssrn.com/abstract=2355778. - DeMiguel, V., Garlappi, L., Nogales, F., & Uppal, R. (2009a). A generalized approach to portfolio optimization: Improving performance by constraining portfolio norm.
*Management Science*,*55*, 798–812.CrossRefGoogle Scholar - DeMiguel, V., Garlappi, L., Nogales, F., & Uppal, R. (2009b). Optimal versus naive diversification: How inefficient is the 1/n portfolio strategy?
*Review of Financial Studies*,*22*(5), 1915–1953.CrossRefGoogle Scholar - Fama, E., & French, K. (2006). Profitability, investment and average returns.
*Journal of Financial Economics*,*82*, 491–518.CrossRefGoogle Scholar - Fama, E. F., & French, K. R. (2007). Migration.
*Financial Analysts Journal*,*63*, 48–58.CrossRefGoogle Scholar - Fan, J., Fan, Y., & Lv, J. (2008). High didimension covariance matrix estimation using a factor model.
*Journal of Econometrics*,*147*(1), 186–197.CrossRefGoogle Scholar - Fan, J., Zhang, J., & Yu, K. (2009)
*Asset allocation and risk assessment with gross exposure constraints for vast portfolios*. Working Paper Princton University, New Jersey, USA.Google Scholar - Fan, J., Zhang, J., & You, K. (2012). Vast portfolio selection with gross-exposure constraint.
*Journal of the American Statistical Association*,*107*(498), 592–606.CrossRefGoogle Scholar - Fastrich, B., Paterlini, S., & Winker, P. (2015). Constructing optimal sparse portfolios using regularization methods.
*Computational Management Science*,*12*(3), 417–434.CrossRefGoogle Scholar - Ferson, W., & Qian, M. (2004)
*Conditional performance evaluation*, revisited. Boston College Working Paper.Google Scholar - Giamouridis, D., & Paterlini, S. (2010). Regular(ized) hedge fund clones.
*Journal of Financial Research*,*33*(3), 223–247.CrossRefGoogle Scholar - Gulpinar, N., & Pachamanova, D. (2013). A robust optimization approach to asset-liability management under time-varying investment opportunities.
*Journal of Banking and Finance*,*36*(6), 2031–2041.CrossRefGoogle Scholar - Hartmann, P., Straetmans, S., & de Vries, C. (2001). Asset market linkages in crisis periods.
*Review of Economics and Statistics*,*86*(1), 313–326.CrossRefGoogle Scholar - Hasanhodzic, J., & Lo, A. (2006). Can hedge-fund returns be replicated?: The linear case.
*Journal of Investment Management*,*5*(2), 5–45.Google Scholar - Haugen, R., & Baker, N. (1991). The efficient market inefficiency of capitalization-weighted stock portfolios.
*Journal of Portfolio Management, 17*(3), 35–40.Google Scholar - Hou, K., Xue, C., & Zhang, L. (2015). Digesting anomalies: An investment approach.
*Review of Financial Studies*,*28*(3), 650–705.Google Scholar - Hsu, J. C. (2006). Cap-weighted portfolios are sub-optimal portfolios.
*Journal of Investment Management*,*4*(3), 1–10.Google Scholar - Ilmanen, A., & Kizer, J. (2012). The death of diversification has been greatly exaggerated.
*Journal of Portfolio Management*,*38*, 15–27.CrossRefGoogle Scholar - Jagannathan, R., & Ma, T. (2003). Risk reduction in large portfolios: Why imposing the wrong constraints helps.
*The Journal of Finance*,*58*(4), 1651–1683.CrossRefGoogle Scholar - Jegadeesh, N., & Titman, S. (1993). Returns to buying winners and selling losers: Implications for stock market efficiency.
*Journal of Finance*,*48*(1), 65–91.CrossRefGoogle Scholar - Kourtis, A., Dotsis, G., & Markellos, R. (2012). Parameter uncertainty in portfolio selection: Shrinking the inverse covariance matrix.
*Journal of Banking and Finance*,*36*, 2522–2531.CrossRefGoogle Scholar - Ledoit, O., & Wolf, M. (2004). Honey, I shrunk the covariance matrix.
*Journal of Portfolio Management*,*30*(4), 110–119.CrossRefGoogle Scholar - Ledoit, O., & Wolf, M. (2008). Robust performance hypothesis testing with the sharpe ratio.
*Journal of Empirical Finance*,*15*, 850–859.CrossRefGoogle Scholar - Liu, L., & Zhang, L. (2008). Momentum profits, factor pricing, and macroeconomic risk.
*Review of Financial Studies*,*21*(6), 2417–2448.CrossRefGoogle Scholar - Lohre, H., Opfer, H., & Orszag, G. (2014). Diversifying risk parity.
*Journal of Risk, 16*(5), 53–79.Google Scholar - Maillard, S., Roncalli, T., & Teiletche, J. (2008) Equally-weighted risk contributions:a new method to build risk balanced diversified portfolios. http://www.thierry-roncalli.com/download/erc-slides.pdf.
- Maillard, S., Roncalli, T., & Teiletche, J. (2010). The properties of equally weighted risk contribution portoflios.
*The Journal of Portfolio Management*,*36*(4), 60–70.CrossRefGoogle Scholar - Martellini, L., Milhau, V., & Tarelli, A. (2014).
*Towards conditional risk parity? Improving risk budgeting techniques in changing economic environments*. ERI Scientific Beta Publication. http://www.edhec-risk.com/edhec-publications/all-publications. - Merton, R. (1973). An intertemporal capital asset pricing model.
*Econometrica*,*41*(5), 867–887.CrossRefGoogle Scholar - Merton, R. C. (1980). On estimating the expected return on the market: An exploratory investigation.
*Journal of Financial Economics*,*8*(4), 323–361.CrossRefGoogle Scholar - Meucci, A., Santangelo, A., & Deguest R. (2014).
*Measuring portfolio diversification based on optimized uncorrelated factors*. SSRN: http://ssrn.com/abstract=2276632. - Michaud, R. (1989). The markowitz optimization enigma: Is ’optimized’ optimal?.
*Financial Analyst Journal, 45*(1), 31–42.Google Scholar - Novy-Marx R. (2013).
*The quality dimension of value investing*. Simon Graduate School of Business. www.simon.rochester.edu. - Platanakis, E., & Sutcliffe, C. (2017). Asset-liability modelling and pension schemes: The application of robust optimization to USS.
*The European Journal of Finance*,*23*(4), 324–352.CrossRefGoogle Scholar - Roncalli, T. (2013).
*Introduction to risk parity and budgeting*. Chapman & Hall/CRC Financial Mathematics Series.Google Scholar - Ross, S. (1976). The arbitrage theory of capital asset pricing.
*Journal of Economic Theory*,*13*(3), 341–360.CrossRefGoogle Scholar - Rouwenhorst, G. (1998). International momentum strategies.
*Journal of Finance*,*53*(1), 267–284.Google Scholar - Titman, S., Wei, K., & Xie, F. (2004). Capital investments and stock returns.
*Journal of Financial and Quantitative Analysis*,*39*, 677–700.CrossRefGoogle Scholar - Van Gelderen, E., & Huij, J. (2013).
*Academic knowledge dissemination in the mutual fund industry: Can mutual funds successfully adopt factor investing strategies?*. sSRN: http://ssrn.com/abstract=2295865. - Weber, V., & Peres, F. (2013). Hedge fund replication: Putting the pieces together.
*Journal of Investment Strategies*,*3*(1), 61–119.Google Scholar - Windcliff, H., & Boyle, P. (2004). The 1/n pension investment puzzle.
*North American Actuarial Journal*,*8*(3), 32–45.Google Scholar