Deutsche BankMarkets Research Global Quantitative Strategy Portfolios Under Construction Date 30 May 2013 Yin Luo, CFA DB Handbook of Portfolio Construction: Part 1
[email protected] Sheng Wang
[email protected] Part 1: Risk-Based Portfolio Construction Techniques Rochester Cahan, CFA Innovative techniques The heart of portfolio construction is about how to achieve better diversification and risk reduction. In this paper, we introduce two innovative techniques to accomplish this goal. These techniques recognize the fact that asset returns are not normally distributed. The minimum tail dependence portfolio attempts to find assets that are less dependent to each other at the tail level to avoid crowded trades. The conditional value-at-risk (or CVaR) defines risk in a different way from volatility, by emphasizing tail risk. Portfolios constructed by minimizing CVaR are ex ante more conservative and have delivered the best ex post performance. Encyclopedic coverage of risk-based portfolio construction for both multi-asset and equity portfolios In this research, we conduct a comprehensive portfolio backtesting of a full suite of risk-based portfolio construction techniques, from simple (equally weighted and inverse volatility), traditional (risk parity, global minimum variance, and maximum diversification), to innovative (minimum tail dependence, minimum CVaR, robust minimum CVaR, etc.) for a broad range of contexts in multi-asset (asset allocation, bonds, commodities, alternative betas), country/sector (countries, economic low risk countries, global sectors, US sectors, European sectors, global industries, and regions x sectors), and equity portfolios (US, Europe, Asia ex Japan, EM, and global).
[email protected] Javed Jussa
[email protected] Zongye Chen
[email protected] Miguel-A Alvarez
[email protected] North America: +1 212 250 8983 Europe: +44 20 754 71684 Asia: +852 2203 6990 The superiority of risk-based allocations In almost all contexts (both multi-asset and equities), risk-based allocations significantly outperform capitalization-weighted benchmarks, with higher Sharpe ratios, lower downside risk, better diversification, and less tail dependence, which makes the portfolio less likely to suffer in a liquidity crisis. Source: gettyimages.com ________________________________________________________________________________________________________________ Deutsche Bank Securities Inc. Note to U.S. investors: US regulators have not approved most foreign listed stock index futures and options for US investors. Eligible investors may be able to get exposure through over-the-counter products. Deutsche Bank does and seeks to do business with companies covered in its research reports. Thus, investors should be aware that the firm may have a conflict of interest that could affect the objectivity of this report. Investors should consider this report as only a single factor in making their investment decision. DISCLOSURES AND ANALYST CERTIFICATIONS ARE LOCATED IN APPENDIX 1.MICA(P) 054/04/2013. 30 May 2013 Portfolios Under Construction A letter to our readers Part I of DB Handbook of Portfolio Construction From our extensive interaction with our institutional clients globally, we have noticed an interesting pattern. Most portfolio managers and research analysts tend to spend most of their time searching for alpha-related ideas1, e.g., the next factor to predict stock returns, the next risk premium in commodities, the next global macro strategy, or the so-called the “Great Rotation” in asset allocation recently2. On the other hand, investors tend to rely on data vendors for their risk models, optimizers, and performance attribution tools. It is only in recent years since the global financial crisis that risk management and portfolio construction have been attracting more attentions than ever before. At the same time, most investors do not feel as comfortable in dealing with portfolio construction as alpha research, due to the lack of easily accessible, educational, and practical reading materials. It is our intention to bring investors up to date with the most cutting edge portfolio construction techniques, in a non-technical manner. Rather than focusing on the mathematical details, we concentrate on the practical issues facing investors every day. This is the first part of a series of handbooks that address this increasingly more important area of portfolio construction. In this paper, we focus on what-so-called “riskbased allocation” or “risk-based portfolio construction techniques”. On the one hand, they are simpler – portfolios are constructed solely based on risk prediction. But risk, of course, can be defined in countless ways, so risk-based allocation can still be extremely complicated. On the other hand, risk-based allocation has become one of the hottest fields in both academia and investing in recent years. Again, this is not surprising given the prolonged global financial crisis started in 2008. Not only is managing risk becoming more paramount than outperforming a benchmark, but also risk-based allocation techniques actually do indeed outperform many (if not most) active strategies that require return prediction. In a series of forthcoming research papers, we will discuss how to incorporate risk and return predictions when constructing portfolios. We will also address how to combine qualitative views with quantitative models in portfolio construction. In total, we hope these DB Handbooks of Portfolio Construction will help our readers better understand portfolio construction and use these techniques to better manage their portfolios. It’s all about diversification and risk reduction Our regular readers would have noticed our serious interest in risk-based portfolio construction in both asset allocation and equity portfolio contexts3. In this paper, we summarize the key theories and findings of our previous research. More importantly, we introduce a few new tools in portfolio analytics. 1 This is indeed the case with us as well. In Luo, et al [2010]. “DB Quant Handbook”, we almost exclusively discussed only alpha related topics. 2 The term “Great Rotation” came to prominence in 2012, referring to the fact that bond yields are at record lows; therefore, bonds offer little upside. Investors are likely to rotate out of bonds and into other risky assets, especially stocks. 3 See a complete list of our previous research in this space in Section XIV on page 104. Page 2 Deutsche Bank Securities Inc. 30 May 2013 Portfolios Under Construction Portfolio construction itself does not introduce new assets (i.e., breadth) or new insights in return prediction (i.e., skill). The heart of portfolio construction is about diversification and risk reduction – both terms are yet to be defined. We all know classic finance theory like Markowiz’s mean-variance optimization heavily depends on the assumption that asset returns are jointly normally distributed. We also know that empirical evidence almost universally rejects the normality assumption. The traditional statistical tools (e.g., Pearson’s correlation coefficient, portfolio volatility, etc.) are mostly based on this false assumption. In this research, we first define an alternative approach to measure comovement in asset returns. Rather than relying on Pearson’s correlation, we measure tail dependence between two assets using a Copula model. In Cahan, et al [2012], we demonstrated an interesting way to measure the crowdedness of plain vanilla types of low risk strategies in the equity space – median tail dependence. In this research, we extend the tail dependence concept by further constructing what we called weighted portfolio tail dependence (WPTD) by taking into account asset weights. More important, we design a strategy that proactively avoids crowded trades in what we call the minimum tail dependent portfolio (MinTailDependence). Second, we give a practical introduction to conditional value at risk (also called expected shortfall) and how to construct a portfolio that minimizes expected CVaR, i.e., the MinCVaR portfolio. We further develop a new algorithm by combining robust optimization and CVaR optimization into what we call robust CVaR optimization, which shows great promise by outperforming all other portfolio construction techniques in terms of both Sharpe ratio and downside risk. The MinTailDependence portfolio (along with maximum diversification) helps us better capture diversification benefits, while the MinCVaR strategy (along with global minimum variance portfolio) attempts to manage risk better. Everything from asset allocation, multi-asset, to equity portfolios In this paper, we empirically backtest seven risk-based allocations (equally weighted, inverse volatility/volatility parity, risk parity/equal risk contribution, global minimum variance, maximum diversification, minimum tail dependence, and minimum CVaR), compared to traditional capitalization-weighted benchmarks in different contexts from asset allocation, multi-asset (bonds, commodities, alternative betas), country/sector portfolios (MSCI ACWI, economically hedged country indices, global sectors, US sectors, European sectors, global industries, region x sector combinations), and equity portfolios (US, European, Asia ex Japan, Japan, emerging markets, and global equities). Interestingly, in almost every single context, risk-based allocations significantly outperform traditional capitalization-weighted benchmarks, with higher Sharpe ratio, lower downside risk, better diversification, and less tail dependence. Yin, Rocky, Miguel, Javed, John, and Sheng Deutsche Bank Quantitative Strategy Team Deutsche Bank Securities Inc. Page 3 .................................. 37 VII.................. 54 IX.................................................................................................................................................. 16 Weighted portfolio tail dependent coefficient (WPTD) ............................................................................................................................................................................ 11 Risk parity (RiskParity) or equal risk contribution (ERP) ..................................................................................................................... 66 Page 4 Deutsche Bank Securities Inc................... 18 Portfolio backtesting setup ..... 57 Sovereign bond portfolios .......................... 7 Risk-adjusted performance ....................................................................... 22 Constraints in portfolio construction ........................................................................................... 11 Inverse volatility (InvVol) ........... 6 Return distribution ............................................................................................................................... 23 V............................................................... 61 Commodities portfolios ...................................................... 32 CVaR optimization theory ................................................................................................ 32 Mean-CVaR efficient frontier and minimum CVaR portfolio ................................................................... 49 Country weight .......... 12 Maximum diversification portfolio (MaxDiversification) ................... Strategy crowding and efficacy ............................................................................. 15 Weighted portfolio correlation ....................................................... Multi-asset portfolio ...................................................................... 18 Choices of risk models .......... 19 Robust covariance estimation techniques ................... 13 III........................................................................................... 19 Country and sector risk models ........ 46 Measure of crowdedness – portfolio tail dependence based .......................................... 34 Robust minimum CVaR optimization ............... 42 Fully capturing the benefit of the low risk anomaly at the stock level .............................................................................................. 11 Global minimum variance portfolio (GlobalMinVar)................................................ Minimum tail dependence portfolio......................... 45 Measure of crowdedness – portfolio correlation based .... Performance measurement ................................. 6 Risk analysis ........... 63 Alternative beta portfolios .......................................................................... 52 MEAM country equity portfolios ............................................................... 27 The choice of copula model – does it really matter? ................................................................................. Traditional risk-based allocations ........................................................................ Country portfolios ............................... 10 II................................................................................................................................................................... 18 Equity portfolios.................................................................. 35 How is RobMinCVaR related to other risk-based portfolio construction techniques......................................................................................................... Numerical and computational issues ................................. 28 An alternative minimum tail dependence strategy ................................................................................................. 40 VIII............................. 57 Asset allocation ........................................... 26 Minimum tail dependence portfolio optimization algorithm ....................................................................................................... 9 Portfolio diversification characteristics .......................... 17 IV.............................................. 26 Introducing the copula model ............................................. Minimum CVaR portfolio ........................................................................................................ Risk estimation and portfolio construction ................... ...................................................................................................... 29 Tail-dependence constraint ...................................................... 34 Choice of alpha parameter ........ 31 VI................................................................................ 15 WPC and portfolio diversification .............. 42 Comparison of risk-based allocation ......................................................................................................................................................... 48 How to group these strategies ..............30 May 2013 Portfolios Under Construction Table Of Contents I................. ..................................................... 94 The philosophy of portfolio construction ............................. 95 Potential diversification benefit ............ Conclusion ............................................................................................................................................ Sector/industry portfolios............................................................................................................... Page 5 .................... 89 Global equity ...... Equity portfolios ......................................................................................... 76 Region x sector portfolios ............................. 81 European equities .......................................................................... 85 Japanese equities ............................................................................................................................................................................................................ 87 Emerging markets equities ............................... 83 Asian equities ............................................................................................. 72 MSCI Europe 10 GICS sector portfolio ........... 70 US S&P 500 10 GICS sector portfolio ................................................. 74 MSCI World 24 GICS level 2 industry group portfolio ............................30 May 2013 Portfolios Under Construction Table Of Contents (Cont'd) X............................................................................. 91 XII.......................................................................................................................................... 104 Deutsche Bank Securities Inc........................................................................................................ 94 A horse race of risk-based allocations ....................................... 81 US equities....................... 78 XI.................................. 70 MSCI World 10 GICS sector portfolio .................................................................................................... Bibliography ....... Review of our previous research .............. 101 XIV..... 99 XIII..................................................................... In portfolio construction. Return distribution Traditional performance measures like the Sharpe ratio are only meaningful if the portfolio return follows a normal distribution. We can further apply statistical normality tests. Jarque-Bera test. the joint multivariate distribution is far more important than univariate distribution.10 0. MSCI. In this research.05 0. asset returns are rarely normally distributed. Performance measurement Regular readers of our research may have noticed that we are strong proponents of incorporating a full suite of performance measurement and attribution tools in investment management. and Spain) using the past five years of daily returns to demonstrate two main multivariate normality tests.05 0.00118 10 Density 5 0 Mean: 0. Shapiro-Wilk’s test. MSCI.10 -0. Almost all the normality tests mentioned above easily reject the Null hypothesis of a normal distribution for US and Greece equity market returns. As shown in Figure 1 and Figure 2.Greece -0.05 0. Deutsche Bank Quantitative Strategy The traditional mean-variance portfolio optimization techniques also heavily depend on the assumption that asset returns jointly follow multivariate normal distribution. which will be used extensively throughout the rest of the paper.15 Source: Bloomberg Finance LLP. As a first test. Greece. e.000322 15 10 20 30 40 50 60 Greece 0 Density US -0.00 0. Figure 1: Density plot – US Figure 2: Density plot . Rather.30 May 2013 Portfolios Under Construction I.05 0. the country returns for US and Greece clearly do not follow normal distribution – they both show negative skewness and excess kurtosis.. we need to assess if the normal distribution assumption is really valid. Deutsche Bank Quantitative Strategy Mean: -0. A book-length introduction can be found in Bacon [2004]. Germany. we briefly summarize a few key metrics. We can calculate skewness and kurtosis. KolmogorovSmirnov test.00 0. Italy.10 Source: Bloomberg Finance LLP. Portugal.g. we do not attempt to provide a complete coverage. . Now. Page 6 Deutsche Bank Securities Inc. As we all know. let’s use a simple example of six countries (US. and D’Agostino normality test. we typically use tracking error.10 0. Deutsche Bank Quantitative Strategy Risk analysis The simplest and most traditional measure of risk is volatility.82. corresponding to a p-value of less than 1%.10 *** 0.e.05 US 0.592 *** -0.467 *** 0.10 Spain Source: Bloomberg Finance LLP. we see some clear nonlinear relationships and some heavy tail dependence. corresponding to a p-value of less than 1%. 34 The E-statistic is 3.808 *** *** 0.10 0.553 0. In addition.10 *** 0.10 0.859 *** *** *** 0.10 -0.579 0.05 *** Germany 0.3 × 10 .590 0..05 0. we will show how we can measure tail dependence.902 Greece 0. also called energy test (see Szekely [1989]). Similarly.05 0. The nonparametric E-statistics test.10 *** -0. Volatility and tracking error are appropriate when the return distribution Deutsche Bank Securities Inc. MSCI.864 *** 0. The multivariate Shapiro test shows a w-statistic of 0. Figure 3: Scatterplot of six countries *** *** -0.545 *** 0.606 Italy 0.10 -0.05 -0.858 0.928 Portugal 0.10 0. i. for the relative risk to a benchmark.313 -0.636 0. the Null hypothesis of a multivariate normal distribution is again easily rejected.30 May 2013 Portfolios Under Construction The multivariate Shapiro test (see Royston [1982]).10 -0.10 0. the standard deviation of active returns. therefore. Page 7 . the Null hypothesis of a multivariate normal distribution is rejected. especially among the peripheral European countries. how to incorporate tail dependence in our portfolio construction process. The correlations among all six countries appear to be very strong. In later sections.10 -0. Figure 3 shows the scatterplot of the returns for the six countries. and more importantly. VaR is defined as a threshold value such that the probability that the loss on the portfolio over the given time horizon exceeds this value is the given probability level. 1 1 CVaRα = VaRn (L )du 1 − α α∫ where. If the return series is skewed and/or has excess kurtosis. L is the loss function. Unlike VaR. which incorporates skewness and kurtosis via an analytical estimation using a Cornish-Fisher expansion. where dependence is defined as a copula-based tail dependent coefficient. Peterson. the returns of most assets do not follow a normal distribution. which typically is set as 1% or 5%. In our research. the CVaR/ES of a return series is the expected value of the return when the return is less than its α -quantile. For example. Empirically. as we know. The strategy will be fully defined in Section V on page 26. Cornish-Fisher estimates of CVaR can be more appropriate (see Boudt. In a later section. The biggest problem when using VaR as a risk measure is that it is not coherent. Figure 4 and Figure 5 show the drawdowns of the MSCI ACWI and our MinTailDependence 4 country portfolio. For a given portfolio. defined as the worst cumulative loss ever sustained by an asset. probability and time horizon. Our MinTailDependence portfolio has less severe drawdowns (i. 4 The MinTailDependence portfolio seeks to build a portfolio of countries that are at least dependent from each other as possible. CVaR has all the properties a risk measure should have to be coherent and is a convex function of the portfolio weights.. you may choose to estimate CVaR with the sample average of all returns that are below the α empirical quantile. The typical VaR (also called “historical VaR”) is based on the RiskMetrics (now part of MSCI) methodology. CTAs). and Croux [2008]). With a sufficiently large data set. Specifically.30 May 2013 Portfolios Under Construction is normal. CVaR can be interpreted as the average VaR when the loss is greater than VaR. meaning the VaR of a portfolio can be greater than the sum of the individual risks of each assets comprising the portfolio. and Croux [2008]).e. we focus on the socalled Modified Cornish-Fisher VaR (see Zangari [1996] for the original paper and the more efficient algorithm introduced by Boudt..e. Peterson. The maximum drawdown measure is particularly popular in the world of commodities trading advisors (i. Page 8 Deutsche Bank Securities Inc. Conditional VaR/expected shortfall (ES) At a preset confidence level denoted α . . we will show how to incorporate CVaR in the portfolio construction process. Drawdown measures A widely used downside risk measure is maximum drawdown. VaR is not sub-additive. Value-at-risk (VaR) Value at risk (or VaR) was developed in the early 1990s. maximum drawdown) and the length of each drawdown is shorter on average. and 2 −∞ is the pdf (probability density function) of the portfolio returns.1 -0.. Sortino ratio The Sortino ratio is the mean return over downside deviation below a user-specified target (i.0 Figure 4: Drawdown – MSCI ACWI Jan 00 Jan 02 Jan 04 Jan 06 Jan 08 Jan 10 Jan 12 Source: Bloomberg Finance LLP. Deutsche Bank Securities Inc. MSCI. a required rate of return). Deutsche Bank Quantitative Strategy Risk-adjusted performance Sharpe ratio The Sharpe ratio is defined as the mean return over volatility.0 0.30 May 2013 Portfolios Under Construction In an upcoming research paper. MSCI. SortinoRatio = rp − MAR DownsideDeviation where.. Figure 5: Drawndown – MinTailDependence portfolio -0. we will show how to build portfolios taking into account drawdown measures.2 -0.2 -0. Page 9 . Despite of the issues with the Sharpe ratio. return over benchmark) divided by active risk (also called tracking error).1 0.5 -0.3 Drawdown -0. MAR DownsideDeviation = f (x ) ∫ (MAR − x ) f (x )dx . MAR or minimum acceptable return is the user-specified target return.4 Drawdown -0. which is typically not met in practice.4 -0.3 -0. Deutsche Bank Quantitative Strategy Jan 00 Jan 02 Jan 04 Jan 06 Jan 08 Jan 10 Jan 12 Source: Bloomberg Finance LLP. it is still the most widely used performance metric today. The Sharpe ratio is an appropriate measure of performance only if the portfolio returns follows a normal distribution.e. Adjusted Sharpe ratio The adjusted Sharpe ratio was introduced by Pezier and White [2006] to account for skewness and kurtosis by incorporating a penalty factor for negative skewness and excess kurtosis: ⎡ ⎛ Skewness ⎞ ⎛ Kurtosis − 3 ⎞ 2⎤ AdjustedSharpeRatio = SharpeRatio × ⎢1 + ⎜ ⎟ × SharpeRatio − ⎜ ⎟ × SharpeRatio ⎥ 6 24 ⎠ ⎠ ⎝ ⎣ ⎝ ⎦ Information ratio The information ratio is the mean active return (i.e. the weighted portfolio correlation (WPC).30 May 2013 Portfolios Under Construction Portfolio diversification characteristics In this paper. we focus on four measures of diversification: the diversification ratio. the concentration ratio. We will discuss this topic in detail in Section III: Strategy crowding and efficacy. Page 10 Deutsche Bank Securities Inc. . and the weighted portfolio tail dependence coefficient (WPTD). t ≥ 0 where.t =1 i =1 ωi . down weighted) simply because it’s relatively more volatile. Risk parity (RiskParity) or equal risk contribution (ERP) Risk parity (RiskParity) or equal risk contribution (ERP) is similar to the InvVol allocation. ωi. Page 11 . while it may provide more diversification benefits should correlations also be considered. also called naïve risk parity or volatility parity. RiskParity takes into account the correlation between pairs of assets.t ∑1 / σ i. rp .e. An asset may be unnecessarily penalized (i. et al [2013] “Independence day”. we briefly review the four traditional risk-based portfolio construction techniques. The purpose of balancing risk equally across the assets in the portfolio is consistent with a "no-alpha" strategy in that it gives each asset "equal opportunity" to contribute to the portfolio’s overall performance. t. et al [2011] “Risk parity and riskbased allocation” and Luo. more volatile assets are relatively down weighted. Traditional risk-based allocations In this section. Mathematically.t is the volatility of asset i i at time at time t. The main drawback of InvVol is that it does not take into account the correlation between assets. And it can be implemented via the following optimization algorithm: n n ωi . InvVol is easy to implement.t cov(ri .t is the weight allocated to asset σ i. in that portfolios are weighted inversely to their volatility.t .t n Subject to: N ∑ω i .t = 1 / σ i .30 May 2013 Portfolios Under Construction II. Inverse volatility (InvVol) The first allocation technique we explore is Inverse Volatility (InvVol).t . More details can be found in Alvarez.t = arg min w ∑∑ [ωi .t cov(r j . As such.t ) − ω j .t )]2 i =1 j =1 Subject to: Deutsche Bank Securities Inc. this can be expressed as follows: ωi .rp . It seeks to give equal risk budget to each asset in the portfolio (subject to constraints). The upward-sloped part of the left boundary of this region. One issue with RiskParity strategy is that. is the portfolio with the lowest risk (subject to constraints) on the efficient frontier. is a vector of 1s.e. The efficient frontier together with the minimum variance locus (i.t ≥ 0 where . No portfolios can be constructed corresponding to the points in this region.. the bottom portion of the hyperbola) form the ‘upper border’ and ‘lower border’ lines of the feasible set. in the sense that they have the best possible expected return for their level of risk. This can be implemented via the following optimization algorithm: arg min ω 1 ωt′ ∑ t ωt 2 subject to: ωt′ι = 1 ωt ≥ 0 where. The GlobalMinVar method aims to weight portfolios such that the overall portfolio risk is minimized without taking any particular view on expected returns. Global minimum variance portfolio (GlobalMinVar) The next portfolio allocation method we look at is the popular global minimum variance portfolio (GlobalMinVar). when the number of assets is large (e. a hyperbola. is then called the efficient frontier..t and rp . it becomes increasingly similar to an EquallyWgted or an InvVol. Thus.t are the returns of asset i and portfolio p at time t. it is also the portfolio with the lowest Page 12 Deutsche Bank Securities Inc. as in most equity portfolios). To the right the feasible set is determined by the envelope of all pairwise asset frontiers. greater than 100. The GlobalMinVar portfolio.t =1 i =1 ωi . volatility) – expected return space.30 May 2013 Portfolios Under Construction N ∑ω i . In theory. Points below the frontier are suboptimal.e. Such portfolios (without including the risk-free asset) can be plotted in risk (i. Efficient frontier Portfolios on the efficient frontier are efficient. ex ante. and Σt is the asset-by-asset covariance matrix at time t.g.. a rational investor will hold a portfolio only on the frontier. ωt ι is the vector of asset weights at time t. The region outside of the feasible set is unachievable by holding risky assets alone. GlobalMinVar portfolios are sensitive to our risk estimates and tend to be concentrated in a few assets. ri . . 020 0. it often tends to outperform many (if not most) active portfolios that try to take on more risk.025 MV | solveRquadprog -5e-04 GlobalMinVar -1e-03 Target Return[mean] Germ any 0. This allocation strategy attempts to create portfolios that are more diversified by maximizing the distance between the weighted average volatility of each underlying portfolio and the overall portfolio volatility. Figure 6 shows the efficient frontier and the GlobalMinVar portfolio.010 0. as we will show you in later sections.t σ i .00028 5e-04 MVO portfolio. and Spain). Portugal. Using our example of six countries (US. Figure 6: Mean-variance efficient frontier Efficient Frontier 0. MSCI.t n ωt′ ∑ t ωt subject to: ωt′ι = 1 ωt ≥ 0 Deutsche Bank Securities Inc.015 0. This can be shown in the following equation: arg max ω ∑ω i . Page 13 . we look at the maximum diversification (MaxDiversification) approach. Germany.000187 0e+00 Spain EWP Portugal Italy Greece 0.005 0. long only US 0. Deutsche Bank Quantitative Strategy Maximum diversification portfolio (MaxDiversification) Last. Greece. Italy.030 Target Risk[Cov] Source: Bloomberg Finance LLP.000 0. however. Ex post.30 May 2013 Portfolios Under Construction expected return. the MaxDiversification optimization is almost the same as GlobalMinVar – the difference is to replace the covariance matrix with the correlation matrix.t N ∑ξ j . ωi . no leverage. the MaxDiversification portfolio can be solved easily by minimizing . ψ ′Ρψ Step 1 1 arg min ω ψ t′Ρtψ t 2 subject to: ψ t′ι = 1 ψt ≥ 0 where. The denominator is the total portfolio volatility which takes into account the correlation between the underlying assets. and σ i2.t = ψ i . To maximize the overall ratio. The numerator is simply the weighted sum of the underlying asset volatilities.e.t = ξ i . Numerically. we rescale the second intermediate asset weight vector of the total weight. .t j =1 Page 14 Deutsche Bank Securities Inc.t σ i . so the sum of the final weights equal to 100%. Therefore.t as its i. Step 2 Then we need to rescale the first intermediate vector of asset weights ψ t by each asset’s volatility σ i.30 May 2013 Portfolios Under Construction The equation above warrants some further explanation.t : ξ t = Dt−1 / 2ψ t or ξ i . ξt is the second intermediate vector of asset weights at time Dt is the diagonal matrix of asset variance at time and zero on all off-diagonal elements t with t . the denominator containing the correlations must be minimized.t where. ψt is the first intermediate vector of asset weights at time Ρt is the asset-by-asset correlation matrix at time t . i th element Step 3 Finally. i. The difference between the two is essentially the correlation terms.. The final weights are then retrieved by rescaling the intermediate weight vector (optimized using the correlation matrix) with the standard deviations of the asset returns. where Ρ is the correlation matrix. and t. This allocation strategy attempts to select assets that minimize the correlation between the underlying assets and hence “maximize diversification” as the name suggests. Therefore. For example.30 May 2013 Portfolios Under Construction III. The median pairwise correlation can remain low. In this paper. One of the key ingredients in the calculation is to use high frequency return data5 to measuring the intraday trading behavior. which may still expose our portfolio into a liquidity crisis. rather than measuring strategy crowding.e. EM credit) in this paper do not have high frequency return data.. it is an indication of strategy crowding. 2) as we deal with global capital markets. Deutsche Bank Securities Inc. Page 15 . et al [2012].g. In this research. the approaches introduced in this paper are more to measure the efficacy or potential diversification opportunity of a particular strategy. The rationale is that when investors increasingly trade a basket of stocks together. we use one-minute return data for US stocks and 15-minute return data for other regions.. Similarly. Weighted portfolio correlation The median pairwise correlation (MPC) in Cahan et al [2012] does not take into account asset weights. we extend Cahan et al [2012] paper by taking into account asset weights. The other key difference is that we use daily return rather than high frequency return in our computation. we develop a more refined measure which we call “weighted portfolio correlation” or WPC. the MPC and MPTD of stocks within this basket will rise compared to the market. A portfolio’s risk is not a simple weighted average of each asset’s volatility. it suggests the potential diversification benefit is shrinking and we are holding a portfolio that is increasingly similar to the index. we have to be aware of unsynchronized trading issues. if the weighted portfolio correlation of a risk-based strategy is approaching the average correlation of the market (i. When the relative MPC (or MPTD) compared to the market is at certain level. but we could have owned a few heavily weighted and highly correlated assets in our portfolio. a capitalization weighted benchmark index). for two reasons: 1) many of the assets (e. we can’t calculate a portfolio’s weighted correlation as the weighted pairwise correlation. because the relationship is nonlinear. and more importantly. The variance of a portfolio can be computed as: N N −1 N i =1 i =1 j >i σ p2 = ∑ ωi2σ i2 + 2∑∑ ωiω jσ iσ j ρ ij where: σ p2 is the variance of the portfolio σi is the volatility of asset 5 i Specifically. we attempted to measure the crowdedness of a strategy using two approaches: median pairwise correlation (MPC) and median pairwise tail dependence (MPTD). Strategy crowding and efficacy In Cahan. then the above equation can be written as: N N −1 N N N −1 N i =1 i =1 j >i i =1 i =1 j >i σ p2 = ∑ ωi2σ i2 + 2∑∑ ωiω jσ iσ j ρ ij = ∑ ωi2σ i2 + 2∑∑ ωiω jσ iσ j ρ average Therefore. as shown in Choueifaty and Coignard [2008]. let’s quickly review the definition of DR: N ∑ω σ i DR = i i =1 σp = weightedAverageVolatility portfolioVolatility On the other hand. let’s assume ρ average (i. WPC is closely related to two other key concepts – diversification ratio (DR) and concentration ratio (CR). First. A fully concentrated long-only portfolio with only one asset has unit CR. the CR is defined as: N CR = ∑ω σ 2 i 2 i i =1 ⎛ N ⎞ ⎜ ∑ ωiσ i ⎟ ⎝ i =1 ⎠ 2 The concentration ratio or CR is a simple measure of portfolio concentration that only takes into account the volatility of each asset. N −1 N ∑∑ ω ω σ σ i j i j ρ ij i =1 j >i N −1 N WPC = ρ average = ∑∑ ω ω σ σ i j i j i =1 j >i Therefore. WPC) is the average pairwise correlation. finally. WPC is essentially the weighted pairwise correlation adjusted for asset volatility. WPC and portfolio diversification We can naturally expect a portfolio with a lower WPC to be more diversified. N −1 N N −1 N N −1 N i =1 j >i i =1 j >i i =1 j >i ∑∑ ωiω jσ iσ j ρij = ∑∑ ωiω jσ iσ j ρ average = ρ average ∑∑ ωiω jσ iσ j And. Indeed. while an InvVol portfolio has the lowest CR (which equals to the inverse of the number of assets it contains).30 May 2013 Portfolios Under Construction ωi is the weight of asset ρij is the correlation coefficient between asset i i and j Now. DR = Page 16 1 ρ average (1 − CR ) + CR Deutsche Bank Securities Inc.. .e. without taking into account asset volatility. MaxDiversification or more strictly speaking. Page 17 . we can also define a portfolio tail dependent coefficient that takes into account of asset weights. WPTD is essentially the weighted pairwise tail dependence. the most diversified portfolio. However. the weighted portfolio tail dependent coefficient (WPTD) is more challenging than WPC. which is why it is also called volatility parity. λij is the tail dependent coefficient between asset i and j. the diversification ratio can be improved by either reducing the concentration of the portfolio or decreasing the WPC. the MaxDiversification portfolio has the highest ex ante diversification ratio. To follow a similar logic to how we define WPC. By construction. The reason is that we can’t calculate a portfolio’s tail risk (based on tail dependence) the same as we calculate a portfolio’s variance. is also the correlation parity portfolio. Deutsche Bank Securities Inc. The InvVol portfolio is by construction the minimum concentration portfolio. we define WPTD as: N −1 N ∑∑ ω ω λ i λaverage = j ij i =1 j >i N −1 N ∑∑ ω ω i j i =1 j >i where.30 May 2013 Portfolios Under Construction Therefore. Weighted portfolio tail dependent coefficient (WPTD) Similar to WPC. Therefore. e. we build both portfolios without a maximum holding constraint (i. most investors simply use a vendor supplied risk model. There are three popular factor risk models: fundamental factor models (based on fundamental factors like value.30 May 2013 Portfolios Under Construction IV. At the asset allocation level. All risk models are also point-intime. All data used in this research are point-in-time. etc. Portfolio performance is also measured daily. CVaR. we use factor-based risk models in equities and some fairly naïve covariance matrices based on either the sample covariance matrix or the exponentially weighted moving averages for other asset classes.. the sample covariance matrix may not be as problematic as in the equity portfolios. Risk estimation and portfolio construction Portfolio backtesting setup For all portfolio backtestings in this paper. etc. one or a few assets may dominate the portfolio) and with a maximum holding constraint. et al [2011]. using the sample covariance matrix (or some simple transformations like an exponentially weighted moving average) is still the common practice by many academics and practitioners. In practice. In some occasions.g. We will discuss the impact of constraints on portfolio performance. using typically a rolling one year (or five years) of daily returns. the estimation error of the sample covariance matrix increases dramatically. etc.). however. All backtestings are completely out-of-sample. fully invested. At the multi-asset portfolio level. In most of our previous research. As discussed in Luo.g. e. et al [2010] and Luo. industrial production. All portfolios in this research paper are long only.. Choices of risk models Note that the above portfolio construction techniques require us to estimate the covariance matrix. either using commercial risk models or our own calculated covariance matrices. However. .) In practice. etc. statistical risk models (based on statistic techniques like principle component analysis or factor analysis). In addition. Northfield.) in particular. Axioma. we have to estimate the covariance matrix with limited data. which could hurt portfolio performance. all risk models are computed daily. therefore. covariance is typically assumed to be a given.. commodities prices. For example. as the number of assets grows. we follow the following procedures. momentum. investors typically use factor models to predict the covariance matrix. In the equity portfolio management space. we typically deal with a limited number of asset classes. factor-based risk models in the asset allocation space are not as well developed as in Page 18 Deutsche Bank Securities Inc. Sample covariance matrices tend to suffer from too much estimation error. and macroeconomic risk models (based on macroeconomic factors like inflation. Barra. quality. tail dependence. we will use the point-intime index constituents as our investment universe. In academic theory. consumer confidence. All portfolios are rebalanced monthly. which is essential for downside risk metrics (e. no wgt constraint 12% Axioma. 6 This is also where factor-based risk models are most well developed and adopted. Deutsche Bank Quantitative Strategy Country and sector risk models When we construct country or sector portfolios. Axioma’s risk model shows much stronger numerical stability. In this research. US. we can also leverage our equity risk models. Asia ex Japan. Let’s use US equities to demonstrate the impact of the risk models. Figure 7: Realized volatility 18% 16% 14% Sample. Our investment universe is the MSCI USA index. We compare the performance of four risk-based allocations: InvVol. Japan. and MaxDiversification using both the sample covariance matrix and Axioma’s medium horizon fundamental risk model. no wgt constraint 10% Sample. 7 We also run the backtesting using sample covariance matrices. As shown in Figure 7. For most of the equity portfolios (e. Countries and sectors are nothing but portfolios of single stocks. we primarily use Axioma’s medium horizon fundamental risk models for our portfolio backtesting in this paper7. Deutsche Bank Securities Inc. However. Page 19 . the GlobalMinVar portfolio constructed using Axioma’s risk model indeed delivers much lower realized volatility.g. GlobalMinVar.. emerging markets. as shown in Luo. Compustat. max wgt constraint Source: Axioma. structured risk models can still reduce portfolio noise and improve performance. Bloomberg Finance LLP. max wgt constraint 8% Axioma. we make extensive comparisons between various risk models in both multi-asset and equity portfolio contexts. and global equities). Please contact us for details. with limited choice for vendor risk models. Russell. et al [2012]. European. Equity portfolios This is where we would expect to see a clear benefit from a factor-based risk model6. RiskParity. Thomson Reuters.30 May 2013 Portfolios Under Construction the equity space. MSCI. especially for some of the more challenging optimizations like RiskParity. j = ωi′Σ i . Since most macro investors do not have access to a factor-based stock risk model. the Axioma-based risk model appear to be able to deliver lower CVaR (see Figure 9 ) and comparable tail dependence (see Figure 11). K N = ∑ Nk K . N 1+1 σ 1. ψ i. j is the N i (number of stocks in country/sector i ) x N j (number of stocks in country/sector j ) block matrix from the big N × N stock-by-stock covariance matrix. we use sample covariance matrices for all country/sector/industry portfolios. i. respectively. The results of using Axioma’s risk model are available upon request. ω2 . However. N 1+ N 2 ⎦⎜⎝ ω N 1+ N 2 ⎟⎠ L For real-world application. N 1+1 = (ω1 . N 1+ N 2 ⎤⎛ ω N 1+1 ⎞ ⎟ ⎥⎜ ω ⎜ ⎟ N + 1 2 ⎥ ⎜ L L ⎥ L ⎟ ⎟ ⎥⎜ L σ N 1. we can show that: ψ i . L . N 1+ N 2 L σ 2. N 1+ 2 ⎢σ σ 2. Performance comparison In terms of Sharpe ratio. jω j where. the Axioma-based risk model and sample covariance matrix produce similar results. ω N 1 )⎢ ⎢ L L ⎢ ⎣σ N 1. In terms of downside risk and diversification benefit. Then the covariance matrix between country (or sector) i and country (or sector) j can be computed as: ψ i. Σ i. and k =1 is the number of countries (or sectors) in our universe. Page 20 Deutsche Bank Securities Inc. portfolios constructed with sample covariance matrix seem to achieve higher diversification ratios consistently (see Figure 10). j ωi is the scalar covariance between country (or sector) i and country (or sector) is the vector that contains the weight of each stock in country (or sector) j.30 May 2013 Portfolios Under Construction Let’s assume there are N1 and N 2 stocks in country (or sector) i and country (or sector) j . where we have multiple countries (or sectors). . for the rest of the paper. GlobalMinVar seems to be more sensitive to covariance matrix and benefit more from the sample covariance matrix. for country portfolios. N 1+ 2 σ 1. N 1+1 σ N 1. j ⎡ σ 1. N 1+2 2 . as shown in Figure 8. Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.0 1.3 0. Compustat.0 0. Russell.0 Sample covariance Axiomaderived Source: Axioma.0 3. Russell.0 2.3 0.9 0. Page 21 .9 0.6 0.0 3.0 1.0 0.0 2. Deutsche Bank Quantitative Strategy Figure 10: Average diversification ratio MSCI ACWI countries US sectors MSCI World sectors European sectors MSCI World industries 3.0 0.6 0. Bloomberg Finance LLP. Deutsche Bank Quantitative Strategy Figure 9: CVaR/expected shortfall MSCI ACWI countries US sectors MSCI World sectors European sectors MSCI World industries 0% 0% 0% 0% 0% -4% -4% -4% -4% -4% -8% -8% -8% -8% -8% -12% -12% -12% -12% -12% Sample covariance -16% -16% -16% -16% -16% Axiomaderived Source: Axioma.0 2.0 0.0 2. MSCI.0 1. Thomson Reuters.0 0. MSCI.0 0.0 0.9 0.3 0.0 0.0 0.30 May 2013 Portfolios Under Construction Figure 8: Sharpe ratio MSCI ACWI countries US sectors MSCI World sectors European sectors MSCI World industries 0. Compustat. Thomson Reuters. Compustat.0 1. Russell.9 0.3 0.6 0. MSCI.6 0.0 3.9 0.0 2.3 0.0 1.6 0.0 Sample covariance Axiomaderived Source: Axioma. Bloomberg Finance LLP.0 3. Thomson Reuters. Bloomberg Finance LLP. As shown in Figure 13. Minimum volume ellipsoid (MVE) is definied in Venables and Ripley [2008]. shrinkage estimator (SHRINK). Deutsche Bank Quantitative Strategy Robust covariance estimation techniques As alternatives to the sample covariance matrix and factor-based risk models. in this paper. Page 22 Deutsche Bank Securities Inc. Shrinkage estimator (SHRINK) is defined in Schaefer. Greece. Thomson Reuters. For example. Bootstrap aggregation (BAGGED) is defined in Breiman [1996]. Thomson Reuters. orthogonalized Gnanadesikan-Kettenring estimator (OGK). Compustat. Portugal. Orthogonalized Gnanadesikan-Kettenring estimator (OGK) is defined Gnanadesikan and Kettenring [1972] and Maronna and Zamar [2002]. Figure 12: Correlation matrix (lower triangle = sample/upper triangle = MVE) US Germany Greece Italy Portugal Spain 100% 70% 31% 65% 52% 60% Germany 64% 100% 51% 90% 78% 86% Greece 31% 58% 100% 53% 52% 55% Italy 55% 90% 59% 100% 82% 90% Portugal 47% 81% 61% 86% 100% 82% Spain 54% 86% 59% 93% 86% 100% US Source: Bloomberg Finance LLP. Minimum covariance determinant estimator (MCD) is defined in Rousseeuw [1985] and Rousseeuw and van Driessen [1999]. Russell. Germany. Bloomberg Finance LLP.30 May 2013 Portfolios Under Construction Figure 11: Average weighted portfolio tail dependence MSCI ACWI countries US sectors MSCI World sectors European sectors MSCI World industries 60% 60% 60% 60% 60% 50% 50% 50% 50% 50% 40% 40% 40% 40% 40% 30% 30% 30% 30% 30% 20% 20% 20% 20% 20% 10% 10% 10% 10% 10% 0% 0% 0% 0% 0% Sample covariance Axiomaderived Source: Axioma. in Figure 12 shows one simple example of the six countries (US. The lower triangle is based on sample covariance and correlation. Deutsche Bank Quantitative Strategy Let’s use our MSCI ACWI country portfolios as an example to show the impact of robust covariance matrices in portfolio performance. MSCI. . and bootstrap aggregation (BAGGED). MSCI. and Spain). Compustat. while the upper triangle is based on the MVE robust covariance and correlation. the MCD and Shrinkage covariance matrices seem to produce the highest Sharpe ratio. Ledoit and Wolf [2003]. and Strimmer [2008]. we investigate five robust techniques to estimate the covariance matrix: minimum volume ellipsoid estimator (MVE). minimum covariance determinant estimator (MCD). Russell. Opgen-Rhein. Italy. the difference in correlation coefficient between US and Italy can differ by more than 10%. We can see some notable differences. 80 -8% Sample covariance 0. Almost all robust covariance matrices outperform the sample covariance matrix in terms of Sharpe ratio for InvVol. Compared to InvVol and RiskParity.6 30% Sample covariance 2.6 Sample covariance 25% Minimum volume ellipsoid Minimum covariance determinant Orthogonalized Gnanadesikan-Kettenring 20% 15% 10% Minimum volume ellipsoid Minimum covariance determinant Orthogonalized Gnanadesikan-Kettenring Shrinkage Shrinkage Bootstrap aggregation Bootstrap aggregation Source: Bloomberg Finance LLP. MaxDiversification. Deutsche Bank Quantitative Strategy Figure 15: Average diversification ratio Figure 16: Average weighted portfolio tail dependence 2. Figure 13: Sharpe ratio Figure 14: CVaR/expected shortfall 0. robust covariance matrices do not seem to help to reduce downside risk (see Figure 14) or tail dependence (see Figure 16). the maximum weight of any single asset can’t exceed a certain limit. RiskParity.30 May 2013 Portfolios Under Construction especially for the GlobalMinVar portfolio. MSCI. Compustat. Deutsche Bank Securities Inc. Compustat. countries. RiskParity. Russell. Thomson Reuters. GlobalMinVar. Deutsche Bank Quantitative Strategy Source: Bloomberg Finance LLP. Russell. Compustat. and equities). MSCI. MSCI. portfolios are assumed to be long only and fully invested. MSCI. Deutsche Bank Quantitative Strategy Constraints in portfolio construction In all portfolio simulations in this research. Compustat. and MinTailDependence) on three universes (multi-assets.e. Robust covariance matrices also help us build more diversified portfolios (see Figure 15). Russell. On the other hand.70 0.8 1. Thomson Reuters.50 0. The purpose of the maximum holding constraint is to reduce the exposure of a few highly concentrated positions. we use three simple examples to show the impact of the maximum holding constraint. i. We compare five portfolio construction techniques (InvVol.40 0.4 2..0 1.30 Sample covariance -10% Minimum volume ellipsoid Minimum covariance determinant Orthogonalized Gnanadesikan-Kettenring Shrinkage -12% -14% -16% Minimum volume ellipsoid Minimum covariance determinant Orthogonalized Gnanadesikan-Kettenring Shrinkage Bootstrap aggregation Bootstrap aggregation Source: Bloomberg Finance LLP. but also seems to be more sensitive to covariance estimation.60 0. One of the most common constraints used in practice is a maximum holding constraint. Thomson Reuters. In this section. Russell. Page 23 .2 2. Thomson Reuters. GlobalMinVar shows the best performance. and GlobalMinVar. Deutsche Bank Quantitative Strategy Source: Bloomberg Finance LLP. we focus on strategies that produce more “diversified” or less crowded portfolios for the rest of the research.0 0. there is no clear benefit or damage in imposing a maximum holding constraint. Compustat.2 0. Therefore. Deutsche Bank Quantitative Strategy Page 24 Deutsche Bank Securities Inc. MSCI.8 1.8 1.30 May 2013 Portfolios Under Construction As show from Figure 17 to Figure 20. Compustat.8 1.3 0.3 0.5 1. Bloomberg Finance LLP.9 0. Thomson Reuters.5 1.9 0.0 0. Russell.3 0.2 1.9 0. Bloomberg Finance LLP.6 0. Figure 17: Sharpe ratio Multi-asset US equities (MSCI USA) MSCI ACWI countries 1. Deutsche Bank Quantitative Strategy Figure 18: CVaR/expected shortfall MSCI ACWI countries Multi-asset US equities (MSCI USA) 0% 0% 0% -4% -4% -4% -8% -8% -8% -12% -12% -12% -16% -16% -16% No constraint Max wgt constraint Source: Axioma.0 No constraint Max wgt constraint Source: Axioma.2 1. . rather than data mining the “optimal” constraint.5 1. Russell.6 0. Thomson Reuters.6 0. MSCI. Russell.0 3.0 No constraint Max wgt constraint Source: Axioma. MSCI.5 1. Bloomberg Finance LLP.0 1. Compustat. Thomson Reuters.5 0. Russell.5 1.5 1. Page 25 . Bloomberg Finance LLP.0 1.0 2.5 2. Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.0 0. Compustat.0 2.0 3.0 0.0 2.5 0. Thomson Reuters. MSCI.0 0.30 May 2013 Portfolios Under Construction Figure 19: Average diversification ratio Multi-asset US equities (MSCI USA) MSCII ACWI countries 3. Deutsche Bank Quantitative Strategy Figure 20: Average weighted portfolio tail dependence Multi-asset US equities (MSCI USA) MSCI ACWI countries 40% 40% 40% 30% 30% 30% 20% 20% 20% 10% 10% 10% 0% 0% 0% No constraint Max wgt constraint Source: Axioma.5 2.5 0.5 2.0 1. which defeats the purpose of using a copula model in the first place. et al [2005]. for any Gaussian copula where correlation coefficient less than 100%. in the early days of applying copula models in credit risk. Copula models gained popularity in the 2000s. we try to build a portfolio that is as “diversified” as possible. with MinTailDependence. A more recent textbook explanation can be found in McNeil. the real beauty of a copula (i. A. which was subsequently blamed for the 2008 financial crisis (see Salmon [2009]). Copula models allow for the combination of multivariate dependence with univariate marginals. In a non-technical sense9. In Cahan. a copula model gives us the flexibility to model joint asset return distributions.. Introducing the copula model The copula model was first introduced by Sklar [1959]. Pfaff [2012] first introduced the minimum tail dependence portfolio (MinTailDependence) concept. Empirically. assets are more likely to fall at the 8 More formally speaking. For example. Gaussian copula) was chosen. model the joint distribution using a tcopula model. where diversification is measured by Pearson’s correlation. Joint Distributi on = Copula + Marginal Distributi on Therefore. Tail dependence coefficients among our six countries are clearly higher than Pearson correlation coefficients – as expected. modeling tail dependence) was ignored. [2011]. The Gaussian copula assumes no tail dependence. Figure 21 shows the difference between Pearson correlation and copula-based tail dependence.e. Li’s [2000] paper and the copula models used in the 2000s are more based on the Gaussian copula. tail dependence equals to zero. 9 See Meucci.g. Ironically. practical guide to copulas. et al [2012].com/sol3/papers. Li [2000] developed a well-known but controversial model for credit risk using exponentially distributed default times with a Gaussian copula.e.ssrn. Similar to MaxDiversification.30 May 2013 Portfolios Under Construction V. A short... Minimum tail dependence portfolio The purpose of MaxDiversification is to build a portfolio that is as diversified as possible. while at the same time. we could fit an exponential GARCH model for each asset’s marginal distribution. which assumes no tail dependence8. One may argue that the dependence in the left tail (e. the chance of both assets suffering extreme losses at the same time) is more relevant in risk management and portfolio construction. where “diversification” is measured by tail dependence. we show how to use a copula model to measure tail dependence. because the wrong type of copula (i. asset returns are almost never jointly multivariate normally distributed.cfm?abstract_id=1847864 Page 26 Deutsche Bank Securities Inc. comprehensive. For example. The Pearson correlation coefficient only measures the dependence between two random variables correctly if they are jointly normally distributed. A copula models the dependence between assets in a multivariate distribution. . http://papers. 30 May 2013 Portfolios Under Construction same time than the average. Some differences are strikingly large. For example, the Pearson correlation between Portugal and Spain is only 56%, which is quite normal compared to other pairs of countries. However, the tail dependent coefficient is 86%, which is clearly on the high end. Figure 21: Correlation versus tail dependence (lower triangle = correlation/upper triangle = copula tail dependence) US Germany Greece Italy Portugal Spain 100% 64% 31% 55% 47% 54% Germany 53% 100% 58% 90% 81% 86% Greece 25% 33% 100% 59% 61% 59% Italy 44% 72% 36% 100% 86% 93% Portugal 28% 53% 36% 58% 100% 86% Spain 44% 64% 36% 81% 56% 100% US Source: Bloomberg Finance LLP, Compustat, MSCI, Russell, Thomson Reuters, Deutsche Bank Quantitative Strategy To visually examine how Pearson’s correlation coefficient underestimates the true dependence, let’s compare the theoretical bivariate normal distribution between Portugal and Spain (see Figure 22), with the empirical distribution (see Figure 23). The empirical distribution between Portugal and Spain is clearly bimodal with two distinct peaks (or modes), i.e., the probabilities of these two countries both move higher or fall lower are much higher than any other combinations. Figure 22: Theoretical bivariate normal distribution Figure 23: Empirical distribution nction Density fu nction Density fu a rtug ain ain Sp Sp Po l Source: Bloomberg Finance LLP, MSCI, Deutsche Bank Quantitative Strategy Po a rtug l Source: Bloomberg Finance LLP, MSCI, Deutsche Bank Quantitative Strategy Minimum tail dependence portfolio optimization algorithm Numerically, the MinTailDependence portfolio can be solved easily by minimizing , where Τ is the tail dependence matrix. Therefore, the MinTailDependence optimization is almost exactly the same as MaxDiversification, by replacing the correlation matrix with tail dependence matrix. The final weights are then retrieved by ψ ′Τψ Deutsche Bank Securities Inc. Page 27 30 May 2013 Portfolios Under Construction rescaling the intermediate weight vector (optimized using the tail dependence matrix) with the standard deviations of the assets’ returns. Step 1 1 arg min w ψ t′Τtψ t 2 subject to: ψ t′ι = 1 ψt ≥ 0 where, ψt is the first intermediate vector of asset weights at time Τt is the asset-by-asset tail dependence matrix at time t , and t. Step 2 Then we need to rescale the first intermediate vector of asset weights ψ t by each asset’s volatility σ i,t : ξt = Dt−1 / 2ψ t or ξ i ,t = ψ i ,t σ i ,t where, ξt is the second intermediate vector of asset weights at time Dt is the diagonal matrix of asset variance at time zero on all off-diagonal elements t with t , and σ i2,t at its i, i element and Step 3 Finally, we rescale the second intermediate asset weight vector of the total weight, so the sum of the final weights equal to 100%, i.e., no leverage. ωi ,t = ξi ,t N ∑ξ j ,t j =1 The choice of copula model – does it really matter? There is a long list of copula models proposed in the academic literature. In this research, we test two copula models: Non-parametric estimation using the empirical tail copula Stable tail function Interestingly, the results of the two copula model are almost identical (see Figure 24 and Figure 25). For the rest of this paper, we use the empirical tail copula as our main approach. Page 28 Deutsche Bank Securities Inc. 30 May 2013 Portfolios Under Construction Empirical copula EVT copula Source: Bloomberg Finance LLP, Compustat, MSCI, Russell, Thomson Reuters, Deutsche Bank Quantitative Strategy Empirical copula Global Average Japan Emerging markets Asia ex Japan US Europe Industries, MSCI Regions x sectors,… Sectors, US Sectors, Europe Multi-assets Global Average Japan Emerging markets Asia ex Japan US Europe Industries, MSCI Regions x sectors,… Sectors, US Sectors, Europe Sectors, MSCI Countries, MSCI… Commodities Alternative betas Multi-assets Sovereign bonds Countris, MEAM 0.00 (0.40) Commodities 0.40 Alternative betas 0.80 Sovereign bonds 0.0% (3.0%) (6.0%) (9.0%) (12.0%) (15.0%) (18.0%) (21.0%) 1.20 Sectors, MSCI 1.60 Countries, MSCI… Figure 25: CVaR/expected shortfall Countris, MEAM Figure 24: Sharpe ratio EVT copula Source: Bloomberg Finance LLP, Compustat, MSCI, Russell, Thomson Reuters, Deutsche Bank Quantitative Strategy An alternative minimum tail dependence strategy In this research, we also propose another form of MinTailDependence (hereafter called AltMinTailDependence). In the AltMinTailDependence set up, it is related to both MinTailDependence and GlobalMinVar. Essentially, we take the variance matrix (a diagonal matrix) and the tail dependence matrix (similar to correlation matrix, but replace pairwise correlation coefficient with tail dependent coefficient) to compute the covariance-tail dependence matrix. Finally, we perform a GlobalMinVar optimization using the covariance-tail dependence matrix. As shown in Figure 26 and Figure 27, the MinTailDependence portfolio is similar to MaxDiversification, while the AltMinTailDependence approximates the GlobalMinVar portfolio. The AltMinTailDependence strategy offers higher Sharpe ratios (see Figure 28) and lower downside risk (see Figure 29). However, the MinTailDependence, which follows the similar philosophy as MaxDiversification, does offer better diversification, as measured by higher diversification ratio (see Figure 30) and lower tail dependence (see Figure 31). For the rest of the paper, we mainly use MinTailDependence. Deutsche Bank Securities Inc. Page 29 30 May 2013 Portfolios Under Construction Figure 26: Cluster analysis Figure 27: Eigenvalue ratio plot Cluster Dendrogram Eigenvalue Ratio Plot 0. Europe Industries.10 0. Compustat. Compustat.25 MinTailDependence 0.… Sectors. US Sectors.00 (0. . Thomson Reuters. Russell. MSCI. MSCI.15 0.0%) (21. Deutsche Bank Quantitative Strategy Countris. Compustat. MSCI. "complete") 0.0%) (12.60 Sectors.80 Multi-assets 1. MSCI Figure 28: Sharpe ratio Countries. Russell.2 0.0 GlobalMinVar pearson -0.4 -0.6 -0. Russell. Deutsche Bank Quantitative Strategy Alt MinTailDependence Source: Bloomberg Finance LLP.0%) (15. Thomson Reuters.0 0. MSCI Countries.0%) (9. MSCI… Source: Bloomberg Finance LLP. MEAM Source: Bloomberg Finance LLP.0%) (18. MSCI Sectors. Compustat.20 Sovereign bonds 0. US Global Average Japan Emerging markets Asia ex Japan US Europe Regions x sectors.20 0.0%) 1.0% (3. Thomson Reuters.40 Alternative betas 0.2 0.4 MaxDiv ersif ication AltMinTailDependence -0.0%) (6. Thomson Reuters.2 Risk-based allocation hclust (*.4 -0.2 Eigenvalue 2 MinTailDependence AltMinTailDependence GlobalMinVar MaxDiversification 0.05 Height 0. MSCI. MSCI Regions x sectors. Deutsche Bank Quantitative Strategy Page 30 MinTailDependence Global Average Japan Emerging markets Asia ex Japan US Europe Industries.… Sectors. MEAM 0.4 Eigenvalue 1 MinTailDependence Alt MinTailDependence Source: Bloomberg Finance LLP.6 -0.40) Commodities 0. Europe Figure 29: CVaR/expected shortfall Sectors. Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc. MSCI… Alternative betas Commodities Sovereign bonds Multi-assets Countris. Russell. MSCI Commodities Alternative betas Multi-assets MinTailDependence Sovereign bonds 0. Figure 32: Cluster analysis Figure 33: Eigenvalue ratio plot Eigenvalue Ratio Plot 0. Thomson Reuters. Thomson Reuters. MSCI… 0. Rather. "complete") Source: Bloomberg Finance LLP. Russell. Russell. As a demonstration.00 1.0% 2. MSCI.14 Cluster Dendrogram 0. Thomson Reuters. Deutsche Bank Quantitative Strategy Tail-dependence constraint We don’t have to strictly construct a MinTailDependence portfolio.00 48.2 Eigenvalue 2 MinTailDependence -0.6 Risk-based allocation hclust (*.2 0. Europe Sectors.… Sectors. Compustat.00 Alt MinTailDependence MinTailDependence Source: Bloomberg Finance LLP.4 0. Thomson Reuters.0% 2. MSCI Regions x sectors. MSCI.0% 0. pearson -0. Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc. Compustat. the constrained MaxDiversification portfolio more behaves like the MinTailDependence portfolio. MSCI. MSCI.0% Countries. the more tilt we add towards the tail dependence. Russell. US Sectors.2 0. Compustat.00 Countris.50 36. we add a low (and a high) tilt in the MaxDiversification optimization towards tail dependence. Russell.0 LowTilt -0.6 LowTilt MaxDiversification MinTailDependence HighTilt 0.0% 1. Compustat.2 0.6 MaxDiv ersif ication -0.30 May 2013 Portfolios Under Construction Figure 30: Average diversification ratio Figure 31: Average portfolio tail dependence 3.4 -0. the MaxDiversification portfolio.0 0. Deutsche Bank Quantitative Strategy Global Average Japan Emerging markets Asia ex Japan US Europe Industries.02 Height 0. for example. Deutsche Bank Quantitative Strategy Page 31 . we could add tail dependence as a constraint to.10 0.06 0.6 Eigenvalue 1 Source: Bloomberg Finance LLP.50 24. As shown in Figure 32 and Figure 33.4 HighTilt -0. MEAM 12.4 0.50 Alt MinTailDependence Source: Bloomberg Finance LLP. The cumulative distribution function of the loss associated with a weight vector ω is therefore: () Ψ (ω . r ) = ( ) ∫ p(r )dr f (ω . the VaRα associated with portfolio W is: VaRα = min{γ ∈ R : Ψ (ω . The asset return distribution is defined by vector r . measured by assessing the likelihood (at a specific confidence level) that a specific loss will exceed the VaR. r . we will apply CVaR optimization in a more alpha-seeking context. In upcoming research. maximizing expected returns subject to CVaR constraints. while VaR is not necessarily convex10.r )≤γ For a given confidence level α .30 May 2013 Portfolios Under Construction VI. On the other hand.e. Therefore. CVaR is derived by taking a weighted average between the VaR and losses exceeding the VaR. while global optimizations tend to be very slow. we could have a better model. VaR or value at risk is criticized as being an incoherent risk measure. our decision variable). as shown in Rockafellar and Uryasev [2001].. Page 32 Deutsche Bank Securities Inc. In portfolio optimization. we only explore a simple application of a minimum CVaR portfolio. [2001]) first introduced portfolio optimization with CVaR or conditional value at risk. let’s define ω as the vector of asset weights (i. Rockafellar and Uryasev ([2000]. Despite the theoretical soundness of the CVaR methodology. i. In the risk management literature. [2001]). More importantly. CVaR. CVaR optimization theory Here we will only give a very brief description of the CVaR optimization problem and corresponding algorithm. Please note that CVaR optimization is very flexible and powerful.. but may have more estimation error. r ) ≥ α } Once VaRα is defined.. in practice.e. In this research. It is the same trade-off between model error versus estimation error. which tends to be easier to solve. CVaR optimization can be transformed into linear optimization. Basic definition of CVaR optimization In the CVaR optimization setup. a global optimizer is typically required. is a coherent risk measure. CVaR is a convex function. we can further define CVaRα as: 10 The problem with non-convex optimization is that we may get local optima instead of global optima. Mathematically speaking. A more detailed exposure can be found in Rockafellar and Uryasev ([2000]. The loss function is further defined as f ω . we have to estimate CVaR empirically and face all the usual problems with estimation errors.e. Minimum CVaR portfolio CVaR or conditional value at risk is a statistical measure of tail risk. . i. We assume random vector r has a probability density function of p r . which is also referred as expected shortfall. r ) − γ )+ = max(( f (ω .2. let’s develop a simple auxiliary function: Fα (ω . r ) − γ ) + s s =1 Now. γ ) with respect to the Discretization It is generally more desirable to approximate the continuous joint density function p r with a number of discrete scenarios. e. Page 33 . S . r ) p(r )dr f (ω . rs for s = 1. rs is a linear function of ω . Then. Deutsche Bank Securities Inc. Fα (ω . r ) − γ ) p(r )dr ∫ 1−α (ω ) γ + f . r ) − γ ).0) The CVaRα optimization can be done by optimizing the weight vector ω and VaRγ . the minimum CVaR problem can be approximated by: arg min ω = γ + 1 (1 − α )S S ∑ ( f (ω . which typically represent historical returns. Then we can transform the Fα ω . r ) − γ ) p(r )dr = γ + 1 ( f (ω .r ≥ where. the minimum CVaR optimization problem can be solved by linear programming algorithms. L . γ ) = γ + 1 (1 − α )S S ) ∑ ( f (ω. ( f (ω .g.. Then the S ∑z s s =1 subject to: z s ≥ f (ω . γ ) = γ + 1 1−α ( f (ω .r )≤VaRα (ω ) The minimum CVaR optimization can therefore be defined as: arg min ω CVaRα (ω ) Subject to: ωt′ι = 1 ωt ≥ 0 The solution to CVaR optimization First.30 May 2013 Portfolios Under Construction CVaRα (ω ) = 1 1−α ∫ f (ω . rs ) − γ zs ≥ 0 ( ) It is typically further assumed that f ω . rs ) − γ )+ . γ function into: () ( Fˆα (ω . r ) − γ ) + s s =1 Let’s further introduce an artificial variable z s to replace minimum CVaR problem can be transformed into: arg min ω = γ + 1 (1 − α )S ( f (ω . 5%. long only US 0.03 0. Deutsche Bank Quantitative Strategy Choice of alpha parameter In MinCVaR optimization. the MinCVaR portfolio). Page 34 Deutsche Bank Securities Inc.01 0.05 CVaR | solveRglpk -5e-04 MinCVaR portfolio -1e-03 Target Return[mean] Germ any 0.e. one of the key input parameters is the confidence level α .30 May 2013 Portfolios Under Construction Mean-CVaR efficient frontier and minimum CVaR portfolio Similar to the mean-variance efficient frontier. We further set the α at 1%. let’s first experiment with three different levels of α and study the impact of parameter sensitivity. . The choice of α is more art than science. the MinCVaR portfolio is not very sensitive to the choice of α . and 10%. all three portfolios significantly outperform the capitalization weighted benchmark. We use our country portfolio as an example by investing in the 45 countries comprising the MSCI ACWI. Figure 34: Mean-CVaR efficient frontier Efficient Frontier 0. MSCI. Therefore.000158 0e+00 Spain EWP Portugal Italy Greece 0. More importantly.00 0. The portfolio that separates the efficient frontier from the lower border of the feasible set is the global minimum CVaR portfolio (i.04 0. As shown in Figure 35 and Figure 36. they have the highest expected returns.. we can define the mean-CVaR efficient frontier as a hyperbola containing portfolios with the following characteristics: for given level of risks (defined as CVaR).02 0.06 Target Risk[CVaR] Source: Bloomberg Finance LLP.00028 5e-04 Mean-CVaR portfolio. The family of multivariate skew-t distributions is an extension of the multivariate Student’s t family. 1% conf Min CVaR. 10% conf Source: Bloomberg Finance LLP. i.30 May 2013 Portfolios Under Construction Figure 35: Sharpe ratio Figure 36: CVaR/expected shortfall . for these 45 countries at each month end. this is probably the first serious attempt at combining the philosophy of robust optimization and CVaR optimization in one integrated framework. we calculate the expected CVaR. and Deutsche Bank Securities Inc.8 -.6 -. Robust minimum CVaR optimization We simulate 50 time series of the same five years of daily country returns with the above fitted multivariate skew-t distribution. Deutsche Bank Quantitative Strategy Robust minimum CVaR optimization In this paper. as our final portfolio. To the best of our knowledge.110 .2 -. Deutsche Bank Quantitative Strategy Benchmark Min CVaR. Page 35 . 5% conf Min CVaR. The mean-CVaR optimization relaxes the distribution assumption and therefore is far more flexible than MVO. Fitting a multivariate skew-t distribution To fully account for the nature of non-multivariate normal distribution in asset return data. 5% conf Min CVaR. MSCI.100 SharpeRatio . 2003]). The most conservative portfolio (RobMinCVaR-Conservative): for each of the 50 MinCVaR portfolios..120 . MSCI. We borrow the ideas from robust optimization (see Michaud [1998]) and traditional CVaR optimization above. We then further construct our final portfolio using three approaches: Average (RobMinCVaR-Avg): we simply average the weights of the 50 asset weight vectors. Then we can construct 50 MinCVaR portfolios – one for each simulated data. 10% conf Source: Bloomberg Finance LLP. As shown in previous sections. then we take the portfolio with the lowest CVaR. we propose a new optimization technique that we shall call “robust minimum CVaR optimization” or RobMinCVaR.115 modifiedCVaR_ES95 -.0 Benchmark Min CVaR. The fits are done using maximum likelihood estimation (see Azzalini and Capitanio [1999. Test of multivariate normal distribution Despite the popularity of mean-variance optimization in academic research and professional investment management. using five-years of rolling daily returns to a multivariate skew-t distribution.4 -. one of the key assumptions under MVO (meanvariance optimization) is the multivariate normal distribution. 1% conf Min CVaR. we fit our 45-country return data at each month end. the worst case scenario portfolio.105 . via the introduction of a shape parameter which regulates skewness.e. financial assets almost never follow a multivariate normal distribution. conservative Source: Bloomberg Finance LLP.30 May 2013 Portfolios Under Construction The most optimistic portfolio (RobMinCVaR-Optimistic): for each of the 50 MinCVaR portfolios. If we run 50 simulations over the past 14 years. The RobMinCVaR-Conservative portfolio. . with higher Sharpe ratios (see Figure 37) and slightly higher downside risks (see Figure 38). 5% conf Robust min CVaR. with 45 countries. Figure 37: Sharpe ratio Figure 38: CVaR/expected shortfall .6 -.8 -. then we take the portfolio with the highest CVaR. the most optimistic case scenario portfolio. 1% conf Min CVaR. it can take around three days for a complete backtesting11. MSCI. avg Robust min CVaR.2 -. 10% conf Robust min CVaR. optimistic Min CVaR. optimistic Min CVaR. even compared to all other risk-based allocation techniques. All three RobMinCVaR portfolios outperform the traditional MinCVaR strategy. it takes about 30 seconds per period.105 . Deutsche Bank Quantitative Strategy Numerical and computational issues will be addressed in Section VII on page 40. i. in particular. we fix α at the 10% level.110 -.. For example. we calculate the expected CVaR.120 . 10% conf Robust min CVaR.0 Benchmark Min CVaR.4 -.e. The biggest challenge in our RobMinCVaR optimization is computational speed. the RobMinCVaR portfolio shows the highest Sharpe ratio (see Figure 39) and second lowest downside risk (see Figure 40). as our final portfolio In the following simulation. Deutsche Bank Quantitative Strategy 11 modifiedCVaR_ES95 Benchmark Min CVaR. Page 36 Deutsche Bank Securities Inc. MSCI. avg Robust min CVaR. It can be slow even when the number of assets is modest. 1% conf Min CVaR.100 SharpeRatio . Indeed. conservative Source: Bloomberg Finance LLP. 5% conf Robust min CVaR.115 . shows a decent Sharpe ratio. Compustat. On the other hand. Within this cluster. MaxDiversification and MinTailDependence portfolios are similar.0 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Robust min CVaR Source: Axioma. Bloomberg Finance LLP. Deutsche Bank Securities Inc.8 -. Bloomberg Finance LLP. which further joined by GlobalMinVar and RobMinCVaR portfolios. These algorithms connect "objects" to form "clusters" based on their distance. EquallyWgted. our clustering algorithm essentially divides our risk-based strategies (and benchmark portfolio) into two broad clusters: Cluster I – low diversification strategies (red).12 . The core idea is that objects are more related to nearby objects than to objects farther away. since both attempt to obtain better diversification. as both try to achieve better risk reduction. At different distances. Page 37 . MinTailDependence resembles MaxDiversification. Compustat. Cluster analysis Cluster analysis is a suite of statistical algorithms that divide data into groups (called clusters) in such a way that objects in the same cluster are more similar (in some sense or another) to each other than to those in other clusters. In a dendrogram. Russell.16 . It is also interesting to note that our RobMinCVaR is more closely linked to GlobalMinVar. InvVol.14 . clearly. One particular form of clustering algorithm that is useful for our purpose is hierarchical clustering. different clusters will form to a dendrogram or an upside-down tree. Russell. As shown in Figure 41. MSCI ACWI).10 SharpeRatio modifiedCVaR_ES95 . and finally joined with the benchmark. Thomson Reuters. Deutsche Bank Quantitative Strategy How is RobMinCVaR related to other risk-based portfolio construction techniques In this section.e. then joined with EquallyWgted. RobMinCVaR emerges to deliver better ex post performance. InvVol and RiskParity are closer to each other. MSCI. while the objects are placed along the horizontal-axis. Deutsche Bank Quantitative Strategy Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Robust min CVaR Source: Axioma. the vertical-axis marks the distance at which the clusters merge. Cluster II – high diversification strategies (green). MSCI. we introduce two interesting statistical tools that can help us better understand various portfolio construction techniques and see how they relate to each other.30 May 2013 Portfolios Under Construction Figure 39: Sharpe ratio – risk-based allocations Figure 40: CVaR/expected shortfall . On the other hand.6 -. On the one hand..2 -. Thomson Reuters. capitalization weighted benchmark (i. and RiskParity portfolios form one cluster.4 -. 30 May 2013 Portfolios Under Construction Figure 41: Cluster analysis InvVol Cluster II – low diversification strategies RiskParity Cluster I – high diversification strategies EquallyWgted RobMinCVaR GlobalMinVar MinTailDependence MaxDiversification 0. Deutsche Bank Quantitative Strategy Eigenvalue ratio test Another approach to group strategies together is based on eigenvalue analysis.40 Cluster Dendrogram Risk-based allocation hclust (*. On the one hand. GlobalMinVar. Cluster II – low diversification strategies (yellow).. . Cluster III – high diversification strategies (green). the eigenvalue ratio chart in Figure 42 classifies our strategies into three buckets: Cluster I – benchmark (red). Page 38 Deutsche Bank Securities Inc. MSCI.30 0.e. "complete") Source: Bloomberg Finance LLP. and MaxDiversification portfolios are similar. which further joined by MinTailDependence portfolio.10 0. RobMinCVaR. The capitalization weighted benchmark (i. MSCI ACWI) seems to form a cluster of its own.05 0. then joined by the RiskParity portfolio. We calculate the first two eigenvectors of the correlation matrix and then plot their components against the x and y directions. On the other hand.35 0.20 Benchmark 0.15 Height 0.25 0. InvVol and EquallyWgted portfolios are closer to each other. Similar to the cluster analysis graph. 6 -0. MSCI.0 RiskParity -0.30 May 2013 Portfolios Under Construction Figure 42: Eigenvalue ratio chart Eigenvalue Ratio Plot 0.2 Eigenvalue 2 MinTailDependence pearson -0.6 Benchmark -0.4 Eigenvalue 1 Source: Bloomberg Finance LLP.0 0. Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.2 RobMinCVaR MaxDiversification 0. Page 39 .2 0.2 0.4 InvVol EquallyWgted -0.4 -0.4 GlobalMinVar 0. initial guess of what the weight vector looks like) is critical for both convergence and computing time.e. minimum weight.. A maximum number of holdings constraint is even more difficult.g. or a global optimization solver). sector.5 hours 14 hours 500 assets 1600 assets Source: Deutsche Bank Quantitative Strategy Among the four optimizations (GlobalMinVar.5G CPU for different optimization under Linux environment using multicore packages in R12. InvVol.org Page 40 Deutsche Bank Securities Inc. a quadratic programming solver. and there are no convergence issues. the maximum turnover constraint is a nonlinear constraint and will require a nonlinear solver..5 hours 6. maximum number of holdings. we know it is a completely different matter. The following table shows the approximate time for different number of assets with monthly rebalance over approximately 14 years. maximum turnover. Risk Parity takes the most time compared to the other optimization techniques. On the other hand. Many of the optimization problems require an asset-by-asset covariance matrix (e. we may need to use a mix of different solvers. finding a set of good initial values (i. using eight-core 2. GlobalMinVar.5 hours 2. Most academic research focuses on unconstrained optimization. Figure 43: Numerical computing time (for complete 14 year backtest) Pptimization MaxDiversification GlobalMinVar MinTailDependence RiskParity 20 minutes 20 minutes 30 minutes 50 minutes 2. RiskParity.. industry. maximum/minimum weight by some grouping (e. which tends to be very slow. As discussed in Luo.r-project.g. For example. MaxDiversification.30 May 2013 Portfolios Under Construction VII.g. because it requires a mixed integer solver. while in practice. Some constraints are easier to handle. a nonlinear programming solver. .. In practice. a linear programming solver. using the Higham [1988] algorithm. a mixed integer programming solver. for those researchers who do code their own optimization algorithms. et al [2011b]. sometimes the covariance matrix is not positive definite. A yet better approach is to use either a factor-based risk model or a robust covariance matrix. Depending on the problem at hand. data is well behaved. country) are typically set as box constraints. we typically have a maximum holding constraint. maximum/minimum weight per asset. We have to choose a solver for a specific task (e. It is typically assumed that there are only a few assets involved in the optimization.g. which can be solved easily. and MinTailDependence). e. If we have to compute the covariance matrix using sample data. There could be more constraints. MaxDiversification). RiskParity. Numerical and computational issues Numerical and computational issues are rarely addressed in academic research. A simple solution is the find the closest matrix that is positive definite. MinTailDependence takes more time than the MaxDiversification and GlobalMinVar. 12 See www.. RiskParity. For example. For example. as portfolio resampling/simulation are required. RiskParity takes 2.. For the more computationally difficult problems.g. The most time consuming optimization is CVaR optimization. Deutsche Bank Securities Inc. the computing time grows even faster as the number of asset increases.30 May 2013 Portfolios Under Construction The computational time grows exponential with the number of assets. but takes more than 5x longer with 1600 assets. for the Risk Parity optimization. e. 1600 assets take more than 15 times longer than 500 assets.5x longer than MaxDiversification for 500 assets. Page 41 . especially our RobMinCVaR optimization. Figure 44: MSCI ACWI Developed countries Americas Canada USA EMEA Asia Austria Australia Belgium Hong Kong Denmark Japan Finland New Zealand France Singapore Germany Greece Ireland Israel Italy Netherlands Norway Portugal Spain Sweden Switzerland UK Emerging markets Americas Brazil EMEA Asia Czech Republic China Chile Egypt India Colombia Hungary Indonesia Mexico Morocco Korea Poland Malaysia Russia Philippines Peru South Africa Taiwan Turkey Thailand Source: MSCI. Results with other risk models are qualitatively similar and available upon request. 16 All analyses below use sample covariance matrix as the risk model.30 May 2013 Portfolios Under Construction VIII. 14 Dividends are re-invested on the ex-dividend dates. 15 The strategies are therefore currency unhedged. Page 42 Deutsche Bank Securities Inc. MSCI ACWI 16 . et al [2013]. let’s compare the performance of various risk-based portfolio construction techniques to the benchmark. We use daily total returns14 in USD15 to perform all portfolio backtests below. et al [2012]. All strategies are monthly rebalanced. . Deutsche Bank Quantitative Strategy Comparison of risk-based allocation First. “Independence day”. In this section. “New insights in country rotation” and Luo. we show all major 13 This is further to extend our previous research in Luo. Our sample includes 45 countries comprising the MSCI All Country World Index 13 (see Figure 44 for the full list of countries). Country portfolios Let’s use country equity portfolio as our first example. Russell. value at risk (see Figure 50). all three performance metrics (Sharpe ratio. Deutsche Bank Quantitative Strategy Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP. Please contact us for details. MaxDiversification. Thomson Reuters. all four measures of downside risk are also consistent. we will only show some key statistics.g.. Deutsche Bank Quantitative Strategy 17 We actually have a much more complete list of statistics for each strategies mentioned in this paper.2 1 . More refined analysis about portfolio crowding and efficacy will be addressed at a later section. Interestingly. Russell.0 0 99 00 01 02 03 04 05 Benchmark Inverse Vol Global min variance Min tail dependence 06 07 08 09 10 11 12 13 Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP. InvVol. RiskParity. Figure 53 and Figure 54 show some basic statistics of portfolio concentration and diversification. and conditional value at risk/expected shortfall (see Figure 51). Deutsche Bank Securities Inc. MSCI.4 2 .30 May 2013 Portfolios Under Construction performance metrics: Sharpe ratio (see Figure 46). EquallyWgted. and MinTailDependence. worst monthly return (see Figure 49). For our universe of 45 countries. Compustat. Page 43 . MSCI. Compustat. for completeness17. For most other portfolio backtests in the next few sections.6 3 . e. Figure 45: Wealth curve Figure 46: Sharpe ratio 5 . Thomson Reuters. CVaR/expected shortfall. Sortino ratio (see Figure 48). indicating GlobalMinVar has the lowest tail risk. adjusted Sharpe ratio. to the more sophisticated techniques like GlobalMinVar.8 4 SharpeRatio . Sharpe ratio. adjusted Sharpe ratio (see Figure 47). and Sortino ratio) are consistent – we see a nice monotonic increase from the benchmark portfolio. MSCI. Russell.35 adjustedSharpe . Compustat. Thomson Reuters.20 . Deutsche Bank Quantitative Strategy Source: Bloomberg Finance LLP. MSCI. Compustat.12 -.28 -. MSCI. Compustat. MSCI.60 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.5 .25 . Compustat. Deutsche Bank Quantitative Strategy Source: Bloomberg Finance LLP.30 . Russell. Deutsche Bank Quantitative Strategy Figure 49: Worst monthly return Figure 50: Maximum drawdown -. Deutsche Bank Quantitative Strategy Figure 51: Value at risk Figure 52: Conditional value at risk/expected shortfall -. MSCI. Thomson Reuters.10 -.06 -. Thomson Reuters.10 modifiedVaR modifiedCVaR_ES95 -. Russell.55 -.08 -.10 Benchmark Inverse Vol Global min variance Min tail dependence SortinoRatio Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.3 .45 -.7 . Russell.16 -.6 . Compustat. Thomson Reuters.14 -. . Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.15 . Russell.16 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.40 minimum maxDD -. Compustat.20 -. Thomson Reuters. Deutsche Bank Quantitative Strategy Page 44 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.50 -.07 -. MSCI. Thomson Reuters. Russell.2 .24 -.09 -.30 May 2013 Portfolios Under Construction Figure 47: Adjusted Sharpe ratio Figure 48: Sortino ratio .4 . 09M12 NO .08M06 NO .99M12 NO .10M06 .08M12 NO . as shown in Figure 58 (and more details in Section X on page 70). A related question is whether we can replicate the same low risk equity portfolios using sectors/industries. MSCI. However.12M12 Benchmark Inverse Vol Global min variance Min tail dependence .00M12 NO .10M06 NO .. which tends to be much easier (operationally simpler) than an equity portfolio.07M12 . our risk-based country portfolios capture almost all the benefit of the socalled traditional low risk anomaly. As shown from Figure 55 to Figure 58.08M12 . sector/industry portfolios tend to be less effective than country portfolios.04M06 NO .10M12 . MSCI..04M12 .10M12 NO .08M06 .06M12 .05M06 NO . the Sharpe ratio of our country portfolios is very much in line with the ones realized using global developed equities (i.07M12 NO .01M06 NO .04M06 .00M06 NO .03M06 NO . Page 45 .12M06 NO .11M12 .02M06 NO . ETFs.04M12 NO . which typically is constructed with single stocks.07M06 . or custom swaps.00M06 .12M12 0 0 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.11M12 NO .e. Compustat.02M12 .02M12 NO . Deutsche Bank Quantitative Strategy MAX MAX MAX MAX MAX MAX MAX MAX MAX MAX MAX MAX MAX MAX MAX MAX MAX MAX MAX MAX MAX MAX MAX MAX MAX MAX MAX NO .e.11M06 NO .03M12 NO . Managing a portfolio of countries can be achieved by investing in country futures.07M06 NO .03M06 .00M12 .01M12 .05M12 NO .09M06 NO .09M06 . Thomson Reuters. Russell.06M06 NO .01M06 . Russell.99M12 .06M12 NO .02M06 .03M12 . Thomson Reuters.05M12 .11M06 . Compustat.05M06 .06M06 .01M12 NO .30 May 2013 Portfolios Under Construction Figure 53: # of assets in the portfolios Figure 54: Highest weight 50 60 40 40 30 20 20 10 Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.09M12 .12M06 . Deutsche Bank Securities Inc. MSCI EM). It appears that country effect still dominates the sector/industry effect. Deutsche Bank Quantitative Strategy Fully capturing the benefit of the low risk anomaly at the stock level Interestingly. MSCI World) and emerging markets (i. 2 . Group II – high diversification strategies. On the other hand. Compustat. the capitalization weighted benchmark. the GlobalMinVar portfolio.6 . and RiskParity form one group.6 0. Compustat.4 0.0 0.8 SharpeRatio . Russell.0 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Axioma. Thomson Reuters. Within this group. Deutsche Bank Quantitative Strategy Figure 57: Sharpe ratio – emerging markets equities Figure 58: Sharpe ratio – global industry groups 1.0 SharpeRatio 1.0 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification SharpeRatio Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.2 0.6 . EquallyWgted.4 . It is very interesting to see from Figure 59 to Figure 60 that the seven portfolios seem to form two groups. MSCI. MSCI. Compustat. MSCI. the MaxDiversification portfolio. Bloomberg Finance LLP. where the weighted portfolio correlation is low and diversification ratio is high. and the minimum tail dependence portfolio form another group.0 . Thomson Reuters.2 1. Deutsche Bank Quantitative Strategy Source: Axioma.8 . Russell. On the one hand. . Page 46 Deutsche Bank Securities Inc.8 0. Thomson Reuters. Thomson Reuters. Compustat. Russell. Group I – low diversification strategies.2 0. then the MaxDiversification is by definition (ex ante) the least crowded portfolio. the weighted portfolio correlation is high and diversification ratio is low.6 0.4 0.4 . MSCI. Russell.30 May 2013 Portfolios Under Construction Figure 55: Sharpe ratio – country portfolios Figure 56: Sharpe ratio – global equities (MSCI World) . Deutsche Bank Quantitative Strategy SharpeRatio Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.2 . Bloomberg Finance LLP.8 0. Deutsche Bank Quantitative Strategy Measure of crowdedness – portfolio correlation based If we measure the crowdedness of a particular strategy based on weighted portfolio correlation (WPC). 11M06 DR .07M12 DR . and MinTailDependence) portfolios clearly offer superior diversification benefit than the Group I (i.e. Compustat. the capitalization weighted benchmark (MSCI 2 ACWI) had a DR2 of 1.e. Russell.5 3.07M12 CORR .30 May 2013 Portfolios Under Construction The diversification ratio (actually.0 . Therefore.05M12 DR . and Reynier [2011] suggest that DR2 is equal to the number of independent risk factors (or degrees of freedom). MSCI.99M12 DR . On the other hand.06M06 DR .4 2.05M06 DR .99M12 CORR .00M06 DR . Figure 59: Weighted portfolio correlation Figure 60: Diversification ratio .10M06 DR .. DR . Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.00M12 CORR .5 CORR . and RiskParity) strategies.2 1.04M12 DR .01M12 CORR .03M06 CORR .09M12 DR .0 .e. Thomson Reuters.5 2. another way to gauge the level of “crowdedness” or the potential of “diversification benefit” can be shown in Figure 61 and Figure 62. The Group II (i.10M12 CORR ..0 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP. benchmark. The market’s WPC and DR also changes over time.12M12 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.8 independent risk factors – a highly undiversified portfolio. MaxDiversification.02M12 CORR . i. Thomson Reuters.08M12 DR .07M06 DR .82 .04M06 DR .00M12 DR .03M12 CORR .11M12 CORR . MSCI.02M12 DR .08M06 CORR . implying that a passive index investor would have been effectively exposure to 1.04M06 CORR . the square of DR – we call it DR2) is a very useful measure.05M12 CORR .15 .06M12 DR . Froidure.6 3.08M06 DR .01M06 CORR .06M12 CORR .12M06 DR . 2013. which suggests that the benchmark misses out on the opportunity to effectively diversify across more than three additional independent risk factors.03M06 DR . Compustat. GlobalMinVar.27 2 = 5.11M12 DR . EquallyWgted.04M12 CORR . as of April 30.09M06 CORR ..12M06 CORR .10M06 CORR .10M12 DR . InvVol. represented in the portfolio.05M06 CORR . Figure 59 and Figure 60 show the weighted portfolio correlation (WPC) and diversification ratio (DR) of various risk-based strategies. Russell. Deutsche Bank Quantitative Strategy Page 47 .01M06 DR .09M12 CORR . Choueifaty.35 = 1.07M06 CORR .03M12 DR .09M06 DR .0 .02M06 CORR .01M12 DR . the relative WPC and DR to the benchmark.12M12 1.08M12 CORR .02M06 DR .00M06 CORR . the MaxDiversification portfolio had a DR2 of 2.06M06 CORR . Therefore.11M06 CORR . 4 1. Deutsche Bank Quantitative Strategy Measure of crowdedness – portfolio tail dependence based If we measure the crowdedness of a particular strategy based on the weighted portfolio tail dependence (WPTD).10M12 CORR .6 2.07M12 CORR .09M06 CORR .02M12 CORR . Thomson Reuters. MSCI.03M12 CORR .05M06 DR . Russell.09M12 DR .11M12 DR .4 1.01M12 CORR .09M06 DR .0 0.05M06 CORR .03M06 CORR . From Figure 63 and Figure 64.00M12 DR . Russell.06M12 CORR .07M06 DR .01M06 DR .04M12 CORR .02M12 DR .08M12 CORR .04M06 DR .11M06 DR . The more diversified strategies (GlobalMinVar. Page 48 Deutsche Bank Securities Inc.10M06 CORR .12M12 Figure 61: Relative WPC Equally w eighted Risk parity Max diversification Inverse Vol Global min variance Min tail dependence Source: Bloomberg Finance LLP.04M12 DR . as shown in Figure 64. MaxDiversification. then the MinTailDependence portfolio is by definition (ex ante) the least crowded portfolio.06M06 DR .00M12 CORR .01M06 CORR .05M12 CORR . MSCI.03M06 DR .99M12 DR .01M12 DR .11M06 CORR . Compustat.00M06 DR .06M12 DR .02M06 CORR .12M06 DR .02M06 DR . Thomson Reuters. we see essentially the same two clusters as measured by WPC or DR.03M12 DR .04M06 CORR .2 2. Deutsche Bank Quantitative Strategy DR .06M06 CORR .07M12 DR .12M12 Figure 62: Relative DR CORR .10M12 DR .30 May 2013 Portfolios Under Construction 1.0 0.11M12 CORR . More interestingly.99M12 CORR . and MinTailDependence portfolios) offer much better chance of avoiding crowded trades (the green circled strategies).8 1.12M06 CORR . Compustat.8 Equally w eighted Risk parity Max diversification Inverse Vol Global min variance Min tail dependence Source: Bloomberg Finance LLP.10M06 DR .08M06 DR .08M06 CORR .08M12 DR .05M12 DR .09M12 CORR .6 0. .07M06 CORR . the EquallyWgted and InvVol at times had even higher tail dependence than the benchmark (the red circled strategies).2 0.00M06 CORR . 02M06 .03M12 .10M06 .01M06 .04M12 .6% 56. Compustat.01M12 .1% 100.6 1.0% GlobalMinVar 37.06M06 . Deutsche Bank Quantitative Strategy How to group these strategies Holding correlation and overlap Another way to investigate correlation is to see how their holdings are correlated.11M06 .06M12 . 2013. Russell. Compustat. As shown in Figure 66. Figure 65: Holding correlation Benchmark InverseVol RiskParity GlobalMinVar MaxDiversification MinTailDependence Benchmark 100.08M06 .12M12 0.6 1.00M06 .99M12 . given the definitions of InvVol and RiskParity.08M12 .0% RiskParity 12.03M12 . MSCI.02M12 . Page 49 ..05M12 .2% 79.08M12 .0% 61. Russell.1% 27.06M12 . However. Thomson Reuters.06M06 .0% InverseVol 30.07M06 .11M12 .09M12 .8% 100. Thomson Reuters. whether they hold the same assets with the same weights.6% 65. i. Deutsche Bank Quantitative Strategy A more robust way to measure portfolio overlap is simply to see whether these strategies hold similar assets.03M06 .3% 100.0% Source: Bloomberg Finance LLP. Compustat. Figure 65 shows the holding correlation as of April 30.8 .09M06 .10M12 . Holding correlation seems to suggest that these strategies are quite different.11M12 . MSCI.8% 73.4 WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM .4 0.08M06 .12M06 .9% 13.11M06 .10M06 . Figure 66 shows the holding overlap as of April 30.2 0.00M06 .01M06 .04M06 .10M12 .07M12 . but this fact is by construction.00M12 .2 .04M12 . and RiskParity portfolios hold essentially the same assets. Thomson Reuters.05M12 .0 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.05M06 .00M12 .03M06 . The other risk-based strategies seem to hold very different assets.12M06 .07M06 . InvVol. these risk-based strategies seem to be uncorrelated to the capitalization weighted benchmark and inverse volatility portfolio.4% 100.12M12 .3% 81.05M06 . the capitalization weighted benchmark.09M12 .4% 61. MSCI.02M12 .07M12 .0% MaxDiversification 38.30 May 2013 Portfolios Under Construction Figure 63: Weighted portfolio tail dependence Figure 64: Relative WPTD .0 . Deutsche Bank Quantitative Strategy Inverse Vol Global min variance Min tail dependence Source: Bloomberg Finance LLP. MaxDiversification. 2013. GlobalMinVar.8% 48.04M06 . There is some modest correlation among the risk-based allocations: RiskParity.09M06 . Deutsche Bank Securities Inc. and MinTailDependence.0% MinTailDependence 32. Russell.02M06 .99M12 .0% 100.01M12 .e. 0% MinTailDependence 37. Page 50 Deutsche Bank Securities Inc.0% InverseVol 100.0% EquallyWgted 95. and finally joined with the benchmark.8% 37.30 May 2013 Portfolios Under Construction Figure 66: Holding overlap Benchmark InverseVol RiskParity GlobalMinVar MaxDiversification MinTailDependence Benchmark 100.0% 20. Since they are all long-only/noleverage portfolios. Deutsche Bank Quantitative Strategy The holding based analysis can. Thomson Reuters.7% 95.8% 36.0% MaxDiversification 37.0% 100.8% 37.8% 100.1% 100.3% 100. MaxDiversification and MinTailDependence portfolios are similar. then further joined by the GlobalMinVar portfolio. On the one hand. MaxDiversification. and MinTailDependence seem to form another group.1% 95. MSCI.0% Source: Bloomberg Finance LLP.0% GlobalMinVar 80. EquallyWgted.6% 91. . clearly. EquallyWgted.8% 54. however.1% 91. our clustering algorithm essentially divides our risk-based strategies (and benchmark portfolio) into two broad clusters: Cluster I – low diversification strategies (red). Compustat.0% 99. Russell. Compustat.0% Source: Bloomberg Finance LLP.6% 100. InvVol and RiskParity are closer to each other. On the other hand. Holding different assets does not automatically indicate that their performances are uncorrelated. and RiskParity portfolios seem to form one group. Cluster II – high diversification strategies (green).0% RiskParity 100. We also see that capitalization weighted benchmark.8% 100.0% GlobalMinVar 20.9% 94.0% MinTailDependence 87.0% 100.0% 100. then joined with EquallyWgted.7% 97..0% 100.5% 100.0% 20.0% RiskParity 93.5% 93. InvVol. MSCI ACWI). be misleading.5% 100.5% 95. Deutsche Bank Quantitative Strategy Cluster analysis As shown in Figure 68. InvVol.7% 87. Figure 67: Performance correlation Benchmark Benchmark EquallyWgted InverseVol RiskParity GlobalMinVar MaxDiversification MinTailDependence 100. MSCI.0% 91. Russell.8% 37.3% 97.2% 100.3% 99. capitalization weighted benchmark (i. Performance correlation A more meaningful way to measure strategy correlation is to show how correlated these strategies perform over time (see Figure 67). while GlobalMinVar. and RiskParity portfolios form one cluster.8% 89. their performances are highly correlated.0% InverseVol 95.8% 37.8% 36. Thomson Reuters.0% MaxDiversification 83. Within this cluster.e.1% 99. Similar to the cluster analysis graph.05 Height 0. Compustat. The capitalization weighted benchmark (i. MSCI ACWI) seems to form a cluster of its own. Deutsche Bank Securities Inc.20 0. InvVol and EquallyWgted portfolio are closer to each other. "complete") Source: Bloomberg Finance LLP.35 Cluster Dendrogram Cluster II – low diversification strategies Risk-based allocation hclust (*. Thomson Reuters. Cluster III – high diversification strategies (green). Deutsche Bank Quantitative Strategy Eigenvalue ratio test Another approach to group strategies together is based on eigenvalue analysis. the eigenvalue ratio chart in Figure 69 classifies our strategies into three buckets: Cluster I – benchmark (red). Page 51 . Cluster II – low diversification strategies (yellow). then further joined by the MinTailDependence portfolio. MSCI.e.30 May 2013 Portfolios Under Construction Figure 68: Cluster analysis Cluster I – high diversification strategies RiskParity InvVol EquallyWgted Benchmark MinTailDependence MaxDiversification GlobalMinVar 0. On the other hand. On the one hand. then joined with RiskParity portfolio. Russell. MaxDiversification and GlobalMinVar portfolios are similar.. 4 0. MSCI. Morocco. Thomson Reuters. .2 RiskParity InvVol EquallyWgted pearson -0.2 MinTailDependence 0. US. Deutsche Bank Quantitative Strategy Country weight The country weight graphs from Figure 70 to Figure 73 again demonstrate that the capitalization weighted benchmark (i..2 0. followed by Japan and UK (see Figure 70).6 -0. The MaxDiversification (see Figure 72) and MinTailDependence (Figure 73) portfolios are more diversified.e. the US equity index dominates the benchmark throughout the backtest.6 -0. MSCI ACWI) is overly concentrated – in fact.0 0. and Canada dominate the weights (see Figure 71).4 Eigenvalue 2 GlobalMinVar Benchmark -0.0 MaxDiversification -0. Russell. Page 52 Deutsche Bank Securities Inc.4 -0.30 May 2013 Portfolios Under Construction Figure 69: Eigenvalue ratio chart Eigenvalue Ratio Plot 0. Compustat. Malaysia.2 0. The GlobalMinVar portfolio is also highly concentrated with high turnover.4 Eigenvalue 1 Source: Bloomberg Finance LLP. Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc. MSCI. Thomson Reuters. Russell. Compustat. Deutsche Bank Quantitative Strategy Page 53 . MSCI. Compustat. Russell. Canada Belgium France Ireland Netherlands Spain UK Japan Brazil Mexico Egypt Poland Turkey Indonesia Philippines USA Denmark Germany Israel Norway Sweden Australia New Zealand Chile Peru Hungary Russia China Korea Taiwan 2013:04 2012:08 2011:12 2011:04 2010:08 2009:12 2009:04 2008:08 2007:12 2007:04 2006:08 2005:12 2005:04 2004:08 2003:12 2003:04 2002:08 2001:12 1999:12 2013:04 2012:08 2011:12 2011:04 2010:08 0% 2009:12 0% 2009:04 10% 2008:08 10% 2007:12 20% 2007:04 30% 20% 2006:08 30% 2005:12 40% 2005:04 40% 2004:08 50% 2003:12 60% 50% 2003:04 70% 60% 2002:08 70% 2001:12 80% 2001:04 90% 80% 2000:08 100% 90% 1999:12 100% 2001:04 Figure 71: GlobalMinVar 2000:08 Figure 70: Benchmark – MSCI ACWI Austria Finland Greece Italy Portugal Switzerland Hong Kong Singapore Colombia Czech Republic Morocco South Africa India Malaysia Thailand Source: Bloomberg Finance LLP. Thomson Reuters.30 May 2013 Portfolios Under Construction Canada Belgium France Ireland Netherlands Spain UK Japan Brazil Mexico Egypt Poland Turkey Indonesia Philippines USA Denmark Germany Israel Norway Sweden Australia New Zealand Chile Peru Hungary Russia China Korea Taiwan Austria Finland Greece Italy Portugal Switzerland Hong Kong Singapore Colombia Czech Republic Morocco South Africa India Malaysia Thailand Source: Bloomberg Finance LLP. we recommend investors to hedge their country equity (and bond) exposures on those days when certain economic indicators are expected to be released – these economic variables are expected to produce negative returns on their announcement days. Russell. Similar to our country portfolio simulation. and RiskParity. Deutsche Bank Quantitative Strategy MEAM country equity portfolios Investment universe In Luo. MSCI. we illustrate that avoiding economic uncertainties can significantly improve equity (and sovereign bond) performance for 45 countries in the MSCI ACWI. Deutsche Bank Quantitative Strategy Canada Belgium France Ireland Netherlands Spain UK Japan Brazil Mexico Egypt Poland Turkey Indonesia Philippines USA Denmark Germany Israel Norway Sweden Australia New Zealand Chile Peru Hungary Russia China Korea Taiwan 2013:04 2012:08 2011:12 2011:04 2010:08 2009:12 2009:04 2008:08 2007:12 2007:04 2006:08 2005:12 2005:04 2004:08 2003:12 2003:04 2002:08 2001:12 1999:12 2013:04 2012:08 2011:12 2011:04 2010:08 2009:12 2009:04 2008:08 0% 2007:12 10% 0% 2007:04 20% 10% 2006:08 20% 2005:12 30% 2005:04 40% 30% 2004:08 50% 40% 2003:12 60% 50% 2003:04 60% 2002:08 70% 2001:12 80% 70% 2001:04 90% 80% 2000:08 100% 90% 1999:12 100% 2001:04 Figure 73: MinTailDependence 2000:08 Figure 72: MaxDiversification Austria Finland Greece Italy Portugal Switzerland Hong Kong Singapore Colombia Czech Republic Morocco South Africa India Malaysia Thailand Source: Bloomberg Finance LLP. Compustat. once economic risk is hedged. we want to understand the impact of constructing MEAM portfolios with further risk-based allocation techniques. the additional risk-based portfolio construction techniques do not help to reduce downside risk (see Figure 77). and GlobalMinVar together (see Figure 80 and Figure 81). Page 54 Deutsche Bank Securities Inc. as measured by the Sharpe ratio (see Figure 76). and GlobalMinVar portfolios are more diversified by both diversification ratio and WPTD (see Figure 78 and Figure 79). et al [2013]. In this research. Thomson Reuters. followed by GlobalMinVar. . Russell.30 May 2013 Portfolios Under Construction Canada Belgium France Ireland Netherlands Spain UK Japan Brazil Mexico Egypt Poland Turkey Indonesia Philippines USA Denmark Germany Israel Norway Sweden Australia New Zealand Chile Peru Hungary Russia China Korea Taiwan Austria Finland Greece Italy Portugal Switzerland Hong Kong Singapore Colombia Czech Republic Morocco South Africa India Malaysia Thailand Source: Bloomberg Finance LLP. MaxDiversification. rebalanced monthly. Thomson Reuters. MinTailDependence. MSCI. MaxDiversification. Interestingly. MinTailDependence is the clear winner. MinTailDependence. Our investment universe is the 45 MEAM country equity indices and our benchmark is based on the each country’s weight in the MSCI ACWI (investing on each country’s MEAM portfolio). Specifically. Both cluster analysis algorithms also group MaxDiversification. We call this strategy the MEAM (macroeconomic announcement model). Compustat. Russell.09M12 NO .08M08 NO . MSCI. Compustat.10M08 NO .95 -.07M12 NO . Deutsche Bank Quantitative Strategy Page 55 .12M08 NO . Compustat. Russell.08M04 NO . Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc. Compustat.06 SharpeRatio .09M04 NO .11M04 NO . Deutsche Bank Quantitative Strategy Source: Bloomberg Finance LLP.10M12 NO .05M08 NO .0 40 2.06M08 NO .08 .12M04 NO . MSCI.5 10 1.09M08 NO .85 -. MSCI.14 .5 NO . Thomson Reuters.11M08 NO .80 -.70 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.05M04 NO .5 30 2.10M04 NO . Russell.07M04 NO .0 0 0. Thomson Reuters. Russell.0 20 1.30 May 2013 Portfolios Under Construction Figure 75: Wealth curve 50 3.06M04 NO .10 . MSCI. Thomson Reuters. Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.07M08 NO .11M12 NO . Thomson Reuters.12 .08M12 NO .90 modifiedCVaR_ES95 -.12M12 Figure 74: Investment universe Benchmark Inverse Vol Global min variance Min tail dependence 2004 2005 2006 2007 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification 2008 2009 2010 2011 2012 Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP. Deutsche Bank Quantitative Strategy Figure 76: Sharpe ratio Figure 77: CVaR/expected shortfall .06M12 NO .75 -.04M12 NO . Compustat.05M12 NO . 08M08 DR .06M08 DR .05M08 .10M12 .05M04 .12M04 DR .07M04 DR .3 2. Thomson Reuters.11M12 DR .4 .4 InvVol RiskParity EquallyWgted RiskParity Benchmark MinTailDependence MaxDiversification MinTailDependence GlobalMinVar 0.06M08 .12M08 DR . .09M08 .2 1. Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc. Deutsche Bank Quantitative Strategy Page 56 pearson -0. Deutsche Bank Quantitative Strategy Source: Bloomberg Finance LLP.6 .11M08 .0 -0.07M08 .12M04 . Compustat.2 InvVol EquallyWgted Benchmark -0.08M12 .08M08 .04M12 DR .05M08 DR .1 1.09M04 DR .00 0. Deutsche Bank Quantitative Strategy Figure 80: Clustering the strategies Figure 81: Grouping the strategies. Russell.10 0. Thomson Reuters. Thomson Reuters.4 2.05M12 DR .10M04 DR .07M08 DR .06M04 DR .10M08 .05M04 DR .07M12 DR .11M12 .8 .2 WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM Benchmark Inverse Vol Global min variance Min tail dependence .07M04 .10M12 DR .11M08 DR . Compustat.4 Source: Bloomberg Finance LLP.4 MaxDiversification GlobalMinVar 0.09M12 DR .06M04 .20 Cluster Dendrogram -0.08M04 .4 Source: Bloomberg Finance LLP.12M08 .0 Equally w eighted Risk parity Max diversification Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP. MSCI. Russell.2 0.0 0.2 0. Russell.09M04 .12M12 . Eigenvalue Ratio Plot 0.07M12 . Russell. Compustat. Compustat. MSCI. MSCI.10M08 DR . Thomson Reuters.04M12 .5 . MSCI.10M04 .06M12 DR .08M12 DR .11M04 .2 0.05M12 .09M08 DR .30 May 2013 Portfolios Under Construction Figure 78: Diversification ratio Figure 79: Weighted portfolio tail dependence 2.12M12 DR .11M04 DR .09M12 .0 .08M04 DR .06M12 . InvVol. the MinTailDependence portfolio. US small cap equity (Russell 2000) 3.. As shown in Figure 82. EquallyWgted. we lay out the impact of risk-based allocations in the multi-asset world. and GlobalMinVar are not as diversified as the RiskParity. A more sophisticated way to measure diversification and concentration is illustrated in Figure 86 using the diversification ratio (Choueifaty and Coignard [2008]). actually shows more tail dependence than GlobalMinVar and MaxDiversification ex post18. EM credit (Deutsche Bank Emerging Markets Bond USD Index) 10. Emerging markets equity (MSCI EM) 5. Deutsche Bank Securities Inc. Page 57 . Asset allocation Investment universe We study 11 commonly used asset classes: 1. Benchmark. and MinTailDependence portfolios. 18 This is most likely due to the estimation errors in tail dependent coefficient matrix. commodities. US high yield bonds (Deutsche Bank US High Yield Index) 8. EquallyWgted. Multi-asset portfolio In this section.e. i. and RiskParity appear to be more exposed to strategy crowding than GlobalMinVar and MaxDiversification. Interestingly. Investment Grade Sovereign (Deutsche Bank USD Investment Grade Sovereign Bond Index) 9. bonds. REITs (S&P Global REITs) 6. MaxDiversification. 60% global equities (MSCI World) and 40% of bonds (US treasuries). Figure 87 shows the weighted portfolio tail dependence (WPTD) – benchmark. US treasuries (Deutsche Bank US All Treasuries Index) 7. Gold (front-end gold futures) We set our investment benchmark as a traditional 60-40 portfolio. International equity (MSCI EAFE) 4. we test our strategies in asset allocation.30 May 2013 Portfolios Under Construction IX. despite being the least crowded strategy ex ante. US large cap equity (Russell 1000) 2. Commodities (S&P/GSCI) 11. and alternative beta (also called risk premia or risk factors) space. Specifically. the benchmark and GlobalMinVar portfolios are highly concentrated. Compustat.08M12 NO . Compustat. They also seem to form one cluster (see Figure 88). Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.00M12 NO .08M06 NO .06 1.00M06 NO .5 NO .09M12 NO . Figure 82: Investment universe Figure 83: Wealth curve 12 3.05M06 NO .10M12 NO . Thomson Reuters.02M12 NO .01M12 NO .03M12 NO .12M12 99 00 Benchmark Inverse Vol Global min variance Min tail dependence 01 02 03 04 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification 05 06 07 08 09 10 11 12 13 Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP. Thomson Reuters. Thomson Reuters.0 2 0.04M12 NO .06M12 NO .11M06 NO .06M06 NO . Deutsche Bank Quantitative Strategy Page 58 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.5 4 1. MSCI.0 10 2. Russell. Deutsche Bank Quantitative Strategy Figure 84: Sharpe ratio Figure 85: CVaR/expected shortfall 1.02M06 NO .4 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP. Russell.07M12 NO .8 -.04 SharpeRatio modifiedCVaR_ES95 -.5 8 2. .10M06 NO .09M06 NO . Russell.08 0. Compustat.30 May 2013 Portfolios Under Construction All risk-based allocations significantly outperform our 60-40 benchmark. Russell.11M12 NO . Deutsche Bank Quantitative Strategy Source: Bloomberg Finance LLP.05M12 NO .6 -. MSCI.03M06 NO .04M06 NO .2 -.0 6 1.12M06 NO . RiskParity. MSCI.07M06 NO . Benchmark and InvVol portfolios show poor diversification (see Figure 89).99M12 NO . and MinTailDependence have the best performance overall (see Figure 84 and Figure 85).12 0. Compustat. MSCI. MaxDiversification. Thomson Reuters.01M06 NO .10 -. 00M12 DR .03M12 DR .09M06 .12M06 .4 3. Russell.09M12 . Deutsche Bank Quantitative Strategy Asset weight Figure 90 to Figure 93 lay out the asset weights over time for four of our strategies (RiskParity.5 Benchmark -0. MaxDiversification. Thomson Reuters. Page 59 .07M12 DR .02M12 DR .5 Source: Bloomberg Finance LLP. GlobalMinVar.09M12 DR .12M12 DR .01M12 DR . US high yield.07M12 .11M12 DR . Thomson Reuters.05M06 .11M06 .03M12 .0 .08M06 .01M06 DR .5 .00M12 .0 WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM Benchmark Inverse Vol Global min variance Min tail dependence .12M06 DR . and EM credit).01M06 .15 0.00 0.02M06 .04M12 DR .5 .10M12 .04M06 .05M06 DR .09M06 DR .08M12 .03M06 DR .12M12 . RiskParity and MinTalDependence appear to be more balanced than GlobalMinVar and MaxDiversification.00M06 DR .06M12 DR .10M06 . All these strategies have been significantly overweighting bonds (US treasuries.06M06 DR .06M06 . Thomson Reuters. Russell.1 1.30 Cluster Dendrogram EquallyWgted 0. Thomson Reuters.3 2.30 May 2013 Portfolios Under Construction Figure 86: Diversification ratio Figure 87: Weighted portfolio tail dependence 3.04M12 .06M12 . MSCI.99M12 DR .10M12 DR .08M06 DR . Eigenvalue Ratio Plot 0.11M06 DR .08M12 DR .5 EquallyWgted Benchmark MinTailDependence RiskParity MaxDiversification GlobalMinVar InvVol 0. MSCI.99M12 . Deutsche Bank Quantitative Strategy 0.2 2. Deutsche Bank Securities Inc. Sovereign bonds. Compustat.05M12 . Deutsche Bank Quantitative Strategy Figure 88: Clustering the strategies Figure 89: Grouping the strategies. and MinTailDependence).0 Equally w eighted Risk parity Max diversification Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP. MSCI.5 Source: Bloomberg Finance LLP.02M12 .10M06 DR .04M06 DR .5 1. Russell. Compustat. Deutsche Bank Quantitative Strategy Source: Bloomberg Finance LLP.00M06 .07M06 DR . Compustat.11M12 . Russell.0 InvVol RiskParity MinTailDependence MaxDiversification GlobalMinVar pearson -0.07M06 .02M06 DR .03M06 . MSCI.05M12 DR . Compustat.01M12 .0 .0 0. However.30 May 2013 Portfolios Under Construction US LargeCap EM Equity US High Yield GSCI US SmallCap Global REITs Sovereign Bonds Gold EAFE US Treasuries EM Credit US LargeCap EM Equity US High Yield GSCI US SmallCap Global REITs Sovereign Bonds Gold 2013:04 2012:08 2011:12 2011:04 2010:08 2009:12 2009:04 2008:08 2007:12 2007:04 2006:08 2005:12 2005:04 2004:08 2003:12 2003:04 2002:08 2001:12 1999:12 2013:04 2012:08 2011:12 2011:04 2010:08 2009:12 2009:04 2008:08 0% 2007:12 10% 0% 2007:04 20% 10% 2006:08 20% 2005:12 30% 2005:04 40% 30% 2004:08 50% 40% 2003:12 60% 50% 2003:04 60% 2002:08 70% 2001:12 80% 70% 2001:04 90% 80% 2000:08 100% 90% 1999:12 100% 2001:04 Figure 91: GlobalMinVar 2000:08 Figure 90: RiskParity EAFE US Treasuries EM Credit US LargeCap EM Equity US High Yield GSCI US SmallCap Global REITs Sovereign Bonds Gold EAFE US Treasuries EM Credit Source: Bloomberg Finance LLP. Now. Thomson Reuters. unconstrained portfolios should produce the best results. Thomson Reuters. As shown in Figure 95. Deutsche Bank Quantitative Strategy 2000:08 Source: Bloomberg Finance LLP. In practice. by reducing the concentration. Thomson Reuters. let’s re-run our portfolio simulation with a maximum weight constraint of 20%.e. adding a maximum weight constraint makes no difference to EquityWgted. MSCI. benchmark and GlobalMinVar portfolios are highly concentrated. MSCI. Deutsche Bank Quantitative Strategy Maximum weight constraint As shown in Figure 94. In theory. Compustat. MSCI. More importantly. Russell. almost everybody has some constraints. no single asset weight can exceed 20%. InvVol and little difference to the RiskParity. because constraints reduce the transfer coefficient and are likely to push portfolios below the efficient frontier.. portfolio Page 60 Deutsche Bank Securities Inc. Russell. Deutsche Bank Quantitative Strategy EAFE US Treasuries EM Credit Source: Bloomberg Finance LLP. Compustat. Deutsche Bank Quantitative Strategy US LargeCap EM Equity US High Yield GSCI US SmallCap Global REITs Sovereign Bonds Gold 2013:04 2012:08 2011:12 2011:04 2010:08 2009:12 2009:04 2008:08 2007:12 2007:04 2006:08 2005:12 2005:04 2004:08 2003:12 2003:04 1999:12 2013:04 2012:08 2011:12 2011:04 2010:08 2009:12 2009:04 2008:08 0% 2007:12 10% 0% 2007:04 20% 10% 2006:08 20% 2005:12 30% 2005:04 40% 30% 2004:08 50% 40% 2003:12 60% 50% 2003:04 70% 60% 2002:08 70% 2001:12 80% 2001:04 90% 80% 2000:08 100% 90% 1999:12 100% 2002:08 Figure 93: MinTailDependence 2001:12 Figure 92: MaxDiversification 2001:04 Source: Bloomberg Finance LLP. Compustat. Russell. The largest weight of a single asset can be over 90% at a given point in time. and MinTailDependence portfolios. Russell. a maximum weight constraint does help GlobalMinVar portfolio. Compustat. MaxDiversification. i. . MSCI. Constraints could be due to investment mandates. Thomson Reuters. 05M12 MAX . Denmark. Jussa.02M12 MAX . a maximum weight constraint is arbitrary.12M12 0 Benchmark Inverse Vol Global min variance Min tail dependence No constraint Max wgt constraint Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP. and MinTailDependence portfolios offer some diversification benefit (see Figure 100 and Figure 101). However.06M12 MAX . Italy.02M06 MAX .10M06 MAX .g. RiskParity.. et al [2011]. Deutsche Bank Quantitative Strategy Source: Bloomberg Finance LLP.09M12 MAX .03M12 MAX .08M06 MAX . et al [2012]. Thomson Reuters.11M12 MAX .06M06 MAX .. followed by MaxDiversification and MinTailDependence. 19 We choose the same investment universe as in our previous research. Constraints could be quite effective in reducing the impact of estimation errors on portfolio performance.08M12 MAX . Compustat. Russell. Deutsche Bank Quantitative Strategy Sovereign bond portfolios Investment universe We use G13 long-term sovereign bond indices in the following 13 countries as our investment universe: US. Jagannathan and Ma [2003] showed that adding some constraints on weights is equivalent to using a shrinkage estimate of the covariance matrix. and New Zealand19.04M06 MAX . The returns are all based on USD.05M06 MAX . Figure 94: Maximum weight Figure 95: Sharpe ratio comparison 100 1. Switzerland.30 May 2013 Portfolios Under Construction managers realize that they have to estimate all the input parameters (e.0 20 MAX . expected returns and covariance matrix).07M12 MAX .09M06 MAX . However.01M06 MAX .11M06 MAX . Russell. as measured by Sharpe ratio (see Figure 98) and downside risk (see Figure 99). Thomson Reuters. Norway. MSCI. therefore. The GlobalMinVar.8 80 1. Germany.12M06 MAX . currencies are unhedged.03M06 MAX . MSCI. and MinTailDependence portfolios are clearly more robust than GlobalMinVar. Compustat. e. Interestingly. the diversification via risk-based allocation is likely to be limited.04M12 MAX . GlobalMinVar outperforms the other risk-based allocation techniques. et al [2013].07M06 MAX . UK.6 40 0. Canada. Japan. In the sovereign bond universe.00M06 MAX . France. MaxDiversification. and Luo.2 60 0. the three techniques all beat the more popular RiskParity scheme. MaxDiversification. in the sovereign bond space.99M12 MAX . Australia.g. which contains estimation errors.00M12 MAX . Sweden. Page 61 .01M12 MAX .10M12 MAX . Mesomeris. Deutsche Bank Securities Inc. Deutsche Bank Quantitative Strategy Source: Bloomberg Finance LLP.6 4 1. MSCI.05M12 NO .09M12 NO .03M06 NO . MSCI.4 2.03M12 NO . Deutsche Bank Quantitative Strategy Page 62 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.01M12 NO . Compustat.01M06 NO . Russell.05M06 NO .024 SharpeRatio modifiedCVaR_ES95 -.12M06 NO .04M06 NO . MSCI.2 -.02M12 NO .048 0. Russell.8 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.06M06 NO .12M12 0 Benchmark Inverse Vol Global min variance Min tail dependence 99 00 01 02 03 04 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification 05 06 07 08 09 10 11 12 13 Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.036 1. Thomson Reuters.0 -.040 1.6 -. Compustat.08M12 NO .10M06 NO .09M06 NO .00M06 NO . Compustat. Russell.032 -. Thomson Reuters.2 2.07M06 NO .02M06 NO .06M12 NO . Deutsche Bank Quantitative Strategy Figure 98: Sharpe ratio Figure 99: CVaR/expected shortfall 1. Thomson Reuters.0 8 1.4 -.11M12 NO .028 1. MSCI.04M12 NO .044 -.99M12 NO . .08M06 NO . Compustat.8 12 2. Thomson Reuters. Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc. Russell.10M12 NO .07M12 NO .00M12 NO .30 May 2013 Portfolios Under Construction Figure 96: Investment universe Figure 97: Wealth curve 16 3.11M06 NO .8 NO .2 0. 12M12 DR .e.10 0.08M12 . Thomson Reuters. Russell.09M06 DR .11M12 DR . DB Liquidity Commodities Indices Optimum Yield family offers a different approach to rolling over futures contracts. MSCI.2 1.02M06 .4 -0.4 . Compustat.11M06 .02M12 . approximately 24 different commodities future indices (see Figure 104) from the DB Liquidity Commodities Indices Optimum Yield family (DBLCI-OY).00 0.03M06 .0 0.10M06 .99M12 .4 0.. The returns are all based on USD. MSCI.10M12 DR .12M06 DR . i.03M12 DR .08M06 .00M12 DR .03M12 . Russell.30 May 2013 Portfolios Under Construction Figure 100: Diversification ratio Figure 101: Weighted portfolio tail dependence 2. Deutsche Bank Quantitative Strategy Commodities portfolios Investment universe In this research.02M06 DR . DBLCI-OY utilizes a rules-based approach when it rolls from one futures contract to another for each commodity in the index.2 Benchmark RiskParity -0. Thomson Reuters.12M06 . Deutsche Bank Quantitative Strategy Figure 102: Clustering the strategies Figure 103: Grouping the strategies. Eigenvalue Ratio Plot Cluster Dendrogram 0.08M12 DR .4 InvVol -0.0 MinTailDependence -0.09M06 .2 0.07M06 .04M06 .00M06 .03M06 DR .06M06 DR .09M12 .10M06 DR .11M12 .06M12 .05M06 DR . et al [2012].05M12 DR .06M12 DR .05M12 .01M12 DR . MSCI.01M06 DR . Russell. Russell. MSCI.2 0.8 WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM Benchmark Inverse Vol Global min variance Min tail dependence .6 2.11M06 DR .6 GlobalMinVar -0.6 Source: Bloomberg Finance LLP.4 0.4 1.10M12 .04M06 DR .07M12 . Thomson Reuters. Deutsche Bank Quantitative Strategy pearson EquallyWgted -0. Compustat.08M06 DR .20 0.2 MaxDiversification 0. Effectively.0 Equally w eighted Risk parity Max diversification Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.99M12 DR .04M12 DR .00M12 . we use the same commodities universe as in Jussa.07M12 DR . Thomson Reuters.01M12 . Compustat. Compustat.01M06 .00M06 DR .12M12 . Page 63 .6 MinTailDependence MaxDiversification Benchmark GlobalMinVar RiskParity InvVol EquallyWgted 0.09M12 DR .0 .6 Source: Bloomberg Finance LLP.2 0. Deutsche Bank Quantitative Strategy Source: Bloomberg Finance LLP. Contrary to conventional approaches of using front-end futures. this enhanced Deutsche Bank Securities Inc.05M06 .02M12 DR .06M06 .07M06 DR .04M12 . This rule stipulates that each new commodity futures contract will be the one with the maximum “implied roll yield” based on the closing price for each eligible contract.6 . and MinTailDependence produce higher Sharpe ratios (see Figure 107) and lower downside risks (see Figure 108) than other strategies. MaxDiversification. GlobalMinVar. Figure 104: List of commodities futures Category Commodities Bloomberg Ticker Energy Light Crude DBLCOCLT Energy Heating Oil DBLCOHOT Energy Rbob Gas DBLCYTRB Energy Natural Gas DBLCYTCO Energy Brent Crude Oil DBLCYTCO Energy Gasoline DBLCYTGO Gold DBLCOGCT Precious Metal Precious Metal Industrial Metal Silver DBLCYTSI Aluminium DBLCOALT Industrial Metal Zinc DBLCYTZN Industrial Metal Copper DBLCYTCU Industrial Metal Nickel DBLCYTNI Industrial Metal Lead DBLCYTPB Agriculture Corn DBLCOCNT Agriculture Wheat DBLCOWTT Agriculture Soybeans DBLCYTSS Agriculture Sugar DBLCYTSB Agriculture Coffee DBLCYTKC Agriculture Cotton DBLCYTCT Agriculture Cocoa DBLCYTCC Agriculture Kansas Wheat DBLCYTKW Livestock Live Cattle DBLCYTLC Livestock Lean Hogs DBLCYTLH Livestock Feeder Cattle DBLCYTFC Source: Deutsche Bank Similar to the bond investment universe.30 May 2013 Portfolios Under Construction rollover process is systematically optimizing the return generated by rolling a short-term contract into a longer-term contract and vice-versa. Page 64 Deutsche Bank Securities Inc. They also appear to form one cluster (see Figure 111 and Figure 112). . 12M06 NO .95 -. Compustat.10M12 NO .99M12 NO .08M12 NO . Russell.06 .14 .01M06 NO . Russell.05M12 NO . Compustat. Deutsche Bank Quantitative Strategy Source: Bloomberg Finance LLP.12M12 0 Benchmark Inverse Vol Global min variance Min tail dependence 99 00 01 02 03 04 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification 05 06 07 08 09 10 11 12 13 Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP. MSCI.11M06 NO .04 SharpeRatio .10M06 NO .00M06 NO .04M06 NO .01M12 NO .90 modifiedCVaR_ES95 -.06M06 NO . Compustat.08M06 NO .30 May 2013 Portfolios Under Construction Figure 105: Investment universe Figure 106: Wealth curve 25 5 4 20 3 15 2 10 1 5 NO . Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.05M06 NO . MSCI.09M06 NO . Thomson Reuters.08 .07M06 NO .80 -.75 -.70 -.06M12 NO .12 . Thomson Reuters.03M06 NO . Deutsche Bank Quantitative Strategy Page 65 .00M12 NO . Russell. Thomson Reuters.02M06 NO .02M12 NO . MSCI.85 -.11M12 NO . Compustat.07M12 NO . Deutsche Bank Quantitative Strategy Figure 107: Sharpe ratio Figure 108: CVaR/expected shortfall .04M12 NO .09M12 NO . MSCI.10 . Thomson Reuters.03M12 NO . Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc. Russell.65 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP. We have indeed also published our own research in this space (see Mesomeris. Thomson Reuters.0 WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM Benchmark Inverse Vol Global min variance Min tail dependence . . Eigenvalue Ratio Plot EquallyWgted Benchmark 0.08M12 DR . Russell.04M06 .5 .06M12 DR .10M12 .10M06 .12M12 . based on 12-month total returns excluding the most recent one month Quality.10M12 DR .01M12 DR .03M12 DR .99M12 . MSCI.07M12 .09M12 DR .4 MaxDiversification GlobalMinVar -0.11M12 . MSCI. Compustat. MSCI.02M06 DR .00M06 DR .99M12 DR .09M12 .4 -0.5 1.2 0.0 -0. based on return on equity Page 66 Deutsche Bank Securities Inc.02M06 .07M06 DR .09M06 .6 MinTailDependence RiskParity MaxDiversification GlobalMinVar RiskParity InvVol EquallyWgted Benchmark 0. Russell.03M12 .6 0.12M06 .04M12 .11M12 DR .05M12 .2 InvVol 0.00M12 .03M06 DR . Thomson Reuters.3 2. Deutsche Bank Quantitative Strategy Source: Bloomberg Finance LLP. et al [2011] for details). Russell.4 0.0 0.04M06 DR .4 Cluster Dendrogram -0.2 MinTailDependence -0.02M12 DR .06M12 .0 .05M06 DR .2 2. MSCI. based on trailing earnings yield Momentum.05M06 .06M06 DR .08M06 DR .6 Source: Bloomberg Finance LLP.4 0.4 3.08M06 .02M12 .0 Equally w eighted Risk parity Max diversification Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.2 0. Deutsche Bank Quantitative Strategy pearson -0.05M12 DR . Compustat.01M12 . Deutsche Bank Quantitative Strategy Figure 111: Clustering the strategies Figure 112: Grouping the strategies.11M06 DR .06M06 . Compustat.6 Source: Bloomberg Finance LLP. Compustat. Deutsche Bank Quantitative Strategy Alternative beta portfolios Investing along the alternative beta portfolios (also called risk premia or risk factors) has become more and more popular in recent years. Thomson Reuters. Thomson Reuters. Russell.03M06 .12M06 DR .2 0.07M12 DR .01M06 DR . we define five simple alternative beta portfolios in global equities: Value.08M12 . In this research.0 .30 May 2013 Portfolios Under Construction Figure 109: Diversification ratio Figure 110: Weighted portfolio tail dependence 3.00M06 .12M12 DR .1 1.07M06 .10M06 DR .01M06 .5 .00M12 DR .0 0.09M06 DR . especially among the pension/endowment/foundation/sovereign wealth funds.11M06 .04M12 DR . 07M06 NO . Compustat. by forming a long/short quintile portfolio. Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc. It is interesting to see that all portfolios constructed on alternative betas significantly outperform the benchmark MSCI World index. and Australia/New Zealand. Figure 113: Investment universe Figure 114: Wealth curve 5 2.0 1. Russell.10M12 NO . Deutsche Bank Quantitative Strategy Page 67 .8 0.06M12 NO .00M06 NO .2 4 0.04M12 NO . Asia ex Japan Developed. with much higher Sharpe ratios (see Figure 115) and much lower downside risks (see Figure 116). Russell. Compustat.03M06 NO . we do not invest in those buckets. the GlobalMinVar and RiskParity strategies are likely to be more diversified.04M06 NO . Japan.09M06 NO . equally weighted within the long and the short portfolios. MSCI.05M06 NO .08M06 NO . we have not seen any sign of crowding yet. Thomson Reuters. Europe ex UK.09M12 NO . We divide the MSCI World into the following regions: US.12M12 3 Benchmark Inverse Vol Global min variance Min tail dependence 01 02 03 04 Benchmark Inverse Vol Global min variance Min tail dependence 05 06 07 08 09 10 11 12 13 Equally w eighted Risk parity Max diversification Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP. For those region/sector buckets where we have less than five stocks.05M12 NO . Based on WPC and WPTD.30 May 2013 Portfolios Under Construction Size.4 NO .02M06 NO .03M12 NO .00M12 NO .01M06 NO .99M12 NO . MSCI.10M06 NO .02M12 NO .01M12 NO . UK.12M06 NO . While our MinTailDependence strategy displays the highest Sharpe ratio and the lowest downside risk.08M12 NO . based on trailing one-year daily realized volatility All portfolios (other than size) are constructed for the MSCI World universe.6 1.11M12 NO . Thomson Reuters.07M12 NO .06M06 NO . Canada.11M06 NO . in a regional and sector neutral way. MSCI World SmallCap total return index – MSCI World LargeCap total return index Low volatility/low risk. 99 00 Source: Bloomberg Finance LLP. We select equal number of stocks from each region in each of the 10 GICS sectors. 02M06 .10M06 DR .03M06 DR .1 1 . MSCI.12M12 Figure 118: Weighted portfolio tail dependence .8 -.10M06 .12 Benchmark Inverse Vol Global min variance Min tail dependence Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.10 0.11M12 DR .07M12 DR .08 0.06M12 DR .30 May 2013 Portfolios Under Construction Figure 115: Sharpe ratio Figure 116: CVaR/expected shortfall 1.10M12 DR .12M06 DR . Thomson Reuters.4 -.11M06 DR . Compustat.05M06 DR .2 -.11M12 .08M12 .0 Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.04M06 .03M12 .11M06 .00M12 DR .06M12 . MSCI.02M06 DR .02 SharpeRatio 0.07M06 DR . Deutsche Bank Quantitative Strategy Page 68 WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM Benchmark Inverse Vol Global min variance Min tail dependence .07M12 . Thomson Reuters.04M12 DR .99M12 . Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.06 0. Deutsche Bank Quantitative Strategy modifiedCVaR_ES95 Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP. Compustat.09M12 DR . Thomson Reuters. MSCI.3 3 .04M12 .01M12 DR .08M12 DR .09M06 DR .99M12 DR . Compustat.6 -.00M12 .0 -.12M12 Figure 117: Diversification ratio 5 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.2 2 .12M06 .09M06 .4 DR .05M06 . Thomson Reuters. .02M12 DR .01M06 . Russell.06M06 DR .08M06 .10M12 .08M06 DR .05M12 .03M06 .02M12 . MSCI. Deutsche Bank Quantitative Strategy 4 .01M06 DR . Russell.01M12 .06M06 . Russell.03M12 DR . Compustat.0 -.04 0.04M06 DR .00M06 .00M06 DR . Russell.09M12 .07M06 .05M12 DR . Russell. MSCI. Deutsche Bank Quantitative Strategy Page 69 . Russell.5 Source: Bloomberg Finance LLP. Thomson Reuters.0 InvVol EquallyWgted pearson -0.3 0.30 May 2013 Portfolios Under Construction Figure 119: Clustering the strategies Figure 120: Grouping the strategies. Compustat. Thomson Reuters.0 0. Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc. MSCI.6 Benchmark -0. Eigenvalue Ratio Plot Cluster Dendrogram 0.5 Source: Bloomberg Finance LLP.5 Benchmark 0.5 MinTailDependence RiskParity GlobalMinVar MaxDiversification RiskParity MinTailDependence InvVol EquallyWgted MaxDiversification GlobalMinVar 0. 0. Compustat.0 0. Thomson Reuters.12M12 Figure 121: Investment universe Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP. Russell. We clearly see less diversification benefit by investing in the sector/industry portfolios compared to the country portfolios MSCI World 10 GICS sector portfolio First.06M12 NO .10M06 NO .02M06 NO .01M12 NO . European sectors.04M12 NO .00M12 NO .08M12 NO .11M12 NO . Figure 122: Wealth curve 10 3. The GlobalMinVar is the clear winner in this simulation with the highest Sharpe ratio (see Figure 123) and lowest downside risk (see Figure 124) and forms a cluster of its own (see Figure 127 and Figure 128).01M06 NO .07M06 NO .0 4 1. Sector/industry portfolios Sector/industry rotation has gained a lot of attention in recent years. MSCI.5 6 2. along with country rotation.09M06 NO . In this section.03M12 NO . let’s look at the 10 GICS sectors in the MSCI World universe. US sectors. Compustat.07M12 NO .00M06 NO .02M12 NO .05M06 NO .05M12 NO . Deutsche Bank Quantitative Strategy Page 70 99 00 01 02 03 04 Benchmark Inverse Vol Global min variance Min tail dependence 05 06 07 08 09 10 11 12 13 Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.0 0 0.99M12 NO .09M12 NO . Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc. MSCI World industries.5 NO . a combined region x sector portfolio. Compustat. we study five sector portfolios: MSCI World sectors.08M06 NO . MSCI.0 8 2.03M06 NO .06M06 NO .30 May 2013 Portfolios Under Construction X. .11M06 NO .5 2 1.10M12 NO . and finally. Russell.04M06 NO . Thomson Reuters.12M06 NO . Deutsche Bank Quantitative Strategy Page 71 . Russell.11M06 .07M12 DR . Compustat.4 0.8 -. MSCI.09 .2 Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.11M12 .12M12 0.09M06 DR . Thomson Reuters. MSCI. Thomson Reuters.06M06 DR .6 -. Russell. Compustat. MSCI.0 2.0 1.6 1.99M12 .00M12 DR .8 1.07M06 .11 .02M06 . Russell. Deutsche Bank Quantitative Strategy Figure 125: Diversification ratio Figure 126: Weighted portfolio tail dependence 2.02M12 DR .0 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.12 Benchmark Inverse Vol Global min variance Min tail dependence Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.02M06 DR .03M12 .10M06 .06M06 .04M06 .07M06 DR .30 May 2013 Portfolios Under Construction Figure 123: Sharpe ratio Figure 124: CVaR/expected shortfall .01M12 DR . Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.00M06 DR .0 1.08M12 DR .03M06 DR .00M12 .12M06 . Deutsche Bank Quantitative Strategy Source: Bloomberg Finance LLP. Thomson Reuters.03M06 .08M06 DR .01M06 DR .08M12 .09M06 .12M12 Benchmark Inverse Vol Global min variance Min tail dependence .10M12 DR .10 .05M06 DR .2 -.04M06 DR .02M12 .00M06 .05M06 .09M12 .4 -.10M12 . Thomson Reuters.04M12 DR .08 .12M06 DR .09M12 DR . Compustat.07 SharpeRatio modifiedCVaR_ES95 -. WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM DR .5 1. Russell.5 1.01M06 .06M12 DR . MSCI.11M06 DR .0 -.08M06 .01M12 .0 1.10M06 DR .11M12 DR .05M12 DR .07M12 .06M12 .05M12 . Compustat.99M12 DR .04M12 .03M12 DR . 12M06 NO . Thomson Reuters.5 2 1.08M12 NO .03M12 NO . Deutsche Bank Quantitative Strategy 0. MSCI.5 RiskParity InvVol EquallyWgted MinTailDependence MaxDiversification Benchmark 0.07M06 NO .11M06 NO .30 Cluster Dendrogram -0.5 Source: Bloomberg Finance LLP.00 0.01M12 NO .03M06 NO .0 0 0.11M12 NO . Russell.02M06 NO .12M12 Figure 129: Investment universe Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.5 6 2.04M12 NO .02M12 NO .07M12 NO .5 NO .09M06 NO . .99M12 NO .01M06 NO .09M12 NO .15 0.5 Source: Bloomberg Finance LLP.04M06 NO .30 May 2013 Portfolios Under Construction Figure 127: Clustering the strategies Figure 128: Grouping the strategies.00M06 NO . Thomson Reuters. Compustat.0 0.06M12 NO . Compustat. Similar to global sector investment.10M06 NO . GlobalMinVar also dominates the other strategies (see Figure 131 and Figure 132) and forms its own cluster (see Figure 135).06M06 NO .00M12 NO . Russell. MSCI. Thomson Reuters.5 GlobalMinVar GlobalMinVar 0. MSCI. MSCI. Deutsche Bank Quantitative Strategy US S&P 500 10 GICS sector portfolio Now. in the US sectors. Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.10M12 NO . Compustat.0 8 2.05M12 NO . Thomson Reuters. Eigenvalue Ratio Plot 0. Deutsche Bank Quantitative Strategy Page 72 99 00 01 02 03 04 Benchmark Inverse Vol Global min variance Min tail dependence 05 06 07 08 09 10 11 12 13 Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.08M06 NO .0 4 1. Russell.0 RiskParity InvVol MinTailDependence MaxDiversification EquallyWgted Benchmark pearson -0. Russell. Figure 130: Wealth curve 10 3. let’s use the 10 GICS sectors in the S&P 500 universe.05M06 NO . Compustat. 8 0.08 . Compustat.05M12 DR .08M06 DR .2 1.2 -.11M06 DR .00M12 DR .11M06 .07M12 .10M06 DR .99M12 .6 0.06M12 DR . Thomson Reuters.08M06 .30 May 2013 Portfolios Under Construction Figure 131: Sharpe ratio Figure 132: CVaR/expected shortfall .09M12 .08M12 .07M06 .06M06 DR . Deutsche Bank Quantitative Strategy Page 73 .03M12 DR . Russell.12M12 Source: Bloomberg Finance LLP. Russell.06M12 . MSCI.11M12 .10 .10M06 .00M06 DR .99M12 DR .04M12 .8 -.10M12 .6 2.4 -.01M06 DR .12M12 Source: Bloomberg Finance LLP.09 . Compustat.02M12 .04M06 DR .03M12 . Deutsche Bank Quantitative Strategy DR .04M06 .10M12 DR .8 1. MSCI. Thomson Reuters. WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM Benchmark Inverse Vol Global min variance Min tail dependence .01M12 . MSCI.02M06 DR .05M06 .00M06 .12M06 DR .02M12 DR . Thomson Reuters.0 -. MSCI.02M06 . Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.2 0. Russell.09M06 .0 Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.03M06 DR .08M12 DR .00M12 .4 0.01M12 DR .06M06 .11M12 DR .09M12 DR . Compustat. Deutsche Bank Quantitative Strategy Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.07 SharpeRatio .6 -.05M06 DR .4 1.07M06 DR .09M06 DR .11 Benchmark Inverse Vol Global min variance Min tail dependence modifiedCVaR_ES95 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Equally w eighted Risk parity Max diversification Figure 133: Diversification ratio Figure 134: Weighted portfolio tail dependence 2. Thomson Reuters.01M06 .05M12 . Compustat.0 1. Russell.12M06 .03M06 .04M12 DR .07M12 DR . 08M12 NO . Thomson Reuters.0 0.11M12 NO .07M12 NO . Deutsche Bank Quantitative Strategy MSCI Europe 10 GICS sector portfolio The results for the European 10 GICS sectors are similar to the results for global and US universes. Deutsche Bank Quantitative Strategy Page 74 99 00 01 02 03 04 Benchmark Inverse Vol Global min variance Min tail dependence 05 06 07 08 09 10 11 12 13 Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.08M06 NO . Russell. Russell. MSCI.05M12 NO .0 Figure 136: Grouping the strategies.0 MinTailDependence EquallyWgted Benchmark pearson -0. Figure 137: Investment universe Figure 138: Wealth curve 10 3 8 2 6 4 1 2 0 NO . Compustat.11M06 NO .06M12 NO .10M12 NO . Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.5 MaxDiversification RiskParity InvVol EquallyWgted Benchmark 0. Thomson Reuters.02M06 NO .30 May 2013 Portfolios Under Construction Figure 135: Clustering the strategies Eigenvalue Ratio Plot Cluster Dendrogram 0. Russell. Russell. Thomson Reuters.01M06 NO .5 GlobalMinVar GlobalMinVar 0.09M12 NO .03M06 NO .12M06 NO .04M12 NO .10M06 NO . MSCI.03M12 NO .00M06 NO .02M12 NO .2 0.04M06 NO .5 Source: Bloomberg Finance LLP. -0.05M06 NO . MSCI. . Deutsche Bank Quantitative Strategy 0.00M12 NO .12M12 0 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.01M12 NO . Compustat.4 MaxDiversification MinTailDependence RiskParity InvVol 0.5 Source: Bloomberg Finance LLP. MSCI. with GlobalMinVar being the dominant strategy. Thomson Reuters.99M12 NO . Compustat.09M06 NO .06M06 NO . Compustat.07M06 NO . 06M06 .5 -. Thomson Reuters.07M06 DR . MSCI.03M12 DR .30 May 2013 Portfolios Under Construction Figure 139: Sharpe ratio Figure 140: CVaR/expected shortfall .01M12 DR .10 .04M06 DR .4 1.0 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.05M06 DR .0 0.00M12 DR .12M12 0.12M06 .11 . Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.2 Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP. Thomson Reuters.02M12 DR .04M12 DR .08M12 .10M06 .3 -.09M12 .02M06 . WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM DR . Compustat. Deutsche Bank Quantitative Strategy Figure 141: Diversification ratio Figure 142: Weighted portfolio tail dependence 1.4 0. Compustat.14 Benchmark Inverse Vol Global min variance Min tail dependence modifiedCVaR_ES95 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.11M12 .6 1.12M06 DR .09M06 DR .99M12 .01M06 .02M12 .04M12 .4 -.12 .11M12 DR .00M06 DR .6 0.11M06 DR .09M06 .06M06 DR .10M12 . Deutsche Bank Quantitative Strategy Page 75 . Deutsche Bank Quantitative Strategy Source: Bloomberg Finance LLP.6 -.08M06 DR .06M12 .02M06 DR . Russell. Compustat.01M12 .03M06 . MSCI.05M06 .12M12 Benchmark Inverse Vol Global min variance Min tail dependence . Thomson Reuters. MSCI.09 SharpeRatio .0 1.8 1.07M06 .05M12 DR .08M12 DR .07M12 DR .10M12 DR .03M12 .03M06 DR .99M12 DR .08M06 .06M12 DR . Thomson Reuters.01M06 DR . MSCI.2 -. Russell.00M06 .10M06 DR . Russell.09M12 DR .05M12 .8 1.11M06 .00M12 . Russell. Compustat.04M06 .1 -.2 0.07M12 .13 . 99M12 NO . Thomson Reuters.5 Source: Bloomberg Finance LLP.05M12 NO .01M12 NO . Compustat.10M12 NO . Russell. . Deutsche Bank Quantitative Strategy Page 76 99 00 01 02 03 04 Benchmark Inverse Vol Global min variance Min tail dependence 05 06 07 08 09 10 11 12 13 Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP. MSCI. Figure 146: Wealth curve 25 3. Russell.1 0. Eigenvalue Ratio Plot 0.2 0.08M12 NO .04M12 NO .02M12 NO .10M06 NO .5 MinTailDependence MaxDiversification EquallyWgted RiskParity InvVol Benchmark 0.5 5 1.12M06 NO . Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.00M12 NO .07M06 NO .05M06 NO .03M12 NO .0 0 0.0 0.0 0.5 Source: Bloomberg Finance LLP.08M06 NO . MSCI.09M06 NO .0 InvVol MinTailDependence RiskParity 0. MSCI.0 10 1. Thomson Reuters.03M06 NO .06M06 NO .12M12 Figure 145: Investment universe Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP. Thomson Reuters. Russell.0 20 2.01M06 NO . Thomson Reuters. Deutsche Bank Quantitative Strategy MSCI World 24 GICS level 2 industry group portfolio Even if we extend to the 24 GICS industry groups.5 NO .11M06 NO .00M06 NO . the results remain the same – GlobalMinVar again dominates.11M12 NO . Russell.3 Cluster Dendrogram GlobalMinVar -0.06M12 NO . Compustat.02M06 NO .04M06 NO . Compustat.07M12 NO . MSCI.5 GlobalMinVar EquallyWgted Benchmark MaxDiversification pearson -0.5 15 2.30 May 2013 Portfolios Under Construction Figure 143: Clustering the strategies Figure 144: Grouping the strategies. Deutsche Bank Quantitative Strategy 0.09M12 NO . Compustat. .00M06 . Deutsche Bank Quantitative Strategy Source: Bloomberg Finance LLP.07M06 .10M12 DR .07M06 DR .06M12 .02M12 .11M06 .5 .08M12 DR .03M12 DR . MSCI.12M12 WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM DR .06 SharpeRatio 0.08M12 .01M06 .03M06 DR .03M12 . Deutsche Bank Quantitative Strategy Page 77 .10M06 DR .10M06 .04M06 DR .04M12 .06M06 DR .30 May 2013 Portfolios Under Construction Figure 147: Sharpe ratio Figure 148: CVaR/expected shortfall 1. MSCI.08M06 .6 2.08 0. Thomson Reuters.11M12 DR .04M12 DR .02M06 DR .10 0.05M06 .06M12 DR .09M12 .0 .4 1.8 -.09M12 DR . Thomson Reuters.8 .07M12 DR . Deutsche Bank Quantitative Strategy Figure 149: Diversification ratio Figure 150: Weighted portfolio tail dependence 2. Compustat.0 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.03M06 .01M12 .08M06 DR . Compustat. Thomson Reuters.02M12 DR .09M06 .2 -.10M12 .01M06 DR .06M06 .07 0.01M12 DR .11M12 . Russell. Compustat.09M06 DR . Compustat.12M12 . Russell. MSCI.11 Benchmark Inverse Vol Global min variance Min tail dependence modifiedCVaR_ES95 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP. MSCI.12M06 DR .11M06 DR .12M06 . Thomson Reuters.00M12 .00M12 DR .07M12 .02M06 .99M12 DR .0 -.5 .4 -.0 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.04M06 .0 -.6 -.00M06 DR .09 0.05M12 .05M12 DR . Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc. Russell.05M06 DR . Russell.2 1.99M12 . . we see decent improvement in Sharpe ratio and reduction in downside risk. MSCI. Campagna. compared to pure sector/industry portfolios. e. Hong Kong. As we add the country/region dimension. Then. UK.e.10 0. we have very limited sector coverage. as long as we have sectors involved. Campagna.5 RiskParity Benchmark InvVol EquallyWgted Benchmark MinTailDependence MaxDiversification 0. and Norman [2013]. Page 78 Deutsche Bank Securities Inc. Australia. Deutsche Bank Quantitative Strategy Region x sector portfolios Investment universe Following the same philosophy as in De Boer. In De Boer. Thomson Reuters. Russell..5 GlobalMinVar 0. Europe ex UK financials. In our experience.g.5 Source: Bloomberg Finance LLP. MSCI. More importantly.30 GlobalMinVar -0. Europe ex UK. Pacific Free ex Japan (i. etc. rather than using countries. we backtest our risk-based allocation strategies in the country and sector two-dimensional grid. Interestingly.20 0. Pacific Free ex Japan information technology. the potential diversification benefit grows significantly. Therefore. Compustat. Singapore.0 0. GlobalMinVar wins again (see Figure 155 and Figure 156). mostly for liquidity concerns) into six broad regions: US. In the end. and Japan. Eigenvalue Ratio Plot Cluster Dendrogram 0. Canada. we first break down the world (and here we focus on developed countries only. we have 6 × 10 = 60 region x sector assets. within each of the six regions. Deutsche Bank Quantitative Strategy 0. . and New Zealand). Thomson Reuters. the authors use the expansion of all countries on the 10 sectors. for many countries. US consumer stables. Russell. Compustat.30 May 2013 Portfolios Under Construction Figure 151: Clustering the strategies Figure 152: Grouping the strategies.00 0.0 MaxDiversification RiskParity InvVol MinTailDependence EquallyWgted pearson -0.5 Source: Bloomberg Finance LLP. we define the 10 GICS sectors. and Norman [2013]. 01M12 NO . Compustat. Compustat.12 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.06 SharpeRatio 0. Compustat. MSCI. Russell.0 -. Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.04M06 NO .08M12 NO .2 -. Deutsche Bank Quantitative Strategy Page 79 .09M06 NO .07M06 NO .05M06 NO . Russell.06M06 NO . Russell.07M12 NO . Compustat.11M06 NO .8 -.09M12 NO .10M06 NO .10M12 NO .03M06 NO .08M06 NO .12M12 0 Benchmark Inverse Vol Global min variance Min tail dependence 99 00 01 02 03 04 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification 05 06 07 08 09 10 11 12 13 Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.04M12 NO . Thomson Reuters. MSCI.08 0.4 -.05M12 NO .06M12 NO .11M12 NO . Deutsche Bank Quantitative Strategy Figure 155: Sharpe ratio Figure 156: CVaR/expected shortfall 1. Thomson Reuters.01M06 NO . Deutsche Bank Quantitative Strategy Source: Bloomberg Finance LLP.02M06 NO .30 May 2013 Portfolios Under Construction Figure 153: Investment universe Figure 154: Wealth curve 60 4 3 40 2 20 1 0 NO . modifiedCVaR_ES95 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP. Thomson Reuters.10 0. MSCI.12M06 NO .00M12 NO . Russell. Thomson Reuters.99M12 NO .02M12 NO .00M06 NO .03M12 NO . MSCI. 4 0. Deutsche Bank Quantitative Strategy Page 80 -0. MSCI.07M12 . Deutsche Bank Quantitative Strategy Figure 159: Clustering the strategies Figure 160: Grouping the strategies. MSCI.2 MaxDiversification 0.04M06 DR . MSCI.0 0.4 EquallyWgted Benchmark pearson -0.0 MinTailDependence -0.11M12 DR .08M12 .03M12 DR .06M12 .05M12 DR .12M12 .4 3 .09M12 .4 -0. Russell.6 MinTailDependence MaxDiversification GlobalMinVar RiskParity InvVol EquallyWgted Benchmark 0.02M12 DR .08M06 .03M06 DR .04M12 .01M06 . Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.6 0.06M06 DR . Thomson Reuters.12M06 .02M06 .2 0.08M12 DR .02M06 DR . Compustat.11M06 .2 2 .11M06 DR . Thomson Reuters.07M12 DR .4 0. Compustat. Russell.30 May 2013 Portfolios Under Construction Figure 157: Diversification ratio Figure 158: Weighted portfolio tail dependence 4 . Thomson Reuters.05M06 .07M06 DR .6 Source: Bloomberg Finance LLP.10M12 .5 .99M12 . Compustat.06M06 . MSCI.04M06 . Eigenvalue Ratio Plot Cluster Dendrogram 0.09M06 DR .05M06 DR .01M12 .10M06 DR .12M06 DR .01M06 DR .03M06 .11M12 . Compustat.09M12 DR .99M12 DR .0 1 Equally w eighted Risk parity Max diversification Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Bloomberg Finance LLP.04M12 DR .09M06 .07M06 .00M06 DR . .6 Source: Bloomberg Finance LLP. Russell.1 Benchmark Inverse Vol Global min variance Min tail dependence WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM DR .2 0.00M06 .03M12 .00M12 DR .08M06 DR . Thomson Reuters.05M12 . Russell.02M12 .10M06 .10M12 DR .4 GlobalMinVar -0. Deutsche Bank Quantitative Strategy Source: Bloomberg Finance LLP.2 0.12M12 .06M12 DR .01M12 DR .00M12 .3 .2 RiskParity InvVol -0.0 0. et al [2013]). Compustat. Russell.08M06 NO . Thomson Reuters. we start to detect increased level of crowding (or less chance of diversification) for EquallyWgted.03M06 NO .06M06 NO . Russell.02M06 NO . Deutsche Bank Quantitative Strategy Page 81 . RiskParity. Bloomberg Finance LLP. Bloomberg Finance LLP.09M06 NO . US equities We use MSCI USA as our investment universe.11M12 NO . If investors are worried about crowding. MaxDiversification and GlobalMinVar significantly outperform the other strategies. emerging markets equities. 99 00 01 02 03 04 Benchmark Inverse Vol Global min variance Min tail dependence 05 06 07 08 09 10 11 12 13 Equally w eighted Risk parity Max diversification Source: Axioma.05M12 NO .01M06 NO . Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc. Asia ex Japan equities. In this section. For US equities. with higher Sharpe ratios (see Figure 163) and lower downside risks (see Figure 164).10M12 NO . MinTailDependence and MaxDiversification strategies are great alternatives to the GlobalMinVar.02M12 NO .12M06 NO .07M12 NO .99M12 NO . Japan equities.11M06 NO .09M12 NO . and finally global equities. we apply our risk-based allocations in six equity universes: US equities.07M06 NO . all riskbased allocations considerably exceed the capitalization weighted benchmark.12M12 0 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Axioma.08M12 NO . InvVol.04M12 NO . MSCI.06M12 NO . Thomson Reuters.00M06 NO . More importantly.03M12 NO .00M12 NO .05M06 NO .10M06 NO . Equity portfolios Applying risk-based allocation in the equity space is probably where we see the most published research.01M12 NO . and GlobalMinVar strategies (see Figure 167 and Figure 168). MSCI. Consistently with what we found recently (see Cahan.04M06 NO . Compustat.30 May 2013 Portfolios Under Construction XI. European equities. Figure 161: Investment universe Figure 162: Wealth curve 600 5 4 400 3 2 200 1 0 NO . 06M12 .05M06 . Thomson Reuters.4 . Bloomberg Finance LLP.02M06 DR . .8 -.08M12 .6 -.03M06 .06M06 DR . Thomson Reuters.08M06 DR . Russell. Russell. Bloomberg Finance LLP.12M12 1.10 . Compustat. Compustat.04M12 DR .11M06 .01M12 . MSCI. Bloomberg Finance LLP.07M12 .07M06 .07 SharpeRatio modifiedCVaR_ES95 -. Thomson Reuters.03M06 DR . MSCI. Deutsche Bank Quantitative Strategy Figure 165: Diversification ratio Figure 166: Weighted portfolio tail dependence 3.08M06 .10M06 .99M12 DR .12M06 DR . Thomson Reuters.10M12 .02M12 DR .8 .07M12 DR .11M12 DR .08M12 DR .00M06 DR .09M12 DR .2 -.12 Benchmark Inverse Vol Global min variance Min tail dependence Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Equally w eighted Risk parity Max diversification Source: Axioma.04M06 DR .03M12 DR .0 1.6 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Axioma.12M12 .00M12 .11 .02M06 .09M12 . Bloomberg Finance LLP. Deutsche Bank Quantitative Strategy Page 82 WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM Benchmark Inverse Vol Global min variance Min tail dependence .06M06 . Russell.10M06 DR .2 2. Russell.0 -.02M12 .00M06 . MSCI.5 3.2 .03M12 .12M06 .05M12 DR .11M12 .00M12 DR .05M12 .09M06 .06M12 DR .09 .11M06 DR . Compustat.04M12 . Compustat.01M06 . Deutsche Bank Quantitative Strategy Source: Axioma.30 May 2013 Portfolios Under Construction Figure 163: Sharpe ratio Figure 164: CVaR/expected shortfall .1 DR .09M06 DR .6 .05M06 DR . MSCI.10M12 DR .08 .4 2.07M06 DR . Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.99M12 .2 .3 2.01M06 DR .0 Equally w eighted Risk parity Max diversification Source: Axioma.4 -.04M06 .01M12 DR . Sweden.10M06 NO . MSCI. Norway. Belgium. Greece. Russell. Interestingly. Bloomberg Finance LLP. All riskbased strategies substantially outpace the capitalization weighted benchmark.30 May 2013 Portfolios Under Construction Figure 167: Clustering the strategies Figure 168: Grouping the strategies.4 GlobalMinVar -0. and RiskParity. Italy. Portugal.03M12 NO . We see much less level of crowding for risk-based strategies in European equities.02M06 NO . Ireland. Austria. and UK). Thomson Reuters. Denmark. Russell.04M12 NO .0 0.06M06 NO . Deutsche Bank Quantitative Strategy European equities We use MSCI Europe as our investment universe (i.11M12 NO .06M12 NO .5 MaxDiversification RiskParity InvVol EquallyWgted GlobalMinVar Benchmark 0.5 0. MSCI. Deutsche Bank Quantitative Strategy Page 83 . Bloomberg Finance LLP. Netherlands. Compustat.05M06 NO .12M12 Figure 169: Investment universe Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Axioma..09M06 NO . Russell.11M06 NO . Figure 170: Wealth curve 800 4 600 3 400 2 200 1 0 0 NO . France. Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc. our MinTailDependence portfolio again forms its own cluster and offers investors a great way to avoid crowded strategies like GlobalMinVar (see Figure 174). Compustat.10M12 NO . in European equities. MSCI.02M12 NO .08M06 NO . Eigenvalue Ratio Plot Cluster Dendrogram 0.03M06 NO . Russell.2 0.00M12 NO .09M12 NO . Thomson Reuters. 99 00 01 02 03 04 Benchmark Inverse Vol Global min variance Min tail dependence 05 06 07 08 09 10 11 12 13 Equally w eighted Risk parity Max diversification Source: Axioma.01M12 NO .05M12 NO . Compustat.5 Source: Axioma.07M06 NO .01M06 NO .12M06 NO . Deutsche Bank Quantitative Strategy 0. Bloomberg Finance LLP. MSCI. Spain. Thomson Reuters.00M06 NO . Switzerland. Israel.07M12 NO .e.08M12 NO . Germany. In addition.0 0. Thomson Reuters. Finland.0 MinTailDependence MinTailDependence MaxDiversification RiskParity InvVol EquallyWgted Benchmark pearson -0. Bloomberg Finance LLP.99M12 NO . Compustat. in terms of higher Sharpe ratios (see Figure 171) and lower downside risks (see Figure 172). it is the simpler strategies that end up winning – EquallyWgted.04M06 NO .5 Source: Axioma. InvVol. 07M06 DR . Thomson Reuters.02M12 DR . Compustat. Bloomberg Finance LLP.2 2 Equally w eighted Risk parity Max diversification Source: Axioma.09M12 . Compustat.03M06 .11M06 DR .01M06 . Bloomberg Finance LLP.30 -. MSCI.01M12 .11M12 .09M06 DR .02M12 .02M06 DR .05M06 DR .07M12 .06M06 DR .00M06 .00M06 DR .99M12 .04M06 DR .16 . MSCI.07M12 DR . Russell.06M12 DR .08M06 DR .08M06 .30 May 2013 Portfolios Under Construction Figure 171: Sharpe ratio Figure 172: CVaR/expected shortfall .10M06 .20 Benchmark Inverse Vol Global min variance Min tail dependence modifiedCVaR_ES95 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Equally w eighted Risk parity Max diversification Source: Axioma.15 -.4 3 .02M06 .00M12 . Bloomberg Finance LLP.09M06 .12 SharpeRatio .99M12 DR .12M12 Benchmark Inverse Vol Global min variance Min tail dependence .03M12 DR .10M12 DR .04M12 DR .00M12 DR . Thomson Reuters. Russell.03M12 . MSCI. Russell. Compustat.03M06 DR .10M12 .0 1 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Axioma.07M06 . Thomson Reuters.20 -. . Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.18 .06M12 .05M12 DR .06M06 .25 -.01M06 DR . Russell.11M06 .6 4 .12M06 DR .10M06 DR . Bloomberg Finance LLP.01M12 DR .35 -. Compustat.04M12 .08M12 .08M12 DR . Thomson Reuters.04M06 .12M06 . Deutsche Bank Quantitative Strategy Figure 173: Diversification ratio Figure 174: Weighted portfolio tail dependence 5 . MSCI.14 .12M12 .05M12 .11M12 DR . Deutsche Bank Quantitative Strategy Source: Axioma. Deutsche Bank Quantitative Strategy Page 84 WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM DR .09M12 DR .05M06 . 05M06 NO . Compustat. In this universe. Korea. Hong Kong. Eigenvalue Ratio Plot Cluster Dendrogram 0. Compustat. China.11M06 NO .01M06 NO . Russell. Singapore. Indonesia. Russell. Malaysia.4 MinTailDependence -0. Thomson Reuters.05M12 NO . Compustat.07M06 NO .08M06 NO . followed by GlobalMinVar. Deutsche Bank Quantitative Strategy 0. Deutsche Bank Quantitative Strategy Asian equities We use the MSCI Pacific ex Japan as our investment universe (i.10M12 NO .03M06 NO . Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.02M06 NO .2 0.04M06 NO .0 0. Bloomberg Finance LLP. MinTailDependence shows the highest Sharpe ratio (see Figure 179) and lowest downside risk (see Figure 180).5 0.12M06 NO .0 0. MSCI. Bloomberg Finance LLP. Figure 177: Investment universe Figure 178: Wealth curve 600 8 6 400 4 200 2 0 NO . Compustat.5 Source: Axioma. Taiwan.08M12 NO . Thomson Reuters. all riskbased allocations significantly beat the capitalization weighted benchmark. Bloomberg Finance LLP. Russell.5 Source: Axioma.11M12 NO .0 MaxDiversification EquallyWgted MaxDiversification RiskParity InvVol GlobalMinVar Benchmark pearson -0. MSCI.03M12 NO . Philippines. MSCI. Bloomberg Finance LLP. Again.00M12 NO .30 May 2013 Portfolios Under Construction Figure 175: Clustering the strategies Figure 176: Grouping the strategies. MinTailDependence again shows its uniqueness (see Figure 183) and is likely to avoid crowded trades (see Figure 182).04M12 NO .07M12 NO . Deutsche Bank Quantitative Strategy Page 85 . Thomson Reuters.99M12 NO .09M12 NO . India.5 GlobalMinVar RiskParity InvVol EquallyWgted Benchmark MinTailDependence 0. Similar to what we find in European equities.02M12 NO .e. Russell.01M12 NO .06M06 NO . Thomson Reuters.09M06 NO .06M12 NO .12M12 0 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Axioma.10M06 NO . and Thailand)..00M06 NO . MSCI. 99 00 01 02 03 04 Benchmark Inverse Vol Global min variance Min tail dependence 05 06 07 08 09 10 11 12 13 Equally w eighted Risk parity Max diversification Source: Axioma. risk-based allocations still offer great diversification benefit in the Asia ex Japan universe. 4 -.99M12 .08M12 .06M06 DR .10M12 DR .00M06 DR . Compustat.13 SharpeRatio modifiedCVaR_ES95 -.00M12 .12M06 .01M12 .05M06 DR .04M12 .09M06 DR .06M06 . Compustat. Deutsche Bank Quantitative Strategy DR . Russell. MSCI.17 . Russell.11M06 DR .99M12 DR . Bloomberg Finance LLP.09M06 .04M06 .09M12 .00M12 DR .10M06 DR .30 May 2013 Portfolios Under Construction Figure 179: Sharpe ratio Figure 180: CVaR/expected shortfall .04M06 DR . Compustat. Bloomberg Finance LLP. MSCI.06M12 .08M06 .12M12 Source: Axioma.11M12 .4 4 .11M06 .15 .06M12 DR .12M12 Source: Axioma. MSCI.02M06 .03M12 .01M06 DR .1 1 .07M12 .09M12 DR .0 Equally w eighted Risk parity Max diversification Source: Axioma. Thomson Reuters.03M12 DR . Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc. .05M12 DR .12M06 DR .11M12 DR .3 3 .08M12 DR . Thomson Reuters.2 2 . Russell.02M06 DR . Thomson Reuters.04M12 DR .03M06 .05M06 . Deutsche Bank Quantitative Strategy Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Axioma. Compustat.08M06 DR . MSCI. Thomson Reuters.2 Benchmark Inverse Vol Global min variance Min tail dependence Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Equally w eighted Risk parity Max diversification Figure 181: Diversification ratio Figure 182: Weighted portfolio tail dependence 5 .16 -.07M12 DR .00M06 .14 . Deutsche Bank Quantitative Strategy Page 86 WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM Benchmark Inverse Vol Global min variance Min tail dependence .10M06 .07M06 DR . Russell.02M12 DR .01M12 DR . Bloomberg Finance LLP.01M06 .10M12 .02M12 . Bloomberg Finance LLP.07M06 .6 -.05M12 .03M06 DR .8 -. 11M12 NO . Eigenvalue Ratio Plot Cluster Dendrogram 0.5 Source: Axioma.5 0.8 0 NO . Figure 185: Investment universe Figure 186: Wealth curve 400 2.0 EquallyWgted InvVol RiskParity Benchmark MaxDiversification GlobalMinVar pearson -0.01M12 NO .0 0. Compustat.02M06 NO . Compustat. GlobalMinVar has the highest Sharpe ratio (see Figure 187).06M12 NO .02M12 NO .4 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Axioma. Bloomberg Finance LLP.09M06 NO .5 Source: Axioma.07M06 NO .0 300 1.11M06 NO . Russell. 99 00 01 02 03 04 Benchmark Inverse Vol Global min variance Min tail dependence 05 06 07 08 09 10 11 12 13 Equally w eighted Risk parity Max diversification Source: Axioma. Russell. while MinTailDependence has the lowest downside risk (see Figure 188).10M12 NO .12M06 NO .30 May 2013 Portfolios Under Construction Figure 183: Clustering the strategies Figure 184: Grouping the strategies. Thomson Reuters. Bloomberg Finance LLP.99M12 NO .00M12 NO .05M06 NO . MSCI. Bloomberg Finance LLP.4 MinTailDependence -0. Compustat.08M06 NO . Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.07M12 NO .00M06 NO .0 0.04M12 NO . while the traditional riskbased strategies appear to be getting crowded in recent years.05M12 NO .6 200 1. Deutsche Bank Quantitative Strategy Japanese equities We use the MSCI Japan as our investment universe. Bloomberg Finance LLP. Thomson Reuters. Thomson Reuters. MinTailDependence again seems to be quite unique (see Figure 191).01M06 NO . Compustat. Deutsche Bank Quantitative Strategy Page 87 . Russell.08M12 NO .06M06 NO .03M12 NO .03M06 NO .5 MaxDiversification GlobalMinVar InvVol EquallyWgted RiskParity Benchmark MinTailDependence 0.4 2.12M12 0.04M06 NO . MSCI.2 0. Deutsche Bank Quantitative Strategy 0. MSCI. with significantly lower chance of crowding (see Figure 190).09M12 NO .2 100 0.10M06 NO . MSCI. Thomson Reuters. Russell. 09M12 DR .10M12 .4 .2 -.04M12 .05M06 DR .2 .04M06 .09M06 DR .10M06 DR .00M06 .4 -.99M12 DR .06M06 .02M12 DR .10M12 DR .0 -.00M12 .07M06 DR .12M06 .07M12 .2 2.11M06 DR . Russell.04M12 DR .1 -.08M06 .3 -. Compustat.10M06 .4 2.01M12 DR .02M12 . Russell.99M12 . MSCI.11M12 . Russell. Bloomberg Finance LLP. Thomson Reuters.08M12 .09M06 .092 .096 . MSCI.06M06 DR .08M06 DR .12M12 1.04M06 DR .05M12 .00M06 DR . Compustat.3 2.03M06 .03M12 DR .6 . Deutsche Bank Quantitative Strategy Page 88 WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM Benchmark Inverse Vol Global min variance Min tail dependence .01M06 DR . Deutsche Bank Quantitative Strategy Figure 189: Diversification ratio Figure 190: Weighted portfolio tail dependence 3.05M06 .100 .01M06 . Thomson Reuters.01M12 .104 modifiedCVaR_ES95 -.1 Benchmark Inverse Vol Global min variance Min tail dependence Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Equally w eighted Risk parity Max diversification Source: Axioma.06M12 . Russell.00M12 DR .02M06 DR . Thomson Reuters. Compustat. Bloomberg Finance LLP.30 May 2013 Portfolios Under Construction Figure 187: Sharpe ratio Figure 188: CVaR/expected shortfall .6 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Axioma. Deutsche Bank Quantitative Strategy Source: Axioma.03M06 DR . Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc. Compustat.088 SharpeRatio .0 1.12M06 DR . Bloomberg Finance LLP.11M06 .2 .06M12 DR .02M06 .108 -.07M12 DR .05M12 DR . MSCI.11M12 DR . Thomson Reuters.08M12 DR . Bloomberg Finance LLP.5 3.09M12 .03M12 .12M12 .8 .07M06 . MSCI.0 Equally w eighted Risk parity Max diversification Source: Axioma. .1 DR . MSCI. Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.06M06 NO .99M12 NO .30 May 2013 Portfolios Under Construction Figure 191: Clustering the strategies Figure 192: Grouping the strategies.00M06 NO .05M06 NO . MSCI. Figure 193: Investment universe Figure 194: Wealth curve 1. Similar to Asia ex Japan. MSCI. Compustat.06M12 NO .07M06 NO .5 Source: Axioma. Deutsche Bank Quantitative Strategy 0. Russell.2 MinTailDependence 0. Compustat.03M06 NO .12M12 0 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Axioma.01M06 NO .5 0.03M12 NO . Russell.000 10 800 8 600 6 400 4 200 2 0 NO . Thomson Reuters. with substantially higher Sharpe ratio (see Figure 195) and lower downside risk (see Figure 196).09M12 NO .07M12 NO . It also appears to be highly unique (see Figure 199 and Figure 200).5 Source: Axioma. Thomson Reuters.09M06 NO . Compustat.11M06 NO .08M06 NO .0 Benchmark EquallyWgted InvVol RiskParity GlobalMinVar MaxDiversification pearson -0. Thomson Reuters. Russell. Deutsche Bank Quantitative Strategy Emerging markets equities We use the MSCI EM as our investment universe.08M12 NO . Bloomberg Finance LLP.12M06 NO .02M06 NO . our MinTailDependence portfolio is the clear winner.10M12 NO . 99 00 01 02 03 04 Benchmark Inverse Vol Global min variance Min tail dependence 05 06 07 08 09 10 11 12 13 Equally w eighted Risk parity Max diversification Source: Axioma. Bloomberg Finance LLP.01M12 NO .0 0. Eigenvalue Ratio Plot Cluster Dendrogram 0. Compustat. Deutsche Bank Quantitative Strategy Page 89 . Russell. MSCI.05M12 NO .5 MaxDiversification GlobalMinVar RiskParity InvVol EquallyWgted Benchmark 0.0 0.04M06 NO .00M12 NO .02M12 NO . Bloomberg Finance LLP. Thomson Reuters.11M12 NO .4 MinTailDependence -0.10M06 NO . with much lower ex post tail dependence among its holdings (see Figure 198).04M12 NO . Bloomberg Finance LLP. 08M12 .1 2 Equally w eighted Risk parity Max diversification Source: Axioma.16 0.07M12 .08M06 .12 SharpeRatio 1. .8 -.30 May 2013 Portfolios Under Construction Figure 195: Sharpe ratio Figure 196: CVaR/expected shortfall 1. Russell.04M12 DR . Deutsche Bank Quantitative Strategy Figure 197: Diversification ratio Figure 198: Weighted portfolio tail dependence 6 .03M12 .11M06 . Deutsche Bank Quantitative Strategy Page 90 WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM DR .02M06 DR .02M12 .12M06 DR .04M06 .00M12 . Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.2 3 .09M12 DR . MSCI.08M12 DR .08M06 DR . MSCI.10M06 DR . Deutsche Bank Quantitative Strategy Source: Axioma.09M06 . MSCI.05M06 .01M12 .09M06 DR .12M06 . Compustat.03M12 DR .2 -.06M12 .10M06 .0 -.15 0.4 -. Thomson Reuters. Bloomberg Finance LLP.10M12 DR .01M06 DR .11M12 DR .11M06 DR .00M06 .02M12 DR .09M12 . Bloomberg Finance LLP.00M06 DR . Thomson Reuters.0 1 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Axioma. Compustat. Russell. Thomson Reuters. Russell.14 0.6 -.03M06 DR .04M06 DR .05M06 DR .11M12 .03M06 .07M06 .00M12 DR .07M12 DR . Bloomberg Finance LLP.12M12 .13 0. Thomson Reuters.02M06 . Compustat.12M12 Benchmark Inverse Vol Global min variance Min tail dependence .06M12 DR . Compustat.01M12 DR .17 Benchmark Inverse Vol Global min variance Min tail dependence modifiedCVaR_ES95 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Equally w eighted Risk parity Max diversification Source: Axioma. Bloomberg Finance LLP.10M12 .04M12 . Russell.2 -. MSCI.06M06 DR .4 5 .07M06 DR .99M12 .99M12 DR .01M06 .05M12 DR .3 4 .06M06 .05M12 . 09M12 NO . Thomson Reuters. Figure 201: Investment universe Figure 202: Wealth curve 2. Russell.10M12 NO ..02M12 NO . 0 99 00 01 02 03 04 Benchmark Inverse Vol Global min variance Min tail dependence 05 06 07 08 09 10 11 12 13 Equally w eighted Risk parity Max diversification Source: Axioma.05M12 NO .4 Cluster Dendrogram MinTailDependence -0. Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc. MaxDiversification.06M12 NO .5 Source: Axioma. Compustat. In the global equity space.10M06 NO .5 Source: Axioma.5 0.11M12 NO .06M06 NO . Bloomberg Finance LLP.04M06 NO .500 3 1. the developed countries. MinTailDependence once again proves to be the least crowded strategy (see Figure 206).12M06 NO . Deutsche Bank Quantitative Strategy 0. GlobalMinVar.2 MinTailDependence 0.0 0. Russell. Eigenvalue Ratio Plot 0. MSCI.0 MaxDiversification GlobalMinVar InvVol EquallyWgted RiskParity MaxDiversification Benchmark 0.04M12 NO . Deutsche Bank Quantitative Strategy Global equity Our investment universe is the MSCI World Index. i.03M12 NO .01M12 NO .11M06 NO .07M06 NO .00M12 NO . Deutsche Bank Quantitative Strategy Page 91 .0 0.07M12 NO . Thomson Reuters.02M06 NO . Compustat.12M12 0 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Axioma. Thomson Reuters.08M12 NO . MSCI.05M06 NO .01M06 NO .00M06 NO . Bloomberg Finance LLP.09M06 NO . MSCI.30 May 2013 Portfolios Under Construction Figure 199: Clustering the strategies Figure 200: Grouping the strategies. Russell.e.99M12 NO . MSCI. Compustat.000 5 4 1.000 2 500 1 NO . Bloomberg Finance LLP.5 GlobalMinVar Benchmark RiskParity InvVol EquallyWgted pearson -0. Russell. Thomson Reuters. and MinTailDependence strategies appear to be very robust.08M06 NO . They also form a unique cluster (see Figure 207). Compustat.03M06 NO . with attractive Sharpe ratios (see Figure 203) and reasonable downside risks (see Figure 204). Bloomberg Finance LLP. 4 -.00M12 .08M06 .01M12 .06M12 .0 1 Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Source: Axioma. Russell.10M06 . Deutsche Bank Quantitative Strategy Figure 205: Diversification ratio Figure 206: Weighted portfolio tail dependence 6 . MSCI.03M12 DR . Compustat.02M12 DR .07M06 DR .10M12 DR . Bloomberg Finance LLP.01M06 DR .04M06 .10M06 DR .12 .02M06 .6 -. Bloomberg Finance LLP. Bloomberg Finance LLP.99M12 DR .02M06 DR .01M06 . Russell. .06M06 . Deutsche Bank Quantitative Strategy Source: Axioma.05M12 .07M06 . MSCI.01M12 DR .00M06 .06M06 DR . Compustat. MSCI. Compustat.03M06 . Thomson Reuters.4 5 .05M06 DR .05M12 DR .2 -.08M12 . Russell.11 .0 -.8 -. Thomson Reuters. MSCI. Compustat. Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.30 May 2013 Portfolios Under Construction Figure 203: Sharpe ratio Figure 204: CVaR/expected shortfall .12M12 Benchmark Inverse Vol Global min variance Min tail dependence .11M12 .09M06 DR .00M06 DR . Thomson Reuters.06M12 DR .07M12 .12M06 .00M12 DR . Thomson Reuters.08M12 DR . Deutsche Bank Quantitative Strategy Page 92 WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM WTD_EM DR .14 Benchmark Inverse Vol Global min variance Min tail dependence Benchmark Inverse Vol Global min variance Min tail dependence Equally w eighted Risk parity Max diversification Equally w eighted Risk parity Max diversification Source: Axioma.12M06 DR .10 .09M12 . Russell.05M06 .11M06 .08M06 DR .04M06 DR .02M12 .04M12 DR .2 3 .3 4 .99M12 .12M12 .03M12 .07M12 DR .11M06 DR .03M06 DR .13 .09 SharpeRatio modifiedCVaR_ES95 -.1 2 Equally w eighted Risk parity Max diversification Source: Axioma.11M12 DR .09M12 DR .10M12 .09M06 .04M12 . Bloomberg Finance LLP. MSCI. Bloomberg Finance LLP.2 RiskParity InvVol EquallyWgted -0. Thomson Reuters.6 Source: Axioma. Thomson Reuters.2 0. Deutsche Bank Quantitative Strategy Page 93 . Compustat.20 0. MSCI.10 0.6 Source: Axioma. Bloomberg Finance LLP. Russell. Russell.4 -0. Eigenvalue Ratio Plot Cluster Dendrogram 0.6 Benchmark -0.6 0.4 pearson MinTailDependence RiskParity InvVol EquallyWgted Benchmark GlobalMinVar MaxDiversification 0.2 MinTailDependence 0.0 MaxDiversification -0. Compustat. -0.30 May 2013 Portfolios Under Construction Figure 207: Clustering the strategies Figure 208: Grouping the strategies.4 0.0 0.30 0. Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc.2 0.4 0.00 GlobalMinVar -0. while GlobalMinVar and MinCVaR are more for risk control 20 Because our RobMinCVaR is very computationally intensive. If we relax the assumption of multivariate normal distribution of asset returns (which is almost always violated with real life data). Diversification based Diversification based strategies primarily try to diversify our investments. to recognize that a linear correlation coefficient can’t fully capture the full extent of comovement.30 May 2013 Portfolios Under Construction XII. Conditional value at risk or CVaR is a more coherent definition of risk and has many nice properties. Conclusion The philosophy of portfolio construction In summary. we only show its application using our global country portfolio as an example. A strategy that tries to reduce CVaR risk is what we called MinCVaR20. For this paper. GlobalMinVar relies on volatility (or variance) as the definition of risk and aims to achieve risk reduction by minimizing a portfolio’s expected variance. Among the seven portfolio construction techniques. we can acknowledge that variance (or volatility) is not the only way to define risk. which should deliver the highest tail diversification (ex ante). then we can design our MinTailDependence strategy. EquallyWgted is probably the most naïve way to reach this goal. based on their main intended goals. RiskParity brings it to the next level by incorporating correlation and asset specific risk. we survey seven risk-based allocations and compare their performance with the more traditional capitalization-weighted benchmark. . If we define asset dependence by tail dependence coefficient. Risk reduction based Strategies in this bucket attempts to reduce risk as the dominant goal. Page 94 Deutsche Bank Securities Inc. Statistical cluster analysis confirms our argument that MaxDiversification and MinTailDependence are more designed to capture diversification. InvVol or volatility parity further takes into account of each asset’s risk in diversification. we can classify them into two categories. by construction is the most diversified strategy (ex ante). we do not show RobMinCVaR in other contexts. MaxDiversification. Figure 209 and Figure 210 reproduce the cluster analysis graphs using our country allocation example in Section VI. European equities.30 0.4 Benchmark -0. economic risk hedged global countries. industry allocation: global countries. commodities. global industries.25 0.4 -0.4 Eigenvalue 1 Risk-based allocation hclust (*.40 Cluster Dendrogram 0. sector. and region x sector combinations Equities: US equities. emerging markets equities. and global equities Deutsche Bank Securities Inc.15 0.6 -0.35 0.0 Source: Deutsche Bank A horse race of risk-based allocations We apply our seven risk-based portfolio construction techniques in three different contexts: Asset allocation: multi-asset allocation.10 Height 0.4 GlobalMinVar RobMinCVaR MaxDiversification -0.0 -0. global sectors. global sovereign bonds. European sectors. US sectors. Asia ex Japan equities.2 pearson -0.30 May 2013 Portfolios Under Construction Figure 209: Cluster analysis Figure 210: Eigenvalue ratio plot Eigenvalue Ratio Plot 0. Page 95 .05 0.2 0. and alternative betas Country.2 0.2 0.6 RiskParity InvVol RobMinCVaR GlobalMinVar MinTailDependence MaxDiversification RiskParity InvVol EquallyWgted EquallyWgted 0. Japanese equities.20 Benchmark Eigenvalue 2 MinTailDependence 0. "complete") Source: Deutsche Bank 0. Bloomberg Finance LLP. avg ranking 6 7 5 4 2 1 3 Multi-assets 7 6 4 2 5 1 3 Sovereign bonds 4 7 6 5 1 2 3 Commodities 6 6 4 5 1 2 3 Alternative betas 7 6 5 4 2 3 1 Country/sector allocation. Thomson Reuters. Deutsche Bank Quantitative Strategy Page 96 Deutsche Bank Securities Inc. Compustat. Figure 211: Sharpe ratio ranking Sharpe ratio Benchmark EquallyWgted InvVol RiskParity GlobalMinVar MaxDiversification MinTailDependence Asset allocation.30 May 2013 Portfolios Under Construction Sharpe ratio ranking In terms of Sharpe ratio (see Figure 211). and MaxDiversification. US 7 5 2 3 1 6 4 Sectors. MSCI 7 5 2 3 1 6 4 Sectors. followed by MinTailDependence. MEAM 7 5 6 4 2 3 1 Sectors. MSCI 7 6 5 3 1 2 4 Equities. avg ranking 7 6 5 3 1 4 2 Countries. . avg ranking 7 6 5 4 3 3 3 US 7 6 5 4 2 1 3 Europe 7 2 1 3 6 5 4 Asia ex Japan 7 6 5 4 2 3 1 Japan 7 5 4 3 1 2 6 Emerging markets 7 4 2 3 5 6 1 Global 7 6 5 4 2 1 3 Overall ranking 7 6 5 4 1 3 2 Source: Axioma. GlobalMinVar achieves the highest Sharpe ratio overall. Russell. Europe 7 6 3 2 1 4 5 Industries. followed by naïve EquallyWgted and InvVol strategies. MSCI 7 6 5 3 1 4 2 Regions x sectors. MSCI. Capitalization weighted benchmark delivers the lowest Sharpe ratio. MSCI ACWI 7 6 5 4 3 2 1 Countris. when we shift our focus to risk reduction. Europe 7 6 4 3 1 2 5 Industries. Bloomberg Finance LLP. US 7 6 4 3 1 5 2 Sectors. MSCI 5 7 6 4 1 2 3 Equities. avg ranking 6 7 5 3 1 2 4 Countries.. avg ranking 4 7 6 5 2 3 1 US 5 7 6 4 1 2 3 Europe 1 4 3 5 6 7 2 Asia ex Japan 3 7 6 5 2 4 1 Japan 7 6 5 4 2 3 1 Emerging markets 6 7 5 4 2 3 1 Global 3 7 6 5 4 1 2 Overall ranking 5 7 6 4 1 2 3 Source: Axioma. InvVol) and benchmark fall behind again with the highest downside risk. MSCI. Deutsche Bank Quantitative Strategy Deutsche Bank Securities Inc. MSCI 6 7 5 4 1 3 2 Sectors. Compustat. EquallyWgted. MSCI 7 6 4 3 1 2 5 Regions x sectors.g.30 May 2013 Portfolios Under Construction Downside risk ranking As shown in Figure 212. The naïve strategies (e. MSCI ACWI 2 7 6 5 1 3 4 Countris. MaxDiversification and MinTailDependence portfolios are not far off. avg ranking 5 7 6 4 2 1 3 Multi-assets 2 7 6 3 4 1 5 Sovereign bonds 3 7 6 5 1 2 4 Commodities 6 6 5 4 1 2 3 Alternative betas 7 6 4 2 5 3 1 Country/sector allocation. Thomson Reuters. Russell. MEAM 1 6 4 5 2 3 7 Sectors. with comparable risk cutback. GlobalMinVar wins again and accomplishes its goal ex post. Figure 212: Downside risk ranking CVaR/expected shortfall Benchmark EquallyWgted InvVol RiskParity GlobalMinVar MaxDiversification MinTailDependence Asset allocation. Page 97 . avg ranking 7 4 5 3 6 1 2 Countries. US 6 4 5 3 7 1 2 Sectors. and InvVol have the lowest diversification. MSCI ACWI 7 5 6 4 3 1 2 Countris. Bloomberg Finance LLP. Deutsche Bank Quantitative Strategy Page 98 Deutsche Bank Securities Inc. defined as linear comovement. MSCI 7 6 5 4 3 1 2 Equities. Compustat. .30 May 2013 Portfolios Under Construction Diversification ranking Now. avg ranking 6 7 5 3 4 1 3 Multi-assets 6 6 4 2 5 1 3 Sovereign bonds 5 7 6 3 4 1 2 Commodities 6 6 5 3 4 1 2 Alternative betas 6 6 5 3 2 1 4 Country/sector allocation. let’s move to diversification. MEAM 7 6 5 4 3 1 2 Sectors. Thomson Reuters. Europe 6 4 5 3 7 1 2 Industries. EquallyWgted. MSCI 7 5 4 3 6 1 2 Regions x sectors. MSCI 6 4 5 3 7 1 2 Sectors. Figure 213: Diversification ranking Diversification ratio Benchmark EquallyWgted InvVol RiskParity GlobalMinVar MaxDiversification MinTailDependence Asset allocation. Russell. MSCI. followed by MinTailDependence and RiskParity. MaxDiversification also earns the highest diversification ratio ex post. avg ranking 7 5 6 3 2 1 5 US 7 5 6 4 3 1 2 Europe 7 4 5 3 2 1 6 Asia ex Japan 7 4 5 3 2 1 6 Japan 7 6 5 3 2 1 4 Emerging markets 6 4 5 3 2 1 7 Global 5 6 7 3 2 1 4 Overall ranking 7 6 6 3 4 1 3 Source: Axioma. The benchmark. As shown in Figure 213. MEAM 5 7 6 4 3 2 1 Sectors. US 7 5 6 4 3 2 1 Sectors. Our MinTailDependence strategy can be an effective diversifier. Similarly. in Figure 216: Weighted portfolio tail dependence. avg ranking 7 5 6 4 3 2 1 Countries. MSCI. We do not observe similar patterns in other regions and asset classes yet. avg ranking 6 7 5 4 2 1 3 Multi-assets 5 5 6 7 2 1 3 Sovereign bonds 4 7 6 5 1 2 3 Commodities 7 7 5 4 3 1 2 Alternative betas 7 7 5 4 1 2 3 Country/sector allocation. country/sector. It is interesting to note that overall multi-asset. avg ranking 7 5 6 4 3 2 1 US 7 4 5 3 6 2 1 Europe 7 5 6 4 3 2 1 Asia ex Japan 7 5 6 4 2 3 1 Japan 7 6 5 4 3 2 1 Emerging markets 7 6 5 4 2 3 1 Global 7 5 6 4 3 2 1 Overall ranking 7 6 5 4 3 2 1 Source: Axioma. Compustat. Deutsche Bank Securities Inc..30 May 2013 Portfolios Under Construction Tail dependence/crowding ranking Finally. while the opportunities in the sector/industry portfolios appear to be limited. where GlobalMinVar-type strategies have attracted great attention and money flow. we compare the weighted portfolio tail dependent parameter for the MinTailDependence and benchmark over the same investment universes. Again. Bloomberg Finance LLP. MSCI 6 4 5 3 7 2 1 Sectors. In Figure 215. especially in the country/sector and equity portfolios. Europe 7 5 6 4 3 2 1 Industries. country. we start to see signs of crowding. Figure 214: Tail dependence/crowding ranking Weighted portfolio tail dependence Benchmark EquallyWgted InvVol RiskParity GlobalMinVar MaxDiversification MinTailDependence Asset allocation. MSCI 7 6 5 4 3 2 1 Equities. MSCI ACWI 7 6 5 4 3 2 1 Countris.e. we focus on strategy crowding and tail dependence (see Figure 214). and equity portfolios offer great diversification potential. multiasset. capitalization weighted benchmark) in our multi-asset. Deutsche Bank Quantitative Strategy Potential diversification benefit We would like to conclude this paper with a summary on potential diversification benefit of various investment universes. Russell. and equity portfolios. Page 99 . Our MinTailDependence portfolio manages to deliver the best diversification ex post. In US equities. MaxDiversification and GlobalMinVar also help avoid crowded trades. we compare the best diversification strategy (MaxDiversification) with the least diversified portfolio (i. country. and equity portfolios offer better diversification upside than sector/industry portfolios. MSCI 7 5 6 4 3 2 1 Regions x sectors. Thomson Reuters. 0 20.0 40.0% 4.0% 2.0 0.0 0. .0% 1.30 May 2013 Portfolios Under Construction Figure 215: Diversification ratio Figure 216: Weighted portfolio tail dependence 60.0 3.0% Benchmark Source: Deutsche Bank Page 100 MaxDiversification Benchmark MinTailDependence Source: Deutsche Bank Deutsche Bank Securities Inc. pp. The Journal of Risk. A. [2011]. Luo. 40-51.. [2013]. Mesomeris.R. Deutsche Bank Quantitative Strategy. 2013. [1999]. Bacon. J. Journal of Royal Statistical Society. Z... Froidure. Jussa. Alvarez.. “Risk parity and riskbased allocation”. 4. and Croux. Journal of Royal Statistical Society. SSRN. No. S. “Disentangling the downside”. 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[2001]. “Conditional value-at-risk for general loss distributions”, Research Report 2001-5, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL. Rousseeuw, P.J. [1985]. “Multivariate estimation with high breakdown point”, in W.e.a. Grossman (ed.), Mathematical Statistics and Applications, vol B, pp. 283-297, Budapest: Akademiai Kiado. Rousseeuw, P.J. and van Driessen, K. [1999]. “A fast algorithm for the minimum covariance determinant estimator”, Technometrics, 41, pp. 212-223. Royston, P. [1982]. “An extension of Shapiro and Wilk’s w test for normality to large samples”, Applied Statistics, 31, pp. 115-124. Schaefer, J., Opgen-Rhein, R., and Strimmer, K. [2008]. “The corpcor package: Efficient estimation of covariance and (partial) correlation, cran.r-project.org. Sklar, A. [1959]. "Fonctions de répartition à n dimensions et leurs marges", Publ. Inst. Statist. Univ. Paris 8: 229–231. Szekely, G.J. [1989]. “Potential and kinetic energy in statistics”, Lecture Notes, Budapest Institute of Technology, Technical University. Venables, W.N., Ripley, B.D. ]2008]. The MASS Package, cran.r-project.org. Zangari, p. [1996]. “How accurate is the delta-gamma methodology?”, RiskMetrics Monitor 1996, Q3. Deutsche Bank Securities Inc. Page 103 30 May 2013 Portfolios Under Construction XIV. Review of our previous research We have been publishing quite intensively in the field of risk-based allocation. Below is a brief summary of each of our recent research. Independence day, Luo, et al [2013], February 6, 2013 In all 45 countries in the MSCI ACWI, for both equities and bonds, we find significant positive (or negative) risk premia can be earned on the days when important economic indicators are released. We find low-risk investing by avoiding macroeconomic uncertainties is different from, additive to, and even dominating the traditional riskbased asset allocation techniques. We further demonstrate that the performance of GTAA and SAA can be improved significantly with the MEAM model. Cross asset class momentum, Jussa, et al [2012], November 5, 2012 We study momentum risk premium in seven asset classes: equities, country/sector indices, hedge fund styles, currencies, commodities, and rates. The momentum risk premia across asset classes are not highly correlated. We then test a wide range of portfolio construction techniques, from naïve equally weighted, past return/Sharpe ratio weighted, to more sophisticated risk-based allocations (e.g., inverse volatility, risk parity, maximum diversification, and minimum variance). We also study a simple regime switching model. Disentangling the downside, Cahan, et al [2012], September 20, 2012 Do investors get compensated for taking on higher risk through higher future returns? This is probably one of the greatest puzzles in finance. In this research, we surveyed a wide range of risk metrics. More importantly, we found an interesting risk-return relationship along the term structure – it seems that investors are only compensated in the longer term. In the end, we proposed a “smart” risk premium factor, by capturing the long-term risk premium, while avoiding the short-term underperformance that comes from holding risky stocks. The risk in low risk?, Cahan, et al [2012], July 19, 2012 This paper tries to answer an important question whether low risk strategies are becoming crowded. We introduce a novel metric for measuring crowdedness called Median Pairwise Tail Dependence (MPTD). This is designed to measure the likelihood that stocks in a portfolio have simultaneous large negative drawdowns. Combined with a few other more conventional techniques, e.g., institutional ownership, we do not find low risk strategies are currently crowded. We also studied alternative low risk strategies in this paper. A new asset allocation paradigm, Mesomeris, et al [2012], July 5, 2012 In this paper, we investigate asset allocation that transcends typical asset class silos to embrace uniquely identified return-producing units, or risk premia. The portfolio including risk factors alongside traditional asset class beta appears to be superior in terms of risk-adjusted return and maximum drawdown across all portfolio construction approaches we have investigated. From Macro to Micro, Luo, et al [2012], May 2, 2012 We further expand our macro quant research (country, sector/industry, and style rotation). We expand our VRP concept globally and find VRP has strong predictive Page 104 Deutsche Bank Securities Inc. 30 May 2013 Portfolios Under Construction power in industry rotation. We study the “sun-of-the-parts” SOP method in country and sector rotation. We also link the country credit and equity markets. Lastly, we demonstrate how to proactively manage tail risk in country portfolios by employing the mean-variance-skewness (MVS) optimization. New insights in country rotation, Luo, et al [2012], February 9, 2012 We study three sets of country selection signals: risk premium, bottom up, and top down. We find two risk premium factors are particularly interesting: Kelly’s tail risk and VRP. We further build a composite country rotation model. We also investigate a range of country risk models and portfolio construction techniques. More importantly, we demonstrate how equity portfolio managers and global macro/GTAA funds can both take advantage of a country rotation model. In the end, we study risk-based country allocation strategies. Low risk strategies, Avettand—Fenoel, et al, [2011], November 16, 2011 Several methodologies are available to investors wishing to allocate money to low risk strategies. We showcase four risk-based portfolio construction techniques which either distribute risk or minimize some kind of risk metric: the dollar risk is distributed with Equally-Weighted (EW) portfolios, the volatility risk is distributed with Inverse Volatility (InvVol) and minimized with Minimum Volatility (MV), and the diversification risk is minimized with Maximum Diversification (MaxDiversification). Risk parity and risk-based allocation, Alvarez, et al [2011], October 13, 2011 This research investigates the mechanics and efficacy of three popular risk-based asset allocation strategies – risk parity, minimum variance, and maximum diversification. We find risk parity and maximum diversification strategies are more robust to asset concentration and market environment. We also demonstrate how to incorporate alpha in risk parity, in the context of quantitative equity portfolio management. Tail risk in optimal signal weighting, Luo, et al [2011], June 7, 2011 Traditional multi-factor stock selection models are built on mean-variance optimization and likely to expose to tail risk. We propose a few structured models to better estimate factor tail distribution. We then design an efficient optimization algorithm to balance the four conflicting goals of maximizing return/skewness and minimizing risk/kurtosis. Our MVSK models are aware of the underlying macroeconomic environment and effective reduce model downside risk, while maintain the same level of IR/Sharpe ratio. Minimum variance: Exposing the “magin”, Alvarez, et al [2011], February 9, 2011 There are some nice properties for minimum variance portfolios, i.e., higher IR than the market portfolios, low turnover, and low correlation with traditional strategies. However, we find MVP is not necessarily a low-risk strategy. In the end, we propose a slight and simple enhancement to the strategy, which significantly improves MVP IR without increasing its risk. We also demonstrate that we can combine the MVP strategy with other active alpha models. Robust factor models, Luo, et al [2011], January 24, 2011 Traditionally, managers focus on selecting factors, while using the sample factor covariance matrix in constructing multifactor models. We compare the performance of the sample factor covariance matrix with 12 structured models (constant correlation, single index, four Bayesian shrinkage estimators, and six multivariate GARCH models). Our backtesting suggests that robust factor models incorporating structured covariance matrices improve portfolio IR significantly. Deutsche Bank Securities Inc. Page 105 and any other expenses that a client would have paid or actually paid. The backtested performance includes hypothetical results that do not reflect the reinvestment of dividends and other earnings or the deduction of advisory fees. . Taking into account historical events the backtesting of performance also differs from actual account performance because an actual investment strategy may be adjusted any time. perhaps materially.db. simulated results are achieved by means of the retroactive application of a backtested model itself designed with the benefit of hindsight. No representation is made that any trading strategy or account will or is likely to achieve profits or losses similar to those shown. Yin Luo/Sheng Wang/Rochester Cahan/Javed Jussa/Zongye Chen/Miguel-A Alvarez Hypothetical Disclaimer Backtested. please see the most recently published company report or visit our global disclosure look-up page on our website at http://gm. In addition. 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