OSSIAM RESEARCH TEAM. The diversification constraint for Minimum Variance portfolios

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1 OSSIAM RESEARCH TEAM November, 24, 2014

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3 1 The diversification constraint for Minimum Variance portfolios Carmine De Franco, PhD and Bruno Monnier, CFA November, 24, 2014 Abstract Carmine De Franco, PhD Quantitative analyst Bruno Monnier, CFA Quantitative analyst We study the quadratic constraint, also known as the Herfindahl, used to improve the diversification in Minimum Variance portfolios. We argue that this measure, based on the Herfindahl and Hirschmann Index (HHI ), provides the most suitable diversification tool for optimization frameworks that use covariance matrix, as it is free from estimation errors in the covariance estimation. This constraint indirectly limits the maximal weight per stock. We provide an explicit upper bound for maximal weight as a function of Herfindahl target and of the dimension of investment universe. We also prove that small perturbations in Herfindahl target have little effect on the optimal Minimum Variance solution. This continuity property also applies to the exante volatility of optimal portfolios: small perturbations in Herfindahl target yield small variations in ex-ante volatility. Finally, volatility reduction achieved by Minimum Variance portfolio with respect to the benchmark is directly dependent (and can be efficiently controlled) by the Herfindahl constraint. We establish an explicit trade-off between the level of diversification given by Herfindahl target and the expected volatility reduction. Key words: Herfindahl-Hirschmann index; Portfolio allocation; Minimum Variance; Diversification

4 2 1 Introduction In the classical portfolio construction framework, diversification allows Investors to lower idiosyncratic risk that is, in theory, not rewarded. If sophisticated Investors are now aware about the full potential of diversified portfolios, many of less sophisticated Investors still do not consider diversification when building portfolios. In an interesting paper, Goetzmann and Kumar (2008) document how the majority of individual Investors in the US hold underdiversified portfolios, according to three different measures of diversification. On the other side, Investors include, more and more, already pre-built portfolios (as index trackers) in their global allocation, so that measuring portfolio diversification becomes a crucial step to understand the overall Investor s hidden risks and concentration. When Investors buy market instruments as index trackers or ETFs, they get an exposure to a specific portfolio. As we will see later on, thinking of portfolio diversification in terms of current number of portfolio s constituent is a poor measure of diversification. Therefore, it is important to control the diversification when buying these index trackers. And this is particularly necessary now as the so called index investing has seen its popularity explode. Investors who chose to consider the Index investing in their allocation essentially track the performance of their benchmark. These benchmarks can be the traditional market-cap weighted indices or more sophisticated strategy indices. A strategy index essentially is a paper portfolio whose components are selected and weighted in order to optimize a given criterion. A periodic index review is scheduled in order to maintain the weights optimality and update parameter estimates according to new available information. These strategies, that go by the name of Smart Beta, became very popular to Asset Managers and Investors who try to move away from traditional index investing, where the index weights are essentially driven by prices. Practitioners are nowadays very familiar with the mechanism of these Smart Beta strategies, especially for the most popular ones: the minimum Variance strategy, the Maximum Sharpe Ratio strategy (Markowitz, 1952), the Equally Weighted strategy or the Equal Risk Contribution strategy (Roncalli, 2010). As we know that the current number of index constituent is not a fair measure of diversification, we consider one of the most established diversification/concentration measure, both for historical and academic reasons: the Herfindahl-Hirschmann index. Definition 1.1 For a given vector of portfolio weights w, the Herfindahl-Hirschmann index is given by HHI(w) := N i=1 Anti-trust authorities have been using this measure to assess industry concentration; fund of funds managers use it to manage their allocations. Empirically, the inverse of the Herfindahl index gives an approximation of the...effective number of assets in the optimal allocation... (Bouchaud and Potters, 2003). For fully invested and non-leveraged portfolios 1 the Herfindahl index varies between 1/N and 1. Low values of Herfindahl index mean that the portfolio is well diversified, high values are instead associated with concentrated portfolios. It is very straightforward to use this index as a diversification measure for investment portfolios. An Investor may calculate the 1 Portfolios for which the weights sum to 1 and each weights is included in [ 1, 1], so that short-sell may be authorized provided that no asset s exposure exceed the portfolio s value. w 2 i is

5 3 Herfindahl index for several portfolios to assess their relative diversification. More important, the Herfindahl index can be employed as a constraint in a portfolio optimization framework to target a desired level of diversification for the optimal solution (DeMiguel et al., 2009). As the inverse of the Herfindahl index represents the effective number of assets in the portfolio, one can enforce a given Herfindahl target in the right hand side of the constraint so that the optimal portfolio s diversification will be guaranteed at the desired level. Optimized allocation strongly depends on the quality of the input parameters. For example, the Minimum Variance portfolio depends on the estimate of the asset covariance matrix, the Maximum Sharpe ratio strategy will depend on estimates of both the covariance matrix and the expected returns. Not surprisingly, estimate errors have negative impact on the optimal allocation. Furthermore, the optimization process may end up with a poorly diversified allocation. This lack of diversification may be, among other things, a consequence of estimation errors. If we combine these two effects - estimation errors and low diversification - we may get unexpected results from the optimal solution. The most common procedure to mitigate these inconveniences is to impose diversification constraints that should 1) limit errors impact on the portfolio allocation and 2) guarantee a certain level of diversification. The use of constraints is a crucial step in portfolio construction. However there is no consensus on the right constraints to impose. Nowadays, many, if not all, index providers promote and compute strategy optimized indices, that differ a lot in terms of constraints applied in the optimization process. For a complete analysis on the use of constraints in an optimization process and their impact on the optimal solution we refer to Jagannathan and Ma (2003) or Rulik (2013) for the particular case of Minimum Variance strategies. The aim of this paper is to provide insights on the Herfindahl constraint, study its properties and the impact on the optimal allocation when imposing a given Herfindahl target 2. We will concentrate on the Minimum Variance portfolio, even if the general ideas apply to other optimized strategies. We first start by discussing several measures of portfolio diversification (Section 2) and we argue that the Herfindahl constraint appears to be the most suitable diversification tool for quadratic optimization process, such as the Minimum Variance. We then provide some empirical evidence that helps to understand the need of diversification constraints in such framework (Section 3). Section 4 collects our main findings and empirical results of using the Herfindahl constraint to manage portfolio diversification. For all numerical examples in the paper we considered stocks from the S&P 500 Index as the investment universe 3. 2 The use of the Herfindahl constraint is also related to cleaning and shrinkage techniques of covariance matrix. We will not go into details but interested readers my refer to Yanou (2010) or Jandrey-Natal and Rulik (2012). 3 The S&P 500 composition data is a courtesy of S&P. The S&P 500 Index ( Index ) is a product of S&P Dow Jones Indices LLC and/or its affiliates and has been licensed for use by Ossiam. Copyright c 2014 by S&P Dow Jones Indices LLC, a subsidiary of the McGraw-Hill Companies, Inc., and/or its affiliates. All rights reserved. Redistribution, reproduction and/or photocopying in whole or in part are prohibited without written permission of S&P Dow Jones Indices LLC. For more information on any of S&P Dow Jones Indices LLC s indices please visit S&P R is a registered trademark of Standard & Poor s Financial Services LLC and Dow Jones R is a registered trademark of Dow Jones Trademark Holdings LLC. Neither S&P Dow Jones Indices LLC, Dow Jones Trademark Holdings LLC, their affiliates nor their third party licensors make any representation or warranty, express

6 4 We show that the use of the Herfindahl constraint in the Minimum Variance optimization automatically limits each stock s exposure and we provide an explicit formula for the maximal weight, which depends on both the size of the investment universe and the Herfindahl target. We show that the optimal Minimum Variance portfolio depends continuously on the Herfindahl target: small perturbations in the Herfindahl target yield small changes in the optimal Minimum Variance portfolio. The same holds true for the exante volatility. We show that the volatility reduction of the Minimum Variance portfolio with respect to a given benchmark directly depends on the Herfindahl target. There is a trade-off between the volatility reduction and diversification: low Herfindahl target give well diversified portfolios but affects the volatility reduction; inversely, high Herfindahl target gives good volatility reduction but poorly diversified portfolios. We establish an explicit lower bound for this trade-off between volatility reduction and diversification. Investors can solve this equation to match their own preferences for diversification versus volatility reduction and find the right Herfindahl target. The Appendix collects the proofs of all our results. or implied, as to the ability of any index to accurately represent the asset class or market sector that it purports to represent and neither S&P Dow Jones Indices LLC, Dow Jones Trademark Holdings LLC, their affiliates nor their third party licensors shall have any liability for any errors, omissions, or interruptions of any index or the data included therein. 2 Diversification constraints in the portfolio construction Intuitively, portfolio diversification is related to the current number of assets in the portfolio the portfolio weighing scheme correlations among the stocks and concentration of risk exposure Many studies have documented that the current number of portfolio assets is a very poor measure of diversification (Goetzmann and Kumar, 2008). For example, a portfolio containing many assets but substantially invested in only a few of them could not be considered well diversified. A standard approach to improve portfolio diversification consists in limiting each asset exposure (the maximum weight constraint). In line with the idea of limiting each asset exposure, maximal weight constraints can also be imposed at an aggregate level (limiting weights of industrial sectors, geographical regions, countries, currencies, or risk groups, such as ratings or durations for bonds portfolios). DeMiguel et al. (2009) proposed a more sophisticated constraint that overcomes the weakness of number-based ones: for a given portfolio weights w, they define the generalized portfolio norm as w α α, α > 1 Remark that for α = 2 we obtain the Herfindahl Index. The inverse of this generalized portfolio norm is a proxy for the effective number of portfolio s constituents. If we consider the above mentioned theoretical portfolio containing many assets, but substantially invested in only few of them, then the

7 5 inverse of the Herfindahl index will approximately give the number of these few assets the portfolio is invested into. As it possesses many interesting geometric interpretations, the Herfindahl index is, among this class of portfolio s norm, well suited for quadratic optimization process. By definition, the diversification measures based on portfolio norms, ignore the correlations among portfolio s assets and concentration of risk exposures. The main drawback of the norm-based diversification measures can be summarized by the following example: even the most diversified portfolio 4, according to these measures, can be highly concentrated on one risk factor. Think, for example, to a 50/50 portfolio made of one stock and one uncorrelated bond. Assume that the annual volatility of this stock is 30% and 5% for the bond. The portfolio s volatility is 17.5%, but the biggest contribution comes from the stock (85.7% of the total portfolio s volatility), while the bond represents only 14.3% of the total volatility. A recent stream of literature has been devoted to these aspects. Deguest et al. (2013) study portfolio diversification across risk factors using the Effective Number of Bets defined in Meucci (2009). Their main finding is that it is possible to build diversified portfolios across different risk factors. Their procedure generalizes the well established Risk Parity approach (Maillard and Teïletche, 2010), that aims to build portfolios that equally spread the overall risk across all assets. There are three main drawbacks to this approach, that are partially highlighted by the authors. The first one relies on the fact that these methods work when one is able to extract uncorrelated factors from the asset price returns: since this is done with the 4 Which turns out to be the equally weighted one, see Remark 3.1 Principal Component Analysis technique, all results suffer from estimation errors. Second, the uncorrelated factors one obtains generally are not fully explainable with established risk factors. Meucci et al. (2014) provide interesting answers to the above drawbacks by avoiding the PCA to extract uncorrelated factors. Their technique, based on the Minimum Torsion approach, is more stable from estimate errors and provides much more explainable risk factors. The third drawback of diversifying among risk factors is that, implicitly, one is assuming that the investment universe is exposed to several risk factors which unable to diversify across them. This is true for multiasset portfolios (ex. portfolios of developed and emerging markets equities, corporate and governments bonds, commodities, currencies, real estate, private equity, etc..) where one can reasonably assume facing different risk factors. But when it comes to building equity only portfolios (or any other main asset class), one generally faces one predominant risk factor - in this case the equity market factor - so there is no point in diversifying across many risk factors. Factor based diversification makes use of the variance-correlation structure and factor exposures of portfolio s assets to spread the overall portfolio s risk in a balanced way. The idea behind is that factor based diversification ensures that the portfolio variability comes from different sources. In the same stock/bond example above, if we consider the stock and the bond as two proxies for uncorrelated factors, then the factor based diversified portfolio will be 14.3% stock and 85.7% bond. By construction, the risk of this portfolio equally comes from the two sources of risk (stock and bond), but it will suffer big losses if the bond defaults. On the other side, the weight based diversification, which makes use only of portfolio s weights, is more adapted to protect from severe losses.

8 6 Diversified portfolios should protect the Investor from severe losses especially in market downturns. It has been documented empirically that classical correlation estimates fail to deliver fair values of the real correlations especially in bear markets (Longin and Solnik, 2001; Butler and Joaquin, 2002). Ang and Chen (2002) find that correlation estimations are significantly away from the one implied by Gaussian distribution in the left-tail of equity return distribution, meaning that when market moves down, assets becomes more correlated than predicted. This has a particular unpleasant consequence: building diversified portfolios based on correlations may fail to protect Investor portfolio from losses especially in bear markets, when Investors are particularly risk averse. On the other hand, many portfolio strategy objectives are based on asset correlation estimates. In practice, one is ready to tolerate potential errors in the correlation estimates affecting its strategy objective, provided that the portfolio diversification constraint does not suffer from these errors and protects the portfolio in case of poor parameter estimations. In this sense, imposing weight-based diversification constraints in portfolio optimization limits the model estimation errors, making the portfolio allocation more stable and robust. Next sections are devoted to illustrating the effect of the Herfindahl diversification constraint in the case of Minimum Variance optimization problem. 3 Minimum Variance constraints Let r R N be the return vector of N stocks and denote C ij = cov(r i, r j ) their covariance matrix. The return of a portfolio with weights w R N is given by R(w) = i w ir i. A portfolio w is said to be admissible if it is long-only and fully invested, i.e. it belongs to the following set: S := { w [0, 1] N and } N w i = 1 i=1 (3.1) Further constraints can also be imposed to obtain a smaller set of admissible portfolios A S: Linear constraints: Limiting portfolio exposure at some aggregate level (industrial sector, country, etc...) A := {w S Ax b, Dx = e} where A is a p N matrix, p > 0, b R p, D is a k N matrix, k > 0, and e R k. Minimum/Maximum weight constraints: A := {w S w min w i w max } where 0 w min < w max 1 Portfolio turnover constraints: { A := w S } w i wi R T 0 i for some reference portfolio w R S and turnover level T 0 > 0 Of course, any mix of the above constraints is possible provided that the set of constraints A remains convex and closed 5. The Minimum Variance strategy is given by the allocation w unc A that minimizes the 5 A closed and convex set is an usual condition that guarantees the existence and uniqueness of the solution of any regular and constrained optimization process. The convexity property assumes that for any couple of points in the set, the entire segment linking the two points also belongs to the set. Geometrically, a convex set is not starred. A set is closed, in this set up, if its own boundary also belongs to the set.

9 7 variance of the portfolio return R(w) among all admissible allocations w A. This portfolio is the unique solution of: w unc = argmin w Cw (3.2) w A provided that the covariance matrix C is positive-definite. With an abuse of language, we call w unc -unconstrained - the solution of Problem (3.2) even if we consider several type of constraints in A. We used the term unconstrained just to distinguish this optimal portfolio from the one where a diversification constraint is also considered. To illustrate the need for diversification constraints, let us focus on the simplified version of the Minimum Variance framework (3.2), where A = S. The allocation is concentrated in only few stocks. Let us consider the S&P 500 Index as of March, as our investment universe. From the initial 500 in the benchmark universe 6 the optimal portfolio w unc is essentially 7 invested in 42 of them; the weight of the two biggest stock exposures reaches nearly 18% of the entire portfolio. In order to improve diversification of the optimal allocation, we need to restrict the set of admissible portfolios. A first attempt can be done by limiting the individual weight that the optimal strategy w unc can allocate to each asset. Consider then the maximum weight constraint: { } N A := w [0, w max ] N and w i = 1 i=1 6 We actually exclude 6 stocks for which the available price history was not long enough to obtain fair covariance estimates. 7 All stocks from the 43 th end up with a weight lower than 10 6, that we simply consider a left-over of the numerical optimization. and solve (3.2). Figure 1 shows the portfolio weights for w max = 8% and w max = 6% together with the optimal solution with w max = 100% (the initial case A = S). Even in these cases we still find that the optimal allocations are essentially invested in 42 stocks: limiting the maximum weight in the portfolio does not bring more stocks in the optimal allocation. Instead the optimal portfolio distributes more weight to the stocks that have been already selected. To evaluate this concentration effect, let us consider the Herfindahl Index as in Definition 1.1: HHI(w) = i We argued that, empirically, 1/HHI(w) gives an estimate of the effective number of assets in the portfolio w. Table 1 lists the inverse of Herfindahl index values for the optimal portfolios shown in Figure 1. One can see that w max Number of Constituents 1/HHI(w) None % % w 2 i Table 1: The Herfindahl index. measuring concentration in number of stocks overstates the actual diversification: there are 42 stocks for the portfolio with w max = 100% (a stock belongs to the portfolio if its weight is bigger than 10 6 ), but if one sums up the weights of the first 20 stocks he will get 85.48%. Similarly, if we only consider the first 21 (resp. 24) stocks in the optimal portfolio with w max = 8% (resp w max = 6%) we obtain 86.69% (resp 88.49%) of the entire portfolio. This example shows that the inverse of the Herfindahl index is a better approximation of the portfolio diversification.

10 8 Figure 1: The Lorentz curve of the portfolio weights profiles for the optimal solution of (3.2) with admissible portfolios with and without max weight. Investment universe is the S&P 500 Index as of March, 24, Covariance matrix computed on 2 year price history window. The Lorentz curve at given n returns the cumulative weights of the n biggest weights. Following this simple example, we may improve the portfolio diversification if we force any admissible portfolio s Herfindahl index equal to a given target: min wcw w A s.t. HHI(w) = ɛ 2 (3.3) for some given ɛ (0, 1). Figure 2 shows the profile of the Minimum Variance weights when adding the Herfindahl constraint: with ɛ 2 = 1/50 the allocation is now much more diversified (84 stocks) and no stock exceeds 4%. As before, if we sum up the weights of the first 20 stocks, we obtain 52.68% of the entire portfolio. Thus the use of the Herfindahl constraint improves the portfolio diversification. The diversification target can be managed with the single parameter ɛ: the smaller ɛ, the higher the diversification. Remark 3.1 If we minimize the Herfindahl index over all admissible portfolios in S defined in (3.1): min w S HHI(w) we obtain the equal weight portfolio EW = (1/N, 1/N,..., 1/N), that is, by definition, the most diversified portfolio. In this sense, the lower the Herfindahl index, the higher the portfolio diversification. 4 Herfindahl and Minimum Variance: main properties and applications In Section 2 we referred to the Herfindahl index as a particular case of the weight-based diversification measures used by DeMiguel et al. (2009) in optimization processes and, more particular, we noticed that the Herfindahl Index is the unique, among these measures, to have a quadratic functional form that makes it particularly suited for the quadratic optimization processes as the Minimum Variance in (3.3). More precisely, with the use of convex

11 9 Figure 2: The Lorenz curve of the portfolio weights profile for the optimal portfolio in (3.3). Investment universe is the S&P 500 Index as of March, 24, Covariance matrix computed on 2 year historical price window. ɛ 2 = 1/50 analysis techniques and geometric interpretations, we can derive many interesting properties for the optimal Minimum Variance portfolio under the Herfindahl constraint. 4.1 Admissible Herfindahl targets Minimum Variance portfolio with Herfindahl constraint in (3.3) is well defined if 1 N ɛ2 (ɛ unc ) 2 where N is the number of assets in the investment universe and unique solution of Minimum Variance problem (3.2) without Herfindahl constraint. Note that when ɛ 2 = 1 N, the set of admissible portfolios reduces to the equal weight portfolio 8. When ɛ 2 = (ɛ unc ) 2, the Herfindahl constraint becomes meaningless. This results is proved in Appendix A. In the rest of the paper we will always consider ɛ ɛ unc. As the Example in Section 3 shows (Table 1), the unconstrained ɛ unc is usually high (1/20.6 = 4.85%), meaning that the typical Minimum Variance portfolio is highly concentrated in few assets. (ɛ unc ) 2 := HHI(w unc ) = i (w unc ) 2 i (4.1) is the Herfindahl index of the unconstrained optimal Minimum Variance problem w unc, the 8 We assume that the equal weight portfolio already belongs to the set A, i.e. it is an admissible portfolio, which is the case in the majority of practical situations.

12 Implicit maximal weight The use of the Herfindahl constraint in the Minimum Variance portfolio (3.3) forces an implicit maximal weight for all admissible portfolios. More precisely, for any admissible portfolio, the maximal possible weight is w max = 1 N 1 N + ɛ N 2 N 1 where N is the number of assets in the investment universe. This result is proven in Appendix B. From a geometrical point of view, the addition of the Herfindahl constraint in an optimization process, as in (3.3), forces the optimal portfolio to lie in the portion of the sphere centered at origin with radius ɛ and contained in A. In other words, one is automatically limiting the maximal exposure to individual assets. Let us note that the above maximal weight is just an upper bound for any covariance matrix: in practice, the realized maximum weight for the optimal portfolio strongly depends on the objective function used. For example, in the Minimum Variance portfolio, the maximal weight will depend on the volatility/correlation structure. Table 2 gives the maximal weight upper bound when varying the size of the investment universe (N) and the Herfindahl target ɛ 2. N \ ɛ 2 1/50 1/80 1/ % 5.97% Empty Set % 9.60% 6.97% % 9.89% 7.39% % 10.44% 8.15% % 10.57% 8.32% Table 2: Maximal weight upper bound as a function of N and ɛ Small increments in the Herfindahl constraint target produce small changes in the Minimum Variance portfolio The Herfindahl constraint satisfies the continuity property: two optimal Minimum Variance portfolios corresponding to two different Herfindahl constraint targets are close as soon as the two Herfindahl constraint targets are close. If ɛ, η (0, ɛ unc ) are two Herfindahl constraint targets then ɛ η 2 i w η,i w ɛ,i 2 η 2 ɛ 2 where w ɛ (resp. w η ) is the optimal solution in (3.3) with Herfindahl constraint targets equal to ɛ 2 (resp. η 2 ). It follows from the above result that perturbing the Herfindahl constraint target ɛ 2 does not produce significant changes in the optimal allocation. In other words, the optimal Minimum Variance allocation is stable under small perturbations of ɛ 2. This result is proven in Appendix C. 4.4 Small increments in the Herfindahl constraint target produce small changes in the Minimum Variance portfolio volatility By using our previous results on the continuity of the optimal Minimum Variance portfolio with respect to the Herfindahl constraint target ɛ, we can derive a similar result for the expected ex-ante portfolio volatility. More precisely, assume ɛ, η (0, ɛ unc ) and let w ɛ, w η be the respective optimal solutions of (3.3). If R ɛ, R η are the respective portfolio returns, then σ (R ɛ ) = (w ɛ Cw ɛ ) 1/2 and σ (R η ) = (w η Cw η ) 1/2 As we proved that small perturbations on the Herfindahl target constraint yield small

13 11 changes in the Minimum Variance portfolios, we can expect that ex-ante volatility also does not change dramatically. Indeed, it can be proven that σ (R η ) σ (R ɛ ) 1 2 max i<j σ (r i r j ) η 2 ɛ 2 1/2 where σ (r i r j ) is the volatility of the spread between the return of stock i and return of stock j. This result is proven in Appendix D. This inequality tells us that the impact on portfolio volatility under small perturbations of the Herfindahl constraint target is quantifiable, and the mapping ɛ σ (R ɛ ) is continuous. Figure 3 shows the application of Proposition D.1 for a Minimum Variance portfolio built on the S&P500 stocks 9 as of March, We fix a reference portfolio with the Herfindahl constraint target ɛ 2 = 1/50, and compare it to portfolios obtained by moving the Herfindahl target constraint within the range [1/25,..., 1/494]. We denote by w 1/50 the weights of the reference Minimum Variance portfolio with ɛ 2 = 1/50, and by w η the optimal Minimum Variance allocations when η varies in the above range. The upper bound for differences in annualized volatilities is given by the expression Figure 3 is in line with the conclusion of Proposition D.1, since the blue line uniformly dominates the red one. 4.5 Volatility reduction of the Minimum Variance portfolio Ex-Ante. It is common to measure the volatility reduction that achieves Minimum Variance portfolio with respect to a given benchmark. We show that the Herfindahl constraint target plays an important role in the volatility reduction of the Minimum Variance portfolio. Usually Investors consider as a benchmark the market-cap weighted portfolio (where weights are proportional to market capitalization of stocks). The equal weight portfolio is also a good benchmark, especially used in academia since, empirically, it is close to the cap-weighted portfolio in terms of volatilities, as showed in Table 3. The equal weight portfolio, EW = (1/N, 1/N,..., 1/N) A is in fact a special (extreme) case of constrained Minimum Variance portfolio with Herfindahl constraint target equal to 1/N, where N is the number of stocks in the dataset. The volatility of the equal weight portfolio is x max i<j σ (r i r j ) 1/x 1/50 1/2 where max i<j σ (r i r j ) 6.09% and 1/x is the variable Herfindahl constraint target, while the annualized volatility difference is given by x 250 ( w 1/x Cw 1/x ) 1/2 ( w1/50 Cw 1/50 ) 1/2 9 As previously explained, from the 500 stocks in the S&P 500 Index, we exclude 6 of them since they do not have a sufficient long price history to obtain fair estimates of volatility and correlations. σ (EW ) = (EW C EW ) 1/2 The ex-ante volatility reduction of the Minimum Variance portfolio with Herfindahl constraint target equal to ɛ 2 with respect to the equal weight portfolio is defined by Red(ɛ) := 1 σ (w ɛ) σ (EW ) where w ɛ is the Minimum Variance solution of (3.3) with Herfindahl constraint target ɛ 2 and σ(w ɛ ) = (w ɛ Cw ɛ ) 1/2. If N is the number of

14 12 Figure 3: The upper bound for the difference of portfolio volatilities Universe Strategy Annualized Index Name Type Volatility S&P 500 Index CW 16.23% EW 17.96% Stoxx Europe 600 Index CW 19.73% EW 19.00% EuroStoxx 50 Index CW 24.10% EW 24.38% FTSE 100 Index CW 12.27% EW 13.43% MSCI Emerging Markets Index CW 19.67% EW 17.36% Table 3: Annualized volatilities for cap-weighted (CW) and equally weighted (EW) version of the index daily returns ending on May Indices are all Net Return, except the FTSE 100 which is Total Return. Data is taken from Bloomberg. Estimation periods depend on the availability of the equally weighted index data: S&P from August, 2011; Stoxx 600 from January, 1999; EuroStoxx from December 1999; FTSE from December 2011 and MSCI from January assets in the investment universe and (ɛ unc ) 2 is the Herfindahl of the unconstrained Minimum Variance strategy defined in (4.1) then Red(ɛ) ɛ 2 1/N (ɛ unc ) 2 1/N Red(ɛunc ) This result is proven in Appendix E. As long as ɛ gets close to 1/N, the lower bound goes to zero: we actually know (Theorem C.1) that the optimal Minimum Variance solution converges to the equal weight Portfolio. When ɛ ɛ unc, the volatility reduction converges to its maximum, achieved by the unconstrained Minimum Variance portfolio, as shown in Figure 4. For intermediate values of Herfindahl constraint target we can estimate the size of

15 13 Figure 4: When ɛ decreases, the optimal Minimum Variance solution is more diversified, and at the limit value ɛ 2 = 1/N we find the equal weight portfolio. When ɛ ɛ unc, the optimal solution gets closer to the unconstrained w unc, and becomes less diversified. the volatility reduction. In the same framework as Figure 3, we consider as investment universe the composition 10 of the S&P 500 Index on March, 24, For each 11 ɛ 2 = 1/25,..., 1/494, we compute the optimal solution of (3.3), its (annualized) exante volatility and the (annualized) volatility of the equal weight portfolio. The equal weight portfolio has an ex-ante annualized volatility of 12.12%. Figure 5-(a) shows how the volatility of the Minimum Variance optimal solution converges to the equal weight one when ɛ 2 1/494. In Figure 5-(b) we compute the lower bound estimations where (ɛ unc ) 2 = and Red(ɛ unc ) = 39.75%. For example, by taking an Herfindahl constraint target ɛ 2 = 1/50, we obtain a theoretical lower bound for the volatility reduction 24.5%, while the realized ex-ante volatility reduction is 36.26%. Of course the real volatility reduction takes into account the covariance structure given in C, while our theoretical lower bound loses precision since it is obtained through several simplifications. Nevertheless the magnitude of the lower bound for the volatility reduction is quite close to the realized reduction. This may allow an Investor to calibrate the diversification target (so, at the end, the value of the Herfindahl target ɛ 2 ) to obtain the targeted volatility reduction. Higher values of ɛ 2 will yield better volatility reduction but weaker diversification; conversely, lower values of ɛ 2 yield good portfolio diversification but smaller volatility reduction. The Investor faces a trade-off between volatility reduction versus diversification and has to strike a balance with the help of the Herfindahl constraint target that suits Investor constraints. 10 In this test we simply consider A = S 11 As before, we exclude 6 stocks with short price history Ex-Post. It is interesting to measure the volatility reduction of the Minimum Variance

16 14 (a) (b) Figure 5: Volatilities of optimal solutions of (3.3) with admissible portfolios according to A. Investment universe is the S&P 500 Index at March, 24, Covariance matrix computed on 2 year price history window. portfolio over time. We still consider the S&P 500 Index as our investment universe and each third Friday of the month, we compute the optimal solution of (3.3) with ɛ 2 = 1/50. Our test starts in January 2001 and ends on March In Figure 6-(a) we can appreciate how volatilities of the equal weight portfolio vary over time, the Minimum Variance and the unconstrained 12 Minimum Variance portfolio. Note that the volatilities of both Minimum Variance portfolios are significantly lower than the volatility of the equal weight portfolio, and also that the differences between the two is small, except during limited periods. In Figure 6-(b) we can appreciate the realized volatility reduction and its theoretical lower bound. 5 Conclusions This paper is devoted to a detailed analysis of the impact of the Herfindahl constraint in 12 Recall that unconstrained is meant with respect to the Herfindahl constraint. This unconstrained solution still belongs to S, so its weights sum to 1 and are all included in [0, 1]. the Minimum Variance framework. Introduced by economists Herfindahl and Hirschman, it is commonly used by anti-trust authorities and government agencies to measure industry concentration and manage potential issues deriving from mergers, acquisitions and other form of corporate activities in a given industrial sector. In the context of portfolio management, it can be efficiently used as a diversification mechanism incorporated into the set of portfolio constraints. Herfindahl constraint gives superior control over optimal portfolio diversification compared to the constraint expressed in terms of number of portfolio constituents. This measure is also better in the context of Minimum Variance, as it does not incorporate quantities coming with estimation errors, such as stock covariance matrix. Our first finding is that the use of the Herfindahl constraint in the Minimum Variance process carries out an implicit limit on maximal exposure for a single portfolio constituent. More precisely, we show that the use of the Herfindahl constraint ensures an explicit maximal weight bound that depends on the number of stocks in the investment universe and the value of the Herfindahl constraint target.

17 15 (a) (b) Figure 6: Volatilities of optimal solutions of Problem (3.3) with admissible portfolios given in A. Investment universe is the S&P 500 Index. The Minimum Variance portfolios rebalance each third Friday of the month. Covariance matrix computed on 2 year price history window. We provide an explicit formula for this maximal bound that is model-free since it does not depend on the covariance matrix. In this sense, we can think of it as a global upper bound. In practice, the maximal weights for the optimal Minimum Variance portfolio are lower, as they are further limited by the structure of the covariance matrix. We then study the impact of the Herfindahl constraint target on the Minimum Variance portfolio composition and its ex-ante volatility. We show that small changes in the Herfindahl constraint target do not have significant impact on both the Minimum Variance portfolio and its ex-ante volatility. This continuity property can help Investor to estimate the difference between two Minimum Variance portfolios by comparing their Herfindahl constraint targets and their volatilities. The explicit expressions for the bounds that we find are quite precise for small variations of the Herfindahl constraint targets but can greatly over- or underestimate the differences when the Herfindahl targets diverge. Finally we establish a relation between the volatility reduction of the Minimum Variance portfolio versus the equal weight one and the Herfindahl constraint parameters. Indeed we show that the volatility reduction can be efficiently controlled with the Herfindahl constraint target, providing a trade-off between portfolio diversification and volatility reduction. Investors can set the Herfindahl constraint target at level that best matches the desired portfolio diversification and volatility reduction (the higher the Herfindahl constraint target and the higher the volatility reduction and the lower the diversification; the lower the Herfindahl constraint target and the higher the diversification with weaker volatility reduction). References Ang, A. and J. Chen (2002). Asymmetric correlations of equity portfolios. Journal of Financial Economics 63 (3), Bouchaud, J.-P. and M. Potters (2003). Theory of financial risk and derivative pricing: from statistical physics to risk management. Cambridge university press. Butler, K. C. and D. C. Joaquin (2002). Are the gains from international portfolio diversification exaggerated? the influence of

18 16 downside risk in bear markets. Journal of International Money and Finance 21 (7), Deguest, R., L. Martellini, and A. Meucci (2013). Risk parity and beyond: From asset allocation to risk allocation decisions. Available at SSRN DeMiguel, V., L. Garlappi, F. J. Nogales, and R. Uppal (2009). A generalized approach to portfolio optimization: Improving performance by constraining portfolio norms. Management Science 55 (5), Goetzmann, W. N. and A. Kumar (2008). Equity portfolio diversification. Review of Finance 12 (3), Roncalli, T. (2010). Understanding the impact of weights constraints in portfolio theory. Munich Personal RePEc Archive. Rulik, K. (2013). Minimum variance portfolio: the art of constraints. Available Research publications/ minimum-varianceportfolio-the-art-of-constraints-v2.pdf. Yanou, G. (2010). Mean-variance framework and diversification objective: Theoretical and empirical implications. Available at SSRN Jagannathan, R. and T. Ma (2003). Risk reduction in large portfolios: Why imposing the wrong constraints helps. The Journal of Finance 58 (4), Jandrey-Natal, P. and K. Rulik (2012). Impact of covariance cleaning techniques on minimum variance portfolios. Longin, F. and B. Solnik (2001). Extreme correlation of international equity markets. The Journal of Finance 56 (2), Maillard, S., T. R. and J. Teïletche (2010). The properties of equally weighted risk contribution portfolios. Journal of Portfolio Management 36 (4), Markowitz, H. (1952). Portfolio selection. The journal of finance 7 (1), Meucci, A. (2009). Managing diversification. Available at Meucci, A., A. Santangelo, and R. Deguest (2014). Measuring portfolio diversification based on optimized uncorrelated factors. Available at SSRN

19 17 A Admissible Herfindahl targets We will proceed in several steps. Step 1. The first thing we will prove is that we can relax the Herfindahl constraint from an equality to an inequality. For sake of simplicity let us redefine problem (3.3) as follows: P (ɛ) := min wcw w A over HHI(w) = ɛ 2 (A.1) and denote by w H the unique solution of the problem above. Theorem A.1 Let min wcw P L w A (ɛ) := over HHI(w) ɛ 2 and w ɛ be the optimal solution of P L (ɛ). (A.2) If ɛ < ɛ unc then P L (ɛ) = P (ɛ) and w ɛ = w H If ɛ ɛ unc then P L (ɛ) = P (ɛ unc ) and w ɛ = w unc where w H is the solution of problem (A.1). Before proving this results, let us deduce that If ɛ 2 < 1/N then the solution w ɛ coincides with the optimal solution of Problem (A.2). But from Remark 3.1, we know that 1/N HHI(w). Hence, the set of admissible portfolios with HHI(w) ɛ 2 is empty. We are forced to assume then ɛ 2 1/N. If ɛ 2 > (ɛ unc ) 2, then the Theorem simply tells us that the Herfindahl constraint is meaningless, as we find the unconstrained w unc to be the optimal solution. The intuition of the Theorem above is clear: one can solve Problem (A.2), according to the Herfindahl constraint target ɛ 2, instead of the same problem with an equality Herfindahl constraint (Problem (A.1)). Using inequality constraints may be preferred from a numerical optimization perspective. Last, as we will see in Appendix C, the problem with inequality constraint is more adapted to obtain some a priori results on the optimal solution. Theorem A.1 also states that the optimal Minimum Variance solution always tend to be highly concentrated: when we allow the problem to select diversified portfolios (Problem (A.2)) the optimum will always lie on the boundary w 2 2 = ɛ2 (Problem (A.1)). Figure 7 gives a nice geometric interpretation of this result: assume that the circle represents the optimal Minimum Variance portfolio without Herfindahl constraint. It lies on the intersection between the budget plane and the circle HHI (w) = ɛ 2 in the N-dimensional space. The set of admissible portfolios corresponds to the gray zone. Theorem A.1 says that any portfolio lying inside the gray zone (the square) is not optimal, since the optimal solution for this problem always belongs to the circle HHI (w) = ɛ 2 (the triangle). Proof. Assume that the unique optimal allocation in Problem (A.2) does not lie on the boundary: HHI(w ɛ ) < ɛ 2 and let π α := (1 α)w ɛ + αw unc It is clear that π A since is it convex. Define Γ(α) := HHI(π α ) (1 α)hhi(w ɛ )+α (ɛ unc ) 2 Since Γ(0) < ɛ 2 and Γ(1) = (ɛ unc ) 2 > ɛ 2, there exists some α max > 0 such that for any 0 < α α max one has π α A and HHI(π α ) ɛ 2

20 18 Figure 7: Minimum Variance solution with Herfindahl constraint i.e., π α is an admissible portfolio for Problem P L in (A.2). The map w wcw being convex, we have π α Cπ α (1 α)w ɛ Cw ɛ + αw unc Cw unc w ɛ Cw ɛ (A.3) simply because w unc Cw unc w ɛ Cw ɛ. Inequality (A.3) contradicts the fact that w ɛ is the unique minimum of problem (A.2). We then conclude HHI(w ɛ ) = ɛ 2 and that w ɛ = w H by uniqueness. The case ɛ > ɛ unc is straightforward. B Implicit Maximal weight Proposition B.1 Let N > 2 be the number of available assets and assume that the set of admissible portfolios is given by T := { w [0, 1] N, } N w i = 1, HHI(w) = ɛ 2 i=1 This set is not empty if and only if ɛ 2 1 N In this case, the maximum weight achievable is given by w max = 1 N 1 N + ɛ N 2 N 1 Proof. Let S denote the set of long only fullinvested portfolios as in (3.1). For each allocation, we can compute the Herfindahl function HHI(w) = wi 2. If for any w S ones has HHI(w) > ɛ 2 then the set T is clearly empty. The unique solution of the problem below min HHI(w) over wi = 1 and w [0, 1] N is given by w = (1/N, 1/N,..., 1/N) and then HHI(w) = 1/N. We conclude that the set T is empty if and only if 1/N > ɛ 2. For the second statement, let start by remarking that for any allocation w A, we must have w max := max i w i < ɛ unless ɛ = 1. If this is not the case, then the allocation must be of the form w = ɛe i, where e i is a vector of the canonical base, to match the Herfindahl constraint HHI(w) = ɛ 2. But in this case wi = 1 will imply ɛ = 1. We can exclude the case ɛ = 1 since this will reduce the set

21 19 of admissible portfolios to those that are 100% invested into only one asset. With this in mind, we may ask how close to ɛ the single stock weight can be. In other words, we ask whenever a portfolio of the form w = (ɛ η, u), u [0, 1] N 1, 0 < η < ɛ (B.1) is admissible, and then take the minimum η that makes w admissible. In this case the maximum weight par stock will be w max = ɛ η. To make w admissible, we must impose N 1 i=1 N 1 i=1 u i = 1 + η ɛ and u 2 i = ɛ 2 (ɛ η) 2 = 2ɛη η 2 By making the change of variable v = u/(1 + η ɛ), we must find a portfolio v [0, 1] N 1 such that N 1 i=1 v i = 1 and N 1 i=1 v 2 i = 2ɛη η2 (1 + η ɛ) 2 Finding such a v is equivalent to ask whenever the set { C := v [0, 1] N 1, N 1 i=1 v i = 1, } 2ɛη η2 HHI(v) = (1 + η ɛ) 2 is not empty, and we already know that this is the case if 2ɛη η 2 (1 + η ɛ) 2 1 N 1 The minimal η that makes the above inequality hold true is given by the η = ɛ 1 N 1 N ɛ N 2 N 1 It follows from (B.1) that the maximum weight is given by ɛ η, i.e. w max = 1 N 1 N + ɛ N 2 N 1 which concludes our proof. C Small increments in the Herfindahl constraint target produce small changes in the Minimum Variance portfolio Theorem C.1 Let ɛ, η (0, ɛ unc ) and consider the Minimum Variance optimization problem P L in (A.2) relatively to ɛ and η. Then ɛ η 2 w η w ɛ 2 2 η2 ɛ 2 Before we provide the proof of this result, remark that this holds true for the optimal Minimum Variance solutions of Problem (A.2). Bur from ɛ, η (0, ɛ unc ) and Theorem A.1 we deduce the same inequality for Problem (3.3). Proof. From ɛ < η and { HHI(w) ɛ 2} { HHI(w) η 2 } we first deduce w η Cw η w ɛ Cw ɛ. Let now α (0, 1) and define Clearly ξ (α) A and ξ (α) := αw η + (1 α)w ɛ ξ (α) Cξ (α) αw η Cw η + (1 α)w ɛ Cw ɛ w ɛ Cw ɛ which automatically yields HHI(ξ (α) ) > ɛ 2 : indeed, if HHI(ξ (α) ) ɛ 2 then we would have built an admissible portfolio whose total variance is lower or equal than the optimal total

22 20 variance of w ɛ, which is impossible. It follows: ɛ 2 < ( HHI ξ (α)) = ξ (α) 2 2 = α 2 η 2 + (1 α) 2 ɛ 2 + 2α(1 α)w ɛw η 0 < (ɛ 2 + η 2 )α 2 2αɛ 2 + 2α(1 α)w ɛw η Dividing by α the last inequality and letting α 0 we obtain this is not true, then ξ (α) in Figure 8b makes a better allocation compared to the optimal w ɛ : since we know that w η Cw η w ɛ Cw ɛ, we build ξ (α) simply by de-leveraging w ɛ and use a portion of the allocation of w η. For the second inequality, we can simply note that (ɛ η) 2 = w ɛ 2 w η 2 wɛ w η 2 ɛ 2 w ɛw η (C.1) This allows us to conclude the first part of the proof since which concludes our proof. w ɛ w η 2 2 = η2 + ɛ 2 2w ɛw η η 2 ɛ 2 Before starting the second part of the proof, let us give some insights on this partial result. The vector of weights w ɛ gives, in some sense, a special direction. Let us call it the Minimum Variance direction MVD. From inequality (C.1) we deduce that the angle θ between the vectors w η and w ɛ is ( ) ( ) w η wɛ wη wɛ θ = cos = cos w η 2 w ɛ 2 ηɛ ( ) ɛ cos 0 η as soon as ɛ η: the optimal Minimum Variance solution w η will not be far from the MVD, i.e. the angle between the two vector is small (Figure 8a). This is the key point to control the L 2 -norm of the difference between the two vectors. We proved that the optimal Minimum Variance solution w η cannot be the one pointing at P 2 on Figure 8a. It must lie on the spherical cap delimited, in the Figure, by the arc AB. This spherical cap is given by the intersection between the η-herfindahl sphere and the tangent plane to the ɛ-herfindahl sphere at w ɛ. If D Small increments in the Herfindahl constraint target produce small changes in the Minimum Variance portfolio volatility Proposition D.1 Let ɛ, η (0, ɛ unc ) and consider the Minimum Variance optimization problem (3.3) relatively to Herfindahl constraint targets ɛ and η. Denote with R ɛ, R η the returns of the optimal portfolios relatively to the allocations w ɛ and w η. The ex-ante volatilities of these portfolios are σ (R ɛ ) = (w ɛ Cw ɛ ) 1/2 σ (R η ) = (w η Cw η ) 1/2 Then σ (R η ) σ (R ɛ ) 1 2 max i<j σ (r i r j ) η 2 ɛ 2 1/2 where r i, r j are the return of stocks i and j. Proof. If we define the new norm w C then := wcw σ (R η ) σ (R ɛ ) = w η C w ɛ C w η w ɛ C

23 21 (a) The angle (b) The vector ξ (α) Remark that (w η w ɛ ) i = 0 i and (w η w ɛ ) 2 i η 2 ɛ 2 i from Theorem C.1. If we now denote α 2 := η 2 ɛ 2 and x the solution of the problem maximize xcx over x i = 0, x 2 i α 2 x i [ 1, 1] then w η w ɛ C x C simply because A is a subset of S defined in (3.1). Let us start by remarking that the map x xcx is convex and then, for any x, y and δ (0, 1) (δx + (1 δ)y)c(δx + (1 δ)y) δxcx + (1 δ)ycy max (xcx, ycy) In other words, diversification does not help to reach the maximum. It follows that the maximum is necessarily attained by the less diversified portfolios x. And the less diversified portfolios are those with only two assets. Note that the portfolios with only one asset do not verify the constraint x i = 0. Such portfolio must be of the form x = γe i γe j, where e i, e j are two elements of the canonical base. From the quadratic constraint we obtain γ α 2 1 since γ is also bounded by 1. However, α = η 2 ɛ 2 1 as long as ɛ, η < 1, so we can simply write Finally γ α 2 xcx α2 2 (C ii + C jj 2C ij ) = α2 2 max V AR(r i r j ) i so then x C = xcx α 2 max i<j σ (r i r j ) which concludes the proof.

24 22 E Volatility reduction of the Minimum Variance portfolio Proposition E.1 Let ɛ (0, ɛ unc ) and consider the Minimum Variance optimization problem relatively to Herfindahl constraint target ɛ 2 in (3.3). Then Red(ɛ) Proof. Let α (0, 1) and Since vol ɛ 2 1/N (ɛ unc ) 2 1/N Red(ɛunc ) ξ (α) = αw unc + (1 α)ew A ( ξ (α)) α σ (w unc ) + (1 α) σ (EW ) and from w unc 2 2 = (ɛunc ) 2 EW 2 2 = 1/N and < w unc, EW >= 1/N we obtain α 2 ( (ɛ unc ) 2 1/N ) = ɛ 2 1/N which concludes our proof. then vol ( ξ (α)) σ (EW ) 1 vol ( ξ (α)) σ (EW ) α σ (wunc ) + (1 α) σ (EW ) αred(w unc ) If we select α (0, 1) such that HHI ( ξ (α)) = ɛ 2 then, by optimality of w ɛ we have σ (w ɛ ) vol from which we obtain (ξ (α)) Red(ɛ) 1 vol ( ξ (α)) σ (EW ) αred(wunc ) Computing this α is straightforward: ( HHI ξ (α)) = ɛ 2 α 2 w unc (1 α)2 EW α(1 α) < w unc, EW >= ɛ 2

25 About Ossiam Ossiam is a research-driven French asset management firm (authorized by the Autorité des Marchés Financiers) and specializes in delivering smart beta* solutions. Efficient indexing is at the core of Ossiam s business model. The firm was founded in response to a post-subprime crisis demand from investors for simplicity, liquidity and transparency. Given the environment, there was a growing need among investors for enhanced beta exposure and risk hedging. Ossiam is focused on the development of innovative investment solutions for investors via a new generation of indices. *'Smart beta' refers to systematically managed, non-market-cap-weighted strategies covering any asset class. Ossiam, a subsidiary of Natixis Global Asset Management, is a French asset manager authorized by the Autorité des Marchés Financiers (Agreement No. GP ). Although information contained herein is from sources believed to be reliable, Ossiam makes no representation or warranty regarding the accuracy of any information of which it is not the source. The information presented in this document is based on market data at a given moment and may change from time to time. This material has been prepared solely for informational purposes only and it is not intended to be and should not be considered as an offer, or a solicitation of an offer, or an invitation or a personal recommendation to buy or sell participating shares in any Ossiam Fund, or any security or financial instrument, or to participate in any investment strategy, directly or indirectly. It is intended for use only by those recipients to whom it is made directly available by Ossiam. Ossiam will not treat recipients of this material as its clients by virtue of their receiving this material. This material reflects the views and opinions of the individual authors at this date and in no way the official position or advices of any kind of these authors or of Ossiam and thus does not engage the responsibility of Ossiam nor of any of its officers or employees. All performance information set forth herein is based on historical data and, in some cases, hypothetical data, and may reflect certain assumptions with respect to fees, expenses, taxes, capital charges, allocations and other factors that affect the computation of the returns. Past performance is not necessarily a guide to future performance. Any opinions expressed herein are statements of our judgment on this date and are subject to change without notice. Ossiam assume no fiduciary responsibility or liability for any consequences, financial or otherwise, arising from, an investment in any security or financial instrument described herein or in any other security, or from the implementation of any investment strategy. This information contained herein is not intended for distribution to, or use by, any person or entity in any country or jurisdiction where to do so would be contrary to law or regulation or which would subject Ossiam to any registration requirements in these jurisdictions. 80, avenue de la Grande Armée Paris France info@ossiam.com