The diversification delta: A different perspective. Author. Published. Journal Title. Version DOI. Copyright Statement.

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1 The diversifiation delta: A different perspetive Author Salazar Flores, Yuri, Bianhi, Robert, Drew, Mihael, Truk, Stefan Published 07 Journal Title Journal of Portfolio Management Version Post-print DOI Copyright Statement 07 Institutional Investor. This is the author-manusript version of this paper. Reprodued in aordane with the opyright poliy of the publisher. Please refer to the journal's website for aess to the definitive, published version. Downloaded from Griffith Researh Online

2 The Diversifiation Delta: A Different Perspetive Yuri Salazar Flores, Robert J. Bianhi, Mihael E. Drew, Stefan Trük Yuri Salazar Flores is a Leturer at the Faulty of Sienes, ational Autonomous University of Mexio, Mexio City, Mexio. yurisf@ienias.unam.mx Robert J. Bianhi is an Assoiate Professor of Finane at Griffith Business Shool and Diretor of the Griffith Centre for Personal Finane and Superannuation at Griffith University in Brisbane, QLD, Australia. r.bianhi@griffith.edu.au Mihael E. Drew is a Professor of Finane at Griffith Business Shool at Griffith University in Brisbane, QLD, Australia. mihael.drew@griffith.edu.au Stefan Trük is a Professor of Finane at the Faulty of Business and Eonomis and Co-Diretor of the Centre for Finanial Risk at Maquarie University in Sydney, SW, Australia. stefan.truek@mq.edu.au Abstrat In a 0 artile published in this Journal, Vermorken, Medda, and Shröder introdue a new measure of diversifiation, the Diversifiation Delta (DD), based on the entropy of the portfolio return distribution. Entropy as a measure of unertainty has been used suessfully in several frameworks and takes into aount the entire statistial distribution rather than just the first two moments. In this artile, the authors highlight some drawbaks of the DD measure and go on to propose an alternative measure based on exponential entropy whih overomes the identified shortomings. The authors present the properties of this new measure and propose it as an alternative for portfolio optimization that inorporates higher moments of asset returns suh as skewness and exess kurtosis. Preprint submitted to Journal of Portfolio Management 6 September 07

3 Vermorken et al. (0) introdue a new measure of diversifiation, whih is based on higher moments of the return distribution of a portfolio. The proposed measure, alled Diversifiation Delta (DD), is based on the onept of Shannon entropy, or information entropy, that an measure the unertainty related to the entire statistial distribution and not just the first two moments of a distribution. Investors typially diversify their portfolios with the aim of reduing their exposure to idiosynrati risks of individual assets, while the orrelation matrix of asset returns is regarded as the ommon metri for measuring portfolio diversifiation. However, the orrelation matrix quantifies the pairwise relation between two or more stohasti proesses. As pointed out by Statman and Sheid (008), the orrelation matrix does not aount for the impat of individual assets on the variane of the portfolio. Furthermore, modern portfolio theory quantifies the level of diversifiation by using only the first two moments of the return distribution. As stated in Vermorken et al. (0), different diversifiation measures have been proposed in the finane literature. Various researhers onsider the use of the orrelation matrix, as well as alternative measures suh as lustering based methods, the portfolio diversifiation index, and the return gaps. For a detailed analysis of these methods, see, for example, Dopfel (003), Brown and Goetzmann (003), Rudin and Morgan (006), and Statman and Sheid (008). These models ome as a response to the lassi portfolio optimization model introdued by Markowitz (95). Other models inlude indexes based on mean-variane analysis (MVA) suh as the reward-to-variability ratio by Sharpe (966), hereafter referred to as SR, and indexes based on risk measures, suh as the diversifiation index defined by Tashe (006), whih is based on the Value-at-Risk (VaR). Although the use of these indexes has been suessful to a large extent, the DD has the advantage of not being restrited to the first two moments of the return distribution. It also aptures diversifiation by omparing return distributions of the assets before and after the portfolio is onstruted. At the same time, it is relatively easy to alulate and interpret. As argued by Vermorken et al. (0), entropy aptures the redution in unertainty as a portfolio of various assets beomes more diversified. More diversifiation redues unertainty and lowers entropy. The proposed DD measure has the advantage of being a straightforward appliation and interpretation for portfolios onsisting of different asset lasses. Based on an empirial example and using returns from different infrastruture indexes, the authors argue that the appliation of DD [..] gives the portfolio manager a muh

4 learer piture of the reality in the market than the orrelation oeffiient. Therefore, DD presents an interesting ontribution to the study of portfolio theory and diversifiation. However, the onstrution of the DD measure based on the exponential of the weighted mean of the entropies of the individual assets leads to some issues that will be examined in this study. For example, it is easy to illustrate that the proposed measure does not provide a range of 0 DD as suggested by Vermorken et al. (0). This is true, in partiular, when assets with a different level of risk or variane are ombined, whih makes an aurate interpretation of the measure quite diffiult. Further, in some instanes, the measure provides results that ontradit what one would intuitively expet from a diversifiation measure. We propose a revised measure that uses the weighted mean of the exponential entropies of the individual assets in a portfolio as a suitable alternative. The remainder of the paper is strutured as follows. We first analyze the DD presented in Vermorken et al. (0) and illustrate some shortomings of this measure. We then present a revised measure that is also based on the onept of Shannon entropy of a random variable but overomes the problems of the original DD. We then illustrate how higher moments suh as skewness and kurtosis impat portfolios that are onstruted when using the new DD measure. The onlusion summarizes our main findings and disusses the appliation of the new DD measure in portfolio management. The Diversifiation Delta The DD is based on the entropy as a measure of unertainty. Originally, entropy was defined in a statistial mehanis framework and introdued in information theory by Shannon (948). Sine then, entropy has been suessfully used in measuring unertainty in several areas suh as applied mathematis, eletrial engineering, omputer siene, physis, neurosiene, among others. In eonometris, disrete entropy is often maximized to fit probability funtions, for example, in the work of Maasoumi (993) and Ullah (996). In the ontext of finanial eonomis, entropy and onditional entropy have been used to define onvex risk measures (Laeven and Stadje, 00; Föllmer and Knispel, 0), while entropy measures have been used to estimate distributions assoiated with finanial data (Kitamura and Stutzer, 997; Robertson et al., 005). Although entropy is not a widespread measure in the portfolio optimization literature or among pratitioners, studies and appliations exist. In the seminal work of Philippatos and 3

5 Wilson (97), entropy is used for the first time in a portfolio optimization framework. Following this, Hua and ingsi (003) and Bera and Park (008) report that entropy an be an effetive alternative to the MVA. More reently, entropy has reeived new attention beause it an apture heavy tails that are often present in finanial return data (Urbanowiz et al., 0; Dey and Juneja, 0). Regarding the use of the onept of entropy to build diversifiation measures, Bera and Park (008) and Meui (009) develop diversifiation measures based on maximizing entropy. The framework by whih entropy is used in the definition of the DD differs from the way it is used in other portfolio analysis methods. In this ase, entropy plays the entral role and is evaluated in both the assets and the portfolio with no further analysis needed. Also, as suggested by Campbell (966), exponential entropy is used to avoid singularities of entropy, while still letting the unertainty speak for itself. Exponential entropy is also mentioned as a risk measure in Rahev et al. (0). For a given portfolio P onsisting of assets (,..., ) and weights ( w,..., w ), with i w i, Vermorken et al. (0) define the DD as exp wh i ( i) exp H w i i i i DD( P), exp wh i ( i) i () H f x f x dx is the differential where f is the density of and ( ) ( ) ( ) x log ( ) entropy that is used as a measure of unertainty. The estimator of entropy they onsider is the one developed in Stowell and Plumbley (009). The higher the level of entropy, the higher the unertainty and vie-versa. The DD is designed to measure the diversifiation effet of a portfolio by onsidering entropy of the individual assets and by omparing it with the entropy of the portfolio. The DD is defined as a ratio that ompares the weighted individual assets and the portfolio. The value of using suh a ratio is that it quantifies the effet of portfolio diversifiation. Furthermore, using entropy to measure unertainty in this ontext is itself an important ontribution. However, the way entropy is used by Vermorken et al. (0) to measure unertainty presents some issues. First, the measure of unertainty in Equation () must satisfy a number of properties for the ratio to adequately measure portfolio 4

6 diversifiation. Artzner et al. (999), Rokafellar et al. (006), and Meil et al. (005), among others, have analyzed desirable properties for measures of risk and unertainty in a portfolio optimization framework. Although entropy has proven to be effetive in measuring unertainty, it is understood that it is not homogeneous, is not left-bounded, and is not subadditive, as noted by Cover and Thomas (99). In the definition of the DD, the exponential funtion is used to aount for these issues; however, it is not ahieved in an adequate framework, as we now disuss in further detail. Realling that the DD ompares the individual assets with the portfolio using a measure of unertainty, the first harateristi we notie is that, for the DD to be positive, the measure of unertainty must satisfy some kind of subadditivity. This will ensure that the numerator in () is positive. As noted by Artzner et al. (999) and Meil et al. (005), subadditivity is a ruial feature for measures of unertainty or risk beause it reflets the fat that risk an be redued by diversifiation. The authors also omment on the shortomings of the use of measures that do not aount for this, espeially VaR (Meil et al., 005). Subadditivity is not satisfied in the definition of the DD, whih an, in fat, be negative and is also not left-bounded. Vermorken et al. (0) state that the DD should exhibit a range of outomes between 0 and. The authors suggest that this is atually one of the great benefits of the measure allowing for straightforward interpretation. While a value of 0 represents no diversifiation, a [...] value of one indiates that only market risk remains in the portfolio and all idiosynrati risk has been diversified (page 68). Therefore, the measure is not meant to be negative, sine this auses some diffiulties with respet to its interpretation. As we will illustrate in an example, negative outomes for DD are likely to our even under very plausible senarios. These negative values are problemati beause they ontravene the essential idea of the DD. Homogeneity is another desirable property for measures of unertainty beause it ensures that hanges in the size of an asset or the portfolio are deteted aording to their magnitude. Given that H(a) H() + log( a ), the left-hand side in the numerator of () is not homogeneous with respet to the asset, see, e.g., Cover and Thomas (99). This means that hanges in the size of the assets are not deteted in the same way as hanges in the portfolio, leading to inonsistenies in the measurement of diversifiation. In the following Examples A funtion F is homogenous if for a onstant a, F(ax) af(x). 5

7 to 3, we highlight the shortomings of the DD we have desribed. For simpliity, we onsider the bivariate ase. In Example, we present a portfolio formed by two assets following a normal distribution. We onsider the partiular ase when one of the assets is riskier than the other, indiated by a signifiantly higher variane. Suh a senario is realisti for real world portfolios, for example when ombining defensive assets (bonds) with growth assets (equities). Unfortunately, Vermorken et al. (0) in their analysis do not onsider suh a ase, but rather fous on the behavior of the DD when assets with idential or similar variane are ombined to reate a portfolio. Example Assume that the expeted returns of Asset are normally distributed with µ 0.05 and 0., while expeted returns of Asset follow a normal distribution with µ 0.0 and 0.0. Thus, the first asset yields a higher expeted return with higher risk, measured by the standard deviation. The seond asset exhibits a lower expeted return, but also a signifiantly lower standard deviation. The DD for a portfolio with normally distributed assets an be expressed as (see Appendix A, equation (6) of the online supplementary materials): DD( P) w w P w w. P w w () Let us first onsider the ase when asset returns are independent. For a portfolio onsisting of Asset and Asset, the DD is then equal to: 6 5 DD( P ) 0.4, 5 whih is less than zero. Furthermore, it is lear that DD(P) beomes inreasingly negative the smaller is. In fat, if 0.w DD( P) , then P w 0.w and, from equation (6), 6

8 Exhibit : Diversifiation Delta as a funtion of the orrelation oeffiient and the portfolio weight exemplary two-asset portfolio. Expeted returns of Asset are normally distributed with for an and, while expeted returns of Asset follow a normal distribution with and. On the left-hand side, we assume and an ex ante determined orrelation oeffiient varying between and. On the right-hand side, we assume the assets are independent and an ex ante determined weight of the seond asset. This issue presents ompliations when interpreting the DD. Exhibit illustrates the results for the onstruted portfolios using Asset and Asset. On the left-hand side, similar to Vermorken et al. (0), we onsider an equal weighted portfolio and determine the oeffiient of orrelation ex ante from to. On the right-hand side we onsider a portfolio onsisting of independent assets by hanging the weight w of Asset from 0 to. The exhibit illustrates that the DD beomes negative one the oeffiient of orrelation is greater than 0.8. It is noteworthy that we still observe that the DD delines one the orrelation oeffiient inreases, i.e., when the diversifiation is redued. Therefore, in a relative way, the proposed measure still onveys information on the benefits of diversifiation when the two assets are ombined into a portfolio. However, the interpretation of DD beomes signifiantly more diffiult in this ase, sine it an take on negative values. The right-hand side shows that the DD exhibits errati behavior when the weights hange, whih further ompliates its interpretation. Moreover, as disussed earlier, the lak of homogeneity redues the effiay of the DD. It is a well-known fat that portfolios onsisting of assets that are a linear ombination of one or more assets offer no diversifiation. However, the DD fails to detet this simple property and 7

9 yields negative values for suh portfolios. For simpliity, we will one more onsider the bivariate ase in our seond example and onstrut a portfolio of assets whih are a linear ombination of eah other. Example Let P be a portfolio onsisting of two assets, and a, where a denotes a positive onstant. Using the property of differential entropy, H(a)H()+log( a ), and w+ w, the diversifiation delta is: Exhibit : Diversifiation Delta as a funtion of portfolio weight w for an exemplary two-asset portfolio with Asset equal to *Asset. The onstruted portfolio will not provide any diversifiation. w a ( + ( a ) w ) DD( P). w a This value does not depend on entropy H() and is only zero when one of the weights is zero and negative in any other ase. Consider Example, whih desribes the ase where a, that is, one of the assets simply equals two times the other asset. In this ase the DD an be alulated as w ( + w ) w DD( P). In Exhibit, we illustrate this ase when the weight for Asset, w, is allowed to vary from w 0 up to w. It beomes obvious that for the illustrated 8

10 example, the DD is negative for all ases exept for either w 0 or w, where the DD takes on a value of zero. The measure reahes its minimum value for w However, while the onstruted portfolio does not provide any diversifiation benefits, it is also neither less nor more diversified than the original assets or. It is the lak of homogeneity in the left-hand side of the numerator of () whih yields different results for different weights. For the speified example, an appropriate measure for diversifiation of a portfolio should be 0 for all onstruted portfolios and should not depend on the hoie of the portfolio weights. Let us finally onsider Example 3, where we illustrate the lak of homogeneity of the original DD, whih leads to inonsistent results. Example 3 Investor is onstruting a portfolio onsisting of assets and. This investor determines that a portfolio with equal weights is optimal, i.e,. P +. In a 3 3 different market, investor onstruts a portfolio from assets Y and Y. 4 This investor determines optimal weights of w and w, yielding the same portfolio 3 3 P +. Given that both portfolios are the same and have the same underlying assets, one would expet that the two portfolios have the same DD. However, this is not the ase (see Appendix B of the online supplementary materials) suh that DD( P ) DD( P ). This follows from the lak of homogeneity of the left-hand side of the numerator in equation (). Given the drawbaks we identified in the DD, we now define an alternative measure. A revised Diversifiation Delta (DD ) measure As we stated before, using a ratio that ompares the unertainty of individual assets with the unertainty of the portfolio is an interesting approah to portfolio analysis. Also, given the ability of entropy to measure unertainty while taking into aount higher moments, we agree with Vermorken et al. (0) that a measure based on entropy will provide a useful tool to quantify portfolio diversifiation. We now fous on the issue of defining a measure that ote that the oeffiients in these equations represent volume, not weights. 9

11 overomes the drawbaks of the original DD, while still relying on entropy to measure unertainty. As we have seen, a measure of unertainty should satisfy ertain properties in order to be well defined. In partiular, it is desirable that the measure is homogeneous and subadditive, see Artzner et al. (999); Meil et al. (005). The measure should also be bounded between 0 and, while refleting the level of portfolio diversifiation. Differential entropy in Vermorken et al. (0) is not subadditive or homogeneous. Moreover, unlike disrete entropy, it an be negative. Differential entropy is maximized by the normal distribution. That is, when is a random variable with finite variane, we have: H ( ) log( π e ), (3) see, for example, Cover and Thomas (99). An interesting ase for analysis is the entropy of a onstant, that is when C and 0. Although the density is not defined in this ase, it is obvious that when a random variable tends to approah a onstant, its differential entropy tends to approah. Although this is onsistent with the idea that the smaller the entropy, the less unertainty there is, it makes entropy more diffiult to deal with. These issues an be addressed by onsidering exponential entropy as a measure of unertainty. Exponential entropy satisfies the following properties for variables and Y and onstants C R and λ > 0, 3 () exp (H ( + C )) exp (H ( )), () exp (H (0)) 0 and exp (H (λ )) λ exp (H ( )), (3) exp (H ( + Y )) exp (H ( )) + exp (H (Y )), These properties imply that for random variables,..., and weights ( w,..., w ), with wi : i exp H wi( i) wiexp ( H( i) ). i i 3 For a proof, see Cover and Thomas (99). 0

12 This is straightforward for and it follows indutively for higher values. Considering this and properties () to (3), we propose the following revised measure of the DD, that is DD of a portfolio P: wiexp ( H( i) ) exp H w i i * i i DD ( P). w exp ( ) i i ( H ) i (4) We note that the differene with the original DD proposed by Vermorken et al. (0) is that we use the weighted mean of the exponential entropies of the individual assets, instead of the exponential of the weighted mean of the entropies of the individual assets, on the left-hand side of the numerator and in the denominator. ote that like in Vermorken et al. (0), the estimator of the entropy H(x) we onsider is also the one found in Stowell and Plumbley (009). We now analyze Examples to 3 using the new DD measure.

13 Exhibit 3: ew Diversifiation Delta DD as a funtion of the oeffiient of orrelation and portfolio weight for an exemplary two-asset portfolio. Expeted returns of Asset are normally distributed with and, while expeted returns of Asset follow a normal distribution with and. On the left-hand side, we assume and an ex ante determined orrelation oeffiient varying between and. On the right-hand side, we assume the assets are independent and an ex ante determined weight of the seond asset. Examples to 3 revisited. Considering Example, in the bivariate Gaussian ase, the new DD is equal to 4 ( w * + w ) P DD ( P) ( w + w ) P, ( w + w ) (5) with w + ρww + w. ote that only when the variane for Asset and P Asset are the same, the new measure and the original DD in Equation () are idential and equal to P. However, in Appendix A of the online supplementary materials, we illustrate that the two measures will yield signifiantly different results when. Thus, for Example, with 0. and 0.0 we obtain quite different results for the original and the revised measure. In Exhibit 3 we repliate Exhibit using the new 4 See Appendix A, equation (7) of the online supplementary materials.

14 diversifiation measure DD. We find that all values for DD are between 0 and and exhibit a less errati behavior in omparison to the original DD. ow, onsidering Example, let P be a portfolio onsisting of assets (,..., ) whih are all positive linear ombinations of an asset. Hene, i a i + bi, for onstants a i > 0 and b i with i {,...,}. Using the properties of differential entropy it is easy to show (see Appendix C of the online supplementary materials) that wexp H( ) exp H w, ( ) i i i i i i suh that the revised measure of diversifiation takes on a value of zero in this ase. Therefore, the revised measure will be equal to zero for this ase, where no diversifiation is ahieved, sine only linear ombinations of an individual asset are being ombined. Finally, for Example 3, let us onsider the same notations as previously, in partiular exp( H( )), exp( H( )) and 3 exp( H( + )). It is easy to show (see Appendix D of the online supplementary materials) that for the new measure, we get + DD ( P ) DD ( P ). * * 3 + Given the homogeneity of exponential entropy, the same holds for similar onstrutions. From the analysis, the DD measure is always between 0 and, therefore, it is equal to when the portfolio is onstant (no risk) and 0 when the assets are a positive linear ombination of a single asset (perfet positive dependene). Also, given the homogeneity of exponential entropy, hanges in the size of the portfolio and the assets are deteted aording to their magnitude. Diversifiation Delta and the Sharpe Ratio Let us now onsider how the original DD and the newly derived diversifiation measure DD relate to the SR. Assume that the expeted returns of Asset are normally distributed with µ 0.05 and 0., while expeted returns of Asset B follow a normal distribution with µ 0.0 and 0.0. So the first asset yields a higher expeted return with higher risk, 3

15 Exhibit 4: Sharpe ratio (upper panel), original Diversifiation Delta (DD) (middle panel) and revised Diversifiation Delta DD * (lower panel) as a funtion of the oeffiient of portfolio weight for the exemplary two-asset portfolio. Expeted returns of Asset are normally distributed with and, while returns of Asset are normally distributed with and. Asset and Asset are ombined into a portfolio with. The oeffiient of orrelation between the returns of Asset and Asset is set at 0.3. Both the DD and the Sharpe ratio are maximized for 0.75 measured by the standard deviation, while the seond asset has a lower expeted return, but also a signifiantly lower standard deviation. We assume that the oeffiient of orrelation equals ρ 0.3 for this example and that the portfolio onsists only of these two assets, i.e., w w +. We now alulate values of the SR 5, the original DD, and the revised DD * for portfolios in whih w and w are determined ex ante. In other words, we express the onsidered diversifiation and SR measures as a funtion of w, the weight for the riskier asset. Exhibit 4 one more illustrates that the original DD does not ontain any information that an be easily interpreted by an investor. The measure takes on negative values for 0.5 w whih makes it almost impossible to reate a meaningful interpretation. On the other hand, we observe that for normally distributed returns, the SR and the revised DD * are 5 ote that for simplifiation, in our Sharpe ratio alulations, we assume a zero risk-free rate. 4

16 highly orrelated. This makes sense, sine entropy of the normal distribution is entirely defined by its variane; therefore, a ombination of assets with the highest SR will also yield a high DD *. Both measures are initially rising to their maximum value for w 0.75 and then they deline. Sine both individual assets provide an SR of 0.5, this is the lowest possible outome for the ratio when either w or w 0. Overall, we find that for the different portfolios we obtain 0.5 SR The revised DD *, on the other hand, takes on a value of 0 when the entire investment is either alloated to Asset or Asset (no diversifiation) and inreases when the two assets are ombined. It reahes a maximum value for w 0.75 where we obtain * DD Overall, we onlude that when returns follow a normal distribution, DD * and the SR yield similar results for the portfolio that maximizes these measures. However, unlike the SR, the DD * onsiders the variane of the different assets (and not only of the portfolio) and is not dependent on the expeted return. In that sense, the DD * measures whether the risk is diversified away by the portfolio and depends only on risk. It must be emphasized that real-world asset returns are not normally distributed but will exhibit skewness, exess kurtosis, and other features that make the empirial return distribution deviate from a normal distribution. Therefore, under real-world senarios, the SR and DD -optimal portfolios may be different. In the following, we will examine the behavior of the DD * measure for skewed and leptokurti returns in more detail. Higher Moments We will now investigate how hanges in higher moments of the return distribution impat the proposed DD measure. So far, the analysis has been restrited to assets with symmetri returns from a Gaussian distribution. Sine the proposed DD is based on the onept of Shannon entropy, it should take into aount the unertainty related to the entire return distribution and not just the first two moments of the distribution. Therefore, we examine the behavior of DD for different levels of skewness and kurtosis for an exemplary two-asset portfolio. We onsider two assets similar to the previous examples: expeted returns of Asset are distributed with µ 0.05 and 0., while expeted returns of Asset follow a distribution with µ 0.0 and 0.0. We first examine a portfolio with equal 5

17 Exhibit 5: ew Diversifiation Delta DD as a funtion of skewness and kurtosis for an exemplary two-asset portfolio. Expeted returns of Asset are distributed with and, while expeted returns of Asset follow a distribution with and. We onsider a portfolio with equal weights in both assets and unorrelated returns. On the left-hand side, we illustrate the behavior of DD for an ex ante determined skewness oeffiient varying between 0.8 skew 0. On the right-hand side, we illustrate the behavior of DD for an ex ante determined kurtosis between 3 kurt 3. Expeted returns, standard deviation, portfolio weights and the oeffiient of orrelation are held onstant. weights in both assets where w w 0.5 and they exhibit unorrelated returns. We relax the assumption of Gaussian returns and onsider skewed and leptokurti asset returns. In eah ase we simulate,000,000 random numbers for Asset and Asset for different levels of skewness and kurtosis, while expeted returns, standard deviation, portfolio weights, and the oeffiient of orrelation are held onstant. Exhibit 5 illustrates the results for DD for the assets and the generated portfolio. The left panel illustrates the behavior of DD for an ex ante determined oeffiient of skewness varying between 0 and ote that a negative oeffiient of skewness refers to returns that are skewed to the left, i.e., have a higher probability to provide more extreme negative outomes in omparison to a symmetri distribution. We observe that when asset returns beome inreasingly left-skewed, lower values for DD are observed, indiating that the diversifiation of the portfolio is redued. Values of DD range from * DD 0.5 for skew 0 to * DD 0.0 for skew 0.8 for the 6

18 equal-weighted portfolio. 6 The right-hand panel illustrates the behavior of DD for an ex ante determined kurtosis between 3 and 3. A kurtosis of 3 is equal to the kurtosis of the Gaussian distribution. We observe that when asset returns beome inreasingly leptokurti, the diversifiation potential of the portfolio is slightly redued and DD dereases. Values of DD range from * DD 0.5 for kurt 3 to * DD for kurt 3. Our results indiate that for the onsidered equal-weighted portfolio, hanges in the skewness seem to affet the proposed DD more than inreased kurtosis does. We also notie that in omparison to the impat of hanges in the asset orrelation (illustrated in Exhibit 3), the effets are learly smaller. However, the diversifiation potential of the portfolio as measured by DD is still signifiantly redued for returns that are left-skewed or leptokurti. In a final step we examine DD and SR optimal portfolios for situations in whih asset returns beome inreasingly volatile and leptokurti as typially happens during a risis or market turmoil. Given that DD also takes into aount higher moments of the return distribution, we would expet a different behavior for DD in omparison to the SR. We are espeially interested in omparing the asset alloation for portfolios that optimize the SR and DD. For the analysis we onsider the same two risky assets as before: expeted returns of Asset again are distributed with Asset follow a distribution with µ 0.05 and µ 0.0 and 0., while expeted returns of 0.0. The orrelation between the returns initially equals ρ 0.3. ow assume that due to a risis period, the volatility for both assets inreases, while at the same time asset returns beome more orrelated. We also assume that returns of the more risky Asset beome inreasingly skewed and leptokurti, while the returns for Asset remain symmetri with only the volatility inreasing. To implement these hanges of the market environment in our simulation study, we stepwise inrease the orrelation between the asset returns from ρ 0.3 to ρ 0.5, while at the same time we also inrease the volatility for Asset from initially 0. to 0. and for Asset from 0.0 to Finally, we also stepwise 6 Similar results for DD ould be observed for asset returns that are skewed to the right, i.e., for an ex ante determined oeffiient of skewness varying between 0 and 0.8. However, we report results for left-skewed asset returns only, sine we believe investors are more worried about the risk of extremely negative returns. 7

19 derease the skewness of Asset returns from skew 0 to skew 0.7 and inrease its kurtosis from kurt 3 to remain symmetri. kurt 0, while we assume that returns from Asset Exhibit 6: Weight w given to Asset for Sharpe ratio optimal (dashed) and DD -optimal (solid) portfolios under different market senarios. Expeted returns of Asset are distributed with µ 0.05, while expeted returns of Asset follow a distribution with µ 0.0. The orrelation between the asset returns is stepwise inreased from ρ 0.3 to ρ 0.5, while at the same time, the volatility for Asset stepwise inreases from initially 0. to 0., for Asset from 0.0 to 0.06, skewness and kurtosis of Asset returns stepwise hange from skew 0 to skew 0.7, and from kurt 3 to kurt 0, respetively. In eah simulation step, we generate,000,000 orrelated returns for Asset and Asset and then alulate the portfolio weights w and w that maximize the SR and DD, i.e., we determine the SR and DD optimal portfolios. Given the worsening market onditions of inreased volatility for both assets and higher orrelation between the asset returns, we find that both the SR and the diversifiation potential deline. The SR for the optimal portfolio gradually dereases from approximately 0.6 under the original senario to 0.6 for the risis senario, while DD gradually dereases from 0.9 to 0.. So both measures are redued, showing the worsening performane of onstruted portfolios with regard to the SR and diversifiation under the hanged market onditions. However, examining Exhibit 6, we find that under the senarios, SR and DD optimal portfolios provide quite different results regarding asset alloation. Reall that optimal SR portfolios take into aount only the first and seond moment of the portfolio distribution. Thus, for µ 0.05 and µ 0.0 and rising from its initial value of 0. to 8

20 0., respetively, from 0.0 to 0.06, the optimal portfolio aording to the SR suggests an inreased investment in Asset. We find w stepwise inreases from approximately w 0.7 to w 0.59 in the risis senario with higher asset orrelation, volatility and inreased skewness and kurtosis for Asset. This is not surprising beause the standard deviation of expeted returns doubles (from 0. ) for Asset, while it atually triples (from 0.0 to 0. to 0.06 for Asset. Thus, portfolios with higher weights in Asset will yield a higher SR under the hanged market onditions. On the other hand, DD also onsiders higher moments of the return distribution of the individual assets and the portfolio. Thus, while Asset provides an inreasingly unfavorable SR in omparison to Asset, DD also takes into aount that returns of Asset beome inreasingly more skewed and leptokurti. While the orrelation and volatility for both assets inreases, we assume that also skewness and kurtosis of Asset returns stepwise inrease from skew 0 to skew 0.7, and from kurt 3 to kurt 0, respetively. Based on the fat that Asset returns also exhibit inreased negative skewness and kurtosis, we find only a relatively minor inrease in alloations to Asset for DD -optimal portfolios from approximately w 0.7 under the original senario to w 0.5 under the hanged market onditions. Overall, our results illustrate how hanged market onditions for orrelation, volatility, and higher moments of the return distribution impat the proposed DD measure. Furthermore, we show that portfolios that are onstruted to maximize the SR may suggest quite different asset alloations in omparison to DD -optimal portfolios. As demonstrated, this differene in asset alloation an be attributed to the fat that DD also onsiders higher moments of the return distribution of the individual assets and the portfolio, while the SR takes into aount only the first and seond moments of the return distribution. Therefore, in future work we reommend further analysis of the empirial performane of the DD regarding portfolio optimization. 9

21 Conlusion Vermorken et al. (0) introdue a new measure of diversifiation, the DD, based on Shannon entropy of finanial returns for individual assets or a portfolio. Entropy as a measure of unertainty has been used suessfully in several frameworks and takes into aount the unertainty related to the entire statistial distribution and not just the first two moments of a distribution. We illustrate that the originally proposed DD measure has some drawbaks, in partiular when risky assets suh as equities are ombined with asset lasses with a lower risk profile. We propose a revised measure DD * that is based on exponential entropy whih overomes some of the identified shortomings of the original DD metri. This study demonstrates the properties of this new entropy-based statisti and illustrates the usefulness of the revised DD * measure. The results show that the revised DD * measure an be applied as an alternative riterion to onstrut optimally diversified portfolios. Portfolios that are optimal with respet to maximizing DD * an yield very different asset alloations ompared to portfolios that are onstruted by optimizing the SR. This is true in partiular when empirial asset returns are asymmetri and exhibit exess kurtosis. Based on our findings, we reommend the entropy based DD * as an alternative measure for portfolio optimization in institutional asset management. Endnotes We would like to thank the Centre for International Finane and Regulation (CIFR) for providing funding support to this projet under grant number SUP

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24 Online Supplementary Materials Appendies A Diversifiation Delta (DD) and revised Diversifiation Delta (DD * ) for a bivariate portfolio normally distributed assets P ( w + w ) be a portfolio where the two assets are normal, ~ (0, ) and Let ~ (0, ) and the weights are positive and satisfy w + w. In this ase P~ (0, ), with P w ww w + ρ +, where ρ is the orrelation between the assets. ote that the entropy of a normally distributed variable, with variane, is log( e ) π (see Cover and Thomas (99)). Therefore, onsidering equations () and (3), the Diversifiation Delta (DD) and the revised Diversifiation Delta (DD * ) in this bivariate normal ase are ( ) DD P ( w ( p e ) + w ( p e )) ( p e P ) exp( wlog ( p e ) + wlog ( p e )) w w ( ( pe )) ( ( pe ) ) pep w w exp( log ( pe ) ) exp( log ( pe ) ) w w ( ) ( ) w ( pe ) ( pe ) w+ w w w ( pe) pe P w+ w w w ( pe ) ( ) exp log log exp log exp log exp log pe pe pe w w P w w P w w P w P (6) 3

25 DD * ( P) ( ( )) ( ) ( p ) + ( w + w ) ( w + w ) ( w + w ) + ( ) ( ( P )) ( ) ( ( p )) wexp log p e wexp log p e exp log p e w exp log e w exp log e w pe + w pe pe w pe + w pe P P P (7) B Original Diversifiation Delta for Example 3 Investor onstruts a portfolio from assets and. This investor determines that a portfolio with equal weights is optimal, i.e. P +. In a different market, investor 3 3 onstruts a portfolio from assets Y and Y. 7 This investor determines 4 optimal weights of w and w, yielding the same portfolio P Let exp( H( )), exp( H( )) and 3 exp( H( + )). Given that the portfolios are the same, from equation () in both ases we have ( ( ) ( ) ) exp wh Y + w H Y exp( H( P)) DD( P) exp( why ( ) + why ( )) ( ( ) ( )) exp wh Y + wh Y 3. exp( why ( ) + why ( )) Error! Bookmark not defined.in the ase of Portfolio, and for Portfolio exp exp ( wh ( Y) + wh( Y) ) H( ) + H( ) ( ) ( ) ( H( ) H( ) ) exp +, 7 ote that the oeffiients in these equations represent volume not weights. 4

26 Therefore, we get 3 exp( wh ( Y) + wh( Y) ) exp H + H exp H exp H ( H( )) 3 3 exp exp ( H( ) ) ( ) DD P ( ) 3 3 DD( P ) ( ) C Revised Diversifiation Delta for ombination of idential assets Let P be a portfolio onsisting of assets (,..., ) whih are all positive linear ombinations of an asset. Hene, i a i + b i, for onstants a i > 0 and b i with i {,...,}. w exp( H( )) wa exp ( H( ) ) exp ( H( ) ) wa exp H wa n n n i i i i i i i i i i i i Therefore, n n n exp H wa i i + wb i i exp H w i i, i i i wiexp ( H( i) ) exp H w i i * i i DD ( P) 0 w exp ( ) i i ( H ) i 5

27 D Revised Diversifiation Delta for Example 3 Again, let exp( H( )), exp( H( )) and 3 exp( H( + )). Using (3), in both ases we have ( ( )) ( ) exp ( ) + exp ( ( )) exp ( ) ( ) + exp + ( ) ( ) ( ( )) + ( ) ( ) ( ( )) wexp H Y * wexp H Y exp( H( P)) DD ( P) w H Y w H Y w exp H Y w H Y w exp H Y w H Y 3 For Portfolio, wexp ( H( Y) ) + wexp ( H( Y) ) ( exp( H( ) ) + exp ( H( ) )) ( + ), and for Portfolio 3 3 wexp ( H( Y) ) + wexp ( H( Y) ) exp 3 H + H exp ( + ). Error! Bookmark not defined. Therefore, we obtain + DD ( P ) DD ( P ). * * 3 + ( ) ( H( ) ) exp H( ) 6