Global Horizons

February 2016This document is intended for institutional investors and investment professionals only and should not be distributed to or relied upon by retail clients.

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Global Horizons comprises part of the Global Series of publications at Standard Life Investments. It allows us to publish some of our more detailed work on topics ranging across asset classes, markets and methodologies.

In this edition, we look at the topic of diversification measurement. Without so much as a basic measure of diversification agreed within the investment management industry, it is difficult for investors to evaluate the portfolio construction methodologies available to them and to make improvements to existing strategies. We introduce a measure of diversification that we term the Effective Portfolio Dimensionality (EPD). We believe EPD to be an original way to assess the number of independent dimensions of portfolio risk in a way that is consistent with standard risk models.

A key element of the measure is to introduce a distinction between diversification and volatility impact. For example, we propose that negatively correlated strategies, traditionally viewed as useful diversifiers, can be most helpfully deconstructed into a perfectly negatively correlated hedge that simply reduces volatility and a perfectly uncorrelated component that provides diversification benefit. We assess a number of traditional portfolio construction methodologies such as Risk Parity and examine the variation in dimensionality of a typical UK pension plan over the last decade. Results appear to match intuition and prompt us to look at some initial results for a Maximum EPD portfolio. These are encouraging and suggest continuing research in this area may prove to be quite fruitful.

Brian Fleming Head of Multi-Asset Risk and Structuring

Jens Kroeske Quantitative Investment Manager

Global Horizons 2

Dimensions of Diversification

3 Global Horizons

Chart 1Factor snakes and ladders

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Source: Standard Life Investments (as of November 2015)E�ective number of bets Minimum torsion bets

Diversification is a surprisingly elusive concept given how often it is discussed in the investment management industry. Primarily this is because, in contrast to another common term - volatility, there is no agreed way to quantify it. This in turn limits our ability to easily monitor and then improve the diversification achieved within portfolios. Diversification means different things to different investors, from spreading of capital and risk across investments, to portfolio robustness under stress testing and scenario analysis. However, the assessment is not trivial even in a basic mean-variance setting due to the nuanced interpretation of myriad volatilities and correlations between investments. Strong positive correlations are generally considered to be bad, but anything short of perfect correlation should improve diversification; negative correlations are generally considered to be good, however, a correlation close to minus one looks akin to hedging, which theoretically negates all risk and return; correlations close to zero are perhaps most desirable but are of limited use if the volatility of one investment dominates.

There is a significant related challenge for investors in terms of their ability to evaluate the efficacy of various methodologies that are available to construct better diversified portfolios. One such popular approach is Risk Parity. While Risk Parity by name and ambition is designed to achieve a balanced exposure to diverse risks, associated calculations produce an asset allocation rather than an explicit assessment of diversification. Choosing between different portfolios based on a preference for greater diversification is therefore not possible. For this reason we believe it would be instructive to have a single number that represents a portfolio’s diversification and fits within a standard risk modelling framework. While caution towards all models and their outputs is warranted, without such a number investors must resign themselves to heavier reliance on qualitative judgement to evaluate the diversity of risks in a portfolio.

Intuitively, many investors believe that being diversified involves holding a balanced collection of investments whose returns behave dissimilarly. This naturally points to looking at measures such as the average correlation between investments. However, correlation alone is not sufficient to understand total portfolio risk. Dissimilar return profiles also

hint at volatility reduction as a measure of diversification; the price fluctuations of individual investments are always expected to negate each other to some extent at the portfolio level. However, we argue that portfolios which achieve significant volatility reduction through combined holdings are not necessarily better diversified (see case study - Misdirection from volatility reduction).

In this article, we introduce a diversification measure that we term Effective Portfolio Dimensionality (EPD). This is complementary to volatility and hence represents a new measure of risk. In accordance with more recent strategies for quantifying diversification [1,2], our initial aim in this area of research is to deconstruct the volatility and correlation structure of a portfolio to evaluate the effective number of independent investments, which we have termed the Dimensionality. This facilitates direct comparison of different asset allocations and allows us to specifically target better diversified portfolios. In the following we also use the EPD to compare well-known portfolio construction techniques such as Risk Parity and Minimum Variancea and demonstrate that this measure produces intuitive results for real world data.

Global Horizons 4

Chart 2A tour through Flatland

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1/N (Equally weighted) Volatility parity Risk parity Most diversi�ed Minimum variance

Hedge ­nds Portfolio allocation to:

Source: Bloomberg, Standard Life Investments (as of Novermber 2015)

Real estate UK credit Index-Linked gilts Gilts Global equities UK equities

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1/N Volatility Parity Risk Parity Most Diversified Minimum Variance

Volatility (%) 6.7 4.2 4.0 3.6 2.6

EPD 1.6 2.2 2.6 2.4 1.7

Misdirection from volatility reductionAn example commonly used to demonstrate the beneficial effect of diversification is to examine an equally weighted portfolio of n uncorrelated assets of equal variance i.e. = , i = 1,…,n.a The portfolio variance is then given by = so that 0 as n ∞, leading to the observation that increasing diversification through the addition of uncorrelated investments significantly reduces portfolio variance. However, if we consider a second portfolio of two assets of equal variance that have a correlation of ρ1,2= –1, then we can achieve a portfolio of zero variance by allocating capital to them equally. This two asset portfolio will therefore always achieve a greater proportionate reduction in variance at the portfolio level, being 100%, when compared to a portfolio containing an arbitrary large and finite number of uncorrelated assets. In fact, we do not require the correlation to be as strong as perfectly negative for this type of argument to hold.

Although the two asset portfolio does achieve complete negation of volatility, we believe most investors would view it as an example of hedging and the uncorrelated portfolio of investments as being more diversified. Volatility reduction does not therefore seem to be a broad enough measure of diversification. Of course, well diversified portfolios may exhibit attractive levels of volatility reduction as a result of having many uncorrelated parts, but those that achieve the greatest reduction can often be very concentrated in terms of capital allocation to different holdings. Using real data and constructing Minimum Variance portfolios we shall see that this can also be true in the situation where correlations are not extremely negative.

Portfolios with extra dimensionsWe have built upon a highly innovative procedure introduced by Meucci et al. [1,2] that led to the introduction of two new ways to quantify diversification: the Effective Number of Bets and Minimum Torsion Bets. The procedure uses statistical factors to decompose the variance of a portfolio of n investments into n uncorrelated components. The natural idea is that if the portfolio variance is concentrated in one component then there is minimal diversification and if the variance is evenly spread over all n uncorrelated components, then this represents maximal diversification. It is a direct mathematical translation of the desire to have balanced exposure to a diverse set of risks, which here are constructed to be perfectly uncorrelated.

The Effective Number of Bets and Minimum Torsion Bets are, respectively, single numbers that vary smoothly between 1 and n depending on the statistical factors used and the balance of exposures that the portfolio has to those factors. However, Meucci et al. [2] have also highlighted certain pathological cases where the Effective Number of Bets and Minimum Torsion bets produce less intuitive answers. It is these cases that we address as part of a new framework through the EPD. We report on the technical details of the EPD elsewhere [3] but here review some underlying concepts and properties:

Concepts¬ Within the linear framework where portfolio volatility is

the primary measure of risk, only zero correlation between investments should be viewed as pure diversification.b

¬ Perfect negative correlation between investments should be viewed as hedging. Hedging can be seen as being beneficial from a volatility reduction perspective, but does not contribute towards diversification.

¬ Using statistical (or named) factors in the assessment of diversification can cause ambiguity. This is most easily seen in considering an individual security. Any security in any risk model will typically have significant exposure to multiple factors which, in and of itself, suggests

5 Global Horizons

Chart 3Escape from Flatland

Source: Bloomberg, Standard Life Investments (as of Novermber 2015)

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Volatility parity Risk parity Max EPD Most divers�ed Minimum variance 1/N (Equally weighted)Hedge �nds Portfolio allocation to: Real estate UK credit Index-Linked gilts Gilts Global equities UK equities

1/N Volatility Parity Risk Parity Maximum EPD Most Diversified Minimum variance

Volatility (%) 6.7 4.2 4.0 3.8 3.6 2.6

EPD 1.6 2.2 2.6 3.3 2.4 1.7

some diversification through exposure to different risks. However, no two models are identical so the assessment will never be the same. For this reason we do not use factors in our approach and consider the investments themselves as fundamental building blocks.

The EPD essentially divides imperfect and non-zero correlations between investments into perfect positive correlation, perfect negative correlation and zero correlation. The first two parts are more concerned with the magnitude of individual asset volatility and the third with diversification, but all are required to compute the EPD. Additionally, the EPD is independent of the level of portfolio volatility and so remains unaffected by a linear scaling of positions, including through the use of leverage.

We later provide examples of the trade-off between volatility and diversification. We also note that rejecting a factor based approach does not preclude the EPD from being consistent with traditional risk modelling approaches, but rather guides us to a choice of yardstick that is independent of any chosen set of factors. This is one property that the EPD has and we list further properties below.

Properties¬ For a portfolio containing n investments, the EPD can only

have a value between 1 and n. This number can be, and typically is, fractional.

¬ The EPD has a value of 1 if all investments are perfectly correlated and an EPD of 1 is also produced in the trivial instance of a portfolio containing only one investment.

¬ The EPD can take a maximum value of n if and only if the investments are uncorrelated and of equal variance i.e. all investments are uncorrelated with each other and the standalone volatilities of each investment are equal. This is, of course, uncommon in practice and the EPD is often much less than n.

The properties described above are intuitive in that a portfolio of perfectly correlated assets has only one effective

dimension of risk, and a portfolio will only appear maximally diversified with n risk dimensions if it is constructed from a perfect balance of uncorrelated investments.

To demonstrate the behaviour of the EPD we consider an equally weighted portfolio of n investments of equal volatility and constant correlation i.e. ρi,j= ρ for all pairs of investments. We choose an arbitrary value of n=10 and recalculate the EPD as we vary the correlation ρ smoothly between 0 and 1. This is depicted in Chart 1 where we observe, as desired, a correspondingly smooth transition in the EPD between 10 and 1.

We also overlay the lines for the Effective Number of Bets and Minimum Torsion Bets, which are horizontal. Meucci et al. [2] demonstrate that, for such a homogeneous portfolio both quantities are, surprisingly, independent of ρ and equal to 1 and n respectively.

Balance it and they will comeWe now compare the following portfolio construction techniques using the EPD to illustrate the efficacy of our framework:

¬ 1/N (equally weighted)

¬ Volatility Parity

¬ Risk Parity

¬ Most Diversified Portfolio™

¬ Minimum Variance

While the Minimum Variance approach specifically targets variance and not diversification, the resultant portfolios have interesting properties that are helpful to include in our analysis. We focus on asset categories that are broadly representative of the holdings of UK pension schemes: UK Equities, Global Equities, Gilts, Index-Linked Gilts, UK Credit, Real Estate, Hedge Funds. Given that there are seven asset categories, the EPD has a maximum possible value of 7.

Global Horizons 6

Chart 4The cusp of good hope

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Source: Bloomberg, Standard Life Investments (as of November 2015)

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Chart 5Weighing it all up

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Hedge �nds Portfolio allocation to: Real estate UK credit Index-Linked gilts Gilts Global equities UK equities

Each of the techniques and the time series used for the asset categories are described in the Appendix.

Using several years of recent data, we calculated the asset allocations produced using each strategy, which are plotted in Chart 2 along with the associated portfolio volatility and dimensionality numbers. The ordering reflects a decreasing trend in volatility as we move from 1/N through to Minimum Variance, but such a monotonic trend is not evident in the dimensionality.

In Chart 2, we see that 1/N has the highest volatility (6.7%) and the lowest dimensionality (1.6) in comparison to the others. However, this ranking is somewhat arbitrary as 1/N, being an equally weighted portfolio, does not use any risk information (volatility or correlation) in the calculation of its asset allocation. As volatility information is introduced in Volatility Parity, both risk characteristics improve, with a decrease in volatility (4.2%) and an increase in dimensionality (2.2). Both risk characteristics improve again with the use of correlation information to construct the Risk Parity portfolio (4.0%, 2.6). The Most Diversified Portfolio™, using the same inputs as Risk Parity, produces a further reduction in portfolio volatility (3.6%). However, it is striking that in the Most Diversified portfolio there is a shift to a high concentration of capital within one asset category (gilts) and an intuitive associated fall in dimensionality (2.4). We therefore see a divergence in terms of volatility improving and diversification deteriorating slightly compared with Risk Parity. This is data set specific, but is generally due to the stronger negative correlation between gilts and the other assets being exploited within the Most Diversified Portfolio™, which blends volatility reduction with diversification in its assessment. Separating these two components using the EPD allows us to understand this.

An exaggeration of this impact can be seen if we ignore diversification as a goal and simply minimise variance, the effect of which we also see in Chart 2. Here the assets end up concentrated in only two categories (gilts and hedge fundsc) and display a significantly lower volatility relative to all other

approaches. This comes at the cost of a significantly worse dimensionality (1.7), which is almost a round-trip to that of the 1/N portfolio (1.6).

With a diversification measure at hand we are now in a position to directly target a more diversified portfolio. In Chart 3 we have added an EPD optimised solution (Max EPD) to our series of allocations, with the asset weightings inserted in between Risk Parity and the Most Diversified Portfolio to maintain the decreasing trend of volatility.

For this particular data set we observe that the Max EPD portfolio has a slightly lower volatility (3.8%) than Risk Parity and a notably higher dimensionality (3.3). The Most Diversified Portfolio in turn has a slightly lower volatility again but the higher dimensionality is lost. The appeal of the Max EPD portfolio can be seen by plotting the relationship between dimensionality and volatility for this series of asset allocations as we have done in Chart 4. This hints at an extended efficient-frontier framework that incorporates diversification; for example, as we move away from the minimum variance portfolio for a given level of target return, diversification and the EPD provide a further attribute and associated measure with which to assess the relative merits of different allocations. The idea of extending the mean-variance efficient frontier to a mean-variance-diversification efficient surface has previously been proposed by Meucci [1].

To sum up, the Max EPD portfolio looks balanced from a capital allocation perspective, and offers noticeably higher dimensionality and a competitive volatility when viewed against Risk Parity and the Most Diversified Portfolio™. As discussed in the case study, the volatility aspect is particularly pleasing as one would hope that a well-diversified portfolio would exhibit attractive volatility levels due to its focus on combining uncorrelated components.

EPD in practiceIn this section we look at the EPD of a typical asset portfolio over time to illustrate the performance of our new approach.

7 Global Horizons

Chart 6Dimension is not momentum

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Source: Bloomberg, Standard Life Investments (as of 21 October 2015)

Using the same asset categories as earlier we examine the dimensionality of the average asset allocation of UK pension schemes taken from the Purple Book published by the UK Pension Protection Fund [4]. This publication reviews the asset allocations of around 6,000 UK private sector defined benefit pension schemes. For comparison purposes we do not incorporate any scheme liabilities into the analysis. This implies that the portfolio analytics are absolute in nature.

Chart 5 presents the average asset allocation, normalised to remove cash positions. In terms of similarity to other techniques, the weightings are closest to 1/N, which are also shown. Interestingly, we observe that in comparison to the previous analysis, the two allocations have the same dimensionality when rounded to one decimal place, although the average UK pension scheme has lower volatility.

The variation of the EPD for this portfolio over time is depicted in Chart 6 with the FTSE All Share index shown to provide market context (for details and calculations see Appendix). As one might expect, the EPD moves broadly up and down with equity market movements but it does not simply follow price levels. The values remain in a relatively low and tight range of between 1 and 2.5 and it seems appropriate that it finds its lowest values around the financial and Eurozone crises in 2008/09 and 2011, reaching a minimum of 1.3 in December 2011. We see that the dimensionality increased substantially through 2005 before declining very sharply in 2006; it then oscillated around a low level of 1.5 going into the financial crisis in 2008. Surprisingly, the EPD then failed to rediscover levels above 2 until 2014, despite the earlier rally in asset prices. More recently we have seen a notable fall in the dimensionality, reflecting concerns about global growth. This recent loss of diversification potential has also been seen across equity markets, suggesting continued systemic fragility. Overall, the dimensionality of a typical UK pension scheme may seem surprisingly low, but we believe this level to be reasonable when one considers the correlation and concentration of typical pension scheme assets.

ConclusionsWe have proposed a new way to measure diversification that we term the Effective Portfolio Dimensionality. This allows us to assess individual portfolios and compare different portfolio construction methodologies. The EPD produces intuitive results and its properties show the EPD to be a risk quantity that is complementary to portfolio volatility. This enables us to extend traditional portfolio construction techniques to explicitly include it for the purposes of monitoring and improvement.

Initial results suggest a maximum diversification portfolio (Max EPD) is relatively attractive: in our example the Max EPD portfolio showed both lower volatility and higher dimensionality than a Risk Parity approach. Future work will examine the longer-term performance of Max EPD portfolios in terms of return, volatility, fat tails and turnover. To do this, extensive work will be required in the area of portfolio optimisation, although initial results calculating simple out-of-sample drawdown properties for higher dimensional portfolios do look promising. In relation to this, we believe there is a demonstrable mathematical link between a higher EPD and reduced drawdowns. We are currently exploring this and plan to publish results soon.

One reason the EPD is attractive is because it is compatible with a standard linear risk-modelling framework; however, this means that it is subject to all of the known limitations of such a structure. As in our day to day use of risk models, we believe the resultant analysis can be highly informative as long as we remain aware of the underlying assumptions. This awareness continues to point us toward the importance of scenario analysis to test the robustness of any portfolio.

Global Horizons 8

References[1] A. Meucci. Managing Diversification. Risk 22, 74-79 (May 2009).

[2] A. Meucci, A. Santangelo, and R. Deguest. Measuring Portfolio Diversification Based on Optimized Uncorrelated Factors (July 2013). Available at: http://www.symmys.com/node/599

[3] B. Fleming and J. Kroeske. An Introduction to Effective Portfolio Dimensionality as a Measure of Diversification. In preparation.

[4] Pension Protection Fund. The Purple Book (2014). Available at PPF: http://www.pensionprotectionfund.org.uk/DocumentLibrary/Documents/purple_book_2014.pdf

[5] Y. Choueifaty and Y. Coignard. Toward Maximum Diversification. Journal of Portfolio Management 34(4), 40-51 (2008).

Footnotesa While it is common to discuss the volatility of a portfolio, the mathematics of risk is often more easily demonstrated using

portfolio variance i.e. = . This is because the variances (and covariances) of investments can simply be added together to arrive at the portfolio variance, whereas the same linear relationship does not hold for volatilities. We shall switch between volatility and variance throughout this article as appropriate; however, higher and lower volatility do equate to higher and lower variance so, for example, minimum variance and minimum volatility portfolios are identical.

b Note that the non-linear extension of this would be full statistical independence of return distributions, which includes higher order cross terms than covariance.

c It is important to note that while our analysis reflects the correct overall expected behaviour, it should not be taken at face value as an evaluation of the relative merits of different asset categories. Hedge fund data in particular is subject to many flattering biases, while the representation of Real Estate using REITs produces a high correlation with equities. Due to their high correlations, UK Equities, Global Equities and Real Estate behave as close substitutes in this analysis.

*

9 Global Horizons

Appendix

Portfolio construction methodologiesFor a portfolio of n assets we denote the weighting of each asset as w > 0, for i = 1,…,n, such that ∑w =1. The volatilities of each asset and the portfolio are given by σ and σ respectively, with the asset standalone risk calculated as w σ . The correlation between an asset and the total portfolio is denoted ρ ,p. We define each of the portfolio construction methodologies by the following.

¬ 1/N: equal allocation of capital across assets such that w = 1⁄ n.

¬ Volatility Parity: equal standalone risk such that w σ = w σ for all i and j.

¬ Risk Parity: equal asset beta to the total portfolio such that w ρ ,p = w ρ ,p .

¬ Most Diversified Portfolio™ [5]: weights are chosen to maximise the Diversification Ratio™ (DR), which is defined as (∑w σ )⁄σ . Notice that the DR is improved by increasing the sum of the asset standalone risks relative to the portfolio volatility. When applied to the examples in our case study, one can see that the DR of the two asset portfolio is only a function of the correlation between the two assets. As ρ1,2 –1, σ 0 and DR ∞, so we can always find a two asset portfolio correlation that has a better DR than a broadly balanced portfolio of uncorrelated investments.

¬ Minimum Variance: weights are chosen to minimise σ = ∑i,jwi wj σi σj ρi, j.

Data sets and window lengthsAll data for the asset categories used in the main text is sourced from Bloomberg for the period 01/04/2003 - 21/10/2015. The descriptions and index codes are shown in Table 1:

Table 1Category Index Description Index CodeUK Equities FTSE All-Share ASXUK Equities FTSE All-Share Total Return (TR) FTPTTALLGlobal Equities MSCI World TR Gross USD (to GBP) GDDUWIGilts FTSE UK Conventional Gilts All Stocks FTFIBGTIndex-Linked gilts FTSE UK Index Linked Gilts All Stocks FTFIILAUK Credit BAML Sterling Non-Gilt UN00Real Estate FTSE EPRA UK Total Return GBP RLUKHedge Funds HFRX Global Hedge Fund Index HFRXGL

Charts 2-5 use data based on an equally weighted 180-week window over the period 31/03/2012 – 30/01/2015. Weekly logarithmic returns are calculated Wednesday to Wednesday, which is typical of many risk models to reduce the impact of time zones and holidays. Specific dates aside, this structure is chosen to match that of our 3rd-party multi-asset risk model.

Chart 6 is calculated using daily data over the period 01/04/2003 - 21/10/2015 and an exponentially weighted rolling window of 104 weeks with a half-life of 52 weeks. The asset category weights are held constant on a weekly basis. The FTSE All Share (Price) Index (ASX) is used only in this chart as the background comparator.

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Contact Details

For further information on Standard Life Investments’ research on longer-term investment themes, please contact:

Frances HudsonGlobal Thematic Strategistfrances_hudson@standardlife.comTelephone: +44 (0)131 245 2787

Visit www.standardlifeinvestments.com or contact us at one of the following offices.

EuropeStandard Life Investments 1 George Street Edinburgh United Kingdom EH2 2LL Telephone: +44 (0)131 225 2345

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Asia PacificStandard Life Investments (Hong Kong) Ltd 30th Floor LHT Tower 31 Queen’s Road Central Hong Kong Telephone: +852 3589 3188

Standard Life Investments Limited Beijing Representative Office Room A902-A903, 9th Floor, New Poly Plaza No.1 Chaoyangmen Beidajie Dongcheng District, Beijing 100010 People’s Republic of China Telephone: +86 10 8419 3400

Standard Life Investments (Hong Kong) Ltd Korea Representative Office 21/F Seoul Finance Center 84 Taepyungro 1-ka, Chung-ku Seoul, 100-101 Korea Telephone: +82 2 3782 4760

Global Horizons 10

Dr Brian FlemingHead of Multi-Asset Risk and StructuringMulti-Asset Investingbrian_fleming@standardlife.comTelephone: +44 (0)131 245 8505

Dr Jens KroeskeQuantitative Investment ManagerMulti-Asset Investingjens_kroeske@standardlife.comTelephone: +44 (0)131 245 0057

Important InformationAll information, opinions and estimates in this document are those of Standard Life Investments, and constitute our best judgement as of the date indicated and may be superseded by subsequent market events or other reasons.

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This material is not to be reproduced in whole or in part without the prior written consent of Standard Life Investments.

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