Portfolio Diversification

By Francois-Serge Lhabitant

Portfolio Diversification provides an update on the practice of combining several risky investments in a portfolio with the goal of reducing the portfolio's overall risk. In this book, readers will find a comprehensive introduction and analysis of various dimensions of portfolio diversification (assets, maturities, industries, countries, etc.), along with time diversification strategies (long term vs. short term diversification) and diversification using other risk measures than variance. Several tools to quantify and implement optimal diversification are discussed and illustrated.

• Focuses on portfolio diversification across all its dimensions
• Includes recent empirical material that was created and developed specifically for this book
• Provides several tools to quantify and implement optimal diversification

• Language: English
• Publisher: Elsevier Science
• Release date: Sep 26, 2017
• ISBN: 9780081017869

Francois-Serge Lhabitant

François-Serge Lhabitant is the C.E.O. and C.I.O of Kedge Capital, a Professor of Finance at the EDHEC Business School, and a visiting Professor of Finance at the Hong Kong University of Science and Technology.

Book preview

Portfolio Diversification

François-Serge Lhabitant

Quantitative Finance Set

coordinated by

Patrick Duvaut and Emmanuelle Jay

Table of Contents

Cover

Title page

Copyright

Introduction

1: Portfolio Size, Weights and Entropy-based Diversification

Abstract

1.1 Mathematical notations

1.2 Portfolio concentration and diversity measures

1.3 Entropy

1.4 Conclusions on pure weights and entropy-based diversification

2: Modern Portfolio Theory and Diversification

Abstract

2.1 The mathematics of return and risk

2.2 Modern Portfolio Theory (MPT)

2.3 Empirical applications

2.4 Using MPT in practice: key issues

2.5 Increasing the diversification of Markowitz portfolios

2.6 Conclusions on MPT

3: Naive Portfolio Diversification

Abstract

3.1 A (very) simplified model

3.2 The law of average covariance

3.3 The relative benefits of naive portfolio diversification

3.4 Empirical tests

3.5 Economic limits and statistical tests

3.6 Naive versus Markowitz diversification

3.7 Conclusions on naive diversification

4: Risk-budgeting and Risk-based Portfolios

Abstract

4.1 Risk measures and their properties

4.2 The toolkit for portfolio risk attribution

4.3 Risk allocation and risk parity approaches

4.4 The maximum diversification approach

4.5 The minimum variance approach

4.6 Revisiting portfolio construction with a risk-based view

4.7 Conclusions on risk-based approaches

5: Factor Models and Portfolio Diversification

Abstract

5.1 Factor models

5.2 Principal component analysis (PCA)

5.3 Conclusion on factor models

6: Non-normal Return Distributions, Multi-period Models and Time Diversification

Abstract

6.1 Non-normal returns

6.2 Multi-period models and time diversification

7: Portfolio Diversification in Practice

Abstract

7.1 Households and the empirical diversification puzzle

7.2 Diversification versus concentration

7.3 Interpreting correlations: history, facts and fallacies

Conclusion

Bibliography

Index

Copyright

First published 2017 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Press Ltd

27-37 St George’s Road

London SW19 4EU

UK

www.iste.co.uk

Elsevier Ltd

The Boulevard, Langford Lane

Kidlington, Oxford, OX5 1GB

UK

www.elsevier.com

Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

For information on all our publications visit our website at http://store.elsevier.com/

© ISTE Press Ltd 2017

The rights of François-Serge Lhabitant to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

Library of Congress Cataloging in Publication Data

A catalog record for this book is available from the Library of Congress

ISBN 978-1-78548-191-8

Printed and bound in the UK and US

Introduction

Portfolio diversification, or the practice of spreading one’s money among many different investments, aims to reduce risk. It has many parallels in common parlance, for instance, in the old saying don’t put all your eggs in one basket. It is also widely advocated in non-financial literature. In William Shakespeare’s The Merchant of Venice (1598), the title character Antonio says: My ventures are not in one bottom trusted, nor to one place; nor is my whole estate upon the fortune of this present year. A similar sentiment was expressed in Robert Louis Stevenson’s Treasure Island (1883) where the main villain Long John Silver comments on how he keeps his wealth: I put it all away, some here, some there, and none too much anywhere, by reason of suspicion.

In Finance, the first official attempt to introduce portfolio diversification goes back to the 18th Century and the development of mutual funds in the Netherlands. The goal was to create diversified pools of securities specifically designed for citizens of modest means. For instance, the mandate of the 1774 "Negotiatie onder de Zinspreuk Eendragt Maakt Magt" (Fund under the Motto Unity is Strength), organized by Abraham van Ketwich, was to hold a portfolio as close to equally weighted as possible of bonds issued by foreign governments and plantation loans in the West Indies. These included bonds from the Bank of Vienna, Russian government bonds, government loans from Mecklenburg and Saxony, Spanish Canal loans, English colonial securities, South American Plantation loans, as well as other securities from various Danish American ventures. All of them were traded on the Amsterdam market.

More formal discussions about portfolio diversification started with Leroy-Beaulieu [LER 06] and his division of capital idea. He described it as ten, fifteen or twenty values, especially values that are not of a similar nature and which were issued by different countries. Similarly, Lowenfeld [LOW 07] stated that the only means for insuring permanent investment success consists of the adoption of a true and systematic method of averaging investment risks and introduced the idea of a geographical distribution of capital. Using price data from global securities traded on the London Exchange around the turn of the century, Lowenfeld carried out quantitative studies of the risk-adjusted performance of equally weighted industry-neutral international portfolios. Later, Neymarck [NEY 13] suggested that a portfolio should be composed of stocks of different sorts which will not be influenced in the same way by a given event and for which, on the contrary, the fall in price of certain stocks would be, as far as possible, counterbalanced by the simultaneous increase of the price of other stocks. He also introduced the notion of general scale risks, which affect all the stocks, and inside scale risks, which affect only one company.

Today, all these ideas seem remarkably familiar and echo what we would call modern portfolio theory. However, the associated works remained purely literary with occasional calculations but no formal mathematical approach to portfolio diversification. This was probably intentional, as most investors at the time had no advanced education in mathematics and/or statistics. Thus, for many years, investors were aware of the notion of portfolio diversification, probably practiced it informally, but only discussed it in very general terms. Simply stated, the question of the underlying common characteristic according to which it would make sense to diversify assets had never been formally addressed. No analysis had been conducted on how to measure the benefits of diversification with respect to this characteristic. By contrast, Markowitz’s [MAR 52] approach was both normative and positive. He provided not only the scientific arguments to support portfolio diversification, but also the tools to measure it and build better portfolios. Many other authors have since built on these foundations and most portfolio construction paradigms are based on the idea that diversification pays.

Warren Buffet once said: Diversification is protection against ignorance. Indeed, from a financial perspective, portfolio diversification seems like a common-sense approach in the presence of uncertainty, or equivalently when future asset returns cannot be forecasted perfectly. In such a case, one of the simplest ways to avoid a financial disaster is by holding several investments, which ideally behave differently from each other. Keep in mind that we live in an increasingly litigious society where financial advisors, financial planners and money managers are justifiably afraid of being sued if a company or issuer blows up, causing client asset values to plunge. It is therefore not surprising that portfolio diversification has become an established tenet of conservative investing. The growth of mutual funds and exchange-traded funds has further facilitated the creation of diversified portfolios for smaller and less sophisticated investors, making diversification inherent in portfolio construction. However, this trend seems to have weakened over the recent past. First, many investors have pushed diversification to an extreme by adding assets to their portfolios just for the sake of dampening volatility, rather than based on their investment quality. They often ended up holding a little bit of everything and, more importantly, disappointed by their portfolio performance. Second, most investors endured a hard lesson in 2008 when what they thought were well-diversified portfolios collapsed with the market and experienced large losses. Thus, many of them started claiming that portfolio diversification no longer works. Reality shows that the question of how to best create a well-diversified portfolio is the one that comes with many answers. Going back to basics, even the definition of diversification and the way it can be measured are not unique. Moreover, these notions are continuously evolving with the progression of financial theory. Therefore, it is probably a good time to carefully revisit the concepts behind portfolio diversification, and see how one may integrate them in the portfolio construction. This is exactly what we plan to do in the following chapters.

The structure of this book is as follows. Chapter 1 introduces portfolio diversification when there is very little information about the risk and return behavior of the underlying assets. In such cases, investors must rely either on asset weights or on the notion of entropy to assess the diversification of their portfolios. Chapter 2 analyzes the well-known parametric modern portfolio theory and its impact on portfolio diversification. Investors often forget that modern portfolio theory does not aim to deliver a diversified portfolio and, worse yet, that it is structurally biased towards generating concentrated portfolios, due to the combination of approximation and estimation errors with an optimization process. There are of course some workarounds to force optimizers to behave in a better way; however, they come with drawbacks. Chapter 3 discusses a very simple heuristic called naive portfolio diversification, which consists of allocating an equal amount of capital to each asset. Despite its simplicity, it generally performs well and will often be used as a benchmark case when testing more sophisticated approaches. Chapter 4 focuses on risk-based portfolios, meaning the portfolios built through an optimization process exclusively focused on risk allocations and ignoring expected return considerations. Specifically, it reviews: (1) risk parity portfolios, which split risk evenly across assets; (2) the so-called most diversified portfolio, whose exclusive focus is to maximize diversification benefits; (3) the minimum variance portfolio, i.e. the portfolio that has the smallest variance of all possible portfolios. Chapter 5 explores factor-based diversification, including principal components analysis, and how the technique can be used to partition efficiently large universes of assets to increase diversification benefits. Chapter 6 discusses a series of more complex models for portfolio diversification, including the case of non-normal return distributions, the diversification of skewness and kurtosis as well as the use of tail risk measures. It also provides an introduction to multiple period models, in which investors can choose a mix of asset diversification (spreading their wealth over several assets) and time diversification (changing their allocation to assets over time). Chapter 7 discusses various empirical observations on how investors effectively diversify – or not – their portfolios. It also challenges the usual practical interpretation of what correlation coefficients effectively measure. Chapter 8 concludes our journey.

1

Portfolio Size, Weights and Entropy-based Diversification

Abstract

Investors willing to diversify their portfolio will typically spread it amongst various assets. Their implicit assumption is that diversification increases as a function of the number of assets they hold. In financial literature, the latter is often referred to as portfolio size or number of lines, and is commonly used as a quick indicator of how well or poorly diversified a portfolio is. Intuitively, we would expect a portfolio made of N2 assets to be more diversified than a portfolio made of N1 assets, if N2 is larger than N1. For instance, Markowitz reports that the adequacy of diversification is not thought by investors to depend solely on the number of different securities held. Sharpe also affirms that the number of securities in a portfolio provides a fairly crude measure of diversification. However, in practice, there are several cases where these statements happen to be wrong. For instance, a 50-stock portfolio can have all its positions equally weighted at 2%, or be 99% invested in one stock and share the remaining 1% between the other 49 stocks. Both portfolios would have an identical size, but their diversification level would obviously be very different. To be meaningful, a measure of portfolio diversification should therefore consider the distribution of asset weights in its calculation.

Keywords

Cross entropy; Entropy-based Diversification; Entropy-based portfolio optimization; Herfindahl–Hirschman index; Lorenz curve and the Gini index; Mathematical notations; Portfolio concentration measure; Portfolio Size; Pure weights; Shannon entropy

Investors willing to diversify their portfolio will typically spread it amongst various assets. Their implicit assumption is that diversification increases as a function of the number of assets they hold. In financial literature, the latter is often referred to as portfolio size or number of lines, and is commonly used as a quick indicator of how well or poorly diversified a portfolio is. Intuitively, we would expect a portfolio made of N2 assets to be more diversified than a portfolio made of N1 assets, if N2 is larger than N1. For instance, Markowitz [MAR 52] reports that the adequacy of diversification is not thought by investors to depend solely on the number of different securities held. Sharpe [SHA 72] also affirms that the number of securities in a portfolio provides a fairly crude measure of diversification. However, in practice, there are several cases where these statements happen to be wrong. For instance, a 50-stock portfolio can have all its positions equally weighted at 2%, or be 99% invested in one stock and share the remaining 1% between the other 49 stocks. Both portfolios would have an identical size, but their diversification level would obviously be very different. To be meaningful, a measure of portfolio diversification should therefore consider the distribution of asset weights in its calculation.

The concept of diversity was introduced by Fernholz [FER 99] as a measure of the distribution of capital in an equity market. It was later extended by Fernholz [FER 02] and Fernholz et al. [FER 05] in the context of stochastic portfolio theory¹. Heuristically, a market is considered as being diverse if its capital is spread amongst a reasonably large number of assets rather than concentrated into a few very large positions. The same notion is applicable in the context of a long-only portfolio of assets, which is a subset of the market. We will say that a portfolio exhibits diversity if no single asset or group of assets dominates it in terms of relative weighting. This definition is both intuitive and simple. At one end of the spectrum, a completely diverse portfolio will have its capital equally distributed across all the assets. At the other end of the spectrum, a completely undiversified portfolio will concentrate all its capital into one single asset. All other portfolios will fall between these two extreme cases. What we need now is a quantitative indicator to measure and compare the degree of diversity for various portfolios, or by symmetry, their degree of concentration ².

A capital distribution curve is useful to visualize the concentrated or diverse nature of a given portfolio. It is essentially a graph showing the portfolio weights versus their respective ranks in descending order, most of the time in a log–log format – it uses logarithmic scales on both the horizontal and vertical axes. Figure 1.1 shows the capital distribution curves for the Swiss Market Index (SMI, 20 stocks), the French CAC 40 index (40 stocks) and the German DAX index (30 stocks). The SMI seems to be the most concentrated of these indices. This was somehow predictable given the very large weighting of its top three components at the end of 2016 (Nestlé: 22.7%; Novartis: 19.5%; Roche: 16.3%) compared with the CAC 40 (Total: 10.9%; Sanofi: 8.4%; BNP Paribas: 6.4%) and the DAX (Siemens: 10%; Bayer AG: 8.8%; and BASF: 8.7%). Keep in mind that the DAX and the CAC 40 have a cap on the weights of their components fixed at 10 and 15%, respectively, while the SMI has no cap.

Figure 1.1 Capital distribution curves calculated for the Swiss Market Index (SMI), the CAC 40 index and the DAX index as of October 2016

Capital distribution curves are useful visual indicators, but they do not summarize all the portfolio weights into one single number or index. It would be helpful to have a descriptive statistic to summarize diversity information. Fortunately, many concentration and diversity measures have already been introduced in various areas of economics such as welfare and monopoly theory, competitive strategy or industrial economics. Such measures have typically been used to assess the competition that, in a given market, country, or group of countries, with a generally accepted view that competition should benefit consumers, workers, entrepreneurs, small businesses, and the economy more generally. By making a few adjustments, these measures can also be used to assess a portfolio’s concentration level.

Other interesting concentration and diversity measures come from the worlds of physics and/or information theory, given the functional parallelisms that exist between nature, the transmission of information, probabilities and financial markets. These measures are also directly applicable in the context of assessing a portfolio’s diversification level. Before reviewing these measures and discussing which might be superior, let us first introduce a few mathematical notations.

1.1 Mathematical notations

In its most general form, a portfolio P can be modeled mathematically as a collection of exposures to each of the N underlying assets in the investment universe. The term exposure should be taken in a broad sense here. It can include any non-negative numerical value, for instance, notional dollars invested, beta-adjusted dollar amounts, duration-adjusted capital, market values, etc., as long as the individual exposures add up to the overall portfolio exposure. We will denote by Ei the portfolio exposure to asset i and by

   [1.1]

the total exposure of the portfolio P. We define the weight of asset i in portfolio P as

   [1.2]

For notation consistency with other chapters, we will store these weights as an N×1 column vector

   [1.3]

In some instances, we will need to sort these weights from the smallest one to the largest one. To avoid confusion, we will use a tilde for sorted weights, so that

   [1.4]

is the vector of sorted weights in ascending order, e.g. i < j . Since all exposures are required to be non-negative, all the weights are also non-negative:

   [1.5]

By construction, in a fully invested portfolio, asset weights must sum to one

   [1.6]

which implies that

   [1.7]

These properties are essential because they will allow us to interpret asset weights as probabilities. Formally, we now have a probability space whose elementary elements are the random selection of a small fractional portfolio exposure out of the overall portfolio exposure EP , and the identification of the exposure E ∈ {Ei} that it belongs to. By construction, any fractional portfolio exposure will be part of an actual exposure and will have a probability pi ≡ wi associated with it. From there, we can define two random variables: E, which corresponds to the size of the exposure it belongs to, and l, which is the ranking order of the exposure it belongs to. While this recasting of portfolio weights as probabilities may look cumbersome at first glance, it will allow for a consistent and uniform interpretation of the various portfolio concentration and diversity measures to be discussed.

1.2 Portfolio concentration and diversity measures

As mentioned above, several metrics are available to capture the concentration of a given portfolio. Most of them take the form of a weighted sum of some function of the portfolio weights. The weighting scheme used in the sum essentially determines the sensitivity of the concentration measure towards changes at the tail end of the asset weights distribution. In practice, there are four types of commonly used weighting schemes to calculate the concentration measure: (1) use weights of one for some assets and zero for others; (2) use portfolio weights as weights; (3) use the rankings of the asset weights as weights and (4) use the negative of the logarithm of the portfolio weights as weights. Each of these approaches will be illustrated shortly. In addition, the concentration measure may also be discrete and focus only on some assets in the portfolio, or it may be cumulative and use the entire set of asset weights. The former is simpler when the portfolio is dominated by a few assets. The latter is better when any asset in the portfolio