Portfolio Diversification

Francois-Serge Lhabitant

• Verlag: ISTE Press - Elsevier
• Jahr: 2017
• ISBN:
  9780081017869
• Thema: Mathematics
• Thema: Game Theory

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

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 portfo...