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Multi-Asset Indices

Written by Pierre Héreil, Julien Laplante and Thierry Roncalli

December 21, 2012

The concept of the market portfolio has a long history and dates back to the seminal work of Markowitz (1952). In his 1952 paper, Markowitz defined precisely what portfolio selection means: “the investor does (or should) consider expected return a desirable thing and variance of return an undesirable thing.” Indeed, he showed that an efficient portfolio is the portfolio that maximises the expected return for a given level of risk (corresponding to the variance of return). Markowitz concluded that there is not only one optimal portfolio, but a set of optimal portfolios called the efficient frontier (represented by the solid orange curve in Figure 1).

By studying investors’ so-called “liquidity preference,” Tobin (1958) showed that the efficient frontier becomes a straight line in the presence of a risk-free asset. If we consider a combination of an optimised portfolio and the risk-free asset, we obtain a straight line (represented by the dashed black line in Figure 1). One straight line dominates all the other straight lines and the efficient frontier. It is the capital market line, which corresponds to the blue dashed line in Figure 1. In this case, optimal portfolios correspond to a combination of the risk-free asset and one particular efficient portfolio, named the tangency portfolio. Sharpe (1964) summarised the results of Markowitz and Tobin as follows “the process of investment choice can be broken down into two phases: first, the choice of a unique optimum combination of risky assets and second, a separate choice concerning the allocation of funds between such a combination and a single riskless asset.”This two-step procedure is today known as the Separation Theorem.

One of the difficulties faced when computing the tangency portfolio is to define precisely the vector of the risky assets’ expected returns and the corresponding covariance matrix of returns. In 1964, Sharpe developed the CAPM theory and highlighted the relationship between the risk premium of the asset (the difference between the expected return and the risk-free rate) and its beta (the systematic risk with respect to the tangency portfolio). By assuming that the market is at equilibrium, he showed that the asset prices  are such that the tangency portfolio is the market portfolio, which is composed by all risky assets in proportion to their market capitalisation. The major contribution of Sharpe generated a flurry of empirical investigations that tended to validate his theory. The ensuing emergence of competing models (APT, Fama-French, etc.) and the increased sophistication of econometric tests then started to cast doubt on the CAPM, making it less unanimously accepted both among the academic community and the asset management industry.

Regardless of the academic infighting, two concepts introduced by the seminal work of Sharpe continue to be extensively used in the industry: beta and the market portfolio. The latter led to the rapid development of indices and passive management. Investing in both equities and bonds is today facilitated by the existence of many indices. As explained by Dimson and Mussavian (1999), the notion of market portfolio has had also a major impact on the theory and practice of investment management. “It is now common to view a managed portfolio as a blend of a passive portfolio (such as index fund) and an active portfolio comprising a series of bets on the relative performance of individual securities.” This is particularly true for portfolios investing in one asset class such as equities or bonds, but certainly not for portfolios investing in multiple asset classes. As Roll (1977) points out, the market portfolio defined in the theoretical CAPM is an index of multiple asset classes including, of course (domestic) equities, but also bonds, foreign assets, etc.

Whereas investors refer to the market portfolio when they consider one specific asset class, they neglect its use when analysing their entire portfolio. This is the case for diversified funds, which are generally benchmarked against a composite, constant-proportion equity and fixed income index. Investors—including long-term ones such as pension funds—express their tactical bets with respect to this constant-mix benchmark. But no reference to the market portfolio is ever made, in large part, due to the absence of multi-asset-class indices. However, the accumulated wealth of data on equity and fixed income markets over the last 10 years now makes the construction of accurate multi-asset-class indices possible. In this article, we restrict our analysis to stocks and bonds, because these two asset classes are the main contributors to the performance of long-term investment portfolios (Brinson et al., 1991).

Computing the Market Portfolio for Multiple Asset Classes

Except for the works of Roger Ibbotson (see for example Ibbotson and Fall, 1979; Ibbotson and Siegel, 1983), the literature on the computation of the global market portfolio is limited. This is partly due to the fact that no single data provider offers the comprehensive data needed to compute it. However, the development of financial databases over the last 10 years has made such computation easier. One of the main challenges is to choose the right sources and to combine them.

The single-currency market portfolio
Let us first consider the single-country case. We assume that we can associate to this country the universe A of the corresponding financial domestic assets. At time t, the share number and the price of asset i are denoted respectively Ni(t) and Pi(t). The market capitalisation for this country is defined as follows:

We assume that we can divide the set A into m disjoint asset classes Aj. We characterise the market portfolio by the weights (w1 ,…,wj ,…,wm ) associated with the different asset classes:

These asset classes may be all-encompassing, such as equities, bonds, commodities, etc. They may also be more granular, such as large cap equities, small cap equities, sovereign bonds, corporate investment grade bonds, high yield bonds, etc. In this paragraph, we only consider the stock/bond market portfolio. We define the equity market capitalisation MC(t;E) of a country by considering all the equity stocks which are traded in the country’s financial market. We define the market capitalisation of bonds MC(t;B) similarly. Therefore, we characterise the market portfolio by the relative proportion of equities w(t;E) in the total market capitalisation. We have:

One difficulty in this exercise is to obtain all the data to compute MC(t;E) and MC(t;B). For example, there are several thousand stocks traded in the US financial markets, making it easier to consider a broad index to approximate the market capitalisation. In the case of the United States, we can use the Wilshire 5000 index. For Japan, the Topix index is a good proxy. However, the problem of these indices is that they do not follow the same construction rules. Using Datastream equity indices and the Datastream sovereign bond indices enables us to circumvent this problem.

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