**Felix Goltz** and **Veronique Le Sourd**, of Edhec, ask if cap-weighted stock market indices, which gave rise to indexation in the 1970s, can provide efficient risk-return portfolios. The proponents of cap-weighted indices frequently refer to the theoretical groundings of this means of index construction. The main argument for capitalisation weighting comes from the capital asset pricing model (CAPM). It has been said that widespread use of index funds began about the time the CAPM was developed and that CAPM is the basis for a number of index models, especially capitalisation-weighted indexes like the S&P 500.

In the 1970s, after the CAPM became widely known, Wells Fargo was among the first to propose index funds.

The two main theoretical predictions of the CAPM are, first, that the market portfolio is mean-variance efficient. In other words, no other portfolio can provide a higher return with the same risk, or lower risk with the same return. Second, it predicts that only systematic risk, ie, the beta of a stock with the market portfolio, is rewarded by higher expected returns. It has been shown that the second prediction follows from the first.

In our research, then, we concentrate on analysing the efficiency of the market portfolio. If the market portfolio is inefficient, the beta pricing relationship usually does not hold.

The recommendation that follows from the theory is that investors should hold the market portfolio. If this portfolio is the most mean-variance efficient, mean-variance investors would obviously prefer it. Likewise, if the only way of achieving higher returns is to increase the beta of one’s investment, the investor can simply choose between a certain fraction invested in the market portfolio and another fraction invested in a risk-free asset to adjust his overall beta to his desired risk and expected return. Since any portfolio that is different from the market portfolio would introduce some unsystematic and hence unrewarded risk only the market portfolio is of interest to the investor.

From these theoretical recommendations, index providers have concluded that the best thing to do is to buy a stock market index. For this conclusion to be of practical relevance, the CAPM theory must hold and stock market indices must be identical to the market portfolio. But should we expect the market portfolio to be efficient? And can stock market indices represent the market portfolio?

As it happens, the CAPM makes many tenuous assumptions. It assumes: investors are rational mean-variance investors; their wealth is entirely described by holdings in tradable securities; it is possible to borrow unlimited amounts at the risk-free rate or that there are no restrictions on short selling; all investors have identical preferences and identical horizons; and that taxes, transaction costs, and other frictions are insignificant for most investors and assets.

Our review of the literature shows that if any one of these assumptions does not bear out the CAPM predictions are no longer valid. Financial theory tells us that under real-world conditions the market portfolio is not necessarily an efficient portfolio and thus is not the optimal portfolio that every investor should hold. In addition, our review of the empirical literature on testing the CAPM highlights the evidence for its invalidity; in view of the model’s great reliance on unrealistic assumptions, this evidence is hardly surprising.

Let us repeat that even if the CAPM theory were valid, it is the market portfolio that is the optimal choice for all investors. The CAPM actually suggests that stock market indices are not efficient portfolios unless they are identical to the market portfolio. What is this market portfolio that the CAPM theory is based on? It is the cap-weighted portfolio of all assets that reflects the aggregate wealth in the economy. Thus, the market portfolio includes all sorts of financial assets, including unlisted securities, as well as other illiquid assets such as private housing.

To draw the conclusion that there is a theoretical basis for holding cap-weighted stock market indices, these indices would have to include all assets in the economy. Clearly, stock market indices, which usually include only a fraction of the stocks listed on an exchange, do not fulfil this requirement.** A brief review of the CAPM**

According to the CAPM, the expected excess return of an asset is linearly proportional to the expected excess market return called the market risk premium. Excess returns refer to returns above the risk-free interest rate. The amount of market risk of an asset is measured by its beta, which reflects the systematic risk of the asset.

The CAPM has been constructed under several hypotheses, such as investor preferences and that investors are risk averse and seek to maximise the expected utility of their wealth at the end of the period.

It has been established, according to the two-fund separation theorem, that in the presence of a risk-free asset, the optimal portfolio choice is always made up of a linear combination of the risk-free asset and a fraction of an optimal risky portfolio. This risky portfolio is the tangency portfolio, the portfolio that provides the highest expected return per unit of risk, or the highest Sharpe ratio. The tangency portfolio is the same for all investors whatever their risk aversion. Investors with a different risk aversion will choose a different fraction invested in the tangency portfolio, rather than choose different portfolios of risky assets.

This shows that the investment decision can be divided into two parts: first, the choice of the tangency portfolio and second the choice of the split between the risk-free asset and the tangency portfolio, depending on the desired level of risk.

Investors should try to find the maximum Sharpe ratio portfolio and then leverage up or down using a risk-free investment, depending on their risk aversion.

The CAPM theory extended this investment perspective to an equilibrium theory. What happens if all investors behave this way and they all have the same beliefs about expected asset returns and covariances? They will all end up with the same portfolio of risky assets though they may give different weightings to this portfolio compared to their holdings in the riskless asset.

Since all investors hold the same tangency portfolio and all assets have to be held in equilibrium, the tangency portfolio will be made up of all assets available weighted by their market value. This is the market portfolio. This theory leads to two central predictions: the efficient market portfolio and the beta pricing relation.** The efficient market portfolio**

The central prediction of this model is that the market portfolio of invested wealth is mean-variance efficient. It is the portfolio that has the highest Sharpe ratio among all possible portfolios. This market portfolio is defined as an asset portfolio that mimics the market: it is made of all assets in the economy and these assets are capital-weighted, ie., each one is weighted according to its percentage of the total value of the entire market for assets. ** Beta pricing relation**

The efficiency of the market portfolio implies that every security can be priced based on its beta with the market portfolio. More precisely expected returns on securities are a positive linear function of their market beta – the slope in the regression of a security’s return on the market’s return – and differences in betas of different stocks completely capture the variation of expected returns across stocks.

The CAPM thus leads to very elegant predictions about the pricing of assets (the beta pricing relationship) and optimal investment strategy (the efficient market portfolio). A strong implication is that if the CAPM holds, investors just have to hold the market portfolio. Thus they can save all effort of investment analysis to predict expected returns and covariances. The best possible portfolio in a mean-variance sense is simply the market portfolio.

• **Felix Goltz**, PhD, is head of applied research, and **Véronique Le Sourd** is senior research engineer at Edhec-Risk Institute **©2010 funds europe**