EDHEC RESEARCH: Are cap-weighted indices justified?

It may not be possible for theory to predict an efficient market portfolio, says Felix Goltz of Edhec…

In last month’s (March 2010) Edhec research article in Funds Europe we reviewed the theoretical background to the argument that cap-weighted stock market indices provide efficient risk/return portfolios. In this month’s article, we will determine whether theory predicts that the market portfolio will be efficient.

The capital asset pricing model (CAPM) makes a range of assumptions that lead to the prediction of a mean-variance-efficient market portfolio at equilibrium. We review each of these assumptions and outline how the result of the efficient market portfolio changes when the assumption does not hold. We will see that the failure to hold any one of these assumptions may mean that theory does not predict an efficient market portfolio. It is then important to ask whether the assumptions are reasonable.

Investor preferences
The model makes bold assumptions about investor preferences; as a result, investors choose mean-variance optimal portfolios. We look now at what happens if these assumptions are incorrect. The CAPM assumes that investors seek to maximise expected utility, as defined in Von Neumann and Morgenstern (1944). The theory of maximum expected utility posits that investors always seek to maximise their terminal wealth. It also assumes that investors can compare alternatives, will establish a preference order, and will make consistent choices. The assumption that investors would rather have more than less underpins the choices it is assumed they make. This theory has been criticised by Allais (1953), Ellsberg (1961), Kahneman and Tversky (1979, 1992), and others.

These critics point out that it is not always consistent with investors’ psychology and that investors often behave in keeping with what Kahneman and Tversky (1979) call prospect theory. Indeed, investors may not consider terminal wealth; instead, they look at gains and losses throughout the investment period. They also appear to be loss averse, a trait not without consequences for the CAPM theory. These consequences have been studied by Levy and Levy (2004), De Giorgi, Hens, and Levy (2004), and others. De Giorgi, Hens, and Levy (2004) established that the cumulative prospect theory, posited by Kahneman and Tversky (1992), does not allow the existence of financial market equilibrium and, consequently, of the CAPM. A modification of utility function forms can, of course, make the theory compatible with financial market equilibrium. However, this compatibility is achieved only if asset returns are normally distributed. For Hearings and Kluber (2000), investor loss aversion may mean that the utility function is not concave and that market equilibria do not exist.

Although the efficiency of the market portfolio relies on an equilibrium in which all investors use an identical and clearly defined strategy of expected utility maximisation, a large and growing body of literature that takes into account investor behaviour concludes that equilibrium may not even be defined in the presence of more complex investor preferences.

No operational friction
The CAPM theory assumes that there is no operational friction, that is, that there are no transaction costs and that investors are not subject to taxation. These are two distinct considerations, and we look now at what the theoretical literature has to say about the efficiency of the market portfolio if these assumptions are false.

1. Taxes
If taxes are ignored, investors are supposed to have no preferences for capital gains or dividends. If, on the contrary, taxes are taken into account, investors subject to different taxation will certainly not hold the same portfolio of risky assets. They will still hold diversified portfolios, not far from the market portfolio, but investors subject to lower taxes will hold more high-dividend stocks than will investors subject to higher taxation. Long (1977) demonstrates that the existence of taxes leads to the disappearance of the mean-variance efficiency. Basak and Gallmeyer (2003) conclude that investors facing different after-tax opportunity sets of investment will require different risk premia and thus will not share the same market equilibrium model.

However, several authors have argued that even in the presence of taxation, the market portfolio may be efficient. Brennan (1970) assumes that investors have homogeneous expectations for returns and derives a modified linear form of the CAPM including dividend yields. Elton and Gruber (1978) extend the analysis and show that investors will choose to hold a portfolio made of a fraction of the market portfolio and a fraction of a dividend-weighted portfolio. These authors do not reject the efficiency of the market portfolio in the presence of taxes, as they conclude that the existence of taxes does not change the structure of the CAPM.

2. Transaction costs
Transaction costs, ignored in the CAPM, can also affect the efficiency of the market portfolio. Transaction costs keep investors from fully diversifying their portfolios, as they choose not to hold all tradable assets. In addition, investors will rebalance only if the increase in expected returns is sufficient to offset the resulting transaction costs. This observation is underlined by Markowitz (1959), Tobin (1965), and Magill and Constantinides (1976), who note an association of transaction costs and less frequent and only partial portfolio revision. As a result, each asset will be held by only a fraction of investors, a phenomenon leading to heterogeneous investor holdings; the investor’s optimal portfolio is no longer the market portfolio.

No short sales constraints
For the market portfolio to be mean-variance efficient, at least one of the two assumptions – about the existence of the risk-free asset or the possibility of unrestricted short sales of risky assets – must hold, as was demonstrated by Black (1972). Black (1972) showed that if there is no risk-free asset, but short selling is allowed, the CAPM theory is still valid. The risk-free asset is replaced by a zero-beta portfolio. Instead of lending or borrowing at the risk-free rate, the investor takes short positions on the risky assets. In that case, investors select mean-variance-efficient portfolios and the aggregate of these individual mean-variance-efficient portfolios is also efficient, so the market portfolio is still efficient.

If there is no risk-free asset and short sales of risky assets are not allowed, mean-variance investors still choose efficient portfolios. But in that case, we lose the convenient property that combinations of efficient portfolios are themselves efficient. So the market portfolio, a portfolio obtained by aggregating the efficient portfolios chosen by investors, is no longer efficient.

This issue is explained in Markowitz (2005). In fact, the market portfolio is the optimal mix of risky securities if and only if each investor can adjust the risk of his portfolio by buying or lending the risk-free asset (or by selling short risky assets, if there is no risk-free asset), in keeping with his risk tolerance. If borrowing capacity is limited and if it is not possible to sell short without restrictions, it is no longer possible to derive a portfolio with a risk suitable for each investor by combining the market portfolio and the risk-free rate. As a consequence, investors will choose a risky portfolio that corresponds to their appetite for risk, and different investors will hold different portfolios. The key conclusion of the CAPM, that all investors hold the same risky portfolio and combine it with holdings in the risk-free asset, no longer holds.

In this case, there are multiple efficient set lines, one for each investor, not a unique efficient set. And the combination of these efficient set lines does not allow us to obtain an efficient portfolio, as this aggregation no longer represents the optimal consensus for all investors. Markowitz (2005) shows that, if there is no risk-free asset and if short selling is restricted, there is no longer a linear relationship between expected asset returns and beta.

Likewise, Sharpe (1991) analyses asset pricing when short sales are not possible and comes to the conclusion that the market portfolio may not be efficient and that there may be no linear relationship between expected returns and CAPM beta.

So Markowitz, who laid the groundwork for the CAPM by introducing the concept of mean-variance portfolio choice in the 1950s, and Sharpe, who published the seminal paper on CAPM theory in the 1960s, have both concluded more recently that the market portfolio may not be efficient if risk-free lending and borrowing and short sales are restricted.

All assets can be traded
The CAPM assumes that all assets are tradable. Assets that cannot be traded, such as claims to labour income, are assumed not to exist. Alternatively, it is assumed that claims to future labour income are tradable. In the real world, of course, human capital is a great source of income for investors, as well as a great source of risk, and this asset is not tradable.

Mayers (1972, 1973) was the first to investigate the consequences of the existence of non-tradable assets on the standard form of the CAPM. He derives a pricing relationship that is still linear, but in which factors other than the covariance with the market portfolio are important. The return on an asset will depend on its covariance with the portfolio of tradable assets, its covariance with the portfolio of non-tradable assets, and the covariance between the two portfolios. The model implies that the portfolios of tradable assets of investors will differ widely, as they will depend on their holdings of non-tradable assets. So it appears that the market portfolio of tradable assets will no longer be the optimal portfolio for individual investors. Instead, different investors will hold different portfolios, depending on their holdings of non-tradable assets.

In the same vein, Athanasoulis and Shiller (2000) point out that the CAPM prescription of holding the market portfolio “disregards the correlation of portfolio returns with other endowments, traded risks with non-traded risks” (303). Put differently, investors will want to hedge the risk posed by the non-traded assets in their portfolio. Van den Goorbergh, de Roon, and Werker (2003) show that, in the presence of non-tradable assets, the optimal portfolio that an investor holds can be split into two components: speculative demand, corresponding to the traditional mean-variance optimal portfolio, and hedging demand owing to the non-tradable risks the investor is exposed to. Such hedging demand will cause a different allocation of the investor’s tradable portfolio. For example, to remain diversified, an investor working in a publicly traded company should hold less of this company’s stock than other investors. Bodie, Merton, and Samuelson (1992) show that, over the long term, an investor receiving labour income, assumed risk free, should invest more of the tradable share of his portfolio in stocks than an investor who receives no labour income. Jagannathan and Kocherlakota (1996) offer the same advice, as long as human capital and stock returns are relatively uncorrelated.

In short, investors will not choose the same optimal portfolio if non-tradable assets play a role, and the market portfolio will not be an efficient portfolio.

Homogeneous beliefs
For the CAPM, investors have homogeneous expectations about the probability distribution of asset returns. Many studies have examined the accuracy of CAPM predictions in the presence of investor heterogeneity. Lintner (1969) was the first to look into heterogeneous beliefs. He assumed that investors had negative exponential utility functions, and he showed that current asset prices depend on a weighted average of individual investors’ expectations and on the covariance matrix of these expectations. He concludes that heterogeneity in expectations would not affect the basic conclusions of the CAPM. Following Lintner’s work, other authors have investigated the consequences of investors’ heterogeneous beliefs on the equilibrium model. Most conclude that the CAPM still holds (Sun and Yang 2003; He and Shi 2009; Chiarella, Dieci, and He 2006a, 2006b; Levy, Levy, and Benita 2006).

Investors may have heterogeneous views not only of asset returns and risk but also of desirable investment periods. Gressis, Philippatos, and Hayya (1976) investigate this case, assuming that all other CAPM hypotheses are valid. Investors are assumed to invest over different horizons. The model shows that an efficient market portfolio is associated with each horizon and the optimal portfolio of each investor will be a combination of the market portfolio that corresponds to the length of his investment horizon and of the risk-free asset. In this context, the overall equilibrium market portfolio is a linear combination of the market portfolios of the various periods, with each of these portfolios efficient relative to its market segment. However, a linear combination of portfolios efficient for different time horizons is not necessarily an efficient portfolio overall.

Hence, if investors have differing time horizons, the market portfolio cannot be expected to be efficient, even if all other CAPM assumptions hold.

The CAPM assumptions in practice
According to the literature referred to above, the violation of even one of the CAPM hypotheses means that the theoretical market portfolio may not be mean-variance efficient. Under the CAPM, the market portfolio must be efficient. But this efficiency is based on many unrealistic assumptions. Finance theory tells us that the market portfolio may not be efficient if investors face restrictions on borrowing or short selling, if they hold non-tradable assets such as human capital, or if they have different time horizons.

So can we reasonably assume homogeneous preferences, the absence of taxes, transaction costs, and non-tradable assets, unrestricted borrowing or short selling, and identical time horizons? The immediate response would seem to be that we cannot. As Merton (1987) puts it: “Financial models based on frictionless markets and complete information are often inadequate to capture the complexity of rationality in action.”

In fact, investors often fail to seek utility maximisation (Barberis and Thaler 2003). This phenomenon has been described by Kahneman and Tversky (1979), who observed that people’s choices do not always tally with those that expected utility theory would suggest. Individual investors may also seek risk after incurring losses, just as they may shed risk after posting gains (Jahnke 2006): investors will keep a losing position too long, hoping trends will reverse; they will also liquidate winning positions too early to secure their gains. Boyer, Mitton, and Vorkink (2009) argue that investor preferences can be described as a gamble in which investors seek exposure to lottery-like payoffs rather than as mean-variance preferences in which investors seek mean-variance efficiency.

The existence of taxes and transaction costs is undeniable. For many investors, taxes matter. Although they may be low for the most liquid and the most heavily traded assets, transaction costs are likewise a fact
of life.

Unlimited borrowing is hardly feasible for most investors. Although short selling is an alternative that would be sufficient to save the CAPM, it is somewhat restricted in most countries. For example, some regulators require that all short sales be executed on an uptick. During the 2008 financial crisis, regulatory authorities around the world adopted a complete ban of short sales for certain stocks for a limited period. In addition, apart from any regulatory considerations, the costs and friction in the market for borrowing stocks also limit short sales.

Lilti, Rainelli-Le Montagner and Gouzerth (2006) indicate that such non-tradable assets as human capital account for a significant percentage of investor wealth, as measured by the gross domestic product (GDP). For the United States, Japan and European Union countries, the share of salaries in GDP ranges from 63% to about 70%. In comparison, the share of dividends in per capita income nowhere exceeds 5%. So, it is clearly unreasonable to assume that all assets are traded. In addition to assets that cannot be traded per se, there are assets, such as housing and claims to social security benefits, that can, in principle, be traded but for which markets are highly concentrated and illiquid or simply do not exist (Athanasoulis and Shiller 2000). Such assets often account for a large share of an individual’s wealth.

Investors also have clearly different investing horizons. A young investor saving for retirement, for example, will have a longer horizon than if he is saving to buy a house or for his children’s education. Nor do all institutional investors have equally distant horizons. Some sovereign wealth funds, for example, are set up to save for future generations, whereas others serve to smooth out short-term fluctuations in the economy (Rozanov 2007). So it is unrealistic to assume that all investors have the same investing period. In addition, many papers point out that there are different groups of investors with long horizons and others with short time horizons (Mankiw, Summers, and Weiss 1984; De Long et al. 1990).

Empirical tests of the CAPM theory
The previous discussion shows that from a purely theoretical standpoint the CAPM predictions do not hold if one makes reasonable assumptions about investors’ borrowing and shorting capacity, the role of non-tradable assets or the heterogeneity of investors’ time horizons. Regardless of the assumptions made by the CAPM, one may also test whether the theoretical predictions of the CAPM can be rejected empirically. Fama and French (2003) provide a comprehensive review of CAPM tests. In most cases, empirical studies reject the validity of the CAPM, hardly surprising in view of the assumptions it makes. However, Fama and French note: “Unfortunately, the empirical record of the [CAPM] model is poor – poor enough to invalidate the way it is used in application.” They also note that “whether the model’s problems reflect weakness in the theory or in its empirical implementation, the failure of the CAPM in empirical tests implies that most applications of the model are invalid”.

• Felix Goltz, PhD, is head of applied research at Edhec-Risk Institute

©2010 Funds Europe



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