# Inferential Tests of the CAPM

# Inferential Tests of the CAPM

While the multivariate tests described above attempted to alleviate the limits of traditional univariate tests, in testing for the mean-variance efficiency of a given proxy, the most direct answers to Roll's critique are contained in studies by Kandel [1984], Shanken [1986a; 1987] and Kandel & Stambaugh [1987]. These considered tests of the mean-variance efficiency of the unobservable market portfolio, conditional on assumptions about the correlation between the proxy and the true market portfolio.

In particular Kandel [1984] showed that, while the efficiency of the market index cannot in general be rejected when the returns on some components of the index (the so called 'missing asset') are not perfectly observable, such rejection becomes possible when perfect or partial information - in the form of an upper bound - is available about the variance of the missing asset. In this case, the CAPM can in principle be positively rejected when the market share of the missing asset is both relatively small and an index portfolio of a subset of assets, with the same relative weights as the market portfolio, is shown to be inefficient on this subset. Thus, when considering the size of real estate as an asset class (see table 1.2), and its correlation with equities and bonds (see table 3.6 on Section 3.4.3), it is clear that both these requirements are satisfied. However, building on the same analytical framework, Shanken [1986a] proceeded to show that rejection of the CAPM is also possible, conditional on placing a lower bound upon the 'missing asset' *β* and an upper bound on its expected return. This result holds irrespective of the size of the market share of the 'missing asset'.

Kandel & Stambaugh [1987] conducted a formal test of the joint hypothesis that the unobservable market portfolio is the Sharpe-Lintner tangential portfolio, and that this has a correlation of at least *ρ*_{0} with the proxy used. Using weekly data for the period January 1963 to December 1981, a sample asset set, consisting of ten portfolios made up of all the equities on the NYSE and AMEX, was compiled. These were then stratified by reference to total market value and two different market proxies, a value-weighted and an equal weighted index of NYSE and AMEX equities. The hypothesis was that the market portfolio was efficient and could be rejected at the 0.01 significance level, conditional on the correlation between the proxy and the market being at least 0.9. For the value-weighted proxy and the last two sub-periods - 2 July 1969 to 1 October 1975, and 8 October 1975 to 23 December 1981 - a correlation of 0.7 was sufficient. If it is believed that the market portfolio has a correlation of at least 0.7 with the value-index of the NYSE and AMEX equities, this result rejects the Sharpe-Lintner CAPM.

Shanken [1987] developed a similar test methodology which allows for multivariate proxies i.e. for proxies consisting of arbitrary combinations of a given set of indices. The tests used:

- monthly data from the CRSP database for the period February 1953 to November 1983 (with January returns omitted);
- an asset set consisting of 20 equally-weighted portfolios stratified by total market value of equity; and
- a multivariate proxy combining the CRSP equally-weighted equity index and the Ibbotson-Sinquefield long-term US government bond index.

Using this test Shanken rejected, at the 0.1 significance level, the joint hypothesis that the Sharpe-Lintner CAPM holds and that the market portfolio has a correlation higher than 0.7 with the proxy used.