## Basic Models

In this section the empirical testing and development of the standard and zero-*β* CAPM has been reviewed, by presenting the models and statistical conditions implied. Several applications of the CAPM to financial decision making are then considered, together with criticisms and alternative models.

### Statistical Framework

The Sharpe-Lintner CAPM, equation (5.5), provides the following relationship between risk and expected return:

where

*R*_{i} is the return on asset *i*, *r*_{f} is the return on the risk-free asset, *R*_{m} is the return on
the mean-variance efficient market portfolio, and *β*_{i} is a measure of the relative risk of
asset *i*. The zero-*β* version of CAPM implies a similar relationship:

where *R*_{z}_{ }is the return on an asset that is uncorrelated with the market portfolio. The hypotheses behind the CAPM suggest that the following should hold for the standard model:

and for the zero-*β* model:

These relationships are not observable, (à la Schrodinger's Cat), but researchers have studied several models under the assumptions that:

- the CAPM holds in every period;
- investors share common rational expectations of ex post distributions;
- the market index is observable; and
- the
*β*s are stationary.

In order to test the CAPM, the statistical problem of estimating the *β*s must first be overcome:

and

The *β*s estimated by using equations (5.14) and (5.15) must then be tested:^{8}

and

*R*_{i} is the observed return for security *i*, *R*_{it} is the observed return for security *i* in time period *t*, *R*_{m}, is the observed return for the market proxy, *R*_{mt} is the observed return for the market proxy at time *t*, and *ε* and *γ* are the corresponding error terms in each equation. Testing the relationships of the CAPM involves testing whether the estimates of the coefficients in equations (5.16) and (5.17) are significantly different from those expected in equations (5.12) and (5.13) under the CAPM, the hypotheses being that

and

The additional assumptions included in these basic models, are that the error terms are assumed to be i.i.d., and that they are also uncorrelated with the explanatory variables. Most of the earlier tests also required normally distributed returns. Several studies have subsequently considered these assumptions.

The following sections review those tests relevant to real estate investment; in particular, how the assumptions were treated. This was undertaken either by refining the model specifications, or by adjusting the data to conform to the assumptions.

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^{8}
where denotes the historical mean return on security* i*, *E*[*R*_{i}] the theoretical expected return.