# The Zero-β Model

# The Zero-β Model

Although these early papers of Lintner [1965a], Douglas [1969], Miller & Scholes [1972], it inter alios, questioned the validity of the CAPM, the general belief was that the CAPM held, and that there was something in the measurement process or statistical estimation that was affecting the tests of the model. Black, Jensen & Scholes [1972] hypothesised that the deviations from the CAPM were due to market constraints; specifically, the absence of risk-free borrowing.

## Black, Jensen & Scholes [1972]

The explicit tests of CAPM cited above were performed using cross-sectional regressions on individual securities. Black, Jensen & Scholes examined the non-stationarity of returns and concluded that their evidence, and Roll's earlier results, indicated that a time series regression (similar to the portfolio evaluation method used by Jensen [1968]; discussed in Chapter 4) would yield a more accurate description of the returns process. They attempted to explain the anomalies observed in earlier studies, but their primary concern was in developing procedures to contrast the standard and zero-*β* CAPM. They did not, for example, test for the importance of residual risk^{12} or *β*^{2}.

They tested the traditional form of the model, equation (5.18), but improved the efficiency of the estimators - previous tests ignored data available on other securities - by splitting the securities into ten risk groups.^{13} This method of examining returns reduced the bias from measurement errors in the *β* estimates, and almost all subsequent tests of the model used this procedure.^{14} To eliminate the bias introduced by using

s generated in the same time period, they used

s calculated from the previous measurement period.

Using 35 years of monthly returns data on ten portfolios,^{15} Black, Jensen & Scholes found that the excess returns expected were lower on high *β* equities and higher on low *β* equities. They then used the two-factor model - equation (5.11) developed by Black [1972] - to correct for the absence of risk-free borrowing opportunities. The intercept a from the corresponding regression represents (*R _{z}* -

*r*) (1 -

_{f}*β*).

_{i}*R*being greater than

_{z}*r*, would explain high,

_{f}*β*securities having negative α's and low

*β*securities having positive α's. They again grouped the securities to increase efficiency and found that the two-factor model, with a non-zero intercept, better described the data. Black, Jensen & Scholes also provided a method for estimating

*R*, and estimated average zero-

_{z}*β*excess returns over ten year periods. They found strong evidence to suggest that they were significantly different from zero. However, a detailed test of the two-factor model was not performed and the development of unbiased estimators was left for future work. They also mentioned the presence of other effects e. g. errors in measurement of

*R*, non-marketable assets, taxes and dividends, but unfortunately did not examine the effect of these on the model. These factors are of increased import when models such as those above are applied to real estate investment decisions.

_{m}## Fama & MacBeth [1973]

Fama & MacBeth [1973] also used the portfolio formation technique - to mitigate measurement error problems in *β* - and two-pass regression procedure to examine the Sharpe-Lintner and zero-*β* versions of the CAPM.

Under the null hypothesis that the CAPM is true, the appropriate formation of the second pass regression is:

If the Sharpe-Lintner version of the CAPM holds:

Under both versions of the CAPM:

Fama & MacBeth also considered the alternative hypothesis:

where denotes the estimated residual risk of company *i*, and *γ*_{1t}, *γ*_{2t}, and *γ*_{3t} are risk premia to be estimated.

If the CAPM is valid, the restrictions outlined above should hold i. e.:

- equation (5.20) if the zero-
*β*version of the CAPM holds; and - both equations (5.19) and (5.20) under the Sharpe-Lintner version of the CAPM.

In addition, the coefficient restrictions should not be capable of being rejected:

These restrictions test for the importance of non-linearities in *β*s (*γ*_{2t}), and the rôle of residual risk (*γ _{3t}*

_{)}.

As discussed earlier, these *γ* coefficients can be interpreted as the returns on particular portfolios.

*γ*_{0t}is the return on a positive investment in a minimum-variance portfolio which has a equal to 0, a weighted-average of squared s equal to 0, and a weighted average of residual risk equal to 0.^{16}

- Similarly,
_{γ1t}is the return on a zero net investment, minimum-variance portfolio which has a weighted-average equal to 1, a weighted-average of squared

- s equal to 0, and a weighted average of residual risk equal to 0.
^{17}

*γ*_{2t}is the return on a zero net investment, minimum-variance portfolio which has a weighted-average equal to 1, and a weighted-average of squared

- s equal to 1, and a weighted average of residual risk equal to 0; and

*γ*_{3t}is the return on a zero net investment, minimum-variance portfolio which has a weighted-average equal to 0, and a weighted-average of squared

- s equal to 0, and a weighted average of residual risk equal to 1.

The main hypotheses being tested were:

- linearity (
*E*[*γ*_{3t}]= 0); - the presence of systemic effects on residual risk (
*E*[*γ*_{1t}] = 0); - a positive expected risk-return trade-off (
*E*[*γ*_{2t}] > 0); and - the standard CAPM, versusthe zero-
*β*model of the market.

The standard deviations included in the model were the average of the least squares residuals from estimates of *β* as in equation (5.15), and were a measure of the portion of the variance not directly related to the *β*s. The quadratic term in the model was the average of the squared individual *β*s for the security, and was thus less than the square of the portfolio's *β*. Therefore, establishing the absence of the first two effects would allow a more detailed test of the CAPM and zero-*β* models.

Fama & MacBeth grouped portfolios based on the previous period's estimates of *β*. This instrumental variables technique was designed to reduce the bias in the *β* estimates as documented in Blume [1970] and Black, Jensen & Scholes [1972].^{18} The ranking of *β*s to form portfolios was performed to reduce the loss of information; inter-group variance was maximised and intra-group variance was minimised. Monthly adjustments to portfolios included the delisting of equities; the estimates of *β* were recalculated annually. The market proxy used was Fisher's Index, which is an equally weighted average; see section 4.2. Nine portfolio formation periods were used with four yeartesting periods. For an equity to be included within a portfolio, it was required to have data for the previous five years. This implicit bias against high *β* equities was subsequently found by many researchers. This issue is further discussed in section 4.5.5.

A strong assumption of the model was the homoscedasticity and independence of the error terms. However, Fama & MacBeth found evidence of interdependencies, especially among the high *β* equities. Thus the value of using grouped portfolios was reduced. They also emphasised the predictive nature of the tests:

' the model was initially developed by Markowitz as a normative theory ... it only has content if there is a relationship between future returns and ... estimates of risk that can be made on the basis of current information.'

An additional caveat regarding their tests, and later tests of the model, was that the *t*-statistics were likely to result in overestimates of the significance of the estimated coefficients. There was therefore a higher probability of rejecting the null hypotheses. The reason for the bias relied on earlier studies suggesting that the distribution of returns was leptokurtic relative to the normal distribution. However, the level of the differences would only affect the extreme tails of the distribution.

Linearity of the market model was supported by their study. In addition they found no evidence of a systemic effect other than *β* affecting expected returns. As in previousstudies they found a positive trade-off between risk and return. They also found that γ0 was significantly different from the risk-free rate, supporting the zero-*β* version of the model. However, they did not test for other variables, for instance the importance of companies debt to equity ratios, earnings-price ratios, or size, by including these variables in the cross-sectional regressions. This was the focus of subsequent work by Fama & French [1992], which is discussed in section 5.3.8.

An additional hypothesis studied by Fama & MacBeth was the efficiency of the market portfolio. Under the homogeneous expectations hypothesis the market portfolio is always considered efficient. Fama & MacBeth studied the coefficients and the residuals to test whether the market operates as a fair game. Correlation was computed between regression coefficient estimates in adjoining periods. Their results confirmed the fair game hypothesis.

____________________________

^{12}Lehmann [1990] uses the basic Black, Jensen & Scholes [1972] regression specification to examine the impact of residual risk.

^{13}In particular, Black, Jensen & Scholes [1972] used the following procedure for forming portfolios and running two-pass regressions:

- Estimate βs for individual securities using five years of monthly data (e.g. 1926-1930). Use the estimated, s to form ten portfolios. Portfolio one consists of the 10% of equities with the highest s, and portfolio ten consists of the 10% of equities with the lowest

- s;
- Compute the returns for each portfolio over the next twelve months (e.g. 1931).
- Using a rolling window, estimate
*β*s for individual equities for a new five-year period (e.g. 1927- 1931). Form ten portfolios based on these estimates. - Compute the returns for each portfolio over the next twelve months (e.g. 1932).
- Repeat steps (iii) and (iv) for the entire sample.

^{14}However, see Roll & Ross [1994], for a discussion on the effect of measurement error on the results of cross-sectional mean-*β* tests.

^{15}Black, Jensen & Scholes [1972] included all securities with at least 24 months of data; see section 4.5.5 on page 102 for a discussion on sample selection bias.

^{18}Outlined in footnote 13 on page 131.