Alternative Models

Alternative Models

The search for improvements to the CAPM continued, as did tests of its efficacy in different markets. The majority of these studies attempted to find an improved algorithm to help explain the discrepancies from the model. This section will discuss the main developments - the CAPM with dividends and taxes, the international asset pricing model, bond pricing, downside-variance, and the incorporation of skewness and ex ante returns. Two other models were also devised - to handle non-marketable assets (Mayers [1973]) and inflation (Chen & Boness [1975]. The common thread between all these models was the linear relationship between expected return and one of the two risk variables. However, it should be noted that as the complexity of the models grew, so did the econometric problems.


Black & Scholes [1974] tested the CAPM relationship in the presence of personal taxes, as in Brennan [1973]; see section 5.2.6. The model specified was a modification of equation (5.16):

where di and dm are, respectively, the dividend yields on security i and on the market portfolio. Black & Scholes estimated equation (5.23) using pooled time-series and cross-sectional monthly data from 1926 to 1966. To avoid the inefficiency associated with non-spherical disturbances, they used a complicated procedure to estimate α and γ. Using 25 intermediate portfolios (thus simplifying the computation of the variance-covariance matrix) Black & Scholes constructed minimum-variance portfolios with expected returns equal to the parameters they wanted to estimate. The mean returns on these portfolios were then used as an estimate of the parameters. They were unable to reject the null hypothesis that γ, the coefficient of the excess dividends, was equal to zero.

This approach was criticised by Rosenberg & Marathe [1979], who demonstrated that it was equivalent to estimating the parameters in equation (5.23) using ordinary least-squares ("OLS"). They argued that the result was simply due to the inefficiency of the procedure used, aggravated by the grouping of equities into portfolios. Indeed, had the null hypothesis been that γ was positive, it could not have been rejected either. Using instrumental variables to solve the problem of unobservable β coefficients and generalised least-squares, Rosenberg & Martthe found that the dividend term was statistically significant. This places particular importance on the proportion of total return derived in the form of income, and the levels of taxation on different forms of return.

The above factor is of particular relevance when considering institutional investment in real estate. This is further discussed in section 2.4.12.


One of the most extensive tests of the effect of taxes on the CAPM relationship was conducted by Litzenberger & Ramaswamy [1979], who tested their generalisation of Brennan's model, equation (5.9). The empirical specification was:

To make the estimation of the variance-covariance matrix of asset returns possible, they assumed that:

  • security returns are serially uncorrelated;
  • the error terms are uncorrelated across securities; and
  • returns are generated by a market model of the form:

Placing these assumptions on the form of the variance-covariance matrix allowed the authors to avoid the grouping of individual equities into portfolios, thereby increasing the power of the subsequent statistical tests. As for the measurement error in the β coefficients, they assumed that the estimates of β were equal to the true βs plus a measurement error vit :

where vit was assumed to be cross-sectionally uncorrelated, and vit and εit were jointly normal and independent. In this case the variance of the measurement error could be consistently estimated by:

and consistent estimators of the coefficients, equation (5.24), could be obtained using maximum-likelihood.

Litzenberger & Ramaswamy estimated equation (5.24) using monthly data on equities for the period 1931-1977. To avoid capturing informational effects that might bias the coefficients, they used specifications for the dividend variable, dit, based on when security i went ex dividend, and when its announcement was made.

The coefficient of the excess dividend yield variable, γ2, was highly significant. The estimated point value was comparable with that obtained by Black & Scholes, but the standard error was about 25% lower, enough to reverse their conclusion about the significance of the estimated value.

Miller & Scholes [1982] argued that the definition of the dividend variable used by Litzenberger & Ramaswamy [1979 did not eliminate spurious correlation, due to the informational rôle of dividends. Specifically, the observed correlation between dividends and returns might have been due to the presence of companies that were expected to pay a dividend in a given month, but suspended the dividend. Miller & Scholes called this the case of the 'dog that didn't bark.'

For example, suppose that an equity trading at £10 has an even chance of either announcing a £2 dividend (which would result in a doubling of the equity price) or suspending the dividend (which would result in the equity price falling to £5). The ex ante rate-of-return is 35%, and the ex ante dividend yield is 10%. However, using Litzenberger & Ramaswamy's definition, the yield is 20% (a 120% return) if the dividend is paid, but is 0% (-50% return) if the dividend is not paid.

Thus the regression tended to show a positive association between returns and dividend yields, although this correlation was not related to the existence of taxes. When Miller & Scholes repeated the test, but included only equities for which dividends were paid in month t and announced in advance, the coefficient γ2 was not significantly different from zero.

Litzenberger & Ramaswamy [1982] responded to Miller-Scholes's criticism by repeating their regression using a 'predicted dividend yield', obtained using an explicit dividend forecasting model. Again, they obtained a statistically significant coefficient for the effect of dividend announcements.

International Asset Pricing

Solnik [1974] studied the international pricing of risk and found that equity prices are based on international systemic risk, but still depended largely on national factors. Solnik's data set consisted of a sample of 299 equities from eight European countries and the US. The construction of the market indices and risk-free rates employed were different for each country. The market-weighted world index was the average of the individual indices weighted by that country's gross national product. Tests of this model, using the grouped-instrumental variable approach, did not reject the CAPM. However, he argued that investors ought to hold the world market portfolio rather than the domestic market portfolio (Solnik [1977]).

At the time of the CAPM's conception it may have been reasonable to assume a closed economy framework, where assets would be priced in a domestic context. In that context, the one normative result of the CAPM - that all investors hold a combination of the market portfolio and risk-free asset - is not as unacceptable as it might appear in an international context. This is as it could be argued that most investors' diversified portfolios are quite similar e. g. they share a common benchmark.

The globalisation of financial markets20 has, however, undermined this assumption. It is no longer reasonable to assume that assets are priced exclusively within their domestic market, particularly when considering centres such as London, which are heavily influenced by international investors. For example, overseas investors took large positions in London's office market e.g. Canary Wharf, in the late 1980's.21 Investors in different countries will not use an overall global benchmark, and tend to invest more heavily at home than abroad, tracking performance in their domestic currencies. This is considered in section 4.5.6. Consequently, global benchmarks vary by base currency and country. This violates one of the CAPM's standard assumptions, and there is no international CAPM which has an empirical content.

Bond Pricing

Friend, Westerfield & Granito [1978] used a data set consisting of corporate bonds listed on the NYSE from 1968 to 1973; 891 bonds had sufficient information to be used. The market index used was a weighted average of the NYSE Composite Index, the RLW index (an equally-weighted quarterly index of market returns on bonds) and Salomon's long-term Government bond index. Average quarterly returns for individual bonds were regressed on their β coefficient and the residuals from the β estimation procedure. This was repeated for equities, bonds, and for equities and bonds combined, as well as with and without grouping. The equation for grouped data was as in Fama & MacBeth [1973]. Similarly Friend, Westerfield & Granito found that the CAPM did not appear to explain the returns observed. The implied zero-β return for bonds was significantly lower than for equities. They also found some evidence of differences in the risk-return relationship for bonds as compared with equities.

Results such as those of Friend, Westerfield & Granito question the validity of applying the CAPM across asset classes, especially as between two markets of varying structure. The real estate and equity markets are a prime example. A study of the real estate market, similar to those above, is not possible due to the quality of data available.


First proposed by Markowitz [1959], Jahankhani [1976] attempted to refine the CAPM so as to provide a better description of equity returns. The usual assumption is that investment is based on mean-variance efficiency. However, Jahankhani developed a model based on mean-semi-variance ("MSV"), where risk was described by semi-variance22 rather than variance. Semi-variance is defined as:

where rf is the return on a portfolio with a semi-variance equal to zero.23

Jahankhani cites empirical studies indicating that many financial decision-makers measure risk in a manner more consistent with semi-variance than variance. The structure of the CAPM equations remains the same. However β is now defined by:

The model was tested by regressing average realised returns against the estimate of βsi

He tested both the standard CAPM and MSV models under four hypotheses:

  • a linear relationship between returns and the estimate of β;
  • the estimate of β is a complete measure of risk in the efficient portfolio;
  • the intercept is equal to the risk-free rate-of-return; and
  • the slope is equal to E[Rm] - rf, and is positive.

The study covered the period from July 1947 to June 1969 and included the analysis of 380 equities. Fisher's Index and 30-day Treasury bills were used as proxies of the market return and risk-free rate respectively.

The MSV model was tested using a two-stage procedure, following Fama & MacBeth [1973]; see section 5.3.3. However, since β and σ(vi) as defined for the semi-variance measure cannot be estimated from a regression, they were estimated using the following procedure:


For the mean-variance model, Jahankhani used past βs24 as a proxy for expected βs and grouped the data into 19 portfolios, each of a two-year period. In both models the relationship was found to be linear and no measure of risk other than ß systemically affected return. In both cases, however, the intercept was greater than the proxy for the risk-free rate-of-return and the slope was less than:

Both models were inconsistent with the data. Errors of measurement of rf, and the assumption of unlimited risk-free borrowing, were cited as possible reasons. However, Jahankani's tests indicated that the two systemic risk measures were similar. Nantell & Price [1979] subsequently demonstrated that under the assumption of the multivariate normal distribution of returns, the CAPM and downside-variance model were equivalent. Their paper, combined with the unacceptable assumption of risk-neutrality above the specified target rate-of-return,25 seems to have stemmed research in this area. Price, Price & Nantell [1982], however, refuted the equivalence of the CAPM and downside-variance βs, arguing that the skewness of the market portfolio returns renders the systemic risk in the downside-variance model a meaningful alternative to the CAPM β. It should therefore be noted that the equivalence of the CAPM with the MSV and downside-variance models, relies heavily on the assumption of the multivariate normal distribution of returns. This is further discussed in Chapter 7.

Skewness Preferences

Another observation of preferences was that investors seemed to prefer positive skewness. A number of studies have examined the question of skewness of equity returns by extending the asset-pricing model to three parameters. While the three-moment CAPM will not be explored in detail, it is useful to highlight the main work in this area.26 Kraus & Litzenberger [1976] extended the CAPM by incorporating systemic skewness (as opposed to total skewness examined by Arditti [1967]). They studied the excess rate-of-return, deflated by 1+rf. β and γ, a measure of the systemic skewness of an asset, were estimated for each equity, and the excess return measure for grouped portfolios was regressed on the β and γ estimates of the portfolio. OLS estimates were calculated and the variances of these estimates were corrected for heteroscedasticity. Kraus & Litzenberger tested the results in their model as compared to that of Fama & MacBeth, and found much higher estimates of the coefficient on the portfolio βs than previously. Furthermore, the coefficient of γ was significant and negative, consistent with the three-moment CAPM. They also found that higher β portfolios had more than proportionately higher γs, and attributed this to investors' aversion to variance and preference for skewness. These results place increasing pressure on the assumption of the multivariate normal distribution of returns, and are further discussed in section 2.3.

Kraus & Litzenberger also questioned the use of quadratic utility, and concluded that the findings of Friend & Blume, Black, Jensen & Scholes, and Fama & MacBeth, mayhave resulted not from restrictions on risk-free borrowing, or divergent borrowing and lending rates, but from a misspecification of the CAPM.

Ex Ante Returns

One of the problems in testing the CAPM involves the fact that expected (ex ante) returns are unobservable. Friend, Westerfield & Granito [1978] obtained ex ante data from financial institutions for the years 1974,1976 and 1977 in order to gain a more accurate picture of the relationship between expected returns and risk. The studies were of 66 equities held by 21 financial institutions in 1974, 49 from 33 in 1976, and 56 from 29 in 1977. The equities all had low β estimates. The expectation used was a spot dividend yield plus the annual growth rate in earnings per share expected over the following five years. The mean expected return for each equity was regressed on its β coefficient alone, then on its β coefficient and residual standard deviation. The mean expected return for each equity was also regressed on β, the standard deviation and the standard deviation of expected returns - a measure of heterogeneity. The coefficient of β in the regressions was negative for the regression solely on β, being significant only in 1977. The coefficient on σri was statistically significant for 1974 and 1977 and the coefficient of hi was significant in 1977. Friend, Westerfield & Granito concluded that the findings support consideration of σ and related variance measures as significant in pricing assets. However, they also noted that the estimates of β and σ contained substantial measurement errors; because of the limited amount of data, grouping could not be utilised. Furthermore, Friend, Westerfield & Granito [1978, p. 908] stated that:

'... tests of capital asset pricing theory that rely only on grouped data, to the exclusion of tests based on individual assets, are not completely satisfactory, since it is the returns on individual assets which the theory is trying to explain and individual asset deviations from linearity may cancel out in the formation of portfolios.'

In response, Sharpe [1978b] cited a study based on Wells Fargo's27 weekly predictions for 300 equities. Expected returns were regressed on predicted β values. For every week over a six-year period the relationship was positive and statistically significant.


20See Fingleton [1992] for an institutional perspective; and Pugh [1991a; 1991b] who specifically deals with real estate.

21See Bank of England [1992] and Debenham Towson & Chinnocks [1988; 1992; 1995].

22The terms semi-variance, downside-variance and second-order lower-partial moment have been used interchangeably in finance literature. Specifically, semi-variance is the partial moment from - ∞ to the mean, and not an arbitrary point; see equation 5.25. Downside-variance is the partial moment from -∞ to an arbitrary point, often a target rate-of-return. If this point of reference is the mean, then downside-variance is equivalent to semi-variance. Where the term downside-variance is used in this thesis, it will include the special case of semi-variance.

23A variation of this model was first proposed by Hogan & Warren [1972]. The systemic and total risk measures of Hogan & Warren [1972; 1974], and Bawa & Lindenberg [1977] can be obtained by substituting rf for τ in equation 5.25, where τ is a target rate-of-return. It is referred to here as downside-variance.

24Price, Price & Nantell [1982] and Harlow & Rao [1989] have compared the CAPM and MMSV βs. Both studies found that the estimated β was on average greater under the MSV model than the standard CAPM.

25The linearity of a portion of the utility function implies that return distributions above the target rate-of-return are inconsequential, thus investors are concerned exclusively with downside risk; further discussed in Chapter 7.

26This includes the work of Arditti & Levy [1975], who argued that investors pay a premium for positive skewness. Francis [1975] found that skewness is not priced, and Lim [1989] indicated that the systemic skewness is priced. There is some, but not conclusive, evidence that either the total or systemic skewness is priced.

27A US bank.