Benchmark Issue in CAPM
Jeffry Merril Liando (2007)
CAPM and efficient portfolio
CAPM assumes that a major stock index can be used as a benchmark to determine risk premium and beta for calculating the required rate of return of a stock. The formulae can be shown as follows, Rs = Rf + Beta (Rm – Rf), or by noticing the future expectation it can be written as this: E(Rs) = Rf + Beta (E(Rm) – Rf). To see the relationship between share and market returns, it can be formulated as follows: E(Rs) – Rf = Beta (E(Rm) – Rf).
The benchmark index assumed in the CAPM is promoted as the market proxy of the efficient portfolio of risky assets. The benchmark index is then set artificially to be a manifestation of a whole market reflecting the acceptable portfolio chosen by investors within the efficient frontier. Markowitz (1952)[1] suggests that rational investors would choose minimum risk and maximum return in diversification and with any combination of weight the optimal portfolio lies on the efficient frontier.
This is such an excellent portfolio theory so that Sharpe (1964)[2], Lintner (1965)[3] and Mossin (1966)[4] further modelled it to include the risk free rate. As the efficient frontier only includes the portfolio of risky assets, a risk free asset with a zero risk return can be combined with risky assets added to the efficient portfolio and investors can then go beyond the frontier by borrowing and lending at risk free rate. A capital market line that used to linearly predict the market return can be drawn by lining up a point of a portfolio with only risk free rate to the other point that touches the efficient frontier, known as a tangency portfolio. The tangency portfolio explains that investors separate their decisions in investing and financing the investment as suggested by Tobin (1958)[5].
In a sense of linear prediction of an individual share return, CAPM is then modelled with a security market line, in which a share’s rate of return can be predicted given the market return, risk free rate and beta. The line is plotted by the expected rate of return of a share with its beta to the market return. The issue of benchmark index can be seen at this point. If there is a benchmark error, CAPM cannot estimate the correct beta and risk premium properly thus cannot calculated the expected rate of share return correctly.
Benchmark error
Benchmark error using CAPM for evaluating portfolio performance according to Roll (1980,1981)[6] can be seen in two ways when the market index produces an incorrect beta for the share and when it produces incorrect estimation for the market premium optimised to the risk free rate (see figure 1 and 2). The problem is not due to statistical variation but rather to the cause that the market index is not a good predictor of mean/variance efficient portfolio.
Figure-1. Incorrect beta
Figure-2. Incorrect market premium
A further study by Green (1986)[7] shows that benchmark errors are continuous behaviour and different for different indexes, thus, share or portfolio performance is sensitive to the choice of benchmark for the market index. We may now say that as beta assumed equals to one, the expected rate of return should be higher as we choose any benchmark that produces a higher market risk premium, and lower for any benchmark that produces a lower premium.
Benchmark errors are also considered in the context of global investment. Reilly and Akhtar (1995)[8] found that there is a variation of beta when using a domestic index, global index or a diversified global stock and bond portfolio. The beta of domestic equity index is lower than the world equity index and much larger for the diversified global stock and bond portfolio.
Market proxy, beta and risk premium problems
The root of benchmark error was fundamentally based on Roll’s critique (1977)[9] that found that market index is efficient per se, not for the individual shares or portfolios. Ross (1978)[10] also added that market proxy is not ex ante mean-variance efficient and individual preference in portfolio selection may be judged with a different market index and then will be penalised by share’s beta according to the different market index. Roll and Ross (1994)[11] found that the market proxy may be located within 22 bps below the efficient frontier (Figure-3).
Figure-3. Market proxy and efficient frontier
Using the true market proxy of the value weighted portfolios of all US
Figure-3. Beta inconsistency.
Figure-3 shows an inconsistency of beta in CAPM. Low beta shares that are predicted to have low returns are in fact too high whereas high beta shares that are predicted to have high returns are in fact too low, as seemingly the line rotates.
The inconsistency of beta was identified a decade before by Fama and French (1992)[13] as suggested a three-factor model by adding size and book to market ratio (B/M) to CAPM. They concluded that value shares with high dividend yield, high B/M, low P/E tend to have bigger expected return than growth shares with low dividend yield, low B/M, high P/E.
Again using the true market proxy and the three-factor CAPM, Fama and French (2006)[14] tested whether the value premium exists in CAPM pricing. The result is rejecting CAPM pricing for portfolio based on size, B/M and beta as concluding that size and B/M, not beta, reward the expected return. The evidence shows that expected return does not compensate beta variation unrelated with size and B/M for both small stocks and big stocks. We may say that if CAPM is fine a good benchmark index should contain both small stocks and big stocks.
Alpha
Market proxy distortion opens an opportunity to have a better portfolio performance than the market itself which was earlier suggested by Jensen (1969)[15] with alpha as a performance indicator: Alpha = Rs - Rf + Beta ( Rm – Rf). However, one can say that even a passive portfolio can beat the benchmark. Bowden (2000)[16] further argued that alpha relates to market timing and cannot be observed by conventional performance measures and suggested ordered mean difference (OMO) as an alternative measure, which is a function of a running mean of the difference between asset/portfolio and benchmark returns that is ordered by values of the benchmark.
Recently, Fama and French (2006)[17] discussed about a portable alpha as a way to add a portfolio consisting risk free rate and index funds with an additional hedge position that generates alpha. Moreover, an alternative indexing has been suggested by Arnott (2005)[18] in fundamental indexation to solve the distortion in value weighted index with some alternative weighting constraints, such as book value, cash flow, revenues, gross sales, gross dividend and total employment.
In order to see the impact of benchmark distortion in the New Zealand context, there are some market characteristics need to be considered (Bowden, 2005, p.133-168)[19], as follows, narrow true market proxy, large foreign capitalisation, high dividend yield, high risk free rate, low holding period return and low prospect dominance.
A consensus for share market proxy in NZ is the NZX-50 index that can be seen in every business page and news and the NZX-All for the true market proxy. However, there is a significant proportion of investment taking in the form of farming shares as the grass root of the NZ economy as a whole. For example, a rich Waikato Fonterra farmer may see a comparable investment choice between investing in Fonterra shares and NZX-50 shares, thus she needs a broader efficient frontier for the true market proxy other than just NZX-All. In fact, Fonterra only issues capital notes and none of its capitalisation in the share market.
The top ten capitalisations in the NZX-50 hold around 37.5% of foreign capitalisation. Since beta domestic equity market after influenced by beta foreign equity market may change, then the return of individual shares may be predicted below or above the original security market line.
High dividend yield and low P/E make NZ shares considered as value shares. Value shares may have a higher expected rate of return, thus a higher cost of capital, than growth shares according to the CAPM test by Fama and French (1992). If the CAPM holds, it is likely that the correct benchmark used would be the gross index, not the capital index which is widely used in the other market. A promotion to use the gross index to international investors may attract them to choose NZ equity.
High risk free rate may increase the cost of capital with a condition that there is no false market risk premium estimation in the benchmark. However, if benchmark error deviates to a lower market return then the market risk premium would be narrow and the security market line may rotate.
Low holding period return for NZ shares may be compensated by high risk free rate so that at some period the return of NZ shares is in fact is lower than government bond return. Under CAPM prediction, if NZ benchmark consists mainly of such shares, of course the market risk premium would be negative and the expected share return would be below the risk free rate.
The NZX-50 benchmark is dominated by larger companies with low growth prospect. That is why investors may have the opportunity of seeking alpha and beat the index with some shares like Michael Hill, Fisher and Paykel, Cavalier, Dorchester, etc. If tested with CAPM, there would be a big deviation from the predicted return of such shares.
Conclusion
CAPM assumes a major share market index as the best market proxy for efficient portfolio to predict the required rate of return. Despite the good theory background of mean (return) – variance (risk) efficient portfolio, the linear CAPM prediction using such proxy could be far from the reality of historical returns. As Fama and French (2004, p.44) suggested, “But we also warn students that despite its seductive simplicity, the CAPM’s empirical problems probably invalidate its use in applications”.
References
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