Validity of CAPM: Security Market Line (SML) can never predict Required Rate of Return for Equity even

Validity of CAPM: Security Market Line (SML) can never predict Required Rate of Return for Equity even if the Markets are Efficient A Simple Intuitive Explanation N Murugesan About the Author: Author is a MBA-CFA -Formerly working as Assistant Professor, VIT Business School, VIT University, Vellore, Tamil Nadu, India. The author can be reached at n_murugs@hotmail.com (mobile 9176553967) Electronic copy available at: http://ssrn.com/abstract=2265712

Abstract Many of the financial literature on empirical testing of validity of CAPM rely on one of the major assumptions behind the model the efficiency of markets and market portfolio. The generally accepted reason for continued usage of CAPM as valid asset pricing theory even though it fails in real world seems to be the conclusion that CAPM can not be practically tested empirically due to the problem of testing the efficiency of market portfolio and hence can neither be proved nor disproved (Richard Roll (1977)). In this paper, it is argued that even if the markets (or market portfolio) are efficient at all points of time, CAPM model will still be not able to predict asset prices. The SML relationship is a mathematical truism that will hold good in any efficient portfolio subject to some specific conditions. When these conditions are not applicable, then the SML relationship will not hold good. It is found that when we extend the mathematical property of SML to market portfolio (as defined by CAPM) for equilibrium conditions, some of the conditions do not hold good. This paper shows intuitively with simple illustration that time varying property of covariance matrix is implied aspect of the efficiency of the markets and directly implies that SML relationship will not hold good even when the market portfolio is efficient. Key words: CAPM, SML, Market Portfolio, Market Efficiency, Time Varying Covariances Electronic copy available at: http://ssrn.com/abstract=2265712

1. Introduction: The Capital Asset Pricing Model is widely used in Financial Text Books for explaining pricing of assets in an efficient paradigm. The Security Market Line (SML) Relationship is used for arriving at expected return for the stocks based on the concept of effect of efficient diversification and systematic risk. In practice, the ex-ante return arrived at using SML Model rarely has any semblance with actual return though these parameters are widely used for arriving at required rate of return equity for cost of capital of companies and for analysing performance of mutual funds. Many of the financial literature on this subject that tries to explain the non-utility or practicality of CAPM model relies on one of the major assumptions behind the model the efficiency of markets. Roll (1977) in his paper asserted that the asset pricing theory is not testable unless the exact composition of the true market portfolio is known. We can use a market proxy which has its own associated problems. The market proxy may be mean variance efficient even when true market portfolio is not mean variance efficient and alternatively the market proxy may be inefficient when market can be either efficient or truly inefficient. This implies that in the absence of testable market portfolio, we can neither prove nor disprove CAPM as an asset pricing theory for equilibrium conditions. In this paper, it is argued that even if the markets are efficient at any point of time, CAPM model will still be useless to predict asset prices. The paper first reviews the generally quoted reasons for the invalidity of CAPM to explain asset prices. The paper then focuses on three important factors in evaluating validity of CAPM: 1. Market Efficiency 2. Ex-Ante Expected Mean-Covariance Matrix and its dynamic nature 3. Constancy of Efficient weights and Rebalancing of an efficient portfolio It is shown next that the SML is a mathematical truism that applies for any efficient portfolio with an illustration of real world data. Later the mathematical property is extended to the market portfolio as would be defined by CAPM for equilibrium conditions and an analysis of its validity is done. This results into a conclusion that SML will not be applicable even when the market is efficient. 2. General Reasons on why CAPM is not valid: Any literature survey on the subject gives one or more of the reasons listed below for nonapplicability of CAPM: 1. Market Portfolio can not be tested for efficiency in practice This is widely accepted reason arising out of Roll s Critique (Roll, 1977). This has resulted into a conclusion that implies since CAPM can not be tested; it will prevail as long as an alternative asset pricing is formulated. 2. Assumptions underlying CAPM are too far fetched in reality There is no debate on CAPM assumptions being too unrealistic for the real world but such simplification of reality is often considered necessary to develop useful scientific models. In the case of CAPM, owing to its mathematical elegance and simple intuitive conclusions, it is generally believed by adherents of the theory that even if the assumptions are relaxed, CAPM would hold good though empirical testing again leads to point 1 above. Electronic copy available at: http://ssrn.com/abstract=2265712

Some of the limitations have been explained using different methodologies while others have resulted in Extended CAPM models e.g. Intertemporal CAPM (Merton, 1973), Consumption based CAPM (Breeden, 1979), Tax based CAPM (Brennan, 1970) etc. 3. CAPM is ex-ante whereas testing can only be done ex-post There is no methodology available to predict ex-ante variance-covariance matrix (or even beta for single index model). It is generally considered that betas are stable and historical betas can be considered for SML. Projecting ex-ante betas would amount to projecting the returns themselves (which may be simpler as they do not depend on other stocks in the portfolio) making SML itself redundant. Many of the empirical testing of CAPM assumes stability of beta. Some research in this area include Sharpe and Cooper (1972) and Miller and Scholes (1972) 4. Betas depend on period of returns and investment horizon and may not be constant: The entire efficient mathematics for optimization depends on the assumption that the expected mean return and Var-Covar (and hence Betas) are stable for the period for which optimization is done. In reality, the betas may not be constant for the duration of the investment horizon. As elaborated in previous point, it is generally assumed that betas are stable as it can not be predicted. The supporting argument usually is that the betas of diversified portfolios are stable. (Note: The betas of the stock are dependent on: a. Period for which returns are calculated (we can verify that daily/monthly betas are different for any stock) b. Investment horizon for which the expected mean returns and var-covar matrix are considered (the Var-Covar or Beta Matrix for the stocks vary depending on the investment horizon (say a quarter or a year). This is applicable whether we consider ex-ante or ex-post betas c. Any change in beta of one stock in an efficient portfolio affects beta of other stocks any change in variance-covariance of any stock affect the betas of other stocks of the market portfolio The above implies that to make an assumption that betas are stable implicitly assumes that the market participants are unanimous on all the above parameters which is highly impractical. If we go by the financial theories on market efficiency, historical returns (or volatility) are no indicator of future returns. It is not correct to assume that the historical betas can be used in place of ex-ante betas. As will be explained in this paper, if the betas are stable, then the efficient market portfolio can not be different on every day basis (e.g. if we consider the major index portfolio consisting of 80% market capitalization to reflect market as a whole, the weightages (i.e., the market capitalizations) of the index stock changes on every day basis. This implies ex-ante betas are changing)

Multivariate testing of CAPM considers the above factors. Some of the early research in this area includes Gibbons [1982] and Shanken [1985] Bollerslev (1988) shows that the conditional covariance of asset returns are strongly autoregressive and the empirical data rejects the assumption that the covariance matrix is constant over time. 5. Problems with assumptions of riskless lending and borrowing The assumption of unrestricted risk-free borrowing and lending is too unrealistic in reality. This has resulted in development of zero-beta CAPM model (Fischer Black, 1972) 6. When short sale restrictions are placed (no negative weights), SML is no longer theoretically valid When any sample market universe is optimized as per Markowitz paradigm, there are possibilities of negative weights for some of the stock in the portfolio. However, as per the CAPM definition of market portfolio, the market portfolio can not contain negative weights. When restrictions are placed for optimization, it leads to constrained optimization process where SML is no more applicable. Adherents of CAPM suggest non-linear SML models. Sharpe (1991) considers the above problem and concludes that SML will not be applicable as the relationship will not be linear when constrained optimization (no negative weights) is performed involving Kuhn-Tucker conditions. 7. Market may not be efficient If the market is not efficient, then CAPM will not hold good as the prices will not reflect their fundamental values resulting in inefficient weightage or market capitalizations. However, there is no point in any asset pricing theory if the markets are considered not efficient in equilibrium. As can be seen, the conclusions of Roll s Critique (Roll, 1977) have had major impact on the literature. The rest of the reasons have been explained away with different methodologies of testing and extension of models. This paper asserts that even if market portfolio is efficient, CAPM will not be valid. This is intuitively understandable when we realize that the betas can never be stable when the markets are efficient. In the next section, it is shown that the SML is based on sound mathematical properties basically a mathematical truism. Subsequently, the mathematical truism is analysed when extended to the equilibrium conditions of CAPM i.e. to market portfolio. It is found that some of the conditions which facilitate SML relationship no longer hold good when extended to market portfolio. 3. CAPM is a Mathematical Truism: In spite of various criticisms on the applicability and assumptions behind it, the CAPM is difficult to disprove mathematically as it is actually based on sound mathematical principles. Unlike what some of the practitioners who discredit CAPM for its non-practicality may think, CAPM is based on sound financial mathematics based on mean-variance optimization principles, and more importantly, for SML relationship to be applicable, not all of the CAPM assumptions are actually required to be valid.

The CAPM assumptions are meant only for market equilibrium conditions whereas SML relationship is a mathematical result. In other words, SML relationship will be applicable for any Optimal and Efficient Markowitz Portfolio. The SML Model can be stated as follows: For any efficient and optimal portfolio (the portfolio at the efficient frontier where the straight line from risk free rate meets it tangentially), the return on any individual stock, compared to other stocks in the portfolio, is directly proportional to the beta of that stock with regard to the portfolio returns The CAPM Model can be stated further as : Since the above mathematical principle is applicable for any efficient and optimal portfolio as stated above, if we assume that the market is efficient (i.e. if we consider the entire market as an efficient portfolio), as the stocks are priced in such a way that the portfolio yields maximum return possible per unit of risk, every stock return will be linearly related to its beta with regard to market portfolio returns (with slope being the difference between market return and risk free return and intercept the risk free return) Thus SML is a mathematical truism wherein if we are given a random set of stocks (or say index portfolio), we can construct an efficient and optimal portfolio using these stocks based on the given set of variance-covariance matrix and mean expected return. For such an efficient and optimal portfolio, SML will still be applicable i.e. the return on any stock within the portfolio will be linearly related to beta: Ri = Rf + (Rm Rf) β (Note that the beta above will obviously however refer to the stocks covariance with the returns of the sample given portfolio and not the entire market returns) An Illustration with SENSEX stocks: The real world illustration below with 10 SENSEX stocks of Bombay Stock Exchange shows an efficient and optimal portfolio and the resulting SML relationship: Consider the following SENSEX stocks with their initial price as at Jan 2011 and their exante expected mean monthly returns over the next two years (A subset of sensex stocks are considered for analysis and illustration with market data from Bombay Stock Exchange (BSE).) Month Bajaj Auto BHEL Bharti Airtel Cipla Coal India Dr Reddy GAIL HDFC Bank Hero Motocorp Hindalco Industries Price at Jan 2011 1249 2218 319 332 304 1624 471 2043 1631 229 Ex-ante Expected Returns 2.57% -5.79% 0.34% 1.19% 0.84% 0.63% -1.04% -0.97% 1.04% -1.95% Consider the following covariance matrix:

Variance-Covariance Matrix Bajaj Auto Bharat Heavy Bharti Airtel Cipla Coal India Dr Reddy GAIL HDFC BankHero MotocoHindalco Bajaj Auto 0.46% 0.07% 0.20% -0.02% 0.06% 0.06% 0.21% 0.12% 0.24% 0.32% Bharat Heavy Electricals 0.07% 3.43% 0.12% -0.05% 0.19% -0.47% 0.14% 0.40% -0.12% 0.31% Bharti Airtel 0.20% 0.12% 0.74% 0.10% 0.22% 0.08% 0.18% -0.10% 0.06% 0.32% Cipla -0.02% -0.05% 0.10% 0.45% 0.06% 0.08% 0.07% 0.42% -0.09% 0.14% Coal India 0.06% 0.19% 0.22% 0.06% 0.33% 0.04% 0.06% 0.28% 0.19% 0.20% Dr Reddy 0.06% -0.47% 0.08% 0.08% 0.04% 0.24% 0.08% 0.01% 0.07% 0.14% GAIL 0.21% 0.14% 0.18% 0.07% 0.06% 0.08% 0.34% -0.05% 0.09% 0.25% HDFC Bank 0.12% 0.40% -0.10% 0.42% 0.28% 0.01% -0.05% 3.25% 0.41% 0.49% Hero Motocorp 0.24% -0.12% 0.06% -0.09% 0.19% 0.07% 0.09% 0.41% 0.74% 0.19% Hindalco Industries 0.32% 0.31% 0.32% 0.14% 0.20% 0.14% 0.25% 0.49% 0.19% 0.98% (The mean-variance matrix is of course derived from ex-post stock returns. We assume that these have been predicted) The above mean expected returns and covariance matrix gives the following optimal efficient portfolio with the risk free rate of 5% p.a. Efficient Portfolio Weightages Bajaj Auto 358.89% Bharat Heavy Electricals -45.07% Bharti Airtel -54.83% Cipla 151.96% Coal India 168.47% Dr Reddy -52.69% GAIL -243.65% HDFC Bank -37.80% Hero Motocorp -36.13% Hindalco Industries -109.15% Total 100.00% For the above efficient weights, the portfolio returns would be as follows:

Period PF Returns Bajaj Auto Bharat HeaBharti AirteCipla Coal India Dr Reddy GAIL HDFC BankHero MotoHindalco Feb-11 48.19% 1.57% -9.78% 3.94% -9.80% 7.94% -4.76% -9.09% 0.34% -10.15% -12.47% Mar-11 32.32% 15.10% 3.01% 7.97% 7.12% 5.77% 5.91% 8.58% 14.31% 8.30% 3.91% Apr-11-2.94% 0.21% -2.92% 5.96% -3.80% 9.51% 1.45% 2.30% -2.15% 7.76% 3.31% May-11 15.63% -8.10% -2.83% -1.23% 5.59% 7.46% -2.77% -6.36% 4.18% 8.39% -8.56% Jun-11 17.95% 4.58% 5.27% 5.64% 1.30% -3.93% -5.13% -0.94% 4.79% 1.33% -8.17% Jul-11 28.18% 4.19% -10.17% 10.56% -6.81% -1.13% 3.53% 4.65% -80.52% -4.82% -6.96% Aug-11 50.66% 7.38% -3.84% -7.51% -9.05% -3.35% -5.90% -11.08% -3.12% 14.58% -10.72% Sep-11-3.10% -2.46% -7.39% -6.46% 0.96% -11.25% -0.87% 0.09% -1.05% -5.18% -12.67% Oct-11 63.23% 12.96% -80.59% 3.52% 4.19% -0.11% 12.87% 2.77% 4.67% 12.04% 3.85% Nov-11 45.51% -3.48% -11.22% -1.56% 11.34% -1.81% -5.79% -7.11% -9.73% -7.93% -10.05% Dec-11-7.20% -4.79% -15.31% -11.00% -2.56% -7.83% 0.21% -2.09% -3.26% -4.89% -5.63% Jan-12-6.44% 0.51% 4.83% 6.42% 9.26% 8.24% 6.96% -2.93% 14.95% -2.17% 26.70% Feb-12 21.62% 12.37% 23.01% -4.29% -9.27% 2.30% -2.91% 0.35% 5.48% 4.64% 1.36% Mar-12-7.92% -6.73% -16.63% -3.58% -3.87% 2.99% 7.32% 0.21% 0.43% 5.35% -12.92% Apr-12 38.09% -3.27% -12.47% -7.85% 2.40% 2.75% 0.08% -11.88% 4.26% 8.76% -6.84% May-12-15.69% -6.78% -5.76% -2.64% -0.71% -8.75% -3.93% -2.41% -6.69% -17.80% -3.23% Jun-12-8.11% 3.90% 9.51% 0.96% 2.18% 7.99% -2.45% 9.06% 11.37% 16.61% 2.74% Jul-12 27.39% 1.87% -7.07% -1.59% 7.00% 3.43% -2.11% 0.64% 4.32% -6.37% 0.00% Aug-12 45.51% 0.88% -0.63% -18.14% 11.70% -1.82% 4.38% -0.65% 1.22% -10.85% -13.47% Sep-12-0.90% 13.44% 15.16% 7.79% 0.65% 1.90% -2.24% 8.94% 5.65% 5.10% 16.14% Oct-12 8.07% -1.01% -8.81% 1.57% -4.49% -3.67% 6.65% -8.94% 0.93% -0.02% -3.36% Nov-12 30.72% 6.43% 3.67% 25.33% 13.89% 5.69% 3.76% 1.02% 10.89% -2.72% -0.09% Dec-12 20.37% 10.38% -2.12% -6.04% 0.02% -2.98% 0.31% 1.06% -3.53% 3.85% 12.25% Mean Returns 19.18% 2.57% -5.79% 0.34% 1.19% 0.84% 0.63% -1.04% -0.97% 1.04% -1.95% Variance 5.02% 0.46% 3.43% 0.74% 0.45% 0.33% 0.24% 0.34% 3.25% 0.74% 0.98% First Pass Regression: The regression of the stock returns with portfolio returns yield the following betas: Stock Beta Bajaj Auto 0.11 BHEL -0.33 Bharti Airtel 0.00 Cipla 0.04 Coal India 0.02 Dr Reddy 0.01 GAIL -0.08 HDFC Bank -0.07 Hero Motocorp 0.03 Hindalco -0.13 Portfolio 1 Second Pass Regression: The regression of betas with mean expected returns would give the following SML relationship which matches the actual values: SML Actual Slope 18.76% Rm-Rf =18.76 % Intercept 0.42% Rf = 0.42% The SML relationship for the portfolio would be: Note the following: Ri = 0.42% + 18.76% β

1. If we had taken any other ten stocks we would get a different portfolio and returns and a different SML relationship. In this case, most of the CAPM assumptions are not applicable. Any investor who performs the above optimization for his portfolio, would only need to bother about his own prediction on mean variance parameters. 2. If we consider the entire index scrips for optimization (and if SENSEX is a market portfolio proxy for Indian stock markets and CAPM equilibrium exists), then the SML relationship would be Market SML for Indian stock markets. Here all the CAPM assumptions should hold good. 4. Conditions for the validity of the mathematical property of SML: Among others, there are three major conditions for validity of SML s mathematical truism to be applicable for any portfolio: 1. Portfolio is efficient and optimal By this we mean that weightage of each stock is such that the returns of the stocks and portfolio are maximum per unit of risk among all possible portfolio combinations with those stocks for the given expected mean return and variances and a risk free asset. Note that every portfolio lying on the efficient frontier is efficient (above global minimum variance portfolio) but need not be optimal. An optimal and efficient portfolio is one that is tangential to the return of the risk free asset (or zero beta asset). 2. There can not be any restrictive conditions for efficient optimization. SML will not be satisfied for an efficient portfolio if the procedure involves constrained optimization. Any constrained optimization procedure like short sale restrictions could result in an efficient portfolio but SML will not necessarily hold in such a portfolio and SML will not be linear (Sharpe, 1991). For example, the market portfolio might result in some of the stocks having negative weights (short sale requirements) which are not a feasible solution for the complete market portfolio. If the portfolio is optimized to make all the weights greater than zero, the resulting portfolio will not abide by SML. For the market as a whole, there can not be any negative weights for any constituent of the market portfolio. But Markowitz optimization procedure (or single index model) need not necessarily arrive at a portfolio where there are no negative weights. Given below are results of constrained optimization for the illustrative portfolio: Efficient Portfolio Weightages Bajaj Auto 71.69% Bharat Heavy Electricals 0.00% Bharti Airtel 0.00% Cipla 28.27% Coal India 0.04% Dr Reddy 0.00% GAIL 0.00%

HDFC Bank 0.00% Hero Motocorp 0.00% Hindalco Industries 0.00% Total 100.00% Portfolio Return 2.18% Sharpe Ratio 34.308% The SML relationship for the constrained optimization turns out to be as follows: Ri = 1.65% + 2.28 % β The actual risk premium is (Rm Rf) = 2.18 % - 0.42 % = 1.76%. As expected, the SML relationship does not hold good for constrained optimization. Note that this condition is necessary only for CAPM to be applicable for the market - the individual investor need not have any short sale restrictions. 3. The mean-variance parameters are constant and the efficient portfolio is rebalanced periodically. For the SML to hold good, for a given constant mean-variance parameters, the portfolio should be balanced periodically so that the efficient weights are maintained constant for the investment horizon. The constantly rebalanced efficient portfolio gives the maximum return per unit of risk among all possible portfolios compared to the same efficient portfolio that was held constant during the investment horizon. Note that this is similar to the rebalancing of index mutual funds who rebalances their portfolio periodically as per index market capitalizations to mimic index returns but the capitalizations (or efficient weights if the index is efficient) are different for each period. But for thee SML relationship to hold good the efficient weights should be constant. 5. The Rebalancing of Portfolios: The SML model assumes that the mean-variance-covariance parameters are constant for the investment horizon. This means if the stock prices of the portfolio changes, then the investor is required to rebalance the portfolio in such a way that the weightages of individual stocks will still represent optimal and efficient weights (with no change in expected var-covar matrix and mean returns, this would mean buying and selling stocks in such a way that the weightages of each stock in the portfolio remains same). The following illustration shows the effect of rebalancing. The following table compares the portfolio returns with buy and hold strategy to show the magnitude of difference:

Period PF Value Bajaj Auto BHEL Bharti AirtelCipla Coal India Dr Reddy GAIL HDFC Bank Hero Motoc Hindalco Prices on Jan 11 1249 2218 319 332 304 1624 471 2043 1631 229 Eff Weights 359% -45% -55% 152% 168% -53% -244% -38% -36% -109% PF Value 10,00,000 35,88,925-4,50,706-5,48,278 15,19,555 16,84,690-5,26,894-24,36,520-3,77,969-3,61,275-10,91,528 Prices on Dec 12 2131 228.4 316.8 414.1 355.05 1828.5 356.1 678.8 1898.35 130.6 Buy and Hold 5792170 6125014-46422 -545266 1893898 1967596-593078 -1841937-125592 -420623-621419 PF Mean Monthly Returns (Buy and Hold) 7.05% PF Mean Monthly Returns as per Markowitz 19.18% The difference in portfolio returns between buy and hold and Markowitz paradigm is due to the rebalancing of portfolio to maintain the efficient weights - Given below is an illustration for the first period: Period PF Value Bajaj Auto BHEL Bharti Airtel Cipla Coal India Dr Reddy GAIL HDFC Bank Hero MotocoHindalco Prices on Jan 11 1249 2218 319 332 304 1624 471 2043 1631 229 Eff Weights 359% -45% -55% 152% 168% -53% -244% -38% -36% -109% PF Value 10,00,000 35,88,925-4,50,706-5,48,278 15,19,555 16,84,690-5,26,894-24,36,520-3,77,969-3,61,275-10,91,528 Prices on Feb 11 1268.3 2000.65 331.1 299.7 328.15 1547.05 428.25 2049.7 1464.95 200.8 % Change (Return) 1.57% -9.78% 3.94% -9.80% 7.94% -4.76% -9.09% 0.34% -10.15% -12.47% PF Value 14,81,905 36,45,404-4,06,632-5,69,879 13,70,687 18,18,523-5,01,789-22,15,135-3,79,237-3,24,593-9,55,444 New Weights 246% -27% -38% 92% 123% -34% -149% -26% -22% -64% Rebalancing Action ((+) Inflow /sell and (-) Outflow/buy ) 56479 44075-21601 -148868 133833 25105 221384-1267 36681 136084 New PF Value 10,00,000 3588925-450706 -548278 1519555 1684690-526894 -2436520-377969 -361275-1091528 New PF Weights 359% -45% -55% 152% 168% -53% -244% -38% -36% -109% PF Return for the period 48.19% Alternatively, this portfolio return for the month of Jan 2011 is nothing but same as the product of stock returns and portfolio weights for the period. The importance of this portfolio returns of 48.19% for the period while keeping the efficient weights constant is its implication for calculation of beta. The beta of each stock is arrived at based on regression of these portfolio returns with the return of each individual stock for the period resulting in the efficient weights remaining same for each stock. Note that the prices of stocks have changed while mean expected return and covariances are expected to remain as before till end of the investment horizon along with efficient weights. This has major implication for market portfolio - we shall come back to this when we explain the implications of rebalancing for market portfolio. 6. A Digression - Market Efficiency and Asset Pricing: Consider any such optimal and efficient portfolio as described in previous section (not equilibrium market portfolio) where SML is applicable. In such an efficient portfolio, is it necessary that stocks are correctly valued? The SML - Markowitz efficient paradigm actually does not bother about whether the stocks are correctly priced or not for any optimal and efficient portfolio as long as we could predict the mean return for the investment horizon and the co-variances. (We shall consider market portfolio subsequently where they should necessarily be priced right). The SML mathematical property will be applicable irrespective of whether the stocks are rightly priced or not. If a stock that has been included in the portfolio is undervalued and if the investor expects that the stock is expected to be priced right by the investment horizon,

and then his expected mean return would be higher to that extent for that stock (compared to other investor who might also have similar var-covar expectations). The SML will hold good for any efficient portfolio whether the portfolio stocks were correctly valued or not when optimization is done as long as the mean expected returns and beta are predicted correctly. This also means the SML property will be applicable whether the market as a whole is undervalued (bearish bottom) or overvalued (bullish top) as long as the investors projections about mean return and betas materializes. The implications are different for when we consider CAPM i.e. for market equilibrium conditions. For every investor to hold a fraction of market portfolio, all investors are supposed to have same mean-variance expectations. In this case, again, market will considered to be rightly valued whether the market is in bullish or bearish period basically the stocks are rightly priced in relative terms. 7. Extending the SML Mathematical Truism to the Market Portfolio: As explained earlier, CAPM relies on the mathematical property of SML by extending the concept to the entire market. If the entire market is efficient and if each stock is priced such that the market portfolio gives maximum return per unit of risk, then SML will be applicable for the entire market. Then we can project the expected return for any stock based on its beta as per SML equation. Assuming that the market portfolio is efficient at any instance as per Markowitz meanvariance paradigm, then the market capitalization of each stock at that instance represents the optimal weight that would give maximum return per unit of risk for the market portfolio. (Note that everybody need not hold the market portfolio it is sufficient if every body holds a fraction of the market portfolio where the constituent stocks are in optimal weights as per the market expected mean-covariance matrix. If the returns of any stock in such partial market portfolio are regressed with market returns, SML will still hold good.) The first condition for the mathematical truism is applicable for the market if we conclude that the market is always efficient. The second condition is that the optimization can not have any restrictive conditions. In all sample universe optimizations, we always encounter negative weights for some of the market constituents. This is obviously not feasible for market portfolio. As explained earlier, short sale restrictions do not allow SML relationship to hold (sharpe, 1991). However, while we can say that SML will not hold good when there are restrictions on negative weights, we can not say that an efficient and optimal portfolio will necessarily have some negative weights markets as a whole does not have negative weights. Hence we shall ignore this condition in the absence of identifiable market portfolio for testing. The third condition is that constant mean-variance parameters and the implication of rebalancing of portfolios for efficient weights. As explained earlier, the portfolio needs to be rebalanced so that the efficient weights as per mean-variance matrix are maintained constant. When we extend this condition to market portfolio, the market portfolio returns should be considered as a result of similar portfolio rebalancing illustrated earlier. The betas are supposedly the regression of stock returns with these portfolio returns. The important point to

note is that the variance-covariance expectations remains same when we calculate betas and as a result, the efficient weights are also supposedly constant for the investment horizon. For market portfolio, the market capitalizations are nothing but efficient weights for an efficient market that is in equilibrium as per CAPM. If all the investors hold a fraction of market portfolio and if they have same investment horizon and expectations, then they would all try to maintain their efficient weights as per Markowitz optimization. However, in reality we find that the market capitalizations change on daily basis. This implies for all the investors the efficient weights are changing on daily basis. In this scenario, for SML to be applicable, this would only mean that the expected return and var-covar matrix for the market itself is undergoing changes on a periodic basis - if not so, it would mean the market is inefficient the changing market capitalizations themselves are efficient as the composition of efficient portfolio changes. Thus either the market is not rebalancing as per mean-variance parameter or the mean-variance parameters themselves are changing. Based on the above discussion, we can conclude that SML will not hold good whether the market is efficient or not. This is further elaborated for each case: 1. Market is not efficient Market is not rebalancing in such a way to yield maximum return per unit of risk for the market portfolio for the ex-ante mean-variance parameters applicable for the stocks. 2. Market is efficient Market is rebalancing as per the changes in mean-variance expectations of the market participants. Case 1: Market is not efficient: As explained earlier, market need not be efficient for SML to hold good for any individual portfolio because of the mathematical truism. However, market needs to be efficient for SML to be applicable for the entire market or for CAPM equilibrium conditions to prevail for the entire market. If the market is not efficient, CAPM will not be found empirically valid as SML will not hold good the mathematical property will not be applicable as the market portfolio will not be optimal and efficient (first condition explained earlier) when the market is not efficient. Case 2: Market is efficient: If the market is efficient and if the market capitalizations change on daily basis, then the market capitalizations themselves should represent efficient weights on each of these periods. There are two possibilities when market capitalizations change: 1. The market capitalization changes but weightage of the stocks in the market portfolio remains same after rebalancing by investors. In this case, overall total market capitalization itself should change.

2. The changes in market capitalizations results in changes in efficient weights of each of the stocks in the market portfolio 1. Changes in Market Capitalizations do not change weightages: If this is true, then the applicability of CAPM and SML relationship will not be affected for the market portfolio. However, we can readily verify with any proxy for market portfolio that the weightages of stocks themselves are changing when the market capitalizations are changing. This will be considered as part of the next case. For the market portfolio as a whole which is not identifiable, we can argue that though capitalizations are changing, the weightages remain same in equilibrium as per the common mean-variance expectations of the market participants and the same can not be proved. But there is a flaw in this argument for the assumptions of the CAPM would imply that the constancy of efficient weights would imply constancy of market capitalizations. With the number of shares being constant, when the market participants try to rebalance their portfolio, everybody will initiate similar actions buy and sell same stocks for the same proportions. This should result in stock prices going back to their original prices (fair price) resulting in same market portfolio unless otherwise there are changes in mean-variance expectations which would change in the efficient weights themselves. We shall illustrate the above scenario with three asset market universe. We consider three stocks from the earlier illustration and the Markowitz Optimization for the same data results in following: Stock PF Eff Weights Beta Mean Returns Bajaj Auto 71.68% 1.22294 2.57% Cipla 28.24% 0.43619 1.19% Coal India 0.08% 0.24079 0.84% Portfolio 100.00% 1.00000 2.18% If we consider the above three assets as entire market portfolio, the rebalancing after the first period will result in following:

Period PF Value Bajaj Auto Cipla Coal India Prices on Jan 11 1249 332 304 Eff Weights 71.68% 28.24% 0.08% PF Value 10,00,000 7,16,825 2,82,409 766 Prices on Feb 11 1268.3 299.7 328.15 % Change (Return) 1.57% -9.80% 7.94% PF Value 9,83,674 7,28,106 2,54,742 827 New Weights 74% 26% 0% Rebalancing Action ((+) Inflow /sell and (-) Outflow/buy ) 11281-27667 61 New PF Value 10,00,000 716825 282409 766 New PF Weights 71.68% 28.24% 0.08% PF Return for the period -1.63% No. of Shares 574 850 3 Market Cap on Jan 11 1000000 716825 282409 766 Market Weights 71.68% 28.24% 0.08% Market Cap on Feb 11 983674 728106 254742 827 Market Weights 74.02% 25.90% 0.08% Now, what happens when the market participants try to rebalance after the first period? Every investor will try to sell 1.57% of Bajaj Auto, buy 9.8 % of Cipla and sell 7.94% of Coal India in order retain the efficient portfolio weights. In the process, the market will regain its initial position without resulting in any transaction at all for the period unless otherwise the mean variance expectations has changed in which case everybody can not hold market portfolio. The above result will also be applicable for n-asset market universe. Hence we can conclude that if the prices changes and the investors retain efficient weights as per CAPM assumptions, then the weightages of each of the stock of the market portfolio has to change else the equilibrium position continues without any change in prices. The logic here is same as that used for deriving CAPM theory itself. 2. Changes in Market Capitalization changes weightages of market portfolio In this scenario, we have found earlier that for SML to hold good for any efficient and optimal portfolio, the investors have to maintain their efficient weights based on meanvariance parameters. If the efficient weights themselves are changing, then the investors expectations about mean-variance parameters are also changing on daily basis. This violates the third condition and results in failure of SML even though market is efficient. When the market weights of stocks of market portfolio is changing, in the terminology of Markowitz paradigm, the mean-variance expectations of the market participants itself is changing on daily basis based on various market, economic and fundamental developments pertaining to the economy and stocks. So, a rational investor would change his investment weights based on the changed mean-variance expectations if he follows the Markowitz s theory of efficient frontier in order to attain maximum return per unit of risk possible. As the investors changes the investment weights, the market changes but remains efficient as it rebalances the market capitalizations as per the expectations of market participants. The market portfolio will be different for everyday but the market will still be efficient.

Thus the market may be efficient with changes in mean-variance parameters but SML relationship will not hold good as the mathematical property will no longer apply (though market portfolio may be efficient an every instant). This above discussion directly implies that the betas of the stocks are changing periodically. Now, it is obvious that when we calculate the ex-post betas of the stocks, they would not satisfy the SML. This is because the betas themselves were changing on periodic basis and the return would not have any relationship with the ex-post beta for the period. Moreover, it is impossible to arrive at ex-ante beta when the expected variance-co variances are changing constantly. But we can take a historical return data and argue that the betas have actually remained constant and whoever have predicted it could have achieved an efficient portfolio with maximum return (irrespective of market being efficient during the period or not). If this so, the ex-post SML should hold good. In reality, this is never the case. Thus though we can argue that the betas have remained constant and the portfolio returns were as per CAPM predictions, in reality the market returns were due to changes in mean-variance frontier itself. If we conclude that the market is not efficient (the changes in capitalizations are not due to actual changes in mean-variance matrix) and if any investor could project the actual ex-ante beta and mean return for the investment horizon (in the above example), he would of course get the maximum return per unit of risk and SML will be satisfied for his portfolio returns. However, this will not be applicable for the entire market when the market is not efficient and market wide SML / CAPM will have no meaning as they would not be able to project the asset prices. Conclusion: The widely accepted Roll s Critique claims that testing CAPM is same as testing efficiency of market which is not practically feasible. This paper shows that CAPM will not be applicable even if the market portfolio is efficient. This is because if the market is efficient, then the changes in market capitalizations (and hence efficient weights) amount to changes in expectations of mean-variance frontier in which case SML will not hold good as it violates the conditions of constant mean-variance frontier for the investment horizon. Hence any expost CAPM testing will always result in non-applicability of SML while ex-ante predictions of betas are practically infeasible. References: Black, Fischer, Michael C. Jensen and Myron Scholes. (1972) The Capital Asset Pricing Model: Some Empirical Tests, in Studies in the Theory of Capital Markets. Michael C. Jensen, ed. New York: Praeger, pp. 79-121. Benninga, Simon, Financial Modelling, 3 rd Edition, MIT Press Bollerslev, Tim, Robert F. Engle, and Jeffrey M. Wooldridge, (1988), A capital asset pricing model with time varying covariances,, The Journal of Political Economy, Vol. 96, No. 1. (1988), pp. 116-131

Markowitz, Harry. (1952) Portfolio Selection. Journal of Finance. 7:1, pp. 77-99. Roll, Richard W. (1977) A Critique of Asset Pricing Theory s Tests, Part 1: On Past and Potential Testability of the Theory. Journal of Financial Economics 4: 129 176. Roll, Richard W., and Stephen A. Ross (1994) On the Cross Sectional Relation between Expected Returns and Betas. Journal of Financial Economics 119: 1101 121. Sharpe, William F. (1964). Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. Journal of Finance. 19:3, pp. 425-442. Sharpe, W. F. (1991), Capital Asset Prices with and without Negative Holdings. The Journal of Finance, 46: 489 509. Sharpe, W.F. and G.M. Cooper (1972): Risk-return classes of New York Stock Exchange common stocks, 1931-1967. Financial Analysts Journal (March-April), 46-54, 81.