Systematic Risk and Corporate Business Performance

,


Introduction
The present paper investigates the relation between systematic risk and its underlying determinants.Systematic or non-diversifiable risk derives from CAPM theory, developed mainly by Sharpe (1964) and Lintner (1965a).CAPM identifies a primary relation between the stock excess return, compared to the risk-free asset return, and the stock market excess return through a variable named β, specific to the common stock, which measures its risk compared to the market portfolio, even if at the outset such a risk was related only to the portfolio to be optimised.
From the ensuing CAPM developments, numerous studies have examined the empirical evidence, the theoretical implications, and multiple practical applications related to the cost of equity and the impact of the capital structure on systematic risk and capital budgeting, to list a few.
CAPM is undoubtedly the most famous and used corporate financial theory, but it has always generated heated debate between its proponents and detractors.It gave rise to doubts about its validity as the empirical evidence did not fully support its theoretical conclusions (Jensen, 1972).Other doubts arise from the absence or tenuousness of the link between β and its underlying determinants, primarily related to corporate business since the relation with the capital structure is due to Hamada (1972) and Rubinstein (1973).The essay by Mandelker et al. (1984) represents the only significant exception.The present paper builds on this essay and attempts to verify the relation between the degree of operative and financial leverage on one side and the systematic risk and stock return on the other.
The paper presents three main Sections, each divided into several subsections.Section 2 defines the degree of operative and financial leverage and recalls their importance as measures of corporate performance.The contribution by Mandelker et al. (1984) is analysed, together with the approach by Miller et al. (1961), in determining the relation between corporate performance and stock values and returns.
short sales and the inability to explain a significant fraction of the stock return variability.Such topics offer the opportunity to propose an alternative approach to explain stock returns determined by the stock market's and corporate business's joint performance.The portfolio risk originates from the mutual combination of the underlying corporate businesses, filtered by the stock market feedback effect, highlighting the correlation between these two explanatory variables, a condition entirely ignored by CAPM.From this perspective, CAPM could prove to be an incomplete theory of stock returns, considering that returns are assumed to be exogenous data.
Section 4 analyses CAPM's asset allocation and security market line using 100 stocks of the S&P 500 Index in the 1991-2020 timespan without using homogeneous stock portfolios.Subsequently, we shall conduct an integration test of the joint performance of the stock market and the corporate business to evaluate the goodness-in-fit of an MLR relation in explaining the stock return variability.
Conclusions follow in Section 5.

The Relation between the DOL-DFL Nexus and Corporate Performance
What role do DOL and DFL play in explaining corporate performance?If we define the degree of operating and financial leverage, namely DOL and DFL, in the following way: (1) where π states the corporate net profit, Δ%S t and Δ%EBIT t represent the percentage change of Revenue and EBIT between two consecutive periods.
For more details on the definitions, determinants, and impact of DOL and DFL, please refer in full to the essays by Paganini (2019Paganini ( , 2021)).

The Mandelker-Ghon Rhee Equation
The essay by Mandelker et al. (1984) represents a starting point for deriving a link between the risky asset β and the underlying corporate business.Starting from the β classic definition as the ratio over a given timespan between the covariance Cov(R i , R P ) of common stock and portfolio returns on one side and the portfolio variance   2 on the other, through a series of algebraic steps, the authors determine its equivalence with two measures of corporate performance, DOL and DFL and an intrinsic β; the equation is the following: where β   definition is the following: This conclusion is adopted in a different form by Fama et al. (2015) in their paper "Five-factor asset pricing model".
The other conclusion Miller et al. (1961) reached is the role of dividends: the dividend is considered a financial illusion.Therefore, the value of the firm and its common stock are determined exclusively by business considerations and not by the method of packaging and distributing the fruits deriving from the income capacity.What matters is the income capacity of corporate assets and its investment policy.Therefore, these factors are reflected in the stock return, even if not easily visible.
At this point, it is evident that the corporate value is determined, in summary, by extracting net profit from the current and future assets, the latter determined by the investment policy.We can identify these factors in equation (3 bis), which describes the corporate economic and financial dynamics.Such dynamics depend on the initial situation, represented by the net profit π t-1 in the previous period t-1, and on how the business evolves in period t, based on the sales growth, DOL and DFL.The Modigliani-Miller equation ( 8) provides the fundamental insight that stock return and net profit are interdependent.
It would be very intriguing and powerful to link corporate performance to systematic risk analytically, but the characteristics of the equations examined up to now do not allow for an explicit, simple, and linear relation among stock return, systematic risk, and corporate performance.

The Dilution
Another element that plays a crucial role in determining the profitability of a common stock, which influences EPS, is the number of shares outstanding or dilution.This element operates exclusively at the price and return level of the common stock.It is possible that when the corporate daily management fails to produce an EPS level considered satisfactory, the Board of Directors could use the dilution to achieve the EPS target, provided the financial resources are available without any legal obstacle to implementing such a program.After all, a temporary reduction in the outstanding shares could also be a good business for the firm.
Moving from net profit to EPS, equation (3 bis) changes to the following form to consider the dilution: EPS t = EPS t-1 * (1+DOL t * DOL t * %S t ) 1+%N t (3 ter) where %N t is the percentage change in the shares outstanding between periods t-1 and t.
A decrease in dilution leads to an increase in EPS, other conditions being equal; the reduction of the outstanding shares certainly impacts DFL, partially offsetting the dilution effect and making the equity contraction perhaps less favourable.

CAPM Based on Lintner and Merton Contributions
The two seminal papers by Sharpe (1964) and Lintner (1965a) laid the foundation for the CAPM.In particular, the latter allows us to systematize simultaneously and analytically the following topics: 1) the role of short sales; 2) the role of risk-free assets; 3) the optimal mix of investments in risky assets, with and without short sales; 4) the portfolio risk; 5) the risky asset contribution to the portfolio risk.
The essay is much richer in additional information than those mentioned above, such as the market price implications of portfolio optimisation, corporate capital budgeting, and corporate project portfolio optimisation.
The optimal portfolio investment mix in risky assets is determined, in the mean-standard deviation plane, through a tangent line to the efficient frontier of the portfolio of risky assets and intercept on the ordinates equal to the return of the risk-free asset R F ̅̅̅ .The procedure followed by Lintner maximizes the slope θ of the market opportunity line by setting its partial derivative to the weight assumed by the i th risky asset equal to zero.We do not know whether Lintner realized that the efficient frontier of the portfolio of risky assets in the mean-standard deviation plane was a shifted hyperbola: Lintner (1965b) stated such a frontier as an envelope.The analytical determination of this frontier was developed by Merton (1972) and published about seven years after Lintner 's paper.The critical difference between Lintner's and Merton's approach concerns the valuation of short sales: for Lintner, going short on a risky asset is an investment like going long, while for Merton, a short position needs offsetting with more long positions.In fact, for Lintner, the sum of the percentage weights of investments in individual risky assets is equal to 1 only if assumed in absolute value, as in equation ( 9), while it is always equal to 1 for Merton, as in equation (10): =1 (10) The immediate consequence of such a difference could involve a translation of Lintner's conic section compared to Merton's towards the southwest along the market opportunity line with slope θ and intercept R F ̅̅̅ .The translation along the abscissa is equal to: 11) where: Mx PT = point of tangency of the market opportunity line with Merton's conic section; Lw i = Lintner's weight in the i th risky asset.
With short sales, the sum of Lintner's weights Lw i is always less than one, and consequently, the translation  on the abscissa of Lintner's point of tangency is negative.The market opportunity line, with intercept R F ̅̅̅ and slope θ, will have the following equation: where the slope θ is: Matrices and vectors are in boldface, while the superscript T indicates the transposed matrix/vector.Merton's conic section linking the return R P to the standard deviation  P of the portfolio of m risky assets in the mean-standard deviation plane is the following: where: = minimum portfolio return, equal to the A/C ratio (minimum of the efficient frontier) The equation ( 17), with only the positive radical, represents the portfolio efficient frontier.
We can locate the point of tangency of Merton's conic section with the market opportunity line as a simple geometric problem of a line passing through R F ̅̅̅ tangent to a given conic.In contrast, the conic descends from an optimisation procedure to minimize the portfolio variance as a measure of risk.The point of tangency Mx PT of Merton's conic with the market opportunity line is equal to the standard deviation resulting from the following equation: The points of tangency of Merton's and Lintner's conics overlap if there are no short sales; they differ in the opposite case, but both are tangent to the same straight line.Such conics are hyperbolas with different geometric centres but the same shape.The matrix representation of conic sections is the reference for more details.
The geometric translation of Lintner's hyperbola is due to the peculiar assessment adopted with equation ( 9) for short positions.This difference has no significant impact if we can borrow or lend at the same interest rate equal to R F ̅̅̅ .The investor can choose the final mix of his portfolio between risky and risk-free assets, determining his position on the market opportunity line based on personal preferences or utility curves.Since Lintner 's optimal portfolio positions are southwest of Merton's, the former portfolio may generate more debt than the latter, notwithstanding its overinvestment required by the presence of short positions.Lintner's assessment of short sales seems orthodox from an economic perspective, and it should be more advisable than the Merton solution, which is algebraically simpler due to relation (10).
If we need to get Merton's weights directly (Nocedal et al., 1999), we must set the following system of equations: where:  = m x m covariance matrix of risky asset returns A = 2 x m Jacobian matrix of constraints (risky asset returns and unit weights) Mw = column vector of m Merton's weights   = column vector of the 2 Lagrangians Z = zero column vector of m first-order necessary condition R P = scalar of the target portfolio return 1 = scalar of the sum of the stock weights in the portfolio From this system of equations, we can obtain Merton's weights Mw, and the Lagrangians   by solving: Using equations ( 12), (13), and (18), we obtain the optimal portfolio return R M , given the risk-free rate R F ̅̅̅ .By entering the R M value as the R P target, we get Merton's weights of the optimal portfolio, while by inserting any other figure, we obtain the corresponding mix of the portfolio frontier, efficient or not.
Interestingly, Lintner (1965a) does not use the β definition in his paper, resorting to an alternative measure λ, defined below: The subscript M to the portfolio return and variance indicates that it is optimised.We get used to the following formulation: Lintner limits himself to writing the first three members of the following equation that correspond, after a series of transformations, to the fifth member, undoubtedly equivalent to the second member of (21): The relation (21 bis) is the necessary and sufficient condition to obtain the weights   that guarantee a single solution at the maximum of θ to   .Lintner interprets the riskiness of an asset within a portfolio based on its variance and covariance with all the other stocks in the portfolio, not based on the standard deviation of its returns.Therefore, λ represents the return/risk required by the investor to maintain a position on the i th asset within the portfolio for any stock, given its risk represented by Cov(R i , R M ); such a risk, therefore, changes according to which stocks are held in the portfolio since it is not an absolute measure of asset riskiness.The required return/risk λ to keep an asset is the same for all stocks in the portfolio but is conceptually different from θ that determines the investment size in the optimal portfolio and risk-free assets.
For reasons we will explain in subsection 4.6, relation ( 21) is exclusively valid ex-post; therefore, we will never use the expression expected returns.
The CAPM standard formulation is extremely assertive in believing that the return of a risky asset is due to its risk profile measured by β.In fact, from (21), we see that the stock excess return is commensurate with the optimal portfolio excess return, considered equivalent to the market portfolio (Fama et al., 1973), through a specific β.This armoured relation does not leave much room for the fundamental determinants of the common stock risk.It is worth reiterating that ( 21) is a portfolio equilibrium condition of a non-deterministic nature.

Flaws in CAPM Theoretical Framework
CAPM standard configuration presents some theoretical inconsistencies.The most significant is the role of short positions within the optimal portfolio.Lintner notices this inconsistency and identifies an alternative solution, constraining the portfolio positions to be exclusively equal to or greater than zero through the KKT condition (Kuhn et al., 2013;Nocedal et al., 1999;Ghojogh et al., 2021).However, such a solution may be suboptimal compared to the optimised portfolio.How do we reconcile the optimised portfolio that needs short sales with the market portfolio that only has long positions?Now, as the number m of risky assets in the portfolio increases, two crucial phenomena occur: 1) the short positions progressively increase towards 50%; 2) some stock weights become extreme, both in long and short positions.Levy et al. (2001) treat such an issue theoretically, concluding that the characteristic that makes an asset good in a sizeable portfolio, even with only 100 risky assets, is not quickly evident.The negative weights that generate short sales depend on the values assumed by z  , that is, the sum of all the stock excess returns multiplied by the corresponding element v  deriving from the inverse of the covariance matrix: Given that   is positive and common to all assets, even assuming that all excess returns are positive, a negative value of    in the optimal portfolio depends on the v  values in the inverse matrix in correspondence with the asset column considered.If the sum of these values based on the excess returns of all m stocks in the portfolio is negative, then we have a short sale.Thus, the specific characteristics of the risky asset do not necessarily determine its positive or negative weight, mainly depending on the property of the inverse matrix.The particular asset combination determines the covariance matrix, and its inverse establishes the weight sign.
Furthermore, we must consider that each row of the covariance matrix is orthonormal to each column of its inverse matrix and vice versa.Given that its product is 1 when i=j and 0 when i≠j, even starting from a matrix with positive covariances, it is inevitable that many v  elements outside the main diagonal are negative, determining the almost automatic presence of negative weights originating short sales.As the number of common stocks in the portfolio increases, the appearance of negative v  is physiological as the number of elements below the main diagonal (while those above the diagonal are the transposition of those below, being the matrix symmetric) is preponderantly compared to those on the diagonal.Such a property seems unrelated to any economic explanation of what stocks are short-sold.
If we wish to go deeper into the topic, it would be necessary to examine the essay by Stevens (1995) that takes its cue from Anderson et al. (1981).All the elements of each specific row of the inverse matrix are the ratio between the same denominator   2 (1 −   ) and a numerator based on −β  when i≠j and 1 when i=j; through a few algebraic steps, (22) becomes the following: where: =  2 of the multiple regression between the i th asset and the other n-1 assets  T =  .T  −1 −1 = vector of multiple regression coefficients β  of the i th asset to the other n-1 assets  −1 −1 = inverse matrix of the n-1 assets obtained by discarding the row and column containing the i th asset  .1 = column vector of the n-1 covariances obtained by discarding the variance   2 of the i th asset From (22 bis), we see that the optimal weight    will be positive only when the excess return of the i th asset exceeds the mean of the excess returns of the other n-1 assets weighted on the specific multiple regression coefficients β  between the i th asset and the other n-1 assets: The result obtained with (22 bis) allows us to understand the sign of the weight in the optimal portfolio: if the stock excess return is not high enough and is correlated positively with the other n-1 stocks with high excess returns through a high β  coefficient, then its weight could be negative.Conversely, a low-excess return asset negatively correlated to the other n-1 stocks could have a positive weight.Lintner (1965a) also reaches the same conclusion logically.In short, the positive correlation between two common stocks, high excess return and a high β  coefficient to the i th asset leads to a decrease in the optimal weight    of the i th stock.If the number of combinations of this kind is high enough, the weight will become negative, and such a change of state occurs faster the lower the excess return of the i th asset is.
Eventually, Levy et al. (2001) observe that the Sharpe ratio tends to halve by banning short sales, implying a high implicit cost for the investor.
The price adjustment process of risky assets is not understandable before a significant discrepancy between the weight assumed in an optimal and the market portfolio, if not a generic down or upward pressure for excess/deficient assets held compared to the optimal portfolio.We will see the implications of asset pricing with empirical evidence.
This perspective leads to a further consideration: Lintner's analysis of the optimal portfolio concerns m assets with m, which need not necessarily tend to infinity.The number of common stocks does not necessarily have to equal the market portfolio.If we limit ourselves to an analysis of m risky assets of which we know the returns, variances and covariances deriving from their time series, we obtain some critical information: 1) the risk-free return R F ̅̅̅ is a datum of the moment in which we carry out the ex-post analysis of the times series; it is an element not entirely extraneous to the computation of β as long as R M is the return on the optimised portfolio of m risky assets.
2) The β weighted mean of the m risky assets always equals 1.
3) The α weighted mean of the m risky assets always equals 0.
4) The regression  2 of each stock against the optimal portfolio build with the same m assets is relatively low.
5) The t-stat measurements confirm the null hypothesis for α and the alternative for β.
6) By increasing the number and frequency of the observations, there is no significant improvement in  2 .
We must assess F and t-stat with caution for the reasons stated in subsection 4.3.
Comparing m risky asset returns to a market index return, such as the S&P 500 Index, we get the same conclusions as in points 4, 5 and 6 above.Conclusions 1, 2, and 3 are not necessarily valid when the benchmark index is not coming from the optimised portfolio for the reasons we will see in subsection 3.3; for the moment, it is enough to observe that these are pure algebraic consequences of having chosen a regression where the explanatory variable, the return on the optimised portfolio, descends from the variable we would explain.Conclusions from 4 to 6 above rely on the hypothesis that the stock return distribution is normal despite showing "fatter tails".
The present paper fully shares the observations by Roll and Ross, separately and jointly, expressed in their multiple essays about CAPM at the level of individual risky assets, which we can summarize: 1) The linearity relation between return and β holds regardless of the chosen market portfolio or a set of m stocks, whether efficient or not (Roll, 1977); the efficiency of the market portfolio and CAPM are equivalent (Ross, 1977).
2) CAPM is not testable without knowing the proper market portfolio mix (Roll, 1977;Roll et al., 1994).Shifting to a market index, we cannot improve its testability; 3) Given the previous points above, the theory is not testable (Roll, 1977;Gibbons et al., 1989) at the risk of turning out to be a tautology; 4) CAPM's ability to explain stock price changes is modest (Roll, 1988).Roll (1988) argues that the  2 regression of the monthly returns of single assets, not a homogeneous asset portfolio in terms of risk, to a market index does not deviate much from 0.30.Adding a sector factor, we reach 0.35, thus leaving 65% of the variance of this return completely unexplained.
We recall that  2 equals: Consequently, the share of the R i variance unexplained equals: while the R i variance unexplained equals: For a demonstration, see Appendix B.2 by Ciech (2016).The i th stock variance   2 is due to a component linked to systematic risk and a residual component ε unrelated to the market return.Suppose now that there exists a fictitious variable X, uncorrelated to the return of the market portfolio R M , such that it can explain the residual variance   2 ; we can then write the following relation: Since the share of systematic risk is lower than the unexplained one, we can write the following relation: from which we get the following condition: It follows that CAPM is unable to explain most of the common stock risk, essentially the correlation between R i and R M is not adequate to explain the variability of the former; it can identify the non-diversifiable part of the risk but leaves the diversifiable part unexplained without explaining to what the first risk component is ascribable.This issue has already been addressed by Lintner (1965b) when he deals with the advantage deriving from diversification: in the case in which all the covariances of the assets are zero, all the risk would be non-systematic, and the benefit of diversification would be substantial; in the opposite case, all the residual variances would be zero; consequently, all the asset returns would be perfectly correlated with each other, and diversification would cease to have effects.Portfolio diversification takes advantage of assets correlated negatively with other common stocks and, above all, from residual variances greater than zero with consequent imperfect correlations between assets.
A polynomial may explain a larger share of common stock return variability.The Mandelker-Ghon Rhee and Modigliani-Miller equations, already mentioned, provide clues that a multiple regression equation like this perhaps is needed: where ψ i is a corporate performance measure,    and    are the stock market and corporate performance coefficients or regressors linking the stock market and corporate performance variables to the common stock return.
CAPM's worth lies in the ability to select an optimal portfolio starting from m risky assets and maximizing the utility to risk-averse investors, given their indifference curves and the return of the risk-free asset.In essence, CAPM states the best risky asset portfolio to invest in and how intensively to use it by combining it with risk-free assets.
Again, Lintner (1965 b) is illuminating: "The goal of diversification is not to avoid or minimize risk per se but to select the best portfolio, i.e., the best combination of risk and expected return from the portfolio mix".
If we want to have an explanation of the behaviour of the risk/return ratio of single assets, we must look elsewhere.

An Alternative Approach: A Model Under Certainty Conditions
Let us imagine for simplicity that we have a common stock whose variance is intelligible at 100% using the following equation: R i =   +   R P +     (29) where the coefficients   ,   , and   are not regressors, at least for the time being.The stock market performance R P as well as the corporate performance ψ i explain the stock return.The variance of the common stock with these characteristics will be the following: where: R i = i th common stock return R P = stock market return Cov(X,Y) = X and Y covariance operator Suppose we have a portfolio made up of m risky assets with the same algebraic characteristics as the previous common stock; the mean and the variance of the portfolio return will vary according to the weight assumed by the investment   in every single asset of the portfolio: From the previous two equations, we obtain the following: Suppose, by reducing to absurd, that both ψ i and Cov(ψ i , R P ) are null, what would happen to such a system?Equation ( 32) would become: from which we can obtain, dividing the portfolio variance by itself, that: What happens in CAPM doing the same operation?Let us examine the portfolio variance   2 : Dividing both members of (36) by the portfolio variance   2 we obtain: Cov(R  ,R P ) The above condition occurs when the weight of each asset is constant across the timespan analysis, i.e., when R P is endogenous to the model.
If this occurs, the following condition occurs: from which we also get the following: (38) Therefore, in CAPM asset allocation, the weighted means of α and β must necessarily converge towards 0 and 1, respectively, and this occurs only when the search for both the regressors refers to the portfolio of m risky assets because the mean portfolio return R P is endogenous to the model.If, on the other hand, we compute the regression against a market index, there is no guarantee that the weight of each asset is constant over time; on the contrary, precisely the opposite occurs without paying attention to the weight of each stock within the market portfolio.In this context, an average weight for the whole period is meaningless.Consequently, both α and β weighted means may diverge from their theoretical values of 0 and 1. Empirical tests should take it into account.Such a result does not depend on the certainty conditions of the hypothesized system; it is a general conclusion that is also valid for CAPM.
We must go back to equation ( 35): in the hypothesis that all the weights of each asset of (35) coincide with those of (37), we can conclude that: It follows that assuming that both ψ i and Cov(ψ i , R P ) are null, equations ( 31) and ( 32) could be compatible with CAPM.However, in such a system, the portfolio return would be infinite as the denominator of (33) would collapse to zero.Indeed, the denominator of (34) would also collapse to zero, like its numerator, leaving the portfolio variance undetermined.In this system, both ψ i and Cov(ψ i , R P ) cannot, by definition, be null.
In short, if we ignore both ψ i and Cov(ψ i , R P ), we create an incomplete system.Hence, CAPM, interpreted through (21), implicitly assumes no relation between the portfolio return and the corporate performance of the risky assets held in the portfolio, so the covariance between these variables is null.The asset returns, variances and covariances are exogenous market variables.Indeed, the approach that CAPM has been adopting over time would appear rudimentary.

Further Development Under Certainty Conditions
At this point, we can take a further step forward.If we perfect the equation ( 34) by inserting the equation (33) in place of R P in Cov(ψ i , R P ), we get the following result: (39) In the system we are illustrating, the portfolio variance   2 is determined by the ratio between: a. the double summation of the corporate performance covariances weighted both on the weight   assumed by each asset in the portfolio and the coefficient   which measures the transferability of the corporate performance on the asset return; b. the square of the difference between 1 and the weighted mean transferability coefficient   of the stock market performance on the asset return based on the weight assumed by each asset in the portfolio.
To fully understand the meaning of equation ( 39), one last algebraic transformation is needed; exchanging the denominator of the second member with the first member, we obtain: from which we get: where: MR = share of portfolio risk arising from the stock market FR = share of the portfolio risk coming from the joint risk of the corporate businesses Equation ( 41) represents the breakdown of portfolio risk in the mean-standard deviation plane between the share ascribable to stock market risk and the residual share from the joint corporate business performance.Therefore, for each portfolio of m risky assets, we can decompose its risk into a share relating to the stock market and the residual share deriving from corporate performance, the one not explained by CAPM.
CAPM assumes that risky asset returns are exogenous data, showing which is the optimal way of building the portfolio, but is unable to explain in depth the asset returns due to its essential incompleteness: the absence of a formalized link between corporate performance ψ i and R i makes it disputable.Instead, CAPM introduces a feedback effect of the stock market on the common stock returns, which we must carefully evaluate to understand the portfolio risk measured by its standard deviation   , easily derivable from (39): The feedback effect comes into play with the denominator of (39 bis), exactly as in a closed-loop system whose operation we will mention in subsection 3.5.We will see later what kind of operation we can obtain with such a denominator; for the time being, we observe that the closer the risk share of the stock market  = ∑    =1 approaches 1, the greater the system instability will be.The meaning of the numerator of (39 bis) is simply the square root of the risk deriving from the covariance matrix of corporate performance multiplied by the column vector obtained with the product of the weight   assumed by the i th asset in the portfolio by the transferability coefficient   of the corporate performance, multiplied again by the transposition of the same column vector.
The stock portfolio risk derives essentially from the joint corporate portfolio businesses, suitably filtered by the feedback effect of the stock market.The effort, which is not entirely trivial, will be to search for the transfer function of corporate performance on the common stock returns, while CAPM provides a bright explanation of the feedback transfer function.Another issue is that the equation system ( 29), ( 30), ( 33) and (39) allows multiple solutions.
It is necessary to quantify the corporate performance ψ i and the parameter   that allows the corporate performance transferability on the stock return to understand the transfer function.All this for every single asset in the portfolio, a gigantic task since it is not predictable a priori which process conveys the corporate performance ψ i nor the transferability parameter   .

Closed-Loop System
In subsection 3.4, we realized a topic underestimated in CAPM, if not completely ignored: the impact each stock in the portfolio has on the portfolio itself and the feedback effect of the latter on any asset in the portfolio.In CAPM R p and R i are independent, thanks to the fact that their values are exogenous variables.In the real stock market R i influences R p and CAPM theorizes the feedback of R p on R i through equation ( 21).In closed-loop system theory, the output signal y t , for control purposes, is the input, via the β stage, into the mixer, which adds or differs from the input signal x t .In the case under analysis, the signal is added to the input signal, originating positive feedback, as in Figure 1:  44) arises the following relation, which typifies the ratio between the output and the input signal: A and β are the transfer functions (Millman et al., 1972).We can point out that the system shows a strong discontinuity if Aβ is equal or close to 1.Typically, a system of this kind is an oscillator characterized by intrinsic instability, just the opposite of systems in which the output signal subtracts from the input signal.An oscillating system is not necessarily unstable: if it simply oscillated between two predetermined states, it would be considered stable.The stock market fluctuates for several reasons: the stream of news relating to the firms, the industries in which firms operate, macroeconomic and political information, and, in general, all the information relevant to the firm participating in the stock market.There is no guarantee that the stock market is stable, and it is difficult to determine the conditions for stabilizing it, provided it is functional.
It follows that, as in closed-loop systems, it is perfectly useless to continuously examine the progress of a signal, i.e., instant by instant, being able to obtain the same result with an appropriate sampling of the input signal and predict the behaviour of the output signal based on the knowledge of the transfer functions.From this point of view, corporate finance is still an immature theory as the transfer function β, which does not coincide with systematic risk, appears sufficiently clear and studied by CAPM, while the transfer function A has not been well turned inside out or does not have a universally accepted and shared solution.
The second members of equation ( 45) and (39 bis) are very similar; first, the denominator represents the feedback effect while the numerator represents the transfer function A, which, in the case of a risky asset, links the corporate performance x t to the return y t of the specific asset.The feedback effect acts on the corporate performance, adding βy t as portfolio return, and giving rise to the signal e t which, suitably transformed, allows us to obtain the return y t .Mutatis mutandis, no logical difference can be deduced by replacing an electrical signal with economic-financial information relating to risky assets, the stock market and corporate performance.

Incompleteness Consequences
We have already examined how the  2 achievable with CAPM is low enough, leaving much of the risky asset return variability unexplained.Furthermore, CAPM does not allow us to decompose systematic risk into a share ascribable to the stock market and corporate performance.CAPM's supposed incompleteness causes both such problems and has some operational implications.In a context where equations ( 29) and ( 30) represent the asset return, try to estimate the parameters   and   , completely ignoring the existence of a second variable and its parameter   , leads to serious estimation errors, downloading the value of the latter parameter on the former two.
Let us examine the error that occurs in the estimation of   defined in a classical way as the ratio between the covariance Cov(R i , R P ) of the risky asset and portfolio returns and the portfolio variance   2 .We identify this estimator with   ̂, assuming we know the true ₂  obtainable with precision through a multiple regression or MLR in which  2 is equal to 1.We can obtain this value of ₂  analytically through the MLR regressor and subtract   ̂ from it, we get the following error: where: = stock market performance regressor Now, this error tends to zero in two conditions, assuming that MLR allows us to reach an  2 equal to 1: 1) when the correlation between the stock market and corporate performance is zero or 2) when the estimator   ̂ is equal to the ratio of the covariances of corporate performance to stock return and stock market performance.
Both these two conditions seem unfeasible; therefore, ignoring the existence of Cov(ψ i , R i ) implies the presence of an error in   ̂ estimation: the higher the correlation between the stock market and corporate performance, the higher the error.Such an error reverberates in the intercept estimation   ̂, resulting in the following error: where: = corporate performance regressor Also, for the intercept, we can point out that the error would tend to zero only if the correlation between the stock market and corporate performance is null, a condition that is not impossible but not readily achievable.
The errors represented by equations ( 46) and ( 47) appear large enough to justify a poor result of  2 .Furthermore, this result should direct research towards a better understanding of the stock return pattern.
The fourth point concerns the  2 partitioning of the MLR regression or commonality analysis (Nathans et al., 2012).We should ask ourselves whether and how  2 can be broken down into shares of the explanatory variables R P and ψ i of the asset return.If we now compare the  2 of the simple regressions of R P and ψ i against R i ,   2 and   2 respectively, with the MLR  + 2 , we realize that their difference will hardly be zero and will give rise to an overlap or bridge effect, depending on whether the sign is positive or negative: From relation (48), we can obtain the net contribution of the variables R P e ψ i on  2 , respectively   2 and   2 , with the overlap/bridge effect  + : Here, it is not as essential to establish how relevant the net contribution or the overlap of the explanatory variables is as to note that  2 could be the result of the effect of a ghost variable that does not appear in the OLS regression.Determining the net and overlapping effects of the explanatory variables on  2 is complex as many variables impact the risky asset return R i , many of which are entirely unknown: even the MLR with three or four regressors fails to reach an  2 equal to 100%.First, it is necessary to resort to  2 ̅̅̅̅ , an adjusted measure of  2 , every time the number of regressors increases.Consequently, already with a single regressor, it is convenient to use immediately  2 ̅̅̅̅ to put the first two explanatory variables on the same level of importance, regardless of which of the two we use first; otherwise, the arbitrary choice of the first regressor can pollute the result.
The overlap effect can be determined, as we have already seen, in a simple way as the difference between the sum of the  2 ̅̅̅̅ of the two OLS regressions and that coming from the MLR; analytically, the following equation represents the overlap: The overlap depends on the correlation between the two explanatory variables R P e ψ i .Indeed, the coefficients of the multiple regressions ₂  and ₂  in the case of covariance Cov(R P , ψ i ) = 0, collapse on the simple regressors   and   , making the overlap null.Consequently, the greater the correlation ρ(R P ψ i ), the greater the overlap is.
We do not know the overlap effect of a second explanatory variable from the CAPM empirical evidence.CAPM ignores such a problem.Consequently, Roll's estimate that CAPM can explain only 30% of a stock return variability could depend on an omitted variable with a substantial overlap effect.
The results are even more complicated to understand because, in CAPM empirical evidence, common stocks are combined in portfolios to avoid EIV problems.We will deal with this issue in subsection 4.5.
The fifth problem concerns the transition from analysing a portfolio of m risky assets to a market portfolio in which the mix changes over time.This assessment will result in the following: 1) A market portfolio return   * different from the optimised portfolio return R M .
2) A variable market portfolio mix   * different from the constant and optimised portfolio mix   .
Using the OLS regression of   * against R i , we reach α and β estimates different from those achievable with an optimised portfolio, where the weights   are constant all along the timespan.

Summary of CAPM Incompleteness
We try to summarize what has been highlighted so far by a simple comparison between a two-variable model to CAPM, essentially based on the optimised or market portfolio return as the only explanatory variable of the risky asset return: 1) The  2 achievable with CAPM is too modest; clearly, there is an external explanatory variable to CAPM that justifies the remaining risky asset return variability, but for now, we do not know what it is.
2) The systematic risk may be ascribable to corporate performance risk rather than stock market risk: CAPM can provide limited clues about such a decomposition, mainly due to the corporate capital structure.
3) Each asset influences the portfolio or the market portfolio return, and the latter affects the former through a feedback effect, with undeniable oscillating consequences.
4) The α and β descending from risky assets and optimal portfolio return regression are not free to assume a correct value since their weighted mean must be constrained to 0 and 1, respectively.The market portfolio does not make this fluctuation unrestrained; on the contrary, it soils it.
5) MLR highlights overlap effects that we can conveniently anatomise to determine the impacts of two or more explanatory variables on  2 ̅̅̅̅ .This effect depends on the existence of a correlation between the explanatory variables.CAPM could reach a modest  2 also thanks to this overlap effect.So, there are one or more ghost variables that limit the CAPM explanatory power on one side and the other the  2 obtained may not be ascribable to CAPM due to such omitted variables.
6) The market portfolio is very different from the Lintner or Merton optimised portfolio.Several essays by Roll (1977Roll ( , 1988)), Ross (1977), and Roll et al. (1994) are enlightening.Even replacing the return of the optimised portfolio of m risky assets with the market portfolio return, albeit represented by a primary market index, does not add more sharpness to CAPM's significance.
CAPM is an essential corporate finance theory, but we should take for what it is worth: a.To determine the efficient frontier of the portfolio of m risky assets with the return vector  and the covariance matrix .
b.To determine the portfolio's optimal mix, given the current level of risk-free asset return R F ̅̅̅ .
c. To evaluate the distance between the current portfolio and the optimal one.d.To evaluate the distance between the optimal portfolio and the market one.
e. To define a different investment allocation by borrowing or lending sums at the R F ̅̅̅ rate.
In the absence of better or equally simple alternatives, CAPM can provide captivating explanations of the stock market operations, even if not always accurate or validated by empirical evidence.Therefore, CAPM is a theory that links the portfolio of m risky assets and allows us to determine the optimal portfolio's correct risk/return profile, given a set of ex-post information, which means the vector of risky assets returns  and their covariances matrix .
Given the risk-free asset return R F ̅̅̅ at time t, the previous set of information allows us to determine the optimal portfolio with the investment share for every single risky asset that allows maximizing the investor utility, who will be able to adjust the risk/return profile of the overall portfolio by borrowing or lending at the rate R F ̅̅̅ , even if this is not strictly necessary due to the presence of the orthogonal portfolio.In this regard, see Black (1972).How close or far the stock market is from an optimal condition can be assessed by comparing the mix of the optimised portfolio with the market portfolio.Some common stocks in the current portfolio will be in excess compared to the optimal portfolio mix and will be sold to invest in stocks that will appear in shortage.These movements will generate price and return changes, which will again modify the optimal portfolio, perhaps accompanying this movement with sensitive changes in the R F ̅̅̅ rate.

Objectives of the Analysis
Having concluded the CAPM theoretical examination, the time has come to analyse empirically some essential topics highlighted in Section 3.
First, the strategy is to verify the asset allocation of 100 common stocks included in the S&P 500 Index in May 2022.We started with creating a 10-stock portfolio with no short sales, gradually expanding the portfolio to 25, 50, 75 and 100 stocks, both with and without short sales, getting nine optimised portfolios, correlating them with the performance of the S&P 500 Index.We shall track and explain the trends of some CAPM parameters of the optimised portfolios and their single common stocks.
Later, the analysis focused on the security market line or β, using the 30-year time series of monthly returns over 5, 10 and 30-year timespan and relating them to the return of the S&P 500 Index as a proxy of the market portfolio.For each common stock compared to the S&P 500, we present the 5-year rolling β for each month from January 1992 to December 2020.
Lastly, to explain stock return, we shall integrate into an MLR the S&P 500 Index return and the corporate performance measurable by a sufficiently large set of business variables, notably some DOL and DFL variables.
We shall present the outputs from these test batteries and draw some preliminary considerations.

Empirical Evidence Data
To empirically verify the previous three topics, i.e., asset allocation, security market line, and integration of stock market return with corporate performance, we use multiple sources of information, easily accessible to even non-professional investors, as CAPM prescribes.
First, we select the 100 nonfinancial risky assets from those in the S&P Index in May 2022, representing over 50% of the index mix.
The monthly market prices of the 100 common stocks and the value of the S&P 500 Index come from Yahoo! Finance.We limit the analysis to 30 years, from January 1991 to December 2020.The S&P 500 Index and 68 stocks are present in each of the 360 months of such a timespan, while the remaining 32 stocks progressively join as they land on the stock exchange.The prices need adjustments for dividend payouts, splits, and other equity operations.Based on these quotations, we computed the monthly returns of 100 risky assets and the market index; we compared such data with those coming from Portfolio Visualizer, not detecting significant differences in terms of mean returns but only modest differences between the monthly returns caused primarily by the presence of dividends.
For the optimised portfolios computation, we used both the tout-court and the lognormal returns, i.e., (1 + R i ), without highlighting appreciable or significant differences in the final outputs.Consequently, we use the lognormal return for the asset allocation, while for the subsequent analyses, we use the return tout-court.
Concerning the financial statements and outstanding shares, we used Bloomberg data and the annual reports (form type 10-K) available on the Security and Exchange Commission's EDGAR website.Bloomberg data are not ideally suited to the purposes of this study as the time series available often include ex-post adjustments.Regarding comparability, Bloomberg's work is impeccable, but we prefer to use the original financial statements without any ex-post adjustments for the current analysis.We use Yahoo!Finance, Bloomberg and EDGAR for dividends, splits, and other equity adjustments, correcting them when and where necessary.When deemed necessary, we resort to the dividend and stock split histories published directly by the companies.

Distribution of Stock Returns
For the distribution of stock returns, we refer to the essays by Mandelbrot (1963), Fama (1963), Fama (1965) and Officer (1972).Briefly, stock prices follow a random-walk behaviour based on two assumptions: 1) successive price changes are independent and 2) they conform to some probability distribution.
While there are no doubts regarding the first point relating to the independence of successive price changes, the distribution does not appear to be perfectly described by a Gaussian; Mandelbrot's hypothesis seems more fitting.
In particular, the study by Officer (1972) believes that a symmetric stable class of distributions better describes the distribution of returns due to tails that are fatter than the Gaussian but with properties inconsistent with the stable hypothesis, such as the behaviour of the sample standard deviation.Fame et al. (1973), based on the results of Fama (1965) and Blume (1970), believe that the interpretation of t-stat, valid for normal distributions, applied to the distribution of stock returns leads to overestimate probabilities and significance levels.The values of F, t-stat and P-Value presented in the following subsections must take this topic into account even though Fama et al. (1973) write that "as one is not concerned with precise estimates of probability levels, interpreting t-statistics in the usual way does not lead to serious errors".

Asset Allocation
The first objective was to verify the behaviour of the optimised portfolios' main parameters as the number of stocks increases, correlating them with single common stocks and the S&P 500 Index return.The analysis starts with a set of ten common stocks to avoid short sales without constraints, and we progressively increase the number of stocks to 25, 50, 75 and 100, first with and then without short sales.We show the data collected from the nine portfolios in Table 1.
Table 1 allows us to observe the following topics as the number of risky assets in the portfolio increases: 1) The return and the standard deviation of the optimised portfolio decrease while in the optimised portfolio sine, without short sales, decreases mainly only the standard deviation.
2) Both the slope θ of the market opportunity line and the return/risk parameter λ of the portfolio increase; it means that the standard deviation is declining faster than the portfolio return, at least; 3) With short sales, the number of active stocks is always equal to the number of selected stocks, while the number of stocks sold short progressively increases up to 46%.
4) Without short sales, not all the selected stocks are active; indeed, the percentage of inactive stocks proliferates.
5) Lintner's conic sections differ from Merton's when short sales are involved.
6) The market opportunity line is always tangent to Merton's and Lintner's conic sections, with and without short sales, but the portfolio optimal mix with the KKT condition active does not lie on the market opportunity line.
7) The A, B, C and D parameters of the conic sections progressively increase in value; they indicate non-degenerate conics and are shifted hyperbolas.We can refer to the matrix representation of conic sections for more details.
8) The centres of Merton's conic sections always have standard deviations equal to zero, but returns seem roughly constant.
9) The centres of Lintner's conic sections with short sales show a progressive reduction in the abscissa and ordinate; the abscissa is always negative, while the ordinate becomes negative as the number of common stocks increases.Since the centre on the ordinates of each conic corresponds to its minimum point   , Lintner's conic section overflows more into the second Cartesian quadrant, generating a portfolio with zero standard deviation and return lower than risk-free assets.Hence, this portfolio lies below the market opportunity line.
10) The orthogonal portfolios always lie on the inefficient part of Merton's conic sections.According to the Lintner and Merton methodologies to assess weights, we have traced in Figure 2 the three primary conic sections, one with the 10-stock portfolio and the others with two 100-stock portfolios.Apart from the southwest shift of Lintner's conic section compared to Merton's, already mentioned, we can note that the conic section tends to move west as the number of stocks in the portfolio increases, with a marked enlargement of the shape, which allows it to have a higher return for the same standard deviation.Consequently, the market opportunity line must have a steeper slope, allowing for intercepting higher indifference curves.
We can see the trajectories of the following points: 1) Merton's point of tangency (squared indicator): it moves west and then heads north; this movement indicates a return increase with the same portfolio risk.
2) Lintner's point of tangency (round indicator): moves in a westerly and slightly southerly direction; such a movement indicates a progressive portfolio risk reduction joint to a less than proportional reduction in the portfolio return.
3) The optimal point with short sales constraints due to the KKT condition (triangle-shaped): it moves westward and is always suboptimal compared to Merton's and Lintner's point of tangency, given that they coincide in the absence of short sales; this shift indicates a progressive reduction of the risk with an almost constant portfolio return.
4) The minimum point of Merton's conic section (diamond shape): it moves westward and slightly southward, indicating a reduction in portfolio risk with roughly the same return.
The main conclusion is that by expunging short sales through the KKT condition, the optimal portfolio does not lie on the market opportunity line and, therefore, must be considered suboptimal.Secondly, whenever the number of common stocks in the portfolio increases, it is necessary to have a growing share of stocks sold short; the market portfolio, to be efficient, should have a negative quotation for many stocks listed on the stock exchange.Since this cannot happen, the market portfolio is not as efficient as an optimised portfolio: better, the market portfolio is neither efficient nor optimal, and the alleged syllogism that the market portfolio is efficient is not confirmed.Thirdly, even having expunged short sales, the optimised portfolio sine has a limited set of active common stocks compared to those selected from the stock market index: more than 70% of the stocks do not enter the 100-stock optimised portfolio sine.See Levy (1983) for such an issue.Finally, we can point out that the increase in the number of stocks entails an appreciable risk reduction and a modest return increase.So, the 100-stock optimised portfolio sine indicates diversification's true potential: risk reduction with substantial return stability.We will first analyse the β behaviour of the optimised portfolio with short sales.The fundamental point is that 100% of the stocks are active; consequently, when the stocks increase from m to n, we witness homogeneous behaviour of all the m stocks already in the portfolio: all the βs increase homogeneously (see Figure 3 on the left), and such growth is related to the trend of the portfolio excess return, R M − R F ̅̅̅ = X M .We analyse the β percentage change of a generic stock, exploiting CAPM equilibrium relations, through a few algebraic passages we obtain: where: The facts are as follows: 1) The β of all common stocks increases as the number of stocks in the portfolio rises; 79 of the 100 stocks in the portfolio have a β greater than two, while only four stocks have a β less than one.
2) The portfolio excess return X M trend involves such an increase, which, as we have seen, decreases as the number of stocks in the portfolio grows, and this does not depend on the absolute risk level of the single stock.
3) Consequently, the increase is homogeneous for all common stocks.What happens if we base the portfolio weights on Merton's method instead of Lintner's?Relations ( 51) and ( 52) are still valid, but using Merton's conic section, there is a tendency to increase the portfolio return with equal standard deviation, implying that β must decrease.With a 100-stock portfolio, the contraction is so powerful that only one of the β exceeds 1.0: see Figure 3 on the right.Such a trend, apparently illogical from an economic perspective, leads us again to favour Lintner's assessment of the portfolio weights.Certainly, Merton's evaluation is more valuable in the optimisation stage but less economically understandable.
At this point, it is unclear why we observe β, a specific indicator of common stock, instead of watching λ, common to all stocks in the portfolio.By precisely defining the market portfolio mix and size, it would be easy to verify its optimisation to discover that the market is perhaps not as optimised as we usually consider it, even if it remains an excellent tool for sharing and containing risks (Lintner, 1970).
Furthermore, the β weighted mean of the optimised portfolio equals one due to many negative weights, just as the α weighted mean is null.These topics are sufficiently substantiated and highlight the characteristics the stock β should have if the market portfolio were optimised: a very high β, even though their weighted mean is 1.

β Behaviour without Short Sales
What characteristics must stocks possess to have a positive or negative weight?Observing the covariance matrix, its inverse, and the stock excess return X i , it does not appear at first sight identifiable what characteristic determines its sign and value.As already extensively treated in subsection 3.2, everything depends on the v  elements of the inverse matrix.The last issue introduces the β behaviour in the optimised portfolio sine.Even for the latter, it is not easy to understand the characteristics that the common stocks excluded must have since it is not sufficient to have a negative weight in the optimised portfolio with short sales to be a candidate for taking on a zero weight in the optimised portfolio sine.The zero-weight choice mainly depends on a nonlinear system where even the v  elements of the inverse matrix do not play an explanatory role.Having said all this, let us see the behaviour of the stocks in the portfolio sine.First, we focus on the dynamic of inclusions and exclusions.From Table 3, it is possible to examine the stratification of the stocks in the five portfolios sine; from 10 to 75 stocks, the dynamic presents exclusively stopping stocks, while moving from 75 to 100 stocks appear assets not previously selected.
In Figure 4 on the left, we can examine the overall β dynamics, assuming that their value nullifies when they leave the portfolio so as not to pollute the overall picture.We will exclude or include stocks from the analysis as they leave or enter the portfolios.Also, in the optimised portfolios sine, we have homogeneity in the β dynamics, which remains confined between 0.3 and 1.8, with some oscillations, without a decisive and constant increase as in the optimised portfolios with short sales.Determining the β dynamic appears problematic as we initially witness a stopping and rising dynamic altogether, subsequently mixed up by repechages of excluded stocks.
The βs follow the same dynamic already highlighted by equation ( 51).All this is for the stocks included in the portfolio.Therefore, the relative β constancy follows the same dynamics of the portfolio excess returns: the portfolio variance shrinks while its return is relatively constant.
Considering that the portfolios sine are sub-optimised compared to those with short sales, it follows that low βs are typical of a sub-optimised stock market.
What happens to the βs of the excluded stocks?From Figure 4 on the right, we can examine such trends.They present the most disparate dynamics; they have no connection with the portfolio return trend or each other.Their values are lower than those of the optimised portfolio with short sales.Apart from one stock, the other 70 stocks show values between 0.50 and 1.90.We must ask ourselves the meaning of common stock βs in a non-optimised portfolio or for stocks excluded stricto sensu from the portfolio, even the market portfolio.

Correlation and Determination Indices
The last issue concerns the trend of the correlation and determination indices of single stock returns compared to the optimised portfolio return.Again, we examine the relations between these indices in a portfolio of n stocks starting from a portfolio of m stocks with n > m.The relation for the correlation index is as follows: From this, we can conclude that in an optimised portfolio of any kind, the correlation between stock and portfolio returns tends to decrease as the covariance reduction exceeds the reduction of the portfolio standard deviation.Although the covariance and standard deviation dynamics depend on the weight assessment by Lintner or Merton, this cannot influence the correlation between the common stock and optimised portfolio returns.As we have already seen above, the equilibrium relation requires that when the number of common stocks in the portfolio increases, there is an inverse relation between the covariance Cov(R i , R M ) and λ: the former decreases as the latter increases, linked to the θ rise.Furthermore, as the number of stocks increases, the portfolio variance decreases, which leads to the standard deviation shrinkage.The covariance reduction is faster than the standard deviation shrinkage.Consequently, the correlation index decreases when we move from m to n stocks, leading to an automatic  2 contraction: In optimised portfolios with short sales, the dynamics of the correlation and determination indices of the stock returns compared to portfolio return are always completely homogeneous for every stock, except for those just included since all the stocks are always active.In optimised portfolios sine, we must exclude the stopping stocks to appreciate the trend homogeneity of correlation and determination indices.We show such dynamics in Table 4.
Within an optimised portfolio with short sales, as the number of stocks increases, we can observe the following events: 1) β tends to increase with Lintner's weights (the opposite with Merton's weights); 2) ρ and  2 tend to decrease.
It follows that trying to explain the common stock return against the optimised portfolio return through OLS is misleading, while the β trend is due to the increase in the risk perception of common stock compared to the optimised portfolio variance that decreases continuously.We can also apply the same considerations to an optimised portfolio sine, albeit in a much softer way.The emphasized tendency towards an  2 reduction is due to portfolio optimisation; consequently, Roll (1988) absurdly should rejoice in having an  2 around 30%; such a parameter should be much lower if the market portfolio were optimised.With a 100-stock optimised portfolio with short sales, the mean  2 is around 4.5%.It is a further clue that the market portfolio is far from optimised; this is a market failure, and CAPM should be cleared, not blamed.

Security Market Lines
After analysing the asset allocation, we must look at the monthly return of the 100 common stocks compared to the market portfolio return using the S&P 500 Index as its proxy.We analysed three periods ending in 2020: 30 years from 1991, 10 years from 2011 and 5 years from 2016.We present the data in Tables 5, 6a and 6b.We replaced the company's names with aliases.Table 5 summarizes the data from the three periods.
Briefly, we can see that using the monthly returns, we get the following results: 1) The intercepts and slopes of the regressions are considered admissible based on the F test, but as the time window analysis shrinks, its goodness-in-fit decreases.
2) While the slopes align with the goodness-in-fit of the regression, measured by t-stat and P-Value, the intercepts fail to pass the null hypothesis easily; the more the time window shrinks, the less they are significant: in the 5-year timespan, only 16 intercepts are significant.
3) The determination index  2 increases on average as the time window decreases and remains at relatively low levels, around 30%, but not insignificant, as expected in an optimised portfolio with short sales.
4) Moving from monthly to weekly returns  2 worse, but this result is not due to better portfolio optimisation but to more erratic price movements.
5) The intercepts have an arithmetic mean value between 0.7% and 1.0%, statistically insignificant, while the slopes have values that fluctuate around 1.
On the one hand, the analysis confirms that if we think of explaining stock excess return through the market portfolio excess return by using β, we risk incurring excessive risks since the relation does not have an outstanding predictive value if measured by  2 , on the other hand, the βs oscillating around one and the  2 s suggest that the market portfolio is not optimised.All this confirms that the security market line of each stock has the meaning of a portfolio optimisation condition purely, without any predictive worth.Table 5 shows the mean rolling β of all common stocks recalculated monthly from 1992 to 2020.The analysis is captivating, but the summary does not do justice to the data detail shown in Table 7.It is necessary to examine the diagrams to understand their meaning.We report in Figure 5 the diagrams of three stocks issued by historical corporations, which can represent what a rolling β means.
In Figure 5, we show in blue with the scale on the left the 5-year rolling β and the 30-year stationary β, while all the other variables have the scale on the right.With a solid blue line, we have the 5-year rolling β while the trend of the relative intercept α,  2 , and the P-Value of the rolling regression have dashed lines.The 30-year stationary β, its intercept α and the  2 of the regression have a dotted line with a double point, shown purely for comparison purposes with the 5-year rolling analysis.

Such analysis highlights:
1) a substantial variability of the rolling β, which contrasts with the presumed constancy of the stationary β, 2) an equally significant variability of the  2 of the rolling β regression, 3) a modest oscillation of the rolling α and 4) a localized P-Value movement of the rolling β.
In the first diagram on top of Figure 5, the first stock highlights a marked oscillation in rolling β corresponding with the 2008 crisis, also felt in the third diagram in the bottom but not in the second in the middle, which is instead affected by a period of the corporate downturn that began in 1992 and ended in 1996.The overall graphical analysis of the 100 common stocks shows abrupt changes in the 5-year rolling β in a period of one or few months, paired with other sharp α,  2 and P-Value movements, not always synchronous with their respective rolling β.
Such trends are not unrelated to stock market trends and corporate performance.The security market line does not appear insensitive to such changes.It is unclear what the transmission mechanism of the corporate or industry performance on the stock return is: CAPM cannot specify it, being persuasive in allocating assets but less effective in explaining the individual stock price and return trends.Furthermore, we must ask ourselves how it is possible to create homogeneous β portfolios if this variable could undergo sudden fluctuations due to internal or external causes to the firm: over a 5-year timespan, we can get the false feeling that β is or has been constant when it can change suddenly and abruptly in one month.The first stock on top of Figure 5, fell from 2.50 to 1.50 between 2007 and 2009 to suddenly rise again towards 2.50 with an  2 in between 20% and 40%; a similar trend would not have emerged by combining this stock in a portfolio of 50/100 other stocks in a regime where β computation takes place annually or biannually.Portfolios can mitigate EIV bias but have other drawbacks that mask factors that characterize individual stocks (Jegadeesh et al., 2015).

First Conclusions on the Empirical Evidence
From the previous subsections 4.4 and 4.5, we have verified that CAPM can perform an essential function in contouring the portfolio efficient frontier of m risky assets and determining the optimal combination that maximizes the investor utility through the slope of the market opportunity line.Once this function is completed, the prospect of using the equation ( 21) reported below falls sharply.
We reiterate that equation ( 21) is an equilibrium condition to minimise portfolio risk.Furthermore, it is a stochastic relation whose only constant element is R F ̅̅̅ .Thinking that ( 21) is verified every instant is a pious illusion, like thinking that β i remains constant in time.Equation ( 21) is a valid relation for a specific portfolio aimed at minimizing its risk.Assigning such a relation to a different task involves fatal errors for the reasons we are about to present.
We have seen that β takes on different meanings depending on its use: within an optimised portfolio or as a generic measure of a stock risk compared to a market index.Let us examine the results of the first kind using Figure 6, which presents the 360 observations of stock monthly returns in the 1991-2020 timespan.If we assumed that such a risky asset was the only one available on the market, the portfolio returns would be equal to the stock returns and the observations would be distributed in Figure 6 as the amaranth points along a bisector of the Cartesian axes: intercept equal to 0, slope equal to 1 and such a distribution would have an  2 equal to 1. Combining the common stock into the 10-stock optimised portfolio analysed already, we know there would be no short sales, and the orange dots represent the distribution of the observations.The optimisation process causes a squeeze of the portfolio returns along the abscissa while there is no change along the ordinate.The security market line increases from 1 to approximately 1.317; there is no significant change in the intercept but  2 plummets to approximately 16.5%.
Moving to the 100-stock optimised portfolio with short sales and Lintner's weights, portfolio returns squeeze further with the same stock return distribution as before: the observations in Figure 6 are in blue.The optimisation is so effective that the portfolio variance collapses, originating a sharp rise in the yellow regression line, with a slope of 4.65, intercept practically zero, and  2 at 4.1%.We have already presented the dynamics of these parameters with equations ( 51) and (54).
In summary, portfolio optimisation is so efficient in containing return variance that it makes the common stock appear riskier: the squeeze effect is more prominent as the number of common stocks increases.At the same time, the relation between portfolio and stock returns becomes less and less significant.
The inclusion of risk-free assets does not change β because R F ̅̅̅ is constant, so it does not affect the variance, covariance, correlation, and determination indices.
In Figure 7 on the left, we replaced the 100-stock portfolio with the S&P 500 Index (turquoise dots).The index composed of 500 stocks has squeezed returns like the 10-stock optimised portfolio, while on the right, we have replaced the S&P 500 Index with the 100-stock optimised portfolio sine, of which only 29 stocks are active (red dots).Now, it appears evident that the effectiveness of the optimisation with short sales constraints exceeds the S&P 500 Index.It follows that the slope of the regression line of the former is steeper than the latter, and, as we have also seen in the previous case, there are algebraic and stochastic reasons which should make us reflect on the optimisation effectiveness of the market portfolio represented by a proxy.If the market portfolio represented by the S&P 500 proxy were as effective as the 100-stock optimised portfolio sine, there would be two numerical consequences: it would have a steeper slope and a lower  2 but unfortunately, this does not happen.Table 8 compares the essential parameters of the five possible portfolios to the sample common stock to assess their internal dynamics: the variable θ allows us to establish their optimisation ranking.Even the 10-stock portfolio sine beats the S&P 500 Index, which is slightly better than the single stock itself, despite having the latter a nine times higher variance.The only element ignored by this analysis is that the five portfolios are based on ex-post decisions while the single stock and S&P 500 Index are potentially resulting from ex-ante choices: if an investor decided to invest in 1991, he could have bought only the single stock and the S&P 500 Index while no one could have invested in the 100-stock portfolios and only with much good luck could it have been possible to opt for the 10-stock portfolio that would initially have been composed by five stocks with five subsequent additions between 1997 and 2012, but probably with different weights from those coming from equation (19 bis).Ex-post, we know the covariance matrix for 1991-2020; ex-ante, we should have had the crystal ball or a better forecast tool than that provided by CAPM.The approach used by Black et al. (1992) is encouraging.It would be interesting to use the data for the first n months to optimise the portfolio for the subsequent month through all the 30 years and compare that result with the 30-year static optimisation and the S&P 500 Index.
The outcome is that equation ( 21), of pure stochastic nature, is instrumental only for portfolio optimisation and performs a magnificent job.It loses worth when the portfolio is not optimised, such as the market portfolio or its proxy, and even when the risky asset, excluded from the market portfolio, is compared to this latter.In any case, its predictive reliability can only be modest, being a stochastic relation with a percentage of the explained variance lower the broader the stock set held in the portfolio is.Of course, it plays an essential role in defining the risk/return ratio of the stock, but alas, it cannot quantify it correctly outside a specifically optimised portfolio.
CAPM empirical evidence studies assume that the market is efficient without any checking about its optimisation, and such a fault has led to nothing.Extremely refined statistical techniques employed to validate CAPMs clash with the empirical evidence that: 1) The market portfolio might be neither efficient nor optimised; 2) The better the portfolio optimisation, with many uncorrelated stocks and short sales, the worse the explained variance of the relation (21); 3) β changes over time, even within a few weeks.
CAPM takes the risky asset prices, returns, variances and covariances as exogenous data.It cannot explain the performance of the capital market by itself owing to its incompleteness.If we want a theory explaining the stock market returns, we must look outside CAPM.

Integration of Stock Market with Corporate Performance
After analysing the asset allocation and security market lines of 100 stocks embedded in the S&P 500 Index, we investigate the possibility of explaining stock returns based on the stock market's and corporate business's joint performance.

Stock Market and Corporate Business Performance Data Sampling
As far as the stock returns are concerned, we said everything already; for the stock market performance, we will use the S&P 500 Index return as a proxy for the market portfolio, while for the corporate business performance, we will use some variables linked to the DOL and DFL definition, already presented in subsection 2.1.Table 9 summarizes the fourteen variables employed according to their kind.Most variables are selected from the income statement except for dividends, partly associated with the net income trend.The ratios between variables are simply ratios between a specific quantity at time t to time t-1 to determine its growth trend.The Risk Rate at time t is the reciprocal of the ratio between EBIT and Total Revenue at time t-1.It indicates the risk of the corporate business, albeit in a rough form: the higher this ratio, the greater the chance that unfavourable changes, even modest, of unit prices, unit variable costs, volumes and sales or manufacturing mixes cause adverse effects on corporate profitability.Instead, DTL is simply the DOL by DFL product.
The first problem in managing a heterogeneous mass of similar financial and market data is their sampling; the stock and stock market index quotations are even at an intra-daily level while listed firms release quarterly financial statements.Even choosing a monthly frequency for market data, as has been done already, would cause a significant mismatch with quarterly financial statement data.Furthermore, the comparison between quarterly corporate data should involve a seasonal adjustment to be meaningful.Even so, there would still be the problem of defining the trend at the quarter level versus the previous quarter or quarter of the last year, avoiding seasonal data adjustment.We discarded the use of quarterly data owing to the consideration that the stock market operates skilful professional investors who can evaluate the overall corporate performance by inferring it from the quarterly data and converting it into an annual projection; an excellent quarterly performance combined with a good corporate knowledge would allow the expert analyst to understand the real corporate trend or even to predict it in advance, without having information other than that available on the market.For this reason, we decided to use stock annual returns by relating them to annual financial statement data.

Regression Procedure
We regress the stock market and corporate performance of the year t on the stock return of the same year: for the S&P 500 Index return, there are no problems of synchronicity with the stock return, but with the corporate performance it is necessary to make a logical leap and link it to the stock return in advance of the moment in which the financial statements for the same year are made available, perhaps several months in advance.We have hypothesized that the market does not have to wait for the end of the financial period to find out how the firm is doing: it knows beforehand its performance.Therefore, we paired the annual corporate performance at time t, relative to the previous 12 months, with the annual stock market and the specific stock returns in one of the 12 months of the same financial period.
But which month in particular?The one which maximizes  2 within the 12 months of the financial period.
Of course, it was necessary to convert the monthly time series from month/year into month/financial period as the start and end date of the financial period do not always coincide with the solar one and are specific for each firm.Fortunately, there are very few cases of firms that have changed the closing date of the financial period by one or more months while scheduling the end of the financial period on a specific variable day within a limited period of a few days around the end of a month is relatively frequent.In this case, we assumed the month-end date as the end of the financial period.
Once we acquired the financial statements, we based the analysis for each stock on the following process: 1) Stage 1: simple regression at the annual level of the S&P 500 Index return as the explanatory variable of the stock return, identifying the month of the financial period in which  2 is maximized on a thirty-year or the available timespan.
2) Stage 2: once defined the month in which there is the maximum variability explained of the stock return against the index return and vice versa, we carried out an analysis of the correlations with all the fourteen variables of Table 9 on the available timespan for the stock to identify the possible candidates for the subsequent multiple linear regression or MLR.
3) Stage 3: MLR to identify the best second regressor for the same month identified in Stage 1 based on the criterion to maximize  2 in the absence of multicollinearity.
4) Stage 4: search for a better month than the one already identified in Stage 1.If we observe a better month, the analysis restarts from Stage 2.

5)
Stage 5: identification of the third and fourth regressor that maximize  2 , in the absence of multicollinearity.In some cases, Stage 5 changed the reference month identified with Stages 1 and 4, restarting the analysis from Stage 2.
The described process makes it possible to identify the best MLR for each stock, using three explanatory variables of corporate performance and the stock market performance over a sufficiently long time.
The method of choosing the month of the financial period in which to match the stock and stock market returns with the financial statement data might seem arbitrary and opportunistic, but if it allows explaining a significant share of the stock return variability over 30 years for 68 stocks out of 100, could prove decisive.Considering that the month of the financial period that shows tremendous significance is between 5 and 6, it seems to indicate that even before the availability of the half-year report, the trends of stock return R i and corporate performance  i are already correlated, in part mitigated by the presence of R M .This undeniable fact should not necessarily lead to insider trading actions or other criminally relevant conduct but only the possibility of skilfully analysing corporate performance in advance based on incomplete information.
For each stock, we will present the details of the simple regressions against the S&P 500 Index return and MLR against the same return and the corporate performance's best variable.For each stock, we will also show the  2 of the simple regressions against each of the corporate performance's fourteen variables, determining an overall ranking based on the highest  2 .We will also analyse the best MLR with two regressors focusing on the overlap or the bridge discovered with the commonality analysis, verifying the absence of multicollinearity.

OLS Results
We can start with the summary of the OLS regression of R M against R i with the annual return data, shown in Table 10; we present the details in   12 for all the  2 of the fourteen variables plus the S&P 500 Index return, while in Table 13, we present the summary.It seems important to underline two topics relating to the annual stock returns: 1) Although the annual stock return is strongly correlated to the S&P 500 Index return, not unrelated to the fact that the latter derives from a weighted mean of the former, at least in the last seven years, the market also seems to appreciate other corporate performance indicators in 39 cases, and Revenue Growth and EBIT Growth play this role in 27 instances.
2) Dividends show a mean  2 of less than 3%, with only six cases exceeding 10%: the market does not appreciate this form of equity remuneration, confirming the thesis by Modigliani-Miller that dividends are a financial illusion.

MLR Summary
Table 14 shows the MLR data analysis summary, while the details for each stock are in Tables 15a and 15b.
The F test appears significant in 94 cases, while only 28 intercepts, 81 stock market regressors, and 68 corporate performance regressors pass the null hypothesis test.

Table 14. Summary of MLR analysis
Compared to the significant 100 β regressors of the 30-year OLS with monthly returns shown in subsection 4.3, it would appear like a significant step backwards, but if we look at the MLR  2 ̅̅̅̅ and  2 , we get a mean value of 51.8% and 55.8%, respectively, starting from 22,5% in Table 5.

The Commonality Analysis
The mean contribution of the stock market explanatory variable through  2 ̅̅̅̅ is 33.4%, while the corporate performance variable reaches 22.0% with a mean overlap of 3.5% (data in Table 14).
We should investigate further such mean results by separating the overlap from bridge cases.From Figure 8 on the left, we can examine the bridge case, where there is no apparent overlap between the  2 ̅̅̅̅ of the stock market and corporate performance variables,   2 ̅̅̅̅ and   2 ̅̅̅̅ , respectively, while on the right, we have the overlap case.First, the transition from the monthly to the annual return is more noticeable in the bridge than in the overlap case by more than 5%.The adjustment due to the transition from one regressor to two regressors results in a reduction of about 3%.
With two regressors, the gross contribution of the stock market is higher than 5% in the bridge case, while the corporate performance contribution is only 6%, the same value as the mean bridge.In the overlap case, the gross contribution of the stock market is reduced by an overlap of 10%, while the corporate performance contributes almost 33%, higher than the contribution of the stock market.The  + 2 is greater than 4% in the overlap than the bridge case.
Such a result mainly depends on the correlation between the stock market and corporate performance: in the overlap case, the mean correlation reaches 17.4%, while in the bridge case, the correlation is -1.7%.There are 36 negative correlations between R M and the corporate performance ψ i selected as the second regressor with a mean of -14.7%, while the mean value for the remaining 64 correlations is 23.2%.The mean value of the correlation for all 100 stocks is 9.5%.

Figure 8. Commonality Analysis: Bridge and Overlap Split
There are no significant correlations between the weights assumed by the various stocks in the optimised portfolios, with or without short sales, and the  2 ̅̅̅̅ value assumed in the various analyses carried out so far.
Bridge and overlap cannot explain short sales or zero weights in the two different kinds of optimised portfolios, and we cannot deduce any correlation between weights on one side and the joint performance of the stock market and the corporate business on the other.
Moving from one to two regressors,  2 ̅̅̅̅ increases by 15.5%, but the impact of the second regressor results in a mean 16% decrease in the β due to stock market Moreover, this reduction is entirely clustered in the overlap cases, resulting in a reduction of the corresponding β by 27.5%; in the bridge case, there is a slight increase in the stock market β of 0.7%.In essence, the traditional β could overestimate the impact of stock market performance on the stock return in the overlap case with a second variable affected by the corporate performance.This effect would happen when the two performances are positively correlated; if the correlation is negative, this effect is less observable.Instead, when we observe the Condition Number, many regressors are affected by high values, much higher than 30, and this occurs when we use a regressor with significantly different values from R i and R M , for example, when using data in dollars or when the percentages of corporate performance are several orders of magnitude, even 1000 times higher in absolute value.Although there is a matrix ill-conditioning problem, it does not appear to be ascribable to multicollinearity.
Table 16 summarises the results of the OLS regressions on monthly and annual returns and MLR on yearly returns, all over 30 years or the shorter available timespan.The second most used regressor is Revenue Growth in 30 cases, followed by EBIT Growth in 14 instances and Total Revenue in 11 cases.These three corporate performance variables stand for 55% of the second regressors.Also, other regressors assume a moderate relevance while the Adjusted Dividends, the Adjusted Basic EPS and the Net Income are absent.From this analysis, it seems that the stock market appraises corporate performance variables linked to real markets and operating profitability rather than variables directly related to net profitability.Corporate business expectancies outclass the ability to generate immediate profit, which is entirely unexpected but an utterly logical behaviour from a perspicuous market.
We point out that in the overlap case with Revenue Growth and EBIT Growth, the  2 ̅̅̅̅ share ascribable to the stock market is smaller than the corporate performance.This effect also occurs in the overlap case where DOL, EBIT, and the Adjusted Diluted EPS appear.
In all cases but one,  2 ̅̅̅̅ grows consistently from one regressor to two regressors with annual returns, especially if the second regressor is Revenue Growth (+24.9%),EBIT Growth (+20.3%) and EBIT (+21.8%).The only case this does not occur is when the EBIT regressor is concerned about a firm that does not communicate such data, which is available only through Bloomberg.
We have already seen that passing from one to two regressors with annual returns entails a β reduction to 1.220 from 1.460 (Table 16) due to the stock market performance by about 16%; if we compare the annual return ₂  (1.220) with the monthly returns β (1.008), we note that it is 21% underestimated.In both cases, the evaluation varies according to the presence of bridges or overlaps, which largely depend on the correlation between the stock market and corporate performance.In general, a high correlation between the stock market and corporate performance implies the presence of an overlap, while a low or negative correlation implies the presence of a bridge.
In essence, the empirical evidence suggests that the stock market performance alone cannot explain stock return; it is necessary to introduce a variable representing the corporate performance to improve  2 ̅̅̅̅ remarkably.This result is not totally in contrast with CAPM, which partly assumes stock prices and returns as exogenous data, without providing a convincing explanation of the market adjustment process facing an essential deviation from its theoretical assumptions.

Conclusions from Empirical Evidence
Having completed the analysis of the empirical evidence topics, which means asset allocation, security market lines, and integration of the stock market's and corporate business's joint performance, the time has come to draw a unitary conclusion.
The asset allocation test has highlighted that portfolio optimisation leads to the physiological presence of short sales, reaching almost 50% of the stock positions as risky assets increase.Such a requirement makes the optimised portfolio inconsistent with the market portfolio.At the same time, comparing the optimised portfolio sine and market portfolio raises further concerns about stock market efficiency and optimisation since it appears sub-optimised compared to a 100-stock portfolio sine, with inactive stocks reaching over 70% of the positions.
Considering that β and  2 of a common stock belonging to an optimised portfolio are parameters which, as a result of the optimisation, assume respectively increasing and decreasing trends as the number of stocks increases, we must ask about the β eligibility to express the risk of a stock not belonging to an optimised portfolio or belonging to a non-optimised portfolio like the market one or even worse when the stock is compared but not belonging to a non-optimised portfolio.
Furthermore, we must reflect on the efficient portfolio frontier: is it an optimal and achievable theoretical framework or unachievable but valuable as a benchmark?We cannot answer that question, but the doubt is legitimate.
The security market line is affected by the problems that emerged with asset allocation, and the possibility that it changes rapidly in a few weeks exacerbates it.On one side, the instability of the   , due to the stock market performance, is not unexpected but makes the relation between the stock market and the stock excess returns less explicative.Furthermore, finding more space in the financial communication for the data relating to the rolling β would be appropriate.
The integration into an MLR relation of the stock market and corporate performance to explain the annual returns of risky assets in a long-term perspective has highlighted a greater explanatory power of the relation (28) compared to (21), measured by  2 ̅̅̅̅ , higher than 51% on average.The transition from the monthly return to the annual return determines a greater explanatory power of equation ( 21).The S&P 500 Index return has the most significant explanatory power.However, this role finds valid competitors in some explanatory variables of the corporate performance, such as Revenue Growth, EBIT Growth and EBIT, which in specific cases play a role even more critical than stock market performance.
Analysing in detail the 100 MLR integrations, we have seen that in the case of a negative correlation between the stock market and corporate performance, a bridge arises, and the transition to the annual return determines a clear improvement of the first variable in explaining the stock return while the second variable remains marginal.
In the presence of a positive correlation, the opposite occurs: first, an overlap arises, and the corporate performance acquires a crucial explanatory power, often superior to the stock market performance.In that case,  2 and   are respectively significantly under and overestimated in the framework of the OLS with annual returns.If the correlation is strongly negative, this outcome is less important.
The hypothesis that relation ( 21), the classic CAPM equation, is not the best possible explanation of the behaviour of the risky asset return appears not entirely implausible considering the empirical evidence presented.
The MLR integration of the stock market and corporate performance with annual returns has greater explanatory power than the CAPM equation, at least in temporal local conditions.

Conclusions
This paper aims to identify a relation between systematic risk and corporate performance represented by DOL and DFL, based on the essay by Mandelker et al. (1984) or through further theoretical analyses that will need empirical evidence.
In Section 2, we examined the Mandelker et al. (1984) contribution, concluding that if such a link exists, it is not in the form presented by the two economists, as DOL and DFL are not static parameters.The study by Miller et al. (1961), also referred to by Fama et al. (2015), permits linking the return of common stock to corporate performance without being able to specify a link with systematic risk.If such a link exists, it would be invisible.Furthermore, we must consider the impact of dilution, which needs more focus.
Section 3 briefly reviewed CAPM, highlighting a divergence in short positions, entirely physiological in a 100-stock optimised portfolio but completely absent in a market portfolio.Another topic concerns the inconsistency of  2 in the CAPM empirical evidence, a significant share of the risky asset return variability is unexplained.Starting from this perspective, we hypothesized that CAPM might be an incomplete theory due to the total absence of corporate performance variables, given that the stock returns are, by assumption, exogenous data.Furthermore, we have observed that neglecting the correlation between the stock market and corporate performance, in addition to determining low  2 values, around 30%, according to Roll (1988), masks the presence of omitted variables since critical overlapping phenomena may be present.Finally, we hypothesized that stock returns derive from the joint performance of the stock market and the corporate business, which could lead to the feedback effect of the former on the latter, both at the level of single stock and portfolio return.From this perspective, the portfolio risk could be due to the joint corporate business portfolio, filtered by the stock market feedback effect.It is an exciting hypothesis, worthy of further study, but not easy to verify empirically due to the presence of an excessive number of unknowns compared to the known variables.
In Section 4, we first analysed two CAPM topics, the asset allocation and the security market line, through the empirical evidence of 100 common stocks included in the S&P 500 Index in the 1991-2020 timespan.
Asset allocation shows that short sales create a notable detachment from the market portfolio, represented by its proxy; unfortunately, this gap persists by using the KKT condition to expunge short sales.The stock market does not appear as efficient and optimised as imagined, even without short sales.Alternatively, Merton's efficient analytical frontier could be a splendid theoretical framework unachievable by the market.In any case, the portfolio optimisation process with short sales, as the number of common stocks increases, leads to increasingly higher β values, which in turn determine an increasingly insignificant  2 value; with short sales constraints, ) while A, B and C are the parameters of the Merton conic: vector of m risky asset returns  −1 = inverse of the covariance matrix of m risky asset returns 1 = unit column vector of size m.

Figure 2 .
Figure 2. Conic sections, market opportunity lines and trajectories

Figure 3 .
Figure 3. β trends inside five optimised portfolios: Lintner's weights on the left and Merton's on the rightThe β continuous increase can only come from reducing the optimised portfolio variance faster than the covariance of the stock return compared to the portfolio return.As the number of stocks grows, both θ and λ increase; if they do not increase, the incremental stock should have zero weight.To keep constant the stock excess return to R F ̅̅̅ , the more λ increases, the more the covariance Cov(R i , R M ) must decrease.But the optimised portfolio variance shrinks even more, and this leads to the β increase, an increase measurable more simply employing the percentage change of the portfolio excess return %X M .Although β is a measure of the risk of the common stock compared to the optimised portfolio, its dynamic depends on the %X M trend via λ.

Figure 4 .
Figure 4. β trends inside five optimised portfolios sine: on the left, stocks included, and on the right, the excluded ones

Figure 5
Figure 5. 5-Year Monthly Rolling β of Three Historical Corporations of the Automotive, IT and Energy Industries

Figure 6 .
Figure 6.Portfolio Optimisation Effect: Portfolio Return on the Abscissa and Stock Return on the Ordinate

Figure 7 .
Figure 7. Portfolio Optimisation Effect: Portfolio Returns on the Abscissa and Stock Returns on the Ordinate

Table 1 .
Comparison of the main portfolio parameters: Lintner's weights with short sales

Table 2 .
Regression between optimised portfolio against S&P 500 Index returns

Table 3 .
Common stock layers in optimised portfolio sine

Table 4 .
Correlation e determination index dynamics referred to the number of stocks in the portfolio

Table 5 .
Summary of β analyses

Table 8 .
Regression of five portfolios against a sample stock (benchmark)

Table 9 .
Corporate performance variables and ratios

Table 11 :
on average, we analysed around 26 periods for every stock, of which 68 stocks had 30 periods, and only four stocks had less than ten periods.Based on the F test, 83 regressions are significant.Based on t-stats, 70 intercepts did not pass the null hypothesis test, while the βs are significant in 83 regressions.We can see that shifting from monthly to annual 30-year returns,  2 increases by more than 13%, to 36.3% in Table10from 22.5% in Table5, with 41 stocks exceeding 40% versus only five stocks with monthly returns.

Table 10 .
Summary of OLS with annual returns

Table 12 .
2 matrix between corporate performance variables and S&P 500 Index return against the stock returns 4.7.4The  2 between Explanatory Variables and Stock Returns Before looking at MLRs, it is worth examining which of the stock market and corporate performance variables shows the highest  2 against each of the 100 stock returns.The details of this analysis are in Table

Table 13 .
Summary of  2 dataThe S&P 500 Index return shows the highest mean  2 with 36.3%, already established in Table10, and in 61 cases, it results in having the highest value of  2 compared to the other fourteen corporate performance variables.Not surprisingly, Revenue Growth and EBIT Growth present 18 and 9 cases, respectively, with the highest  2 value with a rounded mean of 17.3% and 12.9%.Four variables never reached the maximum  2 : Net Income, Adjusted Diluted EPS, Adjusted Dividends and DTL.The other eight variables compete for the remaining 12 places.

Table 16 .
Summary of the MLR regressors and commonality analysis4.7.7 MLR Details and Comparison with OLSThere is no multicollinearity in the chosen 200 explanatory variables of each MLR when VIF measures the test.