Given the paradox between the complexities of the real world, in order to construct good models, those complexities having little effect of the model should be assumed away. A theory is usually validated when it is based on empirical accuracy of its predictions rather than on the realism of its assumptions. The major assumptions of the CAPM are listed as follows: Investors aim at the maximization of utility from holding wealth. Investors' selection criteria of investment opportunities are based on expected return and risk. All investors have a risk adverse attitude and behave rationally. Investors choose investment opportunities set based on expected return and risk. Expected returns follow a normal distribution. The lending and borrowing process is unlimited at a common interest rate. No transactions costs are entailed in the trading of securities. Taxes on dividends and capital gains are at similar rates.

## 3.2 The Capital Market Line

In order to represent the set of portfolios that investors would choose in equilibrium through the stated assumptions above, an opportunity set of all risky portfolios is drawn where with the inclusion of a risk free rate asset, the combination of the risk free asset with any risky portfolios is made possible. Expected return Iii Ii Capital market line Rm Opportunity set ## M

Rf ÃÆ’m Standard deviation Figure 1 In an equilibrium state, investors prefer a composition of the risk free asset and one risky portfolio that provides the maximum expected return for any given level of variance that is hold efficient portfolios. Such a condition is labeled M in the above diagram, where capital market line (the vertical line starting at the risk free rate of return) meets the opportunity set of risky assets. One of the unparalleled characteristic of the CML is that investors would not want to move beyond point M. That is, investors cannot improve upon the alternatives by this set of portfolios otherwise the market would not be in equilibrium and arbitrage would occur. ## 3.3 The Security Market Line

Sharpe and Lintner developed a framework to describe the relationship between expected returns and the risk associated with securities with the following equation which is in its ex-ante form: E (ri) = rf + ÃŽÂ² [E (rm) - rf] (1) In simpler terms, the above equation (1) shows that the expected return on an asset which is equal to the risk free rate of return plus a risk premium. The risk premium is the price of risk (slope of the line) multiplied by the quantity of risk which is the systematic risk (ÃŽÂ²). Equation (1) propounds that in the equilibrium state, an asset with zero systematic risk (ÃŽÂ²=0) will have expected return just equal to that on the riskless asset rf , and expected return on all risky securities (ÃŽÂ² >0) will be higher by the risk premium which is directly proportional to their risk as measured by ÃŽÂ². Such a relation is graphed through the Security Market Line (SML) in the below diagram (figure 2) with expected returns on the vertical axis and beta on the horizontal axis. The SML shows a positive linear relationship between beta and expected return and the intercept is equal to the risk free rate. Expected Return Security market line ## Rm M

## Rf

Figure 2 Beta It important to note that efficient portfolios are usually plot on the CML and Figure 3 provides two diagrams which portrays the relationship between the CML and the SML. Point A represents an efficient portfolio A which lies on the CML and point B is an inefficient portfolio outside the CML. However, both portfolios have similar expected return and beta value. Note that the CML concentrates on portfolio standard deviation rather than beta. E(R) E(R) A A, B E (RA) =E (RB) B Rf Rf Figure 3 ÃÆ’A ÃÆ’B ÃŽÂ²A = ÃŽÂ²B An efficient portfolio is said to have lesser standard deviation of returns than the inefficient portfolio given equal expected returns. The excess standard deviation associated with the inefficient portfolio is called diversifiable risk or unsystematic risk. Investors are not compensated for this kind of risk because in a state of equilibrium, investors only hold efficient portfolios. Hence it can be perceived that the CAPM is effective in the pricing of all assets whether they demonstrate efficiency or not but the CML only prices efficient portfolios. ## 3.4 The transition from the ex-ante to the ex-post model

As incorporated in the overall study, equation (2)Ã¢â‚¬Â¦ is denoted as an ex-ante or a forward-looking model which uses entirely historical data for the testing purposes. However, a contradiction is formed upon this belief of historical data as there is no proof that the rates of return expected in the future will automatically be equal to realized rates of return over the past periods. Moreover, it should be acclaimed that historical beta may or may not mirror expected future risk. Hence the need to traverse from the ex-ante principles to the ex-post is felt in order to better test for CAPM. The ex-post model specifies some return generating process by assuming that the rate of return on an asset follows a "fair game". The "fair game" signifies that, on average, across a large number of samples the expected return on an asset equals its actual return and is explained as follows: ri = E (ri) + ÃŽÂ²iÃŽÂ´m + ÃŽÂµi (..) Where, ÃŽÂ´m = rm - E (rm) E (ÃŽÂ´m) = 0 ÃŽÂµi = a random error term Cov (ÃŽÂµi , ÃŽÂ´m ) = 0 Cov (ÃŽÂµi , ÃŽÂµi t-1 ) = 0 ÃŽÂ²i = systematic risk Note that since CAPM assumes that asset returns are jointly normal, ÃŽÂ²i in the fair game model behaves exactly as in the CAPM model and the market model must hold. Equation (..) assumes that if expected return is taken on both sides, the average realized return is equal to the expected return: E (ri) = E (ri) Substituting E (ri) from the CAPM into (,,) yields: ri = { rf + [ E (rm) ] ÃŽÂ²i + ÃŽÂ²i [ rm - E (rm) ] + ÃŽÂµi ri = rf + (rm - rf) ÃŽÂ²i + ÃŽÂµi Substracting the risk free rate from both sides: ri - rf = (rm - rf) ÃŽÂ²i + ÃŽÂµi (..) Equation (..) represents the ex-post model of the CAPM. One important difference between the ex-ante theoretical model and the ex-post model is that the latter can present a negative slope while the former cannot because the theoretical CAPM must have a higher expected return on the market than the risk free rate of return.